Subgroups — #2223

Order 128.

72 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	#504	C₂	1	1	yes	maximal, characteristic, complemented
3	#473	C₂	1	1	yes	maximal, characteristic, complemented
4	#472	C₂	1	1	yes	maximal, characteristic
5	#368	C₂	1	1	yes	maximal, characteristic, complemented
6	#362	C₂	1	1	yes	maximal, characteristic, complemented
7	#361	C₂	1	1	yes	maximal, characteristic, complemented
8	#137	—	2	1	no	—
9	AGL(1,ℤ/8)	—	2	1	no	—
10	C₂×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×D₈	C₂²	1	1	yes	characteristic
12	#131	C₂²	1	1	yes	characteristic
13	SD₃₂	—	2	1	no	—
14	SD₃₂	—	2	1	no	—
15	M₃₂	—	2	1	no	—
16	M₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₁₆	—	2	1	no	—
18	C₂×C₁₆	C₂²	1	1	yes	characteristic
19	#108	C₂²	1	1	yes	characteristic
20	#108	C₂²	1	1	yes	characteristic
21	#101	—	2	1	no	—
22	#100	—	2	1	no	—
23	#41	D₄	1	1	yes	characteristic
24	C₂×Q₈	—	2	1	no	—
25	C₂×D₄	—	2	1	no	—
26	Q₁₆	—	2	1	no	—
27	Q₁₆	—	2	1	no	—
28	Q₁₆	—	4	1	no	—
29	SD₁₆	—	4	1	no	—
30	SD₁₆	—	4	1	no	—
31	D₈	—	2	1	no	—
32	D₈	—	2	1	no	—
33	D₈	—	4	1	no	—
34	M₁₆	—	2	1	no	—
35	M₁₆	—	2	1	no	—
36	M₁₆	—	2	1	no	—
37	M₁₆	D₄	1	1	yes	characteristic
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	C₂³	1	1	yes	characteristic
40	C₁₆	—	2	1	no	—
41	C₁₆	D₄	1	1	yes	characteristic
42	C₁₆	D₄	1	1	yes	characteristic
43	C₂³	—	4	1	no	—
44	Q₈	—	4	1	no	—
45	Q₈	—	4	1	no	—
46	Q₈	D₈	1	1	yes	characteristic
47	D₄	—	2	1	no	—
48	D₄	—	4	1	no	—
49	D₄	—	4	1	no	—
50	D₄	D₈	1	1	yes	characteristic
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	C₂×D₄	1	1	yes	characteristic
54	C₈	—	2	1	no	—
55	C₈	—	4	1	no	—
56	C₈	—	4	1	no	—
57	C₈	C₂×D₄	1	1	yes	characteristic
58	C₈	#41	1	1	yes	characteristic
59	C₂²	—	2	1	no	—
60	C₂²	—	8	1	no	—
61	C₂²	—	16	1	no	—
62	C₂²	#135	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	8	1	no	—
65	C₄	#121	1	1	yes	characteristic
66	C₄	C₂×D₈	1	1	yes	characteristic
67	C₂	—	2	1	no	—
68	C₂	—	4	1	no	—
69	C₂	—	16	1	no	—
70	C₂	#466	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#466
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	#472	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ 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. #504	C₂	semidirect
3. #473	C₂	semidirect
5. #368	C₂	semidirect
6. #362	C₂	semidirect
7. #361	C₂	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
#504	2
#473	2
#472	2
#368	2
#362	2
#361	2