Subgroups — #27353121

Order 13568.

756 conjugacy classes of subgroups.

The subgroup lattice has 412 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×#16810647	C₂	1	1	yes	maximal, characteristic
3	#20273482	C₂	1	4	yes	maximal, complemented
4	#20273474	C₂	1	2	yes	maximal, complemented
5	#20273471	C₂	1	2	yes	maximal, complemented
6	#20273467	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11719917	—	2	1	no	—
9	#16810680	—	2	1	no	—
10	#16810680	C₂²	1	2	yes	—
11	#16810676	—	2	1	no	—
12	#16810676	C₂²	1	2	yes	—
13	#16810647	—	2	1	no	—
14	#16810647	—	2	1	no	—
15	#16810647	C₂²	1	1	yes	characteristic
16	#16810647	C₂²	1	2	yes	—
17	#16810647	C₂²	1	1	yes	characteristic
18	#16810647	C₂²	1	2	yes	—
19	C₂×C₄⋊#88322	—	2	1	no	—
20	C₂×C₄⋊#88322	C₂²	1	2	yes	—
21	#16810587	—	2	1	no	—
22	C₄×#11719917	—	2	1	no	—
23	#16810376	C₂²	1	8	yes	complemented
24	#16810374	C₂²	1	8	yes	complemented
25	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
26	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
28	C₄×M₁₆×C₅₃	C₄	1	2	yes	—
29	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
30	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
31	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
32	C₂×#97×C₅₃	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
34	C₂×#11719917	—	2	1	no	—
35	C₂×#11719917	—	2	1	no	—
36	C₂×#11719917	—	2	2	no	—
37	C₂×#11719917	—	2	2	no	—
38	C₂×#11719917	—	4	4	no	—
39	C₂²×#88322	—	2	2	no	—
40	C₄⋊#88322	—	2	2	no	—
41	C₄⋊#88322	—	2	2	no	—
42	C₄⋊#88322	—	2	2	no	—
43	C₄⋊#88322	—	2	1	no	—
44	C₄⋊#88322	—	2	1	no	—
45	C₄⋊#88322	C₂³	1	2	yes	—
46	C₄⋊#88322	C₂³	1	4	yes	—
47	C₄⋊#88322	C₂³	1	2	yes	—
48	C₂²⋊#88322	—	2	2	no	—
49	C₂²⋊#88322	—	2	2	no	—
50	#13133128	—	2	2	no	—
51	C₄×#88322	—	2	2	no	—
52	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
55	C₂×M₁₆×C₅₃	—	2	8	no	—
56	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
57	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
58	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
59	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
60	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
72	#97×C₅₃	C₂×C₄	1	2	yes	—
73	#97×C₅₃	C₂×C₄	1	4	yes	—
74	#97×C₅₃	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
76	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₅₃	C₂³	1	4	yes	—
78	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	—	2	4	no	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	—	2	2	no	—
82	C₂²×C₄×C₅₃	—	4	2	no	—
83	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
101	M₁₆×C₅₃	—	2	4	no	—
102	M₁₆×C₅₃	—	2	8	no	—
103	M₁₆×C₅₃	M₁₆	1	8	yes	complemented
104	C₂×C₈×C₅₃	—	2	8	no	—
105	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
106	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
107	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
120	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
121	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
122	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
123	#11719917	—	4	4	no	—
124	#11719917	—	4	2	no	—
125	#11719917	—	4	4	no	—
126	#11719917	—	4	2	no	—
127	C₂×#88322	—	2	2	no	—
128	C₂×#88322	—	2	2	no	—
129	C₂×#88322	—	2	4	no	—
130	C₂×#88322	—	4	4	no	—
131	C₂³×C₅₃	—	2	2	no	—
132	C₂³×C₅₃	—	2	2	no	—
133	C₂³×C₅₃	—	4	1	no	—
134	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	2	1	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	4	1	no	—
153	C₂×C₄×C₅₃	—	4	1	no	—
154	C₂×C₄×C₅₃	—	4	1	no	—
155	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
165	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
166	C₂×C₄×C₅₃	#137	1	2	yes	—
167	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
170	C₈×C₅₃	—	2	4	no	—
171	C₈×C₅₃	C₂×M₁₆	1	8	yes	complemented
172	#88322	—	4	4	no	—
173	#88322	—	4	2	no	—
174	#88322	—	4	2	no	—
175	#17796	—	53	1	no	maximal, Sylow
176	C₂²×C₅₃	—	2	1	no	—
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	—	2	1	no	—
179	C₂²×C₅₃	—	2	1	no	—
180	C₂²×C₅₃	—	2	1	no	—
181	C₂²×C₅₃	—	2	1	no	—
182	C₂²×C₅₃	—	2	1	no	—
183	C₂²×C₅₃	—	4	4	no	—
184	C₂²×C₅₃	#423	1	2	yes	—
185	C₂²×C₅₃	#428	1	2	yes	—
186	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
187	C₄×C₅₃	—	2	1	no	—
188	C₄×C₅₃	—	2	1	no	—
189	C₄×C₅₃	—	2	1	no	—
190	C₄×C₅₃	—	2	1	no	—
191	C₄×C₅₃	—	2	1	no	—
192	C₄×C₅₃	—	2	1	no	—
193	C₄×C₅₃	—	2	1	no	—
194	C₄×C₅₃	#423	1	2	yes	—
195	C₄×C₅₃	#428	1	2	yes	—
196	C₄×C₅₃	#430	1	2	yes	—
197	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
198	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
199	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
200	#2920	—	53	1	no	—
201	C₂×#423	—	53	1	no	—
202	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
203	#1569	—	53	4	no	—
204	#1567	—	53	2	no	—
205	#1566	—	53	2	no	—
206	#1565	—	53	4	no	—
207	C₂×C₅₃	—	2	2	no	—
208	C₂×C₅₃	—	2	2	no	—
209	C₂×C₅₃	—	4	1	no	—
210	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
211	C₂×C₅₃	C₂×#428	1	1	yes	characteristic
212	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
213	C₂²×M₁₆	—	106	1	no	—
214	C₂²×M₁₆	Dic₅₃	1	2	yes	—
215	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
216	#432	—	53	2	no	—
217	#432	—	106	1	no	—
218	#431	—	53	2	no	—
219	#431	—	106	1	no	—
220	#423	—	53	1	no	—
221	#423	—	53	2	no	—
222	#423	—	53	1	no	—
223	#423	—	53	2	no	—
224	#423	—	106	1	no	—
225	#423	—	106	1	no	—
226	C₂×C₄⋊C₈	—	53	2	no	—
227	C₂×C₄⋊C₈	—	106	1	no	—
228	#407	—	106	1	no	—
229	C₄×M₁₆	—	106	1	no	—
230	C₄×M₁₆	Dic₅₃	1	1	yes	characteristic
231	C₄×M₁₆	Dic₅₃	1	2	yes	—
232	C₄×M₁₆	Dic₅₃	1	1	yes	characteristic
233	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
234	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
235	C₂×#97	Dic₅₃	1	2	yes	—
236	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
237	#335	—	53	8	no	—
238	#334	—	53	8	no	—
239	C₅₃	#17796	1	1	yes	characteristic, complemented, Sylow
240	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
241	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
242	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
243	C₂×M₁₆	—	2	8	no	—
244	C₂×M₁₆	—	106	2	no	—
245	C₂×M₁₆	—	106	2	no	—
246	C₂×M₁₆	—	106	1	no	—
247	C₂×M₁₆	—	106	1	no	—
248	C₂×M₁₆	—	212	4	no	—
249	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
250	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
251	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
252	C₂²×C₈	—	106	2	no	—
253	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
254	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
255	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
256	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
257	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
258	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
259	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
260	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
261	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
262	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
263	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
264	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
265	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
266	C₄⋊C₈	—	53	2	no	—
267	C₄⋊C₈	—	53	4	no	—
268	C₄⋊C₈	—	53	2	no	—
269	C₄⋊C₈	—	106	2	no	—
270	C₄⋊C₈	—	106	2	no	—
271	C₄⋊C₈	—	106	2	no	—
272	C₄⋊C₈	—	106	1	no	—
273	C₄⋊C₈	—	106	1	no	—
274	C₂²⋊C₈	—	106	2	no	—
275	C₂²⋊C₈	—	106	2	no	—
276	#97	—	106	2	no	—
277	#97	C₂×Dic₅₃	1	2	yes	—
278	#97	C₂×Dic₅₃	1	4	yes	—
279	#97	C₂×Dic₅₃	1	2	yes	—
280	C₄×C₈	—	106	2	no	—
281	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
282	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
283	C₄×C₈	C₂²×D₅₃	1	4	yes	—
284	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	4	2	no	—
289	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
290	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
291	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
292	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
293	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
294	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
295	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
296	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
297	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
298	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
299	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
300	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
302	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
303	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
304	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
305	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
306	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
307	M₁₆	—	2	4	no	—
308	M₁₆	—	2	8	no	—
309	M₁₆	—	212	4	no	—
310	M₁₆	—	212	4	no	—
311	M₁₆	—	212	2	no	—
312	M₁₆	—	212	2	no	—
313	M₁₆	#11719917	1	8	yes	complemented
314	C₂×C₈	—	2	8	no	—
315	C₂×C₈	—	106	2	no	—
316	C₂×C₈	—	106	2	no	—
317	C₂×C₈	—	106	4	no	—
318	C₂×C₈	—	212	4	no	—
319	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
320	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
321	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
322	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
323	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
324	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
325	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
326	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
327	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
328	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
329	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
330	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
331	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
332	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
333	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
334	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
335	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
336	C₄²	C₂³×D₅₃	1	1	yes	characteristic
337	C₂³	—	2	2	no	—
338	C₂³	—	2	2	no	—
339	C₂³	—	4	1	no	—
340	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
341	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
342	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
362	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
363	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
364	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
365	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
366	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
367	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
368	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
369	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
370	C₂×C₄	C₂×#11719917	1	4	yes	—
371	C₂×C₄	#13133245	1	2	yes	—
372	C₂×C₄	#13133250	1	2	yes	—
373	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
375	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
376	C₈	—	2	4	no	—
377	C₈	—	212	4	no	—
378	C₈	—	212	2	no	—
379	C₈	—	212	2	no	—
380	C₈	C₂×#11719917	1	8	yes	complemented
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	4	4	no	—
389	C₂²	#16810647	1	2	yes	—
390	C₂²	#16810665	1	2	yes	—
391	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	#16810647	1	2	yes	—
400	C₄	#16810665	1	2	yes	—
401	C₄	#16810672	1	2	yes	—
402	C₄	C₂²×#11719917	1	2	yes	—
403	C₄	C₂×#13133245	1	1	yes	characteristic
404	C₄	C₂×#13133250	1	1	yes	characteristic
405	C₂	—	2	2	no	—
406	C₂	—	2	2	no	—
407	C₂	—	4	1	no	—
408	C₂	C₂×#16810647	1	1	yes	characteristic
409	C₂	C₂×#16810665	1	1	yes	characteristic
410	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	#17796
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. #20273474	C₂	semidirect
5. #20273471	C₂	semidirect
6. #20273467	C₂	semidirect
23. #16810376	C₂²	semidirect
24. #16810374	C₂²	semidirect
103. M₁₆×C₅₃	M₁₆	semidirect
171. C₈×C₅₃	C₂×M₁₆	semidirect
239. C₅₃	#17796	semidirect
313. M₁₆	#11719917	semidirect
380. C₈	C₂×#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×#16810647	2
#20273482	2
#20273474	2
#20273471	2
#20273467	2
C₂×C₄×M₁₆×C₅₃	2
#17796	53