Subgroups — C₇×#7531 #13184747

Order 1792.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1709	C₂	1	1	yes	maximal, characteristic
2	C₇×#1411	C₂	1	2	yes	maximal, complemented
3	C₇×#494	C₂²	1	1	yes	characteristic
4	C₇×#493	C₄	1	1	yes	characteristic
5	C₇×#486	—	2	2	no	—
6	C₇×#356	—	2	2	no	—
7	C₇×C₈²	C₄	1	1	yes	characteristic
8	#7531	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₇×Q₁₆	—	4	1	no	—
10	C₂×C₇×D₈	—	4	1	no	—
11	C₇×C₄⋊Q₈	D₄	1	2	yes	—
12	C₇×#127	—	2	1	no	—
13	C₇×#102	—	4	4	no	—
14	C₇×#101	—	4	2	no	—
15	C₄×C₇×C₈	—	2	1	no	—
16	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	#1709	C₂×C₇	1	1	yes	characteristic
18	#1411	C₂×C₇	1	2	yes	complemented
19	C₂×C₇×Q₈	—	2	2	no	—
20	C₂×C₇×D₄	—	4	2	no	—
21	C₂×C₇×D₄	—	4	1	no	—
22	C₇×Q₁₆	—	8	2	no	—
23	C₇×D₈	—	8	2	no	—
24	C₇×M₁₆	—	8	2	no	—
25	C₂×C₇×C₈	—	2	1	no	—
26	C₂×C₇×C₈	—	2	2	no	—
27	C₇×C₄⋊C₄	—	4	2	no	—
28	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
29	#494	C₂²×C₇	1	1	yes	characteristic
30	#493	C₄×C₇	1	1	yes	characteristic
31	#486	—	2	2	no	—
32	#356	—	2	2	no	—
33	C₈²	C₄×C₇	1	1	yes	characteristic
34	C₂³×C₇	—	8	1	no	—
35	C₇×Q₈	—	8	2	no	—
36	C₇×D₄	—	8	2	no	—
37	C₇×D₄	—	8	4	no	—
38	C₂×C₄×C₇	—	2	1	no	—
39	C₂×C₄×C₇	—	4	2	no	—
40	C₂×C₄×C₇	#99	1	1	yes	characteristic
41	C₇×C₈	—	2	2	no	—
42	C₇×C₈	—	4	2	no	—
43	C₇×C₈	—	16	2	no	—
44	C₂×Q₁₆	—	4	1	no	—
45	C₂×D₈	—	4	1	no	—
46	C₄⋊Q₈	C₇×D₄	1	2	yes	—
47	#127	—	2	1	no	—
48	#102	—	4	4	no	—
49	#101	—	4	2	no	—
50	C₄×C₈	—	2	1	no	—
51	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
52	C₂²×C₇	—	16	2	no	—
53	C₂²×C₇	—	16	1	no	—
54	C₂²×C₇	#353	1	1	yes	characteristic
55	C₄×C₇	—	2	2	no	—
56	C₄×C₇	—	2	1	no	—
57	C₄×C₇	—	8	2	no	—
58	C₂×Q₈	—	2	2	no	—
59	C₂×D₄	—	4	2	no	—
60	C₂×D₄	—	4	1	no	—
61	Q₁₆	—	8	2	no	—
62	D₈	—	8	2	no	—
63	M₁₆	—	8	2	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	2	2	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₇	—	2	1	no	—
69	C₂×C₇	—	32	1	no	—
70	C₂×C₇	#1408	1	1	yes	characteristic
71	C₂³	—	8	1	no	—
72	Q₈	—	8	2	no	—
73	D₄	—	8	2	no	—
74	D₄	—	8	4	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	C₇×#99	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₈	—	4	2	no	—
80	C₈	—	16	2	no	—
81	C₇	#7531	1	1	yes	characteristic, direct product factor, Sylow
82	C₂²	—	16	2	no	—
83	C₂²	—	16	1	no	—
84	C₂²	C₇×#353	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	—	2	1	no	—
87	C₄	—	8	2	no	—
88	C₂	—	2	1	no	—
89	C₂	—	32	1	no	—
90	C₂	C₇×#1408	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#1408
commutator	C₄×C₈	C₂×C₄×C₇
Frattini, cosocle	#494	C₂²×C₇
prime radical	C₇×#493	C₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₄²×C₇ ◁ C₇×#494 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#1411	C₂	semidirect
8. #7531	C₇	direct (N × H)
18. #1411	C₂×C₇	semidirect
81. C₇	#7531	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×#1709	2
C₇×#1411	2
#7531	7

Subgroups — C7×#7531 #13184747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7531 #13184747