Subgroups — #27353105

Order 13568.

762 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	2	yes	maximal
2	#20273478	C₂	1	4	yes	maximal, complemented
3	#20273474	C₂	1	2	yes	maximal, complemented
4	#20273471	C₂	1	2	yes	maximal, complemented
5	#20273463	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	#16810680	—	2	2	no	—
8	#16810680	C₂²	1	4	yes	—
9	#16810676	—	2	2	no	—
10	#16810676	C₂²	1	4	yes	—
11	#16810647	C₂²	1	4	yes	—
12	#16810587	—	2	2	no	—
13	C₄×#11719917	—	2	2	no	—
14	#16810376	C₂²	1	8	yes	complemented
15	#16810374	C₂²	1	8	yes	complemented
16	C₂³×C₈×C₅₃	C₄	1	2	yes	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
27	C₂×#11719917	—	2	4	no	—
28	C₄⋊#88322	—	2	4	no	—
29	C₄⋊#88322	C₂³	1	8	yes	—
30	C₂²⋊#88322	—	2	4	no	—
31	C₂²⋊#88322	—	2	4	no	—
32	#13133128	—	2	4	no	—
33	C₄×#88322	—	2	4	no	—
34	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	—	2	8	no	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₅₃	C₂³	1	2	yes	—
65	C₄×C₈×C₅₃	C₂³	1	2	yes	—
66	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	—	2	4	no	—
68	C₂²×C₄×C₅₃	—	2	4	no	—
69	C₂²×C₄×C₅₃	—	2	4	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	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₅₃	—	2	4	no	—
89	C₂×C₈×C₅₃	—	2	4	no	—
90	C₂×C₈×C₅₃	—	2	4	no	—
91	C₂×C₈×C₅₃	—	2	4	no	—
92	C₂×C₈×C₅₃	—	2	4	no	—
93	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
94	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
113	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
114	#11719917	—	4	8	no	—
115	C₂×#88322	—	2	8	no	—
116	C₂³×C₅₃	—	2	2	no	—
117	C₂³×C₅₃	—	2	2	no	—
118	C₂³×C₅₃	—	2	2	no	—
119	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
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	—	2	2	no	—
137	C₂×C₄×C₅₃	#108	1	4	yes	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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×M₁₆	1	4	yes	—
148	C₂×C₄×C₅₃	#135	1	4	yes	—
149	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
152	C₈×C₅₃	—	2	2	no	—
153	C₈×C₅₃	—	2	2	no	—
154	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
155	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
156	#88322	—	4	8	no	—
157	#17793	—	53	1	no	maximal, Sylow
158	C₂²×C₅₃	—	2	8	no	—
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	—	2	1	no	—
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	#423	1	2	yes	—
166	C₂²×C₅₃	#427	1	2	yes	—
167	C₂²×C₅₃	C₂×#108	1	2	yes	—
168	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	#423	1	2	yes	—
176	C₄×C₅₃	#427	1	2	yes	—
177	C₄×C₅₃	C₂×#108	1	2	yes	—
178	C₄×C₅₃	C₂×#108	1	2	yes	—
179	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
180	C₄×C₅₃	C₂×#135	1	2	yes	—
181	#2920	—	53	2	no	—
182	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
183	#1568	—	53	4	no	—
184	#1567	—	53	2	no	—
185	#1566	—	53	2	no	—
186	#1564	—	53	4	no	—
187	C₂×C₅₃	—	2	2	no	—
188	C₂×C₅₃	—	2	2	no	—
189	C₂×C₅₃	—	2	2	no	—
190	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
191	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
192	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
193	C₂³×C₈	Dic₅₃	1	2	yes	—
194	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
195	#432	—	53	4	no	—
196	#432	—	106	2	no	—
197	#431	—	53	4	no	—
198	#431	—	106	2	no	—
199	#423	—	53	4	no	—
200	#407	—	106	2	no	—
201	C₄×M₁₆	—	106	2	no	—
202	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
203	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
204	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
205	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
206	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
207	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
208	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
209	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
210	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
211	#335	—	53	8	no	—
212	#334	—	53	8	no	—
213	C₅₃	#17793	1	1	yes	characteristic, complemented, Sylow
214	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
215	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
216	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
217	C₂×M₁₆	—	106	4	no	—
218	C₂²×C₈	—	2	8	no	—
219	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
220	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
221	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
222	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
223	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
224	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
225	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
226	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
227	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
228	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
229	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
230	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
231	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
232	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
233	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
234	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
237	C₄⋊C₈	—	53	8	no	—
238	C₄⋊C₈	—	106	4	no	—
239	C₂²⋊C₈	—	106	4	no	—
240	C₂²⋊C₈	—	106	4	no	—
241	#97	—	106	4	no	—
242	C₄×C₈	—	106	4	no	—
243	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
244	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
245	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
246	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
247	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
248	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
249	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
250	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
251	C₄×C₈	C₂²×D₅₃	1	2	yes	—
252	C₄×C₈	C₂²×D₅₃	1	2	yes	—
253	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
258	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
259	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
260	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
261	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
262	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
263	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
264	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
266	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
267	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
268	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
272	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
275	M₁₆	—	212	8	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	106	8	no	—
282	C₂×C₈	#11719917	1	8	yes	complemented
283	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
284	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
285	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
286	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
287	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
288	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
289	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
290	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
291	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
292	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
293	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
294	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
295	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
296	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
297	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
298	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
299	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
300	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
301	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
302	C₄²	C₂³×D₅₃	1	1	yes	characteristic
303	C₂³	—	2	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	2	2	no	—
306	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
307	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
308	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
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	#13133153	1	4	yes	—
325	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
326	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
327	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
328	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
329	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
330	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
331	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
333	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂×#11719917	1	4	yes	—
335	C₂×C₄	#13133239	1	4	yes	—
336	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
337	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
338	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	212	8	no	—
342	C₈	C₂×#11719917	1	4	yes	—
343	C₈	C₂×#11719917	1	4	yes	—
344	C₂²	—	2	8	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	#16810647	1	2	yes	—
352	C₂²	#16810661	1	2	yes	—
353	C₂²	C₂×#13133153	1	2	yes	—
354	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	#16810647	1	2	yes	—
362	C₄	#16810661	1	2	yes	—
363	C₄	C₂×#13133153	1	2	yes	—
364	C₄	C₂×#13133153	1	2	yes	—
365	C₄	C₂²×#11719917	1	2	yes	—
366	C₄	C₂×#13133239	1	2	yes	—
367	C₂	—	2	2	no	—
368	C₂	—	2	2	no	—
369	C₂	—	2	2	no	—
370	C₂	C₂×#16810647	1	1	yes	characteristic
371	C₂	C₂×#16810661	1	1	yes	characteristic
372	C₂	C₂²×#13133153	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	#17793
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20273478	C₂	semidirect
3. #20273474	C₂	semidirect
4. #20273471	C₂	semidirect
5. #20273463	C₂	semidirect
14. #16810376	C₂²	semidirect
15. #16810374	C₂²	semidirect
93. C₂×C₈×C₅₃	M₁₆	semidirect
213. C₅₃	#17793	semidirect
282. C₂×C₈	#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
#20273478	2
#20273474	2
#20273471	2
#20273463	2
C₂²×C₄×C₈×C₅₃	2
#17793	53