Group #27171173 — subgroups

Subgroups — #27171173

Order 12928.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×#13108057	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692190	C₂	1	2	yes	maximal, complemented
3	#20265140	C₂	1	2	yes	maximal
4	#20264942	C₂	1	1	yes	maximal, characteristic, complemented
5	#341×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
7	C₂×#13108058	C₂²	1	2	yes	—
8	C₂×#13108057	—	2	4	no	—
9	C₂×#13108057	C₄	1	4	yes	—
10	C₂²×#11717579	C₂²	1	1	yes	characteristic
11	#15692190	—	2	4	no	—
12	#15692167	—	2	4	no	—
13	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂³×D₁₀₁	Q₈	1	1	yes	characteristic
16	C₂²×Dic₁₀₁	Q₈	1	1	yes	characteristic
17	C₂×C₄×D₁₀₁	—	2	2	no	—
18	C₂×C₄×D₁₀₁	—	2	2	no	—
19	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
21	#13108058	—	2	2	no	—
22	#13108058	—	2	2	no	—
23	#13108057	—	2	8	no	—
24	C₂×#11717579	—	2	2	no	—
25	C₂×#11717579	C₂×C₄	1	2	yes	—
26	C₂×#11717579	D₄	1	4	yes	—
27	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₀₁	—	2	2	no	—
29	M₁₆×C₁₀₁	—	2	2	no	—
30	C₂×C₈×C₁₀₁	—	2	2	no	—
31	C₂×C₈×C₁₀₁	—	2	2	no	—
32	C₂×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
33	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
34	C₂²×D₁₀₁	—	2	2	no	—
35	C₂²×D₁₀₁	—	4	2	no	—
36	C₂²×D₁₀₁	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	C₂×Dic₁₀₁	—	2	2	no	—
42	C₂×Dic₁₀₁	C₄⋊C₄	1	2	yes	—
43	C₄×D₁₀₁	—	2	2	no	—
44	C₄×D₁₀₁	—	2	2	no	—
45	C₄×D₁₀₁	—	2	4	no	—
46	C₈×C₁₀₁	—	2	2	no	—
47	C₈×C₁₀₁	—	4	2	no	—
48	#11717579	—	4	2	no	—
49	#11717579	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₁₀₁	—	2	2	no	—
51	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
53	C₂×D₁₀₁	—	2	2	no	—
54	C₂×D₁₀₁	—	2	4	no	—
55	C₂×D₁₀₁	—	4	4	no	—
56	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₁₀₁	#117	1	1	yes	characteristic
58	C₄×C₁₀₁	#121	1	2	yes	—
59	Dic₁₀₁	—	2	2	no	—
60	C₂×C₁₀₁	—	2	2	no	—
61	C₂×C₁₀₁	#380	1	1	yes	characteristic
62	C₂×C₁₀₁	#430	1	2	yes	—
63	D₁₀₁	—	4	2	no	—
64	#1811	—	101	1	no	maximal, Sylow
65	C₁₀₁	#1811	1	1	yes	characteristic, complemented, Sylow
66	C₂²×M₁₆	—	101	1	no	—
67	C₂×#108	—	101	2	no	—
68	#407	—	101	2	no	—
69	#341	—	101	1	no	—
70	#341	D₁₀₁	1	1	yes	characteristic, complemented
71	C₂³×C₄	—	101	1	no	—
72	C₂×M₁₆	—	101	4	no	—
73	C₂×M₁₆	—	101	2	no	—
74	C₂×M₁₆	—	202	4	no	—
75	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
76	C₂²×C₈	—	101	1	no	—
77	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
78	#108	—	202	4	no	—
79	C₂²⋊C₈	—	202	4	no	—
80	C₂⁴	—	101	1	no	—
81	C₂²×C₄	—	101	1	no	—
82	C₂²×C₄	—	101	2	no	—
83	C₂²×C₄	—	101	2	no	—
84	C₂²×C₄	—	202	2	no	—
85	C₂²×C₄	—	202	2	no	—
86	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	202	8	no	—
90	M₁₆	—	202	2	no	—
91	M₁₆	—	202	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	101	2	no	—
95	C₂×C₈	—	101	4	no	—
96	C₂×C₈	—	202	2	no	—
97	C₂×C₈	C₄×D₁₀₁	1	2	yes	—
98	C₂³	—	101	2	no	—
99	C₂³	—	404	2	no	—
100	C₂³	—	202	2	no	—
101	C₂³	Q₈×D₁₀₁	1	1	yes	characteristic
102	C₂×C₄	—	101	2	no	—
103	C₂×C₄	—	202	2	no	—
104	C₂×C₄	—	202	2	no	—
105	C₂×C₄	—	202	2	no	—
106	C₂×C₄	—	202	4	no	—
107	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
108	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
109	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
110	C₂×C₄	#13108074	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	101	4	no	—
114	C₈	—	404	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	404	4	no	—
117	C₂²	—	202	4	no	—
118	C₂²	—	202	2	no	—
119	C₂²	C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
120	C₂²	#15692233	1	2	yes	—
121	C₄	—	202	2	no	—
122	C₄	C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
123	C₄	#15692212	1	1	yes	characteristic
124	C₄	#15692228	1	2	yes	—
125	C₂	—	2	2	no	—
126	C₂	—	404	2	no	—
127	C₂	#20265036	1	1	yes	characteristic
128	C₂	#20265224	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13108057

C₂

yes

maximal, characteristic

C₂×#15692190

C₂

yes

maximal, complemented

#20265140

C₂

yes

maximal

#20264942

C₂

yes

maximal, characteristic, complemented

#341×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₀₁

C₂²

yes

characteristic

C₂×#13108058

C₂²

yes

—

C₂×#13108057

—

C₂×#13108057

C₄

yes

—

C₂²×#11717579

C₂²

yes

characteristic

#15692190

—

#15692167

—

C₂×M₁₆×C₁₀₁

C₂²

yes

—

C₂²×C₈×C₁₀₁

C₂²

yes

characteristic

C₂³×D₁₀₁

Q₈

yes

characteristic

C₂²×Dic₁₀₁

Q₈

yes

characteristic

C₂×C₄×D₁₀₁

—

C₂×C₄×D₁₀₁

—

C₂×C₄×D₁₀₁

C₂×C₄

yes

—

C₂×C₄×D₁₀₁

D₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#11717579

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

C₂×C₄

yes

—

C₂³×C₁₀₁

C₂×Q₈

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

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

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₄⋊C₄

yes

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₈×C₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

#11717579

C₂²⋊C₄

yes

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₂²⋊Q₈

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

C₂×C₂²⋊C₄

yes

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#380

yes

characteristic

C₂×C₁₀₁

#430

yes

—

D₁₀₁

—

#1811

—

101

maximal, Sylow

C₁₀₁

#1811

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

101

—

C₂×#108

—

101

—

#407

—

101

—

#341

—

101

—

#341

D₁₀₁

yes

characteristic, complemented

C₂³×C₄

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

202

—

C₂×M₁₆

C₂×D₁₀₁

yes

—

C₂²×C₈

—

101

—

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

#108

—

202

—

C₂²⋊C₈

—

202

—

C₂⁴

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

202

—

M₁₆

—

202

—

M₁₆

—

202

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

C₄×D₁₀₁

yes

—

C₂³

—

101

—

C₂³

—

404

—

100

C₂³

—

202

—

101

C₂³

Q₈×D₁₀₁

yes

characteristic

102

C₂×C₄

—

101

—

103

C₂×C₄

—

202

—

104

C₂×C₄

—

202

—

105

C₂×C₄

—

202

—

106

C₂×C₄

—

202

—

107

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

108

C₂×C₄

D₄×D₁₀₁

yes

—

109

C₂×C₄

D₄×D₁₀₁

yes

characteristic

110

C₂×C₄

#13108074

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

101

—

114

C₈

—

404

—

115

C₂²

—

116

C₂²

—

404

—

117

C₂²

—

202

—

118

C₂²

—

202

—

119

C₂²

C₄⋊C₄×D₁₀₁

yes

characteristic

120

C₂²

#15692233

yes

—

121

C₄

—

202

—

122

C₄

C₂²⋊C₄×D₁₀₁

yes

characteristic

123

C₄

#15692212

yes

characteristic

124

C₄

#15692228

yes

—

125

C₂

—

126

C₂

—

404

—

127

C₂

#20265036

yes

characteristic

128

C₂

#20265224

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₁
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#341×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₄⋊C₄
prime radical	C₂³×D₁₀₁	Q₈
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#341	D₁₀₁
hypocenter	C₁₀₁	#1811
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₁

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#341×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₄⋊C₄

prime radical

C₂³×D₁₀₁

Q₈

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂×#15692190	C₂	semidirect
4. #20264942	C₂	semidirect
5. #341×C₁₀₁	C₂	semidirect
65. C₁₀₁	#1811	semidirect
70. #341	D₁₀₁	semidirect

H ≅ G/N

type

2. C₂×#15692190

C₂

semidirect

4. #20264942

C₂

semidirect

5. #341×C₁₀₁

C₂

semidirect

65. C₁₀₁

#1811

semidirect

70. #341

D₁₀₁

semidirect

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

subgroup	index
C₂²×#13108057	2
C₂×#15692190	2
#20265140	2
#20264942	2
#341×C₁₀₁	2
#1811	101

subgroup

index

#341×C₁₀₁

101

Subgroups — #27171173

Conjugacy classes of subgroups

Complements

Maximal subgroups