Subgroups — C₁₆×#1161 #13507314

Order 1792.

602 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#1161	C₂	1	1	yes	maximal, characteristic
2	C₇×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#11727031	C₂	1	3	yes	maximal, complemented
5	#11727028	C₂	1	3	yes	maximal, complemented
6	C₁₆×D₂₈	C₂	1	3	yes	maximal, complemented
7	C₄×D₇×C₁₆	C₂	1	3	yes	maximal, complemented
8	C₄×#1161	C₄	1	1	yes	characteristic
9	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
10	Q₈×C₇⋊C₈	C₄	1	1	yes	characteristic
11	C₂×D₇×C₁₆	C₂²	1	3	yes	—
12	C₂×D₇×C₁₆	C₂²	1	3	yes	—
13	#89606	C₄	1	3	yes	—
14	#89605	C₄	1	3	yes	—
15	C₈×D₂₈	C₂²	1	3	yes	—
16	C₄×C₈×D₇	C₂²	1	3	yes	—
17	C₇×C₄⋊C₁₆	C₂²	1	3	yes	—
18	C₄×C₇×C₁₆	C₂²	1	3	yes	—
19	(C₂×D₇)⋊C₁₆	C₂²	1	6	yes	—
20	C₁₆×Dic₇	C₂²	1	3	yes	—
21	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
22	C₄⋊(C₇⋊C₁₆)	C₂²	1	3	yes	—
23	C₄×C₇⋊C₁₆	C₂²	1	3	yes	—
24	C₁₆×#41	—	7	1	no	maximal, Sylow
25	C₂×#1161	C₈	1	1	yes	characteristic
26	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
27	Q₈×Dic₇	C₈	1	1	yes	characteristic
28	C₂×C₈×D₇	C₂×C₄	1	3	yes	—
29	C₂×C₈×D₇	C₂³	1	3	yes	—
30	#6456	C₈	1	3	yes	—
31	#6455	C₈	1	3	yes	—
32	C₄×D₂₈	C₂×C₄	1	3	yes	—
33	C₄²×D₇	C₂×C₄	1	3	yes	—
34	C₂×C₇×C₁₆	C₂³	1	3	yes	—
35	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
36	C₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
37	C₄×C₇×C₈	C₂³	1	3	yes	—
38	(C₂×D₇)⋊C₈	C₂×C₄	1	6	yes	—
39	C₈×Dic₇	C₂×C₄	1	3	yes	—
40	C₈×Dic₇	C₂³	1	1	yes	characteristic
41	C₂×C₇⋊C₁₆	C₂³	1	3	yes	—
42	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
43	C₄⋊(C₇⋊C₈)	C₂×C₄	1	3	yes	—
44	C₄×C₇⋊C₈	C₂×C₄	1	3	yes	—
45	D₇×C₁₆	—	2	6	no	—
46	D₇×C₁₆	—	2	6	no	—
47	C₈×#41	—	7	1	no	—
48	Q₈×C₁₆	—	7	1	no	—
49	Q₈×C₁₆	D₇	1	1	yes	characteristic, complemented
50	D₄×C₁₆	—	7	3	no	—
51	D₄×C₁₆	—	7	3	no	—
52	#2162	—	7	3	no	—
53	C₂×C₄×C₁₆	—	7	3	no	—
54	C₂×C₇×Q₈	C₂×C₈	1	1	yes	characteristic
55	#1161	C₁₆	1	8	yes	direct product factor
56	C₂×D₂₈	C₂×C₈	1	3	yes	—
57	C₂×C₄×D₇	C₂×C₈	1	3	yes	—
58	C₂×C₄×D₇	C₂×C₈	1	3	yes	—
59	C₂×C₄×D₇	C₂²×C₄	1	3	yes	—
60	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
61	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
62	C₇×C₄⋊C₄	C₂×C₈	1	3	yes	—
63	C₄²×C₇	C₂²×C₄	1	3	yes	—
64	(C₂×D₇)⋊C₄	C₂×C₈	1	6	yes	—
65	C₄⋊Dic₇	C₂×C₈	1	3	yes	—
66	C₄×Dic₇	C₂×C₈	1	3	yes	—
67	C₄×Dic₇	C₂×C₈	1	3	yes	—
68	C₄×Dic₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₇⋊C₈	C₂²×C₄	1	3	yes	—
70	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
71	C₈×D₇	—	2	3	no	—
72	C₈×D₇	—	2	3	no	—
73	C₈×D₇	—	2	6	no	—
74	C₇×C₁₆	—	2	3	no	—
75	C₇×C₁₆	#41	1	2	yes	complemented
76	C₇⋊C₁₆	—	2	3	no	—
77	C₇⋊C₁₆	#41	1	2	yes	complemented
78	C₄×#41	—	7	1	no	—
79	C₂²×C₁₆	—	7	3	no	—
80	C₂²×C₁₆	—	7	3	no	—
81	C₈×Q₈	—	7	1	no	—
82	C₈×Q₈	C₂×D₇	1	1	yes	characteristic
83	C₈×D₄	—	7	3	no	—
84	C₈×D₄	—	7	3	no	—
85	#431	—	7	3	no	—
86	C₂×C₄×C₈	—	7	3	no	—
87	C₄⋊C₁₆	—	7	3	no	—
88	C₄⋊C₁₆	C₂×D₇	1	3	yes	—
89	C₂²⋊C₁₆	—	7	6	no	—
90	C₄×C₁₆	—	7	3	no	—
91	C₄×C₁₆	—	7	1	no	—
92	C₄×C₁₆	—	7	3	no	—
93	C₄×C₁₆	C₂×D₇	1	3	yes	—
94	C₂²×D₇	C₂²×C₈	1	3	yes	—
95	C₇×Q₈	C₂×C₁₆	1	4	yes	complemented
96	C₂×C₄×C₇	C₂²×C₈	1	3	yes	—
97	C₂×C₄×C₇	C₂²×C₈	1	3	yes	—
98	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×Dic₇	C₂²×C₈	1	3	yes	—
100	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
101	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
102	D₂₈	C₂×C₁₆	1	12	yes	complemented
103	C₄×D₇	—	2	3	no	—
104	C₄×D₇	—	2	3	no	—
105	C₄×D₇	—	2	6	no	—
106	C₄×D₇	C₂×C₁₆	1	12	yes	complemented
107	C₇×C₈	—	2	3	no	—
108	C₇×C₈	C₂×#41	1	1	yes	characteristic
109	C₇×C₈	C₂×#41	1	1	yes	characteristic
110	C₇⋊C₈	—	2	3	no	—
111	C₇⋊C₈	#131	1	2	yes	—
112	C₂×#41	—	7	1	no	—
113	C₂²×C₈	—	7	3	no	—
114	C₂²×C₈	—	7	3	no	—
115	C₄×Q₈	—	7	1	no	—
116	C₄×Q₈	C₄×D₇	1	1	yes	characteristic
117	C₄×D₄	—	7	3	no	—
118	C₄×D₄	—	7	3	no	—
119	#117	—	7	3	no	—
120	C₂×C₄²	—	7	3	no	—
121	C₂×C₁₆	—	7	1	no	—
122	C₂×C₁₆	—	7	3	no	—
123	C₂×C₁₆	—	14	6	no	—
124	C₂×C₁₆	—	14	6	no	—
125	C₂×C₁₆	C₂²×D₇	1	3	yes	—
126	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
127	C₄⋊C₈	—	7	3	no	—
128	C₄⋊C₈	C₄×D₇	1	3	yes	—
129	C₂²⋊C₈	—	7	6	no	—
130	C₄×C₈	—	7	3	no	—
131	C₄×C₈	—	7	1	no	—
132	C₄×C₈	—	7	3	no	—
133	C₄×C₈	C₂²×D₇	1	3	yes	—
134	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
135	C₂×D₇	—	2	3	no	—
136	C₂×D₇	—	2	3	no	—
137	C₂×D₇	C₂²×C₁₆	1	6	yes	—
138	C₄×C₇	—	2	3	no	—
139	C₄×C₇	C₂²×C₁₆	1	6	yes	—
140	C₄×C₇	C₄×#41	1	1	yes	characteristic
141	C₄×C₇	C₂×#131	1	1	yes	characteristic
142	Dic₇	—	2	3	no	—
143	Dic₇	C₂²×C₁₆	1	2	yes	—
144	Dic₇	#504	1	2	yes	—
145	#41	—	7	8	no	—
146	C₂×Q₈	C₈×D₇	1	1	yes	characteristic
147	C₂×D₄	—	7	3	no	—
148	C₂²×C₄	—	7	3	no	—
149	C₂²×C₄	—	7	3	no	—
150	C₂²×C₄	—	7	3	no	—
151	C₂×C₈	—	7	1	no	—
152	C₂×C₈	—	7	3	no	—
153	C₂×C₈	—	14	6	no	—
154	C₂×C₈	—	14	3	no	—
155	C₂×C₈	—	14	3	no	—
156	C₂×C₈	C₂×C₄×D₇	1	3	yes	—
157	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
158	C₄⋊C₄	—	7	3	no	—
159	C₄⋊C₄	C₈×D₇	1	3	yes	—
160	C₂²⋊C₄	—	7	6	no	—
161	C₄²	—	7	3	no	—
162	C₄²	—	7	3	no	—
163	C₄²	—	7	1	no	—
164	C₄²	C₂×C₄×D₇	1	3	yes	—
165	C₁₆	—	2	3	no	—
166	C₁₆	—	7	2	no	—
167	C₁₆	—	14	3	no	—
168	C₁₆	#1161	1	2	yes	direct product factor
169	C₂×C₇	C₈×#41	1	1	yes	characteristic
170	C₂×C₇	C₂³×C₁₆	1	1	yes	characteristic
171	C₂×C₇	C₂×#504	1	1	yes	characteristic
172	D₇	—	2	6	no	—
173	C₂³	—	7	3	no	—
174	Q₈	D₇×C₁₆	1	4	yes	complemented
175	D₄	—	7	12	no	—
176	C₂×C₄	—	7	1	no	—
177	C₂×C₄	—	7	1	no	—
178	C₂×C₄	—	7	12	no	—
179	C₂×C₄	—	7	3	no	—
180	C₂×C₄	—	14	6	no	—
181	C₂×C₄	—	14	3	no	—
182	C₂×C₄	—	14	3	no	—
183	C₂×C₄	C₂×C₈×D₇	1	3	yes	—
184	C₂×C₄	C₂×C₈×D₇	1	3	yes	—
185	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
186	C₈	—	2	3	no	—
187	C₈	—	7	2	no	—
188	C₈	—	14	3	no	—
189	C₈	C₂×#1161	1	1	yes	characteristic
190	C₈	D₇×#41	1	1	yes	characteristic
191	C₇	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
192	C₂²	—	7	6	no	—
193	C₂²	—	14	3	no	—
194	C₂²	—	14	3	no	—
195	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
196	C₄	—	2	3	no	—
197	C₄	—	7	2	no	—
198	C₄	—	7	2	no	—
199	C₄	—	14	3	no	—
200	C₄	C₂×D₇×C₁₆	1	6	yes	—
201	C₄	C₄×#1161	1	1	yes	characteristic
202	C₄	D₇×#131	1	1	yes	characteristic
203	C₂	—	14	6	no	—
204	C₂	C₈×#1161	1	1	yes	characteristic
205	C₂	C₂²×D₇×C₁₆	1	1	yes	characteristic
206	C₂	D₇×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₇
commutator	C₂×C₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×Q₈×C₁₆	C₂
socle	C₂²×C₇	C₂³×C₈
prime radical	C₂×#1161	C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	Q₈×C₁₆	D₇
hypocenter	C₇	C₁₆×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₇×Q₈×C₁₆	C₂	semidirect
3. Q₈×C₇⋊C₁₆	C₂	semidirect
4. #11727031	C₂	semidirect
5. #11727028	C₂	semidirect
6. C₁₆×D₂₈	C₂	semidirect
7. C₄×D₇×C₁₆	C₂	semidirect
49. Q₈×C₁₆	D₇	semidirect
55. #1161	C₁₆	direct (N × H)
75. C₇×C₁₆	#41	semidirect
77. C₇⋊C₁₆	#41	semidirect
95. C₇×Q₈	C₂×C₁₆	semidirect
102. D₂₈	C₂×C₁₆	semidirect
106. C₄×D₇	C₂×C₁₆	semidirect
168. C₁₆	#1161	direct (N × H)
174. Q₈	D₇×C₁₆	semidirect
191. C₇	C₁₆×#41	semidirect

Maximal subgroups

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

subgroup	index
C₈×#1161	2
C₇×Q₈×C₁₆	2
Q₈×C₇⋊C₁₆	2
#11727031	2
#11727028	2
C₁₆×D₂₈	2
C₄×D₇×C₁₆	2
C₁₆×#41	7

Subgroups — C16×#1161 #13507314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×#1161 #13507314