Subgroups — #27353127

Order 13568.

712 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810680	C₂	1	1	yes	maximal, characteristic
3	#20273482	C₂	1	4	yes	maximal, complemented
4	#20273478	C₂	1	4	yes	maximal, complemented
5	#20273467	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	#16810680	—	2	1	no	—
8	#16810680	—	2	1	no	—
9	#16810680	—	2	1	no	—
10	#16810680	C₂²	1	2	yes	—
11	#16810680	C₂²	1	2	yes	—
12	#16810680	C₂²	1	2	yes	—
13	#16810676	—	2	1	no	—
14	#16810676	C₂²	1	2	yes	—
15	#16810647	C₂²	1	2	yes	—
16	C₂×C₄⋊#88322	C₂²	1	2	yes	—
17	#16810587	—	2	1	no	—
18	C₂×C₂²⋊#88322	—	2	1	no	—
19	C₄×#11719917	—	2	1	no	—
20	C₂×#13133128	—	2	1	no	—
21	#16810376	C₂²	1	8	yes	complemented
22	#16810374	C₂²	1	8	yes	complemented
23	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
24	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
26	C₄×M₁₆×C₅₃	C₄	1	2	yes	—
27	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
28	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
29	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
30	C₂×#97×C₅₃	C₄	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
32	C₂×#11719917	—	2	2	no	—
33	C₂²×#88322	—	2	2	no	—
34	C₄⋊#88322	—	2	2	no	—
35	C₄⋊#88322	—	2	2	no	—
36	C₄⋊#88322	C₂³	1	4	yes	—
37	C₄⋊#88322	C₂³	1	4	yes	—
38	C₂²⋊#88322	—	2	2	no	—
39	C₂²⋊#88322	—	2	2	no	—
40	C₂²⋊#88322	—	2	2	no	—
41	C₂²⋊#88322	—	2	2	no	—
42	#13133128	—	2	2	no	—
43	#13133128	—	2	2	no	—
44	#13133128	—	2	2	no	—
45	C₄×#88322	—	2	2	no	—
46	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
49	C₂×M₁₆×C₅₃	—	2	8	no	—
50	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
51	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	#97×C₅₃	C₂×C₄	1	2	yes	—
67	#97×C₅₃	C₂×C₄	1	4	yes	—
68	#97×C₅₃	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₅₃	C₂³	1	4	yes	—
72	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	—	2	4	no	—
74	C₂²×C₄×C₅₃	—	4	2	no	—
75	C₂²×C₄×C₅₃	—	4	2	no	—
76	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
94	M₁₆×C₅₃	—	2	4	no	—
95	M₁₆×C₅₃	—	2	8	no	—
96	M₁₆×C₅₃	M₁₆	1	8	yes	complemented
97	C₂×C₈×C₅₃	—	2	8	no	—
98	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
100	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
113	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
115	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
116	#11719917	—	4	4	no	—
117	C₂×#88322	—	2	4	no	—
118	C₂×#88322	—	2	4	no	—
119	C₂×#88322	—	4	4	no	—
120	C₂³×C₅₃	—	2	2	no	—
121	C₂³×C₅₃	—	2	2	no	—
122	C₂³×C₅₃	—	4	1	no	—
123	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	4	1	no	—
137	C₂×C₄×C₅₃	—	4	1	no	—
138	C₂×C₄×C₅₃	—	4	1	no	—
139	C₂×C₄×C₅₃	—	4	1	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	—	4	1	no	—
142	C₂×C₄×C₅₃	—	4	1	no	—
143	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
153	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
156	C₈×C₅₃	—	2	4	no	—
157	C₈×C₅₃	C₂×M₁₆	1	8	yes	—
158	#88322	—	4	4	no	—
159	#88322	—	4	4	no	—
160	#17797	—	53	1	no	maximal, Sylow
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	2	1	no	—
164	C₂²×C₅₃	—	2	1	no	—
165	C₂²×C₅₃	—	2	1	no	—
166	C₂²×C₅₃	—	2	1	no	—
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	—	4	4	no	—
169	C₂²×C₅₃	#423	1	2	yes	—
170	C₂²×C₅₃	#430	1	2	yes	—
171	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	#423	1	2	yes	—
182	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
183	C₄×C₅₃	#575	1	2	yes	—
184	#2920	—	53	1	no	—
185	C₂×#432	—	53	1	no	—
186	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
187	#1569	—	53	4	no	—
188	#1568	—	53	4	no	—
189	#1565	—	53	4	no	—
190	C₂×C₅₃	—	2	2	no	—
191	C₂×C₅₃	—	2	2	no	—
192	C₂×C₅₃	—	4	1	no	—
193	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
194	C₂×C₅₃	#2911	1	1	yes	characteristic
195	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
196	C₂²×M₁₆	Dic₅₃	1	2	yes	—
197	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
198	#432	—	53	2	no	—
199	#432	—	53	2	no	—
200	#432	—	53	2	no	—
201	#432	—	106	1	no	—
202	#432	—	106	1	no	—
203	#432	—	106	1	no	—
204	#431	—	53	2	no	—
205	#431	—	106	1	no	—
206	#423	—	53	2	no	—
207	C₂×C₄⋊C₈	—	53	2	no	—
208	#407	—	106	1	no	—
209	C₂×C₂²⋊C₈	—	106	1	no	—
210	C₄×M₁₆	—	106	1	no	—
211	C₄×M₁₆	Dic₅₃	1	1	yes	characteristic
212	C₄×M₁₆	Dic₅₃	1	2	yes	—
213	C₄×M₁₆	Dic₅₃	1	1	yes	characteristic
214	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
215	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
216	C₂×#97	—	106	1	no	—
217	C₂×#97	Dic₅₃	1	2	yes	—
218	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
219	#335	—	53	8	no	—
220	#334	—	53	8	no	—
221	C₅₃	#17797	1	1	yes	characteristic, complemented, Sylow
222	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
223	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
224	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
225	C₂×M₁₆	—	2	8	no	—
226	C₂×M₁₆	—	106	2	no	—
227	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
228	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
229	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
230	C₂²×C₈	—	106	2	no	—
231	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
232	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
233	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
234	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
235	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
236	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
237	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
238	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
239	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
240	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
243	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
244	C₄⋊C₈	—	53	4	no	—
245	C₄⋊C₈	—	53	4	no	—
246	C₄⋊C₈	—	106	2	no	—
247	C₄⋊C₈	—	106	2	no	—
248	C₂²⋊C₈	—	106	2	no	—
249	C₂²⋊C₈	—	106	2	no	—
250	C₂²⋊C₈	—	106	2	no	—
251	C₂²⋊C₈	—	106	2	no	—
252	#97	—	106	2	no	—
253	#97	—	106	2	no	—
254	#97	—	106	2	no	—
255	#97	C₂×Dic₅₃	1	2	yes	—
256	#97	C₂×Dic₅₃	1	4	yes	—
257	#97	C₂×Dic₅₃	1	2	yes	—
258	C₄×C₈	—	106	2	no	—
259	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
260	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
261	C₄×C₈	C₂²×D₅₃	1	4	yes	—
262	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
269	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
276	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
277	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
281	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
284	M₁₆	—	2	4	no	—
285	M₁₆	—	2	8	no	—
286	M₁₆	—	212	4	no	—
287	M₁₆	#11719917	1	8	yes	complemented
288	C₂×C₈	—	2	8	no	—
289	C₂×C₈	—	106	4	no	—
290	C₂×C₈	—	106	4	no	—
291	C₂×C₈	—	212	4	no	—
292	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
293	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
294	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
295	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
296	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
297	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
298	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
299	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
300	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
301	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
302	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
303	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
304	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
305	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
306	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
307	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
308	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
309	C₄²	C₂³×D₅₃	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	—	2	2	no	—
312	C₂³	—	4	1	no	—
313	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
314	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
315	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
335	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
337	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
338	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
339	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
340	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
341	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
342	C₂×C₄	C₂×#11719917	1	4	yes	—
343	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
344	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
345	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
346	C₈	—	2	4	no	—
347	C₈	—	212	4	no	—
348	C₈	—	212	4	no	—
349	C₈	C₂×#11719917	1	8	yes	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	4	4	no	—
358	C₂²	#16810647	1	2	yes	—
359	C₂²	#16810672	1	2	yes	—
360	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	#16810647	1	2	yes	—
371	C₄	C₂²×#11719917	1	2	yes	—
372	C₄	#16811423	1	2	yes	—
373	C₂	—	2	2	no	—
374	C₂	—	2	2	no	—
375	C₂	—	4	1	no	—
376	C₂	C₂×#16810647	1	1	yes	characteristic
377	C₂	#20283305	1	1	yes	characteristic
378	C₂	C₂×#16810672	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	#17797
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20273482	C₂	semidirect
4. #20273478	C₂	semidirect
5. #20273467	C₂	semidirect
21. #16810376	C₂²	semidirect
22. #16810374	C₂²	semidirect
96. M₁₆×C₅₃	M₁₆	semidirect
221. C₅₃	#17797	semidirect
287. M₁₆	#11719917	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
C₂×#16810680	2
#20273482	2
#20273478	2
#20273467	2
C₂×C₄×M₁₆×C₅₃	2
#17797	53