Subgroups — C₂×#20266110 #27187554

Order 12928.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266110	C₂	1	4	yes	maximal, direct product factor
2	C₂×#15692305	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₆×D₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#15692305	C₂²	1	4	yes	complemented
5	C₂×C₈×D₁₀₁	C₄	1	1	yes	characteristic
6	C₂×#13108049	C₂²	1	1	yes	characteristic
7	C₁₆×D₁₀₁	C₄	1	2	yes	—
8	C₁₆×D₁₀₁	C₂²	1	2	yes	—
9	C₂×C₁₆×C₁₀₁	C₄	1	1	yes	characteristic
10	C₂×C₄×D₁₀₁	C₈	1	1	yes	characteristic
11	C₂×#11717579	C₈	1	1	yes	characteristic
12	C₈×D₁₀₁	C₈	1	2	yes	—
13	C₈×D₁₀₁	C₂×C₄	1	2	yes	—
14	#13108049	C₂×C₄	1	1	yes	characteristic
15	#13108049	C₂³	1	1	yes	characteristic
16	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₁₀₁	C₂×C₄	1	2	yes	—
18	C₂²×D₁₀₁	C₁₆	1	1	yes	characteristic
19	C₂×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
20	C₂×Dic₁₀₁	C₁₆	1	1	yes	characteristic
21	C₄×D₁₀₁	C₁₆	1	2	yes	—
22	C₄×D₁₀₁	C₂×C₈	1	2	yes	—
23	C₈×C₁₀₁	C₂×C₈	1	1	yes	characteristic
24	C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
25	#11717579	C₂×C₈	1	2	yes	—
26	C₂²×C₁₀₁	C₂×C₁₆	1	1	yes	characteristic
27	C₂×D₁₀₁	—	2	2	no	—
28	C₂×D₁₀₁	C₂×C₁₆	1	2	yes	—
29	C₄×C₁₀₁	C₂×C₁₆	1	1	yes	characteristic
30	C₄×C₁₀₁	C₂²×C₈	1	1	yes	characteristic
31	Dic₁₀₁	C₂×C₁₆	1	2	yes	—
32	C₂×C₁₀₁	M₆₄	1	2	yes	complemented
33	C₂×C₁₀₁	C₂²×C₁₆	1	1	yes	characteristic
34	D₁₀₁	—	2	2	no	—
35	C₂×M₆₄	—	101	1	no	maximal, Sylow
36	C₁₀₁	C₂×M₆₄	1	1	yes	characteristic, complemented, Sylow
37	C₂²×C₁₆	—	101	1	no	—
38	M₆₄	—	101	4	no	—
39	C₂×C₃₂	—	101	2	no	—
40	C₂²×C₈	—	101	1	no	—
41	C₂×C₁₆	—	101	1	no	—
42	C₂×C₁₆	—	101	2	no	—
43	C₂×C₁₆	—	101	2	no	—
44	C₂×C₁₆	#87938	1	1	yes	characteristic
45	C₃₂	—	101	4	no	—
46	C₂²×C₄	—	101	1	no	—
47	C₂×C₈	—	101	1	no	—
48	C₂×C₈	—	101	2	no	—
49	C₂×C₈	—	101	2	no	—
50	C₂×C₈	C₂×#87938	1	1	yes	characteristic
51	C₁₆	—	101	1	no	—
52	C₁₆	—	101	1	no	—
53	C₁₆	C₂×#87938	1	2	yes	—
54	C₂³	—	101	1	no	—
55	C₂×C₄	—	101	1	no	—
56	C₂×C₄	—	101	2	no	—
57	C₂×C₄	—	101	2	no	—
58	C₂×C₄	D₁₀₁⋊C₈	1	1	yes	characteristic
59	C₈	—	101	2	no	—
60	C₈	D₁₀₁⋊C₈	1	1	yes	characteristic
61	C₈	C₂²×#87938	1	1	yes	characteristic
62	C₂²	—	101	2	no	—
63	C₂²	—	202	2	no	—
64	C₂²	D₁₀₁⋊C₁₆	1	1	yes	characteristic
65	C₄	—	101	2	no	—
66	C₄	D₁₀₁⋊C₁₆	1	1	yes	characteristic
67	C₄	C₂×D₁₀₁⋊C₈	1	1	yes	characteristic
68	C₂	—	202	2	no	—
69	C₂	#20266110	1	2	yes	direct product factor
70	C₂	C₂×D₁₀₁⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×#87938
commutator	C₂×C₁₀₁	C₂²×C₁₆
Frattini	C₈	C₂²×#87938
Fitting	C₂×C₁₆×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂×C₁₆
prime radical	C₂²×D₁₀₁	C₁₆
cosocle	#13108049	C₂³
hypercenter	C₂×C₁₆	#87938
hypocenter	C₁₀₁	C₂×M₆₄
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. #20266110	C₂	direct (N × H)
2. C₂×#15692305	C₂	semidirect
4. #15692305	C₂²	semidirect
32. C₂×C₁₀₁	M₆₄	semidirect
36. C₁₀₁	C₂×M₆₄	semidirect
69. C₂	#20266110	direct (N × H)

Maximal subgroups

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

subgroup	index
#20266110	2
C₂×#15692305	2
C₂×C₁₆×D₁₀₁	2
C₂×M₆₄	101

Subgroups — C2×#20266110 #27187554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20266110 #27187554