Group #27181447 — subgroups

Subgroups — C₂×#20265531 #27181447

Order 12928.

412 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15692247	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265531	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15692179	C₂	1	2	yes	maximal, complemented
4	C₂×#15692176	C₂	1	2	yes	maximal
5	C₂×C₄×#11717579	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#124×C₁₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11717579	C₂²	1	2	yes	—
8	#15692247	—	2	2	no	—
9	#15692247	C₂²	1	4	yes	complemented
10	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
11	C₂×#13108053	—	2	2	no	—
12	#15692179	C₂²	1	8	yes	complemented
13	#15692176	C₂²	1	8	yes	—
14	C₄×#11717579	C₂²	1	4	yes	complemented
15	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
17	#124×C₁₀₁	—	2	2	no	—
18	#124×C₁₀₁	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
20	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
21	C₂²×Dic₁₀₁	—	2	2	no	—
22	C₂×#11717579	—	2	4	no	—
23	C₂×#11717579	C₂³	1	4	yes	—
24	C₄⋊Dic₁₀₁	—	2	2	no	—
25	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
26	#13108053	—	2	8	no	—
27	C₂⁴×C₁₀₁	—	2	1	no	—
28	C₂×Q₈×C₁₀₁	—	2	1	no	—
29	C₂×Q₈×C₁₀₁	—	2	4	no	—
30	C₂×Q₈×C₁₀₁	C₂³	1	2	yes	—
31	C₂×D₄×C₁₀₁	—	2	1	no	—
32	C₂×D₄×C₁₀₁	—	2	4	no	—
33	C₂×D₄×C₁₀₁	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₀₁	—	2	1	no	—
35	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
36	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
37	C₂²⋊C₄×C₁₀₁	—	2	8	no	—
38	C₄²×C₁₀₁	D₄	1	2	yes	—
39	C₄²×C₁₀₁	C₂³	1	2	yes	—
40	C₂³×C₁₀₁	—	2	2	no	—
41	C₂³×C₁₀₁	—	4	2	no	—
42	C₂³×C₁₀₁	—	4	4	no	—
43	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₁₀₁	—	2	4	no	—
45	Q₈×C₁₀₁	—	4	2	no	—
46	D₄×C₁₀₁	—	2	4	no	—
47	D₄×C₁₀₁	—	4	2	no	—
48	C₂×C₄×C₁₀₁	—	2	4	no	—
49	C₂×C₄×C₁₀₁	—	2	2	no	—
50	C₂×C₄×C₁₀₁	—	4	2	no	—
51	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
52	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₁	#41	1	4	yes	—
54	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₁₀₁	—	2	4	no	—
56	C₂×Dic₁₀₁	—	4	4	no	—
57	#11717579	—	2	4	no	—
58	C₂²×C₁₀₁	—	4	2	no	—
59	C₂²×C₁₀₁	—	4	4	no	—
60	C₂²×C₁₀₁	—	8	4	no	—
61	C₂²×C₁₀₁	#124	1	2	yes	—
62	C₂²×C₁₀₁	#135	1	4	yes	—
63	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₁₀₁	—	2	4	no	—
65	C₄×C₁₀₁	—	4	2	no	—
66	C₄×C₁₀₁	#124	1	2	yes	—
67	C₄×C₁₀₁	C₂×#41	1	2	yes	—
68	Dic₁₀₁	—	4	4	no	—
69	C₂×C₁₀₁	—	8	2	no	—
70	C₂×C₁₀₁	#488	1	4	yes	complemented
71	C₂×C₁₀₁	C₂×#124	1	1	yes	characteristic
72	C₂×C₁₀₁	C₂×#135	1	2	yes	—
73	C₂×#488	—	101	1	no	maximal, Sylow
74	C₁₀₁	C₂×#488	1	1	yes	characteristic, complemented, Sylow
75	C₂×#125	—	101	1	no	—
76	C₂×#124	D₁₀₁	1	1	yes	characteristic
77	#488	—	101	8	no	—
78	C₂×#103	—	101	2	no	—
79	C₂×#102	—	101	2	no	—
80	C₂×C₄×C₈	—	101	1	no	—
81	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
82	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
83	C₂²×C₈	—	101	2	no	—
84	#125	—	101	4	no	—
85	#125	—	202	2	no	—
86	#124	—	2	2	no	—
87	#124	C₂×D₁₀₁	1	4	yes	—
88	C₂×C₄⋊C₄	—	101	2	no	—
89	C₂×C₄⋊C₄	—	202	2	no	—
90	C₂×C₂²⋊C₄	—	2	2	no	—
91	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
92	#103	—	101	8	no	—
93	#102	—	101	8	no	—
94	C₄×C₈	—	101	4	no	—
95	C₂⁴	—	2	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	C₂²×D₁₀₁	1	2	yes	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	4	no	—
101	C₂×D₄	C₂²×D₁₀₁	1	2	yes	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	202	2	no	—
104	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
105	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
106	C₂×C₈	—	101	4	no	—
107	C₂×C₈	—	202	4	no	—
108	C₄⋊C₄	—	101	4	no	—
109	C₄⋊C₄	—	202	8	no	—
110	C₄⋊C₄	—	202	2	no	—
111	C₂²⋊C₄	—	2	8	no	—
112	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
113	C₄²	C₂²×D₁₀₁	1	2	yes	—
114	C₂³	—	2	2	no	—
115	C₂³	—	4	2	no	—
116	C₂³	—	4	4	no	—
117	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	4	2	no	—
120	D₄	—	2	4	no	—
121	D₄	—	4	2	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	404	4	no	—
126	C₂×C₄	—	202	4	no	—
127	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
128	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
129	C₂×C₄	#13108073	1	4	yes	—
130	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
131	C₈	—	202	4	no	—
132	C₂²	—	4	2	no	—
133	C₂²	—	4	4	no	—
134	C₂²	—	8	4	no	—
135	C₂²	#15692242	1	2	yes	—
136	C₂²	#15692275	1	4	yes	—
137	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
138	C₄	—	2	4	no	—
139	C₄	—	4	2	no	—
140	C₄	—	404	4	no	—
141	C₄	#15692242	1	2	yes	—
142	C₄	C₂×#13108073	1	2	yes	—
143	C₂	—	8	2	no	—
144	C₂	#20265531	1	4	yes	direct product factor
145	C₂	C₂×#15692242	1	1	yes	characteristic
146	C₂	C₂×#15692275	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692247

C₂

yes

maximal, characteristic, complemented

#20265531

C₂

yes

maximal, direct product factor

C₂×#15692179

C₂

yes

maximal, complemented

C₂×#15692176

C₂

yes

maximal

C₂×C₄×#11717579

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

#15692247

—

#15692247

C₂²

yes

complemented

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×#13108053

—

#15692179

C₂²

yes

complemented

#15692176

C₂²

yes

—

C₄×#11717579

C₂²

yes

complemented

C₂²×Q₈×C₁₀₁

C₂²

yes

characteristic

C₂²×D₄×C₁₀₁

C₂²

yes

characteristic

#124×C₁₀₁

—

#124×C₁₀₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₀₁

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂²×Dic₁₀₁

—

C₂×#11717579

—

C₂×#11717579

C₂³

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂³

yes

—

#13108053

—

C₂⁴×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

C₂³

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

C₂³

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#124

yes

—

C₂²×C₁₀₁

#135

yes

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#124

yes

—

C₄×C₁₀₁

C₂×#41

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×#488

—

101

maximal, Sylow

C₁₀₁

C₂×#488

yes

characteristic, complemented, Sylow

C₂×#125

—

101

—

C₂×#124

D₁₀₁

yes

characteristic

#488

—

101

—

C₂×#103

—

101

—

C₂×#102

—

101

—

C₂×C₄×C₈

—

101

—

C₂²×Q₈

C₂×D₁₀₁

yes

characteristic

C₂²×D₄

C₂×D₁₀₁

yes

characteristic

C₂²×C₈

—

101

—

#125

—

101

—

#125

—

202

—

#124

—

#124

C₂×D₁₀₁

yes

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

202

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#103

—

101

—

#102

—

101

—

C₄×C₈

—

101

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₁₀₁

yes

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

C₂²×D₁₀₁

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

202

—

104

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

105

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

106

C₂×C₈

—

101

—

107

C₂×C₈

—

202

—

108

C₄⋊C₄

—

101

—

109

C₄⋊C₄

—

202

—

110

C₄⋊C₄

—

202

—

111

C₂²⋊C₄

—

112

C₄²

C₂²⋊D₁₀₁

yes

—

113

C₄²

C₂²×D₁₀₁

yes

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

118

Q₈

—

119

Q₈

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

404

—

126

C₂×C₄

—

202

—

127

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

128

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

129

C₂×C₄

#13108073

yes

—

130

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

131

C₈

—

202

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#15692242

yes

—

136

C₂²

#15692275

yes

—

137

C₂²

C₂²×C₂²⋊D₁₀₁

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

404

—

141

C₄

#15692242

yes

—

142

C₄

C₂×#13108073

yes

—

143

C₂

—

144

C₂

#20265531

yes

direct product factor

145

C₂

C₂×#15692242

yes

characteristic

146

C₂

C₂×#15692275

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₂×#124×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂×D₄
prime radical	C₂²×D₄×C₁₀₁	C₂²
hypercenter	C₂×#124	D₁₀₁
hypocenter	C₁₀₁	C₂×#488
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator, cosocle

C₂×C₄×C₁₀₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₁

Fitting

C₂×#124×C₁₀₁

C₂

socle

C₂³×C₁₀₁

C₂×D₄

prime radical

C₂²×D₄×C₁₀₁

C₂²

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×#15692247	C₂	semidirect
2. #20265531	C₂	direct (N × H)
3. C₂×#15692179	C₂	semidirect
5. C₂×C₄×#11717579	C₂	semidirect
9. #15692247	C₂²	semidirect
12. #15692179	C₂²	semidirect
14. C₄×#11717579	C₂²	semidirect
70. C₂×C₁₀₁	#488	semidirect
74. C₁₀₁	C₂×#488	semidirect
144. C₂	#20265531	direct (N × H)

H ≅ G/N

type

1. C₂×#15692247

C₂

semidirect

2. #20265531

C₂

direct (N × H)

3. C₂×#15692179

C₂

semidirect

5. C₂×C₄×#11717579

C₂

semidirect

9. #15692247

C₂²

semidirect

12. #15692179

C₂²

semidirect

14. C₄×#11717579

C₂²

semidirect

70. C₂×C₁₀₁

#488

semidirect

74. C₁₀₁

C₂×#488

semidirect

144. C₂

#20265531

direct (N × H)

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

subgroup	index
C₂×#15692247	2
#20265531	2
C₂×#15692179	2
C₂×#15692176	2
C₂×C₄×#11717579	2
C₂×#124×C₁₀₁	2
C₂×#488	101

subgroup

index

C₂×#124×C₁₀₁

101

Subgroups — C₂×#20265531 #27181447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20265531 #27181447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265531 #27181447