Subgroups — #2168

Order 128.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₄	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#346	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₈	C₂²	1	1	yes	characteristic
10	C₄×D₄	C₄	1	1	yes	characteristic
11	M₃₂	—	2	2	no	—
12	C₂×C₁₆	—	2	2	no	—
13	C₂×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₁₆	C₂²	1	1	yes	characteristic
17	C₄⋊C₈	C₄	1	1	yes	characteristic
18	C₂²⋊C₈	C₄	1	1	yes	characteristic
19	C₂²⋊C₈	C₄	1	1	yes	characteristic
20	C₄×C₈	C₂²	1	1	yes	characteristic
21	C₂×D₄	C₈	1	1	yes	characteristic
22	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈	—	2	1	no	—
25	C₂×C₈	—	2	1	no	—
26	C₂×C₈	—	2	1	no	—
27	C₂×C₈	—	2	1	no	—
28	C₂×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈	C₂³	1	1	yes	characteristic
32	C₄⋊C₄	C₈	1	1	yes	characteristic
33	C₂²⋊C₄	C₈	1	1	yes	characteristic
34	C₂²⋊C₄	C₈	1	1	yes	characteristic
35	C₄²	C₂×C₄	1	1	yes	characteristic
36	C₁₆	—	2	1	no	—
37	C₁₆	—	2	1	no	—
38	C₁₆	—	2	1	no	—
39	C₁₆	D₄	1	2	yes	complemented
40	C₂³	C₂×C₈	1	1	yes	characteristic
41	C₂³	C₂×C₈	1	1	yes	characteristic
42	D₄	—	2	2	no	—
43	C₂×C₄	—	2	1	no	—
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₈	—	2	1	no	—
53	C₈	—	2	1	no	—
54	C₈	—	2	1	no	—
55	C₈	C₂×D₄	1	1	yes	characteristic
56	C₈	#41	1	1	yes	characteristic
57	C₂²	—	2	1	no	—
58	C₂²	—	2	1	no	—
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	M₃₂	1	2	yes	complemented
63	C₂²	C₂²×C₈	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	—	2	1	no	—
68	C₄	C₄×D₄	1	1	yes	characteristic
69	C₄	#131	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	—	4	1	no	—
72	C₂	C₈×D₄	1	1	yes	characteristic
73	C₂	C₂×M₃₂	1	1	yes	characteristic
74	C₂	#504	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂³
commutator, socle	C₂²	C₂²×C₈
prime radical	C₂×D₄	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×M₃₂	C₂	semidirect
2. C₂²×C₁₆	C₂	semidirect
4. C₄⋊C₁₆	C₂	semidirect
5. C₂²⋊C₁₆	C₂	semidirect
6. C₂²⋊C₁₆	C₂	semidirect
7. #346	C₂	semidirect
39. C₁₆	D₄	semidirect
62. C₂²	M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂	2
C₂²×C₁₆	2
C₈×D₄	2
C₄⋊C₁₆	2
C₂²⋊C₁₆	2
#346	2