Group #27181437 — subgroups

Subgroups — C₂×#20265523 #27181437

Order 12928.

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

normal subgroups only

This group has 528 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	C₂×#15692255	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265523	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15692176	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×#11717579	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₂	—	2	1	no	—
7	C₂²×#11717579	C₂²	1	2	yes	—
8	#15692255	—	2	2	no	—
9	#15692255	C₂²	1	4	yes	complemented
10	C₂×C₄⋊Dic₁₀₁	—	2	1	no	—
11	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
12	#15692176	C₂²	1	16	yes	complemented
13	C₄×#11717579	C₂²	1	4	yes	complemented
14	C₂²×D₄×C₁₀₁	—	2	1	no	—
15	C₂²×D₄×C₁₀₁	—	2	1	no	—
16	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
17	#127×C₁₀₁	—	2	2	no	—
18	#127×C₁₀₁	C₂²	1	4	yes	—
19	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₀₂	—	2	4	no	—
21	C₂×Dic₂₀₂	—	4	4	no	—
22	C₂²×Dic₁₀₁	—	2	2	no	—
23	C₂×#11717579	—	2	4	no	—
24	C₂×#11717579	C₂³	1	4	yes	—
25	C₄⋊Dic₁₀₁	—	2	4	no	—
26	C₄⋊Dic₁₀₁	—	2	2	no	—
27	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
28	C₂⁴×C₁₀₁	—	2	2	no	—
29	C₂×D₄×C₁₀₁	—	2	2	no	—
30	C₂×D₄×C₁₀₁	—	2	4	no	—
31	C₂×D₄×C₁₀₁	—	2	4	no	—
32	C₂×D₄×C₁₀₁	—	2	8	no	—
33	C₂×D₄×C₁₀₁	—	4	4	no	—
34	C₂×D₄×C₁₀₁	—	4	4	no	—
35	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
36	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
39	C₄²×C₁₀₁	D₄	1	2	yes	—
40	C₄²×C₁₀₁	C₂³	1	2	yes	—
41	C₂³×C₁₀₁	—	2	4	no	—
42	C₂³×C₁₀₁	—	4	4	no	—
43	C₂³×C₁₀₁	—	4	4	no	—
44	C₂³×C₁₀₁	—	4	4	no	—
45	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
46	D₄×C₁₀₁	—	2	8	no	—
47	D₄×C₁₀₁	—	4	4	no	—
48	D₄×C₁₀₁	—	4	8	no	—
49	D₄×C₁₀₁	—	4	8	no	—
50	C₂×C₄×C₁₀₁	D₈	1	4	yes	—
51	C₂×C₄×C₁₀₁	SD₁₆	1	4	yes	—
52	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₀₁	#41	1	4	yes	—
56	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₁₀₁	—	2	4	no	—
58	C₂×Dic₁₀₁	—	4	4	no	—
59	Dic₂₀₂	—	4	8	no	—
60	#11717579	—	2	4	no	—
61	C₂²×C₁₀₁	—	4	4	no	—
62	C₂²×C₁₀₁	—	4	4	no	—
63	C₂²×C₁₀₁	—	4	4	no	—
64	C₂²×C₁₀₁	—	8	8	no	—
65	C₂²×C₁₀₁	#124	1	2	yes	—
66	C₂²×C₁₀₁	C₂×D₈	1	2	yes	—
67	C₂²×C₁₀₁	C₂×SD₁₆	1	2	yes	—
68	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
69	C₄×C₁₀₁	#124	1	2	yes	—
70	C₄×C₁₀₁	C₂×D₈	1	4	yes	—
71	C₄×C₁₀₁	C₂×SD₁₆	1	4	yes	—
72	C₄×C₁₀₁	C₂×#41	1	2	yes	—
73	Dic₁₀₁	—	4	4	no	—
74	C₂×C₁₀₁	—	8	4	no	—
75	C₂×C₁₀₁	#486	1	4	yes	complemented
76	C₂×C₁₀₁	C₂×#124	1	1	yes	characteristic
77	C₂×C₁₀₁	C₂²×D₈	1	1	yes	characteristic
78	C₂×C₁₀₁	C₂²×SD₁₆	1	1	yes	characteristic
79	C₂×#486	—	101	1	no	maximal, Sylow
80	C₁₀₁	C₂×#486	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₄⋊Q₈	—	101	1	no	—
82	C₂×#127	D₁₀₁	1	1	yes	characteristic
83	#486	—	101	8	no	—
84	C₂×#102	—	101	4	no	—
85	C₂×C₄×C₈	—	101	1	no	—
86	C₂²×Q₈	—	202	1	no	—
87	C₂²×D₄	—	2	1	no	—
88	C₂²×D₄	—	2	1	no	—
89	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
90	C₂²×C₈	—	101	2	no	—
91	C₄⋊Q₈	—	101	4	no	—
92	C₄⋊Q₈	—	202	2	no	—
93	#127	—	2	2	no	—
94	#127	C₂×D₁₀₁	1	4	yes	—
95	C₂×C₄⋊C₄	—	101	2	no	—
96	C₂×C₄⋊C₄	—	202	1	no	—
97	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
98	#102	—	101	16	no	—
99	C₄×C₈	—	101	4	no	—
100	C₂⁴	—	2	2	no	—
101	C₂×Q₈	—	404	4	no	—
102	C₂×Q₈	—	202	4	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	4	no	—
105	C₂×D₄	—	2	4	no	—
106	C₂×D₄	—	2	8	no	—
107	C₂×D₄	—	4	4	no	—
108	C₂×D₄	—	4	4	no	—
109	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
110	C₂²×C₄	—	202	2	no	—
111	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
114	C₂×C₈	—	101	4	no	—
115	C₂×C₈	—	202	4	no	—
116	C₄⋊C₄	—	101	4	no	—
117	C₄⋊C₄	—	202	2	no	—
118	C₄⋊C₄	—	202	4	no	—
119	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
120	C₄²	C₂²×D₁₀₁	1	2	yes	—
121	C₂³	—	2	4	no	—
122	C₂³	—	4	4	no	—
123	C₂³	—	4	4	no	—
124	C₂³	—	4	4	no	—
125	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
126	Q₈	—	404	8	no	—
127	D₄	—	2	8	no	—
128	D₄	—	4	4	no	—
129	D₄	—	4	8	no	—
130	D₄	—	4	8	no	—
131	C₂×C₄	—	404	4	no	—
132	C₂×C₄	—	202	4	no	—
133	C₂×C₄	#13108059	1	4	yes	—
134	C₂×C₄	#13108062	1	4	yes	—
135	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
138	C₂×C₄	#13108073	1	4	yes	—
139	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
140	C₈	—	202	4	no	—
141	C₂²	—	4	4	no	—
142	C₂²	—	4	4	no	—
143	C₂²	—	4	4	no	—
144	C₂²	—	8	8	no	—
145	C₂²	#15692242	1	2	yes	—
146	C₂²	C₂×#13108059	1	2	yes	—
147	C₂²	C₂×#13108062	1	2	yes	—
148	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
149	C₄	—	404	4	no	—
150	C₄	#15692242	1	2	yes	—
151	C₄	C₂×#13108059	1	4	yes	—
152	C₄	C₂×#13108062	1	4	yes	—
153	C₄	C₂×#13108073	1	2	yes	—
154	C₂	—	8	4	no	—
155	C₂	#20265523	1	4	yes	direct product factor
156	C₂	C₂×#15692242	1	1	yes	characteristic
157	C₂	C₂²×#13108059	1	1	yes	characteristic
158	C₂	C₂²×#13108062	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692255

C₂

yes

maximal, characteristic, complemented

#20265523

C₂

yes

maximal, direct product factor

C₂×#15692176

C₂

yes

maximal, complemented

C₂×C₄×#11717579

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

—

C₂²×#11717579

C₂²

yes

—

#15692255

—

#15692255

C₂²

yes

complemented

C₂×C₄⋊Dic₁₀₁

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

#15692176

C₂²

yes

complemented

C₄×#11717579

C₂²

yes

complemented

C₂²×D₄×C₁₀₁

—

C₂²×D₄×C₁₀₁

—

C₂²×D₄×C₁₀₁

C₂²

yes

—

#127×C₁₀₁

—

#127×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂²×Dic₁₀₁

—

C₂×#11717579

—

C₂×#11717579

C₂³

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂³

yes

—

C₂⁴×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄²×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

C₂³

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

D₈

yes

—

C₂×C₄×C₁₀₁

SD₁₆

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#124

yes

—

C₂²×C₁₀₁

C₂×D₈

yes

—

C₂²×C₁₀₁

C₂×SD₁₆

yes

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₄×C₁₀₁

#124

yes

—

C₄×C₁₀₁

C₂×D₈

yes

—

C₄×C₁₀₁

C₂×SD₁₆

yes

—

C₄×C₁₀₁

C₂×#41

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

C₂²×D₈

yes

characteristic

C₂×C₁₀₁

C₂²×SD₁₆

yes

characteristic

C₂×#486

—

101

maximal, Sylow

C₁₀₁

C₂×#486

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

101

—

C₂×#127

D₁₀₁

yes

characteristic

#486

—

101

—

C₂×#102

—

101

—

C₂×C₄×C₈

—

101

—

C₂²×Q₈

—

202

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₀₁

yes

—

C₂²×C₈

—

101

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

—

202

—

#127

—

#127

C₂×D₁₀₁

yes

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#102

—

101

—

C₄×C₈

—

101

—

100

C₂⁴

—

101

C₂×Q₈

—

404

—

102

C₂×Q₈

—

202

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₂²×D₁₀₁

yes

—

110

C₂²×C₄

—

202

—

111

C₂²×C₄

C₂²⋊D₁₀₁

yes

characteristic

112

C₂²×C₄

C₂²⋊D₁₀₁

yes

characteristic

113

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

114

C₂×C₈

—

101

—

115

C₂×C₈

—

202

—

116

C₄⋊C₄

—

101

—

117

C₄⋊C₄

—

202

—

118

C₄⋊C₄

—

202

—

119

C₄²

C₂²⋊D₁₀₁

yes

—

120

C₄²

C₂²×D₁₀₁

yes

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

126

Q₈

—

404

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

404

—

132

C₂×C₄

—

202

—

133

C₂×C₄

#13108059

yes

—

134

C₂×C₄

#13108062

yes

—

135

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

137

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

138

C₂×C₄

#13108073

yes

—

139

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

140

C₈

—

202

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#15692242

yes

—

146

C₂²

C₂×#13108059

yes

—

147

C₂²

C₂×#13108062

yes

—

148

C₂²

C₂²×C₂²⋊D₁₀₁

yes

characteristic

149

C₄

—

404

—

150

C₄

#15692242

yes

—

151

C₄

C₂×#13108059

yes

—

152

C₄

C₂×#13108062

yes

—

153

C₄

C₂×#13108073

yes

—

154

C₂

—

155

C₂

#20265523

yes

direct product factor

156

C₂

C₂×#15692242

yes

characteristic

157

C₂

C₂²×#13108059

yes

characteristic

158

C₂

C₂²×#13108062

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting, prime radical	C₂×#127×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂×D₄
hypercenter	C₂×#127	D₁₀₁
hypocenter	C₁₀₁	C₂×#486
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator, cosocle

C₂×C₄×C₁₀₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₁

Fitting, prime radical

C₂×#127×C₁₀₁

C₂

socle

C₂³×C₁₀₁

C₂×D₄

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#15692255	C₂	semidirect
2. #20265523	C₂	direct (N × H)
3. C₂×#15692176	C₂	semidirect
4. C₂×C₄×#11717579	C₂	semidirect
9. #15692255	C₂²	semidirect
12. #15692176	C₂²	semidirect
13. C₄×#11717579	C₂²	semidirect
75. C₂×C₁₀₁	#486	semidirect
80. C₁₀₁	C₂×#486	semidirect
155. C₂	#20265523	direct (N × H)

H ≅ G/N

type

1. C₂×#15692255

C₂

semidirect

2. #20265523

C₂

direct (N × H)

3. C₂×#15692176

C₂

semidirect

4. C₂×C₄×#11717579

C₂

semidirect

9. #15692255

C₂²

semidirect

12. #15692176

C₂²

semidirect

13. C₄×#11717579

C₂²

semidirect

75. C₂×C₁₀₁

#486

semidirect

80. C₁₀₁

C₂×#486

semidirect

155. C₂

#20265523

direct (N × H)

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

subgroup	index
C₂×#15692255	2
#20265523	2
C₂×#15692176	2
C₂×C₄×#11717579	2
C₂×#127×C₁₀₁	2
C₂×#486	101

subgroup

index

C₂×#127×C₁₀₁

101

Subgroups — C₂×#20265523 #27181437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20265523 #27181437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265523 #27181437