Subgroups — #2159

Order 128.

78 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
2	#427	C₂	1	1	yes	maximal, characteristic
3	C₂×#107	C₂	1	1	yes	maximal, characteristic
4	#367	C₂	1	2	yes	maximal, complemented
5	#366	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈	C₂²	1	1	yes	characteristic
7	#117	—	2	1	no	—
8	C₂×C₄⋊C₄	—	2	1	no	—
9	M₃₂	C₄	1	4	yes	complemented
10	C₂×C₁₆	C₂²	1	2	yes	—
11	#107	—	2	1	no	—
12	#107	C₂²	1	1	yes	characteristic
13	#107	C₂²	1	2	yes	—
14	#107	C₂²	1	1	yes	characteristic
15	#106	—	2	1	no	—
16	C₂²×C₄	—	4	1	no	—
17	C₂²×C₄	D₄	1	1	yes	characteristic
18	C₂×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₈	D₄	1	1	yes	characteristic
20	C₂×C₈	Q₈	1	2	yes	—
21	C₂×C₈	C₂³	1	1	yes	characteristic
22	C₄⋊C₄	—	2	1	no	—
23	C₄⋊C₄	—	2	1	no	—
24	C₄⋊C₄	—	2	2	no	—
25	C₄⋊C₄	—	4	1	no	—
26	C₂²⋊C₄	—	4	1	no	—
27	C₄²	—	4	1	no	—
28	C₁₆	C₂×C₄	1	4	yes	complemented
29	C₂³	D₈	1	1	yes	characteristic
30	C₂×C₄	—	4	1	no	—
31	C₂×C₄	—	4	1	no	—
32	C₂×C₄	—	4	2	no	—
33	C₂×C₄	—	8	2	no	—
34	C₂×C₄	C₄⋊C₄	1	2	yes	—
35	C₂×C₄	D₈	1	1	yes	characteristic
36	C₂×C₄	Q₁₆	1	2	yes	—
37	C₂×C₄	C₂×D₄	1	1	yes	characteristic
38	C₈	C₄⋊C₄	1	2	yes	—
39	C₈	C₂²×C₄	1	1	yes	characteristic
40	C₈	C₂×Q₈	1	1	yes	characteristic
41	C₂²	—	2	1	no	—
42	C₂²	—	2	1	no	—
43	C₂²	#107	1	2	yes	—
44	C₂²	C₂×D₈	1	1	yes	characteristic
45	C₄	—	8	1	no	—
46	C₄	—	8	2	no	—
47	C₄	—	8	1	no	—
48	C₄	#107	1	2	yes	—
49	C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄	C₂×Q₁₆	1	1	yes	characteristic
51	C₂	—	2	2	no	—
52	C₂	C₂×#107	1	1	yes	characteristic
53	C₂	#509	1	1	yes	characteristic
54	C₂	#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator	C₈	C₂²×C₄
Frattini, cosocle	C₂×C₈	C₂³
prime radical	C₂³	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×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
4. #367	C₂	semidirect
5. #366	C₂	semidirect
9. M₃₂	C₄	semidirect
28. C₁₆	C₂×C₄	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
#427	2
C₂×#107	2
#367	2
#366	2