Group #20241978 — subgroups

Subgroups — #20241978

Order 6016.

270 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 63 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	#15689672	C₂	1	1	yes	maximal, characteristic
2	#15689075	C₂	1	2	yes	maximal, complemented
3	#15688863	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688862	C₂	1	2	yes	maximal, complemented
5	C₄₇×#329	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625354	C₄	1	1	yes	characteristic
7	C₂×#10625352	C₄	1	1	yes	characteristic
8	C₂×#10625324	—	2	2	no	—
9	#13094022	C₄	1	1	yes	characteristic
10	#13094020	C₄	1	1	yes	characteristic
11	#13094012	C₂²	1	1	yes	characteristic
12	#13093997	—	2	2	no	—
13	(C₂²⋊D₄₇)⋊C₄	—	2	2	no	—
14	C₄²×D₄₇	C₂²	1	1	yes	characteristic
15	#13093935	C₂²	1	2	yes	—
16	C₈×Dic₄₇	—	2	2	no	—
17	#124×C₄₇	C₂²	1	1	yes	characteristic
18	#97×C₄₇	C₂²	1	2	yes	—
19	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₄₇	—	2	1	no	—
22	C₂×C₂²⋊D₄₇	—	2	2	no	—
23	C₂²×Dic₄₇	—	2	1	no	—
24	#10625354	—	4	2	no	—
25	#10625352	—	4	2	no	—
26	C₂×D₁₈₈	—	2	1	no	—
27	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₄₇	—	2	1	no	—
29	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₄₇	D₄	1	2	yes	—
31	C₂×Dic₉₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₂²⋊C₄×C₄₇	—	2	2	no	—
34	C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	(C₂×D₄₇)⋊C₄	—	2	2	no	—
36	Dic₄₇⋊C₄	—	2	2	no	—
37	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₄₇	D₄	1	2	yes	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	#10625324	—	2	8	no	—
41	C₂³×C₄₇	—	2	1	no	—
42	C₂²×D₄₇	—	2	1	no	—
43	C₂²×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
44	Q₈×C₄₇	—	4	1	no	—
45	D₄×C₄₇	—	4	1	no	—
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
49	C₂²⋊D₄₇	—	4	2	no	—
50	C₂²⋊D₄₇	—	4	4	no	—
51	C₂×Dic₄₇	—	2	1	no	—
52	C₂×Dic₄₇	—	4	2	no	—
53	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₄₇	C₂²⋊C₄	1	2	yes	—
55	D₁₈₈	—	4	1	no	—
56	D₁₈₈	—	4	2	no	—
57	C₄×D₄₇	—	2	2	no	—
58	C₄×D₄₇	—	2	4	no	—
59	C₄×D₄₇	—	4	2	no	—
60	Dic₉₄	—	4	1	no	—
61	C₈×C₄₇	—	2	4	no	—
62	C₄₇⋊C₈	—	2	4	no	—
63	C₂²×C₄₇	—	4	1	no	—
64	C₂²×C₄₇	—	4	2	no	—
65	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×D₄₇	—	2	1	no	—
67	C₂×D₄₇	—	2	2	no	—
68	C₂×D₄₇	—	4	1	no	—
69	C₂×D₄₇	—	4	2	no	—
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	—	2	2	no	—
72	C₄×C₄₇	—	4	1	no	—
73	Dic₄₇	—	4	1	no	—
74	Dic₄₇	#100	1	2	yes	—
75	Dic₄₇	#104	1	4	yes	—
76	#1525	—	47	1	no	maximal, Sylow
77	C₂×C₄₇	—	8	1	no	—
78	C₂×C₄₇	C₂×#100	1	1	yes	characteristic
79	C₂×C₄₇	C₂×#104	1	2	yes	—
80	D₄₇	—	4	1	no	—
81	D₄₇	—	8	1	no	—
82	#532	—	47	1	no	—
83	C₄×M₁₆	—	47	2	no	—
84	#329	—	47	1	no	—
85	#329	—	47	2	no	—
86	#329	D₄₇	1	1	yes	characteristic, complemented
87	C₄₇	#1525	1	1	yes	characteristic, complemented, Sylow
88	C₂×#41	—	47	1	no	—
89	C₂×#41	—	47	1	no	—
90	C₂×M₁₆	—	94	2	no	—
91	C₄⋊Q₈	—	47	1	no	—
92	#127	—	47	1	no	—
93	#124	—	47	1	no	—
94	#124	C₂×D₄₇	1	1	yes	characteristic
95	#121	—	94	2	no	—
96	C₄×D₄	—	94	2	no	—
97	C₂×C₄²	—	47	1	no	—
98	#97	—	47	2	no	—
99	#97	C₂×D₄₇	1	2	yes	—
100	C₄×C₈	—	94	2	no	—
101	#41	—	188	2	no	—
102	#41	—	188	2	no	—
103	C₂×Q₈	—	47	1	no	—
104	C₂×Q₈	C₄×D₄₇	1	1	yes	characteristic
105	C₂×D₄	—	47	1	no	—
106	C₂×D₄	—	94	1	no	—
107	C₂×D₄	—	94	2	no	—
108	C₂×D₄	—	94	1	no	—
109	C₂×D₄	C₄×D₄₇	1	1	yes	characteristic
110	C₂²×C₄	—	47	1	no	—
111	C₂²×C₄	—	47	2	no	—
112	C₂²×C₄	—	94	1	no	—
113	C₂²×C₄	—	94	1	no	—
114	M₁₆	—	94	8	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	94	2	no	—
117	C₄⋊C₄	—	94	2	no	—
118	C₂²⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	94	2	no	—
120	C₄²	—	47	1	no	—
121	C₄²	—	47	2	no	—
122	C₄²	C₂²×D₄₇	1	1	yes	characteristic
123	C₂³	—	2	1	no	—
124	C₂³	—	47	1	no	—
125	C₂³	—	94	1	no	—
126	Q₈	—	4	1	no	—
127	Q₈	—	188	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	188	1	no	—
130	D₄	—	188	2	no	—
131	D₄	—	188	4	no	—
132	D₄	—	188	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	47	2	no	—
135	C₂×C₄	—	47	1	no	—
136	C₂×C₄	—	94	4	no	—
137	C₂×C₄	—	94	2	no	—
138	C₂×C₄	—	94	1	no	—
139	C₂×C₄	—	188	2	no	—
140	C₂×C₄	—	188	2	no	—
141	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
142	C₂×C₄	D₄×D₄₇	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	94	4	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	4	2	no	—
147	C₂²	—	94	1	no	—
148	C₂²	—	94	2	no	—
149	C₂²	—	188	1	no	—
150	C₂²	—	188	2	no	—
151	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	2	no	—
154	C₄	—	4	1	no	—
155	C₄	—	47	2	no	—
156	C₄	—	47	4	no	—
157	C₄	—	188	1	no	—
158	C₂	—	8	1	no	—
159	C₂	—	376	1	no	—
160	C₂	—	188	1	no	—
161	C₂	#100×D₄₇	1	1	yes	characteristic
162	C₂	#104×D₄₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689672

C₂

yes

maximal, characteristic

#15689075

C₂

yes

maximal, complemented

#15688863

C₂

yes

maximal, characteristic, complemented

#15688862

C₂

yes

maximal, complemented

C₄₇×#329

C₂

yes

maximal, characteristic, complemented

C₂×#10625354

C₄

yes

characteristic

C₂×#10625352

C₄

yes

characteristic

C₂×#10625324

—

#13094022

C₄

yes

characteristic

#13094020

C₄

yes

characteristic

#13094012

C₂²

yes

characteristic

#13093997

—

(C₂²⋊D₄₇)⋊C₄

—

C₄²×D₄₇

C₂²

yes

characteristic

#13093935

C₂²

yes

—

C₈×Dic₄₇

—

#124×C₄₇

C₂²

yes

characteristic

#97×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

C₂×C₄

yes

characteristic

C₂×D₄×C₄₇

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₄₇

—

C₂×C₂²⋊D₄₇

—

C₂²×Dic₄₇

—

#10625354

—

#10625352

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

D₄

yes

—

C₂×Dic₉₄

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

(C₂×D₄₇)⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

#10625324

—

C₂³×C₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂²⋊C₄

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₇

C₂²⋊C₄

yes

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

#100

yes

—

Dic₄₇

#104

yes

—

#1525

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#100

yes

characteristic

C₂×C₄₇

C₂×#104

yes

—

D₄₇

—

D₄₇

—

#532

—

C₄×M₁₆

—

#329

—

#329

—

#329

D₄₇

yes

characteristic, complemented

C₄₇

#1525

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#124

—

#124

C₂×D₄₇

yes

characteristic

#121

—

C₄×D₄

—

C₂×C₄²

—

#97

—

#97

C₂×D₄₇

yes

—

100

C₄×C₈

—

101

#41

—

188

—

102

#41

—

188

—

103

C₂×Q₈

—

104

C₂×Q₈

C₄×D₄₇

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₄×D₄₇

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂²×D₄₇

yes

characteristic

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

Q₈

—

127

Q₈

—

188

—

128

D₄

—

129

D₄

—

188

—

130

D₄

—

188

—

131

D₄

—

188

—

132

D₄

—

188

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

188

—

140

C₂×C₄

—

188

—

141

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

142

C₂×C₄

D₄×D₄₇

yes

—

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

188

—

150

C₂²

—

188

—

151

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

188

—

158

C₂

—

159

C₂

—

376

—

160

C₂

—

188

—

161

C₂

#100×D₄₇

yes

characteristic

162

C₂

#104×D₄₇

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#329	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#329	D₄₇
hypocenter	C₄₇	#1525
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #329

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#329

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #329

N	H ≅ G/N	type
2. #15689075	C₂	semidirect
3. #15688863	C₂	semidirect
4. #15688862	C₂	semidirect
5. C₄₇×#329	C₂	semidirect
86. #329	D₄₇	semidirect
87. C₄₇	#1525	semidirect

H ≅ G/N

type

2. #15689075

C₂

semidirect

3. #15688863

C₂

semidirect

4. #15688862

C₂

semidirect

5. C₄₇×#329

C₂

semidirect

86. #329

D₄₇

semidirect

87. C₄₇

#1525

semidirect

Subgroups — #20241978

Conjugacy classes of subgroups

Complements