Group #27179777 — subgroups

Subgroups — #27179777

Order 12928.

400 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₄×#13108074	C₂	1	1	yes	maximal, characteristic
2	#20265239	C₂	1	4	yes	maximal, complemented
3	#20265228	C₂	1	4	yes	maximal, complemented
4	#20265205	C₂	1	2	yes	maximal, complemented
5	#20265145	C₂	1	2	yes	maximal
6	C₂×C₄×#11717579	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13108074	C₂²	1	1	yes	characteristic
9	C₂²×#11717579	C₂²	1	2	yes	—
10	C₄×Dic₂₀₂	C₄	1	2	yes	—
11	C₄×D₄₀₄	C₄	1	2	yes	—
12	C₄×C₂²⋊D₁₀₁	C₄	1	4	yes	—
13	#15692214	C₂²	1	1	yes	characteristic
14	#15692212	C₂²	1	2	yes	—
15	C₄²×D₁₀₁	C₂²	1	2	yes	—
16	#15692184	C₂²	1	8	yes	—
17	#15692167	C₂²	1	8	yes	—
18	C₄×#11717579	C₂²	1	2	yes	—
19	C₄×#11717579	C₂²	1	2	yes	—
20	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
22	#97×C₁₀₁	C₂²	1	2	yes	—
23	C₄×C₈×C₁₀₁	C₂²	1	2	yes	—
24	#13108074	—	2	4	no	—
25	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄₀₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
30	C₂×#11717579	—	2	4	no	—
31	C₂×#11717579	C₂³	1	4	yes	—
32	C₄⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
33	#13108053	C₂×C₄	1	4	yes	—
34	C₂²⋊Dic₁₀₁	C₂×C₄	1	2	yes	—
35	#13108051	C₂×C₄	1	4	yes	—
36	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₀₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
40	M₁₆×C₁₀₁	—	2	4	no	—
41	C₂×C₈×C₁₀₁	C₂³	1	4	yes	—
42	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
43	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
44	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
45	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₀₁	—	2	4	no	—
48	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₁₀₁	#41	1	4	yes	—
52	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
53	C₂²⋊D₁₀₁	—	2	4	no	—
54	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
55	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
56	D₄₀₄	—	2	2	no	—
57	C₄×D₁₀₁	—	2	8	no	—
58	C₄×D₁₀₁	—	2	4	no	—
59	Dic₂₀₂	—	2	2	no	—
60	C₈×C₁₀₁	—	2	4	no	—
61	#11717579	—	2	4	no	—
62	C₂²×C₁₀₁	—	2	2	no	—
63	C₂²×C₁₀₁	#117	1	2	yes	—
64	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₁₀₁	—	2	2	no	—
66	C₂×D₁₀₁	—	2	4	no	—
67	C₄×C₁₀₁	—	2	2	no	—
68	C₄×C₁₀₁	#117	1	2	yes	—
69	C₄×C₁₀₁	#131	1	4	yes	—
70	C₄×C₁₀₁	C₂×#41	1	2	yes	—
71	Dic₁₀₁	—	2	2	no	—
72	Dic₁₀₁	—	2	4	no	—
73	C₂×C₁₀₁	—	2	2	no	—
74	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
75	C₂×C₁₀₁	C₂×#131	1	2	yes	—
76	D₁₀₁	—	4	2	no	—
77	#2922	—	101	1	no	maximal, Sylow
78	C₁₀₁	#2922	1	1	yes	characteristic, complemented, Sylow
79	C₄×#41	—	101	1	no	—
80	#433	—	101	4	no	—
81	#431	—	101	4	no	—
82	#424	—	101	2	no	—
83	#408	—	101	2	no	—
84	C₄×M₁₆	D₁₀₁	1	1	yes	characteristic, complemented
85	C₂×C₄×C₈	—	101	1	no	—
86	C₂×#41	—	101	1	no	—
87	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
88	C₂²×C₈	—	101	2	no	—
89	C₄×Q₈	—	101	2	no	—
90	C₄×D₄	—	101	4	no	—
91	C₄×D₄	—	101	2	no	—
92	#117	—	101	1	no	—
93	#117	—	101	2	no	—
94	C₂×C₄²	—	101	2	no	—
95	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
96	C₄⋊C₈	—	101	8	no	—
97	C₂²⋊C₈	—	101	8	no	—
98	#97	C₂×D₁₀₁	1	2	yes	—
99	C₄×C₈	—	101	2	no	—
100	C₄×C₈	—	101	2	no	—
101	C₄×C₈	C₂×D₁₀₁	1	2	yes	—
102	#41	—	202	4	no	—
103	C₂×Q₈	—	101	1	no	—
104	C₂×D₄	—	101	2	no	—
105	C₂×D₄	—	101	1	no	—
106	C₂²×C₄	—	101	4	no	—
107	C₂²×C₄	—	101	2	no	—
108	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	C₂×C₈	—	101	4	no	—
112	C₂×C₈	—	202	4	no	—
113	C₂×C₈	C₂²×D₁₀₁	1	4	yes	—
114	C₄⋊C₄	—	101	4	no	—
115	C₄⋊C₄	—	101	2	no	—
116	C₂²⋊C₄	—	101	2	no	—
117	C₂²⋊C₄	—	101	4	no	—
118	C₄²	—	101	4	no	—
119	C₄²	—	101	2	no	—
120	C₄²	C₄×D₁₀₁	1	2	yes	—
121	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
122	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
123	C₂³	—	101	2	no	—
124	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
125	Q₈	—	202	2	no	—
126	D₄	—	202	4	no	—
127	D₄	—	202	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	101	4	no	—
130	C₂×C₄	—	101	2	no	—
131	C₂×C₄	—	202	4	no	—
132	C₂×C₄	—	202	8	no	—
133	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
136	C₂×C₄	#13108074	1	4	yes	—
137	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	202	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	202	2	no	—
142	C₂²	—	202	4	no	—
143	C₂²	#15692212	1	2	yes	—
144	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	202	2	no	—
147	C₄	—	202	4	no	—
148	C₄	#15692212	1	2	yes	—
149	C₄	#15692263	1	4	yes	—
150	C₄	C₂×#13108074	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	404	2	no	—
153	C₂	C₂×#15692212	1	1	yes	characteristic
154	C₂	C₂×#15692263	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13108074

C₂

yes

maximal, characteristic

#20265239

C₂

yes

maximal, complemented

#20265228

C₂

yes

maximal, complemented

#20265205

C₂

yes

maximal, complemented

#20265145

C₂

yes

maximal

C₂×C₄×#11717579

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#13108074

C₂²

yes

characteristic

C₂²×#11717579

C₂²

yes

—

C₄×Dic₂₀₂

C₄

yes

—

C₄×D₄₀₄

C₄

yes

—

C₄×C₂²⋊D₁₀₁

C₄

yes

—

#15692214

C₂²

yes

characteristic

#15692212

C₂²

yes

—

C₄²×D₁₀₁

C₂²

yes

—

#15692184

C₂²

yes

—

#15692167

C₂²

yes

—

C₄×#11717579

C₂²

yes

—

C₄×#11717579

C₂²

yes

—

C₂×M₁₆×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

#97×C₁₀₁

C₂²

yes

—

C₄×C₈×C₁₀₁

C₂²

yes

—

#13108074

—

C₂×Dic₂₀₂

C₂×C₄

yes

characteristic

C₂×D₄₀₄

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₁₀₁

C₂×C₄

yes

—

C₂×C₄×D₁₀₁

C₂×C₄

yes

—

C₂×C₄×D₁₀₁

C₂³

yes

—

C₂×#11717579

—

C₂×#11717579

C₂³

yes

—

C₄⋊Dic₁₀₁

C₂×C₄

yes

—

#13108053

C₂×C₄

yes

—

C₂²⋊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

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

C₂³

yes

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂²×D₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

C₂²×C₄

yes

—

C₂×Dic₁₀₁

C₂²×C₄

yes

—

D₄₀₄

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

Dic₂₀₂

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#117

yes

—

C₂²×C₁₀₁

C₂³×C₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#117

yes

—

C₄×C₁₀₁

#131

yes

—

C₄×C₁₀₁

C₂×#41

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#117

yes

characteristic

C₂×C₁₀₁

C₂×#131

yes

—

D₁₀₁

—

#2922

—

101

maximal, Sylow

C₁₀₁

#2922

yes

characteristic, complemented, Sylow

C₄×#41

—

101

—

#433

—

101

—

#431

—

101

—

#424

—

101

—

#408

—

101

—

C₄×M₁₆

D₁₀₁

yes

characteristic, complemented

C₂×C₄×C₈

—

101

—

C₂×#41

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

—

C₂²×C₈

—

101

—

C₄×Q₈

—

101

—

C₄×D₄

—

101

—

C₄×D₄

—

101

—

#117

—

101

—

#117

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

101

—

C₂²⋊C₈

—

101

—

#97

C₂×D₁₀₁

yes

—

C₄×C₈

—

101

—

100

C₄×C₈

—

101

—

101

C₄×C₈

C₂×D₁₀₁

yes

—

102

#41

—

202

—

103

C₂×Q₈

—

101

—

104

C₂×D₄

—

101

—

105

C₂×D₄

—

101

—

106

C₂²×C₄

—

101

—

107

C₂²×C₄

—

101

—

108

C₂²×C₄

C₄×D₁₀₁

yes

—

109

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

101

—

112

C₂×C₈

—

202

—

113

C₂×C₈

C₂²×D₁₀₁

yes

—

114

C₄⋊C₄

—

101

—

115

C₄⋊C₄

—

101

—

116

C₂²⋊C₄

—

101

—

117

C₂²⋊C₄

—

101

—

118

C₄²

—

101

—

119

C₄²

—

101

—

120

C₄²

C₄×D₁₀₁

yes

—

121

C₄²

C₂²×D₁₀₁

yes

characteristic

122

C₄²

C₂²×D₁₀₁

yes

characteristic

123

C₂³

—

101

—

124

C₂³

C₂×C₄×D₁₀₁

yes

characteristic

125

Q₈

—

202

—

126

D₄

—

202

—

127

D₄

—

202

—

128

C₂×C₄

—

129

C₂×C₄

—

101

—

130

C₂×C₄

—

101

—

131

C₂×C₄

—

202

—

132

C₂×C₄

—

202

—

133

C₂×C₄

C₂×C₄×D₁₀₁

yes

—

134

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

135

C₂×C₄

C₂×C₄×D₁₀₁

yes

—

136

C₂×C₄

#13108074

yes

—

137

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

138

C₈

—

139

C₈

—

202

—

140

C₂²

—

141

C₂²

—

202

—

142

C₂²

—

202

—

143

C₂²

#15692212

yes

—

144

C₂²

C₂²×C₄×D₁₀₁

yes

characteristic

145

C₄

—

146

C₄

—

202

—

147

C₄

—

202

—

148

C₄

#15692212

yes

—

149

C₄

#15692263

yes

—

150

C₄

C₂×#13108074

yes

—

151

C₂

—

152

C₂

—

404

—

153

C₂

C₂×#15692212

yes

characteristic

154

C₂

C₂×#15692263

yes

—

	subgroup	G/H
center	C₄²	C₂²×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₄×M₁₆×C₁₀₁	C₂
prime radical	C₂×#13108074	C₂²
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	C₄×M₁₆	D₁₀₁
hypocenter	C₁₀₁	#2922
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²×D₁₀₁

commutator, socle

C₂²×C₁₀₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₀₁

Fitting

C₄×M₁₆×C₁₀₁

C₂

prime radical

C₂×#13108074

C₂²

cosocle

C₂×C₄×C₁₀₁

C₂⁴

hypercenter

C₄×M₁₆

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #20265239	C₂	semidirect
3. #20265228	C₂	semidirect
4. #20265205	C₂	semidirect
6. C₂×C₄×#11717579	C₂	semidirect
7. C₄×M₁₆×C₁₀₁	C₂	semidirect
78. C₁₀₁	#2922	semidirect
84. C₄×M₁₆	D₁₀₁	semidirect

H ≅ G/N

type

2. #20265239

C₂

semidirect

3. #20265228

C₂

semidirect

4. #20265205

C₂

semidirect

6. C₂×C₄×#11717579

C₂

semidirect

7. C₄×M₁₆×C₁₀₁

C₂

semidirect

78. C₁₀₁

#2922

semidirect

84. C₄×M₁₆

D₁₀₁

semidirect

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

subgroup	index
C₄×#13108074	2
#20265239	2
#20265228	2
#20265205	2
#20265145	2
C₂×C₄×#11717579	2
C₄×M₁₆×C₁₀₁	2
#2922	101

subgroup

index

C₄×M₁₆×C₁₀₁

101

Subgroups — #27179777

Conjugacy classes of subgroups

Complements

Maximal subgroups