Group #27172684 — subgroups

Subgroups — #27172684

Order 12928.

210 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 63 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	#20265205	C₂	1	2	yes	maximal, complemented
2	#20265193	C₂	1	2	yes	maximal
3	#20265145	C₂	1	1	yes	maximal, characteristic
4	#20264934	C₂	1	1	yes	maximal, characteristic, complemented
5	#430×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13108074	C₂²	1	1	yes	characteristic
7	C₂²×#11717579	C₂²	1	1	yes	characteristic
8	#15692214	C₂²	1	2	yes	—
9	#15692184	—	2	4	no	—
10	#15692181	—	2	4	no	—
11	#15692167	—	2	2	no	—
12	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
13	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
14	#108×C₁₀₁	—	2	2	no	—
15	#13108074	—	2	2	no	—
16	#13108074	—	2	2	no	—
17	C₂×Dic₂₀₂	Q₈	1	1	yes	characteristic
18	C₂×D₄₀₄	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
20	C₂×C₄×D₁₀₁	—	2	1	no	—
21	C₂×#11717579	—	2	1	no	—
22	C₂×#11717579	—	2	2	no	—
23	C₄⋊Dic₁₀₁	—	2	2	no	—
24	C₂²⋊Dic₁₀₁	—	2	2	no	—
25	C₄×Dic₁₀₁	D₄	1	4	yes	—
26	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₀₁	—	2	2	no	—
28	M₁₆×C₁₀₁	—	2	2	no	—
29	M₁₆×C₁₀₁	—	2	2	no	—
30	C₂×C₈×C₁₀₁	—	2	2	no	—
31	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
32	C₂³×C₁₀₁	#41	1	1	yes	characteristic
33	C₂²×D₁₀₁	—	2	1	no	—
34	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₁₀₁	#41	1	2	yes	—
38	C₂²⋊D₁₀₁	—	4	2	no	—
39	C₂×Dic₁₀₁	—	2	1	no	—
40	C₂×Dic₁₀₁	—	2	4	no	—
41	D₄₀₄	—	2	2	no	—
42	C₄×D₁₀₁	—	4	1	no	—
43	C₄×D₁₀₁	—	4	1	no	—
44	Dic₂₀₂	—	2	2	no	—
45	C₈×C₁₀₁	—	2	2	no	—
46	C₈×C₁₀₁	—	4	2	no	—
47	#11717579	—	4	1	no	—
48	C₂²×C₁₀₁	—	4	1	no	—
49	C₂²×C₁₀₁	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₀₁	#123	1	2	yes	—
51	C₂×D₁₀₁	—	4	2	no	—
52	C₂×D₁₀₁	—	4	1	no	—
53	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
54	C₄×C₁₀₁	#123	1	1	yes	characteristic
55	C₄×C₁₀₁	#127	1	1	yes	characteristic
56	Dic₁₀₁	—	4	4	no	—
57	Dic₁₀₁	—	4	1	no	—
58	C₂×C₁₀₁	—	2	2	no	—
59	C₂×C₁₀₁	—	2	1	no	—
60	C₂×C₁₀₁	#399	1	1	yes	characteristic
61	D₁₀₁	—	8	1	no	—
62	#2098	—	101	1	no	maximal, Sylow
63	C₁₀₁	#2098	1	1	yes	characteristic, complemented, Sylow
64	#430	D₁₀₁	1	1	yes	characteristic, complemented
65	#424	—	101	2	no	—
66	#421	—	101	2	no	—
67	#408	—	101	1	no	—
68	H(3,ℤ/4)	—	101	1	no	—
69	C₂×#41	—	101	1	no	—
70	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
71	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
72	C₂²×C₈	—	101	1	no	—
73	#117	—	101	2	no	—
74	#108	—	2	2	no	—
75	C₄⋊C₈	—	202	4	no	—
76	#104	—	202	4	no	—
77	C₂²⋊C₈	—	202	2	no	—
78	#41	—	202	2	no	—
79	#41	—	202	2	no	—
80	C₂×Q₈	—	101	1	no	—
81	C₂×D₄	—	101	1	no	—
82	C₂×D₄	—	202	1	no	—
83	C₂²×C₄	—	202	1	no	—
84	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	202	2	no	—
90	C₂×C₈	—	202	1	no	—
91	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
92	C₄⋊C₄	—	202	2	no	—
93	C₂²⋊C₄	—	202	2	no	—
94	C₄²	—	101	4	no	—
95	C₂³	—	202	1	no	—
96	C₂³	#13108075	1	1	yes	characteristic
97	Q₈	—	202	2	no	—
98	D₄	—	404	2	no	—
99	D₄	—	202	2	no	—
100	C₂×C₄	—	404	1	no	—
101	C₂×C₄	—	404	1	no	—
102	C₂×C₄	—	202	1	no	—
103	C₂×C₄	—	202	4	no	—
104	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₀₁	1	1	yes	characteristic
107	C₂×C₄	#13108074	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	4	2	no	—
110	C₈	—	404	1	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	404	2	no	—
113	C₂²	—	404	1	no	—
114	C₂²	#15692231	1	1	yes	characteristic
115	C₂²	#15692240	1	2	yes	—
116	C₄	—	404	4	no	—
117	C₄	—	404	1	no	—
118	C₄	#15692233	1	2	yes	—
119	C₄	#15692238	1	1	yes	characteristic
120	C₄	#15692251	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	808	1	no	—
124	C₂	#20265112	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265205

C₂

yes

maximal, complemented

#20265193

C₂

yes

maximal

#20265145

C₂

yes

maximal, characteristic

#20264934

C₂

yes

maximal, characteristic, complemented

#430×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#13108074

C₂²

yes

characteristic

C₂²×#11717579

C₂²

yes

characteristic

#15692214

C₂²

yes

—

#15692184

—

#15692181

—

#15692167

—

C₂×M₁₆×C₁₀₁

C₂²

yes

—

C₂×M₁₆×C₁₀₁

C₂²

yes

characteristic

#108×C₁₀₁

—

#13108074

—

#13108074

—

C₂×Dic₂₀₂

Q₈

yes

characteristic

C₂×D₄₀₄

Q₈

yes

characteristic

C₂×C₂²⋊D₁₀₁

—

C₂×C₄×D₁₀₁

—

C₂×#11717579

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

#41

yes

characteristic

C₂²×D₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

D₄₀₄

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

Dic₂₀₂

—

C₈×C₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₁

#123

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#399

yes

characteristic

D₁₀₁

—

#2098

—

101

maximal, Sylow

C₁₀₁

#2098

yes

characteristic, complemented, Sylow

#430

D₁₀₁

yes

characteristic, complemented

#424

—

101

—

#421

—

101

—

#408

—

101

—

H(3,ℤ/4)

—

101

—

C₂×#41

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

—

C₂×M₁₆

C₂×D₁₀₁

yes

characteristic

C₂²×C₈

—

101

—

#117

—

101

—

#108

—

C₄⋊C₈

—

202

—

#104

—

202

—

C₂²⋊C₈

—

202

—

#41

—

202

—

#41

—

202

—

C₂×Q₈

—

101

—

C₂×D₄

—

101

—

C₂×D₄

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

C₂²⋊D₁₀₁

yes

—

C₄⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₄²

—

101

—

C₂³

—

202

—

C₂³

#13108075

yes

characteristic

Q₈

—

202

—

D₄

—

404

—

D₄

—

202

—

100

C₂×C₄

—

404

—

101

C₂×C₄

—

404

—

102

C₂×C₄

—

202

—

103

C₂×C₄

—

202

—

104

C₂×C₄

D₄×D₁₀₁

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

106

C₂×C₄

Q₈×D₁₀₁

yes

characteristic

107

C₂×C₄

#13108074

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

404

—

111

C₂²

—

112

C₂²

—

404

—

113

C₂²

—

404

—

114

C₂²

#15692231

yes

characteristic

115

C₂²

#15692240

yes

—

116

C₄

—

404

—

117

C₄

—

404

—

118

C₄

#15692233

yes

—

119

C₄

#15692238

yes

characteristic

120

C₄

#15692251

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

808

—

124

C₂

#20265112

yes

characteristic

	subgroup	G/H
center	C₄	#15692251
commutator, cosocle	C₂²×C₄×C₁₀₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#430×C₁₀₁	C₂
socle	C₂×C₁₀₁	#399
prime radical	C₂×#13108074	C₂²
hypercenter	#430	D₁₀₁
hypocenter	C₁₀₁	#2098
LCS	G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₄

#15692251

commutator, cosocle

C₂²×C₄×C₁₀₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#430×C₁₀₁

C₂

C₂×C₁₀₁

C₂²

D₁₀₁

C₁₀₁

G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
1. #20265205	C₂	semidirect
4. #20264934	C₂	semidirect
5. #430×C₁₀₁	C₂	semidirect
63. C₁₀₁	#2098	semidirect
64. #430	D₁₀₁	semidirect

H ≅ G/N

type

1. #20265205

C₂

semidirect

4. #20264934

C₂

semidirect

5. #430×C₁₀₁

C₂

semidirect

63. C₁₀₁

#2098

semidirect

64. #430

D₁₀₁

semidirect

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

subgroup	index
#20265205	2
#20265193	2
#20265145	2
#20264934	2
#430×C₁₀₁	2
#2098	101

subgroup

index

#430×C₁₀₁

101

Subgroups — #27172684

Conjugacy classes of subgroups

Complements

Maximal subgroups