Subgroups — #11728293

Order 896.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#424	C₂	1	1	yes	maximal, characteristic
2	#89897	C₂	1	1	yes	maximal, characteristic
3	C₇⋊#338	C₂	1	1	yes	maximal, characteristic
4	#89347	C₂	1	2	yes	maximal
5	#89328	C₂	1	2	yes	maximal
6	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×#117	C₂²	1	1	yes	characteristic
9	C₂×#1136	C₂²	1	2	yes	—
10	C₂×#1136	C₂²	1	2	yes	—
11	C₇×C₄⋊C₈	—	2	2	no	—
12	#6393	—	2	2	no	—
13	#2103	—	7	1	no	maximal, Sylow
14	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
15	C₇×M₁₆	—	2	2	no	—
16	C₂×C₇×C₈	—	2	1	no	—
17	C₂×C₇×C₈	—	2	2	no	—
18	C₂×C₇×C₈	D₄	1	2	yes	—
19	C₇×C₄⋊C₄	—	2	1	no	—
20	C₇×C₂²⋊C₄	Q₈	1	2	yes	—
21	C₄²×C₇	—	2	1	no	—
22	#1136	—	2	2	no	—
23	#1136	—	2	2	no	—
24	#1136	—	2	2	no	—
25	#1136	—	2	2	no	—
26	C₂×C₇⋊C₈	—	2	2	no	—
27	C₂×C₇⋊C₈	—	2	2	no	—
28	#430	—	7	1	no	—
29	#424	D₇	1	1	yes	characteristic
30	#344	—	7	2	no	—
31	#341	—	7	2	no	—
32	#338	—	7	1	no	—
33	C₂³×C₇	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₇	—	2	2	no	—
35	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₇	#41	1	1	yes	characteristic
37	C₂×C₄×C₇	#41	1	2	yes	—
38	C₂×C₄×C₇	#41	1	2	yes	—
39	C₇×C₈	—	2	2	no	—
40	C₇×C₈	—	4	1	no	—
41	C₇⋊C₈	—	4	2	no	—
42	C₇⋊C₈	—	4	2	no	—
43	C₂×M₁₆	—	7	2	no	—
44	C₂×M₁₆	—	7	2	no	—
45	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic
46	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
47	#117	C₂×D₇	1	1	yes	characteristic
48	#108	—	14	2	no	—
49	C₄⋊C₈	—	2	2	no	—
50	C₂²×C₇	—	4	1	no	—
51	C₂²×C₇	C₂²⋊Q₈	1	1	yes	characteristic
52	C₂²×C₇	#125	1	2	yes	—
53	C₄×C₇	—	4	2	no	—
54	C₄×C₇	#123	1	1	yes	characteristic
55	C₄×C₇	#124	1	1	yes	characteristic
56	C₄×C₇	#126	1	2	yes	—
57	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
58	M₁₆	—	2	2	no	—
59	M₁₆	—	14	2	no	—
60	M₁₆	—	14	2	no	—
61	M₁₆	—	14	2	no	—
62	M₁₆	—	14	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	14	2	no	—
66	C₂×C₈	—	14	2	no	—
67	C₂×C₈	C₂²⋊D₇	1	2	yes	—
68	C₄⋊C₄	—	2	1	no	—
69	C₂²⋊C₄	Dic₁₄	1	2	yes	—
70	C₄²	—	2	1	no	—
71	C₂×C₇	—	2	1	no	—
72	C₂×C₇	—	2	2	no	—
73	C₂×C₇	#400	1	1	yes	characteristic
74	C₂³	C₂×Dic₁₄	1	1	yes	characteristic
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	#1157	1	1	yes	characteristic
77	C₂×C₄	#1159	1	2	yes	—
78	C₂×C₄	#1161	1	2	yes	—
79	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
80	C₈	—	2	2	no	—
81	C₈	—	4	1	no	—
82	C₈	—	28	2	no	—
83	C₈	—	28	2	no	—
84	C₇	#2103	1	1	yes	characteristic, complemented, Sylow
85	C₂²	—	4	1	no	—
86	C₂²	#6453	1	2	yes	—
87	C₂²	#6487	1	1	yes	characteristic
88	C₄	—	4	2	no	—
89	C₄	#6461	1	2	yes	—
90	C₄	#6498	1	1	yes	characteristic
91	C₄	#6510	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	2	2	no	—
94	C₂	#89758	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#6510
commutator, cosocle	C₂²×C₄×C₇	C₂³
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₇×#424	C₂
socle	C₂×C₇	#400
prime radical	C₂³×C₇	C₂×Q₈
hypercenter	#424	D₇
hypocenter	C₇	#2103
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
84. C₇	#2103	semidirect

Maximal subgroups

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

subgroup	index
C₇×#424	2
#89897	2
C₇⋊#338	2
#89347	2
#89328	2
#2103	7