Subgroups — C₂×#555 #3473

Order 128.

620 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#555	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#124	C₂	1	4	yes	maximal, complemented
4	C₂×#121	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×D₄	C₂	1	2	yes	maximal, complemented
6	C₂×#117	C₂	1	4	yes	maximal, complemented
7	C₂²×Q₈	C₂²	1	2	yes	—
8	C₂²×D₄	C₂²	1	2	yes	—
9	C₂²×D₄	C₂²	1	4	yes	—
10	C₂³×C₄	C₂²	1	4	yes	—
11	C₄⋊Q₈	C₂²	1	8	yes	complemented
12	#124	C₂²	1	32	yes	complemented
13	#121	C₂²	1	32	yes	complemented
14	C₄×D₄	C₂²	1	16	yes	complemented
15	#117	C₂²	1	32	yes	complemented
16	C₂×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄	C₂²	1	4	yes	—
18	C₂×C₂²⋊C₄	C₂²	1	8	yes	—
19	C₂×C₂²⋊C₄	C₂²	1	4	yes	—
20	C₂×C₄²	C₂²	1	4	yes	—
21	C₂×C₄²	C₂²	1	1	yes	characteristic
22	C₂⁴	C₂³	1	4	yes	—
23	C₂×Q₈	—	4	4	no	—
24	C₂×Q₈	C₂³	1	8	yes	—
25	C₂×D₄	—	2	8	no	—
26	C₂×D₄	—	4	8	no	—
27	C₂×D₄	C₂³	1	8	yes	—
28	C₂×D₄	C₂³	1	16	yes	—
29	C₂²×C₄	—	2	16	no	—
30	C₂²×C₄	C₂³	1	16	yes	—
31	C₂²×C₄	C₂³	1	4	yes	—
32	C₂²×C₄	C₂³	1	1	yes	characteristic
33	C₂²×C₄	C₂³	1	4	yes	—
34	C₂²×C₄	C₂³	1	2	yes	—
35	C₄⋊C₄	C₂³	1	8	yes	—
36	C₄⋊C₄	C₂³	1	16	yes	—
37	C₂²⋊C₄	C₂³	1	32	yes	—
38	C₂²⋊C₄	C₂³	1	16	yes	—
39	C₄²	C₂³	1	16	yes	—
40	C₄²	C₂³	1	4	yes	—
41	C₂³	—	2	8	no	—
42	C₂³	—	2	8	no	—
43	C₂³	—	2	8	no	—
44	C₂³	C₂⁴	1	8	yes	—
45	C₂³	C₂⁴	1	1	yes	characteristic
46	Q₈	—	4	8	no	—
47	D₄	—	2	16	no	—
48	D₄	—	4	16	no	—
49	C₂×C₄	—	2	8	no	—
50	C₂×C₄	—	2	8	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	32	no	—
53	C₂×C₄	#41	1	8	yes	—
54	C₂×C₄	C₂⁴	1	8	yes	—
55	C₂×C₄	C₂⁴	1	2	yes	—
56	C₂×C₄	C₂⁴	1	8	yes	—
57	C₂×C₄	C₂⁴	1	4	yes	—
58	C₂²	—	2	8	no	—
59	C₂²	—	2	8	no	—
60	C₂²	—	2	8	no	—
61	C₂²	—	4	16	no	—
62	C₂²	C₂×#41	1	4	yes	—
63	C₂²	H(5,𝔽₂)	1	2	yes	—
64	C₂²	C₂⁵	1	1	yes	characteristic
65	C₄	—	2	8	no	—
66	C₄	—	2	8	no	—
67	C₄	—	2	2	no	—
68	C₄	C₂×#41	1	8	yes	—
69	C₂	—	4	8	no	—
70	C₂	#555	1	4	yes	direct product factor
71	C₂	C₂²×#41	1	2	yes	—
72	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. #555	C₂	direct (N × H)
2. C₂×C₄⋊Q₈	C₂	semidirect
3. C₂×#124	C₂	semidirect
4. C₂×#121	C₂	semidirect
5. C₂×C₄×D₄	C₂	semidirect
6. C₂×#117	C₂	semidirect
11. C₄⋊Q₈	C₂²	semidirect
12. #124	C₂²	semidirect
13. #121	C₂²	semidirect
14. C₄×D₄	C₂²	semidirect
15. #117	C₂²	semidirect
70. C₂	#555	direct (N × H)

Maximal subgroups

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

subgroup	index
#555	2
C₂×C₄⋊Q₈	2
C₂×#124	2
C₂×#121	2
C₂×C₄×D₄	2
C₂×#117	2

Subgroups — C2×#555 #3473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#555 #3473