Subgroups — C₇×#2244 #11730713

Order 896.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#493	C₂	1	1	yes	maximal, characteristic
3	C₇×#357	C₂	1	4	yes	maximal, complemented
4	C₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×D₈	—	2	1	no	—
6	C₂×C₇×D₈	C₂²	1	2	yes	—
7	C₇×#127	—	2	1	no	—
8	C₇×#125	—	2	1	no	—
9	C₂×C₇×C₁₆	C₂²	1	2	yes	—
10	C₇×#107	C₂²	1	2	yes	—
11	C₇×#106	—	2	1	no	—
12	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	#2244	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂×C₇×D₄	—	2	2	no	—
15	C₂×C₇×D₄	—	4	2	no	—
16	C₇×D₈	—	2	4	no	—
17	C₇×D₈	—	4	2	no	—
18	C₂×C₇×C₈	D₄	1	1	yes	characteristic
19	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
20	C₇×C₄⋊C₄	—	2	2	no	—
21	C₇×C₄⋊C₄	—	4	2	no	—
22	C₄²×C₇	D₄	1	1	yes	characteristic
23	C₇×C₁₆	—	2	2	no	—
24	#499	C₂×C₇	1	1	yes	characteristic, complemented
25	#493	C₂×C₇	1	1	yes	characteristic
26	#357	C₂×C₇	1	4	yes	complemented
27	#346	C₂×C₇	1	1	yes	characteristic, complemented
28	C₂³×C₇	—	4	2	no	—
29	C₇×D₄	—	4	2	no	—
30	C₇×D₄	—	4	2	no	—
31	C₇×D₄	—	8	4	no	—
32	C₂×C₄×C₇	—	4	2	no	—
33	C₂×C₄×C₇	D₈	1	2	yes	—
34	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
35	C₇×C₈	SD₁₆	1	2	yes	—
36	C₇×C₈	#41	1	2	yes	—
37	C₂×D₈	—	2	1	no	—
38	C₂×D₈	C₂²×C₇	1	2	yes	—
39	#127	—	2	1	no	—
40	#125	—	2	1	no	—
41	C₂×C₁₆	C₂²×C₇	1	2	yes	—
42	#107	C₂²×C₇	1	2	yes	—
43	#106	—	2	1	no	—
44	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
45	C₂²×C₇	—	8	2	no	—
46	C₂²×C₇	—	8	4	no	—
47	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
48	C₄×C₇	—	2	2	no	—
49	C₄×C₇	—	8	2	no	—
50	C₄×C₇	#124	1	1	yes	characteristic
51	C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×D₄	—	2	2	no	—
53	C₂×D₄	—	4	2	no	—
54	D₈	—	2	4	no	—
55	D₈	—	4	2	no	—
56	C₂×C₈	C₇×D₄	1	1	yes	characteristic
57	C₂×C₈	C₂³×C₇	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄²	C₇×D₄	1	1	yes	characteristic
61	C₁₆	—	2	2	no	—
62	C₂×C₇	—	16	2	no	—
63	C₂×C₇	#486	1	1	yes	characteristic
64	C₂×C₇	#509	1	2	yes	—
65	C₂³	—	4	2	no	—
66	D₄	—	4	2	no	—
67	D₄	—	4	2	no	—
68	D₄	—	8	4	no	—
69	C₂×C₄	—	4	2	no	—
70	C₂×C₄	C₇×D₈	1	2	yes	—
71	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
72	C₈	C₇×SD₁₆	1	2	yes	—
73	C₈	C₇×#41	1	2	yes	—
74	C₇	#2244	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²	—	8	2	no	—
76	C₂²	—	8	4	no	—
77	C₂²	C₂×C₇×D₈	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	8	2	no	—
80	C₄	C₇×#124	1	1	yes	characteristic
81	C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
82	C₂	—	16	2	no	—
83	C₂	C₇×#486	1	1	yes	characteristic
84	C₂	C₇×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₇
prime radical	C₇×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#499	C₂	semidirect
3. C₇×#357	C₂	semidirect
4. C₇×#346	C₂	semidirect
13. #2244	C₇	direct (N × H)
24. #499	C₂×C₇	semidirect
26. #357	C₂×C₇	semidirect
27. #346	C₂×C₇	semidirect
74. C₇	#2244	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#499	2
C₇×#493	2
C₇×#357	2
C₇×#346	2
#2244	7

Subgroups — C7×#2244 #11730713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2244 #11730713