Subgroups — C₁₆×#1131 #13262807

Order 1792.

306 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11726806	C₂	1	1	yes	maximal, characteristic
2	C₄×D₇×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×#1131	C₂	1	1	yes	maximal, characteristic
4	C₇×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#11725305	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#11725244	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₇×C₁₆	C₄	1	2	yes	—
9	C₄×#1131	C₄	1	1	yes	characteristic
10	#89457	C₄	1	1	yes	characteristic
11	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
12	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
14	C₇×C₈²	C₂²	1	1	yes	characteristic
15	(C₂×D₇)⋊C₁₆	C₄	1	2	yes	—
16	Dic₇⋊C₁₆	C₄	1	2	yes	—
17	C₁₆×Dic₇	C₄	1	2	yes	—
18	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
20	#89250	C₄	1	1	yes	characteristic
21	#89249	C₄	1	1	yes	characteristic
22	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
23	C₁₆×M₁₆	—	7	1	no	maximal, Sylow
24	C₂×#1131	C₈	1	2	yes	—
25	C₂×C₈×D₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₇	C₂×C₄	1	1	yes	characteristic
27	C₄²×D₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
30	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
33	(C₂×D₇)⋊C₈	C₈	1	2	yes	—
34	#6389	C₈	1	2	yes	—
35	Dic₇⋊C₈	C₈	1	2	yes	—
36	C₈×Dic₇	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
40	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
43	D₇×C₁₆	—	2	4	no	—
44	#2162	—	7	1	no	—
45	C₂×C₄×C₁₆	—	7	1	no	—
46	C₈×M₁₆	—	7	1	no	—
47	#1312	—	7	1	no	—
48	#1312	—	7	1	no	—
49	C₈×C₁₆	—	7	1	no	—
50	C₈×C₁₆	D₇	1	1	yes	characteristic, complemented
51	C₂×C₄×D₇	C₄²	1	1	yes	characteristic
52	C₂×C₄×D₇	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×D₇	C₂×C₈	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
55	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
56	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
59	C₄×Dic₇	C₄²	1	1	yes	characteristic
60	C₄×Dic₇	C₂×C₈	1	1	yes	characteristic
61	C₄×Dic₇	C₂×C₈	1	1	yes	characteristic
62	C₂×C₇⋊C₈	C₄²	1	1	yes	characteristic
63	C₂×C₇⋊C₈	C₄²	1	1	yes	characteristic
64	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
65	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
66	#1131	C₁₆	1	8	yes	direct product factor
67	C₈×D₇	—	2	2	no	—
68	C₈×D₇	—	2	2	no	—
69	C₇×C₁₆	—	2	2	no	—
70	C₇×C₁₆	M₁₆	1	4	yes	complemented
71	C₇⋊C₁₆	—	2	2	no	—
72	C₇⋊C₁₆	—	2	2	no	—
73	C₂²×C₁₆	—	7	2	no	—
74	#431	—	7	1	no	—
75	C₄×M₁₆	—	7	1	no	—
76	C₂×C₄×C₈	—	7	1	no	—
77	C₄⋊C₁₆	—	7	2	no	—
78	C₂²⋊C₁₆	—	7	2	no	—
79	C₄×C₁₆	—	7	2	no	—
80	C₄×C₁₆	—	7	1	no	—
81	C₄×C₁₆	—	7	1	no	—
82	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
83	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
84	#322	—	7	1	no	—
85	#322	—	7	1	no	—
86	C₈²	—	7	1	no	—
87	C₈²	C₂×D₇	1	1	yes	characteristic
88	C₂²×D₇	C₄×C₈	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
92	C₂×Dic₇	C₄×C₈	1	1	yes	characteristic
93	C₂×Dic₇	C₄×C₈	1	1	yes	characteristic
94	C₂×Dic₇	C₄×C₈	1	1	yes	characteristic
95	C₄×D₇	—	2	1	no	—
96	C₄×D₇	—	2	1	no	—
97	C₄×D₇	—	2	2	no	—
98	C₄×D₇	C₂×C₁₆	1	4	yes	—
99	C₇×C₈	—	2	2	no	—
100	C₇×C₈	C₂×C₁₆	1	4	yes	complemented
101	C₇×C₈	C₂×M₁₆	1	2	yes	—
102	C₇×C₈	#131	1	2	yes	—
103	C₇⋊C₈	—	2	1	no	—
104	C₇⋊C₈	—	2	1	no	—
105	C₇⋊C₈	—	2	2	no	—
106	C₇⋊C₈	C₂×C₁₆	1	4	yes	complemented
107	C₂×M₁₆	—	7	2	no	—
108	C₂²×C₈	—	7	1	no	—
109	C₂²×C₈	—	7	1	no	—
110	C₂×C₄²	—	7	1	no	—
111	C₂×C₁₆	—	7	2	no	—
112	C₂×C₁₆	—	7	2	no	—
113	C₂×C₁₆	—	14	4	no	—
114	C₂×C₁₆	C₄×D₇	1	2	yes	—
115	C₂×C₁₆	C₄×D₇	1	2	yes	—
116	C₄⋊C₈	—	7	2	no	—
117	C₂²⋊C₈	—	7	2	no	—
118	#97	—	7	2	no	—
119	C₄×C₈	—	7	1	no	—
120	C₄×C₈	—	7	1	no	—
121	C₄×C₈	—	7	1	no	—
122	C₄×C₈	—	7	1	no	—
123	C₄×C₈	—	7	1	no	—
124	C₄×C₈	C₄×D₇	1	1	yes	characteristic
125	C₄×C₈	C₄×D₇	1	1	yes	characteristic
126	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
127	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
128	C₂×D₇	—	2	1	no	—
129	C₂×D₇	—	2	1	no	—
130	C₂×D₇	C₄×C₁₆	1	2	yes	—
131	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
132	C₄×C₇	#405	1	1	yes	characteristic
133	C₄×C₇	C₂²×C₁₆	1	2	yes	—
134	C₄×C₇	#504	1	2	yes	—
135	Dic₇	—	2	1	no	—
136	Dic₇	—	2	1	no	—
137	Dic₇	C₄×C₁₆	1	2	yes	—
138	C₂²×C₄	—	7	1	no	—
139	C₂²×C₄	—	7	1	no	—
140	C₂²×C₄	—	7	1	no	—
141	M₁₆	—	7	8	no	—
142	C₂×C₈	—	7	2	no	—
143	C₂×C₈	—	7	1	no	—
144	C₂×C₈	—	7	1	no	—
145	C₂×C₈	—	7	2	no	—
146	C₂×C₈	—	14	2	no	—
147	C₂×C₈	—	14	2	no	—
148	C₂×C₈	C₈×D₇	1	2	yes	—
149	C₂×C₈	C₈×D₇	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
151	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
152	C₄²	—	7	1	no	—
153	C₄²	—	7	1	no	—
154	C₄²	—	7	1	no	—
155	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
156	C₁₆	—	2	2	no	—
157	C₁₆	—	14	2	no	—
158	C₁₆	—	14	2	no	—
159	C₁₆	#1131	1	4	yes	direct product factor
160	C₂×C₇	C₈×M₁₆	1	1	yes	characteristic
161	C₂×C₇	C₂×C₄×C₁₆	1	1	yes	characteristic
162	C₂×C₇	#2108	1	1	yes	characteristic
163	D₇	—	2	2	no	—
164	C₂³	—	7	1	no	—
165	C₂×C₄	—	7	1	no	—
166	C₂×C₄	—	7	4	no	—
167	C₂×C₄	—	7	1	no	—
168	C₂×C₄	—	7	1	no	—
169	C₂×C₄	—	14	2	no	—
170	C₂×C₄	—	14	1	no	—
171	C₂×C₄	—	14	1	no	—
172	C₂×C₄	C₄²×D₇	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
174	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	7	4	no	—
177	C₈	—	14	2	no	—
178	C₈	—	14	1	no	—
179	C₈	—	14	1	no	—
180	C₈	D₇×C₁₆	1	4	yes	complemented
181	C₈	C₂×#1131	1	2	yes	—
182	C₈	#6464	1	2	yes	—
183	C₇	C₁₆×M₁₆	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	7	2	no	—
185	C₂²	—	14	1	no	—
186	C₂²	—	14	1	no	—
187	C₂²	C₄×C₈×D₇	1	1	yes	characteristic
188	C₄	—	7	2	no	—
189	C₄	—	14	1	no	—
190	C₄	—	14	1	no	—
191	C₄	C₄×#1131	1	1	yes	characteristic
192	C₄	#89461	1	1	yes	characteristic
193	C₄	C₂×D₇×C₁₆	1	2	yes	—
194	C₄	#89673	1	2	yes	—
195	C₂	—	14	2	no	—
196	C₂	C₈×#1131	1	1	yes	characteristic
197	C₂	C₄×D₇×C₁₆	1	1	yes	characteristic
198	C₂	#11726805	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₆	C₂×D₇
commutator	C₂×C₇	C₂×C₄×C₁₆
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×C₈×C₁₆	C₂
socle	C₂²×C₇	C₂×C₄×C₈
prime radical	C₂²×D₇	C₄×C₈
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₈×C₁₆	D₇
hypocenter	C₇	C₁₆×M₁₆
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄×C₁₆ ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
4. C₇×C₈×C₁₆	C₂	semidirect
5. #11725305	C₂	semidirect
6. C₁₆×C₇⋊C₈	C₂	semidirect
7. #11725244	C₂	semidirect
50. C₈×C₁₆	D₇	semidirect
66. #1131	C₁₆	direct (N × H)
70. C₇×C₁₆	M₁₆	semidirect
100. C₇×C₈	C₂×C₁₆	semidirect
106. C₇⋊C₈	C₂×C₁₆	semidirect
159. C₁₆	#1131	direct (N × H)
180. C₈	D₇×C₁₆	semidirect
183. C₇	C₁₆×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#11726806	2
C₄×D₇×C₁₆	2
C₈×#1131	2
C₇×C₈×C₁₆	2
#11725305	2
C₁₆×C₇⋊C₈	2
#11725244	2
C₁₆×M₁₆	7

Subgroups — C16×#1131 #13262807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×#1131 #13262807