Subgroups — #2108

Order 128.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#405	C₂	1	1	yes	maximal, characteristic
4	#346	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₈	C₂²	1	1	yes	characteristic
8	#117	C₄	1	1	yes	characteristic
9	M₃₂	C₄	1	4	yes	complemented
10	C₂×C₁₆	C₄	1	4	yes	complemented
11	C₂×C₁₆	C₂²	1	2	yes	—
12	C₂×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₆	C₂²	1	1	yes	characteristic
14	#97	C₄	1	2	yes	—
15	C₄×C₈	C₂²	1	2	yes	—
16	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
17	M₁₆	C₈	1	4	yes	—
18	C₂×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₈	C₂×C₄	1	2	yes	—
21	C₂×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₈	C₂³	1	1	yes	characteristic
23	C₄⋊C₄	C₈	1	2	yes	—
24	C₂²⋊C₄	C₈	1	2	yes	—
25	C₄²	C₂×C₄	1	2	yes	—
26	C₁₆	C₂×C₄	1	2	yes	—
27	C₁₆	C₂×C₄	1	2	yes	—
28	C₁₆	C₂×C₄	1	4	yes	—
29	C₂³	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄	C₄²	1	2	yes	—
31	C₂×C₄	C₂×C₈	1	4	yes	—
32	C₂×C₄	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄	C₂×C₈	1	2	yes	—
34	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
35	C₈	C₄²	1	2	yes	—
36	C₈	C₂×C₈	1	4	yes	—
37	C₈	C₂²×C₄	1	1	yes	characteristic
38	C₈	C₂²×C₄	1	1	yes	characteristic
39	C₂²	—	2	2	no	—
40	C₂²	C₄×C₈	1	2	yes	—
41	C₂²	C₂²×C₈	1	1	yes	characteristic
42	C₄	—	2	4	no	—
43	C₄	C₄×C₈	1	2	yes	—
44	C₄	C₂×C₄²	1	1	yes	characteristic
45	C₄	C₂²×C₈	1	1	yes	characteristic
46	C₂	—	2	2	no	—
47	C₂	C₂×C₄×C₈	1	1	yes	characteristic
48	C₂	#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²
commutator	C₂	C₂×C₄×C₈
Frattini, cosocle	C₂×C₈	C₂³
socle	C₂²	C₂²×C₈
prime radical	C₂³	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #346	C₂	semidirect
5. C₄×C₁₆	C₂	semidirect
9. M₃₂	C₄	semidirect
10. C₂×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂	2
C₂²×C₁₆	2
#405	2
#346	2
C₄×C₁₆	2