Subgroups — #2115

Order 128.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×M₃₂	C₂	1	2	yes	maximal, complemented
3	#349	C₂	1	4	yes	maximal, complemented
4	C₂×#41	C₄	1	1	yes	characteristic
5	#131	—	2	2	no	—
6	#131	—	2	2	no	—
7	C₂×M₁₆	C₄	1	1	yes	characteristic
8	C₂×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₈	C₄	1	2	yes	—
10	C₂²×C₈	C₂²	1	1	yes	characteristic
11	M₃₂	—	2	2	no	—
12	M₃₂	C₂²	1	4	yes	—
13	C₂×C₁₆	—	2	2	no	—
14	#41	—	2	2	no	—
15	#41	—	2	2	no	—
16	C₂×Q₈	C₈	1	1	yes	characteristic
17	C₂×D₄	C₈	1	2	yes	—
18	C₂×D₄	C₈	1	1	yes	characteristic
19	C₂²×C₄	C₂×C₄	1	2	yes	—
20	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
21	M₁₆	—	2	2	no	—
22	M₁₆	—	2	4	no	—
23	C₂×C₈	—	2	2	no	—
24	C₂×C₈	—	2	2	no	—
25	C₂×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈	D₄	1	2	yes	—
28	C₂×C₈	D₄	1	2	yes	—
29	C₂×C₈	C₂³	1	1	yes	characteristic
30	C₁₆	—	2	4	no	—
31	C₂³	C₂×C₈	1	2	yes	—
32	C₂³	C₂×C₈	1	1	yes	characteristic
33	Q₈	—	2	2	no	—
34	D₄	—	2	4	no	—
35	D₄	—	2	2	no	—
36	C₂×C₄	—	2	2	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄	C₂×C₈	1	2	yes	—
40	C₂×C₄	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄	M₁₆	1	2	yes	—
42	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
43	C₈	—	2	2	no	—
44	C₈	C₂²⋊C₄	1	1	yes	characteristic
45	C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₈	C₂×D₄	1	2	yes	—
47	C₂²	—	2	2	no	—
48	C₂²	—	4	1	no	—
49	C₂²	—	4	2	no	—
50	C₂²	C₂²⋊C₈	1	2	yes	—
51	C₂²	C₂²×C₈	1	1	yes	characteristic
52	C₄	—	2	2	no	—
53	C₄	C₂²⋊C₈	1	2	yes	—
54	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄	C₂×M₁₆	1	1	yes	characteristic
56	C₂	—	2	1	no	—
57	C₂	—	2	2	no	—
58	C₂	—	4	2	no	—
59	C₂	C₂×C₂²⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊C₄
commutator	C₂²	C₂²×C₈
Frattini, cosocle	C₂×C₈	C₂³
socle	C₂	C₂×C₂²⋊C₈
prime radical	C₂×#41	C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×M₃₂	C₂	semidirect
3. #349	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#131	2
C₂×M₃₂	2
#349	2