Subgroups — C₂²×#16811095 #28290670

Order 13568.

14590 conjugacy classes of subgroups.

The subgroup lattice has 504 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	C₂×#16811095	C₂	1	48	yes	maximal, direct product factor
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	C₂²×#13133175	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#13133174	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×#11719921	C₂	1	2	yes	maximal, complemented
11	C₂²×C₂²⋊C₄×D₅₃	C₂	1	2	yes	maximal, 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	complemented
17	C₂²×D₄×D₅₃	C₂²	1	24	yes	—
18	C₂⁴×Dic₅₃	C₂²	1	2	yes	complemented
19	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
20	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
21	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
22	#16811095	C₂²	1	256	yes	direct product factor
23	C₂×C₄×D₂₁₂	C₂²	1	24	yes	complemented
24	C₂×D₄×Dic₅₃	C₂²	1	24	yes	complemented
25	C₂×#13133179	C₂²	1	24	yes	complemented
26	C₂×C₄×C₂²⋊D₅₃	C₂²	1	48	yes	complemented
27	C₂×#13133177	C₂²	1	48	yes	complemented
28	C₂×#13133175	C₂²	1	24	yes	complemented
29	C₂×#13133174	C₂²	1	24	yes	complemented
30	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
31	C₂²×#11719919	C₂²	1	1	yes	characteristic, complemented
32	C₂²×#11719919	C₂²	1	1	yes	characteristic, complemented
33	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	complemented
34	C₂²×#11719921	C₂²	1	1	yes	characteristic
35	C₂²×#11719921	C₂²	1	2	yes	—
36	C₂²×#11719921	C₂²	1	1	yes	characteristic
37	C₂²×#11719921	C₂²	1	2	yes	complemented
38	C₂²×#11719921	C₂²	1	2	yes	complemented
39	C₂²×#11719921	C₂²	1	48	yes	complemented
40	C₂×C₂²⋊C₄×D₅₃	C₂²	1	48	yes	complemented
41	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
42	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	complemented
43	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
44	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	complemented
45	C₂×C₄×D₄×C₅₃	C₂²	1	24	yes	complemented
46	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
47	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
48	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
49	C₂⁴×D₅₃	—	4	8	no	—
50	C₂⁴×D₅₃	—	4	8	no	—
51	C₂⁴×D₅₃	C₂×C₄	1	32	yes	complemented
52	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
53	C₂⁴×D₅₃	C₂³	1	2	yes	—
54	C₂⁴×D₅₃	C₂³	1	24	yes	—
55	C₂²×D₂₁₂	C₂×C₄	1	16	yes	complemented
56	C₂²×D₂₁₂	C₂³	1	12	yes	—
57	C₂²×C₂²⋊D₅₃	C₂×C₄	1	32	yes	complemented
58	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
59	C₂×D₄×D₅₃	C₂×C₄	1	384	yes	complemented
60	C₂×D₄×D₅₃	C₂³	1	64	yes	—
61	C₂³×Dic₅₃	—	4	8	no	—
62	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
63	C₂³×Dic₅₃	C₂³	1	2	yes	—
64	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
65	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
66	C₂³×Dic₅₃	C₂³	1	24	yes	complemented
67	C₂²×C₄×D₅₃	—	4	8	no	—
68	C₂²×C₄×D₅₃	—	4	8	no	—
69	C₂²×C₄×D₅₃	C₂×C₄	1	16	yes	complemented
70	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
71	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
72	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
73	C₄×D₂₁₂	C₂³	1	64	yes	complemented
74	D₄×Dic₅₃	C₂³	1	64	yes	complemented
75	#13133179	C₂³	1	64	yes	complemented
76	C₄×C₂²⋊D₅₃	C₂³	1	128	yes	complemented
77	#13133177	C₂³	1	128	yes	complemented
78	#13133175	C₂³	1	64	yes	complemented
79	#13133174	C₂³	1	64	yes	complemented
80	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	complemented
81	C₂×#11719919	C₂³	1	12	yes	complemented
82	C₂×#11719919	C₂³	1	12	yes	complemented
83	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	complemented
84	C₂×#11719921	C₂³	1	12	yes	—
85	C₂×#11719921	C₂³	1	24	yes	—
86	C₂×#11719921	C₂³	1	12	yes	—
87	C₂×#11719921	C₂³	1	24	yes	complemented
88	C₂×#11719921	C₂³	1	24	yes	complemented
89	C₂×#11719921	C₂³	1	128	yes	complemented
90	C₂²⋊C₄×D₅₃	C₂³	1	128	yes	complemented
91	C₂×C₄×Dic₅₃	C₂³	1	12	yes	complemented
92	C₂×C₄×Dic₅₃	C₂³	1	24	yes	complemented
93	C₂⁵×C₅₃	C₂³	1	2	yes	—
94	C₂²×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
95	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
96	C₂³×C₄×C₅₃	—	4	8	no	—
97	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
98	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
99	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
100	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
101	C₂³×C₄×C₅₃	C₂³	1	24	yes	complemented
102	C₄×D₄×C₅₃	C₂³	1	64	yes	complemented
103	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	complemented
104	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	complemented
105	C₂×C₄²×C₅₃	C₂³	1	12	yes	complemented
106	C₂⁴×C₅₃	—	2	8	no	—
107	C₂⁴×C₅₃	—	2	8	no	—
108	C₂⁴×C₅₃	C₂²×C₄	1	16	yes	—
109	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
110	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
111	C₂³×D₅₃	—	2	8	no	—
112	C₂³×D₅₃	—	2	8	no	—
113	C₂³×D₅₃	—	2	4	no	—
114	C₂³×D₅₃	—	2	4	no	—
115	C₂³×D₅₃	—	4	128	no	—
116	C₂³×D₅₃	—	4	48	no	—
117	C₂³×D₅₃	—	4	48	no	—
118	C₂³×D₅₃	C₂²×C₄	1	8	yes	—
119	C₂³×D₅₃	C₂²×C₄	1	16	yes	—
120	C₂³×D₅₃	C₂²×C₄	1	192	yes	complemented
121	C₂³×D₅₃	C₂⁴	1	6	yes	—
122	C₂³×D₅₃	C₂⁴	1	12	yes	—
123	C₂³×D₅₃	C₂⁴	1	32	yes	—
124	C₂×D₄×C₅₃	C₂²×C₄	1	96	yes	complemented
125	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
126	C₂²×C₄×C₅₃	—	2	4	no	—
127	C₂²×C₄×C₅₃	—	2	8	no	—
128	C₂²×C₄×C₅₃	—	2	4	no	—
129	C₂²×C₄×C₅₃	—	4	48	no	—
130	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
131	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
132	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
133	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
134	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
135	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	complemented
136	C₂×C₂²⋊D₅₃	C₂²×C₄	1	192	yes	complemented
137	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
138	C₂²×Dic₅₃	—	2	4	no	—
139	C₂²×Dic₅₃	—	2	8	no	—
140	C₂²×Dic₅₃	—	2	4	no	—
141	C₂²×Dic₅₃	—	4	48	no	—
142	C₂²×Dic₅₃	C₂²×C₄	1	8	yes	—
143	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
144	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
145	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
146	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
147	C₂²×Dic₅₃	C₂⁴	1	32	yes	complemented
148	D₄×D₅₃	C₂²×C₄	1	512	yes	complemented
149	C₂×D₂₁₂	C₂²×C₄	1	96	yes	complemented
150	C₂×D₂₁₂	C₂⁴	1	16	yes	—
151	C₂×C₄×D₅₃	—	4	48	no	—
152	C₂×C₄×D₅₃	—	4	48	no	—
153	C₂×C₄×D₅₃	C₂²×C₄	1	96	yes	complemented
154	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
155	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
156	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
157	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
158	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
159	C₄²×C₅₃	C₂⁴	1	16	yes	complemented
160	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
161	#11719921	C₂⁴	1	16	yes	—
162	#11719921	C₂⁴	1	32	yes	—
163	#11719921	C₂⁴	1	16	yes	—
164	#11719921	C₂⁴	1	32	yes	complemented
165	#11719921	C₂⁴	1	32	yes	complemented
166	C₄⋊Dic₅₃	C₂⁴	1	16	yes	complemented
167	#11719919	C₂⁴	1	16	yes	complemented
168	#11719919	C₂⁴	1	16	yes	complemented
169	C₄×Dic₅₃	C₂⁴	1	16	yes	complemented
170	C₄×Dic₅₃	C₂⁴	1	32	yes	complemented
171	C₂³×C₅₃	—	2	64	no	—
172	C₂³×C₅₃	—	2	24	no	—
173	C₂³×C₅₃	—	2	24	no	—
174	C₂³×C₅₃	C₂³×C₄	1	48	yes	—
175	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
176	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
177	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
178	C₂³×C₅₃	C₂⁵	1	8	yes	—
179	C₂³×C₅₃	C₂⁵	1	3	yes	—
180	C₂²×D₅₃	—	2	32	no	—
181	C₂²×D₅₃	—	2	64	no	—
182	C₂²×D₅₃	—	2	24	no	—
183	C₂²×D₅₃	—	2	24	no	—
184	C₂²×D₅₃	—	2	12	no	—
185	C₂²×D₅₃	—	2	12	no	—
186	C₂²×D₅₃	—	4	384	no	—
187	C₂²×D₅₃	—	4	32	no	—
188	C₂²×D₅₃	—	4	32	no	—
189	C₂²×D₅₃	C₂³×C₄	1	24	yes	—
190	C₂²×D₅₃	C₂³×C₄	1	48	yes	—
191	C₂²×D₅₃	C₂³×C₄	1	128	yes	complemented
192	C₂²×D₅₃	C₂⁵	1	4	yes	—
193	C₂²×D₅₃	C₂⁵	1	8	yes	—
194	D₄×C₅₃	C₂³×C₄	1	64	yes	complemented
195	C₂×C₄×C₅₃	—	2	12	no	—
196	C₂×C₄×C₅₃	—	2	24	no	—
197	C₂×C₄×C₅₃	—	2	12	no	—
198	C₂×C₄×C₅₃	—	4	32	no	—
199	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
200	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
201	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
202	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
203	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
204	C₂²⋊D₅₃	C₂³×C₄	1	128	yes	complemented
205	C₂×Dic₅₃	—	2	12	no	—
206	C₂×Dic₅₃	—	2	24	no	—
207	C₂×Dic₅₃	—	2	12	no	—
208	C₂×Dic₅₃	—	4	32	no	—
209	C₂×Dic₅₃	C₂³×C₄	1	24	yes	—
210	C₂×Dic₅₃	C₂⁵	1	4	yes	—
211	C₂×Dic₅₃	C₂⁵	1	8	yes	—
212	C₂×Dic₅₃	C₂⁵	1	4	yes	—
213	C₂×Dic₅₃	C₂⁵	1	4	yes	—
214	D₂₁₂	C₂³×C₄	1	64	yes	complemented
215	C₄×D₅₃	—	4	32	no	—
216	C₄×D₅₃	—	4	32	no	—
217	C₄×D₅₃	C₂³×C₄	1	64	yes	complemented
218	C₂²×#518	—	53	1	no	maximal, Sylow
219	C₂²×C₅₃	—	2	8	no	—
220	C₂²×C₅₃	—	2	8	no	—
221	C₂²×C₅₃	—	2	96	no	—
222	C₂²×C₅₃	#518	1	16	yes	complemented
223	C₂²×C₅₃	C₂⁴×C₄	1	16	yes	—
224	C₂²×C₅₃	C₂⁴×C₄	1	6	yes	—
225	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
226	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
227	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
228	C₂×D₅₃	—	2	8	no	—
229	C₂×D₅₃	—	2	8	no	—
230	C₂×D₅₃	—	2	4	no	—
231	C₂×D₅₃	—	2	4	no	—
232	C₂×D₅₃	—	2	48	no	—
233	C₂×D₅₃	—	2	96	no	—
234	C₂×D₅₃	—	4	128	no	—
235	C₂×D₅₃	C₂⁴×C₄	1	8	yes	—
236	C₂×D₅₃	C₂⁴×C₄	1	16	yes	—
237	C₄×C₅₃	—	2	4	no	—
238	C₄×C₅₃	—	2	8	no	—
239	C₄×C₅₃	—	2	4	no	—
240	C₄×C₅₃	C₂⁴×C₄	1	8	yes	—
241	Dic₅₃	—	2	4	no	—
242	Dic₅₃	—	2	8	no	—
243	Dic₅₃	—	2	4	no	—
244	Dic₅₃	C₂⁴×C₄	1	8	yes	—
245	C₂⁴×D₄	—	53	1	no	—
246	C₂×#518	—	53	48	no	—
247	C₂²×C₄×D₄	—	53	1	no	—
248	C₂²×C₄×D₄	—	53	1	no	—
249	C₂²×C₄×D₄	—	53	2	no	—
250	C₂²×C₄×D₄	—	53	1	no	—
251	C₂²×C₄×D₄	—	53	2	no	—
252	C₂²×C₄×D₄	D₅₃	1	1	yes	characteristic, complemented
253	C₂²×#117	—	53	1	no	—
254	C₂²×#117	—	53	1	no	—
255	C₂³×C₂²⋊C₄	—	53	2	no	—
256	C₂³×C₂²⋊C₄	—	53	2	no	—
257	C₂×C₅₃	—	2	16	no	—
258	C₂×C₅₃	C₂×#518	1	12	yes	complemented
259	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
260	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
261	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
262	D₅₃	—	2	8	no	—
263	D₅₃	—	2	16	no	—
264	C₂⁶	—	53	2	no	—
265	C₂³×D₄	—	53	2	no	—
266	C₂³×D₄	—	53	32	no	—
267	C₂³×D₄	—	53	24	no	—
268	C₂³×D₄	—	53	1	no	—
269	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
270	C₂⁴×C₄	—	53	1	no	—
271	C₂⁴×C₄	—	53	2	no	—
272	C₂⁴×C₄	—	53	2	no	—
273	C₂⁴×C₄	—	53	2	no	—
274	C₂⁴×C₄	C₂×D₅₃	1	2	yes	complemented
275	#518	—	53	256	no	—
276	C₂×C₄×D₄	—	53	24	no	—
277	C₂×C₄×D₄	—	53	48	no	—
278	C₂×C₄×D₄	—	53	24	no	—
279	C₂×C₄×D₄	—	53	48	no	—
280	C₂×C₄×D₄	—	53	24	no	—
281	C₂×C₄×D₄	C₂×D₅₃	1	24	yes	complemented
282	C₂×#117	—	53	24	no	—
283	C₂×#117	—	53	24	no	—
284	C₂²×C₄⋊C₄	—	53	1	no	—
285	C₂²×C₄⋊C₄	—	53	1	no	—
286	C₂²×C₄⋊C₄	—	53	1	no	—
287	C₂²×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
288	C₂²×C₂²⋊C₄	—	53	2	no	—
289	C₂²×C₂²⋊C₄	—	53	1	no	—
290	C₂²×C₂²⋊C₄	—	53	2	no	—
291	C₂²×C₂²⋊C₄	—	53	1	no	—
292	C₂²×C₂²⋊C₄	—	53	2	no	—
293	C₂²×C₂²⋊C₄	—	53	2	no	—
294	C₂²×C₂²⋊C₄	—	53	48	no	—
295	C₂²×C₂²⋊C₄	—	53	48	no	—
296	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
297	C₂²×C₄²	—	53	2	no	—
298	C₂²×C₄²	—	53	1	no	—
299	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
300	C₅₃	C₂²×#518	1	1	yes	characteristic, complemented, Sylow
301	C₂⁵	—	53	1	no	—
302	C₂⁵	—	53	2	no	—
303	C₂⁵	—	53	32	no	—
304	C₂⁵	—	53	24	no	—
305	C₂⁵	—	212	8	no	—
306	C₂⁵	—	212	8	no	—
307	C₂⁵	C₂²×D₅₃	1	2	yes	—
308	C₂²×D₄	—	53	32	no	—
309	C₂²×D₄	—	53	24	no	—
310	C₂²×D₄	—	53	384	no	—
311	C₂²×D₄	—	53	64	no	—
312	C₂²×D₄	—	53	16	no	—
313	C₂²×D₄	—	53	12	no	—
314	C₂²×D₄	C₄×D₅₃	1	16	yes	complemented
315	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
316	C₂³×C₄	—	4	8	no	—
317	C₂³×C₄	—	53	1	no	—
318	C₂³×C₄	—	53	16	no	—
319	C₂³×C₄	—	53	12	no	—
320	C₂³×C₄	—	53	1	no	—
321	C₂³×C₄	—	53	2	no	—
322	C₂³×C₄	—	53	1	no	—
323	C₂³×C₄	—	53	24	no	—
324	C₂³×C₄	—	53	24	no	—
325	C₂³×C₄	—	53	24	no	—
326	C₂³×C₄	—	212	8	no	—
327	C₂³×C₄	—	212	8	no	—
328	C₂³×C₄	—	212	8	no	—
329	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
330	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
331	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
332	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
333	C₂³×C₄	C₂²×D₅₃	1	24	yes	complemented
334	C₄×D₄	—	53	64	no	—
335	C₄×D₄	—	53	128	no	—
336	C₄×D₄	—	53	64	no	—
337	C₄×D₄	—	53	128	no	—
338	C₄×D₄	—	53	64	no	—
339	C₄×D₄	C₂²×D₅₃	1	64	yes	complemented
340	#117	—	53	64	no	—
341	#117	—	53	64	no	—
342	C₂×C₄⋊C₄	—	53	12	no	—
343	C₂×C₄⋊C₄	—	53	12	no	—
344	C₂×C₄⋊C₄	—	53	12	no	—
345	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	complemented
346	C₂×C₂²⋊C₄	—	53	24	no	—
347	C₂×C₂²⋊C₄	—	53	24	no	—
348	C₂×C₂²⋊C₄	—	53	12	no	—
349	C₂×C₂²⋊C₄	—	53	12	no	—
350	C₂×C₂²⋊C₄	—	53	24	no	—
351	C₂×C₂²⋊C₄	—	53	24	no	—
352	C₂×C₂²⋊C₄	—	53	128	no	—
353	C₂×C₂²⋊C₄	—	53	128	no	—
354	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	complemented
355	C₂×C₄²	—	53	24	no	—
356	C₂×C₄²	—	53	12	no	—
357	C₂×C₄²	C₂²×D₅₃	1	12	yes	complemented
358	C₂⁴	—	2	8	no	—
359	C₂⁴	—	2	8	no	—
360	C₂⁴	—	53	8	no	—
361	C₂⁴	—	53	16	no	—
362	C₂⁴	—	53	6	no	—
363	C₂⁴	—	53	12	no	—
364	C₂⁴	—	53	192	no	—
365	C₂⁴	—	53	32	no	—
366	C₂⁴	—	106	8	no	—
367	C₂⁴	—	106	4	no	—
368	C₂⁴	—	106	8	no	—
369	C₂⁴	—	106	4	no	—
370	C₂⁴	—	212	128	no	—
371	C₂⁴	—	212	48	no	—
372	C₂⁴	—	212	48	no	—
373	C₂⁴	C₂×C₄×D₅₃	1	16	yes	—
374	C₂⁴	C₂³×D₅₃	1	12	yes	—
375	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
376	C₂×D₄	—	53	192	no	—
377	C₂×D₄	—	53	32	no	—
378	C₂×D₄	—	53	512	no	—
379	C₂×D₄	—	53	96	no	—
380	C₂×D₄	—	53	16	no	—
381	C₂×D₄	C₂×C₄×D₅₃	1	96	yes	complemented
382	C₂×D₄	C₂³×D₅₃	1	16	yes	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	8	no	—
385	C₂²×C₄	—	2	4	no	—
386	C₂²×C₄	—	4	48	no	—
387	C₂²×C₄	—	53	8	no	—
388	C₂²×C₄	—	53	6	no	—
389	C₂²×C₄	—	53	96	no	—
390	C₂²×C₄	—	53	16	no	—
391	C₂²×C₄	—	53	6	no	—
392	C₂²×C₄	—	53	6	no	—
393	C₂²×C₄	—	53	12	no	—
394	C₂²×C₄	—	53	32	no	—
395	C₂²×C₄	—	53	32	no	—
396	C₂²×C₄	—	53	32	no	—
397	C₂²×C₄	—	106	4	no	—
398	C₂²×C₄	—	106	8	no	—
399	C₂²×C₄	—	106	4	no	—
400	C₂²×C₄	—	212	48	no	—
401	C₂²×C₄	—	212	48	no	—
402	C₂²×C₄	—	212	48	no	—
403	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
404	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
405	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
406	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
407	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
408	C₂²×C₄	C₂³×D₅₃	1	32	yes	complemented
409	C₄⋊C₄	—	53	16	no	—
410	C₄⋊C₄	—	53	16	no	—
411	C₄⋊C₄	—	53	16	no	—
412	C₄⋊C₄	C₂³×D₅₃	1	16	yes	complemented
413	C₂²⋊C₄	—	53	32	no	—
414	C₂²⋊C₄	—	53	32	no	—
415	C₂²⋊C₄	—	53	16	no	—
416	C₂²⋊C₄	—	53	16	no	—
417	C₂²⋊C₄	—	53	32	no	—
418	C₂²⋊C₄	—	53	32	no	—
419	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	complemented
420	C₄²	—	53	32	no	—
421	C₄²	—	53	16	no	—
422	C₄²	C₂³×D₅₃	1	16	yes	complemented
423	C₂³	—	2	64	no	—
424	C₂³	—	2	24	no	—
425	C₂³	—	2	24	no	—
426	C₂³	—	53	24	no	—
427	C₂³	—	53	48	no	—
428	C₂³	—	53	4	no	—
429	C₂³	—	53	8	no	—
430	C₂³	—	53	128	no	—
431	C₂³	—	106	64	no	—
432	C₂³	—	106	32	no	—
433	C₂³	—	106	24	no	—
434	C₂³	—	106	12	no	—
435	C₂³	—	106	24	no	—
436	C₂³	—	106	12	no	—
437	C₂³	—	212	384	no	—
438	C₂³	—	212	32	no	—
439	C₂³	—	212	32	no	—
440	C₂³	C₂²×C₄×D₅₃	1	48	yes	—
441	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
442	C₂³	#13133262	1	4	yes	—
443	C₂³	#13133263	1	4	yes	—
444	C₂³	C₂⁴×D₅₃	1	8	yes	—
445	C₂³	C₂⁴×D₅₃	1	3	yes	—
446	D₄	—	53	128	no	—
447	D₄	—	53	64	no	—
448	D₄	C₂²×C₄×D₅₃	1	64	yes	complemented
449	C₂×C₄	—	2	12	no	—
450	C₂×C₄	—	2	24	no	—
451	C₂×C₄	—	2	12	no	—
452	C₂×C₄	—	4	32	no	—
453	C₂×C₄	—	53	24	no	—
454	C₂×C₄	—	53	4	no	—
455	C₂×C₄	—	53	64	no	—
456	C₂×C₄	—	53	4	no	—
457	C₂×C₄	—	53	8	no	—
458	C₂×C₄	—	53	4	no	—
459	C₂×C₄	—	106	12	no	—
460	C₂×C₄	—	106	24	no	—
461	C₂×C₄	—	106	12	no	—
462	C₂×C₄	—	212	32	no	—
463	C₂×C₄	—	212	32	no	—
464	C₂×C₄	—	212	32	no	—
465	C₂×C₄	C₂²×C₄×D₅₃	1	24	yes	—
466	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
467	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
468	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
469	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
470	C₂²	—	2	96	no	—
471	C₂²	—	2	8	no	—
472	C₂²	—	2	8	no	—
473	C₂²	—	53	8	no	—
474	C₂²	—	53	16	no	—
475	C₂²	—	106	48	no	—
476	C₂²	—	106	96	no	—
477	C₂²	—	106	8	no	—
478	C₂²	—	106	4	no	—
479	C₂²	—	106	8	no	—
480	C₂²	—	106	4	no	—
481	C₂²	—	212	128	no	—
482	C₂²	#16811095	1	16	yes	direct product factor
483	C₂²	C₂³×C₄×D₅₃	1	16	yes	—
484	C₂²	C₂³×C₄×D₅₃	1	6	yes	—
485	C₂²	C₂×#13133262	1	6	yes	—
486	C₂²	C₂×#13133263	1	6	yes	—
487	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
488	C₄	—	2	4	no	—
489	C₄	—	2	8	no	—
490	C₄	—	2	4	no	—
491	C₄	—	53	8	no	—
492	C₄	—	106	4	no	—
493	C₄	—	106	8	no	—
494	C₄	—	106	4	no	—
495	C₄	C₂³×C₄×D₅₃	1	8	yes	—
496	C₂	—	2	16	no	—
497	C₂	—	106	8	no	—
498	C₂	—	106	16	no	—
499	C₂	C₂×#16811095	1	12	yes	direct product factor
500	C₂	C₂⁴×C₄×D₅₃	1	1	yes	characteristic
501	C₂	C₂²×#13133262	1	1	yes	characteristic
502	C₂	C₂²×#13133263	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×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	C₂²×#518
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#16811095	C₂	direct (N × H)
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. C₂²×#13133175	C₂	semidirect
9. C₂²×#13133174	C₂	semidirect
10. C₂³×#11719921	C₂	semidirect
11. C₂²×C₂²⋊C₄×D₅₃	C₂	semidirect
12. C₂²×C₄×D₄×C₅₃	C₂	semidirect
16. C₂²×D₄×D₅₃	C₄	semidirect
18. C₂⁴×Dic₅₃	C₂²	semidirect
20. C₂³×C₄×D₅₃	C₂²	semidirect
21. C₂³×C₄×D₅₃	C₂²	semidirect
22. #16811095	C₂²	direct (N × H)
23. C₂×C₄×D₂₁₂	C₂²	semidirect
24. C₂×D₄×Dic₅₃	C₂²	semidirect
25. C₂×#13133179	C₂²	semidirect
26. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
27. C₂×#13133177	C₂²	semidirect
28. C₂×#13133175	C₂²	semidirect
29. C₂×#13133174	C₂²	semidirect
30. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
31. C₂²×#11719919	C₂²	semidirect
32. C₂²×#11719919	C₂²	semidirect
33. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
37. C₂²×#11719921	C₂²	semidirect
38. C₂²×#11719921	C₂²	semidirect
39. C₂²×#11719921	C₂²	semidirect
40. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
41. C₂²×C₄×Dic₅₃	C₂²	semidirect
42. C₂²×C₄×Dic₅₃	C₂²	semidirect
44. C₂⁴×C₄×C₅₃	C₂²	semidirect
45. C₂×C₄×D₄×C₅₃	C₂²	semidirect
46. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
47. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
48. C₂²×C₄²×C₅₃	C₂²	semidirect
51. C₂⁴×D₅₃	C₂×C₄	semidirect
55. C₂²×D₂₁₂	C₂×C₄	semidirect
57. C₂²×C₂²⋊D₅₃	C₂×C₄	semidirect
59. C₂×D₄×D₅₃	C₂×C₄	semidirect
66. C₂³×Dic₅₃	C₂³	semidirect
69. C₂²×C₄×D₅₃	C₂×C₄	semidirect
71. C₂²×C₄×D₅₃	C₂³	semidirect
72. C₂²×C₄×D₅₃	C₂³	semidirect
73. C₄×D₂₁₂	C₂³	semidirect
74. D₄×Dic₅₃	C₂³	semidirect
75. #13133179	C₂³	semidirect
76. C₄×C₂²⋊D₅₃	C₂³	semidirect
77. #13133177	C₂³	semidirect
78. #13133175	C₂³	semidirect
79. #13133174	C₂³	semidirect
80. C₂×C₄⋊Dic₅₃	C₂³	semidirect
81. C₂×#11719919	C₂³	semidirect
82. C₂×#11719919	C₂³	semidirect
83. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
87. C₂×#11719921	C₂³	semidirect
88. C₂×#11719921	C₂³	semidirect
89. C₂×#11719921	C₂³	semidirect
90. C₂²⋊C₄×D₅₃	C₂³	semidirect
91. C₂×C₄×Dic₅₃	C₂³	semidirect
92. C₂×C₄×Dic₅₃	C₂³	semidirect
94. C₂²×D₄×C₅₃	C₂×C₄	semidirect
101. C₂³×C₄×C₅₃	C₂³	semidirect
102. C₄×D₄×C₅₃	C₂³	semidirect
103. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
104. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
105. C₂×C₄²×C₅₃	C₂³	semidirect
120. C₂³×D₅₃	C₂²×C₄	semidirect
124. C₂×D₄×C₅₃	C₂²×C₄	semidirect
135. C₂²×C₄×C₅₃	C₂⁴	semidirect
136. C₂×C₂²⋊D₅₃	C₂²×C₄	semidirect
147. C₂²×Dic₅₃	C₂⁴	semidirect
148. D₄×D₅₃	C₂²×C₄	semidirect
149. C₂×D₂₁₂	C₂²×C₄	semidirect
153. C₂×C₄×D₅₃	C₂²×C₄	semidirect
155. C₂×C₄×D₅₃	C₂⁴	semidirect
156. C₂×C₄×D₅₃	C₂⁴	semidirect
157. C₄⋊C₄×C₅₃	C₂⁴	semidirect
158. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
159. C₄²×C₅₃	C₂⁴	semidirect
160. C₂²⋊Dic₅₃	C₂⁴	semidirect
164. #11719921	C₂⁴	semidirect
165. #11719921	C₂⁴	semidirect
166. C₄⋊Dic₅₃	C₂⁴	semidirect
167. #11719919	C₂⁴	semidirect
168. #11719919	C₂⁴	semidirect
169. C₄×Dic₅₃	C₂⁴	semidirect
170. C₄×Dic₅₃	C₂⁴	semidirect
191. C₂²×D₅₃	C₂³×C₄	semidirect
194. D₄×C₅₃	C₂³×C₄	semidirect
204. C₂²⋊D₅₃	C₂³×C₄	semidirect
214. D₂₁₂	C₂³×C₄	semidirect
217. C₄×D₅₃	C₂³×C₄	semidirect
222. C₂²×C₅₃	#518	semidirect
252. C₂²×C₄×D₄	D₅₃	semidirect
258. C₂×C₅₃	C₂×#518	semidirect
274. C₂⁴×C₄	C₂×D₅₃	semidirect
281. C₂×C₄×D₄	C₂×D₅₃	semidirect
287. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
296. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
299. C₂²×C₄²	C₂×D₅₃	semidirect
300. C₅₃	C₂²×#518	semidirect
314. C₂²×D₄	C₄×D₅₃	semidirect
333. C₂³×C₄	C₂²×D₅₃	semidirect
339. C₄×D₄	C₂²×D₅₃	semidirect
345. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
354. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
357. C₂×C₄²	C₂²×D₅₃	semidirect
381. C₂×D₄	C₂×C₄×D₅₃	semidirect
408. C₂²×C₄	C₂³×D₅₃	semidirect
412. C₄⋊C₄	C₂³×D₅₃	semidirect
419. C₂²⋊C₄	C₂³×D₅₃	semidirect
422. C₄²	C₂³×D₅₃	semidirect
448. D₄	C₂²×C₄×D₅₃	semidirect
482. C₂²	#16811095	direct (N × H)
499. C₂	C₂×#16811095	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
C₂×#16811095	2
C₂²×C₄×D₂₁₂	2
C₂²×D₄×Dic₅₃	2
C₂²×#13133179	2
C₂²×C₄×C₂²⋊D₅₃	2
C₂²×#13133177	2
C₂²×#13133175	2
C₂²×#13133174	2
C₂³×#11719921	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₂²×#518	53

Subgroups — C22×#16811095 #28290670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811095 #28290670