Subgroups — C₇×#343 #89385

Order 448.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₇	C₄	1	2	yes	—
4	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
5	C₇×M₁₆	—	2	2	no	—
6	C₇×M₁₆	C₄	1	4	yes	complemented
7	C₂×C₇×C₈	—	2	2	no	—
8	C₇×C₄⋊C₄	—	2	2	no	—
9	#343	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×C₇	—	2	2	no	—
12	C₂×C₄×C₇	—	4	2	no	—
13	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₇	D₄	1	1	yes	characteristic
15	C₂×C₄×C₇	D₄	1	2	yes	—
16	C₂×C₄×C₇	Q₈	1	1	yes	characteristic
17	C₇×C₈	—	2	4	no	—
18	C₂×M₁₆	C₂×C₇	1	2	yes	—
19	C₂×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
20	C₂²×C₇	—	2	1	no	—
21	C₂²×C₇	—	2	1	no	—
22	C₂²×C₇	C₄²	1	1	yes	characteristic
23	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
25	C₄×C₇	—	4	2	no	—
26	C₄×C₇	C₂²⋊C₄	1	2	yes	—
27	C₄×C₇	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄	C₄×C₇	1	2	yes	—
29	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
30	M₁₆	—	2	2	no	—
31	M₁₆	C₄×C₇	1	4	yes	complemented
32	C₂×C₈	—	2	2	no	—
33	C₄⋊C₄	—	2	2	no	—
34	C₂×C₇	—	2	1	no	—
35	C₂×C₇	—	2	1	no	—
36	C₂×C₇	#95	1	1	yes	characteristic
37	C₂×C₇	#100	1	1	yes	characteristic
38	C₂×C₇	#101	1	1	yes	characteristic
39	C₂³	C₂×C₄×C₇	1	1	yes	characteristic
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	4	2	no	—
42	C₂×C₄	C₂×C₄×C₇	1	2	yes	—
43	C₂×C₄	C₇×D₄	1	1	yes	characteristic
44	C₂×C₄	C₇×D₄	1	2	yes	—
45	C₂×C₄	C₇×Q₈	1	1	yes	characteristic
46	C₈	—	2	4	no	—
47	C₇	#343	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	C₄²×C₇	1	1	yes	characteristic
51	C₂²	C₇×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²	C₇×C₄⋊C₄	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	C₇×C₂²⋊C₄	1	2	yes	—
55	C₄	C₇×C₄⋊C₄	1	2	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	C₇×#95	1	1	yes	characteristic
59	C₂	C₇×#100	1	1	yes	characteristic
60	C₂	C₇×#101	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²⋊C₄
commutator	C₂²	C₄²×C₇
Frattini, cosocle	C₂²×C₄	C₂²×C₇
prime radical	C₂³×C₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₇×M₁₆	C₄	semidirect
9. #343	C₇	direct (N × H)
31. M₁₆	C₄×C₇	semidirect
47. C₇	#343	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×M₁₆	2
C₂×C₇×C₄⋊C₄	2
#343	7

Subgroups — C7×#343 #89385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#343 #89385