Subgroups — #28318696

Order 13568.

1122 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊#6157	C₂	1	1	yes	maximal, characteristic
3	#20290508	C₂	1	4	yes	maximal, complemented
4	C₈×C₄⋊#6157	C₂	1	4	yes	maximal, complemented
5	C₂²⋊C₈×#6157	C₂	1	4	yes	maximal
6	#431×D₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊#6157	C₄	1	2	yes	—
8	C₂²×C₄×#6157	C₂²	1	2	yes	—
9	#16811642	C₄	1	2	yes	complemented
10	#16811641	C₄	1	2	yes	complemented
11	#16811640	C₄	1	2	yes	complemented
12	#16811639	C₄	1	2	yes	complemented
13	D₅₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
14	D₅₃⋊(C₂²⋊C₈)	C₂²	1	2	yes	—
15	C₄×D₅₃⋊C₈	C₂²	1	2	yes	—
16	C₂×C₈×#6157	C₂²	1	8	yes	—
17	C₄×C₄⋊#6157	C₄	1	2	yes	—
18	C₄×C₄⋊#6157	C₄	1	2	yes	—
19	C₄×C₄⋊#6157	C₂²	1	2	yes	—
20	C₄×C₄⋊#6157	C₂²	1	2	yes	—
21	C₂×(C₂×#6157)⋊C₄	C₄	1	2	yes	—
22	(C₂×#6157)⋊C₈	C₂²	1	8	yes	—
23	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
24	#16810676	C₄	1	1	yes	characteristic, complemented
25	#16810675	C₄	1	2	yes	complemented
26	#16810674	C₄	1	2	yes	complemented
27	#16810673	C₄	1	2	yes	complemented
28	C₄⋊C₈×D₅₃	C₂²	1	2	yes	—
29	C₂²⋊C₈×D₅₃	C₂²	1	2	yes	—
30	C₄×C₈×D₅₃	C₂²	1	2	yes	—
31	C₅₃×#431	C₄	1	1	yes	characteristic, complemented
32	C₂³×#6157	C₂×C₄	1	4	yes	—
33	C₂×D₅₃⋊C₈	C₂³	1	4	yes	—
34	C₂×C₄⋊#6157	C₈	1	16	yes	complemented
35	C₂×C₄⋊#6157	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊#6157	C₂×C₄	1	4	yes	—
37	C₂×C₄×#6157	—	4	4	no	—
38	C₂×C₄×#6157	C₂×C₄	1	4	yes	—
39	C₂×C₄×#6157	C₂³	1	4	yes	—
40	#13133284	C₂×C₄	1	4	yes	complemented
41	C₄⋊#88324	C₂×C₄	1	4	yes	complemented
42	#13133274	C₂×C₄	1	4	yes	—
43	C₂²⋊#88324	C₂×C₄	1	4	yes	—
44	C₄×#88324	C₂×C₄	1	4	yes	complemented
45	C₄×#88324	C₂×C₄	1	4	yes	complemented
46	C₈×#6157	—	4	8	no	—
47	(C₂×#6157)⋊C₄	C₂×C₄	1	4	yes	—
48	(C₂×#6157)⋊C₄	C₂×C₄	1	4	yes	—
49	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×D₅₃	C₂³	1	4	yes	—
53	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄²×D₅₃	C₂³	1	1	yes	characteristic
63	C₄²×D₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊#88322	C₂×C₄	1	2	yes	complemented
65	#13133146	C₂×C₄	1	4	yes	complemented
66	C₂²⋊#88322	C₂×C₄	1	2	yes	—
67	#13133129	C₂×C₄	1	4	yes	—
68	C₄×#88322	C₂×C₄	1	2	yes	complemented
69	C₈×Dic₅₃	C₂×C₄	1	4	yes	complemented
70	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	complemented
72	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	complemented
74	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×#6157	—	4	4	no	—
76	C₂²×#6157	—	4	4	no	—
77	C₂²×#6157	C₂×C₈	1	16	yes	—
78	C₂²×#6157	C₂²×C₄	1	8	yes	—
79	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
83	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
84	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×D₅₃	—	2	2	no	—
86	C₂×C₄×D₅₃	—	2	2	no	—
87	C₂×C₄×D₅₃	—	4	2	no	—
88	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
89	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
90	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
91	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
92	C₂×C₄×D₅₃	C₂×C₈	1	8	yes	—
93	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
101	C₂×#88324	C₂²×C₄	1	8	yes	—
102	C₄⋊#6157	C₂×C₈	1	16	yes	—
103	C₄⋊#6157	C₂×C₈	1	16	yes	—
104	C₄×#6157	—	2	8	no	—
105	C₄×#6157	—	4	4	no	—
106	D₅₃⋊C₈	—	2	4	no	—
107	D₅₃⋊C₈	—	2	4	no	—
108	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
109	C₄²×C₅₃	C₄²	1	1	yes	characteristic
110	C₄²×C₅₃	C₄²	1	1	yes	characteristic
111	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
114	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
115	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
116	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
117	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
118	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
119	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×#88322	C₂²×C₄	1	4	yes	—
126	C₈×D₅₃	—	2	4	no	—
127	C₈×D₅₃	—	2	4	no	—
128	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
129	C₂²×D₅₃	—	2	1	no	—
130	C₂²×D₅₃	—	2	1	no	—
131	C₂²×D₅₃	—	2	2	no	—
132	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
133	C₂²×D₅₃	C₂²×C₈	1	4	yes	—
134	C₂²×D₅₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×#6157	—	2	8	no	—
136	C₂×#6157	—	2	8	no	—
137	C₂×#6157	—	4	16	no	—
138	C₂×#6157	C₂²×C₈	1	16	yes	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	complemented
142	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×Dic₅₃	—	2	2	no	—
149	C₂×Dic₅₃	—	4	1	no	—
150	C₂×Dic₅₃	C₄×C₈	1	4	yes	complemented
151	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
152	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
153	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
154	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
155	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
156	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
157	C₄×D₅₃	—	2	1	no	—
158	C₄×D₅₃	—	2	1	no	—
159	C₄×D₅₃	—	2	2	no	—
160	C₄×D₅₃	—	2	2	no	—
161	C₄×D₅₃	—	2	2	no	—
162	C₄×D₅₃	—	2	2	no	—
163	C₄×D₅₃	—	4	1	no	—
164	C₄×D₅₃	—	4	1	no	—
165	C₄×D₅₃	C₄×C₈	1	4	yes	complemented
166	C₄×D₅₃	C₄×C₈	1	4	yes	complemented
167	C₄×D₅₃	C₂²×C₈	1	4	yes	—
168	C₄×D₅₃	C₂²×C₈	1	4	yes	—
169	C₄×D₅₃	H(5,𝔽₂)	1	2	yes	—
170	C₄×D₅₃	2₋¹⁺⁴	1	2	yes	—
171	#88324	—	2	8	no	—
172	C₈×C₅₃	—	2	4	no	—
173	#88322	—	2	4	no	—
174	#20533	—	53	1	no	maximal, Sylow
175	C₂²×C₅₃	—	2	1	no	—
176	C₂²×C₅₃	—	2	1	no	—
177	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
178	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
179	C₂×D₅₃	—	2	1	no	—
180	C₂×D₅₃	—	2	1	no	—
181	C₂×D₅₃	—	2	4	no	—
182	C₂×D₅₃	—	2	4	no	—
183	C₂×D₅₃	C₂×C₄×C₈	1	2	yes	—
184	C₂×D₅₃	#518	1	1	yes	characteristic
185	C₂×D₅₃	#519	1	1	yes	characteristic
186	C₂×D₅₃	C₂³×C₈	1	2	yes	—
187	C₂×D₅₃	#568	1	2	yes	—
188	#6157	—	2	16	no	—
189	C₄×C₅₃	—	2	1	no	—
190	C₄×C₅₃	—	2	1	no	—
191	C₄×C₅₃	—	2	1	no	—
192	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
193	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
194	C₄×C₅₃	#520	1	2	yes	—
195	Dic₅₃	—	2	1	no	—
196	Dic₅₃	—	2	1	no	—
197	Dic₅₃	—	2	1	no	—
198	Dic₅₃	C₂×C₄×C₈	1	2	yes	—
199	Dic₅₃	C₂×C₄×C₈	1	2	yes	—
200	Dic₅₃	#520	1	2	yes	—
201	C₂×#431	—	53	1	no	—
202	C₂×#431	—	53	1	no	—
203	C₂×C₄×C₄⋊C₄	—	53	1	no	—
204	C₈×C₄⋊C₄	—	53	4	no	—
205	C₈×C₄⋊C₄	—	53	4	no	—
206	C₄×C₂²⋊C₈	—	53	4	no	—
207	C₂×C₅₃	—	2	2	no	—
208	C₂×C₅₃	#2274	1	1	yes	characteristic
209	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
210	C₂×C₅₃	#2873	1	1	yes	characteristic
211	D₅₃	—	2	2	no	—
212	D₅₃	#2972	1	2	yes	—
213	D₅₃	#2982	1	2	yes	—
214	C₂²×C₄⋊C₄	—	53	2	no	—
215	C₂²×C₄²	—	53	1	no	—
216	C₂²×C₄²	—	53	2	no	—
217	#431	—	53	2	no	—
218	#431	—	53	2	no	—
219	#431	—	53	2	no	—
220	#431	—	53	2	no	—
221	#431	—	53	1	no	—
222	#431	—	53	2	no	—
223	#431	—	53	2	no	—
224	#431	—	53	2	no	—
225	#431	#6157	1	1	yes	characteristic, complemented
226	C₂×C₄⋊C₈	—	53	2	no	—
227	C₂×C₄⋊C₈	—	53	2	no	—
228	C₂×C₂²⋊C₈	—	53	2	no	—
229	C₂×C₂²⋊C₈	—	53	2	no	—
230	C₂×C₄×C₈	—	53	2	no	—
231	C₂×C₄×C₈	—	53	2	no	—
232	C₂×C₄×C₈	—	53	8	no	—
233	C₄×C₄⋊C₄	—	53	2	no	—
234	C₄×C₄⋊C₄	—	53	2	no	—
235	C₄×C₄⋊C₄	—	53	2	no	—
236	C₄×C₄⋊C₄	—	53	2	no	—
237	C₂×#95	—	53	2	no	—
238	#336	—	53	8	no	—
239	C₅₃	#20533	1	1	yes	characteristic, complemented, Sylow
240	C₂³×C₄	—	53	1	no	—
241	C₂³×C₄	—	53	4	no	—
242	C₂³×C₄	—	53	1	no	—
243	C₂³×C₄	—	53	1	no	—
244	C₂²×C₈	—	53	4	no	—
245	C₂²×C₈	—	53	4	no	—
246	C₂×C₄⋊C₄	—	53	4	no	—
247	C₂×C₄⋊C₄	—	53	4	no	—
248	C₂×C₄⋊C₄	—	53	16	no	—
249	C₂×C₄²	—	53	1	no	—
250	C₂×C₄²	—	53	1	no	—
251	C₂×C₄²	—	53	1	no	—
252	C₂×C₄²	—	53	1	no	—
253	C₂×C₄²	—	53	1	no	—
254	C₂×C₄²	—	53	1	no	—
255	C₂×C₄²	—	53	1	no	—
256	C₂×C₄²	—	53	1	no	—
257	C₂×C₄²	—	53	1	no	—
258	C₂×C₄²	—	53	1	no	—
259	C₂×C₄²	—	53	1	no	—
260	C₂×C₄²	—	53	4	no	—
261	C₂×C₄²	—	53	4	no	—
262	C₂×C₄²	—	212	4	no	—
263	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
264	C₄⋊C₈	—	53	4	no	—
265	C₄⋊C₈	—	53	4	no	—
266	C₄⋊C₈	—	53	4	no	—
267	C₄⋊C₈	—	53	2	no	—
268	C₄⋊C₈	C₂×#6157	1	2	yes	complemented
269	C₂²⋊C₈	—	53	4	no	—
270	C₂²⋊C₈	—	53	4	no	—
271	C₂²⋊C₈	—	53	2	no	—
272	C₂²⋊C₈	—	53	4	no	—
273	C₂²⋊C₈	C₂×#6157	1	2	yes	—
274	C₄×C₈	—	53	4	no	—
275	C₄×C₈	—	53	4	no	—
276	C₄×C₈	—	53	4	no	—
277	C₄×C₈	—	53	2	no	—
278	C₄×C₈	—	212	8	no	—
279	C₄×C₈	C₂×#6157	1	2	yes	complemented
280	#95	—	53	4	no	—
281	#95	—	53	4	no	—
282	C₂⁴	—	53	1	no	—
283	C₂²×C₄	—	53	1	no	—
284	C₂²×C₄	—	53	8	no	—
285	C₂²×C₄	—	53	1	no	—
286	C₂²×C₄	—	53	1	no	—
287	C₂²×C₄	—	53	1	no	—
288	C₂²×C₄	—	53	1	no	—
289	C₂²×C₄	—	53	1	no	—
290	C₂²×C₄	—	53	8	no	—
291	C₂²×C₄	—	53	1	no	—
292	C₂²×C₄	—	53	16	no	—
293	C₂²×C₄	—	53	1	no	—
294	C₂²×C₄	—	53	1	no	—
295	C₂²×C₄	—	53	1	no	—
296	C₂²×C₄	—	53	1	no	—
297	C₂²×C₄	—	53	1	no	—
298	C₂²×C₄	—	53	1	no	—
299	C₂²×C₄	—	53	1	no	—
300	C₂²×C₄	—	53	1	no	—
301	C₂²×C₄	—	106	2	no	—
302	C₂²×C₄	—	106	2	no	—
303	C₂²×C₄	—	212	2	no	—
304	C₂²×C₄	—	212	4	no	—
305	C₂²×C₄	—	212	4	no	—
306	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
307	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
308	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
309	C₂×C₈	—	53	8	no	—
310	C₂×C₈	—	53	4	no	—
311	C₂×C₈	—	106	4	no	—
312	C₂×C₈	—	106	4	no	—
313	C₂×C₈	—	106	4	no	—
314	C₂×C₈	—	106	4	no	—
315	C₂×C₈	C₂²×#6157	1	4	yes	—
316	C₄⋊C₄	—	53	16	no	—
317	C₄⋊C₄	—	53	16	no	—
318	C₄²	—	53	1	no	—
319	C₄²	—	53	1	no	—
320	C₄²	—	53	1	no	—
321	C₄²	—	53	1	no	—
322	C₄²	—	53	1	no	—
323	C₄²	—	53	1	no	—
324	C₄²	—	53	1	no	—
325	C₄²	—	53	1	no	—
326	C₄²	—	53	1	no	—
327	C₄²	—	53	1	no	—
328	C₄²	—	53	1	no	—
329	C₄²	—	53	1	no	—
330	C₄²	—	106	8	no	—
331	C₄²	—	212	4	no	—
332	C₄²	C₄×#6157	1	1	yes	characteristic
333	C₄²	C₄×#6157	1	1	yes	characteristic
334	C₄²	C₂²×#6157	1	1	yes	characteristic
335	C₄²	C₂²×#6157	1	1	yes	characteristic
336	C₂³	—	53	1	no	—
337	C₂³	—	53	4	no	—
338	C₂³	—	53	1	no	—
339	C₂³	—	106	1	no	—
340	C₂³	—	106	1	no	—
341	C₂³	—	106	2	no	—
342	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	53	4	no	—
346	C₂×C₄	—	53	1	no	—
347	C₂×C₄	—	53	1	no	—
348	C₂×C₄	—	53	1	no	—
349	C₂×C₄	—	53	4	no	—
350	C₂×C₄	—	53	4	no	—
351	C₂×C₄	—	53	4	no	—
352	C₂×C₄	—	53	4	no	—
353	C₂×C₄	—	53	2	no	—
354	C₂×C₄	—	53	2	no	—
355	C₂×C₄	—	53	16	no	—
356	C₂×C₄	—	53	1	no	—
357	C₂×C₄	—	53	1	no	—
358	C₂×C₄	—	53	1	no	—
359	C₂×C₄	—	106	2	no	—
360	C₂×C₄	—	106	1	no	—
361	C₂×C₄	—	106	1	no	—
362	C₂×C₄	—	106	2	no	—
363	C₂×C₄	—	106	2	no	—
364	C₂×C₄	—	106	2	no	—
365	C₂×C₄	—	106	8	no	—
366	C₂×C₄	—	106	8	no	—
367	C₂×C₄	—	106	2	no	—
368	C₂×C₄	—	212	1	no	—
369	C₂×C₄	—	212	1	no	—
370	C₂×C₄	—	212	1	no	—
371	C₂×C₄	—	212	16	no	—
372	C₂×C₄	C₈×#6157	1	4	yes	complemented
373	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
374	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
375	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
376	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
377	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
378	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
379	C₈	—	2	4	no	—
380	C₈	—	106	8	no	—
381	C₈	—	106	4	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	53	1	no	—
385	C₂²	—	53	1	no	—
386	C₂²	—	53	2	no	—
387	C₂²	—	53	2	no	—
388	C₂²	—	53	2	no	—
389	C₂²	—	106	4	no	—
390	C₂²	—	106	4	no	—
391	C₂²	—	106	1	no	—
392	C₂²	—	106	1	no	—
393	C₂²	C₂×C₈×#6157	1	2	yes	—
394	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	53	2	no	—
399	C₄	—	53	2	no	—
400	C₄	—	53	2	no	—
401	C₄	—	106	16	no	—
402	C₄	—	106	1	no	—
403	C₄	—	106	1	no	—
404	C₄	—	106	1	no	—
405	C₄	C₂×C₈×#6157	1	2	yes	—
406	C₄	C₂×C₈×#6157	1	2	yes	—
407	C₄	#16811692	1	2	yes	—
408	C₂	—	2	2	no	—
409	C₂	—	53	2	no	—
410	C₂	—	53	2	no	—
411	C₂	—	106	2	no	—
412	C₂	#20291152	1	1	yes	characteristic
413	C₂	C₂²×C₈×#6157	1	1	yes	characteristic
414	C₂	#20291278	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Fitting	C₅₃×#431	C₄
socle	C₂²×C₅₃	C₂²×C₄²
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	#431	#6157
hypocenter	C₅₃	#20533
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
3. #20290508	C₂	semidirect
4. C₈×C₄⋊#6157	C₂	semidirect
9. #16811642	C₄	semidirect
10. #16811641	C₄	semidirect
11. #16811640	C₄	semidirect
12. #16811639	C₄	semidirect
24. #16810676	C₄	semidirect
25. #16810675	C₄	semidirect
26. #16810674	C₄	semidirect
27. #16810673	C₄	semidirect
31. C₅₃×#431	C₄	semidirect
34. C₂×C₄⋊#6157	C₈	semidirect
40. #13133284	C₂×C₄	semidirect
41. C₄⋊#88324	C₂×C₄	semidirect
44. C₄×#88324	C₂×C₄	semidirect
45. C₄×#88324	C₂×C₄	semidirect
64. C₄⋊#88322	C₂×C₄	semidirect
65. #13133146	C₂×C₄	semidirect
68. C₄×#88322	C₂×C₄	semidirect
69. C₈×Dic₅₃	C₂×C₄	semidirect
71. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
73. C₄×C₈×C₅₃	C₂×C₄	semidirect
141. C₂×C₄×C₅₃	C₄×C₈	semidirect
150. C₂×Dic₅₃	C₄×C₈	semidirect
165. C₄×D₅₃	C₄×C₈	semidirect
166. C₄×D₅₃	C₄×C₈	semidirect
225. #431	#6157	semidirect
239. C₅₃	#20533	semidirect
268. C₄⋊C₈	C₂×#6157	semidirect
279. C₄×C₈	C₂×#6157	semidirect
372. C₂×C₄	C₈×#6157	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊#431	2
C₂×C₄×C₄⋊#6157	2
#20290508	2
C₈×C₄⋊#6157	2
C₂²⋊C₈×#6157	2
#431×D₅₃	2
#20533	53