Group #27171241 — subgroups

Subgroups — #27171241

Order 12928.

272 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 81 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	#20265652	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692172	C₂	1	1	yes	maximal, characteristic
3	#20264949	C₂	1	2	yes	maximal, complemented
4	#20264948	C₂	1	2	yes	maximal, complemented
5	C₂×#100×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108058	C₂²	1	2	yes	—
7	D₄×Dic₁₀₁	—	2	2	no	—
8	D₄×Dic₁₀₁	—	2	2	no	—
9	#15692214	C₂²	1	1	yes	characteristic
10	#15692214	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Dic₁₀₁	C₄	1	4	yes	—
12	#15692172	—	2	2	no	—
13	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
15	#100×C₁₀₁	—	2	2	no	—
16	C₂²×Dic₁₀₁	—	2	4	no	—
17	#13108058	—	2	2	no	—
18	#13108058	—	2	2	no	—
19	C₂×#11717579	—	2	2	no	—
20	C₄⋊Dic₁₀₁	—	2	1	no	—
21	C₄⋊Dic₁₀₁	—	2	1	no	—
22	C₂²⋊Dic₁₀₁	—	2	4	no	—
23	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
24	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₀₁	—	2	1	no	—
26	C₄×Dic₁₀₁	—	2	1	no	—
27	C₂⁴×C₁₀₁	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₁₀₁	—	2	2	no	—
29	C₂×D₄×C₁₀₁	—	2	2	no	—
30	C₂×D₄×C₁₀₁	D₄	1	2	yes	—
31	C₂×D₄×C₁₀₁	D₄	1	2	yes	—
32	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₀₁	—	2	2	no	—
34	M₁₆×C₁₀₁	—	2	2	no	—
35	C₂×C₈×C₁₀₁	—	2	2	no	—
36	C₂³×C₁₀₁	—	2	2	no	—
37	C₂³×C₁₀₁	—	2	2	no	—
38	C₂³×C₁₀₁	—	8	2	no	—
39	C₂³×C₁₀₁	C₂²⋊C₄	1	4	yes	—
40	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
41	D₄×C₁₀₁	—	2	4	no	—
42	D₄×C₁₀₁	—	2	4	no	—
43	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₁	#41	1	2	yes	—
46	C₂×C₄×C₁₀₁	#41	1	2	yes	—
47	C₂×Dic₁₀₁	—	2	2	no	—
48	C₂×Dic₁₀₁	—	2	2	no	—
49	C₂×Dic₁₀₁	—	8	4	no	—
50	C₈×C₁₀₁	—	4	2	no	—
51	#11717579	—	4	2	no	—
52	C₂²×C₁₀₁	—	2	4	no	—
53	C₂²×C₁₀₁	—	4	1	no	—
54	C₂²×C₁₀₁	—	4	4	no	—
55	C₂²×C₁₀₁	—	8	2	no	—
56	C₂²×C₁₀₁	—	8	2	no	—
57	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₀₁	#117	1	1	yes	characteristic
59	C₂²×C₁₀₁	#117	1	1	yes	characteristic
60	C₄×C₁₀₁	#123	1	2	yes	—
61	C₄×C₁₀₁	#123	1	2	yes	—
62	Dic₁₀₁	—	4	2	no	—
63	Dic₁₀₁	—	4	2	no	—
64	C₂×C₁₀₁	—	2	1	no	—
65	C₂×C₁₀₁	—	2	1	no	—
66	C₂×C₁₀₁	—	2	1	no	—
67	C₂×C₁₀₁	—	4	4	no	—
68	C₂×C₁₀₁	#381	1	1	yes	characteristic
69	#1829	—	101	1	no	maximal, Sylow
70	C₁₀₁	#1829	1	1	yes	characteristic, complemented, Sylow
71	#518	—	101	1	no	—
72	C₂×#100	—	101	1	no	—
73	C₂×#100	D₁₀₁	1	1	yes	characteristic
74	#344	—	101	2	no	—
75	#344	—	101	2	no	—
76	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
77	C₂×M₁₆	—	101	2	no	—
78	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
79	C₄×D₄	—	202	2	no	—
80	C₄×D₄	—	202	2	no	—
81	#117	—	101	1	no	—
82	#117	—	101	1	no	—
83	C₂×C₂²⋊C₄	—	101	4	no	—
84	#100	—	2	2	no	—
85	#100	—	202	2	no	—
86	C₂⁴	C₄×D₁₀₁	1	2	yes	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	D₄₀₄	1	2	yes	—
90	C₂×D₄	C₂²⋊D₁₀₁	1	2	yes	—
91	C₂²×C₄	—	202	4	no	—
92	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	202	2	no	—
96	M₁₆	—	202	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	202	2	no	—
99	C₄⋊C₄	—	202	1	no	—
100	C₄⋊C₄	—	202	1	no	—
101	C₂²⋊C₄	—	101	2	no	—
102	C₂²⋊C₄	—	101	2	no	—
103	C₂²⋊C₄	—	202	4	no	—
104	C₄²	—	202	1	no	—
105	C₄²	—	202	1	no	—
106	C₂³	—	2	2	no	—
107	C₂³	—	2	2	no	—
108	C₂³	—	8	2	no	—
109	C₂³	#13108051	1	4	yes	—
110	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
111	D₄	—	2	4	no	—
112	D₄	—	2	4	no	—
113	C₂×C₄	—	808	4	no	—
114	C₂×C₄	—	202	2	no	—
115	C₂×C₄	—	202	2	no	—
116	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
117	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
118	C₂×C₄	#13108073	1	2	yes	—
119	C₂×C₄	#13108073	1	2	yes	—
120	C₈	—	4	2	no	—
121	C₈	—	404	2	no	—
122	C₂²	—	2	4	no	—
123	C₂²	—	4	1	no	—
124	C₂²	—	4	4	no	—
125	C₂²	—	8	2	no	—
126	C₂²	—	8	2	no	—
127	C₂²	C₂×#13108051	1	1	yes	characteristic
128	C₂²	#15692211	1	1	yes	characteristic
129	C₂²	#15692211	1	1	yes	characteristic
130	C₄	—	404	2	no	—
131	C₄	—	404	2	no	—
132	C₄	#15692236	1	2	yes	—
133	C₄	#15692239	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	—	2	1	no	—
137	C₂	—	4	4	no	—
138	C₂	#20265038	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265652

C₂

yes

maximal, characteristic

C₂×#15692172

C₂

yes

maximal, characteristic

#20264949

C₂

yes

maximal, complemented

#20264948

C₂

yes

maximal, complemented

C₂×#100×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

D₄×Dic₁₀₁

—

D₄×Dic₁₀₁

—

#15692214

C₂²

yes

characteristic

#15692214

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₀₁

C₄

yes

—

#15692172

—

C₂²×D₄×C₁₀₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₁

C₂²

yes

—

#100×C₁₀₁

—

C₂²×Dic₁₀₁

—

#13108058

—

#13108058

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

C₂×C₄

yes

—

C₂²⋊Dic₁₀₁

C₂×C₄

yes

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂⁴×C₁₀₁

C₂×C₄

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₄×C₁₀₁

yes

—

C₄×C₁₀₁

#123

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#381

yes

characteristic

#1829

—

101

maximal, Sylow

C₁₀₁

#1829

yes

characteristic, complemented, Sylow

#518

—

101

—

C₂×#100

—

101

—

C₂×#100

D₁₀₁

yes

characteristic

#344

—

101

—

#344

—

101

—

C₂²×D₄

C₂×D₁₀₁

yes

characteristic

C₂×M₁₆

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

—

C₄×D₄

—

202

—

C₄×D₄

—

202

—

#117

—

101

—

#117

—

101

—

C₂×C₂²⋊C₄

—

101

—

#100

—

#100

—

202

—

C₂⁴

C₄×D₁₀₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄₀₄

yes

—

C₂×D₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

202

—

M₁₆

—

202

—

C₂×C₈

—

C₂×C₈

—

202

—

C₄⋊C₄

—

202

—

100

C₄⋊C₄

—

202

—

101

C₂²⋊C₄

—

101

—

102

C₂²⋊C₄

—

101

—

103

C₂²⋊C₄

—

202

—

104

C₄²

—

202

—

105

C₄²

—

202

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

#13108051

yes

—

110

C₂³

C₂×C₄×D₁₀₁

yes

characteristic

111

D₄

—

112

D₄

—

113

C₂×C₄

—

808

—

114

C₂×C₄

—

202

—

115

C₂×C₄

—

202

—

116

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

117

C₂×C₄

C₂×D₄₀₄

yes

characteristic

118

C₂×C₄

#13108073

yes

—

119

C₂×C₄

#13108073

yes

—

120

C₈

—

121

C₈

—

404

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×#13108051

yes

characteristic

128

C₂²

#15692211

yes

characteristic

129

C₂²

#15692211

yes

characteristic

130

C₄

—

404

—

131

C₄

—

404

—

132

C₄

#15692236

yes

—

133

C₄

#15692239

yes

—

134

C₂

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#20265038

yes

characteristic

	subgroup	G/H
center	C₂	#20265038
commutator	C₂³×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×#100×C₁₀₁	C₂
socle	C₂×C₁₀₁	#381
prime radical	C₂²×D₄×C₁₀₁	C₂²
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×#100	D₁₀₁
hypocenter	C₁₀₁	#1829
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#100

subgroup

G/H

center

C₂

#20265038

commutator

C₂³×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂×#100×C₁₀₁

C₂

socle

C₂×C₁₀₁

#381

prime radical

C₂²×D₄×C₁₀₁

C₂²

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#100

N	H ≅ G/N	type
3. #20264949	C₂	semidirect
4. #20264948	C₂	semidirect
70. C₁₀₁	#1829	semidirect

H ≅ G/N

type

3. #20264949

C₂

semidirect

4. #20264948

C₂

semidirect

70. C₁₀₁

#1829

semidirect

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

subgroup	index
#20265652	2
C₂×#15692172	2
#20264949	2
#20264948	2
C₂×#100×C₁₀₁	2
#1829	101

subgroup

index

C₂×#100×C₁₀₁

101

Subgroups — #27171241

Conjugacy classes of subgroups

Complements

Maximal subgroups