Subgroups — C₂²×D₄×C₂²⋊D₅₃ #28291000

Order 13568.

20038 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₂²⋊D₅₃	C₂	1	2	yes	maximal, complemented
2	C₂³×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄×C₂²⋊D₅₃	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#13133213	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13133191	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13133189	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133188	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13133186	C₂	1	2	yes	maximal, complemented
9	C₂²×#13133185	C₂	1	2	yes	maximal, complemented
10	C₂²×D₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₄×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁴×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂⁵×D₅₃	C₂²	1	2	yes	complemented
14	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
15	C₂³×C₂²⋊D₅₃	—	2	32	no	—
16	C₂³×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
17	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
18	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
19	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
20	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
21	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
22	C₂³×C₂²⋊D₅₃	C₂²	1	48	yes	complemented
23	C₂²×D₄×D₅₃	—	2	16	no	—
24	C₂²×D₄×D₅₃	C₂²	1	24	yes	complemented
25	C₂⁴×Dic₅₃	C₂²	1	2	yes	complemented
26	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
27	D₄×C₂²⋊D₅₃	C₂²	1	256	yes	direct product factor
28	C₂×#13133213	C₂²	1	24	yes	complemented
29	C₂×#13133191	C₂²	1	24	yes	complemented
30	C₂×#13133189	C₂²	1	48	yes	complemented
31	C₂×#13133188	C₂²	1	24	yes	complemented
32	C₂×#13133186	C₂²	1	48	yes	complemented
33	C₂×#13133185	C₂²	1	48	yes	complemented
34	C₂×D₄×Dic₅₃	C₂²	1	24	yes	complemented
35	C₂×C₄×C₂²⋊D₅₃	C₂²	1	24	yes	complemented
36	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₂²×#11719919	C₂²	1	1	yes	characteristic, complemented
38	C₂²×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
39	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	complemented
40	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	complemented
41	C₂²×#11719921	C₂²	1	1	yes	characteristic, complemented
42	C₂²×#11719921	C₂²	1	2	yes	complemented
43	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
44	C₂⁶×C₅₃	C₂²	1	2	yes	complemented
45	C₂³×D₄×C₅₃	—	2	16	no	—
46	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
47	C₂³×D₄×C₅₃	C₂²	1	2	yes	complemented
48	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
49	C₂³×D₄×C₅₃	C₂²	1	24	yes	complemented
50	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
51	C₂⁴×D₅₃	—	2	16	no	—
52	C₂⁴×D₅₃	—	2	16	no	—
53	C₂⁴×D₅₃	—	4	8	no	—
54	C₂⁴×D₅₃	C₂³	1	2	yes	—
55	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
56	C₂⁴×D₅₃	C₂³	1	24	yes	complemented
57	C₂²×D₂₁₂	—	2	8	no	—
58	C₂²×D₂₁₂	C₂³	1	12	yes	complemented
59	C₂²×C₂²⋊D₅₃	—	2	16	no	—
60	C₂²×C₂²⋊D₅₃	—	2	16	no	—
61	C₂²×C₂²⋊D₅₃	—	2	16	no	—
62	C₂²×C₂²⋊D₅₃	—	2	384	no	—
63	C₂²×C₂²⋊D₅₃	—	2	16	no	—
64	C₂²×C₂²⋊D₅₃	—	4	8	no	—
65	C₂²×C₂²⋊D₅₃	D₄	1	16	yes	direct product factor
66	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
67	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
68	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
69	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
70	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
71	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
72	C₂²×C₂²⋊D₅₃	C₂³	1	128	yes	complemented
73	C₂×D₄×D₅₃	—	2	192	no	—
74	C₂×D₄×D₅₃	C₂³	1	64	yes	complemented
75	C₂³×Dic₅₃	—	2	16	no	—
76	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
77	C₂³×Dic₅₃	C₂³	1	2	yes	—
78	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
79	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
80	C₂³×Dic₅₃	C₂³	1	24	yes	complemented
81	C₂²×C₄×D₅₃	—	2	8	no	—
82	C₂²×C₄×D₅₃	C₂³	1	12	yes	complemented
83	#13133213	C₂³	1	64	yes	complemented
84	#13133191	C₂³	1	64	yes	complemented
85	#13133189	C₂³	1	128	yes	complemented
86	#13133188	C₂³	1	64	yes	complemented
87	#13133186	C₂³	1	128	yes	complemented
88	#13133185	C₂³	1	128	yes	complemented
89	D₄×Dic₅₃	C₂³	1	64	yes	complemented
90	C₄×C₂²⋊D₅₃	C₂³	1	64	yes	complemented
91	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	complemented
92	C₂×#11719919	C₂³	1	12	yes	complemented
93	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	complemented
94	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	complemented
95	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	complemented
96	C₂×#11719921	C₂³	1	12	yes	complemented
97	C₂×#11719921	C₂³	1	24	yes	complemented
98	C₂×C₄×Dic₅₃	C₂³	1	12	yes	complemented
99	C₂⁵×C₅₃	—	2	16	no	—
100	C₂⁵×C₅₃	—	2	16	no	—
101	C₂⁵×C₅₃	—	4	8	no	—
102	C₂⁵×C₅₃	C₂³	1	2	yes	—
103	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
104	C₂⁵×C₅₃	C₂³	1	2	yes	—
105	C₂⁵×C₅₃	C₂³	1	24	yes	complemented
106	C₂²×D₄×C₅₃	—	2	16	no	—
107	C₂²×D₄×C₅₃	—	2	192	no	—
108	C₂²×D₄×C₅₃	—	2	8	no	—
109	C₂²×D₄×C₅₃	D₄	1	16	yes	complemented
110	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
111	C₂²×D₄×C₅₃	C₂³	1	24	yes	complemented
112	C₂²×D₄×C₅₃	C₂³	1	12	yes	complemented
113	C₂²×D₄×C₅₃	C₂³	1	64	yes	complemented
114	C₂³×C₄×C₅₃	—	2	8	no	—
115	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
116	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
117	C₂³×C₄×C₅₃	C₂³	1	12	yes	complemented
118	C₂⁴×C₅₃	—	2	4	no	—
119	C₂⁴×C₅₃	—	2	8	no	—
120	C₂⁴×C₅₃	—	2	4	no	—
121	C₂⁴×C₅₃	—	2	8	no	—
122	C₂⁴×C₅₃	—	2	8	no	—
123	C₂⁴×C₅₃	—	2	8	no	—
124	C₂⁴×C₅₃	—	2	96	no	—
125	C₂⁴×C₅₃	—	2	96	no	—
126	C₂⁴×C₅₃	—	2	8	no	—
127	C₂⁴×C₅₃	—	4	48	no	—
128	C₂⁴×C₅₃	—	4	128	no	—
129	C₂⁴×C₅₃	C₂×D₄	1	8	yes	complemented
130	C₂⁴×C₅₃	C₂×D₄	1	16	yes	complemented
131	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
132	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
133	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
134	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
135	C₂⁴×C₅₃	C₂⁴	1	32	yes	complemented
136	C₂³×D₅₃	—	2	4	no	—
137	C₂³×D₅₃	—	2	4	no	—
138	C₂³×D₅₃	—	2	8	no	—
139	C₂³×D₅₃	—	2	8	no	—
140	C₂³×D₅₃	—	2	96	no	—
141	C₂³×D₅₃	—	2	8	no	—
142	C₂³×D₅₃	—	2	96	no	—
143	C₂³×D₅₃	—	4	48	no	—
144	C₂³×D₅₃	—	4	128	no	—
145	C₂³×D₅₃	C₂×D₄	1	8	yes	complemented
146	C₂³×D₅₃	C₂⁴	1	12	yes	—
147	C₂³×D₅₃	C₂⁴	1	6	yes	—
148	C₂³×D₅₃	C₂⁴	1	32	yes	complemented
149	C₂×D₄×C₅₃	—	2	96	no	—
150	C₂×D₄×C₅₃	—	2	256	no	—
151	C₂×D₄×C₅₃	—	2	48	no	—
152	C₂×D₄×C₅₃	C₂×D₄	1	96	yes	complemented
153	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
154	C₂×D₄×C₅₃	C₂⁴	1	32	yes	complemented
155	C₂×D₄×C₅₃	C₂⁴	1	16	yes	complemented
156	C₂²×C₄×C₅₃	—	2	4	no	—
157	C₂²×C₄×C₅₃	—	2	48	no	—
158	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	complemented
159	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
160	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
161	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	complemented
162	C₂×C₂²⋊D₅₃	—	2	96	no	—
163	C₂×C₂²⋊D₅₃	—	2	96	no	—
164	C₂×C₂²⋊D₅₃	—	2	96	no	—
165	C₂×C₂²⋊D₅₃	—	2	512	no	—
166	C₂×C₂²⋊D₅₃	—	2	96	no	—
167	C₂×C₂²⋊D₅₃	—	4	48	no	—
168	C₂×C₂²⋊D₅₃	C₂×D₄	1	96	yes	direct product factor
169	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
170	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
171	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
172	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
173	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
174	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	complemented
175	C₂²×Dic₅₃	—	2	4	no	—
176	C₂²×Dic₅₃	—	2	8	no	—
177	C₂²×Dic₅₃	—	2	4	no	—
178	C₂²×Dic₅₃	—	2	96	no	—
179	C₂²×Dic₅₃	C₂×D₄	1	8	yes	complemented
180	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
181	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
182	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
183	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
184	C₂²×Dic₅₃	C₂⁴	1	32	yes	complemented
185	D₄×D₅₃	—	2	256	no	—
186	C₂×D₂₁₂	—	2	48	no	—
187	C₂×D₂₁₂	C₂⁴	1	16	yes	complemented
188	C₂×C₄×D₅₃	—	2	48	no	—
189	C₂×C₄×D₅₃	C₂⁴	1	16	yes	complemented
190	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	complemented
191	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
192	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
193	#11719921	C₂⁴	1	16	yes	complemented
194	#11719921	C₂⁴	1	32	yes	complemented
195	C₄⋊Dic₅₃	C₂⁴	1	16	yes	complemented
196	#11719919	C₂⁴	1	16	yes	complemented
197	C₄×Dic₅₃	C₂⁴	1	16	yes	complemented
198	C₂³×C₅₃	—	2	64	no	—
199	C₂³×C₅₃	—	2	32	no	—
200	C₂³×C₅₃	—	2	24	no	—
201	C₂³×C₅₃	—	2	12	no	—
202	C₂³×C₅₃	—	2	24	no	—
203	C₂³×C₅₃	—	2	12	no	—
204	C₂³×C₅₃	—	2	24	no	—
205	C₂³×C₅₃	—	2	24	no	—
206	C₂³×C₅₃	—	2	64	no	—
207	C₂³×C₅₃	—	2	64	no	—
208	C₂³×C₅₃	—	2	24	no	—
209	C₂³×C₅₃	—	4	32	no	—
210	C₂³×C₅₃	—	4	384	no	—
211	C₂³×C₅₃	—	4	32	no	—
212	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
213	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
214	C₂³×C₅₃	C₂²×D₄	1	24	yes	complemented
215	C₂³×C₅₃	C₂²×D₄	1	48	yes	complemented
216	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
217	C₂³×C₅₃	C₂⁵	1	8	yes	—
218	C₂³×C₅₃	C₂⁵	1	8	yes	—
219	C₂³×C₅₃	C₂⁵	1	4	yes	—
220	C₂³×C₅₃	C₂⁵	1	3	yes	—
221	C₂²×D₅₃	—	2	32	no	—
222	C₂²×D₅₃	—	2	12	no	—
223	C₂²×D₅₃	—	2	12	no	—
224	C₂²×D₅₃	—	2	24	no	—
225	C₂²×D₅₃	—	2	24	no	—
226	C₂²×D₅₃	—	2	64	no	—
227	C₂²×D₅₃	—	2	24	no	—
228	C₂²×D₅₃	—	2	64	no	—
229	C₂²×D₅₃	—	4	32	no	—
230	C₂²×D₅₃	—	4	384	no	—
231	C₂²×D₅₃	—	4	32	no	—
232	C₂²×D₅₃	C₂²×D₄	1	24	yes	complemented
233	C₂²×D₅₃	C₂⁵	1	8	yes	—
234	C₂²×D₅₃	C₂⁵	1	4	yes	—
235	D₄×C₅₃	—	2	64	no	—
236	D₄×C₅₃	—	2	32	no	—
237	D₄×C₅₃	C₂²×D₄	1	64	yes	complemented
238	C₂×C₄×C₅₃	—	2	12	no	—
239	C₂×C₄×C₅₃	—	2	32	no	—
240	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	complemented
241	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
242	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
243	C₂²⋊D₅₃	—	2	64	no	—
244	C₂²⋊D₅₃	—	2	64	no	—
245	C₂²⋊D₅₃	—	2	64	no	—
246	C₂²⋊D₅₃	—	2	64	no	—
247	C₂²⋊D₅₃	—	4	32	no	—
248	C₂²⋊D₅₃	C₂²×D₄	1	64	yes	direct product factor
249	C₂×Dic₅₃	—	2	12	no	—
250	C₂×Dic₅₃	—	2	24	no	—
251	C₂×Dic₅₃	—	2	12	no	—
252	C₂×Dic₅₃	—	2	64	no	—
253	C₂×Dic₅₃	C₂²×D₄	1	24	yes	complemented
254	C₂×Dic₅₃	C₂⁵	1	4	yes	—
255	C₂×Dic₅₃	C₂⁵	1	8	yes	—
256	C₂×Dic₅₃	C₂⁵	1	4	yes	—
257	C₂×Dic₅₃	C₂⁵	1	4	yes	—
258	D₂₁₂	—	2	32	no	—
259	C₄×D₅₃	—	2	32	no	—
260	C₂²×D₄²	—	53	1	no	maximal, Sylow
261	C₂²×C₅₃	—	2	8	no	—
262	C₂²×C₅₃	—	2	4	no	—
263	C₂²×C₅₃	—	2	8	no	—
264	C₂²×C₅₃	—	2	4	no	—
265	C₂²×C₅₃	—	2	96	no	—
266	C₂²×C₅₃	—	2	48	no	—
267	C₂²×C₅₃	—	2	8	no	—
268	C₂²×C₅₃	—	2	8	no	—
269	C₂²×C₅₃	—	2	8	no	—
270	C₂²×C₅₃	—	4	128	no	—
271	C₂²×C₅₃	—	4	48	no	—
272	C₂²×C₅₃	D₄²	1	16	yes	complemented
273	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
274	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
275	C₂²×C₅₃	C₂³×D₄	1	8	yes	complemented
276	C₂²×C₅₃	C₂³×D₄	1	16	yes	complemented
277	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
278	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
279	C₂×D₅₃	—	2	4	no	—
280	C₂×D₅₃	—	2	4	no	—
281	C₂×D₅₃	—	2	48	no	—
282	C₂×D₅₃	—	2	8	no	—
283	C₂×D₅₃	—	2	8	no	—
284	C₂×D₅₃	—	2	8	no	—
285	C₂×D₅₃	—	4	128	no	—
286	C₂×D₅₃	—	4	48	no	—
287	C₂×D₅₃	C₂³×D₄	1	8	yes	complemented
288	C₄×C₅₃	—	2	4	no	—
289	C₄×C₅₃	C₂³×D₄	1	8	yes	complemented
290	Dic₅₃	—	2	4	no	—
291	Dic₅₃	—	2	8	no	—
292	Dic₅₃	—	2	4	no	—
293	Dic₅₃	C₂³×D₄	1	8	yes	complemented
294	C₂⁴×D₄	—	53	2	no	—
295	C₂⁴×D₄	—	53	1	no	—
296	C₂⁴×D₄	D₅₃	1	1	yes	characteristic, complemented
297	C₂×D₄²	—	53	48	no	—
298	C₂²×#127	—	53	1	no	—
299	C₂²×#121	—	53	1	no	—
300	C₂²×#121	—	53	2	no	—
301	C₂²×#121	—	53	1	no	—
302	C₂²×#120	—	53	2	no	—
303	C₂²×#120	—	53	2	no	—
304	C₂²×C₄×D₄	—	53	1	no	—
305	C₂²×C₄×D₄	—	53	1	no	—
306	C₂×C₅₃	—	2	16	no	—
307	C₂×C₅₃	—	2	8	no	—
308	C₂×C₅₃	—	4	8	no	—
309	C₂×C₅₃	C₂×D₄²	1	12	yes	complemented
310	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
311	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
312	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
313	D₅₃	—	2	8	no	—
314	D₅₃	—	4	8	no	—
315	C₂⁶	—	53	2	no	—
316	C₂⁶	C₂×D₅₃	1	2	yes	complemented
317	C₂³×D₄	—	2	16	no	—
318	C₂³×D₄	—	53	1	no	—
319	C₂³×D₄	—	53	2	no	—
320	C₂³×D₄	—	53	2	no	—
321	C₂³×D₄	—	53	2	no	—
322	C₂³×D₄	—	53	2	no	—
323	C₂³×D₄	—	53	48	no	—
324	C₂³×D₄	—	53	2	no	—
325	C₂³×D₄	—	53	24	no	—
326	C₂³×D₄	—	53	1	no	—
327	C₂³×D₄	—	106	32	no	—
328	C₂³×D₄	—	106	16	no	—
329	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
330	C₂³×D₄	C₂×D₅₃	1	2	yes	complemented
331	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
332	C₂³×D₄	C₂×D₅₃	1	24	yes	complemented
333	C₂⁴×C₄	—	53	2	no	—
334	C₂⁴×C₄	—	53	1	no	—
335	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
336	D₄²	—	53	256	no	—
337	C₂×#127	—	53	24	no	—
338	C₂×#121	—	53	24	no	—
339	C₂×#121	—	53	48	no	—
340	C₂×#121	—	53	24	no	—
341	C₂×#120	—	53	48	no	—
342	C₂×#120	—	53	48	no	—
343	C₂×C₄×D₄	—	53	24	no	—
344	C₂×C₄×D₄	—	53	24	no	—
345	C₂²×C₄⋊C₄	—	53	1	no	—
346	C₂²×C₄⋊C₄	—	53	1	no	—
347	C₂²×C₂²⋊C₄	—	53	1	no	—
348	C₂²×C₂²⋊C₄	—	53	2	no	—
349	C₂²×C₂²⋊C₄	—	53	2	no	—
350	C₂²×C₂²⋊C₄	—	53	1	no	—
351	C₂²×C₂²⋊C₄	—	53	2	no	—
352	C₂²×C₄²	—	53	1	no	—
353	C₅₃	C₂²×D₄²	1	1	yes	characteristic, complemented, Sylow
354	C₂⁵	—	2	16	no	—
355	C₂⁵	—	2	16	no	—
356	C₂⁵	—	4	8	no	—
357	C₂⁵	—	53	1	no	—
358	C₂⁵	—	53	2	no	—
359	C₂⁵	—	53	24	no	—
360	C₂⁵	—	106	16	no	—
361	C₂⁵	—	106	16	no	—
362	C₂⁵	—	212	8	no	—
363	C₂⁵	C₂²×D₅₃	1	2	yes	—
364	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
365	C₂⁵	C₂²×D₅₃	1	2	yes	—
366	C₂⁵	C₂²×D₅₃	1	24	yes	complemented
367	C₂²×D₄	—	2	16	no	—
368	C₂²×D₄	—	2	192	no	—
369	C₂²×D₄	—	2	8	no	—
370	C₂²×D₄	—	53	16	no	—
371	C₂²×D₄	—	53	12	no	—
372	C₂²×D₄	—	53	24	no	—
373	C₂²×D₄	—	53	24	no	—
374	C₂²×D₄	—	53	24	no	—
375	C₂²×D₄	—	53	24	no	—
376	C₂²×D₄	—	53	128	no	—
377	C₂²×D₄	—	53	24	no	—
378	C₂²×D₄	—	53	64	no	—
379	C₂²×D₄	—	53	12	no	—
380	C₂²×D₄	—	106	16	no	—
381	C₂²×D₄	—	106	384	no	—
382	C₂²×D₄	—	106	16	no	—
383	C₂²×D₄	—	106	16	no	—
384	C₂²×D₄	—	106	16	no	—
385	C₂²×D₄	—	106	192	no	—
386	C₂²×D₄	—	106	8	no	—
387	C₂²×D₄	—	212	8	no	—
388	C₂²×D₄	C₂²⋊D₅₃	1	16	yes	direct product factor
389	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
390	C₂²×D₄	C₂²×D₅₃	1	24	yes	complemented
391	C₂²×D₄	C₂²×D₅₃	1	12	yes	complemented
392	C₂²×D₄	C₂²×D₅₃	1	64	yes	complemented
393	C₂³×C₄	—	2	8	no	—
394	C₂³×C₄	—	53	1	no	—
395	C₂³×C₄	—	53	24	no	—
396	C₂³×C₄	—	53	1	no	—
397	C₂³×C₄	—	53	2	no	—
398	C₂³×C₄	—	53	1	no	—
399	C₂³×C₄	—	53	12	no	—
400	C₂³×C₄	—	106	16	no	—
401	C₂³×C₄	—	106	8	no	—
402	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
403	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
404	C₂³×C₄	C₂²×D₅₃	1	12	yes	complemented
405	#127	—	53	64	no	—
406	#121	—	53	64	no	—
407	#121	—	53	128	no	—
408	#121	—	53	64	no	—
409	#120	—	53	128	no	—
410	#120	—	53	128	no	—
411	C₄×D₄	—	53	64	no	—
412	C₄×D₄	—	53	64	no	—
413	C₂×C₄⋊C₄	—	53	12	no	—
414	C₂×C₄⋊C₄	—	53	12	no	—
415	C₂×C₂²⋊C₄	—	53	12	no	—
416	C₂×C₂²⋊C₄	—	53	24	no	—
417	C₂×C₂²⋊C₄	—	53	24	no	—
418	C₂×C₂²⋊C₄	—	53	12	no	—
419	C₂×C₂²⋊C₄	—	53	24	no	—
420	C₂×C₄²	—	53	12	no	—
421	C₂⁴	—	2	8	no	—
422	C₂⁴	—	2	4	no	—
423	C₂⁴	—	2	4	no	—
424	C₂⁴	—	2	8	no	—
425	C₂⁴	—	2	8	no	—
426	C₂⁴	—	2	96	no	—
427	C₂⁴	—	2	8	no	—
428	C₂⁴	—	2	96	no	—
429	C₂⁴	—	2	8	no	—
430	C₂⁴	—	4	48	no	—
431	C₂⁴	—	4	128	no	—
432	C₂⁴	—	53	8	no	—
433	C₂⁴	—	53	6	no	—
434	C₂⁴	—	53	12	no	—
435	C₂⁴	—	53	32	no	—
436	C₂⁴	—	106	96	no	—
437	C₂⁴	—	106	4	no	—
438	C₂⁴	—	106	4	no	—
439	C₂⁴	—	106	8	no	—
440	C₂⁴	—	106	8	no	—
441	C₂⁴	—	106	96	no	—
442	C₂⁴	—	106	8	no	—
443	C₂⁴	—	212	48	no	—
444	C₂⁴	—	212	128	no	—
445	C₂⁴	D₄×D₅₃	1	8	yes	complemented
446	C₂⁴	C₂×C₂²⋊D₅₃	1	16	yes	complemented
447	C₂⁴	C₂³×D₅₃	1	12	yes	—
448	C₂⁴	C₂³×D₅₃	1	6	yes	—
449	C₂⁴	C₂³×D₅₃	1	12	yes	—
450	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
451	C₂⁴	C₂³×D₅₃	1	32	yes	complemented
452	C₂×D₄	—	2	96	no	—
453	C₂×D₄	—	2	256	no	—
454	C₂×D₄	—	2	48	no	—
455	C₂×D₄	—	53	96	no	—
456	C₂×D₄	—	53	16	no	—
457	C₂×D₄	—	53	32	no	—
458	C₂×D₄	—	53	32	no	—
459	C₂×D₄	—	53	32	no	—
460	C₂×D₄	—	53	32	no	—
461	C₂×D₄	—	53	32	no	—
462	C₂×D₄	—	53	16	no	—
463	C₂×D₄	—	106	96	no	—
464	C₂×D₄	—	106	512	no	—
465	C₂×D₄	—	106	96	no	—
466	C₂×D₄	—	106	96	no	—
467	C₂×D₄	—	106	96	no	—
468	C₂×D₄	—	106	256	no	—
469	C₂×D₄	—	106	48	no	—
470	C₂×D₄	—	212	48	no	—
471	C₂×D₄	C₂×C₂²⋊D₅₃	1	96	yes	direct product factor
472	C₂×D₄	C₂³×D₅₃	1	16	yes	—
473	C₂×D₄	C₂³×D₅₃	1	32	yes	complemented
474	C₂×D₄	C₂³×D₅₃	1	16	yes	complemented
475	C₂²×C₄	—	2	4	no	—
476	C₂²×C₄	—	2	48	no	—
477	C₂²×C₄	—	53	8	no	—
478	C₂²×C₄	—	53	6	no	—
479	C₂²×C₄	—	53	6	no	—
480	C₂²×C₄	—	53	12	no	—
481	C₂²×C₄	—	53	6	no	—
482	C₂²×C₄	—	53	32	no	—
483	C₂²×C₄	—	53	16	no	—
484	C₂²×C₄	—	106	96	no	—
485	C₂²×C₄	—	106	8	no	—
486	C₂²×C₄	—	106	4	no	—
487	C₂²×C₄	—	106	4	no	—
488	C₂²×C₄	—	106	48	no	—
489	C₂²×C₄	C₂×C₂²⋊D₅₃	1	8	yes	complemented
490	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
491	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
492	C₂²×C₄	C₂³×D₅₃	1	16	yes	complemented
493	C₄⋊C₄	—	53	16	no	—
494	C₄⋊C₄	—	53	16	no	—
495	C₂²⋊C₄	—	53	16	no	—
496	C₂²⋊C₄	—	53	32	no	—
497	C₂²⋊C₄	—	53	32	no	—
498	C₂²⋊C₄	—	53	16	no	—
499	C₂²⋊C₄	—	53	32	no	—
500	C₄²	—	53	16	no	—
501	C₂³	—	2	24	no	—
502	C₂³	—	2	12	no	—
503	C₂³	—	2	12	no	—
504	C₂³	—	2	24	no	—
505	C₂³	—	2	32	no	—
506	C₂³	—	2	64	no	—
507	C₂³	—	2	24	no	—
508	C₂³	—	2	64	no	—
509	C₂³	—	2	24	no	—
510	C₂³	—	2	64	no	—
511	C₂³	—	2	24	no	—
512	C₂³	—	4	32	no	—
513	C₂³	—	4	384	no	—
514	C₂³	—	4	32	no	—
515	C₂³	—	53	24	no	—
516	C₂³	—	53	4	no	—
517	C₂³	—	53	8	no	—
518	C₂³	—	106	64	no	—
519	C₂³	—	106	12	no	—
520	C₂³	—	106	12	no	—
521	C₂³	—	106	32	no	—
522	C₂³	—	106	24	no	—
523	C₂³	—	106	24	no	—
524	C₂³	—	106	64	no	—
525	C₂³	—	106	24	no	—
526	C₂³	—	212	32	no	—
527	C₂³	—	212	384	no	—
528	C₂³	—	212	32	no	—
529	C₂³	C₂×D₄×D₅₃	1	4	yes	—
530	C₂³	C₂×D₄×D₅₃	1	24	yes	complemented
531	C₂³	C₂²×C₂²⋊D₅₃	1	4	yes	—
532	C₂³	C₂²×C₂²⋊D₅₃	1	48	yes	complemented
533	C₂³	#13133262	1	4	yes	—
534	C₂³	C₂⁴×D₅₃	1	8	yes	—
535	C₂³	C₂⁴×D₅₃	1	4	yes	—
536	C₂³	C₂⁴×D₅₃	1	8	yes	—
537	C₂³	C₂⁴×D₅₃	1	3	yes	—
538	D₄	—	2	64	no	—
539	D₄	—	2	32	no	—
540	D₄	—	53	64	no	—
541	D₄	—	106	64	no	—
542	D₄	—	106	64	no	—
543	D₄	—	106	64	no	—
544	D₄	—	106	64	no	—
545	D₄	—	106	32	no	—
546	D₄	—	212	32	no	—
547	D₄	C₂²×C₂²⋊D₅₃	1	64	yes	direct product factor
548	C₂×C₄	—	2	12	no	—
549	C₂×C₄	—	2	32	no	—
550	C₂×C₄	—	53	24	no	—
551	C₂×C₄	—	53	4	no	—
552	C₂×C₄	—	53	4	no	—
553	C₂×C₄	—	53	8	no	—
554	C₂×C₄	—	53	4	no	—
555	C₂×C₄	—	106	64	no	—
556	C₂×C₄	—	106	24	no	—
557	C₂×C₄	—	106	12	no	—
558	C₂×C₄	—	106	12	no	—
559	C₂×C₄	—	106	32	no	—
560	C₂×C₄	C₂²×C₂²⋊D₅₃	1	24	yes	complemented
561	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
562	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
563	C₂²	—	2	48	no	—
564	C₂²	—	2	96	no	—
565	C₂²	—	2	8	no	—
566	C₂²	—	2	4	no	—
567	C₂²	—	2	4	no	—
568	C₂²	—	2	8	no	—
569	C₂²	—	2	8	no	—
570	C₂²	—	2	8	no	—
571	C₂²	—	2	8	no	—
572	C₂²	—	4	128	no	—
573	C₂²	—	4	48	no	—
574	C₂²	—	53	8	no	—
575	C₂²	—	106	48	no	—
576	C₂²	—	106	4	no	—
577	C₂²	—	106	4	no	—
578	C₂²	—	106	8	no	—
579	C₂²	—	106	8	no	—
580	C₂²	—	106	8	no	—
581	C₂²	—	212	128	no	—
582	C₂²	—	212	48	no	—
583	C₂²	D₄×C₂²⋊D₅₃	1	16	yes	direct product factor
584	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
585	C₂²	C₂²×D₄×D₅₃	1	8	yes	complemented
586	C₂²	C₂³×C₂²⋊D₅₃	1	6	yes	—
587	C₂²	C₂³×C₂²⋊D₅₃	1	16	yes	complemented
588	C₂²	C₂×#13133262	1	6	yes	—
589	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
590	C₄	—	2	4	no	—
591	C₄	—	53	8	no	—
592	C₄	—	106	8	no	—
593	C₄	—	106	4	no	—
594	C₄	—	106	4	no	—
595	C₄	C₂³×C₂²⋊D₅₃	1	8	yes	complemented
596	C₂	—	2	8	no	—
597	C₂	—	2	16	no	—
598	C₂	—	4	8	no	—
599	C₂	—	106	8	no	—
600	C₂	—	212	8	no	—
601	C₂	C₂×D₄×C₂²⋊D₅₃	1	12	yes	direct product factor
602	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
603	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic
604	C₂	C₂²×#13133262	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂⁴×D₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂⁴×D₄	D₅₃
hypocenter	C₅₃	C₂²×D₄²
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×D₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂⁴×C₂²⋊D₅₃	C₂	semidirect
2. C₂³×D₄×D₅₃	C₂	semidirect
3. C₂×D₄×C₂²⋊D₅₃	C₂	direct (N × H)
4. C₂²×#13133213	C₂	semidirect
5. C₂²×#13133191	C₂	semidirect
6. C₂²×#13133189	C₂	semidirect
7. C₂²×#13133188	C₂	semidirect
8. C₂²×#13133186	C₂	semidirect
9. C₂²×#13133185	C₂	semidirect
10. C₂²×D₄×Dic₅₃	C₂	semidirect
11. C₂²×C₄×C₂²⋊D₅₃	C₂	semidirect
12. C₂⁴×D₄×C₅₃	C₂	semidirect
13. C₂⁵×D₅₃	C₂²	semidirect
14. C₂³×D₂₁₂	C₂²	semidirect
17. C₂³×C₂²⋊D₅₃	C₂²	semidirect
18. C₂³×C₂²⋊D₅₃	C₂²	semidirect
19. C₂³×C₂²⋊D₅₃	C₂²	semidirect
20. C₂³×C₂²⋊D₅₃	C₂²	semidirect
21. C₂³×C₂²⋊D₅₃	C₂²	semidirect
22. C₂³×C₂²⋊D₅₃	C₂²	semidirect
24. C₂²×D₄×D₅₃	C₂²	semidirect
25. C₂⁴×Dic₅₃	C₂²	semidirect
26. C₂³×C₄×D₅₃	C₂²	semidirect
27. D₄×C₂²⋊D₅₃	C₂²	direct (N × H)
28. C₂×#13133213	C₂²	semidirect
29. C₂×#13133191	C₂²	semidirect
30. C₂×#13133189	C₂²	semidirect
31. C₂×#13133188	C₂²	semidirect
32. C₂×#13133186	C₂²	semidirect
33. C₂×#13133185	C₂²	semidirect
34. C₂×D₄×Dic₅₃	C₂²	semidirect
35. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
36. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
37. C₂²×#11719919	C₂²	semidirect
38. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
39. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
40. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
41. C₂²×#11719921	C₂²	semidirect
42. C₂²×#11719921	C₂²	semidirect
43. C₂²×C₄×Dic₅₃	C₂²	semidirect
44. C₂⁶×C₅₃	C₂²	semidirect
47. C₂³×D₄×C₅₃	C₂²	semidirect
48. C₂³×D₄×C₅₃	C₂²	semidirect
49. C₂³×D₄×C₅₃	C₂²	semidirect
50. C₂⁴×C₄×C₅₃	C₂²	semidirect
56. C₂⁴×D₅₃	C₂³	semidirect
58. C₂²×D₂₁₂	C₂³	semidirect
65. C₂²×C₂²⋊D₅₃	D₄	direct (N × H)
67. C₂²×C₂²⋊D₅₃	C₂³	semidirect
68. C₂²×C₂²⋊D₅₃	C₂³	semidirect
69. C₂²×C₂²⋊D₅₃	C₂³	semidirect
70. C₂²×C₂²⋊D₅₃	C₂³	semidirect
71. C₂²×C₂²⋊D₅₃	C₂³	semidirect
72. C₂²×C₂²⋊D₅₃	C₂³	semidirect
74. C₂×D₄×D₅₃	C₂³	semidirect
80. C₂³×Dic₅₃	C₂³	semidirect
82. C₂²×C₄×D₅₃	C₂³	semidirect
83. #13133213	C₂³	semidirect
84. #13133191	C₂³	semidirect
85. #13133189	C₂³	semidirect
86. #13133188	C₂³	semidirect
87. #13133186	C₂³	semidirect
88. #13133185	C₂³	semidirect
89. D₄×Dic₅₃	C₂³	semidirect
90. C₄×C₂²⋊D₅₃	C₂³	semidirect
91. C₂×C₄⋊Dic₅₃	C₂³	semidirect
92. C₂×#11719919	C₂³	semidirect
93. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
94. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
95. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
96. C₂×#11719921	C₂³	semidirect
97. C₂×#11719921	C₂³	semidirect
98. C₂×C₄×Dic₅₃	C₂³	semidirect
105. C₂⁵×C₅₃	C₂³	semidirect
109. C₂²×D₄×C₅₃	D₄	semidirect
111. C₂²×D₄×C₅₃	C₂³	semidirect
112. C₂²×D₄×C₅₃	C₂³	semidirect
113. C₂²×D₄×C₅₃	C₂³	semidirect
117. C₂³×C₄×C₅₃	C₂³	semidirect
129. C₂⁴×C₅₃	C₂×D₄	semidirect
130. C₂⁴×C₅₃	C₂×D₄	semidirect
135. C₂⁴×C₅₃	C₂⁴	semidirect
145. C₂³×D₅₃	C₂×D₄	semidirect
148. C₂³×D₅₃	C₂⁴	semidirect
152. C₂×D₄×C₅₃	C₂×D₄	semidirect
154. C₂×D₄×C₅₃	C₂⁴	semidirect
155. C₂×D₄×C₅₃	C₂⁴	semidirect
158. C₂²×C₄×C₅₃	C₂×D₄	semidirect
161. C₂²×C₄×C₅₃	C₂⁴	semidirect
168. C₂×C₂²⋊D₅₃	C₂×D₄	direct (N × H)
170. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
171. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
172. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
173. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
174. C₂×C₂²⋊D₅₃	C₂⁴	semidirect
179. C₂²×Dic₅₃	C₂×D₄	semidirect
184. C₂²×Dic₅₃	C₂⁴	semidirect
187. C₂×D₂₁₂	C₂⁴	semidirect
189. C₂×C₄×D₅₃	C₂⁴	semidirect
190. C₂²⋊Dic₅₃	C₂⁴	semidirect
191. C₂²⋊Dic₅₃	C₂⁴	semidirect
192. C₂²⋊Dic₅₃	C₂⁴	semidirect
193. #11719921	C₂⁴	semidirect
194. #11719921	C₂⁴	semidirect
195. C₄⋊Dic₅₃	C₂⁴	semidirect
196. #11719919	C₂⁴	semidirect
197. C₄×Dic₅₃	C₂⁴	semidirect
214. C₂³×C₅₃	C₂²×D₄	semidirect
215. C₂³×C₅₃	C₂²×D₄	semidirect
232. C₂²×D₅₃	C₂²×D₄	semidirect
237. D₄×C₅₃	C₂²×D₄	semidirect
240. C₂×C₄×C₅₃	C₂²×D₄	semidirect
248. C₂²⋊D₅₃	C₂²×D₄	direct (N × H)
253. C₂×Dic₅₃	C₂²×D₄	semidirect
272. C₂²×C₅₃	D₄²	semidirect
275. C₂²×C₅₃	C₂³×D₄	semidirect
276. C₂²×C₅₃	C₂³×D₄	semidirect
287. C₂×D₅₃	C₂³×D₄	semidirect
289. C₄×C₅₃	C₂³×D₄	semidirect
293. Dic₅₃	C₂³×D₄	semidirect
296. C₂⁴×D₄	D₅₃	semidirect
309. C₂×C₅₃	C₂×D₄²	semidirect
316. C₂⁶	C₂×D₅₃	semidirect
330. C₂³×D₄	C₂×D₅₃	semidirect
331. C₂³×D₄	C₂×D₅₃	semidirect
332. C₂³×D₄	C₂×D₅₃	semidirect
335. C₂⁴×C₄	C₂×D₅₃	semidirect
353. C₅₃	C₂²×D₄²	semidirect
366. C₂⁵	C₂²×D₅₃	semidirect
388. C₂²×D₄	C₂²⋊D₅₃	direct (N × H)
390. C₂²×D₄	C₂²×D₅₃	semidirect
391. C₂²×D₄	C₂²×D₅₃	semidirect
392. C₂²×D₄	C₂²×D₅₃	semidirect
404. C₂³×C₄	C₂²×D₅₃	semidirect
445. C₂⁴	D₄×D₅₃	semidirect
446. C₂⁴	C₂×C₂²⋊D₅₃	semidirect
451. C₂⁴	C₂³×D₅₃	semidirect
471. C₂×D₄	C₂×C₂²⋊D₅₃	direct (N × H)
473. C₂×D₄	C₂³×D₅₃	semidirect
474. C₂×D₄	C₂³×D₅₃	semidirect
489. C₂²×C₄	C₂×C₂²⋊D₅₃	semidirect
492. C₂²×C₄	C₂³×D₅₃	semidirect
530. C₂³	C₂×D₄×D₅₃	semidirect
532. C₂³	C₂²×C₂²⋊D₅₃	semidirect
547. D₄	C₂²×C₂²⋊D₅₃	direct (N × H)
560. C₂×C₄	C₂²×C₂²⋊D₅₃	semidirect
583. C₂²	D₄×C₂²⋊D₅₃	direct (N × H)
585. C₂²	C₂²×D₄×D₅₃	semidirect
587. C₂²	C₂³×C₂²⋊D₅₃	semidirect
595. C₄	C₂³×C₂²⋊D₅₃	semidirect
601. C₂	C₂×D₄×C₂²⋊D₅₃	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₂²⋊D₅₃	2
C₂³×D₄×D₅₃	2
C₂×D₄×C₂²⋊D₅₃	2
C₂²×#13133213	2
C₂²×#13133191	2
C₂²×#13133189	2
C₂²×#13133188	2
C₂²×#13133186	2
C₂²×#13133185	2
C₂²×D₄×Dic₅₃	2
C₂²×C₄×C₂²⋊D₅₃	2
C₂⁴×D₄×C₅₃	2
C₂²×D₄²	53

Subgroups — C22×D4×C22⋊D53 #28291000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₄×C₂²⋊D₅₃ #28291000