Subgroups — #3498

Order 128.

874 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	#550	C₂	1	4	yes	maximal, complemented
3	D₄²	C₂	1	12	yes	maximal, complemented
4	#535	C₂	1	6	yes	maximal, complemented
5	#532	C₂	1	3	yes	maximal, complemented
6	C₂×#127	C₂	1	3	yes	maximal, complemented
7	C₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
8	H(5,𝔽₂)	—	2	16	no	—
9	C₂×#41	C₂²	1	1	yes	characteristic
10	C₂×#41	C₂²	1	6	yes	complemented
11	C₂×#41	C₂²	1	4	yes	complemented
12	C₂²×D₄	C₂²	1	6	yes	complemented
13	C₂²×D₄	C₂²	1	12	yes	complemented
14	C₂²×D₄	C₂²	1	12	yes	complemented
15	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
16	#127	C₂²	1	3	yes	complemented
17	#127	C₂²	1	6	yes	complemented
18	#127	C₂²	1	12	yes	complemented
19	#124	C₂²	1	6	yes	complemented
20	#121	C₂²	1	12	yes	complemented
21	#121	C₂²	1	24	yes	complemented
22	#120	C₂²	1	24	yes	complemented
23	C₄×Q₈	C₂²	1	2	yes	complemented
24	C₄×D₄	C₂²	1	6	yes	complemented
25	C₄×D₄	C₂²	1	12	yes	complemented
26	#117	C₂²	1	3	yes	complemented
27	C₂×C₄²	C₂²	1	3	yes	complemented
28	C₂⁴	C₂³	1	12	yes	—
29	#41	—	2	16	no	—
30	#41	—	2	24	no	—
31	#41	D₄	1	8	yes	complemented
32	C₂×Q₈	C₂³	1	2	yes	—
33	C₂×Q₈	C₂³	1	1	yes	characteristic
34	C₂×D₄	—	2	48	no	—
35	C₂×D₄	—	2	24	no	—
36	C₂×D₄	—	4	24	no	—
37	C₂×D₄	C₂³	1	12	yes	—
38	C₂×D₄	C₂³	1	12	yes	—
39	C₂×D₄	C₂³	1	12	yes	—
40	C₂×D₄	C₂³	1	6	yes	—
41	C₂×D₄	C₂³	1	24	yes	—
42	C₂×D₄	C₂³	1	3	yes	—
43	C₂²×C₄	C₂³	1	6	yes	—
44	C₂²×C₄	C₂³	1	12	yes	—
45	C₂²×C₄	C₂³	1	3	yes	—
46	C₄⋊C₄	C₂³	1	6	yes	—
47	C₄⋊C₄	C₂³	1	4	yes	—
48	C₂²⋊C₄	C₂³	1	6	yes	—
49	C₂²⋊C₄	C₂³	1	24	yes	—
50	C₄²	C₂³	1	1	yes	characteristic
51	C₄²	C₂³	1	3	yes	—
52	C₄²	C₂³	1	6	yes	—
53	C₂³	—	2	24	no	—
54	C₂³	—	4	24	no	—
55	C₂³	C₂⁴	1	12	yes	—
56	C₂³	C₂⁴	1	3	yes	—
57	Q₈	—	2	4	no	—
58	Q₈	C₂×D₄	1	4	yes	complemented
59	D₄	—	2	24	no	—
60	D₄	—	2	24	no	—
61	D₄	—	2	12	no	—
62	D₄	—	2	24	no	—
63	D₄	—	4	24	no	—
64	D₄	C₂×D₄	1	12	yes	complemented
65	C₂×C₄	—	2	24	no	—
66	C₂×C₄	—	2	12	no	—
67	C₂×C₄	C₂×D₄	1	12	yes	complemented
68	C₂×C₄	C₂⁴	1	6	yes	—
69	C₂×C₄	C₂⁴	1	4	yes	—
70	C₂×C₄	C₂⁴	1	1	yes	characteristic
71	C₂×C₄	C₂⁴	1	3	yes	—
72	C₂×C₄	C₂⁴	1	2	yes	—
73	C₂²	—	2	6	no	—
74	C₂²	—	2	12	no	—
75	C₂²	—	2	24	no	—
76	C₂²	—	8	24	no	—
77	C₂²	C₂²×D₄	1	6	yes	—
78	C₂²	C₂⁵	1	1	yes	characteristic
79	C₄	—	2	4	no	—
80	C₄	—	2	6	no	—
81	C₄	C₂²×D₄	1	6	yes	—
82	C₄	C₂²×D₄	1	2	yes	—
83	C₄	H(5,𝔽₂)	1	4	yes	complemented
84	C₂	—	2	6	no	—
85	C₂	—	4	12	no	—
86	C₂	C₂³×D₄	1	1	yes	characteristic
87	C₂	C₂×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×H(5,𝔽₂)	C₂	semidirect
2. #550	C₂	semidirect
3. D₄²	C₂	semidirect
4. #535	C₂	semidirect
5. #532	C₂	semidirect
6. C₂×#127	C₂	semidirect
7. C₄×#41	C₂	semidirect
10. C₂×#41	C₂²	semidirect
11. C₂×#41	C₂²	semidirect
12. C₂²×D₄	C₂²	semidirect
13. C₂²×D₄	C₂²	semidirect
14. C₂²×D₄	C₂²	semidirect
15. C₄⋊Q₈	C₂²	semidirect
16. #127	C₂²	semidirect
17. #127	C₂²	semidirect
18. #127	C₂²	semidirect
19. #124	C₂²	semidirect
20. #121	C₂²	semidirect
21. #121	C₂²	semidirect
22. #120	C₂²	semidirect
23. C₄×Q₈	C₂²	semidirect
24. C₄×D₄	C₂²	semidirect
25. C₄×D₄	C₂²	semidirect
26. #117	C₂²	semidirect
27. C₂×C₄²	C₂²	semidirect
31. #41	D₄	semidirect
58. Q₈	C₂×D₄	semidirect
64. D₄	C₂×D₄	semidirect
67. C₂×C₄	C₂×D₄	semidirect
83. C₄	H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₂×H(5,𝔽₂)	2
#550	2
D₄²	2
#535	2
#532	2
C₂×#127	2
C₄×#41	2