Subgroups — C₇×#1830 #11730299

Order 896.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#519	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#101	C₂	1	2	yes	maximal
3	C₇×#344	C₂	1	4	yes	maximal
4	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
5	C₂×C₇×M₁₆	C₂²	1	4	yes	—
6	C₄×C₇×Q₈	—	2	4	no	—
7	C₇×#117	C₄	1	4	yes	—
8	C₇×#117	C₂²	1	2	yes	—
9	C₇×#101	—	2	4	no	—
10	#1830	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₇×Q₈	—	2	4	no	—
12	C₂×C₇×Q₈	D₄	1	4	yes	—
13	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
15	C₇×M₁₆	—	2	4	no	—
16	C₇×M₁₆	—	2	4	no	—
17	C₂×C₇×C₈	—	2	4	no	—
18	C₇×C₄⋊C₄	—	2	4	no	—
19	C₇×C₄⋊C₄	—	2	2	no	—
20	C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
21	C₄²×C₇	—	2	4	no	—
22	C₄²×C₇	—	2	2	no	—
23	#519	C₂×C₇	1	1	yes	characteristic
24	C₂×#101	C₂×C₇	1	2	yes	—
25	#344	C₂×C₇	1	4	yes	—
26	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
27	C₇×Q₈	—	2	8	no	—
28	C₂×C₄×C₇	—	2	4	no	—
29	C₂×C₄×C₇	—	2	4	no	—
30	C₂×C₄×C₇	C₂²⋊C₄	1	4	yes	—
31	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₇	#41	1	4	yes	—
33	C₇×C₈	—	4	4	no	—
34	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
35	C₂×M₁₆	C₂²×C₇	1	4	yes	—
36	C₄×Q₈	—	2	4	no	—
37	#117	C₄×C₇	1	4	yes	—
38	#117	C₂²×C₇	1	2	yes	—
39	#101	—	2	4	no	—
40	C₂²×C₇	—	4	1	no	—
41	C₂²×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₇	#117	1	2	yes	—
43	C₄×C₇	—	2	4	no	—
44	C₄×C₇	—	4	4	no	—
45	C₄×C₇	#123	1	4	yes	—
46	C₂×Q₈	—	2	4	no	—
47	C₂×Q₈	C₇×D₄	1	4	yes	—
48	C₂²×C₄	C₂×C₄×C₇	1	2	yes	—
49	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
50	M₁₆	—	2	4	no	—
51	M₁₆	—	2	4	no	—
52	C₂×C₈	—	2	4	no	—
53	C₄⋊C₄	—	2	4	no	—
54	C₄⋊C₄	—	2	2	no	—
55	C₂²⋊C₄	C₂×C₄×C₇	1	4	yes	—
56	C₄²	—	2	4	no	—
57	C₄²	—	2	2	no	—
58	C₂×C₇	—	2	1	no	—
59	C₂×C₇	—	2	2	no	—
60	C₂×C₇	#381	1	1	yes	characteristic
61	C₂³	C₂²×C₄×C₇	1	1	yes	characteristic
62	Q₈	—	2	8	no	—
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	C₇×C₂²⋊C₄	1	4	yes	—
66	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
67	C₂×C₄	C₇×#41	1	4	yes	—
68	C₈	—	4	4	no	—
69	C₇	#1830	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²	—	4	1	no	—
71	C₂²	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²	C₇×#117	1	2	yes	—
73	C₄	—	2	4	no	—
74	C₄	—	4	4	no	—
75	C₄	C₇×#123	1	4	yes	—
76	C₂	—	2	1	no	—
77	C₂	—	2	2	no	—
78	C₂	C₇×#381	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#381
commutator	C₂³	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂³×C₇
prime radical	C₂³×C₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ 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
10. #1830	C₇	direct (N × H)
69. C₇	#1830	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×#519	2
C₂×C₇×#101	2
C₇×#344	2
#1830	7

Subgroups — C7×#1830 #11730299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#1830 #11730299