Subgroups — C₂×#405 #2872

Order 128.

296 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#117	C₂	1	1	yes	maximal, characteristic
4	#405	C₂	1	8	yes	maximal, direct product factor
5	C₂×#97	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₈	C₂	1	2	yes	maximal, complemented
7	C₂³×C₄	C₂²	1	1	yes	characteristic
8	C₂×M₁₆	C₄	1	8	yes	complemented
9	C₂×M₁₆	C₂²	1	4	yes	—
10	C₂²×C₈	C₄	1	8	yes	complemented
11	C₂²×C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₈	C₂²	1	2	yes	—
13	C₂²×C₈	C₂²	1	4	yes	—
14	C₂²×C₈	C₂²	1	1	yes	characteristic
15	#117	C₄	1	4	yes	—
16	#117	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄	C₄	1	2	yes	—
18	C₂×C₂²⋊C₄	C₄	1	2	yes	—
19	C₂×C₄²	C₂²	1	2	yes	—
20	#97	C₂²	1	8	yes	complemented
21	C₄×C₈	C₂²	1	8	yes	complemented
22	C₂⁴	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄	C₂×C₄	1	4	yes	—
24	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄	C₂×C₄	1	2	yes	—
26	C₂²×C₄	C₂×C₄	1	4	yes	—
27	C₂²×C₄	C₂×C₄	1	4	yes	—
28	C₂²×C₄	C₂³	1	1	yes	characteristic
29	C₂²×C₄	C₂³	1	2	yes	—
30	M₁₆	C₂×C₄	1	16	yes	complemented
31	C₂×C₈	C₂×C₄	1	4	yes	—
32	C₂×C₈	C₂×C₄	1	4	yes	—
33	C₂×C₈	C₂×C₄	1	8	yes	—
34	C₂×C₈	C₂×C₄	1	16	yes	complemented
35	C₂×C₈	C₂³	1	2	yes	—
36	C₂×C₈	C₂³	1	4	yes	—
37	C₂×C₈	C₂³	1	2	yes	—
38	C₄⋊C₄	C₂×C₄	1	8	yes	—
39	C₂²⋊C₄	C₂×C₄	1	8	yes	—
40	C₄²	C₂×C₄	1	4	yes	—
41	C₄²	C₂³	1	4	yes	—
42	C₂³	—	2	4	no	—
43	C₂³	C₄²	1	4	yes	—
44	C₂³	C₂²×C₄	1	2	yes	—
45	C₂³	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄	—	2	8	no	—
47	C₂×C₄	C₄²	1	8	yes	—
48	C₂×C₄	C₄²	1	4	yes	—
49	C₂×C₄	C₂²×C₄	1	8	yes	—
50	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄	C₂²×C₄	1	2	yes	—
52	C₂×C₄	C₂²×C₄	1	2	yes	—
53	C₂×C₄	C₂²×C₄	1	2	yes	—
54	C₂×C₄	C₂²×C₄	1	4	yes	—
55	C₂×C₄	C₂²×C₄	1	4	yes	—
56	C₂×C₄	C₂⁴	1	1	yes	characteristic
57	C₈	C₂²×C₄	1	4	yes	—
58	C₈	C₂²×C₄	1	4	yes	—
59	C₈	C₂²×C₄	1	8	yes	—
60	C₂²	—	2	4	no	—
61	C₂²	—	2	8	no	—
62	C₂²	C₂×C₄²	1	4	yes	—
63	C₂²	C₂×C₄²	1	2	yes	—
64	C₂²	#131	1	4	yes	—
65	C₂²	C₂³×C₄	1	1	yes	characteristic
66	C₄	—	2	8	no	—
67	C₄	C₂×C₄²	1	2	yes	—
68	C₄	C₂×C₄²	1	4	yes	—
69	C₄	C₂³×C₄	1	1	yes	characteristic
70	C₄	C₂³×C₄	1	1	yes	characteristic
71	C₂	—	2	4	no	—
72	C₂	#405	1	4	yes	direct product factor
73	C₂	C₂²×C₄²	1	1	yes	characteristic
74	C₂	C₂×#131	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²
commutator	C₂	C₂²×C₄²
Frattini, cosocle	C₂×C₄	C₂⁴
socle	C₂³	C₂²×C₄
prime radical	C₂⁴	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #405	C₂	direct (N × H)
5. C₂×#97	C₂	semidirect
6. C₂×C₄×C₈	C₂	semidirect
8. C₂×M₁₆	C₄	semidirect
10. C₂²×C₈	C₄	semidirect
20. #97	C₂²	semidirect
21. C₄×C₈	C₂²	semidirect
30. M₁₆	C₂×C₄	semidirect
34. C₂×C₈	C₂×C₄	semidirect
72. C₂	#405	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×M₁₆	2
C₂³×C₈	2
C₂×#117	2
#405	2
C₂×#97	2
C₂×C₄×C₈	2

Subgroups — C2×#405 #2872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#405 #2872