Group #20253695 — subgroups

Subgroups — C₂×#15689471 #20253695

Order 6016.

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×#13094021	C₂	1	1	yes	maximal, characteristic
2	#15689471	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13093944	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₈₈	—	2	1	no	—
7	C₂²×D₁₈₈	—	2	1	no	—
8	C₂²×D₁₈₈	C₂²	1	2	yes	—
9	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
10	#13094021	—	2	2	no	—
11	#13094021	C₂²	1	4	yes	—
12	#13093944	C₂²	1	16	yes	complemented
13	C₄×C₄₇⋊C₈	C₂²	1	4	yes	complemented
14	C₂²×Q₈×C₄₇	—	2	1	no	—
15	C₄⋊Q₈×C₄₇	—	2	2	no	—
16	C₄⋊Q₈×C₄₇	C₂²	1	4	yes	complemented
17	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
18	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂³×D₄₇	—	2	2	no	—
21	C₂×Q₈×C₄₇	—	2	4	no	—
22	C₂×Q₈×C₄₇	—	4	4	no	—
23	C₂²×C₄×C₄₇	—	2	2	no	—
24	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×D₁₈₈	—	2	8	no	—
28	C₂×D₁₈₈	—	2	2	no	—
29	C₂×D₁₈₈	—	2	4	no	—
30	C₂×D₁₈₈	—	2	4	no	—
31	C₂×D₁₈₈	—	4	4	no	—
32	C₂×D₁₈₈	—	4	4	no	—
33	C₂×D₁₈₈	C₂³	1	4	yes	—
34	C₄⋊C₄×C₄₇	—	2	4	no	—
35	C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
37	C₄²×C₄₇	D₄	1	2	yes	—
38	C₄²×C₄₇	C₂³	1	2	yes	—
39	C₂×C₄₇⋊C₈	—	2	4	no	—
40	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
41	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₄₇	—	2	4	no	—
43	C₂²×D₄₇	—	4	4	no	—
44	C₂²×D₄₇	—	4	4	no	—
45	C₂²×D₄₇	—	4	4	no	—
46	Q₈×C₄₇	—	4	8	no	—
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	—	4	4	no	—
49	C₂×C₄×C₄₇	D₈	1	4	yes	—
50	C₂×C₄×C₄₇	SD₁₆	1	4	yes	—
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₄₇	#41	1	4	yes	—
55	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
56	D₁₈₈	—	2	8	no	—
57	D₁₈₈	—	4	4	no	—
58	D₁₈₈	—	4	8	no	—
59	D₁₈₈	—	4	8	no	—
60	C₄₇⋊C₈	—	2	4	no	—
61	C₂²×C₄₇	#124	1	2	yes	—
62	C₂²×C₄₇	C₂×D₈	1	2	yes	—
63	C₂²×C₄₇	C₂×SD₁₆	1	2	yes	—
64	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₄₇	—	4	4	no	—
66	C₂×D₄₇	—	4	4	no	—
67	C₂×D₄₇	—	4	4	no	—
68	C₂×D₄₇	—	8	8	no	—
69	C₄×C₄₇	—	4	4	no	—
70	C₄×C₄₇	#124	1	2	yes	—
71	C₄×C₄₇	C₂×D₈	1	4	yes	—
72	C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
73	C₄×C₄₇	C₂×#41	1	2	yes	—
74	C₂×#486	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	#486	1	4	yes	complemented
76	C₂×C₄₇	C₂×#124	1	1	yes	characteristic
77	C₂×C₄₇	C₂²×D₈	1	1	yes	characteristic
78	C₂×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
79	D₄₇	—	8	4	no	—
80	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
81	C₂×#127	—	47	1	no	—
82	#486	—	47	8	no	—
83	C₂×#102	—	47	4	no	—
84	C₂×C₄×C₈	—	47	1	no	—
85	C₄₇	C₂×#486	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	2	1	no	—
87	C₂²×D₄	—	47	2	no	—
88	C₂²×D₄	—	94	1	no	—
89	C₂²×D₄	—	94	1	no	—
90	C₂²×C₈	—	47	2	no	—
91	C₄⋊Q₈	—	2	2	no	—
92	C₄⋊Q₈	C₂×D₄₇	1	4	yes	complemented
93	#127	—	47	4	no	—
94	#127	—	94	2	no	—
95	C₂×C₄⋊C₄	—	2	1	no	—
96	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
97	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
98	#102	—	47	16	no	—
99	C₄×C₈	—	47	4	no	—
100	C₂⁴	—	94	2	no	—
101	C₂×Q₈	—	2	4	no	—
102	C₂×Q₈	—	4	4	no	—
103	C₂×D₄	—	47	4	no	—
104	C₂×D₄	—	94	8	no	—
105	C₂×D₄	—	94	4	no	—
106	C₂×D₄	—	94	4	no	—
107	C₂×D₄	—	94	2	no	—
108	C₂×D₄	—	188	4	no	—
109	C₂×D₄	—	188	4	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
114	C₂×C₈	—	47	4	no	—
115	C₂×C₈	—	94	4	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
119	C₄²	C₂²⋊D₄₇	1	2	yes	—
120	C₄²	C₂²×D₄₇	1	2	yes	—
121	C₂³	—	94	4	no	—
122	C₂³	—	188	4	no	—
123	C₂³	—	188	4	no	—
124	C₂³	—	188	4	no	—
125	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
126	Q₈	—	4	8	no	—
127	D₄	—	94	8	no	—
128	D₄	—	188	4	no	—
129	D₄	—	188	8	no	—
130	D₄	—	188	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	4	4	no	—
133	C₂×C₄	#10625334	1	4	yes	—
134	C₂×C₄	#10625336	1	4	yes	—
135	C₂×C₄	#10625354	1	4	yes	—
136	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
139	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
140	C₈	—	94	4	no	—
141	C₂²	—	376	8	no	—
142	C₂²	—	188	4	no	—
143	C₂²	—	188	4	no	—
144	C₂²	—	188	4	no	—
145	C₂²	#13094013	1	2	yes	—
146	C₂²	C₂×#10625334	1	2	yes	—
147	C₂²	C₂×#10625336	1	2	yes	—
148	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
149	C₄	—	4	4	no	—
150	C₄	#13094013	1	2	yes	—
151	C₄	C₂×#10625334	1	4	yes	—
152	C₄	C₂×#10625336	1	4	yes	—
153	C₄	C₂×#10625354	1	2	yes	—
154	C₂	—	376	4	no	—
155	C₂	#15689471	1	4	yes	direct product factor
156	C₂	C₂×#13094013	1	1	yes	characteristic
157	C₂	C₂²×#10625334	1	1	yes	characteristic
158	C₂	C₂²×#10625336	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094021

C₂

yes

maximal, characteristic

#15689471

C₂

yes

maximal, direct product factor

C₂×#13093944

C₂

yes

maximal, complemented

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

—

C₂²×D₁₈₈

—

C₂²×D₁₈₈

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

#13094021

—

#13094021

C₂²

yes

—

#13093944

C₂²

yes

complemented

C₄×C₄₇⋊C₈

C₂²

yes

complemented

C₂²×Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

complemented

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

D₈

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

#124

yes

—

C₂²×C₄₇

C₂×D₈

yes

—

C₂²×C₄₇

C₂×SD₁₆

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#124

yes

—

C₄×C₄₇

C₂×D₈

yes

—

C₄×C₄₇

C₂×SD₁₆

yes

—

C₄×C₄₇

C₂×#41

yes

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

C₂²×D₈

yes

characteristic

C₂×C₄₇

C₂²×SD₁₆

yes

characteristic

D₄₇

—

C₂×C₄⋊Q₈

D₄₇

yes

characteristic, complemented

C₂×#127

—

#486

—

C₂×#102

—

C₂×C₄×C₈

—

C₄₇

C₂×#486

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

complemented

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#102

—

C₄×C₈

—

100

C₂⁴

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

188

—

109

C₂×D₄

—

188

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

112

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

113

C₂²×C₄

C₂²×D₄₇

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂²×D₄₇

yes

—

119

C₄²

C₂²⋊D₄₇

yes

—

120

C₄²

C₂²×D₄₇

yes

—

121

C₂³

—

122

C₂³

—

188

—

123

C₂³

—

188

—

124

C₂³

—

188

—

125

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

126

Q₈

—

127

D₄

—

128

D₄

—

188

—

129

D₄

—

188

—

130

D₄

—

188

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#10625334

yes

—

134

C₂×C₄

#10625336

yes

—

135

C₂×C₄

#10625354

yes

—

136

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

138

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

139

C₂×C₄

C₂³×D₄₇

yes

characteristic

140

C₈

—

141

C₂²

—

376

—

142

C₂²

—

188

—

143

C₂²

—

188

—

144

C₂²

—

188

—

145

C₂²

#13094013

yes

—

146

C₂²

C₂×#10625334

yes

—

147

C₂²

C₂×#10625336

yes

—

148

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

149

C₄

—

150

C₄

#13094013

yes

—

151

C₄

C₂×#10625334

yes

—

152

C₄

C₂×#10625336

yes

—

153

C₄

C₂×#10625354

yes

—

154

C₂

—

376

—

155

C₂

#15689471

yes

direct product factor

156

C₂

C₂×#13094013

yes

characteristic

157

C₂

C₂²×#10625334

yes

characteristic

158

C₂

C₂²×#10625336

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂×#486
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₄₇

hypocenter

C₄₇

C₂×#486

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #15689471	C₂	direct (N × H)
3. C₂×#13093944	C₂	semidirect
4. C₂×C₄×C₄₇⋊C₈	C₂	semidirect
5. C₂×C₄⋊Q₈×C₄₇	C₂	semidirect
12. #13093944	C₂²	semidirect
13. C₄×C₄₇⋊C₈	C₂²	semidirect
16. C₄⋊Q₈×C₄₇	C₂²	semidirect
75. C₂×C₄₇	#486	semidirect
80. C₂×C₄⋊Q₈	D₄₇	semidirect
85. C₄₇	C₂×#486	semidirect
92. C₄⋊Q₈	C₂×D₄₇	semidirect
155. C₂	#15689471	direct (N × H)

H ≅ G/N

type

2. #15689471

C₂

direct (N × H)

3. C₂×#13093944

C₂

semidirect

4. C₂×C₄×C₄₇⋊C₈

C₂

semidirect

5. C₂×C₄⋊Q₈×C₄₇

C₂

semidirect

12. #13093944

C₂²

semidirect

13. C₄×C₄₇⋊C₈

C₂²

semidirect

16. C₄⋊Q₈×C₄₇

C₂²

semidirect

75. C₂×C₄₇

#486

semidirect

80. C₂×C₄⋊Q₈

D₄₇

semidirect

85. C₄₇

C₂×#486

semidirect

92. C₄⋊Q₈

C₂×D₄₇

semidirect

155. C₂

#15689471

direct (N × H)

Subgroups — C₂×#15689471 #20253695

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689471 #20253695

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689471 #20253695