Subgroups — #1776

Order 128.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×#97	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊C₄	C₂	1	2	yes	maximal
5	#340	C₂	1	2	yes	maximal
6	C₂²×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₄²	—	2	2	no	—
11	C₂×C₄²	C₂²	1	1	yes	characteristic
12	#107	C₄	1	4	yes	complemented
13	#106	C₄	1	4	yes	complemented
14	#97	C₄	1	4	yes	complemented
15	#95	—	2	2	no	—
16	C₂²×C₄	—	2	2	no	—
17	C₂²×C₄	—	2	2	no	—
18	C₂²×C₄	D₄	1	2	yes	—
19	C₂²×C₄	C₂³	1	1	yes	characteristic
20	C₂×C₈	—	2	1	no	—
21	C₂×C₈	—	2	1	no	—
22	C₂×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₈	C₂×C₄	1	2	yes	—
25	C₂×C₈	C₂×C₄	1	2	yes	—
26	C₄⋊C₄	—	2	2	no	—
27	C₄⋊C₄	—	2	2	no	—
28	C₄⋊C₄	C₂×C₄	1	4	yes	—
29	C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₄²	—	4	4	no	—
31	C₄²	C₂×C₄	1	2	yes	—
32	C₄²	Q₈	1	2	yes	—
33	C₂³	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄	—	2	2	no	—
35	C₂×C₄	—	2	2	no	—
36	C₂×C₄	—	2	4	no	—
37	C₂×C₄	—	2	4	no	—
38	C₂×C₄	—	4	2	no	—
39	C₂×C₄	—	4	2	no	—
40	C₂×C₄	—	4	2	no	—
41	C₂×C₄	—	4	2	no	—
42	C₂×C₄	C₄⋊C₄	1	4	yes	—
43	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄	#41	1	1	yes	characteristic
48	C₂×C₄	#41	1	1	yes	characteristic
49	C₈	—	2	2	no	—
50	C₈	C₄²	1	4	yes	complemented
51	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²	C₄×D₄	1	1	yes	characteristic
53	C₂²	C₄×D₄	1	1	yes	characteristic
54	C₂²	AGL(1,ℤ/8)	1	2	yes	—
55	C₂²	#137	1	2	yes	—
56	C₄	—	2	4	no	—
57	C₄	—	4	4	no	—
58	C₄	—	4	4	no	—
59	C₄	C₂×C₄²	1	1	yes	characteristic
60	C₄	#117	1	1	yes	characteristic
61	C₄	C₄×Q₈	1	1	yes	characteristic
62	C₄	C₄×Q₈	1	1	yes	characteristic
63	C₂	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂	#428	1	2	yes	—
65	C₂	#440	1	2	yes	—
66	C₂	#441	1	1	yes	characteristic
67	C₂	#442	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₄
commutator	C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄	C₂³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
12. #107	C₄	semidirect
13. #106	C₄	semidirect
14. #97	C₄	semidirect
50. C₈	C₄²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#107	2
C₂×#106	2
C₂×#97	2
C₄×C₄⋊C₄	2
#340	2