Subgroups — #27392426

Order 13568.

674 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	C₄×#13133209	C₂	1	1	yes	maximal, characteristic
2	#20275433	C₂	1	1	yes	maximal, characteristic
3	#20275433	C₂	1	1	yes	maximal, characteristic
4	#20275279	C₂	1	2	yes	maximal
5	#20275279	C₂	1	2	yes	maximal
6	#20274832	C₂	1	1	yes	maximal, characteristic
7	#20274591	C₂	1	2	yes	maximal
8	#20274583	C₂	1	2	yes	maximal
9	#20274582	C₂	1	1	yes	maximal, characteristic
10	C₈×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
11	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133209	C₄	1	1	yes	characteristic
13	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
15	C₂×#13133146	C₂²	1	2	yes	—
16	C₂×#13133146	C₂²	1	2	yes	—
17	C₂×#13133127	C₂²	1	2	yes	—
18	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
19	#16810521	C₄	1	1	yes	characteristic
20	#16810521	C₄	1	1	yes	characteristic
21	#16810507	C₄	1	2	yes	—
22	#16810507	C₄	1	2	yes	—
23	#16810489	C₄	1	1	yes	characteristic
24	C₄×#11719919	C₂²	1	2	yes	—
25	C₄×#11719919	C₂²	1	2	yes	—
26	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
28	#16810378	C₂²	1	2	yes	—
29	#16810378	C₂²	1	2	yes	—
30	#16810377	C₂²	1	2	yes	—
31	#16810377	C₂²	1	2	yes	—
32	#16810377	C₂²	1	2	yes	—
33	#16810377	C₂²	1	2	yes	—
34	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₄³×C₅₃	C₂²	1	1	yes	characteristic
41	C₂²×#88322	C₂³	1	2	yes	—
42	C₂²×#88322	C₂³	1	2	yes	—
43	#13133209	—	2	4	no	—
44	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×#11719919	C₂×C₄	1	2	yes	—
49	C₂×#11719919	C₂×C₄	1	2	yes	—
50	C₂×#11719919	C₂×C₄	1	2	yes	—
51	C₂×#11719919	C₂×C₄	1	2	yes	—
52	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
53	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
54	C₄⋊#88322	—	2	2	no	—
55	C₄⋊#88322	—	2	2	no	—
56	#13133146	—	2	4	no	—
57	#13133146	—	2	4	no	—
58	#13133127	—	2	4	no	—
59	C₈×Dic₅₃	—	2	4	no	—
60	#13133124	C₂×C₄	1	2	yes	—
61	#13133124	C₂×C₄	1	2	yes	—
62	#13133124	C₂×C₄	1	2	yes	—
63	#13133124	C₂×C₄	1	2	yes	—
64	#13133124	C₂×C₄	1	2	yes	—
65	#13133124	C₂×C₄	1	2	yes	—
66	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	#97×C₅₃	—	2	2	no	—
78	#97×C₅₃	—	2	2	no	—
79	#97×C₅₃	—	2	2	no	—
80	#97×C₅₃	—	2	2	no	—
81	C₄×C₈×C₅₃	—	2	2	no	—
82	C₄×C₈×C₅₃	Q₈	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	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	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
91	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
92	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
93	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₅₃	—	2	1	no	—
95	C₂×C₈×C₅₃	—	2	1	no	—
96	C₂×C₈×C₅₃	—	2	1	no	—
97	C₂×C₈×C₅₃	—	2	1	no	—
98	C₂×C₈×C₅₃	—	2	1	no	—
99	C₂×C₈×C₅₃	—	2	1	no	—
100	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
101	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
102	C₂×C₈×C₅₃	#41	1	2	yes	—
103	C₂×C₈×C₅₃	#41	1	2	yes	—
104	C₂×C₈×C₅₃	#41	1	2	yes	—
105	C₂×C₈×C₅₃	#41	1	2	yes	—
106	C₄²×C₅₃	—	2	2	no	—
107	C₄²×C₅₃	—	2	2	no	—
108	C₄²×C₅₃	—	2	1	no	—
109	C₄²×C₅₃	—	2	1	no	—
110	C₄²×C₅₃	—	2	1	no	—
111	C₄²×C₅₃	—	2	1	no	—
112	C₄²×C₅₃	—	4	1	no	—
113	C₄²×C₅₃	—	4	1	no	—
114	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
115	C₄²×C₅₃	#41	1	2	yes	—
116	C₄⋊Dic₅₃	—	2	2	no	—
117	C₄⋊Dic₅₃	—	2	2	no	—
118	C₄⋊Dic₅₃	—	2	2	no	—
119	C₄⋊Dic₅₃	M₁₆	1	4	yes	—
120	#11719919	—	2	4	no	—
121	#11719919	—	2	4	no	—
122	#11719919	—	4	2	no	—
123	#11719919	—	4	2	no	—
124	C₄×Dic₅₃	—	2	2	no	—
125	C₄×Dic₅₃	—	2	2	no	—
126	C₄×Dic₅₃	—	2	2	no	—
127	C₄×Dic₅₃	—	2	2	no	—
128	C₂×#88322	—	2	2	no	—
129	C₂×#88322	—	2	2	no	—
130	C₂×#88322	—	2	2	no	—
131	C₂×#88322	—	2	2	no	—
132	C₂×#88322	—	2	2	no	—
133	C₂×#88322	—	2	2	no	—
134	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	1	no	—
139	C₂×C₄×C₅₃	—	2	1	no	—
140	C₂×C₄×C₅₃	—	2	1	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	1	no	—
146	C₂×C₄×C₅₃	—	2	1	no	—
147	C₂×C₄×C₅₃	—	2	1	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
150	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
151	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
152	C₂×C₄×C₅₃	#131	1	2	yes	—
153	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
159	C₂×Dic₅₃	—	2	2	no	—
160	C₂×Dic₅₃	—	2	2	no	—
161	C₂×Dic₅₃	—	2	2	no	—
162	C₂×Dic₅₃	—	2	2	no	—
163	C₂×Dic₅₃	—	2	2	no	—
164	C₂×Dic₅₃	—	2	2	no	—
165	C₂×Dic₅₃	—	2	2	no	—
166	C₂×Dic₅₃	—	2	2	no	—
167	C₂×Dic₅₃	—	2	2	no	—
168	C₂×Dic₅₃	—	2	2	no	—
169	C₂×Dic₅₃	—	2	2	no	—
170	C₂×Dic₅₃	C₂×M₁₆	1	4	yes	—
171	C₈×C₅₃	—	2	2	no	—
172	C₈×C₅₃	—	2	2	no	—
173	C₈×C₅₃	—	2	2	no	—
174	C₈×C₅₃	#125	1	4	yes	complemented
175	#88322	—	4	2	no	—
176	#88322	—	4	2	no	—
177	#21675	—	53	1	no	maximal, Sylow
178	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
179	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
180	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
181	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
182	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
183	C₂²×C₅₃	#568	1	1	yes	characteristic
184	C₂²×C₅₃	#568	1	1	yes	characteristic
185	C₄×C₅₃	—	2	2	no	—
186	C₄×C₅₃	—	4	1	no	—
187	C₄×C₅₃	—	4	1	no	—
188	C₄×C₅₃	—	4	1	no	—
189	C₄×C₅₃	—	4	1	no	—
190	C₄×C₅₃	#446	1	4	yes	—
191	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
192	C₄×C₅₃	#529	1	1	yes	characteristic
193	C₄×C₅₃	#529	1	1	yes	characteristic
194	C₄×C₅₃	#533	1	1	yes	characteristic
195	Dic₅₃	—	2	4	no	—
196	Dic₅₃	—	4	2	no	—
197	Dic₅₃	—	4	2	no	—
198	Dic₅₃	—	4	2	no	—
199	C₄×#125	—	53	1	no	—
200	#1939	—	53	1	no	—
201	#1939	—	53	1	no	—
202	#1917	—	53	2	no	—
203	#1917	—	53	2	no	—
204	#1831	—	53	1	no	—
205	#1771	—	53	2	no	—
206	#1770	—	53	2	no	—
207	#1770	—	53	1	no	—
208	C₈×C₄⋊C₄	—	53	1	no	—
209	C₄×#97	D₅₃	1	1	yes	characteristic
210	C₂×C₅₃	C₄×#125	1	1	yes	characteristic
211	C₂×C₅₃	C₂×#446	1	1	yes	characteristic
212	C₂×C₅₃	#2961	1	1	yes	characteristic
213	C₂×C₅₃	#2969	1	1	yes	characteristic
214	C₂×C₅₃	#2969	1	1	yes	characteristic
215	C₂×C₅₃	#2970	1	1	yes	characteristic
216	C₂×C₅₃	#2971	1	1	yes	characteristic
217	C₂×#125	—	53	1	no	—
218	C₂×C₄⋊C₈	—	53	2	no	—
219	C₂×C₄⋊C₈	—	53	2	no	—
220	C₂×C₄⋊C₈	—	53	1	no	—
221	C₂×C₄⋊C₈	—	53	1	no	—
222	C₂×#97	—	53	2	no	—
223	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
224	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
225	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
226	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
227	C₂×C₄×C₈	—	53	2	no	—
228	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
229	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
230	#389	—	53	1	no	—
231	#389	—	53	1	no	—
232	#387	—	53	2	no	—
233	#387	—	53	2	no	—
234	#382	—	53	1	no	—
235	C₄×C₄⋊C₄	—	53	2	no	—
236	C₄×C₄⋊C₄	—	53	2	no	—
237	C₄×C₄⋊C₄	—	53	1	no	—
238	C₄×C₄⋊C₄	—	53	1	no	—
239	C₄³	C₂×D₅₃	1	1	yes	characteristic
240	#336	—	53	2	no	—
241	#336	—	53	2	no	—
242	#336	—	53	2	no	—
243	#336	—	53	2	no	—
244	#336	—	53	2	no	—
245	#336	—	53	2	no	—
246	C₅₃	#21675	1	1	yes	characteristic, complemented, Sylow
247	C₂²×C₈	—	53	2	no	—
248	C₂²×C₈	—	53	2	no	—
249	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
250	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
251	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
252	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
253	#125	—	106	4	no	—
254	C₂×C₄⋊C₄	—	53	1	no	—
255	C₂×C₄⋊C₄	—	53	2	no	—
256	C₂×C₄⋊C₄	—	53	2	no	—
257	C₂×C₄⋊C₄	—	53	1	no	—
258	C₂×C₄⋊C₄	—	53	1	no	—
259	C₂×C₄⋊C₄	—	53	1	no	—
260	C₂×C₄⋊C₄	—	53	2	no	—
261	C₂×C₄⋊C₄	—	53	2	no	—
262	C₂×C₄²	—	53	2	no	—
263	C₂×C₄²	—	53	2	no	—
264	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
265	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
266	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
267	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
268	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
269	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
270	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
271	C₄⋊C₈	—	106	4	no	—
272	C₄⋊C₈	—	106	4	no	—
273	C₄⋊C₈	—	106	2	no	—
274	C₄⋊C₈	—	106	2	no	—
275	#97	—	2	2	no	—
276	#97	—	2	2	no	—
277	#97	—	2	2	no	—
278	#97	—	2	2	no	—
279	#97	—	106	4	no	—
280	C₄×C₈	—	2	2	no	—
281	C₄×C₈	—	106	4	no	—
282	C₄×C₈	Dic₁₀₆	1	4	yes	—
283	#95	—	53	2	no	—
284	#95	—	53	2	no	—
285	#95	—	53	2	no	—
286	#95	—	53	2	no	—
287	#95	—	53	2	no	—
288	#95	—	53	2	no	—
289	C₂²×C₄	—	53	2	no	—
290	C₂²×C₄	—	53	2	no	—
291	C₂²×C₄	—	53	2	no	—
292	C₂²×C₄	—	53	2	no	—
293	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
294	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
295	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
296	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
297	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
298	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
299	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	106	2	no	—
307	C₂×C₈	—	106	2	no	—
308	C₂×C₈	—	106	2	no	—
309	C₂×C₈	—	106	2	no	—
310	C₂×C₈	—	106	2	no	—
311	C₂×C₈	—	106	2	no	—
312	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
313	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
314	C₂×C₈	#11719945	1	2	yes	—
315	C₂×C₈	#11719945	1	2	yes	—
316	C₂×C₈	#11719945	1	2	yes	—
317	C₂×C₈	#11719945	1	2	yes	—
318	C₄⋊C₄	—	53	4	no	—
319	C₄⋊C₄	—	106	4	no	—
320	C₄⋊C₄	—	106	4	no	—
321	C₄⋊C₄	—	106	2	no	—
322	C₄⋊C₄	—	106	2	no	—
323	C₄⋊C₄	—	106	2	no	—
324	C₄⋊C₄	—	212	2	no	—
325	C₄⋊C₄	—	212	2	no	—
326	C₄²	—	2	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	2	1	no	—
329	C₄²	—	2	1	no	—
330	C₄²	—	2	1	no	—
331	C₄²	—	2	1	no	—
332	C₄²	—	4	1	no	—
333	C₄²	—	4	1	no	—
334	C₄²	—	106	2	no	—
335	C₄²	—	106	2	no	—
336	C₄²	—	106	2	no	—
337	C₄²	—	106	2	no	—
338	C₄²	C₂×Dic₁₀₆	1	2	yes	—
339	C₄²	#11719945	1	2	yes	—
340	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	53	4	no	—
356	C₂×C₄	—	106	2	no	—
357	C₂×C₄	—	106	2	no	—
358	C₂×C₄	—	106	2	no	—
359	C₂×C₄	—	106	2	no	—
360	C₂×C₄	—	106	2	no	—
361	C₂×C₄	—	106	2	no	—
362	C₂×C₄	—	106	2	no	—
363	C₂×C₄	—	106	2	no	—
364	C₂×C₄	—	106	2	no	—
365	C₂×C₄	—	106	2	no	—
366	C₂×C₄	—	106	2	no	—
367	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
368	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
369	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
370	C₂×C₄	#13133225	1	2	yes	—
371	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
372	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
373	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
374	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
375	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
376	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
377	C₈	—	2	2	no	—
378	C₈	—	2	2	no	—
379	C₈	—	2	2	no	—
380	C₈	—	212	2	no	—
381	C₈	—	212	2	no	—
382	C₈	#13133209	1	4	yes	complemented
383	C₂²	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
384	C₂²	C₄×#11719945	1	1	yes	characteristic
385	C₂²	C₄×#11719945	1	1	yes	characteristic
386	C₂²	C₂×M₁₆×D₅₃	1	1	yes	characteristic
387	C₂²	C₂×#13133225	1	1	yes	characteristic
388	C₂²	#16811389	1	1	yes	characteristic
389	C₂²	#16811389	1	1	yes	characteristic
390	C₄	—	2	2	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	1	no	—
394	C₄	—	4	1	no	—
395	C₄	—	106	4	no	—
396	C₄	—	212	2	no	—
397	C₄	—	212	2	no	—
398	C₄	—	212	2	no	—
399	C₄	#16810751	1	4	yes	—
400	C₄	C₂×#13133209	1	1	yes	characteristic
401	C₄	#16811157	1	1	yes	characteristic
402	C₄	#16811157	1	1	yes	characteristic
403	C₄	#16811180	1	1	yes	characteristic
404	C₂	C₄×#13133209	1	1	yes	characteristic
405	C₂	C₂×#16810751	1	1	yes	characteristic
406	C₂	#20283655	1	1	yes	characteristic
407	C₂	#20283719	1	1	yes	characteristic
408	C₂	#20283719	1	1	yes	characteristic
409	C₂	#20283730	1	1	yes	characteristic
410	C₂	#20283733	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂³×C₄
Fitting	C₄×#97×C₅₃	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#97	D₅₃
hypocenter	C₅₃	#21675
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
174. C₈×C₅₃	#125	semidirect
246. C₅₃	#21675	semidirect
382. C₈	#13133209	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133209	2
#20275433	2
#20275279	2
#20274832	2
#20274591	2
#20274583	2
#20274582	2
C₈×C₄⋊Dic₅₃	2
C₄×#97×C₅₃	2
#21675	53