Subgroups — #27273745

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
3	#20272563	C₂	1	4	yes	maximal
4	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810676	—	2	1	no	—
6	#16810676	—	2	1	no	—
7	#16810676	—	2	1	no	—
8	#16810676	—	2	1	no	—
9	#16810676	—	2	1	no	—
10	#16810676	—	2	1	no	—
11	#16810676	C₄	1	2	yes	complemented
12	#16810676	C₄	1	2	yes	complemented
13	C₂×C₄⋊#88322	—	2	1	no	—
14	C₂×C₄⋊#88322	C₄	1	2	yes	complemented
15	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
16	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
17	C₂×C₄×#88322	—	2	1	no	—
18	C₂×C₄×#88322	C₄	1	2	yes	complemented
19	C₂×#117×C₅₃	C₄	1	2	yes	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
22	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
23	C₄×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
24	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₅₃×#376	—	2	1	no	—
26	C₄³×C₅₃	—	2	1	no	—
27	C₂²×#88322	—	2	2	no	—
28	C₂²×#88322	C₂×C₄	1	4	yes	—
29	C₄⋊#88322	—	2	2	no	—
30	C₄⋊#88322	—	2	2	no	—
31	C₄⋊#88322	—	2	2	no	—
32	C₄⋊#88322	—	2	2	no	—
33	C₂²⋊#88322	—	2	2	no	—
34	C₂²⋊#88322	—	2	2	no	—
35	C₂²⋊#88322	—	2	2	no	—
36	C₂²⋊#88322	C₂×C₄	1	4	yes	—
37	C₄×#88322	—	2	2	no	—
38	C₄×#88322	—	2	2	no	—
39	C₄×#88322	—	2	2	no	—
40	C₄×#88322	—	2	2	no	—
41	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	#117×C₅₃	—	2	4	no	—
45	#117×C₅₃	C₈	1	8	yes	complemented
46	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	—	2	2	no	—
50	C₂×C₄²×C₅₃	—	2	2	no	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	D₄	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	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
63	#95×C₅₃	—	2	2	no	—
64	#95×C₅₃	—	2	2	no	—
65	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	—	2	4	no	—
67	C₂²×C₄×C₅₃	—	4	2	no	—
68	C₂²×C₄×C₅₃	—	4	2	no	—
69	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²⋊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	C₂×C₈	1	4	yes	—
82	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
89	C₄⋊C₄×C₅₃	—	2	8	no	—
90	C₂²⋊C₄×C₅₃	—	2	4	no	—
91	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
92	C₄²×C₅₃	—	2	8	no	—
93	C₄²×C₅₃	—	4	4	no	—
94	C₄²×C₅₃	—	4	4	no	—
95	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
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	C₂×#88322	—	2	4	no	—
112	C₂×#88322	—	2	4	no	—
113	C₂×#88322	—	4	4	no	—
114	C₂×#88322	—	4	4	no	—
115	C₂³×C₅₃	—	4	1	no	—
116	C₂³×C₅₃	—	4	1	no	—
117	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
118	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
121	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
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	8	no	—
129	C₂×C₄×C₅₃	—	4	1	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	4	1	no	—
132	C₂×C₄×C₅₃	—	4	1	no	—
133	C₂×C₄×C₅₃	—	4	1	no	—
134	C₂×C₄×C₅₃	—	4	1	no	—
135	C₂×C₄×C₅₃	—	4	4	no	—
136	C₂×C₄×C₅₃	—	4	4	no	—
137	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
138	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
139	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
140	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
146	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
147	C₂×C₄×C₅₃	#97	1	2	yes	—
148	C₂×C₄×C₅₃	#97	1	2	yes	—
149	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
150	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
151	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
152	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
153	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
157	#88322	—	4	4	no	—
158	#88322	—	4	4	no	—
159	#7924	—	53	1	no	maximal, Sylow
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	—	2	1	no	—
166	C₂²×C₅₃	—	4	2	no	—
167	C₂²×C₅₃	—	4	2	no	—
168	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
169	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
170	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
171	C₂²×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
172	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
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	—	4	8	no	—
180	C₄×C₅₃	#336	1	2	yes	—
181	C₄×C₅₃	#336	1	2	yes	—
182	C₄×C₅₃	#336	1	2	yes	—
183	C₄×C₅₃	#336	1	2	yes	—
184	C₄×C₅₃	#344	1	2	yes	—
185	C₄×C₅₃	#410	1	2	yes	—
186	C₂×#431	—	53	1	no	—
187	C₂×#431	—	53	1	no	—
188	C₄×#117	D₅₃	1	1	yes	characteristic
189	#1282	—	53	4	no	—
190	C₂×C₅₃	—	2	2	no	—
191	C₂×C₅₃	—	2	1	no	—
192	C₂×C₅₃	—	2	1	no	—
193	C₂×C₅₃	—	2	2	no	—
194	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
195	C₂×C₅₃	#1740	1	1	yes	characteristic
196	C₂×C₅₃	C₂×#344	1	1	yes	characteristic
197	C₂×#117	Dic₅₃	1	2	yes	—
198	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
199	#431	—	53	2	no	—
200	#431	—	53	2	no	—
201	#431	—	106	1	no	—
202	#431	—	106	1	no	—
203	#431	—	106	1	no	—
204	#431	—	106	1	no	—
205	#431	—	106	1	no	—
206	#431	—	106	1	no	—
207	C₂×C₄⋊C₈	—	53	2	no	—
208	C₂×C₄⋊C₈	—	106	1	no	—
209	C₂×C₂²⋊C₈	—	53	2	no	—
210	C₂×C₂²⋊C₈	—	53	2	no	—
211	C₂×C₄×C₈	—	53	2	no	—
212	C₂×C₄×C₈	—	106	1	no	—
213	C₄×C₄⋊C₄	—	2	1	no	—
214	C₄×C₄⋊C₄	—	2	1	no	—
215	C₄×C₂²⋊C₄	Dic₅₃	1	2	yes	—
216	C₄×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
217	#376	—	2	1	no	—
218	C₄³	—	2	1	no	—
219	C₅₃	#7924	1	1	yes	characteristic, complemented, Sylow
220	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
221	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
222	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
223	C₂²×C₈	—	53	4	no	—
224	C₂²×C₈	—	106	2	no	—
225	#117	—	2	4	no	—
226	#117	#88322	1	8	yes	complemented
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	4	yes	—
229	C₂×C₄²	—	2	2	no	—
230	C₂×C₄²	—	2	2	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
233	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
234	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
235	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
236	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
237	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
238	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
239	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
240	C₂×C₄²	C₂²⋊D₅₃	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	—	106	2	no	—
245	C₄⋊C₈	—	106	2	no	—
246	C₄⋊C₈	—	106	2	no	—
247	C₄⋊C₈	—	106	2	no	—
248	C₂²⋊C₈	—	53	4	no	—
249	C₂²⋊C₈	—	106	2	no	—
250	C₂²⋊C₈	—	106	2	no	—
251	C₂²⋊C₈	—	106	2	no	—
252	C₄×C₈	—	106	2	no	—
253	C₄×C₈	—	106	2	no	—
254	C₄×C₈	—	106	2	no	—
255	C₄×C₈	—	106	2	no	—
256	#95	—	2	2	no	—
257	#95	—	2	2	no	—
258	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	C₂×#88322	1	4	yes	—
263	C₂²×C₄	#11719917	1	4	yes	—
264	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
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	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	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
280	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
281	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
282	C₂×C₈	—	106	4	no	—
283	C₂×C₈	—	106	4	no	—
284	C₂×C₈	—	212	4	no	—
285	C₂×C₈	—	212	4	no	—
286	C₄⋊C₄	—	2	8	no	—
287	C₂²⋊C₄	—	2	4	no	—
288	C₂²⋊C₄	C₂×#88322	1	8	yes	—
289	C₄²	—	2	8	no	—
290	C₄²	—	4	4	no	—
291	C₄²	—	4	4	no	—
292	C₄²	#11719919	1	1	yes	characteristic
293	C₄²	#11719919	1	1	yes	characteristic
294	C₄²	#11719919	1	1	yes	characteristic
295	C₄²	#11719919	1	1	yes	characteristic
296	C₄²	#11719919	1	1	yes	characteristic
297	C₄²	#11719919	1	1	yes	characteristic
298	C₄²	#11719919	1	1	yes	characteristic
299	C₄²	#11719919	1	1	yes	characteristic
300	C₄²	#11719921	1	1	yes	characteristic
301	C₄²	#11719921	1	1	yes	characteristic
302	C₄²	#11719921	1	1	yes	characteristic
303	C₄²	#11719921	1	1	yes	characteristic
304	C₄²	#11719921	1	1	yes	characteristic
305	C₄²	#11719921	1	1	yes	characteristic
306	C₄²	#11719921	1	1	yes	characteristic
307	C₄²	#11719921	1	1	yes	characteristic
308	C₂³	—	4	1	no	—
309	C₂³	—	4	1	no	—
310	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
311	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
312	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
313	C₂³	C₂²×#88322	1	2	yes	—
314	C₂³	C₂×#11719917	1	2	yes	—
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	8	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	4	4	no	—
330	C₂×C₄	#13133124	1	1	yes	characteristic
331	C₂×C₄	#13133124	1	1	yes	characteristic
332	C₂×C₄	#13133124	1	1	yes	characteristic
333	C₂×C₄	#13133124	1	1	yes	characteristic
334	C₂×C₄	#13133124	1	1	yes	characteristic
335	C₂×C₄	#13133124	1	1	yes	characteristic
336	C₂×C₄	#13133124	1	1	yes	characteristic
337	C₂×C₄	#13133124	1	1	yes	characteristic
338	C₂×C₄	C₄×#88322	1	2	yes	—
339	C₂×C₄	C₄×#88322	1	2	yes	—
340	C₂×C₄	#13133128	1	2	yes	—
341	C₂×C₄	#13133128	1	2	yes	—
342	C₂×C₄	C₂²⋊#88322	1	4	yes	—
343	C₂×C₄	C₂²⋊#88322	1	4	yes	—
344	C₂×C₄	C₄⋊#88322	1	4	yes	—
345	C₂×C₄	C₄⋊#88322	1	4	yes	—
346	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
347	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
348	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
349	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
350	C₈	—	212	4	no	—
351	C₈	—	212	4	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	—	2	1	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	C₂×#13133124	1	1	yes	characteristic
361	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
362	C₂²	C₂×#13133128	1	1	yes	characteristic
363	C₂²	C₂×C₂²⋊#88322	1	2	yes	—
364	C₂²	C₂×C₄⋊#88322	1	2	yes	—
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	—	2	1	no	—
371	C₄	—	4	8	no	—
372	C₄	#16810378	1	2	yes	—
373	C₄	#16810378	1	2	yes	—
374	C₄	#16810378	1	2	yes	—
375	C₄	#16810378	1	2	yes	—
376	C₄	#16810393	1	2	yes	—
377	C₄	#16810596	1	2	yes	—
378	C₂	—	2	2	no	—
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	2	2	no	—
382	C₂	C₂×#16810378	1	1	yes	characteristic
383	C₂	#20274447	1	1	yes	characteristic
384	C₂	C₂×#16810393	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#11719921
commutator	C₂²×C₅₃	C₂×C₄×C₈
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₄×#117×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₄×#117	D₅₃
hypocenter	C₅₃	#7924
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
11. #16810676	C₄	semidirect
12. #16810676	C₄	semidirect
14. C₂×C₄⋊#88322	C₄	semidirect
18. C₂×C₄×#88322	C₄	semidirect
45. #117×C₅₃	C₈	semidirect
219. C₅₃	#7924	semidirect
226. #117	#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
#20272563	2
C₄×#117×C₅₃	2
#7924	53