Subgroups — M₁₆×C₄⋊#6157 #28318327

Order 13568.

1242 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊#423	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×#6157	C₂	1	2	yes	maximal
3	C₂×C₄×C₄⋊#6157	C₂	1	1	yes	maximal, characteristic
4	#20290513	C₂	1	4	yes	maximal, complemented
5	#20290509	C₂	1	2	yes	maximal, complemented
6	C₈×C₄⋊#6157	C₂	1	2	yes	maximal, complemented
7	#20290377	C₂	1	2	yes	maximal
8	C₄×M₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×D₅₃⋊M₁₆	C₂²	1	2	yes	—
10	C₂²×C₄⋊#6157	C₄	1	2	yes	—
11	C₂²×C₄×#6157	C₂²	1	2	yes	—
12	#16811628	C₄	1	2	yes	complemented
13	#16811627	C₄	1	2	yes	complemented
14	#16811626	C₄	1	2	yes	complemented
15	#16811626	C₄	1	2	yes	complemented
16	D₅₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
17	D₅₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
18	M₁₆×#6157	—	2	8	no	—
19	C₂×#13133271	C₂²	1	4	yes	—
20	C₂×C₈×#6157	C₂²	1	4	yes	—
21	C₄×C₄⋊#6157	C₄	1	2	yes	—
22	C₄×C₄⋊#6157	C₄	1	2	yes	—
23	C₄×C₄⋊#6157	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊#6157	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊#6157	C₂²	1	2	yes	—
26	C₂×(C₂×#6157)⋊C₄	C₄	1	2	yes	—
27	(C₂×#6157)⋊C₈	C₂²	1	4	yes	—
28	(C₂×#6157)⋊C₈	C₂²	1	4	yes	—
29	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
30	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
32	C₄×#11719917	C₄	1	1	yes	characteristic, complemented
33	#16810575	C₄	1	2	yes	complemented
34	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic, complemented
35	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic, complemented
36	C₄×#11719912	C₄	1	2	yes	complemented
37	#97×D₅₃	C₂²	1	2	yes	—
38	C₄×C₈×D₅₃	C₂²	1	2	yes	—
39	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
40	C₂³×#6157	C₂×C₄	1	4	yes	—
41	C₂×#11719936	C₂×C₄	1	4	yes	—
42	D₅₃⋊M₁₆	—	2	4	no	—
43	D₅₃⋊M₁₆	—	2	4	no	—
44	C₂×#11719940	C₂×C₄	1	4	yes	—
45	C₂×D₅₃⋊C₈	C₂³	1	4	yes	—
46	C₂×C₄⋊#6157	—	2	8	no	—
47	C₂×C₄⋊#6157	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊#6157	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊#6157	C₂×C₄	1	2	yes	—
50	C₂×C₄×#6157	—	2	4	no	—
51	C₂×C₄×#6157	—	2	4	no	—
52	C₂×C₄×#6157	C₂×C₄	1	4	yes	—
53	C₂×C₄×#6157	C₂³	1	2	yes	—
54	C₂×C₄×#6157	C₂³	1	2	yes	—
55	#13133284	C₂×C₄	1	4	yes	complemented
56	#13133284	C₂×C₄	1	4	yes	complemented
57	C₄⋊#88324	C₂×C₄	1	4	yes	complemented
58	C₄⋊#88324	C₂×C₄	1	4	yes	complemented
59	#13133271	—	2	8	no	—
60	C₈×#6157	—	2	8	no	—
61	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
62	(C₂×#6157)⋊C₄	C₂×C₄	1	4	yes	—
63	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
64	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
67	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
68	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
69	C₂×#11719912	C₂×C₄	1	2	yes	—
70	C₂×#11719912	C₂×C₄	1	2	yes	—
71	M₁₆×D₅₃	—	2	2	no	—
72	M₁₆×D₅₃	—	2	2	no	—
73	M₁₆×D₅₃	D₄	1	4	yes	—
74	M₁₆×D₅₃	Q₈	1	4	yes	—
75	C₂×C₈×D₅₃	C₂³	1	2	yes	—
76	C₂×C₈×D₅₃	C₂³	1	2	yes	—
77	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
80	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
81	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
83	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
84	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄²×D₅₃	C₂³	1	1	yes	characteristic
87	C₄²×D₅₃	C₂³	1	1	yes	characteristic
88	#13133128	C₂×C₄	1	2	yes	complemented
89	#13133127	C₂×C₄	1	2	yes	complemented
90	#13133127	C₂×C₄	1	2	yes	complemented
91	C₄×#88322	C₂×C₄	1	2	yes	complemented
92	C₈×Dic₅₃	C₂×C₄	1	2	yes	complemented
93	C₈×Dic₅₃	C₂×C₄	1	2	yes	complemented
94	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
95	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
97	#97×C₅₃	C₂×C₄	1	2	yes	complemented
98	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	complemented
99	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×#6157	—	2	8	no	—
101	C₂²×#6157	—	2	8	no	—
102	C₂²×#6157	—	4	4	no	—
103	C₂²×#6157	C₂²×C₄	1	4	yes	—
104	C₂²×#6157	C₂²×C₄	1	4	yes	—
105	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
109	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
110	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×D₅₃	—	2	4	no	—
112	C₂×C₄×D₅₃	—	4	2	no	—
113	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
114	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
115	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
116	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
117	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
125	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
126	C₂×C₄×D₅₃	#41	1	2	yes	—
127	C₂×C₄×D₅₃	#41	1	2	yes	—
128	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
129	#11719940	—	2	8	no	—
130	C₂×#88324	C₂²×C₄	1	8	yes	—
131	C₄⋊#6157	—	2	8	no	—
132	C₄⋊#6157	—	2	4	no	—
133	C₄⋊#6157	M₁₆	1	8	yes	direct product factor
134	C₄×#6157	—	2	8	no	—
135	C₄×#6157	—	2	2	no	—
136	C₄×#6157	—	2	2	no	—
137	C₄×#6157	—	2	2	no	—
138	C₄×#6157	—	2	2	no	—
139	#11719936	—	2	8	no	—
140	D₅₃⋊C₈	—	2	4	no	—
141	D₅₃⋊C₈	—	2	4	no	—
142	M₁₆×C₅₃	—	2	2	no	—
143	M₁₆×C₅₃	C₄⋊C₄	1	4	yes	complemented
144	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
145	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
146	C₄²×C₅₃	C₄²	1	1	yes	characteristic
147	C₄²×C₅₃	C₄²	1	1	yes	characteristic
148	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
151	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
152	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
153	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
154	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
155	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
156	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
157	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
158	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
159	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
162	#11719917	—	2	2	no	—
163	#11719917	C₄⋊C₄	1	4	yes	complemented
164	C₂×#88322	C₂²×C₄	1	2	yes	—
165	C₂×#88322	C₂²×C₄	1	2	yes	—
166	#11719912	—	2	4	no	—
167	#11719912	C₄⋊C₄	1	8	yes	complemented
168	C₈×D₅₃	—	2	2	no	—
169	C₈×D₅₃	—	2	2	no	—
170	C₈×D₅₃	C₂×D₄	1	4	yes	—
171	C₈×D₅₃	C₂×Q₈	1	4	yes	—
172	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
173	C₂²×D₅₃	—	2	2	no	—
174	C₂²×D₅₃	—	2	2	no	—
175	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
176	C₂²×D₅₃	C₄×D₄	1	2	yes	—
177	C₂²×D₅₃	C₄×Q₈	1	2	yes	—
178	C₂²×D₅₃	C₂³×C₄	1	1	yes	characteristic
179	C₂×#6157	—	2	4	no	—
180	C₂×#6157	—	2	4	no	—
181	C₂×#6157	—	2	4	no	—
182	C₂×#6157	—	2	4	no	—
183	C₂×#6157	—	2	4	no	—
184	C₂×#6157	—	4	16	no	—
185	C₂×#6157	C₂×M₁₆	1	8	yes	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	4	1	no	—
188	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
191	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
192	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
193	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
194	C₂×C₄×C₅₃	#117	1	2	yes	—
195	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
196	C₂×Dic₅₃	—	2	2	no	—
197	C₂×Dic₅₃	—	4	1	no	—
198	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
199	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
200	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
201	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
202	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
203	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
204	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
205	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
206	C₄×D₅₃	—	2	2	no	—
207	C₄×D₅₃	—	2	2	no	—
208	C₄×D₅₃	—	2	2	no	—
209	C₄×D₅₃	—	2	2	no	—
210	C₄×D₅₃	—	2	2	no	—
211	C₄×D₅₃	—	4	1	no	—
212	C₄×D₅₃	—	4	1	no	—
213	C₄×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
214	C₄×D₅₃	#117	1	2	yes	—
215	C₄×D₅₃	C₄×D₄	1	2	yes	—
216	C₄×D₅₃	C₄×D₄	1	2	yes	—
217	C₄×D₅₃	C₄×Q₈	1	2	yes	—
218	C₄×D₅₃	C₄×Q₈	1	2	yes	—
219	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
220	C₄×D₅₃	C₂²×D₄	1	1	yes	characteristic
221	C₄×D₅₃	C₂²×Q₈	1	1	yes	characteristic
222	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
223	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
224	#88324	—	2	8	no	—
225	C₈×C₅₃	—	2	2	no	—
226	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	complemented
227	#88322	—	2	2	no	—
228	#88322	C₂×C₄⋊C₄	1	4	yes	complemented
229	C₄⋊C₄×M₁₆	—	53	1	no	maximal, Sylow
230	C₂²×C₅₃	—	2	1	no	—
231	C₂²×C₅₃	—	2	1	no	—
232	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
233	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
234	C₂×D₅₃	—	2	1	no	—
235	C₂×D₅₃	—	2	1	no	—
236	C₂×D₅₃	—	2	4	no	—
237	C₂×D₅₃	—	2	4	no	—
238	C₂×D₅₃	C₄×C₄⋊C₄	1	2	yes	—
239	C₂×D₅₃	C₂×C₄×D₄	1	1	yes	characteristic
240	C₂×D₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
241	C₂×D₅₃	C₂²×M₁₆	1	2	yes	—
242	C₂×D₅₃	#568	1	2	yes	—
243	#6157	—	2	8	no	—
244	#6157	—	4	4	no	—
245	C₄×C₅₃	—	2	1	no	—
246	C₄×C₅₃	—	2	1	no	—
247	C₄×C₅₃	—	2	1	no	—
248	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
249	C₄×C₅₃	C₄×M₁₆	1	2	yes	complemented
250	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
251	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
252	Dic₅₃	—	2	1	no	—
253	Dic₅₃	—	2	1	no	—
254	Dic₅₃	—	2	1	no	—
255	Dic₅₃	C₄×C₄⋊C₄	1	2	yes	—
256	Dic₅₃	C₄×M₁₆	1	2	yes	complemented
257	Dic₅₃	C₂×C₄×D₄	1	1	yes	characteristic
258	Dic₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
259	C₂×#423	—	53	1	no	—
260	C₂×C₄×M₁₆	—	53	1	no	—
261	C₂×C₄×M₁₆	—	53	2	no	—
262	C₂×C₄×C₄⋊C₄	—	53	1	no	—
263	#1771	—	53	4	no	—
264	#1770	—	53	2	no	—
265	C₈×C₄⋊C₄	—	53	2	no	—
266	#1728	—	53	2	no	—
267	C₂×C₅₃	—	2	2	no	—
268	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
269	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
270	C₂×C₅₃	#2873	1	1	yes	characteristic
271	D₅₃	—	2	2	no	—
272	D₅₃	D₄×M₁₆	1	2	yes	—
273	D₅₃	Q₈×M₁₆	1	2	yes	—
274	C₂²×M₁₆	—	53	1	no	—
275	C₂²×M₁₆	—	53	2	no	—
276	C₂²×M₁₆	—	53	1	no	—
277	C₂²×C₄⋊C₄	—	53	2	no	—
278	C₂²×C₄²	—	53	1	no	—
279	C₂²×C₄²	—	53	2	no	—
280	#423	—	53	2	no	—
281	#423	—	53	2	no	—
282	#423	—	53	2	no	—
283	#423	—	53	2	no	—
284	C₂×C₄⋊C₈	—	53	2	no	—
285	C₂×C₄⋊C₈	—	53	2	no	—
286	C₄×M₁₆	—	53	1	no	—
287	C₄×M₁₆	—	53	1	no	—
288	C₄×M₁₆	—	53	1	no	—
289	C₄×M₁₆	—	53	2	no	—
290	C₄×M₁₆	—	53	2	no	—
291	C₄×M₁₆	—	106	8	no	—
292	C₄×M₁₆	#6157	1	1	yes	characteristic, complemented
293	C₂×#97	—	53	2	no	—
294	C₂×#97	—	53	4	no	—
295	C₂×C₄×C₈	—	53	2	no	—
296	C₂×C₄×C₈	—	53	4	no	—
297	C₄×C₄⋊C₄	—	53	2	no	—
298	C₄×C₄⋊C₄	—	53	2	no	—
299	C₄×C₄⋊C₄	—	53	1	no	—
300	C₄×C₄⋊C₄	—	53	1	no	—
301	C₄×C₄⋊C₄	—	53	2	no	—
302	C₂×#95	—	53	2	no	—
303	#336	—	53	4	no	—
304	#336	—	53	4	no	—
305	C₅₃	C₄⋊C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
306	C₂³×C₄	—	53	1	no	—
307	C₂³×C₄	—	53	1	no	—
308	C₂³×C₄	—	53	1	no	—
309	C₂³×C₄	—	53	4	no	—
310	C₂×M₁₆	—	53	1	no	—
311	C₂×M₁₆	—	53	4	no	—
312	C₂×M₁₆	—	53	4	no	—
313	C₂×M₁₆	—	53	4	no	—
314	C₂×M₁₆	—	53	4	no	—
315	C₂×M₁₆	—	53	1	no	—
316	C₂×M₁₆	—	53	2	no	—
317	C₂×M₁₆	—	53	2	no	—
318	C₂×M₁₆	—	106	2	no	—
319	C₂×M₁₆	—	106	2	no	—
320	C₂×M₁₆	—	106	4	no	—
321	C₂×M₁₆	—	106	4	no	—
322	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
323	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
324	C₂²×C₈	—	53	2	no	—
325	C₂²×C₈	—	53	4	no	—
326	C₂²×C₈	—	53	2	no	—
327	C₂×C₄⋊C₄	—	53	2	no	—
328	C₂×C₄⋊C₄	—	53	2	no	—
329	C₂×C₄⋊C₄	—	53	4	no	—
330	C₂×C₄⋊C₄	—	106	8	no	—
331	C₂×C₄²	—	53	1	no	—
332	C₂×C₄²	—	53	1	no	—
333	C₂×C₄²	—	53	1	no	—
334	C₂×C₄²	—	53	1	no	—
335	C₂×C₄²	—	53	1	no	—
336	C₂×C₄²	—	53	1	no	—
337	C₂×C₄²	—	53	1	no	—
338	C₂×C₄²	—	53	1	no	—
339	C₂×C₄²	—	53	1	no	—
340	C₂×C₄²	—	53	1	no	—
341	C₂×C₄²	—	53	1	no	—
342	C₂×C₄²	—	53	2	no	—
343	C₂×C₄²	—	53	2	no	—
344	C₂×C₄²	—	53	4	no	—
345	C₂×C₄²	—	106	4	no	—
346	C₂×C₄²	—	106	4	no	—
347	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
348	C₄⋊C₈	—	53	4	no	—
349	C₄⋊C₈	—	53	4	no	—
350	C₄⋊C₈	—	53	4	no	—
351	C₄⋊C₈	—	53	4	no	—
352	#97	—	53	2	no	—
353	#97	—	53	2	no	—
354	#97	—	53	2	no	—
355	#97	—	106	8	no	—
356	#97	C₂×#6157	1	2	yes	complemented
357	C₄×C₈	—	53	2	no	—
358	C₄×C₈	—	53	2	no	—
359	C₄×C₈	—	53	2	no	—
360	C₄×C₈	—	106	8	no	—
361	C₄×C₈	C₂×#6157	1	2	yes	complemented
362	#95	—	53	2	no	—
363	#95	—	53	4	no	—
364	#95	—	53	2	no	—
365	C₂⁴	—	53	1	no	—
366	C₂²×C₄	—	53	1	no	—
367	C₂²×C₄	—	53	1	no	—
368	C₂²×C₄	—	53	1	no	—
369	C₂²×C₄	—	53	1	no	—
370	C₂²×C₄	—	53	2	no	—
371	C₂²×C₄	—	53	2	no	—
372	C₂²×C₄	—	53	2	no	—
373	C₂²×C₄	—	53	2	no	—
374	C₂²×C₄	—	53	1	no	—
375	C₂²×C₄	—	53	1	no	—
376	C₂²×C₄	—	53	4	no	—
377	C₂²×C₄	—	53	1	no	—
378	C₂²×C₄	—	53	1	no	—
379	C₂²×C₄	—	53	1	no	—
380	C₂²×C₄	—	53	1	no	—
381	C₂²×C₄	—	53	1	no	—
382	C₂²×C₄	—	53	1	no	—
383	C₂²×C₄	—	53	1	no	—
384	C₂²×C₄	—	53	1	no	—
385	C₂²×C₄	—	53	1	no	—
386	C₂²×C₄	—	53	4	no	—
387	C₂²×C₄	—	106	4	no	—
388	C₂²×C₄	—	106	8	no	—
389	C₂²×C₄	—	106	8	no	—
390	C₂²×C₄	—	212	2	no	—
391	C₂²×C₄	—	212	4	no	—
392	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
393	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
394	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
395	M₁₆	—	2	2	no	—
396	M₁₆	—	53	4	no	—
397	M₁₆	—	53	8	no	—
398	M₁₆	—	106	4	no	—
399	M₁₆	—	106	8	no	—
400	M₁₆	—	106	8	no	—
401	M₁₆	—	106	2	no	—
402	M₁₆	C₄⋊#6157	1	4	yes	direct product factor
403	C₂×C₈	—	53	2	no	—
404	C₂×C₈	—	53	4	no	—
405	C₂×C₈	—	53	4	no	—
406	C₂×C₈	—	53	8	no	—
407	C₂×C₈	—	53	2	no	—
408	C₂×C₈	—	106	2	no	—
409	C₂×C₈	—	106	2	no	—
410	C₂×C₈	—	106	4	no	—
411	C₂×C₈	—	106	4	no	—
412	C₂×C₈	C₂²×#6157	1	2	yes	—
413	C₂×C₈	C₂²×#6157	1	2	yes	—
414	C₄⋊C₄	—	53	8	no	—
415	C₄⋊C₄	—	106	4	no	—
416	C₄⋊C₄	—	106	8	no	—
417	C₄²	—	53	1	no	—
418	C₄²	—	53	1	no	—
419	C₄²	—	53	1	no	—
420	C₄²	—	53	1	no	—
421	C₄²	—	53	1	no	—
422	C₄²	—	53	1	no	—
423	C₄²	—	53	1	no	—
424	C₄²	—	53	1	no	—
425	C₄²	—	53	1	no	—
426	C₄²	—	53	1	no	—
427	C₄²	—	53	1	no	—
428	C₄²	—	53	1	no	—
429	C₄²	—	106	2	no	—
430	C₄²	—	106	2	no	—
431	C₄²	—	106	8	no	—
432	C₄²	—	106	2	no	—
433	C₄²	—	106	2	no	—
434	C₄²	C₄×#6157	1	1	yes	characteristic
435	C₄²	C₄×#6157	1	1	yes	characteristic
436	C₄²	C₂²×#6157	1	1	yes	characteristic
437	C₄²	C₂²×#6157	1	1	yes	characteristic
438	C₂³	—	53	1	no	—
439	C₂³	—	53	2	no	—
440	C₂³	—	53	2	no	—
441	C₂³	—	53	1	no	—
442	C₂³	—	106	2	no	—
443	C₂³	—	106	2	no	—
444	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	53	2	no	—
448	C₂×C₄	—	53	2	no	—
449	C₂×C₄	—	53	1	no	—
450	C₂×C₄	—	53	1	no	—
451	C₂×C₄	—	53	1	no	—
452	C₂×C₄	—	53	2	no	—
453	C₂×C₄	—	53	2	no	—
454	C₂×C₄	—	53	2	no	—
455	C₂×C₄	—	53	2	no	—
456	C₂×C₄	—	53	4	no	—
457	C₂×C₄	—	53	2	no	—
458	C₂×C₄	—	53	1	no	—
459	C₂×C₄	—	53	1	no	—
460	C₂×C₄	—	53	2	no	—
461	C₂×C₄	—	53	1	no	—
462	C₂×C₄	—	53	1	no	—
463	C₂×C₄	—	53	8	no	—
464	C₂×C₄	—	53	1	no	—
465	C₂×C₄	—	53	1	no	—
466	C₂×C₄	—	53	1	no	—
467	C₂×C₄	—	106	2	no	—
468	C₂×C₄	—	106	2	no	—
469	C₂×C₄	—	106	2	no	—
470	C₂×C₄	—	106	2	no	—
471	C₂×C₄	—	106	2	no	—
472	C₂×C₄	—	106	4	no	—
473	C₂×C₄	—	106	4	no	—
474	C₂×C₄	—	106	2	no	—
475	C₂×C₄	—	106	4	no	—
476	C₂×C₄	—	106	4	no	—
477	C₂×C₄	—	106	4	no	—
478	C₂×C₄	—	212	1	no	—
479	C₂×C₄	—	212	1	no	—
480	C₂×C₄	—	212	1	no	—
481	C₂×C₄	—	212	16	no	—
482	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
483	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
484	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
485	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
486	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
487	C₂×C₄	C₂×C₄⋊#6157	1	2	yes	—
488	C₂×C₄	#13133296	1	2	yes	—
489	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
490	C₈	—	2	2	no	—
491	C₈	—	53	4	no	—
492	C₈	—	106	8	no	—
493	C₈	—	106	2	no	—
494	C₈	C₂×C₄⋊#6157	1	4	yes	complemented
495	C₂²	—	2	1	no	—
496	C₂²	—	2	1	no	—
497	C₂²	—	53	1	no	—
498	C₂²	—	53	1	no	—
499	C₂²	—	53	2	no	—
500	C₂²	—	53	2	no	—
501	C₂²	—	53	2	no	—
502	C₂²	—	106	4	no	—
503	C₂²	—	106	4	no	—
504	C₂²	—	106	1	no	—
505	C₂²	—	106	1	no	—
506	C₂²	C₄×C₄⋊#6157	1	2	yes	—
507	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
508	C₄	—	2	1	no	—
509	C₄	—	2	1	no	—
510	C₄	—	2	1	no	—
511	C₄	—	53	1	no	—
512	C₄	—	53	1	no	—
513	C₄	—	53	2	no	—
514	C₄	—	53	2	no	—
515	C₄	—	106	8	no	—
516	C₄	—	106	1	no	—
517	C₄	—	106	1	no	—
518	C₄	—	106	1	no	—
519	C₄	—	212	4	no	—
520	C₄	C₄×C₄⋊#6157	1	2	yes	—
521	C₄	M₁₆×#6157	1	2	yes	complemented
522	C₄	C₂²×C₄⋊#6157	1	1	yes	characteristic
523	C₄	C₂×#13133296	1	1	yes	characteristic
524	C₂	—	2	2	no	—
525	C₂	—	53	2	no	—
526	C₂	—	53	2	no	—
527	C₂	—	106	2	no	—
528	C₂	C₂×C₄×C₄⋊#6157	1	1	yes	characteristic
529	C₂	C₂×M₁₆×#6157	1	1	yes	characteristic
530	C₂	#20291278	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₄×M₁₆×C₅₃	C₄
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	C₄×M₁₆	#6157
hypocenter	C₅₃	C₄⋊C₄×M₁₆
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #20290513	C₂	semidirect
5. #20290509	C₂	semidirect
6. C₈×C₄⋊#6157	C₂	semidirect
12. #16811628	C₄	semidirect
13. #16811627	C₄	semidirect
14. #16811626	C₄	semidirect
15. #16811626	C₄	semidirect
32. C₄×#11719917	C₄	semidirect
33. #16810575	C₄	semidirect
34. M₁₆×Dic₅₃	C₄	semidirect
35. M₁₆×Dic₅₃	C₄	semidirect
36. C₄×#11719912	C₄	semidirect
39. C₄×M₁₆×C₅₃	C₄	semidirect
55. #13133284	C₂×C₄	semidirect
56. #13133284	C₂×C₄	semidirect
57. C₄⋊#88324	C₂×C₄	semidirect
58. C₄⋊#88324	C₂×C₄	semidirect
88. #13133128	C₂×C₄	semidirect
89. #13133127	C₂×C₄	semidirect
90. #13133127	C₂×C₄	semidirect
91. C₄×#88322	C₂×C₄	semidirect
92. C₈×Dic₅₃	C₂×C₄	semidirect
93. C₈×Dic₅₃	C₂×C₄	semidirect
97. #97×C₅₃	C₂×C₄	semidirect
98. C₄×C₈×C₅₃	C₂×C₄	semidirect
133. C₄⋊#6157	M₁₆	direct (N × H)
143. M₁₆×C₅₃	C₄⋊C₄	semidirect
163. #11719917	C₄⋊C₄	semidirect
167. #11719912	C₄⋊C₄	semidirect
226. C₈×C₅₃	C₂×C₄⋊C₄	semidirect
228. #88322	C₂×C₄⋊C₄	semidirect
249. C₄×C₅₃	C₄×M₁₆	semidirect
256. Dic₅₃	C₄×M₁₆	semidirect
292. C₄×M₁₆	#6157	semidirect
305. C₅₃	C₄⋊C₄×M₁₆	semidirect
356. #97	C₂×#6157	semidirect
361. C₄×C₈	C₂×#6157	semidirect
402. M₁₆	C₄⋊#6157	direct (N × H)
494. C₈	C₂×C₄⋊#6157	semidirect
521. C₄	M₁₆×#6157	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊#423	2
C₂×M₁₆×#6157	2
C₂×C₄×C₄⋊#6157	2
#20290513	2
#20290509	2
C₈×C₄⋊#6157	2
#20290377	2
C₄×M₁₆×D₅₃	2
C₄⋊C₄×M₁₆	53

Subgroups — M16×C4⋊#6157 #28318327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×C₄⋊#6157 #28318327