Group #27171115 — subgroups

Subgroups — #27171115

Order 12928.

312 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 101 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×#15692212	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692184	C₂	1	2	yes	maximal, complemented
3	C₂×#15692167	C₂	1	2	yes	maximal
4	#20264932	C₂	1	1	yes	maximal, characteristic, complemented
5	#336×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
7	C₂²×#11717579	C₂²	1	2	yes	—
8	#15692212	—	2	4	no	—
9	C₂×#13108053	C₄	1	2	yes	—
10	C₂×#13108051	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
12	#15692184	—	2	4	no	—
13	#15692167	—	2	4	no	—
14	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
18	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₁₀₁	—	2	2	no	—
20	C₂×C₄×D₁₀₁	—	2	2	no	—
21	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
22	C₂×C₄×D₁₀₁	Q₈	1	2	yes	—
23	C₂×#11717579	—	2	2	no	—
24	C₂×#11717579	—	2	2	no	—
25	C₂×#11717579	—	2	2	no	—
26	#13108053	—	2	4	no	—
27	#13108051	—	2	4	no	—
28	C₄×Dic₁₀₁	D₄	1	4	yes	—
29	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₀₁	—	2	2	no	—
32	C₂×C₈×C₁₀₁	—	2	2	no	—
33	C₂×C₈×C₁₀₁	—	2	2	no	—
34	C₄²×C₁₀₁	—	2	2	no	—
35	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
36	C₂²×D₁₀₁	—	2	1	no	—
37	C₂²×D₁₀₁	—	2	1	no	—
38	C₂²×D₁₀₁	—	2	2	no	—
39	C₂²×D₁₀₁	—	2	2	no	—
40	C₂²×D₁₀₁	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₀₁	—	2	2	no	—
42	C₂×C₄×C₁₀₁	—	2	2	no	—
43	C₂×C₄×C₁₀₁	—	2	2	no	—
44	C₂×C₄×C₁₀₁	C₂×D₄	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	1	no	—
49	C₂×Dic₁₀₁	—	2	1	no	—
50	C₂×Dic₁₀₁	—	2	2	no	—
51	C₂×Dic₁₀₁	—	2	2	no	—
52	C₂×Dic₁₀₁	C₂²⋊C₄	1	4	yes	—
53	C₂×Dic₁₀₁	C₄⋊C₄	1	2	yes	—
54	C₄×D₁₀₁	—	2	4	no	—
55	C₄×D₁₀₁	—	2	4	no	—
56	C₈×C₁₀₁	—	4	2	no	—
57	#11717579	—	4	2	no	—
58	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₀₁	#117	1	1	yes	characteristic
61	C₂²×C₁₀₁	#131	1	2	yes	—
62	C₂²×C₁₀₁	#131	1	2	yes	—
63	C₂×D₁₀₁	—	2	2	no	—
64	C₂×D₁₀₁	—	2	4	no	—
65	C₂×D₁₀₁	—	4	2	no	—
66	C₂×D₁₀₁	—	4	1	no	—
67	C₂×D₁₀₁	—	4	1	no	—
68	C₄×C₁₀₁	—	4	2	no	—
69	C₄×C₁₀₁	#121	1	2	yes	—
70	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
71	Dic₁₀₁	—	2	2	no	—
72	Dic₁₀₁	—	2	4	no	—
73	C₂×C₁₀₁	#380	1	1	yes	characteristic
74	C₂×C₁₀₁	#408	1	2	yes	—
75	C₂×C₁₀₁	#424	1	2	yes	—
76	C₂×C₁₀₁	#433	1	1	yes	characteristic
77	C₂×C₁₀₁	#433	1	1	yes	characteristic
78	D₁₀₁	—	4	2	no	—
79	#1798	—	101	1	no	maximal, Sylow
80	C₁₀₁	#1798	1	1	yes	characteristic, complemented, Sylow
81	C₂×#117	—	101	1	no	—
82	C₂×C₄⋊C₈	—	101	2	no	—
83	C₂×C₂²⋊C₈	—	101	2	no	—
84	#336	—	101	1	no	—
85	#336	D₁₀₁	1	1	yes	characteristic, complemented
86	C₂³×C₄	—	101	1	no	—
87	C₂²×C₈	—	101	2	no	—
88	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
89	#117	—	202	4	no	—
90	C₂×C₄⋊C₄	—	101	2	no	—
91	C₂×C₂²⋊C₄	—	101	2	no	—
92	C₂×C₄²	—	101	1	no	—
93	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
94	C₄⋊C₈	—	202	4	no	—
95	C₂²⋊C₈	—	202	4	no	—
96	C₂⁴	—	101	1	no	—
97	C₂²×C₄	—	101	2	no	—
98	C₂²×C₄	—	101	1	no	—
99	C₂²×C₄	—	101	2	no	—
100	C₂²×C₄	—	101	2	no	—
101	C₂²×C₄	—	202	2	no	—
102	C₂²×C₄	—	202	2	no	—
103	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
104	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	202	2	no	—
109	C₂×C₈	—	202	2	no	—
110	C₂×C₈	—	202	2	no	—
111	C₄⋊C₄	—	202	4	no	—
112	C₂²⋊C₄	—	202	4	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	101	4	no	—
115	C₂³	—	101	2	no	—
116	C₂³	—	202	1	no	—
117	C₂³	—	202	1	no	—
118	C₂³	—	202	2	no	—
119	C₂³	—	202	2	no	—
120	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	101	4	no	—
125	C₂×C₄	—	101	2	no	—
126	C₂×C₄	—	202	1	no	—
127	C₂×C₄	—	202	1	no	—
128	C₂×C₄	—	202	2	no	—
129	C₂×C₄	—	202	2	no	—
130	C₂×C₄	—	202	4	no	—
131	C₂×C₄	—	202	4	no	—
132	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
133	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
134	C₂×C₄	Q₈×D₁₀₁	1	1	yes	characteristic
135	C₂×C₄	#13108074	1	2	yes	—
136	C₈	—	4	2	no	—
137	C₈	—	404	2	no	—
138	C₂²	—	404	1	no	—
139	C₂²	—	404	1	no	—
140	C₂²	—	404	2	no	—
141	C₂²	—	202	2	no	—
142	C₂²	—	202	4	no	—
143	C₂²	C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
144	C₂²	C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
145	C₂²	#15692212	1	1	yes	characteristic
146	C₂²	#15692261	1	2	yes	—
147	C₂²	#15692263	1	2	yes	—
148	C₄	—	4	2	no	—
149	C₄	—	202	2	no	—
150	C₄	—	202	4	no	—
151	C₄	#15692228	1	2	yes	—
152	C₄	#15692233	1	2	yes	—
153	C₂	—	404	2	no	—
154	C₂	#20265036	1	1	yes	characteristic
155	C₂	#20265142	1	2	yes	—
156	C₂	#20265202	1	2	yes	—
157	C₂	#20265236	1	1	yes	characteristic
158	C₂	#20265239	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692212

C₂

yes

maximal, characteristic

C₂×#15692184

C₂

yes

maximal, complemented

C₂×#15692167

C₂

yes

maximal

#20264932

C₂

yes

maximal, characteristic, complemented

#336×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₀₁

C₂²

yes

characteristic

C₂²×#11717579

C₂²

yes

—

#15692212

—

C₂×#13108053

C₄

yes

—

C₂×#13108051

C₄

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

#15692184

—

#15692167

—

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂³×D₁₀₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂²×Dic₁₀₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₁

—

C₂×C₄×D₁₀₁

—

C₂×C₄×D₁₀₁

D₄

yes

—

C₂×C₄×D₁₀₁

Q₈

yes

—

—

—

—

—

—

C₄×Dic₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

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

C₂²⋊C₄

yes

—

C₂×Dic₁₀₁

C₄⋊C₄

yes

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₂²×C₁₀₁

#131

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#121

yes

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

#380

yes

characteristic

C₂×C₁₀₁

#408

yes

—

C₂×C₁₀₁

#424

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

D₁₀₁

—

#1798

—

101

maximal, Sylow

C₁₀₁

#1798

yes

characteristic, complemented, Sylow

C₂×#117

—

101

—

C₂×C₄⋊C₈

—

101

—

C₂×C₂²⋊C₈

—

101

—

#336

—

101

—

#336

D₁₀₁

yes

characteristic, complemented

C₂³×C₄

—

101

—

C₂²×C₈

—

101

—

C₂²×C₈

C₂×D₁₀₁

yes

—

#117

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

202

—

C₂²⋊C₈

—

202

—

C₂⁴

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

100

C₂²×C₄

—

101

—

101

C₂²×C₄

—

202

—

102

C₂²×C₄

—

202

—

103

C₂²×C₄

C₄×D₁₀₁

yes

—

104

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

202

—

109

C₂×C₈

—

202

—

110

C₂×C₈

—

202

—

111

C₄⋊C₄

—

202

—

112

C₂²⋊C₄

—

202

—

113

C₄²

—

114

C₄²

—

101

—

115

C₂³

—

101

—

116

C₂³

—

202

—

117

C₂³

—

202

—

118

C₂³

—

202

—

119

C₂³

—

202

—

120

C₂³

C₂×C₄×D₁₀₁

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

101

—

125

C₂×C₄

—

101

—

126

C₂×C₄

—

202

—

127

C₂×C₄

—

202

—

128

C₂×C₄

—

202

—

129

C₂×C₄

—

202

—

130

C₂×C₄

—

202

—

131

C₂×C₄

—

202

—

132

C₂×C₄

D₄×D₁₀₁

yes

characteristic

133

C₂×C₄

D₄×D₁₀₁

yes

—

134

C₂×C₄

Q₈×D₁₀₁

yes

characteristic

135

C₂×C₄

#13108074

yes

—

136

C₈

—

137

C₈

—

404

—

138

C₂²

—

404

—

139

C₂²

—

404

—

140

C₂²

—

404

—

141

C₂²

—

202

—

142

C₂²

—

202

—

143

C₂²

C₂²⋊C₄×D₁₀₁

yes

characteristic

144

C₂²

C₄⋊C₄×D₁₀₁

yes

characteristic

145

C₂²

#15692212

yes

characteristic

146

C₂²

#15692261

yes

—

147

C₂²

#15692263

yes

—

148

C₄

—

149

C₄

—

202

—

150

C₄

—

202

—

151

C₄

#15692228

yes

—

152

C₄

#15692233

yes

—

153

C₂

—

404

—

154

C₂

#20265036

yes

characteristic

155

C₂

#20265142

yes

—

156

C₂

#20265202

yes

—

157

C₂

#20265236

yes

characteristic

158

C₂

#20265239

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₁
commutator, socle	C₂³×C₁₀₁	C₂²×C₄
Fitting	#336×C₁₀₁	C₂
prime radical	C₂³×D₁₀₁	C₂×C₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#336	D₁₀₁
hypocenter	C₁₀₁	#1798
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₁

commutator, socle

C₂³×C₁₀₁

C₂²×C₄

Fitting

#336×C₁₀₁

C₂

prime radical

C₂³×D₁₀₁

C₂×C₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
2. C₂×#15692184	C₂	semidirect
4. #20264932	C₂	semidirect
5. #336×C₁₀₁	C₂	semidirect
80. C₁₀₁	#1798	semidirect
85. #336	D₁₀₁	semidirect

H ≅ G/N

type

2. C₂×#15692184

C₂

semidirect

4. #20264932

C₂

semidirect

5. #336×C₁₀₁

C₂

semidirect

80. C₁₀₁

#1798

semidirect

85. #336

D₁₀₁

semidirect

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

subgroup	index
C₂×#15692212	2
C₂×#15692184	2
C₂×#15692167	2
#20264932	2
#336×C₁₀₁	2
#1798	101

subgroup

index

#336×C₁₀₁

101

Subgroups — #27171115

Conjugacy classes of subgroups

Complements

Maximal subgroups