Group #27169844 — subgroups

Subgroups — #27169844

Order 12928.

332 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 67 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	#127×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	#20265233	C₂	1	2	yes	maximal, complemented
3	#20264921	C₂	1	1	yes	maximal, characteristic, complemented
4	#20264920	C₂	1	2	yes	maximal, complemented
5	#331×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×D₁₀₁	—	2	2	no	—
7	C₂×D₄×D₁₀₁	C₄	1	2	yes	—
8	#15692252	C₂²	1	1	yes	characteristic
9	#15692251	—	2	2	no	—
10	#15692251	C₄	1	2	yes	—
11	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
12	C₄⋊#11717579	C₂²	1	2	yes	complemented
13	#15692167	—	2	2	no	—
14	#15692165	—	2	2	no	—
15	#127×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₀₁	C₂²	1	2	yes	complemented
17	C₂³×D₁₀₁	—	2	2	no	—
18	C₂×D₄₀₄	—	2	2	no	—
19	C₂×D₄₀₄	C₂×C₄	1	2	yes	—
20	C₂×C₂²⋊D₁₀₁	—	2	4	no	—
21	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
22	D₄×D₁₀₁	—	4	4	no	—
23	D₄×D₁₀₁	—	4	4	no	—
24	D₄×D₁₀₁	—	4	4	no	—
25	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
27	C₂×#11717579	—	2	2	no	—
28	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₁₀₁	D₄	1	2	yes	—
30	C₂×D₄×C₁₀₁	—	2	2	no	—
31	C₂×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₀₁	—	2	2	no	—
33	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
34	C₂³×C₁₀₁	—	2	2	no	—
35	C₂²×D₁₀₁	—	2	2	no	—
36	C₂²×D₁₀₁	—	4	8	no	—
37	C₂²×D₁₀₁	—	4	4	no	—
38	C₂²×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
39	D₄×C₁₀₁	—	4	2	no	—
40	D₄×C₁₀₁	—	4	2	no	—
41	D₄×C₁₀₁	—	4	2	no	—
42	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
44	C₂²⋊D₁₀₁	—	4	4	no	—
45	C₂²⋊D₁₀₁	—	4	8	no	—
46	C₂×Dic₁₀₁	C₂²⋊C₄	1	2	yes	—
47	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
48	D₄₀₄	—	4	2	no	—
49	D₄₀₄	—	4	2	no	—
50	D₄₀₄	—	4	2	no	—
51	C₄×D₁₀₁	—	2	2	no	—
52	C₄×D₁₀₁	—	2	2	no	—
53	C₄×D₁₀₁	—	2	2	no	—
54	C₈×C₁₀₁	—	4	2	no	—
55	#11717579	—	4	2	no	—
56	C₂²×C₁₀₁	—	4	2	no	—
57	C₂²×C₁₀₁	—	4	4	no	—
58	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×D₁₀₁	—	2	2	no	—
60	C₂×D₁₀₁	—	4	2	no	—
61	C₂×D₁₀₁	—	4	4	no	—
62	C₂×D₁₀₁	—	8	8	no	—
63	C₂×D₁₀₁	#100	1	2	yes	—
64	C₄×C₁₀₁	—	2	1	no	—
65	C₄×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
66	C₄×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
67	Dic₁₀₁	—	2	1	no	—
68	Dic₁₀₁	—	2	2	no	—
69	C₂×C₁₀₁	—	8	2	no	—
70	C₂×C₁₀₁	C₂×#100	1	1	yes	characteristic
71	C₂×C₁₀₁	#418	1	2	yes	—
72	D₁₀₁	—	2	2	no	—
73	D₁₀₁	—	8	2	no	—
74	#1555	—	101	1	no	maximal, Sylow
75	C₁₀₁	#1555	1	1	yes	characteristic, complemented, Sylow
76	C₂×#127	—	101	1	no	—
77	#432	—	101	2	no	—
78	#331	—	101	1	no	—
79	#331	—	101	2	no	—
80	#331	D₁₀₁	1	1	yes	characteristic, complemented
81	C₂²×D₄	—	101	2	no	—
82	C₂²×D₄	—	202	2	no	—
83	#127	—	101	2	no	—
84	#127	—	101	1	no	—
85	#127	—	202	2	no	—
86	#127	C₂×D₁₀₁	1	1	yes	characteristic
87	C₂×C₄²	—	101	1	no	—
88	C₄⋊C₈	—	101	2	no	—
89	C₄⋊C₈	C₂×D₁₀₁	1	2	yes	complemented
90	C₂²⋊C₈	—	202	2	no	—
91	#97	—	202	2	no	—
92	C₂⁴	—	202	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	101	2	no	—
95	C₂×D₄	—	404	4	no	—
96	C₂×D₄	—	404	4	no	—
97	C₂×D₄	—	404	4	no	—
98	C₂×D₄	—	202	4	no	—
99	C₂×D₄	—	202	2	no	—
100	C₂×D₄	—	202	2	no	—
101	C₂×D₄	C₄×D₁₀₁	1	2	yes	—
102	C₂²×C₄	—	101	1	no	—
103	C₂²×C₄	—	101	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	202	2	no	—
106	C₄²	—	101	1	no	—
107	C₄²	—	101	2	no	—
108	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
109	C₂³	—	2	2	no	—
110	C₂³	—	101	1	no	—
111	C₂³	—	404	4	no	—
112	C₂³	—	404	8	no	—
113	C₂³	—	202	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	D₄	—	4	2	no	—
117	D₄	—	404	2	no	—
118	D₄	—	404	4	no	—
119	D₄	—	404	8	no	—
120	D₄	—	404	2	no	—
121	D₄	—	404	2	no	—
122	C₂×C₄	—	101	1	no	—
123	C₂×C₄	—	101	2	no	—
124	C₂×C₄	—	202	2	no	—
125	C₂×C₄	—	202	2	no	—
126	C₂×C₄	—	202	2	no	—
127	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
128	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
129	C₈	—	4	2	no	—
130	C₈	—	404	2	no	—
131	C₂²	—	4	2	no	—
132	C₂²	—	4	4	no	—
133	C₂²	—	808	8	no	—
134	C₂²	—	101	2	no	—
135	C₂²	—	404	2	no	—
136	C₂²	—	404	4	no	—
137	C₂²	—	202	2	no	—
138	C₂²	C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	202	1	no	—
141	C₄	—	202	2	no	—
142	C₄	#15692279	1	2	yes	—
143	C₄	#15692282	1	2	yes	—
144	C₂	—	8	2	no	—
145	C₂	—	808	2	no	—
146	C₂	—	202	2	no	—
147	C₂	#100×D₁₀₁	1	1	yes	characteristic
148	C₂	#20265180	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#127×D₁₀₁

C₂

yes

maximal, characteristic

#20265233

C₂

yes

maximal, complemented

#20264921

C₂

yes

maximal, characteristic, complemented

#20264920

C₂

yes

maximal, complemented

#331×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₁₀₁

—

C₂×D₄×D₁₀₁

C₄

yes

—

#15692252

C₂²

yes

characteristic

#15692251

—

#15692251

C₄

yes

—

C₄²×D₁₀₁

C₂²

yes

characteristic

C₄⋊#11717579

C₂²

yes

complemented

#15692167

—

#15692165

—

#127×C₁₀₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₁

C₂²

yes

complemented

C₂³×D₁₀₁

—

C₂×D₄₀₄

—

C₂×D₄₀₄

C₂×C₄

yes

—

C₂×C₂²⋊D₁₀₁

—

C₂×C₂²⋊D₁₀₁

—

D₄×D₁₀₁

—

D₄×D₁₀₁

—

D₄×D₁₀₁

—

C₂×C₄×D₁₀₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₁

D₄

yes

—

C₂×#11717579

—

C₄×Dic₁₀₁

C₂×C₄

yes

characteristic

C₄×Dic₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

C₂×C₄

yes

—

C₂×C₈×C₁₀₁

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

C₂²⋊C₄

yes

characteristic

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂²⋊D₁₀₁

—

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

characteristic

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

#100

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₁

AGL(1,ℤ/8)

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#100

yes

characteristic

C₂×C₁₀₁

#418

yes

—

D₁₀₁

—

D₁₀₁

—

#1555

—

101

maximal, Sylow

C₁₀₁

#1555

yes

characteristic, complemented, Sylow

C₂×#127

—

101

—

#432

—

101

—

#331

—

101

—

#331

—

101

—

#331

D₁₀₁

yes

characteristic, complemented

C₂²×D₄

—

101

—

C₂²×D₄

—

202

—

#127

—

101

—

#127

—

101

—

#127

—

202

—

#127

C₂×D₁₀₁

yes

characteristic

C₂×C₄²

—

101

—

C₄⋊C₈

—

101

—

C₄⋊C₈

C₂×D₁₀₁

yes

complemented

C₂²⋊C₈

—

202

—

#97

—

202

—

C₂⁴

—

202

—

C₂×D₄

—

C₂×D₄

—

101

—

C₂×D₄

—

404

—

C₂×D₄

—

404

—

C₂×D₄

—

404

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

100

C₂×D₄

—

202

—

101

C₂×D₄

C₄×D₁₀₁

yes

—

102

C₂²×C₄

—

101

—

103

C₂²×C₄

—

101

—

104

C₂×C₈

—

105

C₂×C₈

—

202

—

106

C₄²

—

101

—

107

C₄²

—

101

—

108

C₄²

C₂²×D₁₀₁

yes

characteristic

109

C₂³

—

110

C₂³

—

101

—

111

C₂³

—

404

—

112

C₂³

—

404

—

113

C₂³

—

202

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

404

—

118

D₄

—

404

—

119

D₄

—

404

—

120

D₄

—

404

—

121

D₄

—

404

—

122

C₂×C₄

—

101

—

123

C₂×C₄

—

101

—

124

C₂×C₄

—

202

—

125

C₂×C₄

—

202

—

126

C₂×C₄

—

202

—

127

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

128

C₂×C₄

D₄×D₁₀₁

yes

—

129

C₈

—

130

C₈

—

404

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

808

—

134

C₂²

—

101

—

135

C₂²

—

404

—

136

C₂²

—

404

—

137

C₂²

—

202

—

138

C₂²

C₂²⋊C₄×D₁₀₁

yes

characteristic

139

C₄

—

140

C₄

—

202

—

141

C₄

—

202

—

142

C₄

#15692279

yes

—

143

C₄

#15692282

yes

—

144

C₂

—

145

C₂

—

808

—

146

C₂

—

202

—

147

C₂

#100×D₁₀₁

yes

characteristic

148

C₂

#20265180

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₀₁
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#331×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	#127×D₁₀₁	C₂
cosocle	C₄²×C₁₀₁	C₂³
hypercenter	#331	D₁₀₁
hypocenter	C₁₀₁	#1555
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₀₁

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₁

Fitting

#331×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

#127×D₁₀₁

C₂

cosocle

C₄²×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #20265233	C₂	semidirect
3. #20264921	C₂	semidirect
4. #20264920	C₂	semidirect
5. #331×C₁₀₁	C₂	semidirect
12. C₄⋊#11717579	C₂²	semidirect
16. C₄⋊C₈×C₁₀₁	C₂²	semidirect
75. C₁₀₁	#1555	semidirect
80. #331	D₁₀₁	semidirect
89. C₄⋊C₈	C₂×D₁₀₁	semidirect

H ≅ G/N

type

2. #20265233

C₂

semidirect

3. #20264921

C₂

semidirect

4. #20264920

C₂

semidirect

5. #331×C₁₀₁

C₂

semidirect

12. C₄⋊#11717579

C₂²

semidirect

16. C₄⋊C₈×C₁₀₁

C₂²

semidirect

75. C₁₀₁

#1555

semidirect

80. #331

D₁₀₁

semidirect

89. C₄⋊C₈

C₂×D₁₀₁

semidirect

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

subgroup	index
#127×D₁₀₁	2
#20265233	2
#20264921	2
#20264920	2
#331×C₁₀₁	2
#1555	101

subgroup

index

#127×D₁₀₁

#331×C₁₀₁

101

Subgroups — #27169844

Conjugacy classes of subgroups

Complements

Maximal subgroups