Subgroups — #13262811

Order 1792.

306 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2162	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×#1136	C₂	1	1	yes	maximal, characteristic
4	#11725305	C₂	1	2	yes	maximal, complemented
5	C₁₆×C₇⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₇⋊C₁₆	C₄	1	2	yes	—
8	#89697	C₄	1	1	yes	characteristic
9	C₄×#1136	C₄	1	1	yes	characteristic
10	C₇×C₄⋊C₁₆	C₄	1	2	yes	—
11	C₇×C₂²⋊C₁₆	C₄	1	2	yes	—
12	C₄×C₇×C₁₆	C₂²	1	2	yes	—
13	C₄×C₇⋊C₁₆	C₄	1	2	yes	—
14	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#89249	C₄	1	2	yes	—
17	C₈×C₇⋊C₈	C₂²	1	2	yes	—
18	C₁₆×M₁₆	—	7	1	no	maximal, Sylow
19	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×#1136	C₈	1	2	yes	—
23	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
24	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₂²⋊(C₇⋊C₈)	C₈	1	2	yes	—
29	C₂×C₇⋊C₁₆	—	2	4	no	—
30	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
32	C₄⋊(C₇⋊C₈)	C₈	1	2	yes	—
33	#6377	C₈	1	2	yes	—
34	C₄×C₇⋊C₈	C₂×C₄	1	2	yes	—
35	C₄×C₇⋊C₈	C₂×C₄	1	2	yes	—
36	#2162	D₇	1	1	yes	characteristic
37	C₂×C₄×C₁₆	—	7	1	no	—
38	C₈×M₁₆	—	7	1	no	—
39	#1312	—	7	2	no	—
40	C₈×C₁₆	—	7	2	no	—
41	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
44	C₂×C₇×C₈	—	2	2	no	—
45	C₂×C₇×C₈	—	2	2	no	—
46	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
47	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
48	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₇	C₄²	1	1	yes	characteristic
51	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
54	#1136	C₁₆	1	8	yes	complemented
55	C₂×C₇⋊C₈	C₂×C₈	1	4	yes	—
56	C₂×C₇⋊C₈	C₂×C₈	1	4	yes	—
57	C₇×C₁₆	—	2	4	no	—
58	C₇⋊C₁₆	—	2	2	no	—
59	C₇⋊C₁₆	M₁₆	1	4	yes	complemented
60	C₂²×C₁₆	—	7	2	no	—
61	#431	—	7	1	no	—
62	C₄×M₁₆	—	7	1	no	—
63	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
64	C₄⋊C₁₆	Dic₇	1	2	yes	—
65	C₂²⋊C₁₆	Dic₇	1	2	yes	—
66	C₄×C₁₆	—	7	2	no	—
67	C₄×C₁₆	—	7	1	no	—
68	C₄×C₁₆	—	7	1	no	—
69	C₄×C₁₆	C₂×D₇	1	2	yes	—
70	#322	—	7	2	no	—
71	C₈²	—	7	2	no	—
72	C₂³×C₇	C₄×C₈	1	1	yes	characteristic
73	C₂×C₄×C₇	—	2	1	no	—
74	C₂×C₄×C₇	—	2	1	no	—
75	C₂×C₄×C₇	—	2	2	no	—
76	C₂×C₄×C₇	C₄×C₈	1	1	yes	characteristic
77	C₂×C₄×C₇	C₄×C₈	1	1	yes	characteristic
78	C₂×C₄×C₇	C₄×C₈	1	1	yes	characteristic
79	C₂×C₄×C₇	C₂×C₁₆	1	4	yes	—
80	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
83	C₇×C₈	—	2	1	no	—
84	C₇×C₈	—	2	1	no	—
85	C₇×C₈	C₂×M₁₆	1	2	yes	—
86	C₇×C₈	#131	1	2	yes	—
87	C₇⋊C₈	—	2	4	no	—
88	C₇⋊C₈	C₂×C₁₆	1	8	yes	complemented
89	C₂×M₁₆	—	7	2	no	—
90	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
91	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
92	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
93	C₂×C₁₆	—	7	2	no	—
94	C₂×C₁₆	—	7	2	no	—
95	C₂×C₁₆	—	14	4	no	—
96	C₂×C₁₆	C₂×Dic₇	1	4	yes	—
97	C₄⋊C₈	—	7	2	no	—
98	C₂²⋊C₈	—	7	2	no	—
99	#97	—	7	2	no	—
100	C₄×C₈	—	7	2	no	—
101	C₄×C₈	—	7	2	no	—
102	C₄×C₈	C₄×D₇	1	1	yes	characteristic
103	C₄×C₈	C₄×D₇	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
105	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
106	C₂²×C₇	—	2	1	no	—
107	C₂²×C₇	—	2	1	no	—
108	C₂²×C₇	C₄×C₁₆	1	2	yes	—
109	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
110	C₄×C₇	—	2	1	no	—
111	C₄×C₇	—	2	1	no	—
112	C₄×C₇	C₄×C₁₆	1	2	yes	—
113	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
114	C₄×C₇	#405	1	1	yes	characteristic
115	C₄×C₇	C₂²×C₁₆	1	2	yes	—
116	C₄×C₇	#504	1	2	yes	—
117	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
118	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
119	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
120	M₁₆	—	7	8	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	7	4	no	—
124	C₂×C₈	—	7	4	no	—
125	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
126	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
127	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
128	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
129	C₄²	C₈×D₇	1	1	yes	characteristic
130	C₄²	C₈×D₇	1	1	yes	characteristic
131	C₄²	C₄×Dic₇	1	1	yes	characteristic
132	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
133	C₁₆	—	2	4	no	—
134	C₁₆	—	7	4	no	—
135	C₁₆	—	14	2	no	—
136	C₂×C₇	—	2	2	no	—
137	C₂×C₇	C₈×M₁₆	1	1	yes	characteristic
138	C₂×C₇	C₂×C₄×C₁₆	1	1	yes	characteristic
139	C₂×C₇	#2108	1	1	yes	characteristic
140	C₂³	C₈×Dic₇	1	1	yes	characteristic
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	D₇×C₁₆	1	4	yes	—
145	C₂×C₄	C₈×Dic₇	1	1	yes	characteristic
146	C₂×C₄	C₈×Dic₇	1	1	yes	characteristic
147	C₂×C₄	C₈×Dic₇	1	1	yes	characteristic
148	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
149	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	7	8	no	—
154	C₈	—	14	4	no	—
155	C₈	D₇×M₁₆	1	2	yes	—
156	C₈	#6511	1	2	yes	—
157	C₇	C₁₆×M₁₆	1	1	yes	characteristic, complemented, Sylow
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₁₆×Dic₇	1	2	yes	—
161	C₂²	C₂×C₈×Dic₇	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	C₁₆×Dic₇	1	2	yes	—
165	C₄	C₂×D₇×C₁₆	1	2	yes	—
166	C₄	#89679	1	2	yes	—
167	C₄	#89769	1	1	yes	characteristic
168	C₄	M₁₆×Dic₇	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	#11728493	1	1	yes	characteristic
171	C₂	C₂×C₁₆×Dic₇	1	1	yes	characteristic
172	C₂	#11729173	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₇
commutator	C₂×C₇	C₂×C₄×C₁₆
Fitting	C₇×#2162	C₂
socle	C₂²×C₇	C₂×C₄×C₈
prime radical	C₂³×C₇	C₄×C₈
cosocle	C₄×C₇×C₈	C₂³
hypercenter	#2162	D₇
hypocenter	C₇	C₁₆×M₁₆
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
4. #11725305	C₂	semidirect
5. C₁₆×C₇⋊C₈	C₂	semidirect
54. #1136	C₁₆	semidirect
59. C₇⋊C₁₆	M₁₆	semidirect
88. C₇⋊C₈	C₂×C₁₆	semidirect
157. 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
C₇×#2162	2
C₂×C₄×C₇⋊C₁₆	2
C₈×#1136	2
#11725305	2
C₁₆×C₇⋊C₈	2
C₁₆×M₁₆	7