Subgroups — #2113

Order 128.

114 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	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂×#41	C₄	1	1	yes	characteristic
6	#131	—	2	2	no	—
7	#131	—	2	2	no	—
8	C₂×M₁₆	C₄	1	1	yes	characteristic
9	C₂×M₁₆	C₄	1	2	yes	—
10	C₂²×C₈	C₂²	1	2	yes	—
11	C₂²×C₈	C₂²	1	1	yes	characteristic
12	M₃₂	—	2	2	no	—
13	C₂×C₁₆	—	2	2	no	—
14	C₂×C₁₆	C₂²	1	2	yes	—
15	C₂×C₁₆	C₂²	1	2	yes	—
16	#41	—	2	2	no	—
17	#41	—	2	2	no	—
18	C₂×Q₈	C₈	1	1	yes	characteristic
19	C₂×D₄	C₈	1	2	yes	—
20	C₂×D₄	C₈	1	1	yes	characteristic
21	C₂²×C₄	C₂×C₄	1	2	yes	—
22	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
23	M₁₆	—	2	2	no	—
24	M₁₆	—	2	4	no	—
25	C₂×C₈	—	2	2	no	—
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈	D₄	1	2	yes	—
30	C₂×C₈	D₄	1	2	yes	—
31	C₂×C₈	C₂³	1	1	yes	characteristic
32	C₁₆	—	2	2	no	—
33	C₁₆	—	2	2	no	—
34	C₂³	C₂×C₈	1	2	yes	—
35	C₂³	C₂×C₈	1	1	yes	characteristic
36	Q₈	—	2	2	no	—
37	D₄	—	2	4	no	—
38	D₄	—	2	2	no	—
39	C₂×C₄	—	2	2	no	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄	C₂×C₈	1	2	yes	—
43	C₂×C₄	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄	M₁₆	1	2	yes	—
45	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₈	—	2	2	no	—
47	C₈	C₂²⋊C₄	1	2	yes	—
48	C₈	C₂×D₄	1	1	yes	characteristic
49	C₈	C₂×D₄	1	1	yes	characteristic
50	C₂²	—	2	2	no	—
51	C₂²	—	2	2	no	—
52	C₂²	—	2	4	no	—
53	C₂²	C₂²⋊C₈	1	2	yes	—
54	C₂²	C₂²×C₈	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	C₂²⋊C₈	1	2	yes	—
57	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄	C₂×M₁₆	1	1	yes	characteristic
59	C₂	—	2	2	no	—
60	C₂	—	4	2	no	—
61	C₂	C₂×C₂²⋊C₈	1	1	yes	characteristic
62	C₂	#504	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂³
commutator, socle	C₂²	C₂²×C₈
prime radical	C₂×#41	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×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. C₂²×C₁₆	C₂	semidirect
4. C₂²⋊C₁₆	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
C₂²×C₁₆	2
C₂²⋊C₁₆	2