Subgroups — C₇×#2868 #11742177

Order 896.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊Q₈	C₂	1	6	yes	maximal
2	C₇×#384	C₂	1	8	yes	maximal
3	C₄³×C₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇×Q₈	C₂²	1	4	yes	—
5	C₇×C₄⋊Q₈	—	2	24	no	—
6	C₂×C₇×C₄⋊C₄	C₂²	1	24	yes	—
7	C₂×C₄²×C₇	C₂²	1	3	yes	—
8	C₂×C₄²×C₇	C₂²	1	4	yes	—
9	#2868	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₇×Q₈	—	2	24	no	—
11	C₂²×C₄×C₇	C₂³	1	8	yes	—
12	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
13	C₂²×C₄×C₇	C₂³	1	6	yes	—
14	C₇×C₄⋊C₄	—	2	48	no	—
15	C₄²×C₇	D₄	1	12	yes	—
16	C₄²×C₇	Q₈	1	16	yes	complemented
17	C₂×C₄⋊Q₈	C₂×C₇	1	6	yes	—
18	#384	C₂×C₇	1	8	yes	—
19	C₄³	C₂×C₇	1	1	yes	characteristic
20	C₂³×C₇	C₂⁴	1	1	yes	characteristic
21	C₇×Q₈	—	4	16	no	—
22	C₂×C₄×C₇	—	2	24	no	—
23	C₂×C₄×C₇	C₂×D₄	1	6	yes	—
24	C₂×C₄×C₇	C₂×D₄	1	12	yes	—
25	C₂×C₄×C₇	C₂×Q₈	1	24	yes	—
26	C₂²×Q₈	C₂²×C₇	1	4	yes	—
27	C₄⋊Q₈	—	2	24	no	—
28	C₂×C₄⋊C₄	C₂²×C₇	1	24	yes	—
29	C₂×C₄²	C₂²×C₇	1	3	yes	—
30	C₂×C₄²	C₂²×C₇	1	4	yes	—
31	C₂²×C₇	C₂²×D₄	1	3	yes	—
32	C₂²×C₇	C₂²×Q₈	1	4	yes	—
33	C₄×C₇	—	4	8	no	—
34	C₄×C₇	#127	1	4	yes	—
35	C₄×C₇	C₄⋊Q₈	1	24	yes	complemented
36	C₂×Q₈	—	2	24	no	—
37	C₂²×C₄	C₂³×C₇	1	8	yes	—
38	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
39	C₂²×C₄	C₂³×C₇	1	6	yes	—
40	C₄⋊C₄	—	2	48	no	—
41	C₄²	C₇×D₄	1	12	yes	—
42	C₄²	C₇×Q₈	1	16	yes	complemented
43	C₂×C₇	C₂×#127	1	1	yes	characteristic
44	C₂×C₇	C₂×C₄⋊Q₈	1	6	yes	—
45	C₂³	C₂⁴×C₇	1	1	yes	characteristic
46	Q₈	—	4	16	no	—
47	C₂×C₄	—	2	24	no	—
48	C₂×C₄	C₂×C₇×D₄	1	6	yes	—
49	C₂×C₄	C₂×C₇×D₄	1	12	yes	—
50	C₂×C₄	C₂×C₇×Q₈	1	24	yes	—
51	C₇	#2868	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²	C₂²×C₇×D₄	1	3	yes	—
53	C₂²	C₂²×C₇×Q₈	1	4	yes	—
54	C₄	—	4	8	no	—
55	C₄	C₇×#127	1	4	yes	—
56	C₄	C₇×C₄⋊Q₈	1	24	yes	complemented
57	C₂	C₂×C₇×#127	1	1	yes	characteristic
58	C₂	C₂×C₇×C₄⋊Q₈	1	6	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
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
9. #2868	C₇	direct (N × H)
16. C₄²×C₇	Q₈	semidirect
35. C₄×C₇	C₄⋊Q₈	semidirect
42. C₄²	C₇×Q₈	semidirect
51. C₇	#2868	direct (N × H)
56. C₄	C₇×C₄⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₄⋊Q₈	2
C₇×#384	2
C₄³×C₇	2
#2868	7

Subgroups — C7×#2868 #11742177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2868 #11742177