Subgroups — C₂²×#16811133 #28290748

Order 13568.

14334 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719947	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16811133	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#13133199	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133197	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13133196	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#13133188	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13133185	C₂	1	2	yes	maximal, complemented
9	C₂²×D₄×Dic₅₃	C₂	1	2	yes	maximal, complemented
10	C₂²×#13133177	C₂	1	2	yes	maximal, complemented
11	C₂²×#13133175	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#121×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
14	C₂²×#11719947	—	2	16	no	—
15	C₂²×#11719947	C₂²	1	24	yes	complemented
16	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
18	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
19	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₂⁴×Dic₅₃	C₂²	1	2	yes	complemented
21	C₂³×C₄×D₅₃	—	2	16	no	—
22	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
23	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
24	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
25	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
26	C₂³×C₄×D₅₃	C₂²	1	24	yes	complemented
27	#16811133	C₂²	1	256	yes	direct product factor
28	C₂×#13133199	C₂²	1	48	yes	complemented
29	C₂×#13133197	C₂²	1	24	yes	complemented
30	C₂×#13133196	C₂²	1	24	yes	complemented
31	C₂×#13133188	C₂²	1	24	yes	complemented
32	C₂×#13133185	C₂²	1	48	yes	complemented
33	C₂×D₄×Dic₅₃	C₂²	1	48	yes	complemented
34	C₂×#13133177	C₂²	1	48	yes	complemented
35	C₂×#13133175	C₂²	1	24	yes	complemented
36	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
38	C₂²×#11719919	C₂²	1	2	yes	complemented
39	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
40	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	complemented
41	C₂²×#11719921	C₂²	1	2	yes	—
42	C₂²×#11719921	C₂²	1	2	yes	—
43	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	complemented
44	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
45	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
46	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×#121×C₅₃	C₂²	1	24	yes	complemented
48	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
49	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
50	C₂⁴×D₅₃	—	2	8	no	—
51	C₂⁴×D₅₃	—	2	8	no	—
52	C₂⁴×D₅₃	—	2	8	no	—
53	C₂⁴×D₅₃	C₂³	1	12	yes	—
54	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
55	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
56	C₂×#11719947	—	2	192	no	—
57	C₂×#11719947	C₂³	1	64	yes	complemented
58	C₂²×Dic₁₀₆	—	2	8	no	—
59	C₂²×Dic₁₀₆	C₂³	1	12	yes	complemented
60	C₂²×C₂²⋊D₅₃	—	2	16	no	—
61	C₂²×C₂²⋊D₅₃	—	2	16	no	—
62	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
63	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
64	C₂³×Dic₅₃	—	2	8	no	—
65	C₂³×Dic₅₃	—	2	16	no	—
66	C₂³×Dic₅₃	C₂³	1	2	yes	—
67	C₂³×Dic₅₃	C₂³	1	2	yes	—
68	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
69	C₂³×Dic₅₃	C₂³	1	12	yes	—
70	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
71	C₂³×Dic₅₃	C₂³	1	24	yes	complemented
72	C₂²×C₄×D₅₃	—	2	8	no	—
73	C₂²×C₄×D₅₃	—	2	8	no	—
74	C₂²×C₄×D₅₃	—	2	8	no	—
75	C₂²×C₄×D₅₃	—	2	192	no	—
76	C₂²×C₄×D₅₃	D₄	1	16	yes	complemented
77	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
78	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
79	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
80	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
81	C₂²×C₄×D₅₃	C₂³	1	64	yes	complemented
82	#13133199	C₂³	1	128	yes	complemented
83	#13133197	C₂³	1	64	yes	complemented
84	#13133196	C₂³	1	64	yes	complemented
85	#13133188	C₂³	1	64	yes	complemented
86	#13133185	C₂³	1	128	yes	complemented
87	D₄×Dic₅₃	C₂³	1	128	yes	complemented
88	#13133177	C₂³	1	128	yes	complemented
89	#13133175	C₂³	1	64	yes	complemented
90	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	complemented
91	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	complemented
92	C₂×#11719919	C₂³	1	24	yes	complemented
93	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	—
94	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	complemented
95	C₂×#11719921	C₂³	1	24	yes	—
96	C₂×#11719921	C₂³	1	24	yes	—
97	C₂×C₄×Dic₅₃	C₂³	1	24	yes	complemented
98	C₂⁵×C₅₃	C₂³	1	2	yes	—
99	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
100	C₂²×D₄×C₅₃	—	2	16	no	—
101	C₂²×D₄×C₅₃	—	2	8	no	—
102	C₂²×D₄×C₅₃	C₂³	1	24	yes	—
103	C₂²×D₄×C₅₃	C₂³	1	12	yes	complemented
104	C₂³×C₄×C₅₃	—	2	8	no	—
105	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
106	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
107	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
108	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
109	#121×C₅₃	C₂³	1	64	yes	complemented
110	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	complemented
111	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	complemented
112	C₂⁴×C₅₃	—	2	4	no	—
113	C₂⁴×C₅₃	—	2	4	no	—
114	C₂⁴×C₅₃	—	2	8	no	—
115	C₂⁴×C₅₃	—	2	8	no	—
116	C₂⁴×C₅₃	—	2	8	no	—
117	C₂⁴×C₅₃	#41	1	8	yes	complemented
118	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
119	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
120	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
121	C₂³×D₅₃	—	2	128	no	—
122	C₂³×D₅₃	—	2	4	no	—
123	C₂³×D₅₃	—	2	4	no	—
124	C₂³×D₅₃	—	2	4	no	—
125	C₂³×D₅₃	—	2	4	no	—
126	C₂³×D₅₃	—	2	48	no	—
127	C₂³×D₅₃	—	2	48	no	—
128	C₂³×D₅₃	—	2	48	no	—
129	C₂³×D₅₃	C₂×D₄	1	8	yes	—
130	C₂³×D₅₃	#41	1	8	yes	complemented
131	C₂³×D₅₃	C₂⁴	1	16	yes	—
132	C₂³×D₅₃	C₂⁴	1	6	yes	—
133	C₂³×D₅₃	C₂⁴	1	6	yes	—
134	C₂×D₄×C₅₃	—	2	96	no	—
135	C₂×D₄×C₅₃	—	2	48	no	—
136	C₂×D₄×C₅₃	C₂⁴	1	32	yes	—
137	C₂×D₄×C₅₃	C₂⁴	1	16	yes	complemented
138	C₂²×C₄×C₅₃	—	2	4	no	—
139	C₂²×C₄×C₅₃	—	2	8	no	—
140	C₂²×C₄×C₅₃	—	2	48	no	—
141	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
142	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
143	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
144	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
145	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
146	C₂×C₂²⋊D₅₃	—	2	96	no	—
147	C₂×C₂²⋊D₅₃	—	2	96	no	—
148	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
149	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
150	C₂²×Dic₅₃	—	2	4	no	—
151	C₂²×Dic₅₃	—	2	8	no	—
152	C₂²×Dic₅₃	—	2	8	no	—
153	C₂²×Dic₅₃	—	2	48	no	—
154	C₂²×Dic₅₃	—	2	96	no	—
155	C₂²×Dic₅₃	C₂×D₄	1	8	yes	—
156	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
157	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
158	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
159	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
160	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
161	C₂²×Dic₅₃	C₂⁴	1	32	yes	complemented
162	#11719947	—	2	256	no	—
163	C₂×C₄×D₅₃	—	2	48	no	—
164	C₂×C₄×D₅₃	—	2	48	no	—
165	C₂×C₄×D₅₃	—	2	48	no	—
166	C₂×C₄×D₅₃	—	2	256	no	—
167	C₂×C₄×D₅₃	C₂×D₄	1	96	yes	complemented
168	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
169	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
170	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
171	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
172	C₂×Dic₁₀₆	—	2	48	no	—
173	C₂×Dic₁₀₆	C₂⁴	1	16	yes	complemented
174	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
175	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
176	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
177	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
178	#11719921	C₂⁴	1	32	yes	—
179	#11719921	C₂⁴	1	32	yes	—
180	C₄⋊Dic₅₃	C₂⁴	1	16	yes	complemented
181	C₄⋊Dic₅₃	C₂⁴	1	32	yes	complemented
182	#11719919	C₂⁴	1	32	yes	complemented
183	C₄×Dic₅₃	C₂⁴	1	32	yes	complemented
184	C₂³×C₅₃	—	2	32	no	—
185	C₂³×C₅₃	—	2	12	no	—
186	C₂³×C₅₃	—	2	12	no	—
187	C₂³×C₅₃	—	2	24	no	—
188	C₂³×C₅₃	—	2	24	no	—
189	C₂³×C₅₃	—	2	24	no	—
190	C₂³×C₅₃	—	4	32	no	—
191	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
192	C₂³×C₅₃	C₂×#41	1	4	yes	—
193	C₂³×C₅₃	C₂×#41	1	4	yes	—
194	C₂³×C₅₃	C₂×#41	1	24	yes	complemented
195	C₂³×C₅₃	C₂⁵	1	8	yes	—
196	C₂³×C₅₃	C₂⁵	1	4	yes	—
197	C₂³×C₅₃	C₂⁵	1	3	yes	—
198	C₂²×D₅₃	—	2	384	no	—
199	C₂²×D₅₃	—	2	32	no	—
200	C₂²×D₅₃	—	2	32	no	—
201	C₂²×D₅₃	—	2	12	no	—
202	C₂²×D₅₃	—	2	12	no	—
203	C₂²×D₅₃	—	2	12	no	—
204	C₂²×D₅₃	—	2	12	no	—
205	C₂²×D₅₃	—	2	32	no	—
206	C₂²×D₅₃	—	2	32	no	—
207	C₂²×D₅₃	—	2	32	no	—
208	C₂²×D₅₃	C₂²×D₄	1	24	yes	—
209	C₂²×D₅₃	C₂×#41	1	24	yes	complemented
210	C₂²×D₅₃	C₂⁵	1	4	yes	—
211	C₂²×D₅₃	C₂⁵	1	4	yes	—
212	D₄×C₅₃	—	2	64	no	—
213	D₄×C₅₃	—	2	32	no	—
214	C₂×C₄×C₅₃	—	2	12	no	—
215	C₂×C₄×C₅₃	—	2	24	no	—
216	C₂×C₄×C₅₃	—	2	32	no	—
217	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
218	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
219	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
220	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
221	C₂²⋊D₅₃	—	2	64	no	—
222	C₂²⋊D₅₃	—	2	64	no	—
223	C₂×Dic₅₃	—	2	24	no	—
224	C₂×Dic₅₃	—	2	12	no	—
225	C₂×Dic₅₃	—	2	24	no	—
226	C₂×Dic₅₃	—	2	32	no	—
227	C₂×Dic₅₃	—	2	64	no	—
228	C₂×Dic₅₃	C₂²×D₄	1	24	yes	—
229	C₂×Dic₅₃	C₂⁵	1	8	yes	—
230	C₂×Dic₅₃	C₂⁵	1	4	yes	—
231	C₂×Dic₅₃	C₂⁵	1	8	yes	—
232	C₂×Dic₅₃	C₂⁵	1	4	yes	—
233	C₄×D₅₃	—	2	32	no	—
234	C₄×D₅₃	—	2	32	no	—
235	C₄×D₅₃	—	2	32	no	—
236	C₄×D₅₃	C₂²×D₄	1	64	yes	complemented
237	Dic₁₀₆	—	2	32	no	—
238	C₂²×#525	—	53	1	no	maximal, Sylow
239	C₂²×C₅₃	—	2	4	no	—
240	C₂²×C₅₃	—	2	4	no	—
241	C₂²×C₅₃	—	2	48	no	—
242	C₂²×C₅₃	—	2	8	no	—
243	C₂²×C₅₃	—	2	8	no	—
244	C₂²×C₅₃	—	2	8	no	—
245	C₂²×C₅₃	—	4	48	no	—
246	C₂²×C₅₃	#525	1	16	yes	complemented
247	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
248	C₂²×C₅₃	C₂²×#41	1	6	yes	—
249	C₂²×C₅₃	C₂²×#41	1	6	yes	—
250	C₂²×C₅₃	C₂²×#41	1	8	yes	complemented
251	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
252	C₂×D₅₃	—	2	128	no	—
253	C₂×D₅₃	—	2	4	no	—
254	C₂×D₅₃	—	2	4	no	—
255	C₂×D₅₃	—	2	4	no	—
256	C₂×D₅₃	—	2	4	no	—
257	C₂×D₅₃	—	2	48	no	—
258	C₂×D₅₃	—	2	48	no	—
259	C₂×D₅₃	C₂³×D₄	1	8	yes	—
260	C₂×D₅₃	C₂²×#41	1	8	yes	complemented
261	C₄×C₅₃	—	2	4	no	—
262	C₄×C₅₃	—	2	8	no	—
263	C₄×C₅₃	C₂³×D₄	1	8	yes	—
264	Dic₅₃	—	2	8	no	—
265	Dic₅₃	—	2	4	no	—
266	Dic₅₃	—	2	8	no	—
267	Dic₅₃	C₂³×D₄	1	8	yes	—
268	C₂³×#41	—	53	1	no	—
269	C₂⁵×C₄	—	53	1	no	—
270	C₂×#525	—	53	48	no	—
271	C₂²×#123	—	53	2	no	—
272	C₂²×C₂²⋊Q₈	—	53	1	no	—
273	C₂²×C₂²⋊Q₈	—	53	1	no	—
274	C₂²×#121	—	53	1	no	—
275	C₂²×#121	D₅₃	1	1	yes	characteristic, complemented
276	C₂²×#120	—	53	2	no	—
277	C₂²×C₄×D₄	—	53	2	no	—
278	C₂²×C₄×D₄	—	53	2	no	—
279	C₂²×#117	—	53	1	no	—
280	C₂×C₅₃	—	2	8	no	—
281	C₂×C₅₃	—	4	8	no	—
282	C₂×C₅₃	C₂×#525	1	12	yes	complemented
283	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
284	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
285	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
286	D₅₃	—	2	8	no	—
287	D₅₃	—	2	8	no	—
288	C₂⁶	—	53	1	no	—
289	C₂²×#41	—	53	24	no	—
290	C₂²×#41	—	106	16	no	—
291	C₂³×Q₈	—	53	1	no	—
292	C₂³×D₄	—	53	2	no	—
293	C₂³×D₄	—	53	2	no	—
294	C₂³×D₄	C₂×D₅₃	1	2	yes	—
295	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
296	C₂⁴×C₄	—	53	1	no	—
297	C₂⁴×C₄	—	53	1	no	—
298	C₂⁴×C₄	—	53	2	no	—
299	C₂⁴×C₄	—	53	1	no	—
300	C₂⁴×C₄	—	53	1	no	—
301	C₂⁴×C₄	—	53	1	no	—
302	C₂⁴×C₄	—	53	24	no	—
303	C₂⁴×C₄	—	106	16	no	—
304	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
305	#525	—	53	256	no	—
306	C₂×#123	—	53	48	no	—
307	C₂×C₂²⋊Q₈	—	53	24	no	—
308	C₂×C₂²⋊Q₈	—	53	24	no	—
309	C₂×#121	—	53	24	no	—
310	C₂×#121	C₂×D₅₃	1	24	yes	complemented
311	C₂×#120	—	53	48	no	—
312	C₂×C₄×D₄	—	53	48	no	—
313	C₂×C₄×D₄	—	53	48	no	—
314	C₂×#117	—	53	24	no	—
315	C₂²×C₄⋊C₄	—	53	2	no	—
316	C₂²×C₄⋊C₄	—	53	1	no	—
317	C₂²×C₄⋊C₄	—	53	2	no	—
318	C₂²×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
319	C₂²×C₂²⋊C₄	—	53	2	no	—
320	C₂²×C₂²⋊C₄	—	53	2	no	—
321	C₂²×C₂²⋊C₄	—	53	2	no	—
322	C₂²×C₂²⋊C₄	—	53	2	no	—
323	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
324	C₂²×C₄²	—	53	2	no	—
325	C₅₃	C₂²×#525	1	1	yes	characteristic, complemented, Sylow
326	C₂⁵	—	53	1	no	—
327	C₂⁵	—	53	1	no	—
328	C₂⁵	—	53	12	no	—
329	C₂⁵	—	106	8	no	—
330	C₂⁵	—	106	8	no	—
331	C₂⁵	—	106	8	no	—
332	C₂⁵	C₂²×D₅₃	1	2	yes	—
333	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
334	C₂×#41	—	53	64	no	—
335	C₂×#41	—	106	192	no	—
336	C₂²×Q₈	—	53	12	no	—
337	C₂²×Q₈	—	106	8	no	—
338	C₂²×D₄	—	2	16	no	—
339	C₂²×D₄	—	2	8	no	—
340	C₂²×D₄	—	53	24	no	—
341	C₂²×D₄	—	53	24	no	—
342	C₂²×D₄	—	106	16	no	—
343	C₂²×D₄	—	106	16	no	—
344	C₂²×D₄	C₂²×D₅₃	1	24	yes	—
345	C₂²×D₄	C₂²×D₅₃	1	12	yes	complemented
346	C₂³×C₄	—	2	8	no	—
347	C₂³×C₄	—	53	1	no	—
348	C₂³×C₄	—	53	16	no	—
349	C₂³×C₄	—	53	12	no	—
350	C₂³×C₄	—	53	12	no	—
351	C₂³×C₄	—	53	24	no	—
352	C₂³×C₄	—	53	2	no	—
353	C₂³×C₄	—	53	1	no	—
354	C₂³×C₄	—	53	2	no	—
355	C₂³×C₄	—	53	12	no	—
356	C₂³×C₄	—	53	12	no	—
357	C₂³×C₄	—	53	12	no	—
358	C₂³×C₄	—	53	64	no	—
359	C₂³×C₄	—	106	8	no	—
360	C₂³×C₄	—	106	192	no	—
361	C₂³×C₄	—	106	8	no	—
362	C₂³×C₄	—	106	8	no	—
363	C₂³×C₄	—	106	16	no	—
364	C₂³×C₄	—	106	8	no	—
365	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
366	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
367	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
368	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
369	#123	—	53	128	no	—
370	C₂²⋊Q₈	—	53	64	no	—
371	C₂²⋊Q₈	—	53	64	no	—
372	#121	—	53	64	no	—
373	#121	C₂²×D₅₃	1	64	yes	complemented
374	#120	—	53	128	no	—
375	C₄×D₄	—	53	128	no	—
376	C₄×D₄	—	53	128	no	—
377	#117	—	53	64	no	—
378	C₂×C₄⋊C₄	—	53	12	no	—
379	C₂×C₄⋊C₄	—	53	24	no	—
380	C₂×C₄⋊C₄	—	53	24	no	—
381	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	complemented
382	C₂×C₂²⋊C₄	—	53	24	no	—
383	C₂×C₂²⋊C₄	—	53	24	no	—
384	C₂×C₂²⋊C₄	—	53	24	no	—
385	C₂×C₂²⋊C₄	—	53	24	no	—
386	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	complemented
387	C₂×C₄²	—	53	24	no	—
388	C₂⁴	—	2	4	no	—
389	C₂⁴	—	2	4	no	—
390	C₂⁴	—	2	8	no	—
391	C₂⁴	—	2	8	no	—
392	C₂⁴	—	2	8	no	—
393	C₂⁴	—	53	8	no	—
394	C₂⁴	—	53	8	no	—
395	C₂⁴	—	53	6	no	—
396	C₂⁴	—	53	6	no	—
397	C₂⁴	—	53	16	no	—
398	C₂⁴	—	106	48	no	—
399	C₂⁴	—	106	48	no	—
400	C₂⁴	—	106	128	no	—
401	C₂⁴	—	106	4	no	—
402	C₂⁴	—	106	4	no	—
403	C₂⁴	—	106	4	no	—
404	C₂⁴	—	106	4	no	—
405	C₂⁴	—	106	48	no	—
406	C₂⁴	#11719947	1	8	yes	complemented
407	C₂⁴	C₂³×D₅₃	1	12	yes	—
408	C₂⁴	C₂³×D₅₃	1	6	yes	—
409	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
410	#41	—	106	256	no	—
411	C₂×Q₈	—	53	16	no	—
412	C₂×Q₈	—	106	48	no	—
413	C₂×D₄	—	2	96	no	—
414	C₂×D₄	—	2	48	no	—
415	C₂×D₄	—	53	32	no	—
416	C₂×D₄	—	53	32	no	—
417	C₂×D₄	—	106	96	no	—
418	C₂×D₄	—	106	96	no	—
419	C₂×D₄	C₂³×D₅₃	1	32	yes	—
420	C₂×D₄	C₂³×D₅₃	1	16	yes	complemented
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	8	no	—
423	C₂²×C₄	—	2	48	no	—
424	C₂²×C₄	—	53	8	no	—
425	C₂²×C₄	—	53	6	no	—
426	C₂²×C₄	—	53	96	no	—
427	C₂²×C₄	—	53	16	no	—
428	C₂²×C₄	—	53	12	no	—
429	C₂²×C₄	—	53	6	no	—
430	C₂²×C₄	—	53	12	no	—
431	C₂²×C₄	—	53	16	no	—
432	C₂²×C₄	—	53	32	no	—
433	C₂²×C₄	—	53	16	no	—
434	C₂²×C₄	—	53	16	no	—
435	C₂²×C₄	—	53	16	no	—
436	C₂²×C₄	—	106	48	no	—
437	C₂²×C₄	—	106	256	no	—
438	C₂²×C₄	—	106	48	no	—
439	C₂²×C₄	—	106	48	no	—
440	C₂²×C₄	—	106	96	no	—
441	C₂²×C₄	—	106	4	no	—
442	C₂²×C₄	—	106	8	no	—
443	C₂²×C₄	—	106	8	no	—
444	C₂²×C₄	—	106	48	no	—
445	C₂²×C₄	D₄×D₅₃	1	8	yes	—
446	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
447	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
448	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
449	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
450	C₄⋊C₄	—	53	16	no	—
451	C₄⋊C₄	—	53	32	no	—
452	C₄⋊C₄	—	53	32	no	—
453	C₄⋊C₄	C₂³×D₅₃	1	16	yes	complemented
454	C₂²⋊C₄	—	53	32	no	—
455	C₂²⋊C₄	—	53	32	no	—
456	C₂²⋊C₄	—	53	32	no	—
457	C₂²⋊C₄	—	53	32	no	—
458	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	complemented
459	C₄²	—	53	32	no	—
460	C₂³	—	2	32	no	—
461	C₂³	—	2	12	no	—
462	C₂³	—	2	12	no	—
463	C₂³	—	2	24	no	—
464	C₂³	—	2	24	no	—
465	C₂³	—	2	24	no	—
466	C₂³	—	4	32	no	—
467	C₂³	—	53	24	no	—
468	C₂³	—	53	24	no	—
469	C₂³	—	53	4	no	—
470	C₂³	—	53	4	no	—
471	C₂³	—	106	32	no	—
472	C₂³	—	106	32	no	—
473	C₂³	—	106	384	no	—
474	C₂³	—	106	32	no	—
475	C₂³	—	106	32	no	—
476	C₂³	—	106	12	no	—
477	C₂³	—	106	12	no	—
478	C₂³	—	106	12	no	—
479	C₂³	—	106	12	no	—
480	C₂³	—	106	32	no	—
481	C₂³	C₂×D₄×D₅₃	1	4	yes	—
482	C₂³	#41×D₅₃	1	4	yes	—
483	C₂³	C₂×#11719947	1	4	yes	—
484	C₂³	C₂×#11719947	1	24	yes	complemented
485	C₂³	C₂⁴×D₅₃	1	8	yes	—
486	C₂³	C₂⁴×D₅₃	1	4	yes	—
487	C₂³	C₂⁴×D₅₃	1	3	yes	—
488	Q₈	—	106	32	no	—
489	D₄	—	2	64	no	—
490	D₄	—	2	32	no	—
491	D₄	—	106	64	no	—
492	D₄	—	106	64	no	—
493	C₂×C₄	—	2	12	no	—
494	C₂×C₄	—	2	24	no	—
495	C₂×C₄	—	2	32	no	—
496	C₂×C₄	—	53	24	no	—
497	C₂×C₄	—	53	4	no	—
498	C₂×C₄	—	53	64	no	—
499	C₂×C₄	—	53	8	no	—
500	C₂×C₄	—	53	4	no	—
501	C₂×C₄	—	53	8	no	—
502	C₂×C₄	—	106	32	no	—
503	C₂×C₄	—	106	32	no	—
504	C₂×C₄	—	106	32	no	—
505	C₂×C₄	—	106	64	no	—
506	C₂×C₄	—	106	12	no	—
507	C₂×C₄	—	106	24	no	—
508	C₂×C₄	—	106	24	no	—
509	C₂×C₄	—	106	32	no	—
510	C₂×C₄	C₂×D₄×D₅₃	1	24	yes	—
511	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
512	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
513	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
514	C₂²	—	2	48	no	—
515	C₂²	—	2	4	no	—
516	C₂²	—	2	4	no	—
517	C₂²	—	2	8	no	—
518	C₂²	—	2	8	no	—
519	C₂²	—	2	8	no	—
520	C₂²	—	4	48	no	—
521	C₂²	—	53	8	no	—
522	C₂²	—	53	8	no	—
523	C₂²	—	106	128	no	—
524	C₂²	—	106	48	no	—
525	C₂²	—	106	48	no	—
526	C₂²	—	106	4	no	—
527	C₂²	—	106	4	no	—
528	C₂²	—	106	4	no	—
529	C₂²	—	106	4	no	—
530	C₂²	#16811133	1	16	yes	direct product factor
531	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
532	C₂²	C₂×#41×D₅₃	1	6	yes	—
533	C₂²	C₂²×#11719947	1	6	yes	—
534	C₂²	C₂²×#11719947	1	8	yes	complemented
535	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
536	C₄	—	2	4	no	—
537	C₄	—	2	8	no	—
538	C₄	—	53	8	no	—
539	C₄	—	106	4	no	—
540	C₄	—	106	8	no	—
541	C₄	—	106	8	no	—
542	C₄	C₂²×D₄×D₅₃	1	8	yes	—
543	C₂	—	2	8	no	—
544	C₂	—	4	8	no	—
545	C₂	—	106	8	no	—
546	C₂	—	106	8	no	—
547	C₂	C₂×#16811133	1	12	yes	direct product factor
548	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
549	C₂	C₂²×#41×D₅₃	1	1	yes	characteristic
550	C₂	C₂³×#11719947	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#121×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×#121	D₅₃
hypocenter	C₅₃	C₂²×#525
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#121

Complements

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

N	H ≅ G/N	type
1. C₂³×#11719947	C₂	semidirect
2. C₂⁴×C₄×D₅₃	C₂	semidirect
3. C₂×#16811133	C₂	direct (N × H)
4. C₂²×#13133199	C₂	semidirect
5. C₂²×#13133197	C₂	semidirect
6. C₂²×#13133196	C₂	semidirect
7. C₂²×#13133188	C₂	semidirect
8. C₂²×#13133185	C₂	semidirect
9. C₂²×D₄×Dic₅₃	C₂	semidirect
10. C₂²×#13133177	C₂	semidirect
11. C₂²×#13133175	C₂	semidirect
12. C₂²×#121×C₅₃	C₂	semidirect
15. C₂²×#11719947	C₂²	semidirect
16. C₂³×Dic₁₀₆	C₂²	semidirect
20. C₂⁴×Dic₅₃	C₂²	semidirect
26. C₂³×C₄×D₅₃	C₂²	semidirect
27. #16811133	C₂²	direct (N × H)
28. C₂×#13133199	C₂²	semidirect
29. C₂×#13133197	C₂²	semidirect
30. C₂×#13133196	C₂²	semidirect
31. C₂×#13133188	C₂²	semidirect
32. C₂×#13133185	C₂²	semidirect
33. C₂×D₄×Dic₅₃	C₂²	semidirect
34. C₂×#13133177	C₂²	semidirect
35. C₂×#13133175	C₂²	semidirect
36. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
37. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
38. C₂²×#11719919	C₂²	semidirect
40. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
43. C₂²×C₄×Dic₅₃	C₂²	semidirect
45. C₂³×D₄×C₅₃	C₂²	semidirect
47. C₂×#121×C₅₃	C₂²	semidirect
48. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
49. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
57. C₂×#11719947	C₂³	semidirect
59. C₂²×Dic₁₀₆	C₂³	semidirect
71. C₂³×Dic₅₃	C₂³	semidirect
76. C₂²×C₄×D₅₃	D₄	semidirect
81. C₂²×C₄×D₅₃	C₂³	semidirect
82. #13133199	C₂³	semidirect
83. #13133197	C₂³	semidirect
84. #13133196	C₂³	semidirect
85. #13133188	C₂³	semidirect
86. #13133185	C₂³	semidirect
87. D₄×Dic₅₃	C₂³	semidirect
88. #13133177	C₂³	semidirect
89. #13133175	C₂³	semidirect
90. C₂×C₄⋊Dic₅₃	C₂³	semidirect
91. C₂×C₄⋊Dic₅₃	C₂³	semidirect
92. C₂×#11719919	C₂³	semidirect
94. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
97. C₂×C₄×Dic₅₃	C₂³	semidirect
103. C₂²×D₄×C₅₃	C₂³	semidirect
109. #121×C₅₃	C₂³	semidirect
110. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
111. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
117. C₂⁴×C₅₃	#41	semidirect
130. C₂³×D₅₃	#41	semidirect
137. C₂×D₄×C₅₃	C₂⁴	semidirect
161. C₂²×Dic₅₃	C₂⁴	semidirect
167. C₂×C₄×D₅₃	C₂×D₄	semidirect
173. C₂×Dic₁₀₆	C₂⁴	semidirect
174. C₄⋊C₄×C₅₃	C₂⁴	semidirect
175. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
177. C₂²⋊Dic₅₃	C₂⁴	semidirect
180. C₄⋊Dic₅₃	C₂⁴	semidirect
181. C₄⋊Dic₅₃	C₂⁴	semidirect
182. #11719919	C₂⁴	semidirect
183. C₄×Dic₅₃	C₂⁴	semidirect
194. C₂³×C₅₃	C₂×#41	semidirect
209. C₂²×D₅₃	C₂×#41	semidirect
236. C₄×D₅₃	C₂²×D₄	semidirect
246. C₂²×C₅₃	#525	semidirect
250. C₂²×C₅₃	C₂²×#41	semidirect
260. C₂×D₅₃	C₂²×#41	semidirect
275. C₂²×#121	D₅₃	semidirect
282. C₂×C₅₃	C₂×#525	semidirect
295. C₂³×D₄	C₂×D₅₃	semidirect
310. C₂×#121	C₂×D₅₃	semidirect
318. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
323. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
325. C₅₃	C₂²×#525	semidirect
345. C₂²×D₄	C₂²×D₅₃	semidirect
373. #121	C₂²×D₅₃	semidirect
381. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
386. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
406. C₂⁴	#11719947	semidirect
420. C₂×D₄	C₂³×D₅₃	semidirect
453. C₄⋊C₄	C₂³×D₅₃	semidirect
458. C₂²⋊C₄	C₂³×D₅₃	semidirect
484. C₂³	C₂×#11719947	semidirect
530. C₂²	#16811133	direct (N × H)
534. C₂²	C₂²×#11719947	semidirect
547. C₂	C₂×#16811133	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×#11719947	2
C₂⁴×C₄×D₅₃	2
C₂×#16811133	2
C₂²×#13133199	2
C₂²×#13133197	2
C₂²×#13133196	2
C₂²×#13133188	2
C₂²×#13133185	2
C₂²×D₄×Dic₅₃	2
C₂²×#13133177	2
C₂²×#13133175	2
C₂²×#121×C₅₃	2
C₂²×#525	53

Subgroups — C22×#16811133 #28290748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811133 #28290748