Group #27171349 — subgroups

Subgroups — #27171349

Order 12928.

272 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20265655	C₂	1	1	yes	maximal, characteristic
2	#20265219	C₂	1	2	yes	maximal
3	#20265084	C₂	1	1	yes	maximal, characteristic
4	#20264938	C₂	1	2	yes	maximal
5	C₂×#101×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₀₁	—	2	2	no	—
7	Q₈×Dic₁₀₁	—	2	2	no	—
8	#15692214	—	2	2	no	—
9	#15692214	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
12	#15692189	—	2	2	no	—
13	#15692187	—	2	2	no	—
14	#15692162	—	2	2	no	—
15	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
17	#101×C₁₀₁	C₄	1	4	yes	complemented
18	C₂²×Dic₁₀₁	—	2	1	no	—
19	C₂²×Dic₁₀₁	—	2	1	no	—
20	C₄⋊Dic₁₀₁	—	2	2	no	—
21	C₄⋊Dic₁₀₁	—	2	2	no	—
22	C₄⋊Dic₁₀₁	—	2	2	no	—
23	C₄⋊Dic₁₀₁	D₄	1	2	yes	—
24	C₄⋊Dic₁₀₁	D₄	1	2	yes	—
25	C₂²⋊Dic₁₀₁	—	2	2	no	—
26	C₄×Dic₁₀₁	—	2	4	no	—
27	C₄×Dic₁₀₁	—	2	2	no	—
28	C₄×Dic₁₀₁	—	2	2	no	—
29	C₂×Q₈×C₁₀₁	—	2	2	no	—
30	C₂×Q₈×C₁₀₁	—	2	2	no	—
31	C₂×Q₈×C₁₀₁	C₂×C₄	1	2	yes	—
32	C₂×Q₈×C₁₀₁	Q₈	1	2	yes	—
33	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
34	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₀₁	—	2	2	no	—
36	M₁₆×C₁₀₁	C₂×C₄	1	4	yes	—
37	C₂×C₈×C₁₀₁	—	2	2	no	—
38	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₁₀₁	—	2	4	no	—
40	Q₈×C₁₀₁	—	2	4	no	—
41	C₂×C₄×C₁₀₁	—	2	2	no	—
42	C₂×C₄×C₁₀₁	—	2	2	no	—
43	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
44	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₀₁	#41	1	2	yes	—
48	C₂×Dic₁₀₁	—	2	2	no	—
49	C₂×Dic₁₀₁	—	2	2	no	—
50	C₂×Dic₁₀₁	—	2	2	no	—
51	C₂×Dic₁₀₁	—	4	1	no	—
52	C₂×Dic₁₀₁	—	4	1	no	—
53	C₂×Dic₁₀₁	—	4	2	no	—
54	C₂×Dic₁₀₁	—	4	1	no	—
55	C₈×C₁₀₁	—	2	4	no	—
56	C₂²×C₁₀₁	—	2	1	no	—
57	C₂²×C₁₀₁	—	2	1	no	—
58	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₀₁	#120	1	1	yes	characteristic
60	C₂²×C₁₀₁	#123	1	1	yes	characteristic
61	C₄×C₁₀₁	—	2	4	no	—
62	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
63	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
64	Dic₁₀₁	—	4	2	no	—
65	Dic₁₀₁	—	4	2	no	—
66	Dic₁₀₁	—	4	2	no	—
67	Dic₁₀₁	—	8	1	no	—
68	C₂×C₁₀₁	—	2	1	no	—
69	C₂×C₁₀₁	—	2	1	no	—
70	C₂×C₁₀₁	#385	1	1	yes	characteristic
71	C₂×C₁₀₁	#454	1	1	yes	characteristic
72	C₂×C₁₀₁	#456	1	1	yes	characteristic
73	#1857	—	101	1	no	maximal, Sylow
74	C₁₀₁	#1857	1	1	yes	characteristic, complemented, Sylow
75	#519	—	101	1	no	—
76	#428	—	101	2	no	—
77	C₂×#101	D₁₀₁	1	1	yes	characteristic
78	#391	—	101	1	no	—
79	#339	—	101	2	no	—
80	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
81	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
82	C₄×Q₈	—	202	2	no	—
83	C₄×Q₈	—	202	2	no	—
84	#117	—	101	2	no	—
85	#117	—	202	2	no	—
86	C₂×C₄⋊C₄	—	101	1	no	—
87	C₂×C₄²	—	101	1	no	—
88	#107	—	202	2	no	—
89	#106	—	202	2	no	—
90	#101	Dic₁₀₁	1	4	yes	complemented
91	#95	—	202	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	Dic₂₀₂	1	2	yes	—
95	C₂×Q₈	C₂×Dic₁₀₁	1	2	yes	—
96	C₂²×C₄	—	202	1	no	—
97	C₂²×C₄	—	202	1	no	—
98	C₂²×C₄	D₄₀₄	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
100	M₁₆	—	2	2	no	—
101	M₁₆	C₂×Dic₁₀₁	1	4	yes	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	101	2	no	—
104	C₄⋊C₄	—	101	2	no	—
105	C₄⋊C₄	—	202	2	no	—
106	C₄⋊C₄	—	202	2	no	—
107	C₄⋊C₄	—	202	2	no	—
108	C₂²⋊C₄	—	202	2	no	—
109	C₄²	—	202	4	no	—
110	C₄²	—	202	2	no	—
111	C₄²	—	202	2	no	—
112	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	404	2	no	—
118	C₂×C₄	—	404	1	no	—
119	C₂×C₄	—	404	1	no	—
120	C₂×C₄	—	404	1	no	—
121	C₂×C₄	—	202	2	no	—
122	C₂×C₄	—	202	2	no	—
123	C₂×C₄	—	202	2	no	—
124	C₂×C₄	C₄⋊Dic₁₀₁	1	4	yes	—
125	C₂×C₄	C₂²×Dic₁₀₁	1	1	yes	characteristic
126	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
127	C₂×C₄	C₂×Dic₂₀₂	1	1	yes	characteristic
128	C₂×C₄	#13108073	1	2	yes	—
129	C₈	—	2	4	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×C₄⋊Dic₁₀₁	1	1	yes	characteristic
133	C₂²	#15692225	1	1	yes	characteristic
134	C₂²	#15692239	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	808	1	no	—
137	C₄	—	404	2	no	—
138	C₄	—	404	2	no	—
139	C₄	—	404	2	no	—
140	C₄	D₄×Dic₁₀₁	1	2	yes	—
141	C₄	#15692232	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#20265052	1	1	yes	characteristic
145	C₂	#20265352	1	1	yes	characteristic
146	C₂	#20265362	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265655

C₂

yes

maximal, characteristic

#20265219

C₂

yes

maximal

#20265084

C₂

yes

maximal, characteristic

#20264938

C₂

yes

maximal

C₂×#101×C₁₀₁

C₂

yes

maximal, characteristic

Q₈×Dic₁₀₁

—

Q₈×Dic₁₀₁

—

#15692214

—

#15692214

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₁

C₂²

yes

—

—

—

C₂²×Q₈×C₁₀₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₁

C₂²

yes

—

#101×C₁₀₁

C₄

yes

complemented

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

D₄

yes

—

C₄⋊Dic₁₀₁

D₄

yes

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

C₂×C₄

yes

—

C₂×Q₈×C₁₀₁

Q₈

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

C₂×C₄

yes

—

C₂×C₈×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

Q₈×C₁₀₁

—

Q₈×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×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×D₄

yes

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#1857

yes

characteristic, complemented, Sylow

#519

—

101

—

#428

—

101

—

C₂×#101

D₁₀₁

yes

characteristic

#391

—

101

—

#339

—

101

—

C₂²×Q₈

C₂×D₁₀₁

yes

characteristic

C₂×M₁₆

C₂×D₁₀₁

yes

—

C₄×Q₈

—

202

—

C₄×Q₈

—

202

—

#117

—

101

—

#117

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄²

—

101

—

#107

—

202

—

#106

—

202

—

#101

Dic₁₀₁

yes

complemented

#95

—

202

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₂₀₂

yes

—

C₂×Q₈

C₂×Dic₁₀₁

yes

—

C₂²×C₄

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

D₄₀₄

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

100

M₁₆

—

101

M₁₆

C₂×Dic₁₀₁

yes

—

102

C₂×C₈

—

103

C₄⋊C₄

—

101

—

104

C₄⋊C₄

—

101

—

105

C₄⋊C₄

—

202

—

106

C₄⋊C₄

—

202

—

107

C₄⋊C₄

—

202

—

108

C₂²⋊C₄

—

202

—

109

C₄²

—

202

—

110

C₄²

—

202

—

111

C₄²

—

202

—

112

C₂³

C₂×D₄₀₄

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

404

—

118

C₂×C₄

—

404

—

119

C₂×C₄

—

404

—

120

C₂×C₄

—

404

—

121

C₂×C₄

—

202

—

122

C₂×C₄

—

202

—

123

C₂×C₄

—

202

—

124

C₂×C₄

C₄⋊Dic₁₀₁

yes

—

125

C₂×C₄

C₂²×Dic₁₀₁

yes

characteristic

126

C₂×C₄

D₄×D₁₀₁

yes

—

127

C₂×C₄

C₂×Dic₂₀₂

yes

characteristic

128

C₂×C₄

#13108073

yes

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×C₄⋊Dic₁₀₁

yes

characteristic

133

C₂²

#15692225

yes

characteristic

134

C₂²

#15692239

yes

characteristic

135

C₄

—

136

C₄

—

808

—

137

C₄

—

404

—

138

C₄

—

404

—

139

C₄

—

404

—

140

C₄

D₄×Dic₁₀₁

yes

—

141

C₄

#15692232

yes

—

142

C₂

—

143

C₂

—

144

C₂

#20265052

yes

characteristic

145

C₂

#20265352

yes

characteristic

146

C₂

#20265362

yes

characteristic

	subgroup	G/H
center	C₂²	#15692225
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×#101×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#120
prime radical	C₂³×C₁₀₁	C₂×D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×#101	D₁₀₁
hypocenter	C₁₀₁	#1857
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#15692225

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂×#101×C₁₀₁

C₂

socle

C₂²×C₁₀₁

#120

prime radical

C₂³×C₁₀₁

C₂×D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
17. #101×C₁₀₁	C₄	semidirect
74. C₁₀₁	#1857	semidirect
90. #101	Dic₁₀₁	semidirect

H ≅ G/N

type

17. #101×C₁₀₁

C₄

semidirect

74. C₁₀₁

#1857

semidirect

90. #101

Dic₁₀₁

semidirect

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

subgroup	index
#20265655	2
#20265219	2
#20265084	2
#20264938	2
C₂×#101×C₁₀₁	2
#1857	101

subgroup

index

C₂×#101×C₁₀₁

101

Subgroups — #27171349

Conjugacy classes of subgroups

Complements

Maximal subgroups