Subgroups — C₇×#2842 #11742151

Order 896.

548 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#399	C₂	1	4	yes	maximal, complemented
2	C₇×#396	C₂	1	8	yes	maximal, complemented
3	C₇×#379	C₂	1	3	yes	maximal
4	C₂⁵×C₇	C₂²	1	1	yes	characteristic
5	C₂×C₇×C₄⋊C₄	C₂²	1	12	yes	complemented
6	C₂×C₇×C₂²⋊C₄	C₂²	1	12	yes	—
7	C₂×C₇×C₂²⋊C₄	C₂²	1	6	yes	—
8	C₇×#95	C₂²	1	4	yes	complemented
9	#2842	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂⁴×C₇	—	4	6	no	—
11	C₂⁴×C₇	Q₈	1	4	yes	—
12	C₂⁴×C₇	C₂³	1	2	yes	—
13	C₂⁴×C₇	C₂³	1	1	yes	characteristic
14	C₂²×C₄×C₇	C₂³	1	12	yes	—
15	C₇×C₄⋊C₄	—	4	12	no	—
16	C₇×C₂²⋊C₄	—	4	12	no	—
17	C₇×C₂²⋊C₄	—	4	6	no	—
18	#399	C₂×C₇	1	4	yes	complemented
19	#396	C₂×C₇	1	8	yes	complemented
20	#379	C₂×C₇	1	3	yes	—
21	C₂³×C₇	—	2	12	no	—
22	C₂³×C₇	—	2	6	no	—
23	C₂³×C₇	—	4	12	no	—
24	C₂³×C₇	—	4	16	no	—
25	C₂³×C₇	C₂×Q₈	1	4	yes	—
26	C₂³×C₇	C₂×Q₈	1	2	yes	—
27	C₂³×C₇	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₇	—	2	24	no	—
29	C₂×C₄×C₇	—	2	12	no	—
30	C₂⁵	C₂²×C₇	1	1	yes	characteristic
31	C₂×C₄⋊C₄	C₂²×C₇	1	12	yes	complemented
32	C₂×C₂²⋊C₄	C₂²×C₇	1	12	yes	—
33	C₂×C₂²⋊C₄	C₂²×C₇	1	6	yes	—
34	#95	C₂²×C₇	1	4	yes	complemented
35	C₂²×C₇	—	2	12	no	—
36	C₂²×C₇	—	2	6	no	—
37	C₂²×C₇	—	4	16	no	—
38	C₂²×C₇	—	4	8	no	—
39	C₂²×C₇	—	4	4	no	—
40	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
41	C₂²×C₇	H(5,𝔽₂)	1	6	yes	—
42	C₄×C₇	—	4	12	no	—
43	C₂⁴	—	4	6	no	—
44	C₂⁴	C₇×Q₈	1	4	yes	—
45	C₂⁴	C₂³×C₇	1	2	yes	—
46	C₂⁴	C₂³×C₇	1	1	yes	characteristic
47	C₂²×C₄	C₂³×C₇	1	12	yes	—
48	C₄⋊C₄	—	4	12	no	—
49	C₂²⋊C₄	—	4	12	no	—
50	C₂²⋊C₄	—	4	6	no	—
51	C₂×C₇	—	4	4	no	—
52	C₂×C₇	—	4	2	no	—
53	C₂×C₇	#543	1	3	yes	—
54	C₂×C₇	#560	1	4	yes	—
55	C₂³	—	2	12	no	—
56	C₂³	—	2	6	no	—
57	C₂³	—	4	12	no	—
58	C₂³	—	4	16	no	—
59	C₂³	C₂×C₇×Q₈	1	4	yes	—
60	C₂³	C₂×C₇×Q₈	1	2	yes	—
61	C₂³	C₂⁴×C₇	1	1	yes	characteristic
62	C₂×C₄	—	2	24	no	—
63	C₂×C₄	—	2	12	no	—
64	C₇	#2842	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	—	2	12	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	4	16	no	—
68	C₂²	—	4	8	no	—
69	C₂²	—	4	4	no	—
70	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
71	C₂²	C₇×H(5,𝔽₂)	1	6	yes	—
72	C₄	—	4	12	no	—
73	C₂	—	4	4	no	—
74	C₂	—	4	2	no	—
75	C₂	C₇×#543	1	3	yes	—
76	C₂	C₇×#560	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂⁵×C₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#399	C₂	semidirect
2. C₇×#396	C₂	semidirect
5. C₂×C₇×C₄⋊C₄	C₂²	semidirect
8. C₇×#95	C₂²	semidirect
9. #2842	C₇	direct (N × H)
18. #399	C₂×C₇	semidirect
19. #396	C₂×C₇	semidirect
31. C₂×C₄⋊C₄	C₂²×C₇	semidirect
34. #95	C₂²×C₇	semidirect
64. C₇	#2842	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×#399	2
C₇×#396	2
C₇×#379	2
#2842	7

Subgroups — C7×#2842 #11742151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2842 #11742151