Subgroups — C₂²×C₄×D₄×D₅₃ #28290653

Order 13568.

19998 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×D₅₃	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13133179	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₂²⋊D₅₃	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133177	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄×D₅₃	C₂	1	48	yes	maximal, direct product factor
9	C₂²×C₄⋊C₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₂²⋊C₄×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₂²×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂⁵×D₅₃	C₂²	1	2	yes	—
14	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
15	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
16	C₂²×D₄×D₅₃	C₄	1	32	yes	direct product factor
17	C₂²×D₄×D₅₃	C₂²	1	24	yes	—
18	C₂⁴×Dic₅₃	C₂²	1	2	yes	complemented
19	C₂³×C₄×D₅₃	—	2	32	no	—
20	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
21	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
22	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
23	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
24	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
25	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
26	C₂³×C₄×D₅₃	C₂²	1	48	yes	complemented
27	C₂×C₄×D₂₁₂	C₂²	1	24	yes	complemented
28	C₂×D₄×Dic₅₃	C₂²	1	24	yes	complemented
29	C₂×#13133179	C₂²	1	24	yes	complemented
30	C₂×C₄×C₂²⋊D₅₃	C₂²	1	48	yes	complemented
31	C₂×#13133177	C₂²	1	48	yes	complemented
32	C₄×D₄×D₅₃	C₂²	1	256	yes	direct product factor
33	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
34	C₂²×#11719919	C₂²	1	2	yes	complemented
35	C₂×C₄⋊C₄×D₅₃	C₂²	1	24	yes	complemented
36	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	complemented
37	C₂²×#11719921	C₂²	1	2	yes	complemented
38	C₂²×#11719921	C₂²	1	2	yes	complemented
39	C₂×C₂²⋊C₄×D₅₃	C₂²	1	48	yes	complemented
40	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
41	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
42	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₂×C₄²×D₅₃	C₂²	1	24	yes	complemented
44	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
45	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	complemented
46	C₂×C₄×D₄×C₅₃	C₂²	1	24	yes	complemented
47	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
48	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
49	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
50	C₂⁴×D₅₃	—	2	16	no	—
51	C₂⁴×D₅₃	—	2	16	no	—
52	C₂⁴×D₅₃	C₂×C₄	1	32	yes	complemented
53	C₂⁴×D₅₃	C₂³	1	24	yes	—
54	C₂⁴×D₅₃	C₂³	1	2	yes	—
55	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
56	C₂²×D₂₁₂	C₂×C₄	1	16	yes	complemented
57	C₂²×D₂₁₂	C₂³	1	12	yes	—
58	C₂²×C₂²⋊D₅₃	C₂×C₄	1	32	yes	complemented
59	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
60	C₂×D₄×D₅₃	C₂×C₄	1	384	yes	direct product factor
61	C₂×D₄×D₅₃	C₂³	1	64	yes	—
62	C₂³×Dic₅₃	—	2	16	no	—
63	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
64	C₂³×Dic₅₃	C₂³	1	2	yes	—
65	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
66	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₂³×Dic₅₃	C₂³	1	24	yes	complemented
68	C₂²×C₄×D₅₃	—	2	16	no	—
69	C₂²×C₄×D₅₃	—	2	8	no	—
70	C₂²×C₄×D₅₃	—	2	16	no	—
71	C₂²×C₄×D₅₃	—	2	16	no	—
72	C₂²×C₄×D₅₃	—	2	384	no	—
73	C₂²×C₄×D₅₃	C₂×C₄	1	16	yes	complemented
74	C₂²×C₄×D₅₃	D₄	1	16	yes	direct product factor
75	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
76	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
77	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
78	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
79	C₂²×C₄×D₅₃	C₂³	1	128	yes	complemented
80	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
81	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
82	C₄×D₂₁₂	C₂³	1	64	yes	complemented
83	D₄×Dic₅₃	C₂³	1	64	yes	complemented
84	#13133179	C₂³	1	64	yes	complemented
85	C₄×C₂²⋊D₅₃	C₂³	1	128	yes	complemented
86	#13133177	C₂³	1	128	yes	complemented
87	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	complemented
88	C₂×#11719919	C₂³	1	24	yes	complemented
89	C₄⋊C₄×D₅₃	C₂³	1	64	yes	complemented
90	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	complemented
91	C₂×#11719921	C₂³	1	24	yes	complemented
92	C₂×#11719921	C₂³	1	24	yes	complemented
93	C₂²⋊C₄×D₅₃	C₂³	1	128	yes	complemented
94	C₂×C₄×Dic₅₃	C₂³	1	12	yes	complemented
95	C₂×C₄×Dic₅₃	C₂³	1	12	yes	complemented
96	C₂×C₄×Dic₅₃	C₂³	1	12	yes	complemented
97	C₄²×D₅₃	C₂³	1	64	yes	complemented
98	C₂⁵×C₅₃	C₂³	1	2	yes	—
99	C₂²×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
100	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
101	C₂³×C₄×C₅₃	—	2	16	no	—
102	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
103	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
104	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
105	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
106	C₂³×C₄×C₅₃	C₂³	1	24	yes	complemented
107	C₄×D₄×C₅₃	C₂³	1	64	yes	complemented
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	complemented
109	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	complemented
110	C₂×C₄²×C₅₃	C₂³	1	12	yes	complemented
111	C₂⁴×C₅₃	—	2	8	no	—
112	C₂⁴×C₅₃	—	2	8	no	—
113	C₂⁴×C₅₃	C₂²×C₄	1	16	yes	complemented
114	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
115	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
116	C₂³×D₅₃	—	2	8	no	—
117	C₂³×D₅₃	—	2	8	no	—
118	C₂³×D₅₃	—	2	256	no	—
119	C₂³×D₅₃	—	2	96	no	—
120	C₂³×D₅₃	—	2	96	no	—
121	C₂³×D₅₃	C₂²×C₄	1	8	yes	—
122	C₂³×D₅₃	C₂²×C₄	1	192	yes	complemented
123	C₂³×D₅₃	C₂²×C₄	1	16	yes	complemented
124	C₂³×D₅₃	C₂×D₄	1	8	yes	—
125	C₂³×D₅₃	#41	1	8	yes	—
126	C₂³×D₅₃	C₂⁴	1	32	yes	—
127	C₂³×D₅₃	C₂⁴	1	12	yes	—
128	C₂³×D₅₃	C₂⁴	1	6	yes	—
129	C₂×D₄×C₅₃	C₂²×C₄	1	96	yes	complemented
130	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
131	C₂²×C₄×C₅₃	—	2	8	no	—
132	C₂²×C₄×C₅₃	—	2	4	no	—
133	C₂²×C₄×C₅₃	—	2	96	no	—
134	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	complemented
135	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	complemented
136	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
137	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
138	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
139	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
140	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	complemented
141	C₂×C₂²⋊D₅₃	C₂²×C₄	1	192	yes	complemented
142	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
143	C₂²×Dic₅₃	—	2	8	no	—
144	C₂²×Dic₅₃	—	2	4	no	—
145	C₂²×Dic₅₃	—	2	96	no	—
146	C₂²×Dic₅₃	C₂²×C₄	1	8	yes	complemented
147	C₂²×Dic₅₃	C₂×D₄	1	8	yes	complemented
148	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
149	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
150	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
151	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
152	C₂²×Dic₅₃	C₂⁴	1	32	yes	complemented
153	D₄×D₅₃	C₂²×C₄	1	512	yes	direct product factor
154	C₂×D₂₁₂	C₂²×C₄	1	96	yes	complemented
155	C₂×D₂₁₂	C₂⁴	1	16	yes	—
156	C₂×C₄×D₅₃	—	2	96	no	—
157	C₂×C₄×D₅₃	—	2	48	no	—
158	C₂×C₄×D₅₃	—	2	96	no	—
159	C₂×C₄×D₅₃	—	2	96	no	—
160	C₂×C₄×D₅₃	—	2	512	no	—
161	C₂×C₄×D₅₃	C₂²×C₄	1	96	yes	complemented
162	C₂×C₄×D₅₃	C₂×D₄	1	96	yes	direct product factor
163	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
164	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
165	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
166	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
167	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
168	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
169	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
170	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
171	C₄²×C₅₃	C₂⁴	1	16	yes	complemented
172	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
173	#11719921	C₂⁴	1	32	yes	complemented
174	#11719921	C₂⁴	1	32	yes	complemented
175	C₄⋊Dic₅₃	C₂⁴	1	16	yes	complemented
176	#11719919	C₂⁴	1	32	yes	complemented
177	C₄×Dic₅₃	C₂⁴	1	16	yes	complemented
178	C₄×Dic₅₃	C₂⁴	1	16	yes	complemented
179	C₄×Dic₅₃	C₂⁴	1	16	yes	complemented
180	C₂³×C₅₃	—	2	64	no	—
181	C₂³×C₅₃	—	2	24	no	—
182	C₂³×C₅₃	—	2	24	no	—
183	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
184	C₂³×C₅₃	C₂³×C₄	1	48	yes	complemented
185	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
186	C₂³×C₅₃	C₂×#41	1	4	yes	—
187	C₂³×C₅₃	C₂⁵	1	8	yes	—
188	C₂³×C₅₃	C₂⁵	1	3	yes	—
189	C₂²×D₅₃	—	2	64	no	—
190	C₂²×D₅₃	—	2	24	no	—
191	C₂²×D₅₃	—	2	24	no	—
192	C₂²×D₅₃	—	2	768	no	—
193	C₂²×D₅₃	—	2	64	no	—
194	C₂²×D₅₃	—	2	64	no	—
195	C₂²×D₅₃	C₄×D₄	1	64	yes	direct product factor
196	C₂²×D₅₃	C₂³×C₄	1	24	yes	—
197	C₂²×D₅₃	C₂³×C₄	1	128	yes	complemented
198	C₂²×D₅₃	C₂³×C₄	1	48	yes	complemented
199	C₂²×D₅₃	C₂²×D₄	1	24	yes	—
200	C₂²×D₅₃	C₂×#41	1	24	yes	—
201	C₂²×D₅₃	C₂⁵	1	8	yes	—
202	C₂²×D₅₃	C₂⁵	1	4	yes	—
203	D₄×C₅₃	C₂³×C₄	1	64	yes	complemented
204	C₂×C₄×C₅₃	—	2	24	no	—
205	C₂×C₄×C₅₃	—	2	12	no	—
206	C₂×C₄×C₅₃	—	2	64	no	—
207	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	complemented
208	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	complemented
209	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
210	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
211	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
212	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
213	C₂²⋊D₅₃	C₂³×C₄	1	128	yes	complemented
214	C₂×Dic₅₃	—	2	24	no	—
215	C₂×Dic₅₃	—	2	12	no	—
216	C₂×Dic₅₃	—	2	64	no	—
217	C₂×Dic₅₃	C₂³×C₄	1	24	yes	complemented
218	C₂×Dic₅₃	C₂²×D₄	1	24	yes	complemented
219	C₂×Dic₅₃	C₂⁵	1	8	yes	—
220	C₂×Dic₅₃	C₂⁵	1	4	yes	—
221	C₂×Dic₅₃	C₂⁵	1	4	yes	—
222	C₂×Dic₅₃	C₂⁵	1	4	yes	—
223	D₂₁₂	C₂³×C₄	1	64	yes	complemented
224	C₄×D₅₃	—	2	64	no	—
225	C₄×D₅₃	—	2	32	no	—
226	C₄×D₅₃	—	2	64	no	—
227	C₄×D₅₃	—	2	64	no	—
228	C₄×D₅₃	C₂³×C₄	1	64	yes	complemented
229	C₄×D₅₃	C₂²×D₄	1	64	yes	direct product factor
230	C₂³×C₄×D₄	—	53	1	no	maximal, Sylow
231	C₂²×C₅₃	—	2	8	no	—
232	C₂²×C₅₃	—	2	8	no	—
233	C₂²×C₅₃	—	2	96	no	—
234	C₂²×C₅₃	C₂×C₄×D₄	1	16	yes	complemented
235	C₂²×C₅₃	C₂⁴×C₄	1	6	yes	—
236	C₂²×C₅₃	C₂⁴×C₄	1	16	yes	complemented
237	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
238	C₂²×C₅₃	C₂²×#41	1	6	yes	—
239	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
240	C₂×D₅₃	—	2	8	no	—
241	C₂×D₅₃	—	2	8	no	—
242	C₂×D₅₃	—	2	96	no	—
243	C₂×D₅₃	—	2	256	no	—
244	C₂×D₅₃	C₂×C₄×D₄	1	96	yes	direct product factor
245	C₂×D₅₃	C₂⁴×C₄	1	8	yes	—
246	C₂×D₅₃	C₂⁴×C₄	1	16	yes	complemented
247	C₂×D₅₃	C₂³×D₄	1	8	yes	—
248	C₂×D₅₃	C₂²×#41	1	8	yes	—
249	C₄×C₅₃	—	2	8	no	—
250	C₄×C₅₃	—	2	4	no	—
251	C₄×C₅₃	C₂⁴×C₄	1	8	yes	complemented
252	C₄×C₅₃	C₂³×D₄	1	8	yes	complemented
253	Dic₅₃	—	2	8	no	—
254	Dic₅₃	—	2	4	no	—
255	Dic₅₃	C₂⁴×C₄	1	8	yes	complemented
256	Dic₅₃	C₂³×D₄	1	8	yes	complemented
257	C₂⁴×D₄	—	53	1	no	—
258	C₂⁵×C₄	—	53	2	no	—
259	C₂²×C₄×D₄	—	53	1	no	—
260	C₂²×C₄×D₄	—	53	1	no	—
261	C₂²×C₄×D₄	—	53	2	no	—
262	C₂²×C₄×D₄	—	53	48	no	—
263	C₂²×C₄×D₄	—	53	1	no	—
264	C₂²×C₄×D₄	—	53	2	no	—
265	C₂²×C₄×D₄	D₅₃	1	1	yes	characteristic, direct product factor
266	C₂³×C₄⋊C₄	—	53	1	no	—
267	C₂³×C₂²⋊C₄	—	53	2	no	—
268	C₂³×C₄²	—	53	1	no	—
269	C₂×C₅₃	—	2	16	no	—
270	C₂×C₅₃	C₂²×C₄×D₄	1	12	yes	complemented
271	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
272	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
273	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
274	D₅₃	—	2	16	no	—
275	D₅₃	C₂²×C₄×D₄	1	16	yes	direct product factor
276	C₂⁶	—	53	2	no	—
277	C₂³×D₄	—	53	2	no	—
278	C₂³×D₄	—	53	32	no	—
279	C₂³×D₄	—	53	24	no	—
280	C₂³×D₄	—	53	1	no	—
281	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
282	C₂⁴×C₄	—	53	1	no	—
283	C₂⁴×C₄	—	53	1	no	—
284	C₂⁴×C₄	—	53	2	no	—
285	C₂⁴×C₄	—	53	2	no	—
286	C₂⁴×C₄	—	53	2	no	—
287	C₂⁴×C₄	—	53	1	no	—
288	C₂⁴×C₄	—	53	2	no	—
289	C₂⁴×C₄	—	53	48	no	—
290	C₂⁴×C₄	—	106	32	no	—
291	C₂⁴×C₄	C₂×D₅₃	1	2	yes	complemented
292	C₂×C₄×D₄	—	53	24	no	—
293	C₂×C₄×D₄	—	53	48	no	—
294	C₂×C₄×D₄	—	53	256	no	—
295	C₂×C₄×D₄	—	53	24	no	—
296	C₂×C₄×D₄	—	53	48	no	—
297	C₂×C₄×D₄	—	53	24	no	—
298	C₂×C₄×D₄	C₂×D₅₃	1	24	yes	direct product factor
299	C₂²×C₄⋊C₄	—	53	2	no	—
300	C₂²×C₄⋊C₄	—	53	1	no	—
301	C₂²×C₄⋊C₄	—	53	24	no	—
302	C₂²×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
303	C₂²×C₂²⋊C₄	—	53	2	no	—
304	C₂²×C₂²⋊C₄	—	53	2	no	—
305	C₂²×C₂²⋊C₄	—	53	2	no	—
306	C₂²×C₂²⋊C₄	—	53	48	no	—
307	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
308	C₂²×C₄²	—	53	1	no	—
309	C₂²×C₄²	—	53	1	no	—
310	C₂²×C₄²	—	53	1	no	—
311	C₂²×C₄²	—	53	24	no	—
312	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
313	C₅₃	C₂³×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
314	C₂⁵	—	53	1	no	—
315	C₂⁵	—	53	32	no	—
316	C₂⁵	—	53	24	no	—
317	C₂⁵	—	53	2	no	—
318	C₂⁵	—	106	16	no	—
319	C₂⁵	—	106	16	no	—
320	C₂⁵	C₂²×D₅₃	1	2	yes	—
321	C₂²×D₄	—	53	32	no	—
322	C₂²×D₄	—	53	24	no	—
323	C₂²×D₄	—	53	384	no	—
324	C₂²×D₄	—	53	64	no	—
325	C₂²×D₄	—	53	16	no	—
326	C₂²×D₄	—	53	12	no	—
327	C₂²×D₄	C₄×D₅₃	1	16	yes	direct product factor
328	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
329	C₂³×C₄	—	2	16	no	—
330	C₂³×C₄	—	53	1	no	—
331	C₂³×C₄	—	53	1	no	—
332	C₂³×C₄	—	53	16	no	—
333	C₂³×C₄	—	53	16	no	—
334	C₂³×C₄	—	53	12	no	—
335	C₂³×C₄	—	53	12	no	—
336	C₂³×C₄	—	53	1	no	—
337	C₂³×C₄	—	53	2	no	—
338	C₂³×C₄	—	53	24	no	—
339	C₂³×C₄	—	53	12	no	—
340	C₂³×C₄	—	53	24	no	—
341	C₂³×C₄	—	53	24	no	—
342	C₂³×C₄	—	53	24	no	—
343	C₂³×C₄	—	53	128	no	—
344	C₂³×C₄	—	106	16	no	—
345	C₂³×C₄	—	106	384	no	—
346	C₂³×C₄	—	106	16	no	—
347	C₂³×C₄	—	106	8	no	—
348	C₂³×C₄	—	106	16	no	—
349	C₂³×C₄	—	106	16	no	—
350	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
351	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
352	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
353	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
354	C₂³×C₄	C₂²×D₅₃	1	24	yes	complemented
355	C₄×D₄	—	53	64	no	—
356	C₄×D₄	—	53	128	no	—
357	C₄×D₄	—	53	64	no	—
358	C₄×D₄	—	53	128	no	—
359	C₄×D₄	—	53	64	no	—
360	C₄×D₄	C₂²×D₅₃	1	64	yes	direct product factor
361	C₂×C₄⋊C₄	—	53	24	no	—
362	C₂×C₄⋊C₄	—	53	12	no	—
363	C₂×C₄⋊C₄	—	53	64	no	—
364	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	complemented
365	C₂×C₂²⋊C₄	—	53	24	no	—
366	C₂×C₂²⋊C₄	—	53	24	no	—
367	C₂×C₂²⋊C₄	—	53	24	no	—
368	C₂×C₂²⋊C₄	—	53	128	no	—
369	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	complemented
370	C₂×C₄²	—	53	12	no	—
371	C₂×C₄²	—	53	12	no	—
372	C₂×C₄²	—	53	12	no	—
373	C₂×C₄²	—	53	64	no	—
374	C₂×C₄²	C₂²×D₅₃	1	12	yes	complemented
375	C₂⁴	—	2	8	no	—
376	C₂⁴	—	2	8	no	—
377	C₂⁴	—	53	8	no	—
378	C₂⁴	—	53	8	no	—
379	C₂⁴	—	53	8	no	—
380	C₂⁴	—	53	6	no	—
381	C₂⁴	—	53	16	no	—
382	C₂⁴	—	53	192	no	—
383	C₂⁴	—	53	32	no	—
384	C₂⁴	—	53	12	no	—
385	C₂⁴	—	106	96	no	—
386	C₂⁴	—	106	256	no	—
387	C₂⁴	—	106	96	no	—
388	C₂⁴	—	106	8	no	—
389	C₂⁴	—	106	8	no	—
390	C₂⁴	C₂×C₄×D₅₃	1	16	yes	complemented
391	C₂⁴	C₂³×D₅₃	1	12	yes	—
392	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
393	C₂×D₄	—	53	192	no	—
394	C₂×D₄	—	53	32	no	—
395	C₂×D₄	—	53	512	no	—
396	C₂×D₄	—	53	96	no	—
397	C₂×D₄	—	53	16	no	—
398	C₂×D₄	C₂×C₄×D₅₃	1	96	yes	direct product factor
399	C₂×D₄	C₂³×D₅₃	1	16	yes	—
400	C₂²×C₄	—	2	8	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	2	96	no	—
403	C₂²×C₄	—	53	8	no	—
404	C₂²×C₄	—	53	8	no	—
405	C₂²×C₄	—	53	6	no	—
406	C₂²×C₄	—	53	6	no	—
407	C₂²×C₄	—	53	96	no	—
408	C₂²×C₄	—	53	96	no	—
409	C₂²×C₄	—	53	16	no	—
410	C₂²×C₄	—	53	16	no	—
411	C₂²×C₄	—	53	6	no	—
412	C₂²×C₄	—	53	12	no	—
413	C₂²×C₄	—	53	32	no	—
414	C₂²×C₄	—	53	16	no	—
415	C₂²×C₄	—	53	32	no	—
416	C₂²×C₄	—	53	32	no	—
417	C₂²×C₄	—	53	32	no	—
418	C₂²×C₄	—	106	96	no	—
419	C₂²×C₄	—	106	512	no	—
420	C₂²×C₄	—	106	48	no	—
421	C₂²×C₄	—	106	96	no	—
422	C₂²×C₄	—	106	96	no	—
423	C₂²×C₄	—	106	8	no	—
424	C₂²×C₄	—	106	4	no	—
425	C₂²×C₄	—	106	96	no	—
426	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	complemented
427	C₂²×C₄	D₄×D₅₃	1	8	yes	direct product factor
428	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
429	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
430	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
431	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
432	C₂²×C₄	C₂³×D₅₃	1	32	yes	complemented
433	C₄⋊C₄	—	53	32	no	—
434	C₄⋊C₄	—	53	16	no	—
435	C₄⋊C₄	C₂³×D₅₃	1	16	yes	complemented
436	C₂²⋊C₄	—	53	32	no	—
437	C₂²⋊C₄	—	53	32	no	—
438	C₂²⋊C₄	—	53	32	no	—
439	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	complemented
440	C₄²	—	53	16	no	—
441	C₄²	—	53	16	no	—
442	C₄²	—	53	16	no	—
443	C₄²	C₂³×D₅₃	1	16	yes	complemented
444	C₂³	—	2	64	no	—
445	C₂³	—	2	24	no	—
446	C₂³	—	2	24	no	—
447	C₂³	—	53	64	no	—
448	C₂³	—	53	24	no	—
449	C₂³	—	53	24	no	—
450	C₂³	—	53	24	no	—
451	C₂³	—	53	4	no	—
452	C₂³	—	53	48	no	—
453	C₂³	—	53	128	no	—
454	C₂³	—	53	8	no	—
455	C₂³	—	106	64	no	—
456	C₂³	—	106	768	no	—
457	C₂³	—	106	64	no	—
458	C₂³	—	106	64	no	—
459	C₂³	—	106	24	no	—
460	C₂³	—	106	24	no	—
461	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
462	C₂³	C₂²×C₄×D₅₃	1	48	yes	complemented
463	C₂³	C₂×D₄×D₅₃	1	4	yes	—
464	C₂³	#41×D₅₃	1	4	yes	—
465	C₂³	C₂⁴×D₅₃	1	8	yes	—
466	C₂³	C₂⁴×D₅₃	1	3	yes	—
467	D₄	—	53	128	no	—
468	D₄	—	53	64	no	—
469	D₄	C₂²×C₄×D₅₃	1	64	yes	direct product factor
470	C₂×C₄	—	2	24	no	—
471	C₂×C₄	—	2	12	no	—
472	C₂×C₄	—	2	64	no	—
473	C₂×C₄	—	53	24	no	—
474	C₂×C₄	—	53	24	no	—
475	C₂×C₄	—	53	4	no	—
476	C₂×C₄	—	53	4	no	—
477	C₂×C₄	—	53	64	no	—
478	C₂×C₄	—	53	64	no	—
479	C₂×C₄	—	53	4	no	—
480	C₂×C₄	—	53	8	no	—
481	C₂×C₄	—	106	64	no	—
482	C₂×C₄	—	106	32	no	—
483	C₂×C₄	—	106	64	no	—
484	C₂×C₄	—	106	64	no	—
485	C₂×C₄	—	106	24	no	—
486	C₂×C₄	—	106	12	no	—
487	C₂×C₄	—	106	64	no	—
488	C₂×C₄	C₂²×C₄×D₅₃	1	24	yes	complemented
489	C₂×C₄	C₂×D₄×D₅₃	1	24	yes	direct product factor
490	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
491	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
492	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
493	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
494	C₂²	—	2	96	no	—
495	C₂²	—	2	8	no	—
496	C₂²	—	2	8	no	—
497	C₂²	—	53	96	no	—
498	C₂²	—	53	8	no	—
499	C₂²	—	53	8	no	—
500	C₂²	—	53	8	no	—
501	C₂²	—	53	16	no	—
502	C₂²	—	106	256	no	—
503	C₂²	—	106	96	no	—
504	C₂²	—	106	8	no	—
505	C₂²	—	106	8	no	—
506	C₂²	C₄×D₄×D₅₃	1	16	yes	direct product factor
507	C₂²	C₂³×C₄×D₅₃	1	6	yes	—
508	C₂²	C₂³×C₄×D₅₃	1	16	yes	complemented
509	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
510	C₂²	C₂×#41×D₅₃	1	6	yes	—
511	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
512	C₄	—	2	8	no	—
513	C₄	—	2	4	no	—
514	C₄	—	53	8	no	—
515	C₄	—	53	8	no	—
516	C₄	—	106	8	no	—
517	C₄	—	106	4	no	—
518	C₄	C₂³×C₄×D₅₃	1	8	yes	complemented
519	C₄	C₂²×D₄×D₅₃	1	8	yes	direct product factor
520	C₂	—	2	16	no	—
521	C₂	—	53	16	no	—
522	C₂	—	106	16	no	—
523	C₂	C₂×C₄×D₄×D₅₃	1	12	yes	direct product factor
524	C₂	C₂⁴×C₄×D₅₃	1	1	yes	characteristic
525	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
526	C₂	C₂²×#41×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×C₄×D₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×D₄×D₅₃	C₂
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂²×C₄×D₄	D₅₃
hypocenter	C₅₃	C₂³×C₄×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×D₅₃	C₂	semidirect
3. C₂²×C₄×D₂₁₂	C₂	semidirect
4. C₂²×D₄×Dic₅₃	C₂	semidirect
5. C₂²×#13133179	C₂	semidirect
6. C₂²×C₄×C₂²⋊D₅₃	C₂	semidirect
7. C₂²×#13133177	C₂	semidirect
8. C₂×C₄×D₄×D₅₃	C₂	direct (N × H)
9. C₂²×C₄⋊C₄×D₅₃	C₂	semidirect
10. C₂²×C₂²⋊C₄×D₅₃	C₂	semidirect
11. C₂²×C₄²×D₅₃	C₂	semidirect
12. C₂²×C₄×D₄×C₅₃	C₂	semidirect
16. C₂²×D₄×D₅₃	C₄	direct (N × H)
18. C₂⁴×Dic₅₃	C₂²	semidirect
24. C₂³×C₄×D₅₃	C₂²	semidirect
25. C₂³×C₄×D₅₃	C₂²	semidirect
26. C₂³×C₄×D₅₃	C₂²	semidirect
27. C₂×C₄×D₂₁₂	C₂²	semidirect
28. C₂×D₄×Dic₅₃	C₂²	semidirect
29. C₂×#13133179	C₂²	semidirect
30. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
31. C₂×#13133177	C₂²	semidirect
32. C₄×D₄×D₅₃	C₂²	direct (N × H)
33. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
34. C₂²×#11719919	C₂²	semidirect
35. C₂×C₄⋊C₄×D₅₃	C₂²	semidirect
36. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
37. C₂²×#11719921	C₂²	semidirect
38. C₂²×#11719921	C₂²	semidirect
39. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
40. C₂²×C₄×Dic₅₃	C₂²	semidirect
41. C₂²×C₄×Dic₅₃	C₂²	semidirect
42. C₂²×C₄×Dic₅₃	C₂²	semidirect
43. C₂×C₄²×D₅₃	C₂²	semidirect
45. C₂⁴×C₄×C₅₃	C₂²	semidirect
46. C₂×C₄×D₄×C₅₃	C₂²	semidirect
47. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
48. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
49. C₂²×C₄²×C₅₃	C₂²	semidirect
52. C₂⁴×D₅₃	C₂×C₄	semidirect
56. C₂²×D₂₁₂	C₂×C₄	semidirect
58. C₂²×C₂²⋊D₅₃	C₂×C₄	semidirect
60. C₂×D₄×D₅₃	C₂×C₄	direct (N × H)
67. C₂³×Dic₅₃	C₂³	semidirect
73. C₂²×C₄×D₅₃	C₂×C₄	semidirect
74. C₂²×C₄×D₅₃	D₄	direct (N × H)
79. C₂²×C₄×D₅₃	C₂³	semidirect
80. C₂²×C₄×D₅₃	C₂³	semidirect
81. C₂²×C₄×D₅₃	C₂³	semidirect
82. C₄×D₂₁₂	C₂³	semidirect
83. D₄×Dic₅₃	C₂³	semidirect
84. #13133179	C₂³	semidirect
85. C₄×C₂²⋊D₅₃	C₂³	semidirect
86. #13133177	C₂³	semidirect
87. C₂×C₄⋊Dic₅₃	C₂³	semidirect
88. C₂×#11719919	C₂³	semidirect
89. C₄⋊C₄×D₅₃	C₂³	semidirect
90. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
91. C₂×#11719921	C₂³	semidirect
92. C₂×#11719921	C₂³	semidirect
93. C₂²⋊C₄×D₅₃	C₂³	semidirect
94. C₂×C₄×Dic₅₃	C₂³	semidirect
95. C₂×C₄×Dic₅₃	C₂³	semidirect
96. C₂×C₄×Dic₅₃	C₂³	semidirect
97. C₄²×D₅₃	C₂³	semidirect
99. C₂²×D₄×C₅₃	C₂×C₄	semidirect
106. C₂³×C₄×C₅₃	C₂³	semidirect
107. C₄×D₄×C₅₃	C₂³	semidirect
108. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
109. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
110. C₂×C₄²×C₅₃	C₂³	semidirect
113. C₂⁴×C₅₃	C₂²×C₄	semidirect
122. C₂³×D₅₃	C₂²×C₄	semidirect
123. C₂³×D₅₃	C₂²×C₄	semidirect
129. C₂×D₄×C₅₃	C₂²×C₄	semidirect
134. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
135. C₂²×C₄×C₅₃	C₂×D₄	semidirect
140. C₂²×C₄×C₅₃	C₂⁴	semidirect
141. C₂×C₂²⋊D₅₃	C₂²×C₄	semidirect
146. C₂²×Dic₅₃	C₂²×C₄	semidirect
147. C₂²×Dic₅₃	C₂×D₄	semidirect
152. C₂²×Dic₅₃	C₂⁴	semidirect
153. D₄×D₅₃	C₂²×C₄	direct (N × H)
154. C₂×D₂₁₂	C₂²×C₄	semidirect
161. C₂×C₄×D₅₃	C₂²×C₄	semidirect
162. C₂×C₄×D₅₃	C₂×D₄	direct (N × H)
167. C₂×C₄×D₅₃	C₂⁴	semidirect
168. C₂×C₄×D₅₃	C₂⁴	semidirect
169. C₄⋊C₄×C₅₃	C₂⁴	semidirect
170. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
171. C₄²×C₅₃	C₂⁴	semidirect
172. C₂²⋊Dic₅₃	C₂⁴	semidirect
173. #11719921	C₂⁴	semidirect
174. #11719921	C₂⁴	semidirect
175. C₄⋊Dic₅₃	C₂⁴	semidirect
176. #11719919	C₂⁴	semidirect
177. C₄×Dic₅₃	C₂⁴	semidirect
178. C₄×Dic₅₃	C₂⁴	semidirect
179. C₄×Dic₅₃	C₂⁴	semidirect
184. C₂³×C₅₃	C₂³×C₄	semidirect
195. C₂²×D₅₃	C₄×D₄	direct (N × H)
197. C₂²×D₅₃	C₂³×C₄	semidirect
198. C₂²×D₅₃	C₂³×C₄	semidirect
203. D₄×C₅₃	C₂³×C₄	semidirect
207. C₂×C₄×C₅₃	C₂³×C₄	semidirect
208. C₂×C₄×C₅₃	C₂²×D₄	semidirect
213. C₂²⋊D₅₃	C₂³×C₄	semidirect
217. C₂×Dic₅₃	C₂³×C₄	semidirect
218. C₂×Dic₅₃	C₂²×D₄	semidirect
223. D₂₁₂	C₂³×C₄	semidirect
228. C₄×D₅₃	C₂³×C₄	semidirect
229. C₄×D₅₃	C₂²×D₄	direct (N × H)
234. C₂²×C₅₃	C₂×C₄×D₄	semidirect
236. C₂²×C₅₃	C₂⁴×C₄	semidirect
244. C₂×D₅₃	C₂×C₄×D₄	direct (N × H)
246. C₂×D₅₃	C₂⁴×C₄	semidirect
251. C₄×C₅₃	C₂⁴×C₄	semidirect
252. C₄×C₅₃	C₂³×D₄	semidirect
255. Dic₅₃	C₂⁴×C₄	semidirect
256. Dic₅₃	C₂³×D₄	semidirect
265. C₂²×C₄×D₄	D₅₃	direct (N × H)
270. C₂×C₅₃	C₂²×C₄×D₄	semidirect
275. D₅₃	C₂²×C₄×D₄	direct (N × H)
291. C₂⁴×C₄	C₂×D₅₃	semidirect
298. C₂×C₄×D₄	C₂×D₅₃	direct (N × H)
302. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
307. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
312. C₂²×C₄²	C₂×D₅₃	semidirect
313. C₅₃	C₂³×C₄×D₄	semidirect
327. C₂²×D₄	C₄×D₅₃	direct (N × H)
354. C₂³×C₄	C₂²×D₅₃	semidirect
360. C₄×D₄	C₂²×D₅₃	direct (N × H)
364. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
369. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
374. C₂×C₄²	C₂²×D₅₃	semidirect
390. C₂⁴	C₂×C₄×D₅₃	semidirect
398. C₂×D₄	C₂×C₄×D₅₃	direct (N × H)
426. C₂²×C₄	C₂×C₄×D₅₃	semidirect
427. C₂²×C₄	D₄×D₅₃	direct (N × H)
432. C₂²×C₄	C₂³×D₅₃	semidirect
435. C₄⋊C₄	C₂³×D₅₃	semidirect
439. C₂²⋊C₄	C₂³×D₅₃	semidirect
443. C₄²	C₂³×D₅₃	semidirect
462. C₂³	C₂²×C₄×D₅₃	semidirect
469. D₄	C₂²×C₄×D₅₃	direct (N × H)
488. C₂×C₄	C₂²×C₄×D₅₃	semidirect
489. C₂×C₄	C₂×D₄×D₅₃	direct (N × H)
506. C₂²	C₄×D₄×D₅₃	direct (N × H)
508. C₂²	C₂³×C₄×D₅₃	semidirect
518. C₄	C₂³×C₄×D₅₃	semidirect
519. C₄	C₂²×D₄×D₅₃	direct (N × H)
523. C₂	C₂×C₄×D₄×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
C₂³×D₄×D₅₃	2
C₂⁴×C₄×D₅₃	2
C₂²×C₄×D₂₁₂	2
C₂²×D₄×Dic₅₃	2
C₂²×#13133179	2
C₂²×C₄×C₂²⋊D₅₃	2
C₂²×#13133177	2
C₂×C₄×D₄×D₅₃	2
C₂²×C₄⋊C₄×D₅₃	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂²×C₄²×D₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₂³×C₄×D₄	53

Subgroups — C22×C4×D4×D53 #28290653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×D₄×D₅₃ #28290653