Subgroups — C₇×#393 #90039

Order 448.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₇×#95	C₂	1	3	yes	maximal, complemented
4	C₂⁴×C₇	C₂²	1	1	yes	characteristic
5	C₂×C₇×Q₈	—	2	6	no	—
6	C₂²×C₄×C₇	C₂²	1	3	yes	—
7	C₂²×C₄×C₇	C₂²	1	3	yes	—
8	C₇×C₂²⋊C₄	—	2	6	no	—
9	#393	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂³×C₇	—	2	3	no	—
11	C₂³×C₇	—	2	3	no	—
12	C₂³×C₇	Q₈	1	2	yes	complemented
13	C₂³×C₇	C₂³	1	1	yes	characteristic
14	C₇×Q₈	—	2	8	no	—
15	C₂×C₄×C₇	—	2	6	no	—
16	C₂×C₄×C₇	—	2	3	no	—
17	C₂×C₄×C₇	—	2	6	no	—
18	C₂×C₄×C₇	D₄	1	6	yes	—
19	C₂²×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
20	C₂×C₂²⋊C₄	C₂×C₇	1	3	yes	—
21	#95	C₂×C₇	1	3	yes	complemented
22	C₂²×C₇	—	2	6	no	—
23	C₂²×C₇	—	4	1	no	—
24	C₂²×C₇	—	4	3	no	—
25	C₂²×C₇	C₂×D₄	1	3	yes	—
26	C₂²×C₇	C₂×Q₈	1	1	yes	characteristic
27	C₂²×C₇	#41	1	3	yes	—
28	C₄×C₇	—	2	6	no	—
29	C₄×C₇	—	4	3	no	—
30	C₂⁴	C₂²×C₇	1	1	yes	characteristic
31	C₂×Q₈	—	2	6	no	—
32	C₂²×C₄	C₂²×C₇	1	3	yes	—
33	C₂²×C₄	C₂²×C₇	1	3	yes	—
34	C₂²⋊C₄	—	2	6	no	—
35	C₂×C₇	—	4	2	no	—
36	C₂×C₇	#120	1	1	yes	characteristic
37	C₂×C₇	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₇	#124	1	3	yes	—
39	C₂³	—	2	3	no	—
40	C₂³	—	2	3	no	—
41	C₂³	C₇×Q₈	1	2	yes	complemented
42	C₂³	C₂³×C₇	1	1	yes	characteristic
43	Q₈	—	2	8	no	—
44	C₂×C₄	—	2	6	no	—
45	C₂×C₄	—	2	3	no	—
46	C₂×C₄	—	2	6	no	—
47	C₂×C₄	C₇×D₄	1	6	yes	—
48	C₇	#393	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²	—	2	6	no	—
50	C₂²	—	4	1	no	—
51	C₂²	—	4	3	no	—
52	C₂²	C₂×C₇×D₄	1	3	yes	—
53	C₂²	C₂×C₇×Q₈	1	1	yes	characteristic
54	C₂²	C₇×#41	1	3	yes	—
55	C₄	—	2	6	no	—
56	C₄	—	4	3	no	—
57	C₂	—	4	2	no	—
58	C₂	C₇×#120	1	1	yes	characteristic
59	C₂	C₇×C₂²⋊Q₈	1	3	yes	—
60	C₂	C₇×#124	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂³
commutator, Frattini, cosocle	C₂³	C₂³×C₇
prime radical	C₂⁴×C₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×Q₈	C₂	semidirect
3. C₇×#95	C₂	semidirect
9. #393	C₇	direct (N × H)
12. C₂³×C₇	Q₈	semidirect
19. C₂²×Q₈	C₂×C₇	semidirect
21. #95	C₂×C₇	semidirect
41. C₂³	C₇×Q₈	semidirect
48. C₇	#393	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×Q₈	2
C₂×C₇×C₂²⋊C₄	2
C₇×#95	2
#393	7

Subgroups — C7×#393 #90039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#393 #90039