Subgroups — #1847

Order 128.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#107	C₂	1	2	yes	maximal
2	C₂×#106	C₂	1	2	yes	maximal
3	C₂×#97	C₂	1	1	yes	maximal, characteristic
4	#384	C₂	1	1	yes	maximal, characteristic
5	#382	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊C₄	—	2	2	no	—
8	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₄²	C₂²	1	1	yes	characteristic
12	#107	—	2	4	no	—
13	#106	—	2	4	no	—
14	#97	C₄	1	4	yes	complemented
15	#95	—	2	2	no	—
16	C₂²×C₄	—	2	1	no	—
17	C₂²×C₄	—	2	1	no	—
18	C₂²×C₄	—	2	2	no	—
19	C₂²×C₄	D₄	1	2	yes	—
20	C₂²×C₄	C₂³	1	1	yes	characteristic
21	C₂×C₈	C₂×C₄	1	4	yes	—
22	C₂×C₈	D₄	1	4	yes	—
23	C₂×C₈	Q₈	1	4	yes	—
24	C₄⋊C₄	—	2	2	no	—
25	C₄⋊C₄	—	2	2	no	—
26	C₄⋊C₄	—	2	2	no	—
27	C₄⋊C₄	—	2	2	no	—
28	C₄⋊C₄	—	4	4	no	—
29	C₄²	C₂×C₄	1	2	yes	—
30	C₄²	Q₈	1	2	yes	—
31	C₂³	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄	—	2	2	no	—
33	C₂×C₄	—	2	2	no	—
34	C₂×C₄	—	4	2	no	—
35	C₂×C₄	—	4	1	no	—
36	C₂×C₄	—	4	1	no	—
37	C₂×C₄	—	4	4	no	—
38	C₂×C₄	—	4	2	no	—
39	C₂×C₄	—	4	2	no	—
40	C₂×C₄	C₄⋊C₄	1	4	yes	—
41	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄	C₂×D₄	1	2	yes	—
43	C₂×C₄	C₂×Q₈	1	2	yes	—
44	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
45	C₈	C₄⋊C₄	1	8	yes	complemented
46	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²	#127	1	1	yes	characteristic
48	C₂²	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²	#137	1	2	yes	—
51	C₄	—	2	4	no	—
52	C₄	—	8	2	no	—
53	C₄	—	8	1	no	—
54	C₄	—	8	1	no	—
55	C₄	C₂×C₄⋊C₄	1	2	yes	—
56	C₄	C₄⋊Q₈	1	2	yes	—
57	C₂	#384	1	1	yes	characteristic
58	C₂	#428	1	2	yes	—
59	C₂	#496	1	1	yes	characteristic
60	C₂	#497	1	1	yes	characteristic
61	C₂	#501	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
LCS	G ▷ C₂×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
14. #97	C₄	semidirect
45. C₈	C₄⋊C₄	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
#384	2
#382	2