Subgroups — C₂²×#135 #3577

Order 128.

752 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41	C₂	1	2	yes	maximal, complemented
2	C₂×#135	C₂	1	24	yes	maximal, direct product factor
3	C₂²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₂²×D₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41	—	2	12	no	—
8	C₂×#41	C₂²	1	24	yes	complemented
9	C₂²×Q₈	C₂²	1	2	yes	—
10	C₂²×D₄	—	2	2	no	—
11	C₂²×D₄	C₂²	1	2	yes	—
12	C₂³×C₄	—	2	2	no	—
13	C₂³×C₄	C₂²	1	1	yes	characteristic
14	#135	C₂²	1	64	yes	direct product factor
15	C₂×Q₁₆	C₂²	1	12	yes	complemented
16	C₂×SD₁₆	C₂²	1	24	yes	complemented
17	C₂×D₈	C₂²	1	12	yes	complemented
18	C₂²×C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₈	C₂²	1	1	yes	characteristic
20	C₂²×C₈	C₂²	1	12	yes	complemented
21	C₂⁴	—	2	2	no	—
22	C₂⁴	D₄	1	1	yes	characteristic
23	#41	—	2	48	no	—
24	#41	C₂³	1	32	yes	complemented
25	C₂×Q₈	—	2	6	no	—
26	C₂×Q₈	C₂³	1	12	yes	—
27	C₂×D₄	—	2	24	no	—
28	C₂×D₄	—	2	6	no	—
29	C₂×D₄	C₂³	1	12	yes	—
30	C₂²×C₄	—	2	2	no	—
31	C₂²×C₄	—	2	24	no	—
32	C₂²×C₄	D₄	1	6	yes	—
33	C₂²×C₄	D₄	1	1	yes	characteristic
34	C₂²×C₄	C₂³	1	1	yes	characteristic
35	C₂²×C₄	C₂³	1	6	yes	—
36	Q₁₆	C₂³	1	16	yes	complemented
37	SD₁₆	C₂³	1	32	yes	complemented
38	D₈	C₂³	1	16	yes	complemented
39	C₂×C₈	C₂³	1	6	yes	—
40	C₂×C₈	C₂³	1	6	yes	—
41	C₂×C₈	C₂³	1	16	yes	complemented
42	C₂³	—	2	4	no	—
43	C₂³	—	2	12	no	—
44	C₂³	—	4	8	no	—
45	C₂³	C₂×D₄	1	6	yes	—
46	C₂³	C₂×D₄	1	1	yes	characteristic
47	Q₈	—	2	12	no	—
48	Q₈	C₂⁴	1	8	yes	—
49	D₄	—	2	32	no	—
50	D₄	—	2	12	no	—
51	D₄	C₂⁴	1	8	yes	—
52	C₂×C₄	—	2	12	no	—
53	C₂×C₄	—	2	32	no	—
54	C₂×C₄	C₂×D₄	1	6	yes	—
55	C₂×C₄	C₂×D₄	1	3	yes	—
56	C₂×C₄	C₂×D₄	1	12	yes	—
57	C₂×C₄	C₂⁴	1	3	yes	—
58	C₂×C₄	C₂⁴	1	4	yes	—
59	C₈	C₂⁴	1	4	yes	—
60	C₈	C₂⁴	1	4	yes	—
61	C₂²	—	2	12	no	—
62	C₂²	—	2	8	no	—
63	C₂²	—	4	24	no	—
64	C₂²	#135	1	4	yes	direct product factor
65	C₂²	C₂²×D₄	1	4	yes	—
66	C₂²	C₂²×D₄	1	3	yes	—
67	C₄	—	2	8	no	—
68	C₄	C₂²×D₄	1	4	yes	—
69	C₄	C₂²×D₄	1	3	yes	—
70	C₄	C₂⁵	1	1	yes	characteristic
71	C₂	—	2	4	no	—
72	C₂	—	4	8	no	—
73	C₂	C₂×#135	1	6	yes	direct product factor
74	C₂	C₂³×D₄	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×#41	C₂	semidirect
2. C₂×#135	C₂	direct (N × H)
3. C₂²×Q₁₆	C₂	semidirect
4. C₂²×SD₁₆	C₂	semidirect
5. C₂²×D₈	C₂	semidirect
6. C₂³×C₈	C₂	semidirect
8. C₂×#41	C₂²	semidirect
14. #135	C₂²	direct (N × H)
15. C₂×Q₁₆	C₂²	semidirect
16. C₂×SD₁₆	C₂²	semidirect
17. C₂×D₈	C₂²	semidirect
20. C₂²×C₈	C₂²	semidirect
24. #41	C₂³	semidirect
36. Q₁₆	C₂³	semidirect
37. SD₁₆	C₂³	semidirect
38. D₈	C₂³	semidirect
41. C₂×C₈	C₂³	semidirect
64. C₂²	#135	direct (N × H)
73. C₂	C₂×#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
C₂²×#41	2
C₂×#135	2
C₂²×Q₁₆	2
C₂²×SD₁₆	2
C₂²×D₈	2
C₂³×C₈	2

Subgroups — C22×#135 #3577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#135 #3577