Subgroups — C₇²×#135 #13096615

Order 1568.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×#41	C₂	1	2	yes	maximal, complemented
2	C₇²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇²×SD₁₆	C₂	1	2	yes	maximal, complemented
4	C₇²×D₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇²×Q₈	C₂²	1	2	yes	—
7	C₇²×D₄	—	2	2	no	—
8	C₇²×D₄	C₂²	1	2	yes	—
9	C₂×C₄×C₇²	—	2	2	no	—
10	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
11	C₇²×C₈	C₂²	1	1	yes	characteristic
12	C₇²×C₈	C₂²	1	1	yes	characteristic
13	C₇×#135	C₇	1	8	yes	maximal, direct product factor
14	C₂²×C₇²	—	2	2	no	—
15	C₂²×C₇²	D₄	1	1	yes	characteristic
16	C₄×C₇²	—	2	2	no	—
17	C₄×C₇²	D₄	1	1	yes	characteristic
18	C₄×C₇²	C₂³	1	1	yes	characteristic
19	C₇×#41	C₂×C₇	1	16	yes	complemented
20	C₇×Q₁₆	C₂×C₇	1	8	yes	complemented
21	C₇×SD₁₆	C₂×C₇	1	16	yes	complemented
22	C₇×D₈	C₂×C₇	1	8	yes	complemented
23	C₂×C₇×C₈	C₂×C₇	1	8	yes	complemented
24	C₂×C₇²	—	2	1	no	—
25	C₂×C₇²	—	4	2	no	—
26	C₂×C₇²	C₂×D₄	1	1	yes	characteristic
27	C₇×Q₈	C₂²×C₇	1	16	yes	—
28	C₇×D₄	—	2	16	no	—
29	C₇×D₄	C₂²×C₇	1	16	yes	—
30	C₂×C₄×C₇	—	2	16	no	—
31	C₂×C₄×C₇	C₂²×C₇	1	8	yes	—
32	C₇×C₈	C₂²×C₇	1	8	yes	—
33	C₇×C₈	C₂²×C₇	1	8	yes	—
34	C₇²	#135	1	1	yes	characteristic, direct product factor, Sylow
35	#135	C₇²	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²×C₇	—	2	16	no	—
37	C₂²×C₇	C₇×D₄	1	8	yes	—
38	C₄×C₇	—	2	16	no	—
39	C₄×C₇	C₇×D₄	1	8	yes	—
40	C₄×C₇	C₂³×C₇	1	8	yes	—
41	#41	C₂×C₇²	1	2	yes	complemented
42	Q₁₆	C₂×C₇²	1	1	yes	characteristic, complemented
43	SD₁₆	C₂×C₇²	1	2	yes	complemented
44	D₈	C₂×C₇²	1	1	yes	characteristic, complemented
45	C₂×C₈	C₂×C₇²	1	1	yes	characteristic, complemented
46	C₂×C₇	—	2	8	no	—
47	C₂×C₇	—	4	16	no	—
48	C₂×C₇	C₂×C₇×D₄	1	8	yes	—
49	Q₈	C₂²×C₇²	1	2	yes	—
50	D₄	—	2	2	no	—
51	D₄	C₂²×C₇²	1	2	yes	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	C₂²×C₇²	1	1	yes	characteristic
54	C₈	C₂²×C₇²	1	1	yes	characteristic
55	C₈	C₂²×C₇²	1	1	yes	characteristic
56	C₇	C₇×#135	1	8	yes	direct product factor
57	C₂²	—	2	2	no	—
58	C₂²	C₇²×D₄	1	1	yes	characteristic
59	C₄	—	2	2	no	—
60	C₄	C₇²×D₄	1	1	yes	characteristic
61	C₄	C₂³×C₇²	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	4	2	no	—
64	C₂	C₂×C₇²×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇²	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₇²
socle	C₂×C₇²	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇² ◁ C₂×C₄×C₇² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇²×#41	C₂	semidirect
2. C₇²×Q₁₆	C₂	semidirect
3. C₇²×SD₁₆	C₂	semidirect
4. C₇²×D₈	C₂	semidirect
5. C₂×C₇²×C₈	C₂	semidirect
13. C₇×#135	C₇	direct (N × H)
19. C₇×#41	C₂×C₇	semidirect
20. C₇×Q₁₆	C₂×C₇	semidirect
21. C₇×SD₁₆	C₂×C₇	semidirect
22. C₇×D₈	C₂×C₇	semidirect
23. C₂×C₇×C₈	C₂×C₇	semidirect
34. C₇²	#135	direct (N × H)
35. #135	C₇²	direct (N × H)
41. #41	C₂×C₇²	semidirect
42. Q₁₆	C₂×C₇²	semidirect
43. SD₁₆	C₂×C₇²	semidirect
44. D₈	C₂×C₇²	semidirect
45. C₂×C₈	C₂×C₇²	semidirect
56. C₇	C₇×#135	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇²×#41	2
C₇²×Q₁₆	2
C₇²×SD₁₆	2
C₇²×D₈	2
C₂×C₇²×C₈	2
C₇×#135	7

Subgroups — C72×#135 #13096615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇²×#135 #13096615