Subgroups — #2117

Order 128.

110 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	C₂×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#350	C₂	1	4	yes	maximal, complemented
5	C₂×#41	C₄	1	1	yes	characteristic
6	#131	—	2	1	no	—
7	#131	—	2	1	no	—
8	#131	C₄	1	2	yes	—
9	#131	C₂²	1	2	yes	—
10	C₂×M₁₆	—	2	1	no	—
11	C₂×M₁₆	C₄	1	1	yes	characteristic
12	C₂²×C₈	—	2	1	no	—
13	C₂²×C₈	C₂²	1	1	yes	characteristic
14	M₃₂	—	2	1	no	—
15	M₃₂	—	2	1	no	—
16	M₃₂	—	2	1	no	—
17	M₃₂	C₂²	1	2	yes	complemented
18	C₂×C₁₆	—	2	1	no	—
19	C₂×C₁₆	C₂²	1	2	yes	complemented
20	#41	—	2	1	no	—
21	#41	—	2	1	no	—
22	#41	C₈	1	2	yes	—
23	#41	C₂×C₄	1	2	yes	—
24	C₂×Q₈	C₈	1	1	yes	characteristic
25	C₂×D₄	—	2	1	no	—
26	C₂×D₄	C₈	1	1	yes	characteristic
27	C₂²×C₄	—	2	1	no	—
28	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
29	M₁₆	—	2	1	no	—
30	M₁₆	—	2	2	no	—
31	M₁₆	—	2	2	no	—
32	M₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	2	2	no	—
35	C₂×C₈	—	2	2	no	—
36	C₂×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈	D₄	1	1	yes	characteristic
38	C₂×C₈	D₄	1	1	yes	characteristic
39	C₂×C₈	D₄	1	1	yes	characteristic
40	C₂×C₈	D₄	1	1	yes	characteristic
41	C₂×C₈	C₂³	1	1	yes	characteristic
42	C₁₆	—	2	2	no	—
43	C₁₆	—	2	2	no	—
44	C₂³	—	2	1	no	—
45	C₂³	M₁₆	1	1	yes	characteristic
46	Q₈	—	2	1	no	—
47	Q₈	C₂×C₈	1	2	yes	—
48	D₄	—	2	2	no	—
49	D₄	—	2	2	no	—
50	D₄	—	2	1	no	—
51	D₄	C₂×C₈	1	2	yes	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄	C₂×C₈	1	1	yes	characteristic
58	C₂×C₄	C₂×C₈	1	1	yes	characteristic
59	C₂×C₄	M₁₆	1	1	yes	characteristic
60	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₈	—	2	2	no	—
62	C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₈	C₂×D₄	1	1	yes	characteristic
65	C₈	C₂×D₄	1	1	yes	characteristic
66	C₂²	—	2	2	no	—
67	C₂²	—	4	1	no	—
68	C₂²	—	4	2	no	—
69	C₂²	C₂²⋊C₈	1	1	yes	characteristic
70	C₂²	C₂²⋊C₈	1	1	yes	characteristic
71	C₂²	C₂×M₁₆	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	C₂²⋊C₈	1	1	yes	characteristic
74	C₄	C₂²⋊C₈	1	1	yes	characteristic
75	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄	C₂²×C₈	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	2	1	no	—
79	C₂	—	2	1	no	—
80	C₂	—	4	2	no	—
81	C₂	C₂×C₂²⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄
commutator	C₄	C₂²×C₈
Frattini, cosocle	C₂×C₈	C₂³
socle	C₂	C₂×C₂²⋊C₈
prime radical	C₂×#41	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ 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×M₃₂	C₂	semidirect
4. #350	C₂	semidirect
17. M₃₂	C₂²	semidirect
19. C₂×C₁₆	C₂²	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
#350	2