Subgroups — #2220

Order 128.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#467	C₂	1	1	yes	maximal, characteristic, complemented
4	#466	C₂	1	1	yes	maximal, characteristic
5	#366	C₂	1	1	yes	maximal, characteristic, complemented
6	#358	C₂	1	1	yes	maximal, characteristic, complemented
7	#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂×D₈	C₂²	1	1	yes	characteristic
10	C₂²×C₈	C₂²	1	1	yes	characteristic
11	C₂²⋊Q₈	—	2	1	no	—
12	#121	—	2	1	no	—
13	SD₃₂	—	2	2	no	—
14	M₃₂	—	2	2	no	—
15	C₂×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₁₆	C₂²	1	1	yes	characteristic
17	#107	C₂²	1	1	yes	characteristic
18	#107	C₂²	1	1	yes	characteristic
19	#103	—	2	1	no	—
20	#102	—	2	1	no	—
21	C₂×Q₈	—	2	1	no	—
22	C₂×D₄	—	2	1	no	—
23	C₂×D₄	—	4	1	no	—
24	C₂²×C₄	D₄	1	1	yes	characteristic
25	Q₁₆	—	2	1	no	—
26	Q₁₆	—	2	1	no	—
27	D₈	—	2	1	no	—
28	D₈	—	2	1	no	—
29	C₂×C₈	—	2	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	D₄	1	1	yes	characteristic
32	C₂×C₈	C₂³	1	1	yes	characteristic
33	C₄⋊C₄	—	2	1	no	—
34	C₄⋊C₄	—	2	1	no	—
35	C₄⋊C₄	—	4	1	no	—
36	C₂²⋊C₄	—	4	1	no	—
37	C₂²⋊C₄	—	4	1	no	—
38	C₁₆	—	2	1	no	—
39	C₁₆	D₄	1	2	yes	complemented
40	C₂³	—	4	1	no	—
41	C₂³	D₈	1	1	yes	characteristic
42	Q₈	—	4	1	no	—
43	Q₈	—	4	1	no	—
44	D₄	—	4	1	no	—
45	D₄	—	4	1	no	—
46	D₄	—	8	2	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	4	1	no	—
50	C₂×C₄	—	4	1	no	—
51	C₂×C₄	—	4	1	no	—
52	C₂×C₄	D₈	1	1	yes	characteristic
53	C₂×C₄	C₂×D₄	1	1	yes	characteristic
54	C₈	—	2	1	no	—
55	C₈	C₂×D₄	1	1	yes	characteristic
56	C₈	#41	1	1	yes	characteristic
57	C₂²	—	2	1	no	—
58	C₂²	—	2	1	no	—
59	C₂²	—	2	1	no	—
60	C₂²	—	8	1	no	—
61	C₂²	—	8	1	no	—
62	C₂²	—	8	1	no	—
63	C₂²	C₂×D₈	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	8	1	no	—
66	C₄	—	8	1	no	—
67	C₄	—	8	1	no	—
68	C₄	#121	1	1	yes	characteristic
69	C₄	#135	1	1	yes	characteristic
70	C₂	—	4	1	no	—
71	C₂	—	16	1	no	—
72	C₂	#466	1	1	yes	characteristic
73	C₂	#509	1	1	yes	characteristic
74	C₂	#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	#466	C₂
LCS	G ▷ C₂×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
1. C₂×SD₃₂	C₂	semidirect
2. C₂×M₃₂	C₂	semidirect
3. #467	C₂	semidirect
5. #366	C₂	semidirect
6. #358	C₂	semidirect
7. #357	C₂	semidirect
39. C₁₆	D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₃₂	2
C₂×M₃₂	2
#467	2
#466	2
#366	2
#358	2
#357	2