Subgroups — #3428

Order 128.

770 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41	C₂	1	1	yes	maximal, characteristic
2	#520	C₂	1	8	yes	maximal, complemented
3	#519	C₂	1	2	yes	maximal, complemented
4	#518	C₂	1	6	yes	maximal, complemented
5	C₄×#41	C₂	1	8	yes	maximal, complemented
6	C₂×#117	C₂	1	6	yes	maximal, complemented
7	C₂×#41	C₄	1	16	yes	complemented
8	C₂×#41	C₂²	1	4	yes	—
9	C₂×#41	C₂²	1	4	yes	—
10	C₂²×Q₈	C₂²	1	1	yes	characteristic
11	C₂²×D₄	C₂²	1	3	yes	—
12	C₂³×C₄	C₂²	1	3	yes	—
13	C₄×Q₈	C₂²	1	16	yes	complemented
14	C₄×D₄	C₂²	1	48	yes	complemented
15	#117	C₂²	1	4	yes	—
16	#117	C₂²	1	12	yes	—
17	#117	C₂²	1	12	yes	complemented
18	#117	C₂²	1	12	yes	complemented
19	C₂×C₄⋊C₄	C₂²	1	12	yes	complemented
20	C₂×C₂²⋊C₄	C₂²	1	12	yes	—
21	C₂×C₄²	C₂²	1	12	yes	complemented
22	C₂⁴	C₂³	1	3	yes	—
23	#41	C₂×C₄	1	32	yes	complemented
24	#41	C₂×C₄	1	32	yes	complemented
25	C₂×Q₈	C₂×C₄	1	8	yes	complemented
26	C₂×Q₈	C₂³	1	4	yes	—
27	C₂×D₄	C₂×C₄	1	24	yes	complemented
28	C₂×D₄	C₂³	1	12	yes	—
29	C₂²×C₄	C₂×C₄	1	24	yes	complemented
30	C₂²×C₄	C₂³	1	24	yes	—
31	C₂²×C₄	C₂³	1	6	yes	—
32	C₂²×C₄	C₂³	1	3	yes	—
33	C₂²×C₄	C₂³	1	6	yes	—
34	C₂²×C₄	C₂³	1	1	yes	characteristic
35	C₄⋊C₄	C₂³	1	8	yes	—
36	C₄⋊C₄	C₂³	1	24	yes	—
37	C₂²⋊C₄	C₂³	1	8	yes	—
38	C₂²⋊C₄	C₂³	1	24	yes	—
39	C₄²	C₂³	1	24	yes	—
40	C₄²	C₂³	1	8	yes	—
41	C₂³	—	4	6	no	—
42	C₂³	C₂²×C₄	1	12	yes	—
43	C₂³	C₂⁴	1	1	yes	characteristic
44	C₂³	C₂⁴	1	6	yes	—
45	Q₈	C₂²×C₄	1	16	yes	—
46	D₄	C₂²×C₄	1	48	yes	—
47	C₂×C₄	—	4	24	no	—
48	C₂×C₄	C₂²×C₄	1	4	yes	—
49	C₂×C₄	C₂²×C₄	1	12	yes	—
50	C₂×C₄	C₂²×C₄	1	24	yes	—
51	C₂×C₄	C₂²×C₄	1	24	yes	—
52	C₂×C₄	C₂⁴	1	16	yes	—
53	C₂×C₄	C₂⁴	1	6	yes	—
54	C₂×C₄	C₂⁴	1	1	yes	characteristic
55	C₂×C₄	C₂⁴	1	1	yes	characteristic
56	C₂²	—	2	12	no	—
57	C₂²	—	2	2	no	—
58	C₂²	—	4	12	no	—
59	C₂²	C₂³×C₄	1	2	yes	—
60	C₂²	C₂³×C₄	1	12	yes	—
61	C₂²	C₂⁵	1	1	yes	characteristic
62	C₄	—	2	16	no	—
63	C₄	C₂³×C₄	1	2	yes	—
64	C₄	C₂³×C₄	1	2	yes	—
65	C₄	C₂³×C₄	1	12	yes	—
66	C₂	—	2	2	no	—
67	C₂	—	2	12	no	—
68	C₂	C₂⁴×C₄	1	1	yes	characteristic
69	C₂	#585	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂⁴
commutator	C₂	C₂⁴×C₄
Frattini, socle, cosocle	C₂²	C₂⁵
prime radical	C₂²×#41	C₂
LCS	G ▷ C₂ ▷ 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
2. #520	C₂	semidirect
3. #519	C₂	semidirect
4. #518	C₂	semidirect
5. C₄×#41	C₂	semidirect
6. C₂×#117	C₂	semidirect
7. C₂×#41	C₄	semidirect
13. C₄×Q₈	C₂²	semidirect
14. C₄×D₄	C₂²	semidirect
17. #117	C₂²	semidirect
18. #117	C₂²	semidirect
19. C₂×C₄⋊C₄	C₂²	semidirect
21. C₂×C₄²	C₂²	semidirect
23. #41	C₂×C₄	semidirect
24. #41	C₂×C₄	semidirect
25. C₂×Q₈	C₂×C₄	semidirect
27. C₂×D₄	C₂×C₄	semidirect
29. C₂²×C₄	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41	2
#520	2
#519	2
#518	2
C₄×#41	2
C₂×#117	2