Subgroups — #3529

Order 128.

529 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563	C₂	1	3	yes	maximal, complemented
2	#562	C₂	1	6	yes	maximal, complemented
3	H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
4	#560	C₂	1	6	yes	maximal, complemented
5	#532	C₂	1	3	yes	maximal, complemented
6	#529	C₂	1	6	yes	maximal, complemented
7	#525	C₂	1	6	yes	maximal, complemented
8	C₂×#41	C₂²	1	3	yes	—
9	C₂×#41	C₂²	1	1	yes	characteristic, complemented
10	C₂³×C₄	C₂²	1	2	yes	complemented
11	C₄⋊Q₈	C₂²	1	3	yes	complemented
12	#127	C₂²	1	3	yes	—
13	#126	C₂²	1	12	yes	complemented
14	#125	C₂²	1	6	yes	complemented
15	#124	C₂²	1	6	yes	—
16	#124	C₂²	1	3	yes	—
17	#124	C₂²	1	3	yes	complemented
18	#123	C₂²	1	6	yes	—
19	#123	C₂²	1	6	yes	complemented
20	#123	C₂²	1	12	yes	complemented
21	C₂²⋊Q₈	C₂²	1	12	yes	complemented
22	C₂²⋊Q₈	C₂²	1	6	yes	complemented
23	#121	C₂²	1	12	yes	—
24	#121	C₂²	1	6	yes	—
25	#121	C₂²	1	12	yes	complemented
26	#120	C₂²	1	6	yes	—
27	#120	C₂²	1	6	yes	complemented
28	C₄×Q₈	C₂²	1	2	yes	complemented
29	C₄×D₄	C₂²	1	6	yes	—
30	C₄×D₄	C₂²	1	12	yes	complemented
31	#117	C₂²	1	6	yes	complemented
32	C₂×C₄²	C₂²	1	3	yes	complemented
33	C₂⁴	C₂³	1	2	yes	—
34	#41	—	4	6	no	—
35	#41	—	4	2	no	—
36	C₂×Q₈	C₂³	1	2	yes	—
37	C₂×Q₈	C₂³	1	3	yes	—
38	C₂×Q₈	C₂³	1	1	yes	characteristic
39	C₂×D₄	C₂³	1	6	yes	—
40	C₂×D₄	C₂³	1	12	yes	—
41	C₂×D₄	C₂³	1	3	yes	—
42	C₂×D₄	C₂³	1	6	yes	—
43	C₂×D₄	C₂³	1	3	yes	—
44	C₂²×C₄	—	4	4	no	—
45	C₂²×C₄	C₂³	1	3	yes	—
46	C₂²×C₄	C₂³	1	6	yes	—
47	C₂²×C₄	C₂³	1	6	yes	—
48	C₂²×C₄	C₂³	1	6	yes	—
49	C₄⋊C₄	C₂³	1	12	yes	—
50	C₄⋊C₄	C₂³	1	6	yes	—
51	C₄⋊C₄	C₂³	1	12	yes	—
52	C₄⋊C₄	C₂³	1	6	yes	—
53	C₂²⋊C₄	C₂³	1	6	yes	—
54	C₂²⋊C₄	C₂³	1	6	yes	—
55	C₂²⋊C₄	C₂³	1	12	yes	—
56	C₂²⋊C₄	C₂³	1	12	yes	—
57	C₂²⋊C₄	C₂³	1	12	yes	—
58	C₄²	C₂³	1	3	yes	—
59	C₄²	C₂³	1	3	yes	—
60	C₄²	C₂³	1	6	yes	—
61	C₂³	—	4	2	no	—
62	C₂³	—	4	4	no	—
63	C₂³	C₂⁴	1	6	yes	—
64	C₂³	C₂⁴	1	3	yes	—
65	Q₈	—	4	2	no	—
66	Q₈	—	4	3	no	—
67	Q₈	—	4	1	no	—
68	D₄	—	4	12	no	—
69	D₄	—	4	6	no	—
70	D₄	—	4	6	no	—
71	D₄	—	4	3	no	—
72	D₄	—	4	3	no	—
73	C₂×C₄	—	2	12	no	—
74	C₂×C₄	—	2	6	no	—
75	C₂×C₄	—	4	6	no	—
76	C₂×C₄	—	4	6	no	—
77	C₂×C₄	C₂⁴	1	6	yes	—
78	C₂×C₄	C₂⁴	1	6	yes	—
79	C₂×C₄	C₂⁴	1	3	yes	—
80	C₂×C₄	C₂⁴	1	6	yes	—
81	C₂×C₄	C₂⁴	1	1	yes	characteristic
82	C₂²	—	2	6	no	—
83	C₂²	—	2	3	no	—
84	C₂²	—	2	6	no	—
85	C₂²	—	2	12	no	—
86	C₂²	—	4	8	no	—
87	C₂²	C₂⁵	1	1	yes	characteristic
88	C₄	—	2	6	no	—
89	C₄	—	2	3	no	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	H(5,𝔽₂)	1	2	yes	—
93	C₂	—	4	3	no	—
94	C₂	—	4	6	no	—
95	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
96	C₂	#585	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #563	C₂	semidirect
2. #562	C₂	semidirect
3. H(3,𝔽₄)	C₂	semidirect
4. #560	C₂	semidirect
5. #532	C₂	semidirect
6. #529	C₂	semidirect
7. #525	C₂	semidirect
9. C₂×#41	C₂²	semidirect
10. C₂³×C₄	C₂²	semidirect
11. C₄⋊Q₈	C₂²	semidirect
13. #126	C₂²	semidirect
14. #125	C₂²	semidirect
17. #124	C₂²	semidirect
19. #123	C₂²	semidirect
20. #123	C₂²	semidirect
21. C₂²⋊Q₈	C₂²	semidirect
22. C₂²⋊Q₈	C₂²	semidirect
25. #121	C₂²	semidirect
27. #120	C₂²	semidirect
28. C₄×Q₈	C₂²	semidirect
30. C₄×D₄	C₂²	semidirect
31. #117	C₂²	semidirect
32. C₂×C₄²	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#563	2
#562	2
H(3,𝔽₄)	2
#560	2
#532	2
#529	2
#525	2