Subgroups — #1387

Order 128.

68 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	C₂×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×#107	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₆	—	2	1	no	—
5	C₂²×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊C₄	—	2	1	no	—
7	M₃₂	—	2	1	no	—
8	M₃₂	C₄	1	2	yes	complemented
9	C₂×C₁₆	—	2	1	no	—
10	#108	—	2	1	no	—
11	#108	C₄	1	2	yes	complemented
12	#107	—	2	1	no	—
13	#107	C₄	1	2	yes	—
14	C₂²×C₄	—	4	1	no	—
15	C₂²×C₄	D₄	1	1	yes	characteristic
16	M₁₆	—	2	2	no	—
17	M₁₆	—	4	1	no	—
18	C₂×C₈	—	4	1	no	—
19	C₂×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈	D₄	1	1	yes	characteristic
23	C₂×C₈	D₄	1	1	yes	characteristic
24	C₂×C₈	Q₈	1	1	yes	characteristic
25	C₄⋊C₄	—	2	2	no	—
26	C₄⋊C₄	—	4	1	no	—
27	C₁₆	—	2	2	no	—
28	C₂³	SD₁₆	1	1	yes	characteristic
29	C₂×C₄	—	4	2	no	—
30	C₂×C₄	—	8	2	no	—
31	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄	D₈	1	1	yes	characteristic
35	C₂×C₄	SD₁₆	1	1	yes	characteristic
36	C₂×C₄	Q₁₆	1	1	yes	characteristic
37	C₈	—	4	2	no	—
38	C₈	C₄²	1	1	yes	characteristic
39	C₈	C₂²⋊C₄	1	1	yes	characteristic
40	C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂²	—	2	1	no	—
43	C₂²	—	2	1	no	—
44	C₂²	#102	1	1	yes	characteristic
45	C₂²	#103	1	1	yes	characteristic
46	C₂²	#106	1	1	yes	characteristic
47	C₄	—	8	2	no	—
48	C₄	#95	1	1	yes	characteristic
49	C₄	#102	1	1	yes	characteristic
50	C₄	#103	1	1	yes	characteristic
51	C₄	#107	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	2	1	no	—
54	C₂	#340	1	1	yes	characteristic
55	C₂	#361	1	1	yes	characteristic
56	C₂	#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#102
commutator	C₈	C₄²
Frattini, cosocle	C₂²×C₈	C₂²
prime radical	C₂³	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
8. M₃₂	C₄	semidirect
11. #108	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
C₂×#108	2
C₂×#107	2