Subgroups — #3584

Order 128.

712 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#585	C₂	1	2	yes	maximal, complemented
2	#578	C₂	1	4	yes	maximal, complemented
3	#577	C₂	1	8	yes	maximal, complemented
4	#576	C₂	1	4	yes	maximal, complemented
5	#575	C₂	1	6	yes	maximal, complemented
6	C₂×#135	C₂	1	6	yes	maximal, complemented
7	C₂×#131	C₂	1	1	yes	maximal, characteristic, complemented
8	2₋¹⁺⁴	—	2	2	no	—
9	2₋¹⁺⁴	C₂²	1	8	yes	complemented
10	H(5,𝔽₂)	—	2	6	no	—
11	H(5,𝔽₂)	C₂²	1	8	yes	complemented
12	C₂×#41	—	2	8	no	—
13	C₂×#41	C₂²	1	1	yes	characteristic
14	C₂×#41	C₂²	1	12	yes	complemented
15	#137	C₂²	1	24	yes	complemented
16	AGL(1,ℤ/8)	C₂²	1	24	yes	complemented
17	#135	C₂²	1	4	yes	complemented
18	#135	C₂²	1	12	yes	complemented
19	#135	C₂²	1	12	yes	complemented
20	#135	C₂²	1	12	yes	complemented
21	C₂×Q₁₆	C₂²	1	6	yes	complemented
22	C₂×SD₁₆	C₂²	1	12	yes	complemented
23	C₂×D₈	C₂²	1	6	yes	complemented
24	#131	C₂²	1	4	yes	complemented
25	#131	C₂²	1	4	yes	complemented
26	C₂×M₁₆	C₂²	1	3	yes	complemented
27	C₂²×C₈	C₂²	1	3	yes	complemented
28	#41	—	2	8	no	—
29	#41	—	2	24	no	—
30	#41	—	2	12	no	—
31	#41	—	2	12	no	—
32	#41	D₄	1	4	yes	—
33	#41	C₂³	1	8	yes	—
34	#41	C₂³	1	24	yes	—
35	#41	C₂³	1	4	yes	—
36	C₂×Q₈	—	2	8	no	—
37	C₂×Q₈	D₄	1	1	yes	characteristic
38	C₂×Q₈	C₂³	1	12	yes	—
39	C₂×D₄	—	2	12	no	—
40	C₂×D₄	—	2	24	no	—
41	C₂×D₄	D₄	1	3	yes	—
42	C₂×D₄	C₂³	1	12	yes	—
43	C₂²×C₄	—	2	12	no	—
44	C₂²×C₄	C₂³	1	3	yes	—
45	Q₁₆	C₂³	1	4	yes	—
46	Q₁₆	C₂³	1	12	yes	—
47	SD₁₆	C₂³	1	8	yes	—
48	SD₁₆	C₂³	1	24	yes	—
49	D₈	C₂³	1	4	yes	—
50	D₈	C₂³	1	12	yes	—
51	M₁₆	C₂³	1	12	yes	—
52	C₂×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₈	C₂³	1	3	yes	—
54	C₂×C₈	C₂³	1	6	yes	—
55	C₂×C₈	C₂³	1	6	yes	—
56	C₂³	—	2	12	no	—
57	C₂³	C₂×D₄	1	3	yes	—
58	Q₈	—	2	12	no	—
59	Q₈	C₂×D₄	1	4	yes	—
60	Q₈	C₂⁴	1	8	yes	—
61	D₄	—	2	24	no	—
62	D₄	—	2	8	no	—
63	D₄	—	2	12	no	—
64	D₄	C₂×D₄	1	12	yes	—
65	D₄	C₂⁴	1	8	yes	—
66	C₂×C₄	—	2	12	no	—
67	C₂×C₄	—	2	24	no	—
68	C₂×C₄	—	2	8	no	—
69	C₂×C₄	C₂×D₄	1	3	yes	—
70	C₂×C₄	C₂×D₄	1	6	yes	—
71	C₂×C₄	C₂⁴	1	6	yes	—
72	C₂×C₄	C₂⁴	1	1	yes	characteristic
73	C₈	C₂⁴	1	1	yes	characteristic
74	C₈	C₂⁴	1	1	yes	characteristic
75	C₈	C₂⁴	1	6	yes	—
76	C₂²	—	2	8	no	—
77	C₂²	—	4	3	no	—
78	C₂²	—	8	12	no	—
79	C₂²	C₂²×D₄	1	6	yes	—
80	C₂²	C₂²×D₄	1	1	yes	characteristic
81	C₄	—	2	8	no	—
82	C₄	C₂²×D₄	1	6	yes	—
83	C₄	C₂²×D₄	1	1	yes	characteristic
84	C₄	C₂⁵	1	1	yes	characteristic
85	C₂	—	2	1	no	—
86	C₂	—	2	6	no	—
87	C₂	—	4	8	no	—
88	C₂	C₂³×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁵
socle	C₂	C₂³×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #585	C₂	semidirect
2. #578	C₂	semidirect
3. #577	C₂	semidirect
4. #576	C₂	semidirect
5. #575	C₂	semidirect
6. C₂×#135	C₂	semidirect
7. C₂×#131	C₂	semidirect
9. 2₋¹⁺⁴	C₂²	semidirect
11. H(5,𝔽₂)	C₂²	semidirect
14. C₂×#41	C₂²	semidirect
15. #137	C₂²	semidirect
16. AGL(1,ℤ/8)	C₂²	semidirect
17. #135	C₂²	semidirect
18. #135	C₂²	semidirect
19. #135	C₂²	semidirect
20. #135	C₂²	semidirect
21. C₂×Q₁₆	C₂²	semidirect
22. C₂×SD₁₆	C₂²	semidirect
23. C₂×D₈	C₂²	semidirect
24. #131	C₂²	semidirect
25. #131	C₂²	semidirect
26. C₂×M₁₆	C₂²	semidirect
27. C₂²×C₈	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#585	2
#578	2
#577	2
#576	2
#575	2
C₂×#135	2
C₂×#131	2