Subgroups — C₇×#2070 #11730539

Order 896.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#430	C₂	1	3	yes	maximal
2	C₂×C₇×#101	C₂	1	3	yes	maximal
3	C₇×#338	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
5	C₂×C₇×M₁₆	C₂²	1	3	yes	—
6	C₂×C₇×M₁₆	C₂²	1	3	yes	—
7	C₇×#108	—	2	6	no	—
8	C₇×#101	—	2	6	no	—
9	#2070	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₇×Q₈	—	2	3	no	—
11	C₂×C₇×Q₈	—	2	3	no	—
12	C₂²×C₄×C₇	Q₈	1	2	yes	—
13	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
14	C₇×M₁₆	—	2	3	no	—
15	C₇×M₁₆	—	2	3	no	—
16	C₇×M₁₆	—	2	3	no	—
17	C₇×M₁₆	D₄	1	6	yes	—
18	C₂×C₇×C₈	—	2	3	no	—
19	C₂×C₇×C₈	—	2	3	no	—
20	#430	C₂×C₇	1	3	yes	—
21	C₂×#101	C₂×C₇	1	3	yes	—
22	#338	C₂×C₇	1	1	yes	characteristic
23	C₂³×C₇	C₂×Q₈	1	1	yes	characteristic
24	C₇×Q₈	—	4	3	no	—
25	C₇×Q₈	—	4	1	no	—
26	C₂×C₄×C₇	—	2	6	no	—
27	C₂×C₄×C₇	C₂×D₄	1	3	yes	—
28	C₂×C₄×C₇	#41	1	3	yes	—
29	C₇×C₈	—	2	6	no	—
30	C₇×C₈	—	4	3	no	—
31	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
32	C₂×M₁₆	C₂²×C₇	1	3	yes	—
33	C₂×M₁₆	C₂²×C₇	1	3	yes	—
34	#108	—	2	6	no	—
35	#101	—	2	6	no	—
36	C₂²×C₇	—	4	1	no	—
37	C₂²×C₇	C₂²⋊Q₈	1	3	yes	—
38	C₄×C₇	—	4	2	no	—
39	C₄×C₇	#121	1	3	yes	—
40	C₄×C₇	#126	1	1	yes	characteristic
41	C₂×Q₈	—	2	3	no	—
42	C₂×Q₈	—	2	3	no	—
43	C₂²×C₄	C₇×Q₈	1	2	yes	—
44	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
45	M₁₆	—	2	3	no	—
46	M₁₆	—	2	3	no	—
47	M₁₆	—	2	3	no	—
48	M₁₆	C₇×D₄	1	6	yes	—
49	C₂×C₈	—	2	3	no	—
50	C₂×C₈	—	2	3	no	—
51	C₂×C₇	—	2	3	no	—
52	C₂×C₇	#396	1	1	yes	characteristic
53	C₂³	C₂×C₇×Q₈	1	1	yes	characteristic
54	Q₈	—	4	3	no	—
55	Q₈	—	4	1	no	—
56	C₂×C₄	—	2	6	no	—
57	C₂×C₄	C₂×C₇×D₄	1	3	yes	—
58	C₂×C₄	C₇×#41	1	3	yes	—
59	C₈	—	2	6	no	—
60	C₈	—	4	3	no	—
61	C₇	#2070	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²	—	4	1	no	—
63	C₂²	C₇×C₂²⋊Q₈	1	3	yes	—
64	C₄	—	4	2	no	—
65	C₄	C₇×#121	1	3	yes	—
66	C₄	C₇×#126	1	1	yes	characteristic
67	C₂	—	2	3	no	—
68	C₂	C₇×#396	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#396
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₇
prime radical	C₂³×C₇	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
9. #2070	C₇	direct (N × H)
61. C₇	#2070	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×#430	2
C₂×C₇×#101	2
C₇×#338	2
#2070	7

Subgroups — C7×#2070 #11730539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2070 #11730539