Subgroups — C₇×#2844 #11742153

Order 896.

356 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#400	C₂	1	6	yes	maximal
2	C₇×#395	C₂	1	6	yes	maximal
3	C₇×#382	C₂	1	3	yes	maximal
4	C₂²×C₇×Q₈	C₂²	1	2	yes	—
5	C₂×C₇×C₄⋊C₄	C₂²	1	3	yes	—
6	C₂×C₇×C₄⋊C₄	C₂²	1	3	yes	—
7	C₂×C₇×C₄⋊C₄	C₂²	1	6	yes	—
8	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₇×#95	C₂²	1	12	yes	—
10	C₇×#95	C₂²	1	6	yes	—
11	C₇×#95	C₂²	1	2	yes	—
12	#2844	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₇×Q₈	—	4	6	no	—
14	C₂²×C₄×C₇	C₂³	1	6	yes	—
15	C₂²×C₄×C₇	C₂³	1	6	yes	—
16	C₂²×C₄×C₇	C₂³	1	3	yes	—
17	C₇×C₄⋊C₄	—	4	3	no	—
18	C₇×C₄⋊C₄	—	4	3	no	—
19	C₇×C₄⋊C₄	—	4	6	no	—
20	C₄²×C₇	Q₈	1	4	yes	complemented
21	#400	C₂×C₇	1	6	yes	—
22	#395	C₂×C₇	1	6	yes	—
23	#382	C₂×C₇	1	3	yes	—
24	C₂³×C₇	C₂⁴	1	1	yes	characteristic
25	C₇×Q₈	—	4	8	no	—
26	C₂×C₄×C₇	—	2	3	no	—
27	C₂×C₄×C₇	—	2	3	no	—
28	C₂×C₄×C₇	—	2	6	no	—
29	C₂×C₄×C₇	—	2	6	no	—
30	C₂×C₄×C₇	—	2	12	no	—
31	C₂×C₄×C₇	—	2	12	no	—
32	C₂×C₄×C₇	C₂×Q₈	1	6	yes	—
33	C₂²×Q₈	C₂²×C₇	1	2	yes	—
34	C₂×C₄⋊C₄	C₂²×C₇	1	3	yes	—
35	C₂×C₄⋊C₄	C₂²×C₇	1	3	yes	—
36	C₂×C₄⋊C₄	C₂²×C₇	1	6	yes	—
37	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
38	#95	C₂²×C₇	1	12	yes	—
39	#95	C₂²×C₇	1	6	yes	—
40	#95	C₂²×C₇	1	2	yes	—
41	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
42	C₂²×C₇	H(5,𝔽₂)	1	3	yes	—
43	C₂²×C₇	2₋¹⁺⁴	1	3	yes	—
44	C₄×C₇	—	2	6	no	—
45	C₄×C₇	—	4	6	no	—
46	C₄×C₇	—	4	6	no	—
47	C₂×Q₈	—	4	6	no	—
48	C₂²×C₄	C₂³×C₇	1	6	yes	—
49	C₂²×C₄	C₂³×C₇	1	6	yes	—
50	C₂²×C₄	C₂³×C₇	1	3	yes	—
51	C₄⋊C₄	—	4	3	no	—
52	C₄⋊C₄	—	4	3	no	—
53	C₄⋊C₄	—	4	6	no	—
54	C₄²	C₇×Q₈	1	4	yes	complemented
55	C₂×C₇	#544	1	3	yes	—
56	C₂×C₇	H(3,𝔽₄)	1	1	yes	characteristic
57	C₂×C₇	#563	1	3	yes	—
58	C₂³	C₂⁴×C₇	1	1	yes	characteristic
59	Q₈	—	4	8	no	—
60	C₂×C₄	—	2	3	no	—
61	C₂×C₄	—	2	3	no	—
62	C₂×C₄	—	2	6	no	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	—	2	12	no	—
65	C₂×C₄	—	2	12	no	—
66	C₂×C₄	C₂×C₇×Q₈	1	6	yes	—
67	C₇	#2844	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
69	C₂²	C₇×H(5,𝔽₂)	1	3	yes	—
70	C₂²	C₇×2₋¹⁺⁴	1	3	yes	—
71	C₄	—	2	6	no	—
72	C₄	—	4	6	no	—
73	C₄	—	4	6	no	—
74	C₂	C₇×#544	1	3	yes	—
75	C₂	C₇×H(3,𝔽₄)	1	1	yes	characteristic
76	C₂	C₇×#563	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
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
12. #2844	C₇	direct (N × H)
20. C₄²×C₇	Q₈	semidirect
54. C₄²	C₇×Q₈	semidirect
67. C₇	#2844	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×#400	2
C₇×#395	2
C₇×#382	2
#2844	7

Subgroups — C7×#2844 #11742153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2844 #11742153