Subgroups — C₂×#550 #3467

Order 128.

868 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	#550	C₂	1	16	yes	maximal, direct product factor
3	C₂×#127	C₂	1	3	yes	maximal, complemented
4	C₂×#121	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄	C₂	1	3	yes	maximal, complemented
7	C₂×#41	—	2	16	no	—
8	C₂×#41	C₂²	1	16	yes	complemented
9	C₂²×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×D₄	C₂²	1	3	yes	—
11	C₂²×D₄	C₂²	1	6	yes	—
12	C₂²×D₄	C₂²	1	6	yes	—
13	C₂³×C₄	C₂²	1	6	yes	—
14	#127	C₂²	1	24	yes	complemented
15	#121	C₂²	1	48	yes	complemented
16	C₄×Q₈	C₂²	1	8	yes	complemented
17	C₄×D₄	C₂²	1	24	yes	complemented
18	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄	C₂²	1	3	yes	—
20	C₂×C₂²⋊C₄	C₂²	1	6	yes	—
21	C₂×C₄²	C₂²	1	3	yes	—
22	C₂⁴	C₂³	1	6	yes	—
23	#41	—	2	64	no	—
24	C₂×Q₈	D₄	1	8	yes	complemented
25	C₂×Q₈	C₂³	1	4	yes	—
26	C₂×D₄	—	2	24	no	—
27	C₂×D₄	—	2	12	no	—
28	C₂×D₄	—	4	12	no	—
29	C₂×D₄	C₂³	1	12	yes	—
30	C₂×D₄	C₂³	1	24	yes	—
31	C₂×D₄	C₂³	1	24	yes	—
32	C₂²×C₄	—	2	24	no	—
33	C₂²×C₄	C₂³	1	24	yes	—
34	C₂²×C₄	C₂³	1	3	yes	—
35	C₂²×C₄	C₂³	1	2	yes	—
36	C₂²×C₄	C₂³	1	3	yes	—
37	C₂²×C₄	C₂³	1	1	yes	characteristic
38	C₄⋊C₄	C₂³	1	4	yes	—
39	C₄⋊C₄	C₂³	1	12	yes	—
40	C₂²⋊C₄	C₂³	1	24	yes	—
41	C₄²	C₂³	1	12	yes	—
42	C₂³	—	2	12	no	—
43	C₂³	—	2	12	no	—
44	C₂³	—	2	12	no	—
45	C₂³	C₂⁴	1	12	yes	—
46	C₂³	C₂⁴	1	1	yes	characteristic
47	Q₈	C₂×D₄	1	16	yes	complemented
48	D₄	—	2	48	no	—
49	D₄	—	2	24	no	—
50	D₄	—	4	24	no	—
51	C₂×C₄	—	2	4	no	—
52	C₂×C₄	—	2	6	no	—
53	C₂×C₄	—	2	48	no	—
54	C₂×C₄	C₂×D₄	1	12	yes	—
55	C₂×C₄	#41	1	4	yes	complemented
56	C₂×C₄	C₂⁴	1	6	yes	—
57	C₂×C₄	C₂⁴	1	4	yes	—
58	C₂×C₄	C₂⁴	1	6	yes	—
59	C₂×C₄	C₂⁴	1	2	yes	—
60	C₂²	—	2	12	no	—
61	C₂²	—	2	12	no	—
62	C₂²	—	2	12	no	—
63	C₂²	—	4	24	no	—
64	C₂²	C₂²×D₄	1	2	yes	—
65	C₂²	C₂×#41	1	2	yes	—
66	C₂²	H(5,𝔽₂)	1	2	yes	—
67	C₂²	C₂⁵	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	6	no	—
70	C₄	C₂²×D₄	1	12	yes	—
71	C₄	C₂×#41	1	4	yes	complemented
72	C₂	—	4	12	no	—
73	C₂	#550	1	4	yes	direct product factor
74	C₂	C₂³×D₄	1	1	yes	characteristic
75	C₂	C₂²×#41	1	1	yes	characteristic
76	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×#41	C₂	semidirect
2. #550	C₂	direct (N × H)
3. C₂×#127	C₂	semidirect
4. C₂×#121	C₂	semidirect
5. C₂×C₄×Q₈	C₂	semidirect
6. C₂×C₄×D₄	C₂	semidirect
8. C₂×#41	C₂²	semidirect
14. #127	C₂²	semidirect
15. #121	C₂²	semidirect
16. C₄×Q₈	C₂²	semidirect
17. C₄×D₄	C₂²	semidirect
24. C₂×Q₈	D₄	semidirect
47. Q₈	C₂×D₄	semidirect
55. C₂×C₄	#41	semidirect
71. C₄	C₂×#41	semidirect
73. C₂	#550	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
#550	2
C₂×#127	2
C₂×#121	2
C₂×C₄×Q₈	2
C₂×C₄×D₄	2

Subgroups — C2×#550 #3467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#550 #3467