Subgroups — C₁₆×C₂²⋊#6157 #28310519

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊(C₂²⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×#6157	C₂	1	2	yes	maximal, complemented
3	C₈×C₂²⋊#6157	C₂	1	1	yes	maximal, characteristic
4	(C₂×#6157)⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊C₁₆	C₂²	1	2	yes	—
7	D₅₃⋊(C₂²⋊C₈)	C₄	1	1	yes	characteristic
8	C₂×C₈×#6157	C₂²	1	2	yes	—
9	C₄×C₂²⋊#6157	C₄	1	1	yes	characteristic
10	#16811518	C₄	1	2	yes	complemented
11	C₂²⋊#11719910	C₄	1	2	yes	complemented
12	C₁₆×#6157	—	2	4	no	—
13	(C₂×#6157)⋊C₈	C₄	1	2	yes	—
14	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	C₂²×#11719908	C₄	1	1	yes	characteristic, complemented
16	C₂×C₁₆×D₅₃	—	2	4	no	—
17	C₂×C₁₆×D₅₃	C₄	1	1	yes	characteristic, complemented
18	C₂×C₁₆×D₅₃	C₄	1	1	yes	characteristic, complemented
19	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
22	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₂²⋊#6157	C₈	1	2	yes	—
24	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
25	D₅₃⋊C₁₆	—	2	2	no	—
26	D₅₃⋊C₁₆	—	2	2	no	—
27	C₂×#11719910	C₂×C₄	1	4	yes	—
28	#13133274	C₈	1	2	yes	—
29	C₂²⋊#88324	C₈	1	2	yes	—
30	C₈×#6157	—	2	2	no	—
31	C₈×#6157	—	2	2	no	—
32	(C₂×#6157)⋊C₄	C₈	1	2	yes	—
33	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₅₃	—	2	2	no	—
36	C₂×C₈×D₅₃	—	2	2	no	—
37	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
41	C₂×#11719908	—	2	2	no	—
42	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
43	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
44	C₁₆×D₅₃	—	2	2	no	—
45	C₁₆×D₅₃	—	2	2	no	—
46	C₁₆×D₅₃	—	2	2	no	—
47	C₁₆×D₅₃	D₄	1	4	yes	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₅₃	—	2	2	no	—
50	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×#6157	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
55	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
56	C₂×C₄×D₅₃	—	2	2	no	—
57	C₂×C₄×D₅₃	—	2	2	no	—
58	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
59	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²⋊#6157	C₁₆	1	8	yes	direct product factor
63	C₂×#88324	C₂×C₈	1	4	yes	—
64	C₄×#6157	—	2	2	no	—
65	C₄×#6157	—	2	2	no	—
66	D₅₃⋊C₈	—	2	2	no	—
67	D₅₃⋊C₈	—	2	2	no	—
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×#88322	—	2	2	no	—
73	C₂×#88322	C₄²	1	1	yes	characteristic
74	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
75	C₈×D₅₃	—	2	2	no	—
76	C₈×D₅₃	—	2	1	no	—
77	C₈×D₅₃	—	2	1	no	—
78	C₈×D₅₃	—	2	2	no	—
79	C₈×D₅₃	C₂×D₄	1	2	yes	—
80	C₈×D₅₃	#41	1	2	yes	—
81	#11719910	—	2	4	no	—
82	C₁₆×C₅₃	—	2	1	no	—
83	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	complemented
84	#11719908	—	2	1	no	—
85	#11719908	C₂²⋊C₄	1	2	yes	complemented
86	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
87	C₂²×D₅₃	—	2	2	no	—
88	C₂²×D₅₃	—	2	2	no	—
89	C₂²×D₅₃	C₄×C₈	1	1	yes	characteristic
90	C₂²×D₅₃	C₂×C₁₆	1	4	yes	—
91	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×#6157	—	2	4	no	—
93	C₂×#6157	—	2	4	no	—
94	C₂×#6157	C₂×C₁₆	1	8	yes	complemented
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
99	C₂×Dic₅₃	—	2	2	no	—
100	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
101	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
102	C₄×D₅₃	—	2	2	no	—
103	C₄×D₅₃	—	2	1	no	—
104	C₄×D₅₃	—	2	1	no	—
105	C₄×D₅₃	—	2	2	no	—
106	C₄×D₅₃	C₄×D₄	1	2	yes	—
107	C₄×D₅₃	#131	1	2	yes	—
108	#88324	—	2	4	no	—
109	C₈×C₅₃	—	2	1	no	—
110	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₈×C₅₃	#117	1	1	yes	characteristic
112	#88322	—	2	1	no	—
113	#88322	C₄×D₄	1	2	yes	—
114	C₁₆×C₂²⋊C₄	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	C₄×C₁₆	1	2	yes	complemented
118	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
119	C₂×D₅₃	—	2	1	no	—
120	C₂×D₅₃	—	2	1	no	—
121	C₂×D₅₃	—	2	8	no	—
122	C₂×D₅₃	C₄×C₁₆	1	2	yes	complemented
123	C₂×D₅₃	C₈×D₄	1	2	yes	—
124	C₂×D₅₃	C₂²×C₁₆	1	2	yes	—
125	C₂×D₅₃	#504	1	2	yes	—
126	#6157	—	2	8	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₅₃	#405	1	1	yes	characteristic
130	Dic₅₃	—	2	1	no	—
131	Dic₅₃	C₈×D₄	1	2	yes	—
132	C₂³×C₁₆	—	53	1	no	—
133	C₂×C₂²⋊C₁₆	—	53	1	no	—
134	C₂×C₄×C₁₆	—	53	2	no	—
135	C₈×C₂²⋊C₄	—	53	1	no	—
136	#1374	—	53	2	no	—
137	C₂×C₅₃	—	2	2	no	—
138	C₂×C₅₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
140	C₂×C₅₃	#2108	1	1	yes	characteristic
141	D₅₃	—	2	2	no	—
142	D₅₃	D₄×C₁₆	1	4	yes	—
143	C₂³×C₈	—	53	1	no	—
144	C₂²×C₁₆	—	53	1	no	—
145	C₂²×C₁₆	—	53	2	no	—
146	C₂²×C₁₆	—	53	1	no	—
147	C₂²×C₁₆	—	53	1	no	—
148	C₂²×C₁₆	—	53	1	no	—
149	C₂²×C₁₆	—	53	1	no	—
150	C₂²×C₁₆	—	106	4	no	—
151	C₂²×C₁₆	#6157	1	1	yes	characteristic, complemented
152	C₂×C₂²⋊C₈	—	53	1	no	—
153	C₂×C₄×C₈	—	53	2	no	—
154	C₄×C₂²⋊C₄	—	53	1	no	—
155	C₂²⋊C₁₆	—	53	2	no	—
156	C₂²⋊C₁₆	—	53	2	no	—
157	C₄×C₁₆	—	106	4	no	—
158	#336	—	53	2	no	—
159	C₅₃	C₁₆×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₄	—	53	1	no	—
161	C₂²×C₈	—	53	1	no	—
162	C₂²×C₈	—	53	2	no	—
163	C₂²×C₈	—	53	1	no	—
164	C₂²×C₈	—	53	1	no	—
165	C₂²×C₈	—	53	1	no	—
166	C₂²×C₈	—	53	1	no	—
167	C₂²×C₈	—	106	2	no	—
168	C₂²×C₈	—	106	2	no	—
169	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	53	2	no	—
171	C₂×C₄²	—	53	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	53	1	no	—
174	C₂×C₁₆	—	53	4	no	—
175	C₂×C₁₆	—	53	4	no	—
176	C₂×C₁₆	—	53	1	no	—
177	C₂×C₁₆	—	106	2	no	—
178	C₂×C₁₆	—	106	2	no	—
179	C₂×C₁₆	—	106	2	no	—
180	C₂×C₁₆	—	106	2	no	—
181	C₂×C₁₆	—	106	2	no	—
182	C₂×C₁₆	—	106	2	no	—
183	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
184	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
185	C₂²⋊C₈	—	53	2	no	—
186	C₂²⋊C₈	—	53	2	no	—
187	C₄×C₈	—	106	2	no	—
188	C₄×C₈	—	106	2	no	—
189	#95	—	53	2	no	—
190	C₂⁴	—	53	1	no	—
191	C₂²×C₄	—	53	1	no	—
192	C₂²×C₄	—	53	4	no	—
193	C₂²×C₄	—	53	1	no	—
194	C₂²×C₄	—	53	1	no	—
195	C₂²×C₄	—	53	1	no	—
196	C₂²×C₄	—	53	1	no	—
197	C₂²×C₄	—	106	2	no	—
198	C₂²×C₄	—	106	2	no	—
199	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	53	1	no	—
203	C₂×C₈	—	53	2	no	—
204	C₂×C₈	—	53	2	no	—
205	C₂×C₈	—	53	4	no	—
206	C₂×C₈	—	53	1	no	—
207	C₂×C₈	—	106	2	no	—
208	C₂×C₈	—	106	2	no	—
209	C₂×C₈	—	106	2	no	—
210	C₂×C₈	—	106	2	no	—
211	C₂×C₈	—	106	2	no	—
212	C₂×C₈	—	106	1	no	—
213	C₂×C₈	—	106	1	no	—
214	C₂×C₈	C₄×#6157	1	1	yes	characteristic
215	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
216	C₂²⋊C₄	—	53	8	no	—
217	C₄²	—	106	2	no	—
218	C₄²	—	106	2	no	—
219	C₁₆	—	2	1	no	—
220	C₁₆	—	53	2	no	—
221	C₁₆	—	106	4	no	—
222	C₁₆	—	106	1	no	—
223	C₁₆	C₂²⋊#6157	1	2	yes	direct product factor
224	C₂³	—	53	1	no	—
225	C₂³	—	53	1	no	—
226	C₂³	—	53	4	no	—
227	C₂³	—	106	2	no	—
228	C₂³	—	106	2	no	—
229	C₂³	C₈×#6157	1	1	yes	characteristic
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	53	1	no	—
233	C₂×C₄	—	53	2	no	—
234	C₂×C₄	—	53	2	no	—
235	C₂×C₄	—	53	8	no	—
236	C₂×C₄	—	53	1	no	—
237	C₂×C₄	—	106	2	no	—
238	C₂×C₄	—	106	2	no	—
239	C₂×C₄	—	106	2	no	—
240	C₂×C₄	—	106	4	no	—
241	C₂×C₄	—	106	4	no	—
242	C₂×C₄	—	106	1	no	—
243	C₂×C₄	—	106	1	no	—
244	C₂×C₄	C₈×#6157	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
246	C₈	—	2	1	no	—
247	C₈	—	53	2	no	—
248	C₈	—	106	4	no	—
249	C₈	—	106	1	no	—
250	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
251	C₈	#13133296	1	1	yes	characteristic
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	53	2	no	—
255	C₂²	—	53	2	no	—
256	C₂²	—	53	2	no	—
257	C₂²	—	53	2	no	—
258	C₂²	—	106	8	no	—
259	C₂²	—	106	1	no	—
260	C₂²	—	106	1	no	—
261	C₂²	C₁₆×#6157	1	2	yes	complemented
262	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	53	2	no	—
265	C₄	—	106	8	no	—
266	C₄	—	106	1	no	—
267	C₄	C₄×C₂²⋊#6157	1	1	yes	characteristic
268	C₄	#16811586	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	53	4	no	—
271	C₂	—	106	2	no	—
272	C₂	C₈×C₂²⋊#6157	1	1	yes	characteristic
273	C₂	C₂×C₁₆×#6157	1	1	yes	characteristic
274	C₂	#20290923	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×#6157
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	C₁₆×C₂²⋊C₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₆×#6157	C₂	semidirect
4. (C₂×#6157)⋊C₁₆	C₂	semidirect
10. #16811518	C₄	semidirect
11. C₂²⋊#11719910	C₄	semidirect
15. C₂²×#11719908	C₄	semidirect
17. C₂×C₁₆×D₅₃	C₄	semidirect
18. C₂×C₁₆×D₅₃	C₄	semidirect
21. C₂²×C₁₆×C₅₃	C₄	semidirect
62. C₂²⋊#6157	C₁₆	direct (N × H)
83. C₁₆×C₅₃	C₂²⋊C₄	semidirect
85. #11719908	C₂²⋊C₄	semidirect
94. C₂×#6157	C₂×C₁₆	semidirect
117. C₂²×C₅₃	C₄×C₁₆	semidirect
122. C₂×D₅₃	C₄×C₁₆	semidirect
151. C₂²×C₁₆	#6157	semidirect
159. C₅₃	C₁₆×C₂²⋊C₄	semidirect
223. C₁₆	C₂²⋊#6157	direct (N × H)
261. C₂²	C₁₆×#6157	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊(C₂²⋊C₁₆)	2
C₂×C₁₆×#6157	2
C₈×C₂²⋊#6157	2
(C₂×#6157)⋊C₁₆	2
C₂²×C₁₆×D₅₃	2
C₁₆×C₂²⋊C₄	53

Subgroups — C16×C22⋊#6157 #28310519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×C₂²⋊#6157 #28310519