Group #27171174 — subgroups

Subgroups — #27171174

Order 12928.

272 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692191	C₂	1	2	yes	maximal, complemented
2	#20265141	C₂	1	2	yes	maximal
3	#20264942	C₂	1	2	yes	maximal, complemented
4	C₂²×M₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#13108058	C₂²	1	4	yes	—
6	#15692191	—	2	4	no	—
7	C₂²⋊#11717579	—	2	4	no	—
8	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₁₀₁	—	2	4	no	—
10	C₂×M₁₆×C₁₀₁	C₄	1	4	yes	—
11	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
13	#13108058	—	2	4	no	—
14	#13108058	—	2	4	no	—
15	C₂×#11717579	—	2	4	no	—
16	C₂⁴×C₁₀₁	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₁₀₁	—	2	2	no	—
18	C₂²×C₄×C₁₀₁	—	2	2	no	—
19	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
21	C₂²×C₄×C₁₀₁	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
23	M₁₆×C₁₀₁	—	2	8	no	—
24	C₂×C₈×C₁₀₁	—	2	2	no	—
25	C₂×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₁₀₁	D₄	1	4	yes	—
28	C₂³×C₁₀₁	—	2	2	no	—
29	C₂³×C₁₀₁	—	4	2	no	—
30	C₂³×C₁₀₁	C₄⋊C₄	1	2	yes	—
31	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₁₀₁	—	2	2	no	—
33	C₂×C₄×C₁₀₁	—	2	2	no	—
34	C₂×C₄×C₁₀₁	—	2	2	no	—
35	C₂×C₄×C₁₀₁	—	2	4	no	—
36	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₁	#41	1	2	yes	—
41	C₈×C₁₀₁	—	2	2	no	—
42	C₈×C₁₀₁	C₂²⋊C₄	1	4	yes	—
43	#11717579	—	4	4	no	—
44	C₂²×C₁₀₁	—	2	2	no	—
45	C₂²×C₁₀₁	—	2	2	no	—
46	C₂²×C₁₀₁	—	2	4	no	—
47	C₂²×C₁₀₁	—	4	4	no	—
48	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₁₀₁	—	2	2	no	—
51	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₁₀₁	#117	1	1	yes	characteristic
53	C₄×C₁₀₁	#121	1	2	yes	—
54	C₂×C₁₀₁	—	2	2	no	—
55	C₂×C₁₀₁	—	4	2	no	—
56	C₂×C₁₀₁	#380	1	1	yes	characteristic
57	C₂×C₁₀₁	#430	1	2	yes	—
58	#1811	—	101	1	no	maximal, Sylow
59	C₁₀₁	#1811	1	1	yes	characteristic, complemented, Sylow
60	C₂²×M₁₆	D₁₀₁	1	1	yes	characteristic
61	C₂×#108	—	101	2	no	—
62	#407	—	101	2	no	—
63	#341	—	101	2	no	—
64	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
65	C₂×M₁₆	—	2	4	no	—
66	C₂×M₁₆	—	101	4	no	—
67	C₂×M₁₆	Dic₁₀₁	1	4	yes	—
68	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
70	#108	—	202	4	no	—
71	C₂²⋊C₈	—	202	4	no	—
72	C₂⁴	Dic₂₀₂	1	1	yes	characteristic
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	Dic₂₀₂	1	1	yes	characteristic
76	C₂²×C₄	D₄₀₄	1	2	yes	—
77	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
78	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
79	M₁₆	—	2	8	no	—
80	M₁₆	—	202	4	no	—
81	M₁₆	—	202	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	202	4	no	—
84	C₂×C₈	C₂×Dic₁₀₁	1	2	yes	—
85	C₂×C₈	C₂×Dic₁₀₁	1	2	yes	—
86	C₂×C₈	C₂²⋊D₁₀₁	1	4	yes	—
87	C₂³	—	2	2	no	—
88	C₂³	—	4	2	no	—
89	C₂³	C₄⋊Dic₁₀₁	1	2	yes	—
90	C₂³	C₂×Dic₂₀₂	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	C₄⋊Dic₁₀₁	1	2	yes	—
96	C₂×C₄	C₂²×Dic₁₀₁	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
98	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
99	C₂×C₄	#13108074	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	404	4	no	—
102	C₈	C₂²⋊Dic₁₀₁	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	4	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	C₂×C₄⋊Dic₁₀₁	1	1	yes	characteristic
108	C₂²	#15692234	1	2	yes	—
109	C₄	—	2	2	no	—
110	C₄	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
111	C₄	#15692214	1	1	yes	characteristic
112	C₄	#15692229	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	—	4	2	no	—
115	C₂	#20265037	1	1	yes	characteristic
116	C₂	#20265226	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692191

C₂

yes

maximal, complemented

#20265141

C₂

yes

maximal

#20264942

C₂

yes

maximal, complemented

C₂²×M₁₆×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

#15692191

—

C₂²⋊#11717579

—

C₂³×C₄×C₁₀₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₁

—

C₂×M₁₆×C₁₀₁

C₄

yes

—

C₂²×C₈×C₁₀₁

C₂²

yes

characteristic

C₂²×C₈×C₁₀₁

C₂²

yes

—

—

—

C₂⁴×C₁₀₁

Q₈

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

Q₈

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

C₂×C₄

yes

—

C₂×C₈×C₁₀₁

C₂×C₄

yes

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₄⋊C₄

yes

—

C₂³×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₈×C₁₀₁

—

C₈×C₁₀₁

C₂²⋊C₄

yes

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₂²⋊Q₈

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×C₂²⋊C₄

yes

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#380

yes

characteristic

C₂×C₁₀₁

#430

yes

—

#1811

—

101

maximal, Sylow

C₁₀₁

#1811

yes

characteristic, complemented, Sylow

C₂²×M₁₆

D₁₀₁

yes

characteristic

C₂×#108

—

101

—

#407

—

101

—

#341

—

101

—

C₂³×C₄

C₂×D₁₀₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

101

—

C₂×M₁₆

Dic₁₀₁

yes

—

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

#108

—

202

—

C₂²⋊C₈

—

202

—

C₂⁴

Dic₂₀₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₂₀₂

yes

characteristic

C₂²×C₄

D₄₀₄

yes

—

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

M₁₆

—

202

—

M₁₆

—

202

—

C₂×C₈

—

C₂×C₈

—

202

—

C₂×C₈

C₂×Dic₁₀₁

yes

—

C₂×C₈

C₂×Dic₁₀₁

yes

—

C₂×C₈

C₂²⋊D₁₀₁

yes

—

C₂³

—

C₂³

—

C₂³

C₄⋊Dic₁₀₁

yes

—

C₂³

C₂×Dic₂₀₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₁₀₁

yes

—

C₂×C₄

C₂²×Dic₁₀₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

C₂×C₄

C₂×D₄₀₄

yes

characteristic

C₂×C₄

#13108074

yes

—

100

C₈

—

101

C₈

—

404

—

102

C₈

C₂²⋊Dic₁₀₁

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₄⋊Dic₁₀₁

yes

characteristic

108

C₂²

#15692234

yes

—

109

C₄

—

110

C₄

C₂×C₂²⋊Dic₁₀₁

yes

characteristic

111

C₄

#15692214

yes

characteristic

112

C₄

#15692229

yes

—

113

C₂

—

114

C₂

—

115

C₂

#20265037

yes

characteristic

116

C₂

#20265226

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₀₄
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂²×M₁₆×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₀₁	Q₈
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂²×M₁₆	D₁₀₁
hypocenter	C₁₀₁	#1811
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₄₀₄

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂²×M₁₆×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₄⋊C₄

prime radical

C₂⁴×C₁₀₁

Q₈

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₂²×M₁₆

D₁₀₁

hypocenter

C₁₀₁

#1811

LCS

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
1. C₂×#15692191	C₂	semidirect
3. #20264942	C₂	semidirect
59. C₁₀₁	#1811	semidirect

H ≅ G/N

type

1. C₂×#15692191

C₂

semidirect

3. #20264942

C₂

semidirect

59. C₁₀₁

#1811

semidirect

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

subgroup	index
C₂×#15692191	2
#20265141	2
#20264942	2
C₂²×M₁₆×C₁₀₁	2
#1811	101

subgroup

index

C₂²×M₁₆×C₁₀₁

101

Subgroups — #27171174

Conjugacy classes of subgroups

Complements

Maximal subgroups