Group #20252991 — subgroups

Subgroups — C₂×#15689382 #20252991

Order 6016.

492 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 157 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	C₂²×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#10625324	C₂	1	1	yes	maximal, characteristic
3	C₂×#13094004	C₂	1	2	yes	maximal
4	#15689382	C₂	1	8	yes	maximal, direct product factor
5	C₂×#13093947	C₂	1	2	yes	maximal, complemented
6	C₂×#106×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₉₄	C₂²	1	2	yes	—
8	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₈₈	—	2	4	no	—
10	C₂×Dic₁₈₈	C₂²	1	4	yes	complemented
11	C₂×#10625324	—	2	4	no	—
12	C₂×#10625324	C₂²	1	4	yes	—
13	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	#13094004	—	2	4	no	—
15	#13094004	C₂²	1	8	yes	—
16	C₂×C₄⋊Dic₄₇	—	2	2	no	—
17	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
18	#13093947	C₂²	1	8	yes	complemented
19	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	#106×C₄₇	C₂²	1	4	yes	complemented
22	C₂³×D₄₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	—	2	2	no	—
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₇	—	2	2	no	—
26	C₂²×Dic₄₇	D₄	1	1	yes	characteristic
27	C₂×C₄×D₄₇	—	2	2	no	—
28	C₂×C₄×D₄₇	—	2	2	no	—
29	C₂×C₄×D₄₇	D₄	1	2	yes	—
30	C₂×C₄×D₄₇	C₂³	1	2	yes	—
31	C₂×Dic₉₄	—	2	4	no	—
32	C₂×Dic₉₄	—	2	4	no	—
33	C₂×Dic₉₄	—	2	2	no	—
34	C₂×Dic₉₄	C₂³	1	4	yes	—
35	C₂×C₈×C₄₇	D₄	1	4	yes	—
36	C₂×C₈×C₄₇	C₂³	1	2	yes	—
37	C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
39	(C₂×D₄₇)⋊C₄	—	2	8	no	—
40	C₄⋊Dic₄₇	—	2	8	no	—
41	C₂×C₄₇⋊C₈	—	2	2	no	—
42	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
43	Dic₁₈₈	—	2	8	no	—
44	#10625324	—	2	8	no	—
45	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₄₇	—	2	2	no	—
47	C₂²×D₄₇	—	2	4	no	—
48	C₂²×D₄₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₄₇	—	2	4	no	—
50	C₂×C₄×C₄₇	—	4	4	no	—
51	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#41	1	2	yes	—
54	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₄₇	—	2	2	no	—
56	C₂×Dic₄₇	—	2	4	no	—
57	C₂×Dic₄₇	—	4	4	no	—
58	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
59	C₄×D₄₇	—	2	4	no	—
60	C₄×D₄₇	—	2	2	no	—
61	C₄×D₄₇	—	2	2	no	—
62	Dic₉₄	—	2	4	no	—
63	Dic₉₄	—	2	4	no	—
64	Dic₉₄	—	4	4	no	—
65	C₈×C₄₇	C₂×D₄	1	4	yes	—
66	C₄₇⋊C₈	—	2	2	no	—
67	C₂²×C₄₇	#121	1	2	yes	—
68	C₂²×C₄₇	#137	1	4	yes	—
69	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
70	C₂×D₄₇	—	2	2	no	—
71	C₂×D₄₇	—	2	4	no	—
72	C₂×D₄₇	—	4	4	no	—
73	C₄×C₄₇	—	4	4	no	—
74	C₄×C₄₇	#121	1	2	yes	—
75	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
76	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
77	Dic₄₇	—	2	2	no	—
78	Dic₄₇	—	4	4	no	—
79	C₂×#470	—	47	1	no	maximal, Sylow
80	C₂×C₄₇	#470	1	4	yes	complemented
81	C₂×C₄₇	C₂×#121	1	1	yes	characteristic
82	C₂×C₄₇	C₂×#137	1	2	yes	—
83	D₄₇	—	4	2	no	—
84	C₂²×Q₁₆	—	47	1	no	—
85	C₂²×M₁₆	—	47	1	no	—
86	C₂×C₂²⋊Q₈	—	47	2	no	—
87	#470	—	47	8	no	—
88	C₂×#106	D₄₇	1	1	yes	characteristic, complemented
89	C₂×#103	—	47	2	no	—
90	C₄₇	C₂×#470	1	1	yes	characteristic, complemented, Sylow
91	C₂²×Q₈	—	47	2	no	—
92	C₂³×C₄	—	47	1	no	—
93	C₂×Q₁₆	—	47	4	no	—
94	C₂×Q₁₆	—	94	4	no	—
95	C₂×M₁₆	—	47	4	no	—
96	C₂×M₁₆	—	94	4	no	—
97	C₂²×C₈	—	47	1	no	—
98	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
99	C₂²⋊Q₈	—	47	8	no	—
100	C₂²⋊Q₈	—	94	4	no	—
101	C₂×C₄⋊C₄	—	94	2	no	—
102	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
103	C₂×C₂²⋊C₄	—	94	2	no	—
104	#106	C₂×D₄₇	1	4	yes	complemented
105	#103	—	47	8	no	—
106	C₂⁴	—	47	1	no	—
107	C₂×Q₈	—	47	4	no	—
108	C₂×Q₈	—	94	2	no	—
109	C₂×Q₈	—	94	4	no	—
110	C₂×Q₈	—	94	4	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	47	1	no	—
113	C₂²×C₄	—	47	2	no	—
114	C₂²×C₄	—	47	2	no	—
115	C₂²×C₄	—	94	2	no	—
116	C₂²×C₄	—	94	2	no	—
117	C₂²×C₄	—	94	2	no	—
118	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
119	Q₁₆	—	94	8	no	—
120	M₁₆	—	94	8	no	—
121	C₂×C₈	—	47	2	no	—
122	C₂×C₈	—	94	2	no	—
123	C₂×C₈	D₁₈₈	1	4	yes	—
124	C₂×C₈	C₂²×D₄₇	1	2	yes	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	94	8	no	—
127	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
128	C₂²⋊C₄	—	94	8	no	—
129	C₂³	—	47	2	no	—
130	C₂³	—	94	2	no	—
131	C₂³	—	94	4	no	—
132	C₂³	D₄×D₄₇	1	1	yes	characteristic
133	Q₈	—	94	4	no	—
134	Q₈	—	94	4	no	—
135	Q₈	—	188	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	47	2	no	—
139	C₂×C₄	—	94	2	no	—
140	C₂×C₄	—	94	4	no	—
141	C₂×C₄	—	94	2	no	—
142	C₂×C₄	—	94	2	no	—
143	C₂×C₄	—	94	4	no	—
144	C₂×C₄	—	188	4	no	—
145	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
146	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
147	C₂×C₄	#10625354	1	2	yes	—
148	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
149	C₈	—	94	2	no	—
150	C₈	C₂×D₁₈₈	1	4	yes	—
151	C₂²	—	94	2	no	—
152	C₂²	—	94	4	no	—
153	C₂²	—	188	4	no	—
154	C₂²	#13093999	1	2	yes	—
155	C₂²	#13094053	1	4	yes	—
156	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
157	C₄	—	4	4	no	—
158	C₄	—	94	2	no	—
159	C₄	—	188	4	no	—
160	C₄	#13093999	1	2	yes	—
161	C₄	C₂²×D₁₈₈	1	1	yes	characteristic
162	C₄	C₂×#10625354	1	1	yes	characteristic
163	C₂	—	188	2	no	—
164	C₂	#15689382	1	4	yes	direct product factor
165	C₂	C₂×#13093999	1	1	yes	characteristic
166	C₂	C₂×#13094053	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₈₈

C₂

yes

maximal, characteristic, complemented

C₂²×#10625324

C₂

yes

maximal, characteristic

C₂×#13094004

C₂

yes

maximal

#15689382

C₂

yes

maximal, direct product factor

C₂×#13093947

C₂

yes

maximal, complemented

C₂×#106×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₉₄

C₂²

yes

—

C₂²×C₄×D₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

C₂²

yes

complemented

C₂×#10625324

—

C₂×#10625324

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

#13094004

—

#13094004

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×(C₂×D₄₇)⋊C₄

—

#13093947

C₂²

yes

complemented

C₂²×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

#106×C₄₇

C₂²

yes

complemented

C₂³×D₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

D₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

D₄

yes

—

C₂×C₄×D₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

—

C₂×C₈×C₄₇

D₄

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂³

yes

—

Dic₁₈₈

—

#10625324

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

C₂×D₄

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

#137

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#121

yes

—

C₄×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

C₂×#41

yes

characteristic

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

—

C₂²×Q₁₆

—

C₂²×M₁₆

—

C₂×C₂²⋊Q₈

—

#470

—

C₂×#106

D₄₇

yes

characteristic, complemented

C₂×#103

—

C₄₇

C₂×#470

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂³×C₄

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

characteristic

C₂²⋊Q₈

—

100

C₂²⋊Q₈

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

103

C₂×C₂²⋊C₄

—

104

#106

C₂×D₄₇

yes

complemented

105

#103

—

106

C₂⁴

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

C₂²×D₄₇

yes

characteristic

119

Q₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

D₁₈₈

yes

—

124

C₂×C₈

C₂²×D₄₇

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

C₂²×D₄₇

yes

—

128

C₂²⋊C₄

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

D₄×D₄₇

yes

characteristic

133

Q₈

—

134

Q₈

—

135

Q₈

—

188

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

188

—

145

C₂×C₄

C₂×D₁₈₈

yes

—

146

C₂×C₄

D₄×D₄₇

yes

characteristic

147

C₂×C₄

#10625354

yes

—

148

C₂×C₄

C₂³×D₄₇

yes

characteristic

149

C₈

—

150

C₈

C₂×D₁₈₈

yes

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

188

—

154

C₂²

#13093999

yes

—

155

C₂²

#13094053

yes

—

156

C₂²

C₂×D₄×D₄₇

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

188

—

160

C₄

#13093999

yes

—

161

C₄

C₂²×D₁₈₈

yes

characteristic

162

C₄

C₂×#10625354

yes

characteristic

163

C₂

—

188

—

164

C₂

#15689382

yes

direct product factor

165

C₂

C₂×#13093999

yes

characteristic

166

C₂

C₂×#13094053

yes

—

	subgroup	G/H
center	C₂³	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×#106×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×#106	D₄₇
hypocenter	C₄₇	C₂×#470
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×#106×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
1. C₂²×Dic₁₈₈	C₂	semidirect
4. #15689382	C₂	direct (N × H)
5. C₂×#13093947	C₂	semidirect
6. C₂×#106×C₄₇	C₂	semidirect
10. C₂×Dic₁₈₈	C₂²	semidirect
18. #13093947	C₂²	semidirect
21. #106×C₄₇	C₂²	semidirect
80. C₂×C₄₇	#470	semidirect
88. C₂×#106	D₄₇	semidirect
90. C₄₇	C₂×#470	semidirect
104. #106	C₂×D₄₇	semidirect
164. C₂	#15689382	direct (N × H)

H ≅ G/N

type

1. C₂²×Dic₁₈₈

C₂

semidirect

4. #15689382

C₂

direct (N × H)

5. C₂×#13093947

C₂

semidirect

6. C₂×#106×C₄₇

C₂

semidirect

10. C₂×Dic₁₈₈

C₂²

semidirect

18. #13093947

C₂²

semidirect

21. #106×C₄₇

C₂²

semidirect

80. C₂×C₄₇

#470

semidirect

88. C₂×#106

D₄₇

semidirect

90. C₄₇

C₂×#470

semidirect

104. #106

C₂×D₄₇

semidirect

164. C₂

#15689382

direct (N × H)

Subgroups — C₂×#15689382 #20252991

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689382 #20252991

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689382 #20252991