Subgroups — #2139

Order 128.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	#358	C₂	1	2	yes	maximal, complemented
5	#357	C₂	1	2	yes	maximal, complemented
6	C₂×#41	—	2	1	no	—
7	#135	—	2	2	no	—
8	#135	C₄	1	4	yes	complemented
9	C₂×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×SD₁₆	—	2	1	no	—
11	C₂×D₈	C₂²	1	1	yes	characteristic
12	C₂²×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄	—	2	1	no	—
14	M₃₂	—	2	2	no	—
15	C₂×C₁₆	C₂²	1	2	yes	—
16	#107	—	2	1	no	—
17	#107	C₂²	1	2	yes	—
18	#41	—	2	4	no	—
19	#41	—	4	2	no	—
20	C₂×Q₈	—	2	1	no	—
21	C₂×D₄	—	2	1	no	—
22	C₂×D₄	—	4	1	no	—
23	C₂²×C₄	—	4	1	no	—
24	C₂²×C₄	—	4	1	no	—
25	C₂²×C₄	D₄	1	1	yes	characteristic
26	Q₁₆	—	2	1	no	—
27	Q₁₆	C₂×C₄	1	2	yes	complemented
28	SD₁₆	—	2	4	no	—
29	D₈	—	2	1	no	—
30	D₈	C₂×C₄	1	2	yes	complemented
31	C₂×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₈	D₄	1	1	yes	characteristic
33	C₂×C₈	D₄	1	2	yes	—
34	C₂×C₈	C₂³	1	1	yes	characteristic
35	C₄⋊C₄	—	2	2	no	—
36	C₄⋊C₄	—	4	1	no	—
37	C₁₆	—	2	2	no	—
38	C₂³	—	4	1	no	—
39	C₂³	D₈	1	1	yes	characteristic
40	Q₈	—	2	2	no	—
41	Q₈	—	4	1	no	—
42	D₄	—	2	2	no	—
43	D₄	—	4	4	no	—
44	D₄	—	4	1	no	—
45	C₂×C₄	—	4	1	no	—
46	C₂×C₄	—	4	4	no	—
47	C₂×C₄	—	4	2	no	—
48	C₂×C₄	—	8	2	no	—
49	C₂×C₄	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄	D₈	1	1	yes	characteristic
51	C₂×C₄	SD₁₆	1	2	yes	—
52	C₂×C₄	C₂×D₄	1	1	yes	characteristic
53	C₈	C₂²⋊C₄	1	2	yes	—
54	C₈	C₂²×C₄	1	1	yes	characteristic
55	C₈	C₂×D₄	1	1	yes	characteristic
56	C₂²	—	2	1	no	—
57	C₂²	—	2	1	no	—
58	C₂²	—	4	2	no	—
59	C₂²	—	8	1	no	—
60	C₂²	—	8	1	no	—
61	C₂²	#102	1	2	yes	—
62	C₂²	C₂×D₈	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	—	8	2	no	—
65	C₄	#102	1	2	yes	—
66	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄	C₂×SD₁₆	1	1	yes	characteristic
68	C₂	—	2	2	no	—
69	C₂	—	8	2	no	—
70	C₂	C₂×#102	1	1	yes	characteristic
71	C₂	#509	1	1	yes	characteristic
72	C₂	#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator	C₈	C₂²×C₄
Frattini, cosocle	C₂×C₈	C₂³
prime radical	C₂×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ 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. C₂×#107	C₂	semidirect
4. #358	C₂	semidirect
5. #357	C₂	semidirect
8. #135	C₄	semidirect
27. Q₁₆	C₂×C₄	semidirect
30. D₈	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#135	2
C₂×M₃₂	2
C₂×#107	2
#358	2
#357	2