Subgroups — #27290028

Order 13568.

774 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810391	C₂	1	2	yes	maximal
4	#20274435	C₂	1	2	yes	maximal
5	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	1	yes	characteristic
7	C₂×#13133176	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	C₂×#13133151	—	2	1	no	—
10	C₂×#13133151	—	2	1	no	—
11	C₂×#13133151	—	2	2	no	—
12	C₂×#13133151	C₄	1	4	yes	complemented
13	#16810578	—	2	2	no	—
14	#16810578	C₄	1	4	yes	complemented
15	C₂×#13133127	—	2	1	no	—
16	C₂×C₈×Dic₅₃	—	2	1	no	—
17	#16810391	—	2	4	no	—
18	#16810386	—	2	4	no	—
19	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#108×C₅₃	—	2	2	no	—
22	C₂×#108×C₅₃	—	2	2	no	—
23	C₂×#108×C₅₃	C₄	1	4	yes	complemented
24	C₂³×Dic₅₃	—	2	2	no	—
25	C₂×#11719917	—	2	1	no	—
26	C₂×#11719917	—	2	1	no	—
27	C₂×#11719917	—	2	1	no	—
28	C₂×#11719917	C₂×C₄	1	2	yes	—
29	C₂×#11719917	D₄	1	2	yes	—
30	C₂×#11719917	Q₈	1	2	yes	—
31	C₂²×#88322	—	2	1	no	—
32	#13133176	—	2	1	no	—
33	#13133176	—	2	1	no	—
34	#13133176	C₂×C₄	1	2	yes	—
35	#13133176	Q₈	1	2	yes	—
36	C₂×C₄⋊Dic₅₃	—	2	2	no	—
37	C₂×C₄⋊Dic₅₃	—	2	2	no	—
38	C₂×C₄⋊Dic₅₃	—	2	2	no	—
39	C₂×C₄⋊Dic₅₃	—	2	2	no	—
40	C₂×C₄⋊Dic₅₃	—	2	2	no	—
41	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
43	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
44	C₂×C₄×Dic₅₃	—	2	1	no	—
45	#13133151	—	2	4	no	—
46	#13133151	—	2	4	no	—
47	#13133127	—	2	4	no	—
48	C₈×Dic₅₃	—	2	4	no	—
49	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₅₃	—	2	2	no	—
51	C₂×M₁₆×C₅₃	—	2	2	no	—
52	C₂×M₁₆×C₅₃	—	2	2	no	—
53	C₂×M₁₆×C₅₃	—	2	2	no	—
54	C₂×M₁₆×C₅₃	—	2	2	no	—
55	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
56	C₂²×C₈×C₅₃	—	2	2	no	—
57	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
58	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	D₄	1	2	yes	—
63	C₂²×C₈×C₅₃	D₄	1	2	yes	—
64	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
65	#108×C₅₃	—	2	8	no	—
66	C₂⁴×C₅₃	#41	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
81	C₂²×Dic₅₃	—	2	2	no	—
82	C₂²×Dic₅₃	—	2	4	no	—
83	C₂²×Dic₅₃	—	4	2	no	—
84	C₂²×Dic₅₃	—	4	2	no	—
85	C₂²×Dic₅₃	—	8	4	no	—
86	M₁₆×C₅₃	—	2	4	no	—
87	M₁₆×C₅₃	—	2	4	no	—
88	M₁₆×C₅₃	—	4	2	no	—
89	M₁₆×C₅₃	—	4	2	no	—
90	C₂×C₈×C₅₃	—	2	4	no	—
91	C₂×C₈×C₅₃	—	4	2	no	—
92	C₂×C₈×C₅₃	—	4	2	no	—
93	C₂×C₈×C₅₃	C₄²	1	4	yes	—
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	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
99	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
100	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
101	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
102	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
103	C₂²⋊Dic₅₃	—	2	2	no	—
104	C₂²⋊Dic₅₃	—	2	2	no	—
105	C₄⋊Dic₅₃	—	2	2	no	—
106	C₄⋊Dic₅₃	—	2	4	no	—
107	C₄⋊Dic₅₃	—	2	4	no	—
108	C₄⋊Dic₅₃	—	4	2	no	—
109	C₄⋊Dic₅₃	—	4	2	no	—
110	C₄⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
111	C₄×Dic₅₃	—	2	2	no	—
112	C₄×Dic₅₃	—	2	2	no	—
113	#11719917	—	2	2	no	—
114	#11719917	—	2	2	no	—
115	#11719917	—	2	2	no	—
116	#11719917	C₄⋊C₄	1	4	yes	—
117	C₂×#88322	—	2	2	no	—
118	C₂×#88322	—	2	2	no	—
119	C₂×#88322	—	2	2	no	—
120	C₂³×C₅₃	—	2	2	no	—
121	C₂³×C₅₃	—	2	2	no	—
122	C₂³×C₅₃	#117	1	1	yes	characteristic
123	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
124	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
125	C₂³×C₅₃	#121	1	1	yes	characteristic
126	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂³×C₅₃	#124	1	1	yes	characteristic
128	C₂³×C₅₃	#125	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
157	C₂×Dic₅₃	—	2	4	no	—
158	C₂×Dic₅₃	—	4	2	no	—
159	C₂×Dic₅₃	—	4	2	no	—
160	C₂×Dic₅₃	—	4	4	no	—
161	C₂×Dic₅₃	—	8	4	no	—
162	C₂×Dic₅₃	—	8	4	no	—
163	C₂×Dic₅₃	—	8	4	no	—
164	C₈×C₅₃	—	4	4	no	—
165	C₈×C₅₃	#95	1	4	yes	—
166	C₈×C₅₃	#95	1	4	yes	—
167	#88322	—	2	4	no	—
168	#10870	—	53	1	no	maximal, Sylow
169	C₂²×C₅₃	—	2	2	no	—
170	C₂²×C₅₃	—	2	2	no	—
171	C₂²×C₅₃	—	2	2	no	—
172	C₂²×C₅₃	—	2	2	no	—
173	C₂²×C₅₃	—	2	1	no	—
174	C₂²×C₅₃	—	2	1	no	—
175	C₂²×C₅₃	—	2	1	no	—
176	C₂²×C₅₃	—	2	1	no	—
177	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₅₃	#380	1	1	yes	characteristic
179	C₂²×C₅₃	#382	1	1	yes	characteristic
180	C₂²×C₅₃	#389	1	1	yes	characteristic
181	C₂²×C₅₃	#389	1	1	yes	characteristic
182	C₂²×C₅₃	#390	1	1	yes	characteristic
183	C₂²×C₅₃	#391	1	1	yes	characteristic
184	C₂²×C₅₃	#472	1	2	yes	—
185	C₂²×C₅₃	#473	1	2	yes	—
186	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
187	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
188	C₄×C₅₃	#380	1	1	yes	characteristic
189	C₄×C₅₃	#384	1	1	yes	characteristic
190	C₄×C₅₃	#389	1	1	yes	characteristic
191	C₄×C₅₃	#389	1	1	yes	characteristic
192	C₄×C₅₃	#390	1	1	yes	characteristic
193	C₄×C₅₃	#391	1	1	yes	characteristic
194	Dic₅₃	—	4	4	no	—
195	Dic₅₃	—	8	4	no	—
196	C₂²×#108	D₅₃	1	1	yes	characteristic
197	C₂²×#107	—	53	1	no	—
198	C₂×#405	—	53	1	no	—
199	C₂×#343	—	53	2	no	—
200	#1736	—	53	2	no	—
201	C₂×C₅₃	—	2	1	no	—
202	C₂×C₅₃	—	2	1	no	—
203	C₂×C₅₃	—	2	1	no	—
204	C₂×C₅₃	—	2	1	no	—
205	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
206	C₂×C₅₃	#1816	1	2	yes	—
207	C₂×C₅₃	#1952	1	2	yes	—
208	C₂×C₅₃	#1978	1	1	yes	characteristic
209	C₂×C₅₃	#1979	1	1	yes	characteristic
210	C₂²×M₁₆	—	53	1	no	—
211	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
212	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
213	C₂×#117	—	53	1	no	—
214	C₂²×C₄⋊C₄	—	53	2	no	—
215	C₂×#108	—	2	2	no	—
216	C₂×#108	—	2	2	no	—
217	C₂×#108	Dic₅₃	1	4	yes	complemented
218	C₂×#107	—	53	4	no	—
219	C₂×#107	—	106	1	no	—
220	C₂×#107	—	106	1	no	—
221	C₂×#107	—	106	2	no	—
222	#405	—	53	4	no	—
223	#405	—	106	2	no	—
224	C₂×#97	—	106	1	no	—
225	C₂×C₄×C₈	—	106	1	no	—
226	#343	—	106	4	no	—
227	#340	—	106	4	no	—
228	C₅₃	#10870	1	1	yes	characteristic, complemented, Sylow
229	C₂³×C₄	—	106	2	no	—
230	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
231	C₂×M₁₆	—	2	2	no	—
232	C₂×M₁₆	—	2	2	no	—
233	C₂×M₁₆	—	2	2	no	—
234	C₂×M₁₆	—	2	2	no	—
235	C₂×M₁₆	—	2	2	no	—
236	C₂×M₁₆	—	53	2	no	—
237	C₂×M₁₆	—	53	2	no	—
238	C₂×M₁₆	—	53	2	no	—
239	C₂×M₁₆	—	106	1	no	—
240	C₂×M₁₆	—	106	1	no	—
241	C₂×M₁₆	—	106	1	no	—
242	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
243	C₂²×C₈	—	2	2	no	—
244	C₂²×C₈	—	106	1	no	—
245	C₂²×C₈	Dic₁₀₆	1	2	yes	—
246	C₂²×C₈	C₄×D₅₃	1	2	yes	—
247	C₂²×C₈	C₄×D₅₃	1	2	yes	—
248	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
249	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
250	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
251	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
252	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
253	#117	—	53	2	no	—
254	#117	—	53	2	no	—
255	#117	—	106	1	no	—
256	#117	—	106	1	no	—
257	C₂×C₄⋊C₄	—	53	2	no	—
258	C₂×C₄⋊C₄	—	53	4	no	—
259	C₂×C₄⋊C₄	—	106	2	no	—
260	C₂×C₄⋊C₄	—	106	2	no	—
261	C₂×C₄⋊C₄	—	106	2	no	—
262	C₂×C₄⋊C₄	—	106	2	no	—
263	C₂×C₄⋊C₄	—	106	2	no	—
264	C₂×C₂²⋊C₄	—	106	1	no	—
265	C₂×C₄²	—	106	1	no	—
266	#108	—	2	8	no	—
267	#107	—	106	4	no	—
268	#107	—	106	4	no	—
269	#97	—	106	4	no	—
270	C₄×C₈	—	106	4	no	—
271	C₂⁴	#11719949	1	1	yes	characteristic
272	C₂²×C₄	—	106	2	no	—
273	C₂²×C₄	—	106	4	no	—
274	C₂²×C₄	—	424	4	no	—
275	C₂²×C₄	—	212	2	no	—
276	C₂²×C₄	—	212	2	no	—
277	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
281	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
282	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
283	C₂²×C₄	#11719947	1	1	yes	characteristic
284	C₂²×C₄	#11719947	1	1	yes	characteristic
285	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
286	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
287	C₂²×C₄	#11719949	1	1	yes	characteristic
288	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
289	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
291	M₁₆	—	2	4	no	—
292	M₁₆	—	2	4	no	—
293	M₁₆	—	4	2	no	—
294	M₁₆	—	4	2	no	—
295	M₁₆	—	53	4	no	—
296	M₁₆	—	106	2	no	—
297	M₁₆	—	106	2	no	—
298	M₁₆	—	106	2	no	—
299	C₂×C₈	—	2	4	no	—
300	C₂×C₈	—	4	2	no	—
301	C₂×C₈	—	4	2	no	—
302	C₂×C₈	—	106	2	no	—
303	C₂×C₈	—	106	2	no	—
304	C₂×C₈	—	106	2	no	—
305	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
306	C₂×C₈	#11719919	1	4	yes	—
307	C₂×C₈	#11719919	1	4	yes	—
308	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
309	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
310	C₂×C₈	#11719921	1	2	yes	—
311	C₂×C₈	#11719921	1	2	yes	—
312	C₂×C₈	#11719921	1	2	yes	—
313	C₂×C₈	#11719921	1	2	yes	—
314	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
315	C₄⋊C₄	—	53	4	no	—
316	C₄⋊C₄	—	106	2	no	—
317	C₄⋊C₄	—	106	4	no	—
318	C₄⋊C₄	—	106	4	no	—
319	C₄⋊C₄	—	212	2	no	—
320	C₄⋊C₄	—	212	2	no	—
321	C₂²⋊C₄	—	106	2	no	—
322	C₂²⋊C₄	—	106	2	no	—
323	C₄²	—	106	2	no	—
324	C₄²	—	106	2	no	—
325	C₂³	—	2	2	no	—
326	C₂³	—	2	2	no	—
327	C₂³	#13133175	1	1	yes	characteristic
328	C₂³	#13133179	1	1	yes	characteristic
329	C₂³	Q₈×Dic₅₃	1	1	yes	characteristic
330	C₂³	#13133192	1	1	yes	characteristic
331	C₂³	#13133193	1	1	yes	characteristic
332	C₂³	#13133206	1	1	yes	characteristic
333	C₂³	#13133208	1	1	yes	characteristic
334	C₂×C₄	—	106	4	no	—
335	C₂×C₄	—	424	4	no	—
336	C₂×C₄	—	424	4	no	—
337	C₂×C₄	—	424	4	no	—
338	C₂×C₄	—	212	4	no	—
339	C₂×C₄	—	212	2	no	—
340	C₂×C₄	—	212	2	no	—
341	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
342	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
343	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
344	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
345	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
346	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
347	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
348	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
349	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
350	C₂×C₄	#13133175	1	1	yes	characteristic
351	C₂×C₄	#13133179	1	1	yes	characteristic
352	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
353	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
354	C₂×C₄	#13133182	1	1	yes	characteristic
355	C₂×C₄	#13133182	1	1	yes	characteristic
356	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
357	C₂×C₄	#13133188	1	1	yes	characteristic
358	C₂×C₄	#13133188	1	1	yes	characteristic
359	C₂×C₄	#13133192	1	1	yes	characteristic
360	C₂×C₄	#13133193	1	1	yes	characteristic
361	C₂×C₄	#13133197	1	1	yes	characteristic
362	C₂×C₄	#13133197	1	1	yes	characteristic
363	C₂×C₄	#13133204	1	1	yes	characteristic
364	C₂×C₄	#13133208	1	1	yes	characteristic
365	C₂×C₄	#13133213	1	1	yes	characteristic
366	C₂×C₄	#13133215	1	1	yes	characteristic
367	C₂×C₄	#13133216	1	1	yes	characteristic
368	C₂×C₄	#13133216	1	1	yes	characteristic
369	C₈	—	4	4	no	—
370	C₈	—	106	4	no	—
371	C₈	#13133124	1	4	yes	—
372	C₈	#13133124	1	4	yes	—
373	C₂²	—	2	2	no	—
374	C₂²	—	2	2	no	—
375	C₂²	—	2	2	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
382	C₂²	#16810480	1	1	yes	characteristic
383	C₂²	#16810488	1	1	yes	characteristic
384	C₂²	#16810519	1	1	yes	characteristic
385	C₂²	#16810522	1	1	yes	characteristic
386	C₂²	#16810524	1	1	yes	characteristic
387	C₂²	#16810530	1	1	yes	characteristic
388	C₂²	#16810894	1	2	yes	—
389	C₂²	#16810899	1	2	yes	—
390	C₄	—	424	4	no	—
391	C₄	—	212	4	no	—
392	C₄	C₂×#13133124	1	1	yes	characteristic
393	C₄	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
394	C₄	#16810480	1	1	yes	characteristic
395	C₄	#16810492	1	1	yes	characteristic
396	C₄	#16810519	1	1	yes	characteristic
397	C₄	#16810522	1	1	yes	characteristic
398	C₄	#16810526	1	1	yes	characteristic
399	C₄	#16810528	1	1	yes	characteristic
400	C₂	—	2	1	no	—
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	2	1	no	—
404	C₂	#20272944	1	1	yes	characteristic
405	C₂	#20274776	1	2	yes	—
406	C₂	#20275501	1	2	yes	—
407	C₂	#20275623	1	1	yes	characteristic
408	C₂	#20275628	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133192
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	#10870
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
12. C₂×#13133151	C₄	semidirect
14. #16810578	C₄	semidirect
23. C₂×#108×C₅₃	C₄	semidirect
217. C₂×#108	Dic₅₃	semidirect
228. C₅₃	#10870	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133151	2
C₂×#16810578	2
C₂×#16810391	2
#20274435	2
C₂²×#108×C₅₃	2
#10870	53