Subgroups — C₇×#340 #89382

Order 448.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₂²×C₄×C₇	—	2	2	no	—
4	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
5	C₂×C₇×C₈	—	2	2	no	—
6	C₂×C₇×C₈	C₄	1	2	yes	complemented
7	C₇×C₄⋊C₄	—	2	2	no	—
8	C₇×C₄⋊C₄	C₄	1	4	yes	complemented
9	#340	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂³×C₇	D₄	1	1	yes	characteristic
11	C₂×C₄×C₇	—	2	4	no	—
12	C₂×C₄×C₇	—	4	2	no	—
13	C₂×C₄×C₇	—	4	2	no	—
14	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₇	D₄	1	2	yes	—
17	C₂×C₄×C₇	Q₈	1	1	yes	characteristic
18	C₇×C₈	—	2	2	no	—
19	C₂²×C₈	C₂×C₇	1	1	yes	characteristic
20	C₂×C₄⋊C₄	C₂×C₇	1	2	yes	—
21	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
22	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×C₇	D₈	1	1	yes	characteristic
24	C₂²×C₇	SD₁₆	1	2	yes	—
25	C₂²×C₇	Q₁₆	1	1	yes	characteristic
26	C₄×C₇	—	4	4	no	—
27	C₄×C₇	C₄²	1	1	yes	characteristic
28	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₄×C₇	C₄⋊C₄	1	2	yes	—
30	C₂²×C₄	—	2	2	no	—
31	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	C₄×C₇	1	2	yes	complemented
34	C₄⋊C₄	—	2	2	no	—
35	C₄⋊C₄	C₄×C₇	1	4	yes	complemented
36	C₂×C₇	#95	1	1	yes	characteristic
37	C₂×C₇	#102	1	2	yes	—
38	C₂×C₇	#103	1	2	yes	—
39	C₂×C₇	#106	1	1	yes	characteristic
40	C₂×C₇	#107	1	1	yes	characteristic
41	C₂³	C₇×D₄	1	1	yes	characteristic
42	C₂×C₄	—	2	4	no	—
43	C₂×C₄	—	4	2	no	—
44	C₂×C₄	—	4	2	no	—
45	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
46	C₂×C₄	C₂×C₄×C₇	1	2	yes	—
47	C₂×C₄	C₇×D₄	1	2	yes	—
48	C₂×C₄	C₇×Q₈	1	1	yes	characteristic
49	C₈	—	2	2	no	—
50	C₇	#340	1	1	yes	characteristic, direct product factor, Sylow
51	C₂²	C₇×C₂²⋊C₄	1	2	yes	—
52	C₂²	C₇×C₄⋊C₄	1	1	yes	characteristic
53	C₂²	C₇×D₈	1	1	yes	characteristic
54	C₂²	C₇×SD₁₆	1	2	yes	—
55	C₂²	C₇×Q₁₆	1	1	yes	characteristic
56	C₄	—	4	4	no	—
57	C₄	C₄²×C₇	1	1	yes	characteristic
58	C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
59	C₄	C₇×C₄⋊C₄	1	2	yes	—
60	C₂	C₇×#95	1	1	yes	characteristic
61	C₂	C₇×#102	1	2	yes	—
62	C₂	C₇×#103	1	2	yes	—
63	C₂	C₇×#106	1	1	yes	characteristic
64	C₂	C₇×#107	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	D₄
commutator	C₄	C₄²×C₇
Frattini, cosocle	C₂²×C₄	C₂²×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₂×C₇×C₈	C₄	semidirect
8. C₇×C₄⋊C₄	C₄	semidirect
9. #340	C₇	direct (N × H)
33. C₂×C₈	C₄×C₇	semidirect
35. C₄⋊C₄	C₄×C₇	semidirect
50. C₇	#340	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₈	2
C₂×C₇×C₄⋊C₄	2
#340	7

Subgroups — C7×#340 #89382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#340 #89382