Group #27187776 — subgroups

Subgroups — #27187776

Order 12928.

484 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 202 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×D₁₀₁⋊M₁₆	C₂	1	2	yes	maximal
2	#20266184	C₂	1	8	yes	maximal, complemented
3	#20266180	C₂	1	4	yes	maximal, complemented
4	C₂×#15692213	C₂	1	1	yes	maximal, characteristic
5	C₂×#13108084	C₂²	1	4	yes	—
6	D₁₀₁⋊M₁₆	—	2	8	no	—
7	C₂×#13108083	C₂²	1	4	yes	—
8	C₂×D₁₀₁⋊C₈	C₂²	1	4	yes	—
9	#15692322	C₂²	1	8	yes	complemented
10	C₄⋊#11717581	C₂²	1	8	yes	complemented
11	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
12	#15692213	C₄	1	4	yes	—
13	#15692213	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₁₀₁	C₄	1	1	yes	characteristic
15	C₂×#13108051	C₄	1	2	yes	—
16	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₀₁	C₄	1	1	yes	characteristic
18	#13108084	—	2	8	no	—
19	#13108083	—	2	8	no	—
20	D₁₀₁⋊C₈	—	2	8	no	—
21	C₂×#11717581	C₂³	1	8	yes	—
22	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
24	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
25	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
27	C₂×C₄×D₁₀₁	Q₈	1	4	yes	—
28	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
29	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
30	#13108051	C₂×C₄	1	8	yes	—
31	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
32	C₄×Dic₁₀₁	C₂³	1	4	yes	—
33	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
36	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
37	C₂²×D₁₀₁	—	4	1	no	—
38	C₂²×D₁₀₁	—	4	1	no	—
39	C₂²×D₁₀₁	C₄⋊C₄	1	4	yes	—
40	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₁₀₁	—	4	2	no	—
42	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
43	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
45	C₂×Dic₁₀₁	—	4	2	no	—
46	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
47	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
49	C₂×Dic₁₀₁	C₂×Q₈	1	2	yes	—
50	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
51	C₄×D₁₀₁	C₄⋊C₄	1	8	yes	—
52	C₄×D₁₀₁	C₂×D₄	1	4	yes	—
53	C₄×D₁₀₁	C₂×Q₈	1	4	yes	—
54	#11717581	—	2	8	no	—
55	C₂²×C₁₀₁	—	2	1	no	—
56	C₂²×C₁₀₁	—	2	1	no	—
57	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
58	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
59	C₂×D₁₀₁	—	2	2	no	—
60	C₂×D₁₀₁	—	2	2	no	—
61	C₂×D₁₀₁	—	4	4	no	—
62	C₂×D₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
63	C₄×C₁₀₁	—	2	4	no	—
64	C₄×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
65	Dic₁₀₁	—	2	4	no	—
66	Dic₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
67	Dic₁₀₁	C₂²×D₄	1	1	yes	characteristic
68	Dic₁₀₁	C₂²×Q₈	1	1	yes	characteristic
69	C₂×C₁₀₁	—	2	2	no	—
70	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₁₀₁	#568	1	1	yes	characteristic
72	C₂×C₁₀₁	#568	1	1	yes	characteristic
73	D₁₀₁	—	2	4	no	—
74	#2905	—	101	1	no	maximal, Sylow
75	C₁₀₁	#2905	1	1	yes	characteristic, complemented, Sylow
76	C₂²×M₁₆	—	101	2	no	—
77	C₂×#117	—	101	1	no	—
78	#424	—	101	8	no	—
79	#423	—	101	4	no	—
80	C₂³×C₄	—	101	1	no	—
81	C₂×M₁₆	—	101	4	no	—
82	C₂×M₁₆	—	101	4	no	—
83	C₂×M₁₆	—	202	8	no	—
84	C₂²×C₈	—	101	4	no	—
85	#117	—	101	4	no	—
86	#117	—	101	4	no	—
87	C₂×C₄⋊C₄	—	101	1	no	—
88	C₂×C₄⋊C₄	#87938	1	1	yes	characteristic
89	C₂×C₂²⋊C₄	—	101	2	no	—
90	C₂×C₄²	—	101	2	no	—
91	C₄⋊C₈	—	101	8	no	—
92	C₄⋊C₈	—	101	8	no	—
93	C₂⁴	—	101	1	no	—
94	C₂²×C₄	—	101	1	no	—
95	C₂²×C₄	—	101	4	no	—
96	C₂²×C₄	—	101	4	no	—
97	C₂²×C₄	—	101	2	no	—
98	C₂²×C₄	—	101	2	no	—
99	C₂²×C₄	—	101	2	no	—
100	C₂²×C₄	C₂×#87938	1	2	yes	—
101	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
102	M₁₆	—	202	8	no	—
103	M₁₆	—	202	8	no	—
104	C₂×C₈	—	101	8	no	—
105	C₂×C₈	—	202	8	no	—
106	C₄⋊C₄	—	101	4	no	—
107	C₄⋊C₄	C₂×#87938	1	4	yes	—
108	C₂²⋊C₄	—	101	8	no	—
109	C₄²	—	101	4	no	—
110	C₄²	—	101	4	no	—
111	C₂³	—	101	4	no	—
112	C₂³	—	101	2	no	—
113	C₂³	—	404	1	no	—
114	C₂³	—	404	1	no	—
115	C₂³	C₂²×#87938	1	1	yes	characteristic
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	101	2	no	—
118	C₂×C₄	—	101	2	no	—
119	C₂×C₄	—	101	1	no	—
120	C₂×C₄	—	101	1	no	—
121	C₂×C₄	—	101	4	no	—
122	C₂×C₄	—	101	8	no	—
123	C₂×C₄	—	101	4	no	—
124	C₂×C₄	—	101	4	no	—
125	C₂×C₄	—	404	2	no	—
126	C₂×C₄	C₄⋊#87938	1	4	yes	—
127	C₂×C₄	C₂²×#87938	1	4	yes	—
128	C₂×C₄	C₂²×#87938	1	2	yes	—
129	C₈	—	202	8	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	101	4	no	—
133	C₂²	—	404	4	no	—
134	C₂²	—	202	2	no	—
135	C₂²	—	202	2	no	—
136	C₂²	C₂×C₄⋊#87938	1	2	yes	—
137	C₂²	C₂³×#87938	1	1	yes	characteristic
138	C₄	—	2	4	no	—
139	C₄	—	101	1	no	—
140	C₄	—	101	2	no	—
141	C₄	—	101	1	no	—
142	C₄	—	202	4	no	—
143	C₄	C₂×C₄⋊#87938	1	4	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	202	4	no	—
146	C₂	C₂²×C₄⋊#87938	1	1	yes	characteristic
147	C₂	#20266255	1	1	yes	characteristic
148	C₂	#20266257	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₀₁⋊M₁₆

C₂

yes

maximal

#20266184

C₂

yes

maximal, complemented

#20266180

C₂

yes

maximal, complemented

C₂×#15692213

C₂

yes

maximal, characteristic

C₂×#13108084

C₂²

yes

—

D₁₀₁⋊M₁₆

—

C₂×#13108083

C₂²

yes

—

C₂×D₁₀₁⋊C₈

C₂²

yes

—

#15692322

C₂²

yes

complemented

C₄⋊#11717581

C₂²

yes

complemented

C₂²×C₄×D₁₀₁

C₂²

yes

characteristic

#15692213

C₄

yes

—

#15692213

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₄

yes

characteristic

C₂×#13108051

C₄

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₁

C₄

yes

characteristic

#13108084

—

#13108083

—

D₁₀₁⋊C₈

—

C₂×#11717581

C₂³

yes

—

C₂³×D₁₀₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂²×Dic₁₀₁

C₂³

yes

characteristic

C₂×C₄×D₁₀₁

C₂×C₄

yes

—

C₂×C₄×D₁₀₁

D₄

yes

—

C₂×C₄×D₁₀₁

Q₈

yes

—

C₂×C₄×D₁₀₁

C₂³

yes

—

C₄⋊Dic₁₀₁

C₂×C₄

yes

—

#13108051

C₂×C₄

yes

—

C₄×Dic₁₀₁

C₂×C₄

yes

—

C₄×Dic₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

C₄⋊C₄

yes

—

C₂²×D₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²×C₄

yes

—

C₂×Dic₁₀₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₁

C₂×D₄

yes

—

C₂×Dic₁₀₁

C₂×Q₈

yes

—

C₂×Dic₁₀₁

C₂⁴

yes

characteristic

C₄×D₁₀₁

C₄⋊C₄

yes

—

C₄×D₁₀₁

C₂×D₄

yes

—

C₄×D₁₀₁

C₂×Q₈

yes

—

#11717581

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

—

C₂²×C₁₀₁

C₂³×C₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×C₄⋊C₄

yes

—

Dic₁₀₁

—

Dic₁₀₁

C₂×C₄⋊C₄

yes

—

Dic₁₀₁

C₂²×D₄

yes

characteristic

Dic₁₀₁

C₂²×Q₈

yes

characteristic

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂²×C₄⋊C₄

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

D₁₀₁

—

#2905

—

101

maximal, Sylow

C₁₀₁

#2905

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

101

—

C₂×#117

—

101

—

#424

—

101

—

#423

—

101

—

C₂³×C₄

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

202

—

C₂²×C₈

—

101

—

#117

—

101

—

#117

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

#87938

yes

characteristic

C₂×C₂²⋊C₄

—

101

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

101

—

C₄⋊C₈

—

101

—

C₂⁴

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

100

C₂²×C₄

C₂×#87938

yes

—

101

C₂²×C₄

C₂×#87938

yes

characteristic

102

M₁₆

—

202

—

103

M₁₆

—

202

—

104

C₂×C₈

—

101

—

105

C₂×C₈

—

202

—

106

C₄⋊C₄

—

101

—

107

C₄⋊C₄

C₂×#87938

yes

—

108

C₂²⋊C₄

—

101

—

109

C₄²

—

101

—

110

C₄²

—

101

—

111

C₂³

—

101

—

112

C₂³

—

101

—

113

C₂³

—

404

—

114

C₂³

—

404

—

115

C₂³

C₂²×#87938

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

101

—

118

C₂×C₄

—

101

—

119

C₂×C₄

—

101

—

120

C₂×C₄

—

101

—

121

C₂×C₄

—

101

—

122

C₂×C₄

—

101

—

123

C₂×C₄

—

101

—

124

C₂×C₄

—

101

—

125

C₂×C₄

—

404

—

126

C₂×C₄

C₄⋊#87938

yes

—

127

C₂×C₄

C₂²×#87938

yes

—

128

C₂×C₄

C₂²×#87938

yes

—

129

C₈

—

202

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

101

—

133

C₂²

—

404

—

134

C₂²

—

202

—

135

C₂²

—

202

—

136

C₂²

C₂×C₄⋊#87938

yes

—

137

C₂²

C₂³×#87938

yes

characteristic

138

C₄

—

139

C₄

—

101

—

140

C₄

—

101

—

141

C₄

—

101

—

142

C₄

—

202

—

143

C₄

C₂×C₄⋊#87938

yes

—

144

C₂

—

145

C₂

—

202

—

146

C₂

C₂²×C₄⋊#87938

yes

characteristic

147

C₂

#20266255

yes

characteristic

148

C₂

#20266257

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂×C₄⋊C₄×C₁₀₁	C₄
prime radical	C₂³×D₁₀₁	C₂×C₄
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	C₂×C₄⋊C₄	#87938
hypocenter	C₁₀₁	#2905
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂³×#87938

commutator, socle

C₂²×C₁₀₁

C₂³×C₄

Fitting

C₂×C₄⋊C₄×C₁₀₁

C₄

prime radical

C₂³×D₁₀₁

C₂×C₄

cosocle

C₂×Dic₁₀₁

C₂⁴

C₂×C₄⋊C₄

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
2. #20266184	C₂	semidirect
3. #20266180	C₂	semidirect
9. #15692322	C₂²	semidirect
10. C₄⋊#11717581	C₂²	semidirect
75. C₁₀₁	#2905	semidirect

H ≅ G/N

type

2. #20266184

C₂

semidirect

3. #20266180

C₂

semidirect

9. #15692322

C₂²

semidirect

10. C₄⋊#11717581

C₂²

semidirect

75. C₁₀₁

#2905

semidirect

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

subgroup	index
C₂×D₁₀₁⋊M₁₆	2
#20266184	2
#20266180	2
C₂×#15692213	2
#2905	101

subgroup

index

C₂×D₁₀₁⋊M₁₆

101

Subgroups — #27187776

Conjugacy classes of subgroups

Complements

Maximal subgroups