Subgroups — C₇²×#103 #13096164

Order 1568.

210 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇²×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₇²×Q₈	—	2	1	no	—
5	C₇²×Q₈	C₄	1	2	yes	complemented
6	C₂×C₄×C₇²	—	2	1	no	—
7	C₂×C₄×C₇²	—	2	1	no	—
8	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
9	C₇²×C₈	—	2	1	no	—
10	C₇×#103	C₇	1	8	yes	maximal, direct product factor
11	C₂²×C₇²	D₄	1	1	yes	characteristic
12	C₄×C₇²	—	2	2	no	—
13	C₄×C₇²	—	4	1	no	—
14	C₄×C₇²	C₂×C₄	1	1	yes	characteristic
15	C₄×C₇²	D₄	1	1	yes	characteristic
16	C₂×C₇×Q₈	C₂×C₇	1	8	yes	—
17	C₂×C₇×C₈	C₂×C₇	1	8	yes	—
18	C₇×C₄⋊C₄	C₂×C₇	1	8	yes	—
19	C₂×C₇²	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₇²	SD₁₆	1	1	yes	characteristic
21	C₂×C₇²	Q₁₆	1	1	yes	characteristic
22	C₇×Q₈	—	2	8	no	—
23	C₇×Q₈	C₄×C₇	1	16	yes	complemented
24	C₂×C₄×C₇	—	2	8	no	—
25	C₂×C₄×C₇	—	2	8	no	—
26	C₂×C₄×C₇	C₂²×C₇	1	8	yes	—
27	C₇×C₈	—	2	8	no	—
28	C₇²	#103	1	1	yes	characteristic, direct product factor, Sylow
29	#103	C₇²	1	1	yes	characteristic, direct product factor, Sylow
30	C₂²×C₇	C₇×D₄	1	8	yes	—
31	C₄×C₇	—	2	16	no	—
32	C₄×C₇	—	4	8	no	—
33	C₄×C₇	C₂×C₄×C₇	1	8	yes	—
34	C₄×C₇	C₇×D₄	1	8	yes	—
35	C₂×Q₈	C₂×C₇²	1	1	yes	characteristic
36	C₂×C₈	C₂×C₇²	1	1	yes	characteristic
37	C₄⋊C₄	C₂×C₇²	1	1	yes	characteristic
38	C₂×C₇	C₇×C₂²⋊C₄	1	8	yes	—
39	C₂×C₇	C₇×SD₁₆	1	8	yes	—
40	C₂×C₇	C₇×Q₁₆	1	8	yes	—
41	Q₈	—	2	1	no	—
42	Q₈	C₄×C₇²	1	2	yes	complemented
43	C₂×C₄	—	2	1	no	—
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	C₂²×C₇²	1	1	yes	characteristic
46	C₈	—	2	1	no	—
47	C₇	C₇×#103	1	8	yes	direct product factor
48	C₂²	C₇²×D₄	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	4	1	no	—
51	C₄	C₂×C₄×C₇²	1	1	yes	characteristic
52	C₄	C₇²×D₄	1	1	yes	characteristic
53	C₂	C₇²×C₂²⋊C₄	1	1	yes	characteristic
54	C₂	C₇²×SD₁₆	1	1	yes	characteristic
55	C₂	C₇²×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₇²	D₄
commutator	C₄	C₂×C₄×C₇²
Frattini, cosocle	C₂×C₄	C₂²×C₇²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇² ◁ C₂×C₄×C₇² ◁ G

Complements

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

N	H ≅ G/N	type
5. C₇²×Q₈	C₄	semidirect
10. C₇×#103	C₇	direct (N × H)
23. C₇×Q₈	C₄×C₇	semidirect
28. C₇²	#103	direct (N × H)
29. #103	C₇²	direct (N × H)
42. Q₈	C₄×C₇²	semidirect
47. C₇	C₇×#103	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×C₇²×Q₈	2
C₂×C₇²×C₈	2
C₇²×C₄⋊C₄	2
C₇×#103	7

Subgroups — C72×#103 #13096164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇²×#103 #13096164