Subgroups — #28319493

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291164	C₂	1	2	yes	maximal
2	#20290651	C₂	1	2	yes	maximal, complemented
3	#20290636	C₂	1	2	yes	maximal, complemented
4	#20290592	C₂	1	4	yes	maximal
5	C₄×#13133286	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133296	—	2	2	no	—
8	C₂×#13133286	C₂²	1	8	yes	—
9	#16811569	C₂²	1	4	yes	—
10	#16811567	—	2	2	no	—
11	#16811565	—	2	2	no	—
12	#16811565	C₂²	1	4	yes	—
13	C₄×C₄⋊#6157	—	2	2	no	—
14	C₄×C₄⋊#6157	C₂²	1	4	yes	—
15	#16811497	C₂²	1	8	yes	—
16	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
17	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×#88322	C₄	1	1	yes	characteristic, complemented
19	C₂×C₈×Dic₅₃	C₄	1	2	yes	complemented
20	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic, complemented
21	C₄×C₈×D₅₃	C₄	1	2	yes	complemented
22	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic, complemented
23	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₅₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
27	#13133296	—	4	8	no	—
28	C₂×C₄⋊#6157	—	2	4	no	—
29	C₂×C₄⋊#6157	—	2	4	no	—
30	C₂×C₄⋊#6157	C₂³	1	8	yes	—
31	C₂×C₂²⋊#6157	—	2	4	no	—
32	C₂×C₄×#6157	—	2	4	no	—
33	#13133286	—	2	16	no	—
34	(C₂×#6157)⋊C₄	—	2	4	no	—
35	(C₂×#6157)⋊C₄	—	2	4	no	—
36	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
39	C₂²×#88322	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₅₃	—	2	8	no	—
41	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₅₃	C₂³	1	2	yes	—
44	C₂×C₈×D₅₃	C₂³	1	2	yes	—
45	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
47	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
48	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄²×D₅₃	D₄	1	1	yes	characteristic
51	C₄²×D₅₃	D₄	1	1	yes	characteristic
52	C₄²×D₅₃	Q₈	1	1	yes	characteristic
53	C₄²×D₅₃	Q₈	1	1	yes	characteristic
54	C₄²×D₅₃	C₂³	1	1	yes	characteristic
55	C₄²×D₅₃	C₂³	1	1	yes	characteristic
56	C₄×#88322	C₂×C₄	1	1	yes	characteristic
57	C₄×#88322	C₂×C₄	1	2	yes	—
58	C₄×#88322	C₂×C₄	1	1	yes	characteristic
59	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
60	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
61	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
62	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
63	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂³×D₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×#6157	—	2	8	no	—
70	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₄×D₅₃	—	2	4	no	—
77	C₂×C₄×D₅₃	—	2	4	no	—
78	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×D₅₃	D₈	1	4	yes	—
83	C₂×C₄×D₅₃	Q₁₆	1	4	yes	—
84	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
92	C₂²⋊#6157	—	4	8	no	—
93	C₄⋊#6157	—	2	16	no	—
94	C₄⋊#6157	—	4	8	no	—
95	C₄⋊#6157	—	4	8	no	—
96	C₄×#6157	—	4	8	no	—
97	C₂×C₈×C₅₃	—	2	4	no	—
98	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
113	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
114	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
115	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
116	C₂×#88322	—	2	4	no	—
117	C₂×#88322	C₂²×C₄	1	2	yes	—
118	C₂×#88322	C₂²×C₄	1	2	yes	—
119	C₈×D₅₃	—	2	4	no	—
120	C₈×D₅₃	—	2	4	no	—
121	C₈×D₅₃	—	2	4	no	—
122	C₈×D₅₃	#41	1	8	yes	—
123	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×D₅₃	—	2	2	no	—
125	C₂²×D₅₃	—	2	2	no	—
126	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×D₅₃	C₂×D₈	1	2	yes	—
128	C₂²×D₅₃	C₂×Q₁₆	1	2	yes	—
129	C₂²×D₅₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×#6157	—	4	8	no	—
131	C₂×#6157	—	4	8	no	—
132	C₂×#6157	—	4	8	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	#107	1	4	yes	complemented
136	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×Dic₅₃	—	2	2	no	—
143	C₂×Dic₅₃	—	2	2	no	—
144	C₂×Dic₅₃	#107	1	4	yes	complemented
145	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
151	C₄×D₅₃	—	2	2	no	—
152	C₄×D₅₃	—	2	2	no	—
153	C₄×D₅₃	—	2	2	no	—
154	C₄×D₅₃	—	2	2	no	—
155	C₄×D₅₃	—	2	2	no	—
156	C₄×D₅₃	—	2	2	no	—
157	C₄×D₅₃	#107	1	4	yes	complemented
158	C₄×D₅₃	#107	1	4	yes	complemented
159	C₄×D₅₃	C₂×D₈	1	2	yes	—
160	C₄×D₅₃	C₂×D₈	1	2	yes	—
161	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
162	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
163	C₄×D₅₃	#135	1	4	yes	—
164	C₄×D₅₃	C₂×#41	1	4	yes	—
165	C₈×C₅₃	—	2	2	no	—
166	C₈×C₅₃	#117	1	4	yes	complemented
167	#88322	—	2	2	no	—
168	#88322	#117	1	4	yes	complemented
169	#20817	—	53	1	no	maximal, Sylow
170	C₂²×C₅₃	—	2	1	no	—
171	C₂²×C₅₃	—	2	1	no	—
172	C₂²×C₅₃	C₂×#107	1	2	yes	—
173	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
174	C₂×D₅₃	—	2	1	no	—
175	C₂×D₅₃	—	2	1	no	—
176	C₂×D₅₃	—	2	4	no	—
177	C₂×D₅₃	—	2	4	no	—
178	C₂×D₅₃	C₂×#107	1	2	yes	—
179	C₂×D₅₃	#532	1	2	yes	—
180	C₂×D₅₃	C₂²×D₈	1	1	yes	characteristic
181	C₂×D₅₃	C₂²×Q₁₆	1	1	yes	characteristic
182	C₂×D₅₃	C₂×#135	1	2	yes	—
183	#6157	—	8	8	no	—
184	C₄×C₅₃	—	2	1	no	—
185	C₄×C₅₃	—	2	1	no	—
186	C₄×C₅₃	C₂×#107	1	2	yes	—
187	C₄×C₅₃	C₂×#107	1	2	yes	—
188	C₄×C₅₃	#427	1	2	yes	—
189	C₄×C₅₃	C₂×#117	1	2	yes	—
190	Dic₅₃	—	2	1	no	—
191	Dic₅₃	—	2	1	no	—
192	Dic₅₃	C₂×#107	1	2	yes	—
193	Dic₅₃	C₂×#107	1	2	yes	—
194	Dic₅₃	#427	1	2	yes	—
195	Dic₅₃	#533	1	2	yes	—
196	C₂²×C₄×C₈	—	53	1	no	—
197	#2285	—	53	2	no	—
198	#1845	—	53	2	no	—
199	#1834	—	53	2	no	—
200	#1808	—	53	4	no	—
201	C₄×#107	—	53	4	no	—
202	C₂×C₅₃	—	2	2	no	—
203	C₂×C₅₃	#2285	1	1	yes	characteristic
204	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
205	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
206	D₅₃	—	2	2	no	—
207	D₅₃	#3169	1	2	yes	—
208	D₅₃	#3170	1	2	yes	—
209	C₂³×C₈	—	53	2	no	—
210	C₂×#117	—	106	2	no	—
211	C₂²×C₄²	—	53	1	no	—
212	C₂×#107	—	53	8	no	—
213	C₂×C₄×C₈	—	53	2	no	—
214	C₂×C₄×C₈	—	53	1	no	—
215	C₂×C₄×C₈	—	53	1	no	—
216	C₂×C₄×C₈	—	53	1	no	—
217	C₂×C₄×C₈	—	53	1	no	—
218	C₂×C₄×C₈	—	53	2	no	—
219	C₂×C₄×C₈	—	53	1	no	—
220	C₂×C₄×C₈	—	53	2	no	—
221	C₂×C₄×C₈	#6157	1	1	yes	characteristic, complemented
222	#384	—	53	4	no	—
223	#382	—	106	2	no	—
224	#380	—	53	4	no	—
225	#380	—	106	2	no	—
226	C₄×C₄⋊C₄	—	53	4	no	—
227	C₄×C₄⋊C₄	—	106	2	no	—
228	#340	—	53	8	no	—
229	C₅₃	#20817	1	1	yes	characteristic, complemented, Sylow
230	C₂³×C₄	—	53	1	no	—
231	C₂³×C₄	—	53	1	no	—
232	C₂³×C₄	—	53	1	no	—
233	C₂²×C₈	—	53	2	no	—
234	C₂²×C₈	—	53	2	no	—
235	C₂²×C₈	—	53	2	no	—
236	C₂²×C₈	—	53	2	no	—
237	C₂²×C₈	—	53	2	no	—
238	C₂²×C₈	—	106	8	no	—
239	C₂²×C₈	C₂×#6157	1	2	yes	—
240	#117	—	212	8	no	—
241	C₂×C₄⋊C₄	—	53	8	no	—
242	C₂×C₄⋊C₄	—	106	4	no	—
243	C₂×C₄⋊C₄	—	106	4	no	—
244	C₂×C₂²⋊C₄	—	106	4	no	—
245	C₂×C₄²	—	53	1	no	—
246	C₂×C₄²	—	53	1	no	—
247	C₂×C₄²	—	53	1	no	—
248	C₂×C₄²	—	53	1	no	—
249	C₂×C₄²	—	53	1	no	—
250	C₂×C₄²	—	53	1	no	—
251	C₂×C₄²	—	53	1	no	—
252	C₂×C₄²	—	53	1	no	—
253	C₂×C₄²	—	53	1	no	—
254	C₂×C₄²	—	53	1	no	—
255	C₂×C₄²	—	53	1	no	—
256	C₂×C₄²	—	106	4	no	—
257	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
258	#107	—	106	16	no	—
259	C₄×C₈	—	53	2	no	—
260	C₄×C₈	—	53	2	no	—
261	C₄×C₈	—	53	2	no	—
262	C₄×C₈	—	53	2	no	—
263	C₄×C₈	—	53	1	no	—
264	C₄×C₈	—	53	1	no	—
265	C₄×C₈	—	53	2	no	—
266	C₄×C₈	C₂×#6157	1	1	yes	characteristic
267	C₄×C₈	C₂×#6157	1	1	yes	characteristic
268	C₄×C₈	C₂×#6157	1	2	yes	—
269	#95	—	106	4	no	—
270	#95	—	106	4	no	—
271	C₂⁴	—	53	1	no	—
272	C₂²×C₄	—	53	1	no	—
273	C₂²×C₄	—	53	4	no	—
274	C₂²×C₄	—	53	4	no	—
275	C₂²×C₄	—	53	1	no	—
276	C₂²×C₄	—	53	1	no	—
277	C₂²×C₄	—	53	1	no	—
278	C₂²×C₄	—	53	1	no	—
279	C₂²×C₄	—	53	1	no	—
280	C₂²×C₄	—	53	1	no	—
281	C₂²×C₄	—	53	1	no	—
282	C₂²×C₄	—	53	1	no	—
283	C₂²×C₄	—	53	1	no	—
284	C₂²×C₄	—	53	1	no	—
285	C₂²×C₄	—	53	1	no	—
286	C₂²×C₄	—	53	1	no	—
287	C₂²×C₄	—	53	1	no	—
288	C₂²×C₄	—	53	1	no	—
289	C₂²×C₄	—	106	4	no	—
290	C₂²×C₄	—	106	4	no	—
291	C₂²×C₄	—	106	8	no	—
292	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
293	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
294	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
295	C₂×C₈	—	2	4	no	—
296	C₂×C₈	—	53	2	no	—
297	C₂×C₈	—	53	8	no	—
298	C₂×C₈	—	53	2	no	—
299	C₂×C₈	—	106	4	no	—
300	C₂×C₈	—	106	4	no	—
301	C₂×C₈	—	106	4	no	—
302	C₂×C₈	—	106	4	no	—
303	C₂×C₈	C₂²×#6157	1	2	yes	—
304	C₂×C₈	C₂²×#6157	1	2	yes	—
305	C₄⋊C₄	—	106	16	no	—
306	C₄⋊C₄	—	212	8	no	—
307	C₄⋊C₄	—	212	8	no	—
308	C₂²⋊C₄	—	212	8	no	—
309	C₄²	—	53	1	no	—
310	C₄²	—	53	1	no	—
311	C₄²	—	53	1	no	—
312	C₄²	—	53	1	no	—
313	C₄²	—	53	1	no	—
314	C₄²	—	53	1	no	—
315	C₄²	—	53	1	no	—
316	C₄²	—	53	1	no	—
317	C₄²	—	53	1	no	—
318	C₄²	—	53	1	no	—
319	C₄²	—	53	1	no	—
320	C₄²	—	53	1	no	—
321	C₄²	—	212	8	no	—
322	C₄²	C₄⋊#6157	1	1	yes	characteristic
323	C₄²	C₄⋊#6157	1	1	yes	characteristic
324	C₄²	C₂²×#6157	1	1	yes	characteristic
325	C₄²	C₂²×#6157	1	1	yes	characteristic
326	C₂³	—	53	1	no	—
327	C₂³	—	53	2	no	—
328	C₂³	—	53	2	no	—
329	C₂³	—	53	1	no	—
330	C₂³	—	106	2	no	—
331	C₂³	—	106	2	no	—
332	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	53	4	no	—
336	C₂×C₄	—	53	1	no	—
337	C₂×C₄	—	53	1	no	—
338	C₂×C₄	—	53	1	no	—
339	C₂×C₄	—	53	1	no	—
340	C₂×C₄	—	53	4	no	—
341	C₂×C₄	—	53	4	no	—
342	C₂×C₄	—	53	2	no	—
343	C₂×C₄	—	53	2	no	—
344	C₂×C₄	—	53	2	no	—
345	C₂×C₄	—	53	2	no	—
346	C₂×C₄	—	53	4	no	—
347	C₂×C₄	—	53	4	no	—
348	C₂×C₄	—	53	1	no	—
349	C₂×C₄	—	53	1	no	—
350	C₂×C₄	—	106	2	no	—
351	C₂×C₄	—	106	2	no	—
352	C₂×C₄	—	106	2	no	—
353	C₂×C₄	—	106	2	no	—
354	C₂×C₄	—	106	2	no	—
355	C₂×C₄	—	106	2	no	—
356	C₂×C₄	—	106	2	no	—
357	C₂×C₄	—	106	2	no	—
358	C₂×C₄	—	212	8	no	—
359	C₂×C₄	—	212	8	no	—
360	C₂×C₄	—	212	8	no	—
361	C₂×C₄	#13133286	1	4	yes	complemented
362	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
363	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
364	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
365	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
366	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
367	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	53	4	no	—
370	C₈	—	106	2	no	—
371	C₈	#13133296	1	4	yes	complemented
372	C₂²	—	2	1	no	—
373	C₂²	—	2	1	no	—
374	C₂²	—	53	1	no	—
375	C₂²	—	53	1	no	—
376	C₂²	—	53	2	no	—
377	C₂²	—	53	2	no	—
378	C₂²	—	53	2	no	—
379	C₂²	—	106	4	no	—
380	C₂²	—	106	4	no	—
381	C₂²	—	106	1	no	—
382	C₂²	—	106	1	no	—
383	C₂²	C₂×#13133286	1	2	yes	—
384	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
385	C₄	—	2	1	no	—
386	C₄	—	2	1	no	—
387	C₄	—	53	2	no	—
388	C₄	—	53	2	no	—
389	C₄	—	53	2	no	—
390	C₄	—	53	2	no	—
391	C₄	—	106	1	no	—
392	C₄	—	106	1	no	—
393	C₄	—	424	8	no	—
394	C₄	C₂×#13133286	1	2	yes	—
395	C₄	C₂×#13133286	1	2	yes	—
396	C₄	#16811633	1	2	yes	—
397	C₄	C₂×#13133296	1	2	yes	—
398	C₂	—	2	2	no	—
399	C₂	—	53	2	no	—
400	C₂	—	53	2	no	—
401	C₂	—	106	2	no	—
402	C₂	#20291164	1	1	yes	characteristic
403	C₂	C₂²×#13133286	1	1	yes	characteristic
404	C₂	C₂×#16811633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×D₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#20817
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20290651	C₂	semidirect
3. #20290636	C₂	semidirect
5. C₄×#13133286	C₂	semidirect
18. C₂×C₄×#88322	C₄	semidirect
19. C₂×C₈×Dic₅₃	C₄	semidirect
20. C₄×C₈×D₅₃	C₄	semidirect
21. C₄×C₈×D₅₃	C₄	semidirect
22. C₄×C₈×D₅₃	C₄	semidirect
26. C₂×C₄×C₈×C₅₃	C₄	semidirect
135. C₂×C₄×C₅₃	#107	semidirect
144. C₂×Dic₅₃	#107	semidirect
157. C₄×D₅₃	#107	semidirect
158. C₄×D₅₃	#107	semidirect
166. C₈×C₅₃	#117	semidirect
168. #88322	#117	semidirect
221. C₂×C₄×C₈	#6157	semidirect
229. C₅₃	#20817	semidirect
361. C₂×C₄	#13133286	semidirect
371. C₈	#13133296	semidirect

Maximal subgroups

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

subgroup	index
#20291164	2
#20290651	2
#20290636	2
#20290592	2
C₄×#13133286	2
C₂×C₄×C₈×D₅₃	2
#20817	53