Subgroups — #1992

Order 128.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×#97	C₂	1	1	yes	maximal, characteristic
3	#384	C₂	1	1	yes	maximal, characteristic
4	#340	C₂	1	4	yes	maximal
5	C₂²×C₈	C₂²	1	2	yes	—
6	#125	—	2	2	no	—
7	#125	C₄	1	4	yes	complemented
8	C₂×C₄⋊C₄	—	2	2	no	—
9	C₂×C₄⋊C₄	—	2	2	no	—
10	C₂×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄²	C₂²	1	1	yes	characteristic
13	#97	—	2	2	no	—
14	C₂²×C₄	—	2	2	no	—
15	C₂²×C₄	—	2	2	no	—
16	C₂²×C₄	D₄	1	2	yes	—
17	C₂²×C₄	C₂³	1	1	yes	characteristic
18	C₂×C₈	—	2	2	no	—
19	C₂×C₈	—	2	2	no	—
20	C₂×C₈	Q₈	1	4	yes	—
21	C₄⋊C₄	—	2	8	no	—
22	C₄⋊C₄	—	2	2	no	—
23	C₄⋊C₄	—	2	4	no	—
24	C₄⋊C₄	—	4	4	no	—
25	C₄⋊C₄	C₂×C₄	1	4	yes	—
26	C₄²	C₂×C₄	1	2	yes	—
27	C₄²	D₄	1	2	yes	—
28	C₂³	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄	—	2	2	no	—
30	C₂×C₄	—	2	2	no	—
31	C₂×C₄	—	2	4	no	—
32	C₂×C₄	—	4	2	no	—
33	C₂×C₄	—	4	2	no	—
34	C₂×C₄	—	4	2	no	—
35	C₂×C₄	—	4	4	no	—
36	C₂×C₄	C₂²⋊C₄	1	4	yes	—
37	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄	C₂×Q₈	1	2	yes	—
40	C₂×C₄	#41	1	2	yes	—
41	C₈	—	2	4	no	—
42	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²	#124	1	1	yes	characteristic
44	C₂²	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²	#137	1	2	yes	—
47	C₄	—	2	4	no	—
48	C₄	—	4	4	no	—
49	C₄	—	8	2	no	—
50	C₄	C₄×Q₈	1	2	yes	—
51	C₄	C₂²⋊Q₈	1	2	yes	—
52	C₂	#391	1	1	yes	characteristic
53	C₂	#417	1	2	yes	—
54	C₂	#490	1	1	yes	characteristic
55	C₂	#491	1	1	yes	characteristic
56	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
7. #125	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#125	2
C₂×#97	2
#384	2
#340	2