Subgroups — C₇×#1829 #11730298

Order 896.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#518	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#100	C₂	1	2	yes	maximal
3	C₇×#344	C₂	1	4	yes	maximal, complemented
4	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
5	C₂×C₇×M₁₆	C₂²	1	4	yes	—
6	C₄×C₇×D₄	—	2	4	no	—
7	C₇×#117	C₂²	1	2	yes	—
8	C₂×C₇×C₂²⋊C₄	C₄	1	4	yes	—
9	C₇×#100	—	2	4	no	—
10	#1829	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂⁴×C₇	C₂×C₄	1	2	yes	—
12	C₂×C₇×D₄	—	2	4	no	—
13	C₂×C₇×D₄	D₄	1	4	yes	—
14	C₂²×C₄×C₇	—	2	4	no	—
15	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
16	C₇×M₁₆	—	2	4	no	—
17	C₇×M₁₆	—	2	4	no	—
18	C₂×C₇×C₈	—	2	4	no	—
19	C₇×C₄⋊C₄	—	2	2	no	—
20	C₇×C₂²⋊C₄	—	2	4	no	—
21	C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
22	C₄²×C₇	—	2	2	no	—
23	#518	C₂×C₇	1	1	yes	characteristic
24	C₂×#100	C₂×C₇	1	2	yes	—
25	#344	C₂×C₇	1	4	yes	complemented
26	C₂³×C₇	—	2	4	no	—
27	C₂³×C₇	—	8	2	no	—
28	C₂³×C₇	C₂²⋊C₄	1	4	yes	—
29	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
30	C₇×D₄	—	2	8	no	—
31	C₂×C₄×C₇	—	2	4	no	—
32	C₂×C₄×C₇	—	8	4	no	—
33	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₇	#41	1	4	yes	—
35	C₇×C₈	—	4	4	no	—
36	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
37	C₂×M₁₆	C₂²×C₇	1	4	yes	—
38	C₄×D₄	—	2	4	no	—
39	#117	C₂²×C₇	1	2	yes	—
40	C₂×C₂²⋊C₄	C₄×C₇	1	4	yes	—
41	#100	—	2	4	no	—
42	C₂²×C₇	—	2	4	no	—
43	C₂²×C₇	—	4	1	no	—
44	C₂²×C₇	—	4	4	no	—
45	C₂²×C₇	—	8	4	no	—
46	C₂²×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₇	#117	1	2	yes	—
48	C₄×C₇	—	4	4	no	—
49	C₄×C₇	#123	1	4	yes	—
50	C₂⁴	C₂×C₄×C₇	1	2	yes	—
51	C₂×D₄	—	2	4	no	—
52	C₂×D₄	C₇×D₄	1	4	yes	—
53	C₂²×C₄	—	2	4	no	—
54	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
55	M₁₆	—	2	4	no	—
56	M₁₆	—	2	4	no	—
57	C₂×C₈	—	2	4	no	—
58	C₄⋊C₄	—	2	2	no	—
59	C₂²⋊C₄	—	2	4	no	—
60	C₂²⋊C₄	C₂×C₄×C₇	1	4	yes	—
61	C₄²	—	2	2	no	—
62	C₂×C₇	—	2	1	no	—
63	C₂×C₇	—	2	2	no	—
64	C₂×C₇	—	4	4	no	—
65	C₂×C₇	#381	1	1	yes	characteristic
66	C₂³	—	2	4	no	—
67	C₂³	—	8	2	no	—
68	C₂³	C₇×C₂²⋊C₄	1	4	yes	—
69	C₂³	C₂²×C₄×C₇	1	1	yes	characteristic
70	D₄	—	2	8	no	—
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	8	4	no	—
73	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
74	C₂×C₄	C₇×#41	1	4	yes	—
75	C₈	—	4	4	no	—
76	C₇	#1829	1	1	yes	characteristic, direct product factor, Sylow
77	C₂²	—	2	4	no	—
78	C₂²	—	4	1	no	—
79	C₂²	—	4	4	no	—
80	C₂²	—	8	4	no	—
81	C₂²	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
82	C₂²	C₇×#117	1	2	yes	—
83	C₄	—	4	4	no	—
84	C₄	C₇×#123	1	4	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	2	2	no	—
87	C₂	—	4	4	no	—
88	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	C₂²×C₇×D₄	C₂²
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
3. C₇×#344	C₂	semidirect
10. #1829	C₇	direct (N × H)
25. #344	C₂×C₇	semidirect
76. C₇	#1829	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×#518	2
C₂×C₇×#100	2
C₇×#344	2
#1829	7

Subgroups — C7×#1829 #11730298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#1829 #11730298