Subgroups — #1953

Order 128.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#430	C₂	1	2	yes	maximal
2	C₂×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×#108	C₂	1	1	yes	maximal, characteristic
4	#405	C₂	1	1	yes	maximal, characteristic
5	#344	C₂	1	2	yes	maximal
6	C₂×M₁₆	C₂²	1	2	yes	—
7	C₂×M₁₆	C₂²	1	2	yes	—
8	C₂×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₈	C₂²	1	1	yes	characteristic
10	#117	C₂²	1	1	yes	characteristic
11	#108	—	2	4	no	—
12	#108	—	2	1	no	—
13	#108	—	2	1	no	—
14	#108	C₄	1	4	yes	complemented
15	#97	—	2	1	no	—
16	C₄×C₈	—	2	1	no	—
17	C₂²×C₄	C₂³	1	1	yes	characteristic
18	M₁₆	—	2	2	no	—
19	M₁₆	—	2	1	no	—
20	M₁₆	—	2	2	no	—
21	M₁₆	—	2	2	no	—
22	M₁₆	C₂×C₄	1	4	yes	—
23	M₁₆	D₄	1	2	yes	—
24	C₂×C₈	—	2	1	no	—
25	C₂×C₈	—	2	2	no	—
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₈	D₄	1	1	yes	characteristic
29	C₂×C₈	D₄	1	1	yes	characteristic
30	C₂×C₈	Q₈	1	2	yes	—
31	C₄⋊C₄	—	2	1	no	—
32	C₂²⋊C₄	Q₈	1	2	yes	—
33	C₄²	—	2	1	no	—
34	C₂³	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄	—	2	2	no	—
36	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄	#41	1	1	yes	characteristic
41	C₂×C₄	#41	1	1	yes	characteristic
42	C₈	—	2	4	no	—
43	C₈	—	2	2	no	—
44	C₈	—	4	2	no	—
45	C₈	C₄⋊C₄	1	2	yes	—
46	C₈	C₄⋊C₄	1	2	yes	—
47	C₂²	—	4	1	no	—
48	C₂²	C₄×Q₈	1	1	yes	characteristic
49	C₂²	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²	C₄⋊Q₈	1	1	yes	characteristic
51	C₄	—	4	2	no	—
52	C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄	C₄×D₄	1	1	yes	characteristic
54	C₄	#121	1	1	yes	characteristic
55	C₄	#125	1	1	yes	characteristic
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	—	2	1	no	—
59	C₂	#389	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#121
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	#389
prime radical	C₂³	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
14. #108	C₄	semidirect

Maximal subgroups

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

subgroup	index
#430	2
C₂×#108	2
#405	2
#344	2