Subgroups — #27537296

Order 13568.

692 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811040	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276547	C₂	1	2	yes	maximal, complemented
4	#20276541	C₂	1	2	yes	maximal, complemented
5	#20276541	C₂	1	2	yes	maximal, complemented
6	D₄×#11719908	C₂	1	2	yes	maximal, complemented
7	#20276474	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276329	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276312	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2929	C₂	1	1	yes	maximal, characteristic
12	#16811040	—	2	4	no	—
13	C₂×#13133157	—	2	4	no	—
14	C₂×#13133157	C₂²	1	2	yes	—
15	C₂×#13133157	C₂²	1	1	yes	characteristic
16	C₂×#13133157	C₂²	1	1	yes	characteristic
17	C₂×#13133157	C₂²	1	2	yes	complemented
18	C₂²×#11719908	C₂²	1	2	yes	—
19	C₂²×#11719908	C₂²	1	1	yes	characteristic
20	C₂²×#11719908	C₂²	1	1	yes	characteristic
21	C₂²×#11719908	C₂²	1	2	yes	complemented
22	C₄⋊#11719908	C₂²	1	2	yes	complemented
23	C₄⋊#11719908	C₂²	1	2	yes	complemented
24	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
25	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
26	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
27	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
28	C₂²⋊#11719908	C₂²	1	2	yes	complemented
29	C₂²⋊#11719908	C₂²	1	2	yes	complemented
30	#16810399	C₂²	1	2	yes	complemented
31	C₄×#11719908	C₂²	1	2	yes	complemented
32	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
33	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#525	C₄	1	1	yes	characteristic
35	C₅₃×#435	C₄	1	2	yes	—
36	C₅₃×#435	C₄	1	2	yes	—
37	C₈×D₄×C₅₃	C₂²	1	2	yes	—
38	C₈×D₄×C₅₃	C₂²	1	2	yes	—
39	C₅₃×#424	C₄	1	1	yes	characteristic
40	C₅₃×#408	C₄	1	1	yes	characteristic
41	C₅₃×#407	C₄	1	1	yes	characteristic
42	C₅₃×#405	C₂²	1	1	yes	characteristic
43	#13133157	—	2	2	no	—
44	#13133157	—	2	4	no	—
45	#13133157	—	2	4	no	—
46	#13133157	D₄	1	4	yes	complemented
47	C₂×#11719908	—	2	2	no	—
48	C₂×#11719908	—	2	2	no	—
49	C₂×#11719908	—	2	4	no	—
50	C₂×#11719908	—	2	4	no	—
51	C₂×#11719908	C₂³	1	1	yes	characteristic
52	C₂×#11719908	C₂³	1	2	yes	—
53	C₂×#11719908	C₂³	1	1	yes	characteristic
54	C₂×#11719908	C₂³	1	1	yes	characteristic
55	C₂×#11719908	C₂³	1	1	yes	characteristic
56	C₂×#11719908	C₂³	1	2	yes	—
57	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
59	#131×C₅₃	—	2	2	no	—
60	#131×C₅₃	—	2	2	no	—
61	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
63	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	—	2	2	no	—
65	C₂²×C₈×C₅₃	—	2	2	no	—
66	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
71	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	#123×C₅₃	C₈	1	2	yes	—
74	C₂²⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
75	C₂²⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
76	#121×C₅₃	C₈	1	1	yes	characteristic
77	#121×C₅₃	C₈	1	1	yes	characteristic
78	#120×C₅₃	C₈	1	2	yes	—
79	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
80	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
81	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
83	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
84	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
85	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
86	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
87	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
88	#97×C₅₃	C₂×C₄	1	2	yes	—
89	C₄×C₈×C₅₃	C₂³	1	2	yes	—
90	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
91	#41×C₅₃	—	2	2	no	—
92	#41×C₅₃	—	2	2	no	—
93	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
94	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
95	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
96	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
97	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
98	C₂²×C₄×C₅₃	—	2	2	no	—
99	C₂²×C₄×C₅₃	—	2	2	no	—
100	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	M₁₆×C₅₃	—	2	2	no	—
108	M₁₆×C₅₃	—	2	2	no	—
109	M₁₆×C₅₃	—	2	4	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	—	2	2	no	—
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	2	1	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	1	no	—
117	C₂×C₈×C₅₃	—	2	1	no	—
118	C₂×C₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₅₃	#41	1	2	yes	—
126	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
128	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
129	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
130	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
131	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
132	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
133	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
134	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
135	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
136	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
137	#11719908	—	2	1	no	—
138	#11719908	—	2	1	no	—
139	#11719908	—	2	2	no	—
140	#11719908	—	2	1	no	—
141	#11719908	—	2	1	no	—
142	#11719908	C₂×D₄	1	4	yes	—
143	C₂³×C₅₃	—	2	2	no	—
144	C₂³×C₅₃	—	2	2	no	—
145	C₂³×C₅₃	—	4	1	no	—
146	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
147	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
148	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
149	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
150	Q₈×C₅₃	—	2	2	no	—
151	D₄×C₅₃	—	2	4	no	—
152	D₄×C₅₃	—	2	4	no	—
153	D₄×C₅₃	—	2	2	no	—
154	D₄×C₅₃	—	4	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	1	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	2	1	no	—
160	C₂×C₄×C₅₃	—	2	1	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
165	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
168	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
169	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
170	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
171	C₂×C₄×C₅₃	#131	1	2	yes	—
172	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
173	C₈×C₅₃	—	2	1	no	—
174	C₈×C₅₃	—	2	1	no	—
175	C₈×C₅₃	—	2	2	no	—
176	C₈×C₅₃	—	2	2	no	—
177	C₈×C₅₃	C₄×D₄	1	2	yes	—
178	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
179	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
180	#33436	—	53	1	no	maximal, Sylow
181	C₂²×C₅₃	—	2	1	no	—
182	C₂²×C₅₃	—	2	1	no	—
183	C₂²×C₅₃	—	2	1	no	—
184	C₂²×C₅₃	—	2	1	no	—
185	C₂²×C₅₃	—	2	2	no	—
186	C₂²×C₅₃	—	2	1	no	—
187	C₂²×C₅₃	—	2	1	no	—
188	C₂²×C₅₃	—	2	2	no	—
189	C₂²×C₅₃	—	2	1	no	—
190	C₂²×C₅₃	—	2	2	no	—
191	C₂²×C₅₃	—	4	4	no	—
192	C₂²×C₅₃	C₈×D₄	1	2	yes	—
193	C₂²×C₅₃	#504	1	2	yes	complemented
194	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
195	C₄×C₅₃	—	2	2	no	—
196	C₄×C₅₃	—	2	2	no	—
197	C₄×C₅₃	—	2	1	no	—
198	C₄×C₅₃	—	2	1	no	—
199	C₄×C₅₃	—	2	2	no	—
200	C₄×C₅₃	C₈×D₄	1	2	yes	—
201	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
202	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
203	C₂×#504	—	53	1	no	—
204	C₂²×M₃₂	—	53	1	no	—
205	#2929	D₅₃	1	1	yes	characteristic
206	#2169	—	53	2	no	—
207	#2168	—	53	2	no	—
208	#2168	—	53	2	no	—
209	D₄×C₁₆	—	53	2	no	—
210	#2151	—	53	1	no	—
211	#2113	—	53	1	no	—
212	C₂×C₂²⋊C₁₆	—	53	1	no	—
213	#2108	—	53	1	no	—
214	C₂×C₅₃	—	2	2	no	—
215	C₂×C₅₃	—	2	2	no	—
216	C₂×C₅₃	—	4	2	no	—
217	C₂×C₅₃	—	4	1	no	—
218	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
219	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
220	C₂×C₅₃	#3407	1	1	yes	characteristic
221	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
222	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
223	#525	Dic₅₃	1	1	yes	characteristic
224	#504	—	106	4	no	—
225	C₂×M₃₂	—	53	1	no	—
226	C₂×M₃₂	—	53	1	no	—
227	C₂×M₃₂	—	53	2	no	—
228	C₂×M₃₂	—	53	2	no	—
229	C₂×M₃₂	—	106	4	no	—
230	C₂²×C₁₆	—	53	1	no	—
231	C₂²×C₁₆	—	53	1	no	—
232	C₂²×C₁₆	—	53	2	no	—
233	C₂²×C₁₆	—	53	2	no	—
234	#435	Dic₅₃	1	2	yes	—
235	#435	Dic₅₃	1	2	yes	—
236	C₈×D₄	C₂×D₅₃	1	2	yes	—
237	C₈×D₄	C₂×D₅₃	1	2	yes	—
238	#424	Dic₅₃	1	1	yes	characteristic
239	#408	Dic₅₃	1	1	yes	characteristic
240	#407	Dic₅₃	1	1	yes	characteristic
241	#405	C₂×D₅₃	1	1	yes	characteristic
242	C₄⋊C₁₆	—	53	2	no	—
243	C₄⋊C₁₆	—	53	2	no	—
244	C₂²⋊C₁₆	—	53	1	no	—
245	C₂²⋊C₁₆	—	53	1	no	—
246	C₂²⋊C₁₆	—	53	1	no	—
247	C₂²⋊C₁₆	—	53	1	no	—
248	C₂²⋊C₁₆	—	53	2	no	—
249	C₂²⋊C₁₆	—	53	2	no	—
250	#346	—	53	2	no	—
251	C₄×C₁₆	—	53	2	no	—
252	C₅₃	#33436	1	1	yes	characteristic, complemented, Sylow
253	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
254	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
255	#131	—	2	2	no	—
256	#131	—	2	2	no	—
257	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
258	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
259	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
260	C₂²×C₈	—	2	2	no	—
261	C₂²×C₈	—	2	2	no	—
262	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
263	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
264	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
265	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
266	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
267	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
268	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
269	#123	#88322	1	2	yes	—
270	C₂²⋊Q₈	#88322	1	1	yes	characteristic
271	C₂²⋊Q₈	#88322	1	1	yes	characteristic
272	#121	#88322	1	1	yes	characteristic
273	#121	#88322	1	1	yes	characteristic
274	#120	#88322	1	2	yes	—
275	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
276	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
277	#117	C₂×Dic₅₃	1	1	yes	characteristic
278	M₃₂	—	53	4	no	—
279	M₃₂	—	106	4	no	—
280	M₃₂	—	106	4	no	—
281	M₃₂	—	106	2	no	—
282	C₂×C₁₆	—	53	2	no	—
283	C₂×C₁₆	—	53	1	no	—
284	C₂×C₁₆	—	53	1	no	—
285	C₂×C₁₆	—	53	1	no	—
286	C₂×C₁₆	—	53	1	no	—
287	C₂×C₁₆	—	53	2	no	—
288	C₂×C₁₆	—	106	4	no	—
289	C₂×C₁₆	—	106	4	no	—
290	C₂×C₁₆	—	106	2	no	—
291	C₂×C₁₆	—	106	2	no	—
292	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
293	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
294	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
295	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
296	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
297	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
298	#97	C₂×Dic₅₃	1	2	yes	—
299	C₄×C₈	C₂²×D₅₃	1	2	yes	—
300	C₂⁴	C₂×#88322	1	1	yes	characteristic
301	#41	—	2	2	no	—
302	#41	—	2	2	no	—
303	C₂×Q₈	C₂×#88322	1	1	yes	characteristic
304	C₂×D₄	C₂×#88322	1	2	yes	—
305	C₂×D₄	C₂×#88322	1	2	yes	—
306	C₂×D₄	C₂×#88322	1	1	yes	characteristic
307	C₂×D₄	C₂×#88322	1	2	yes	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
311	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
312	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
313	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
314	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
315	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
316	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
317	M₁₆	—	2	2	no	—
318	M₁₆	—	2	2	no	—
319	M₁₆	—	2	4	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	D₄×D₅₃	1	2	yes	—
330	C₂×C₈	#11719947	1	2	yes	—
331	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
332	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
333	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
334	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
335	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
336	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
337	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
338	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
339	C₄⋊C₄	C₂×#88322	1	2	yes	—
340	C₄⋊C₄	C₂×#88322	1	2	yes	—
341	C₂²⋊C₄	C₂×#88322	1	2	yes	—
342	C₂²⋊C₄	C₂×#88322	1	2	yes	—
343	C₂²⋊C₄	C₂×#88322	1	2	yes	—
344	C₂²⋊C₄	C₂×#88322	1	2	yes	—
345	C₂²⋊C₄	C₂×#88322	1	2	yes	—
346	C₄²	C₂²×Dic₅₃	1	2	yes	—
347	C₁₆	—	53	4	no	—
348	C₁₆	—	106	1	no	—
349	C₁₆	—	106	2	no	—
350	C₁₆	—	106	1	no	—
351	C₁₆	—	106	1	no	—
352	C₁₆	—	106	1	no	—
353	C₂³	—	2	2	no	—
354	C₂³	—	2	2	no	—
355	C₂³	—	4	1	no	—
356	C₂³	C₂²×#88322	1	1	yes	characteristic
357	C₂³	C₂²×#88322	1	2	yes	—
358	C₂³	C₂²×#88322	1	1	yes	characteristic
359	C₂³	C₂²×#88322	1	1	yes	characteristic
360	Q₈	—	2	2	no	—
361	D₄	—	2	4	no	—
362	D₄	—	2	4	no	—
363	D₄	—	2	2	no	—
364	D₄	—	4	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
375	C₂×C₄	C₂²×#88322	1	2	yes	—
376	C₂×C₄	C₂²×#88322	1	2	yes	—
377	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
378	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
379	C₂×C₄	C₂²×#88322	1	2	yes	—
380	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
381	C₂×C₄	#13133224	1	2	yes	—
382	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
383	C₈	—	2	1	no	—
384	C₈	—	2	1	no	—
385	C₈	—	2	2	no	—
386	C₈	—	2	2	no	—
387	C₈	D₄×Dic₅₃	1	2	yes	—
388	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
389	C₈	C₂×#11719947	1	1	yes	characteristic
390	C₂²	—	2	1	no	—
391	C₂²	—	2	1	no	—
392	C₂²	—	2	1	no	—
393	C₂²	—	2	1	no	—
394	C₂²	—	2	2	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	2	1	no	—
397	C₂²	—	2	2	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	2	2	no	—
400	C₂²	—	4	4	no	—
401	C₂²	D₄×#88322	1	2	yes	—
402	C₂²	#16811040	1	2	yes	complemented
403	C₂²	C₂³×#88322	1	1	yes	characteristic
404	C₄	—	2	2	no	—
405	C₄	—	2	2	no	—
406	C₄	—	2	1	no	—
407	C₄	—	2	1	no	—
408	C₄	—	2	2	no	—
409	C₄	D₄×#88322	1	2	yes	—
410	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
411	C₄	C₂×#13133224	1	1	yes	characteristic
412	C₂	—	2	2	no	—
413	C₂	—	2	2	no	—
414	C₂	—	4	1	no	—
415	C₂	—	4	2	no	—
416	C₂	C₂×D₄×#88322	1	1	yes	characteristic
417	C₂	C₂×#16811040	1	1	yes	characteristic
418	C₂	#20287818	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2929	C₂
prime radical	C₅₃×#525	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2929	D₅₃
hypocenter	C₅₃	#33436
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2929

Complements

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

N	H ≅ G/N	type
1. C₂×#16811040	C₂	semidirect
2. C₂²×#13133157	C₂	semidirect
3. #20276547	C₂	semidirect
4. #20276541	C₂	semidirect
5. #20276541	C₂	semidirect
6. D₄×#11719908	C₂	semidirect
7. #20276474	C₂	semidirect
8. #20276329	C₂	semidirect
9. C₂×C₂²⋊#11719908	C₂	semidirect
10. #20276312	C₂	semidirect
17. C₂×#13133157	C₂²	semidirect
21. C₂²×#11719908	C₂²	semidirect
22. C₄⋊#11719908	C₂²	semidirect
23. C₄⋊#11719908	C₂²	semidirect
28. C₂²⋊#11719908	C₂²	semidirect
29. C₂²⋊#11719908	C₂²	semidirect
30. #16810399	C₂²	semidirect
31. C₄×#11719908	C₂²	semidirect
46. #13133157	D₄	semidirect
193. C₂²×C₅₃	#504	semidirect
252. C₅₃	#33436	semidirect
402. C₂²	#16811040	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811040	2
C₂²×#13133157	2
#20276547	2
#20276541	2
D₄×#11719908	2
#20276474	2
#20276329	2
C₂×C₂²⋊#11719908	2
#20276312	2
C₅₃×#2929	2
#33436	53