Group #20243597 — subgroups

Subgroups — #20243597

Order 6016.

324 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 89 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	#15689598	C₂	1	1	yes	maximal, characteristic
2	#15689163	C₂	1	2	yes	maximal, complemented
3	C₂×#13093941	C₂	1	1	yes	maximal, characteristic
4	#15688883	C₂	1	2	yes	maximal, complemented
5	C₄₇×#390	C₂	1	1	yes	maximal, characteristic
6	C₂×#10625329	C₂²	1	2	yes	—
7	D₄×Dic₄₇	—	2	2	no	—
8	D₄×Dic₄₇	—	2	2	no	—
9	#13093983	C₂²	1	2	yes	—
10	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
11	#13093957	—	2	2	no	—
12	#13093955	—	2	2	no	—
13	#13093941	C₄	1	4	yes	complemented
14	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
17	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄₇	D₄	1	2	yes	—
19	C₂×D₄×C₄₇	—	2	2	no	—
20	C₂×D₄×C₄₇	—	2	2	no	—
21	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
22	C₂×D₄×C₄₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₄₇	—	2	1	no	—
24	C₂²×C₄×C₄₇	—	2	1	no	—
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₇	—	2	2	no	—
27	C₂²×Dic₄₇	—	2	2	no	—
28	C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₂²⋊C₄×C₄₇	—	4	4	no	—
30	C₄²×C₄₇	—	2	2	no	—
31	C₂²⋊Dic₄₇	—	2	4	no	—
32	C₂²⋊Dic₄₇	—	2	2	no	—
33	C₄⋊Dic₄₇	D₄	1	2	yes	—
34	C₄⋊Dic₄₇	D₄	1	2	yes	—
35	C₄×Dic₄₇	—	2	2	no	—
36	#10625329	—	2	2	no	—
37	#10625329	C₂×C₄	1	4	yes	complemented
38	C₂×C₄₇⋊C₈	—	2	2	no	—
39	C₂³×C₄₇	—	2	2	no	—
40	C₂³×C₄₇	—	2	2	no	—
41	C₂³×C₄₇	—	4	2	no	—
42	C₂³×C₄₇	—	4	2	no	—
43	C₂³×C₄₇	C₄⋊C₄	1	4	yes	—
44	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
45	D₄×C₄₇	—	2	4	no	—
46	D₄×C₄₇	—	2	4	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	4	1	no	—
49	C₂×C₄×C₄₇	—	4	1	no	—
50	C₂×C₄×C₄₇	—	4	2	no	—
51	C₂×C₄×C₄₇	—	4	1	no	—
52	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#41	1	2	yes	—
56	C₂×Dic₄₇	—	2	2	no	—
57	C₂×Dic₄₇	—	2	2	no	—
58	C₂×Dic₄₇	—	8	2	no	—
59	C₂×Dic₄₇	—	8	2	no	—
60	C₄₇⋊C₈	—	2	4	no	—
61	C₂²×C₄₇	—	2	1	no	—
62	C₂²×C₄₇	—	2	1	no	—
63	C₂²×C₄₇	—	2	4	no	—
64	C₂²×C₄₇	—	4	4	no	—
65	C₂²×C₄₇	—	4	2	no	—
66	C₂²×C₄₇	—	4	2	no	—
67	C₂²×C₄₇	—	8	2	no	—
68	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄₇	#120	1	1	yes	characteristic
70	C₂²×C₄₇	#123	1	1	yes	characteristic
71	C₄×C₄₇	—	4	2	no	—
72	C₄×C₄₇	—	8	1	no	—
73	C₄×C₄₇	C₄×D₄	1	2	yes	—
74	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
75	Dic₄₇	—	4	2	no	—
76	Dic₄₇	—	4	2	no	—
77	#1856	—	47	1	no	maximal, Sylow
78	C₂×C₄₇	—	2	1	no	—
79	C₂×C₄₇	—	2	1	no	—
80	C₂×C₄₇	—	4	4	no	—
81	C₂×C₄₇	#385	1	1	yes	characteristic
82	C₂×C₄₇	#453	1	1	yes	characteristic
83	C₂×C₄₇	#455	1	1	yes	characteristic
84	#518	—	47	1	no	—
85	#428	—	47	2	no	—
86	C₂×#100	—	47	1	no	—
87	#390	D₄₇	1	1	yes	characteristic
88	#339	—	47	2	no	—
89	C₄₇	#1856	1	1	yes	characteristic, complemented, Sylow
90	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
91	C₂×M₁₆	—	47	2	no	—
92	C₄×D₄	—	94	2	no	—
93	C₄×D₄	—	94	2	no	—
94	#117	—	47	2	no	—
95	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
96	C₂×C₂²⋊C₄	—	2	2	no	—
97	C₂×C₂²⋊C₄	—	94	2	no	—
98	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
99	#107	—	94	2	no	—
100	#106	—	94	2	no	—
101	#100	—	47	4	no	—
102	C₂⁴	C₂²⋊D₄₇	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	Dic₉₄	1	2	yes	—
106	C₂×D₄	C₄×D₄₇	1	2	yes	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	94	2	no	—
110	C₂²×C₄	—	94	2	no	—
111	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
112	M₁₆	—	47	4	no	—
113	M₁₆	—	94	2	no	—
114	C₂×C₈	—	94	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	47	2	no	—
117	C₄⋊C₄	—	47	2	no	—
118	C₂²⋊C₄	—	4	4	no	—
119	C₂²⋊C₄	—	94	4	no	—
120	C₂²⋊C₄	—	94	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	94	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	4	2	no	—
126	C₂³	—	4	2	no	—
127	C₂³	Dic₄₇⋊C₄	1	4	yes	—
128	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
129	D₄	—	2	4	no	—
130	D₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	94	2	no	—
137	C₂×C₄	—	94	2	no	—
138	C₂×C₄	—	376	2	no	—
139	C₂×C₄	—	376	2	no	—
140	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
142	C₂×C₄	D₄×D₄₇	1	2	yes	—
143	C₂×C₄	#10625352	1	2	yes	—
144	C₈	—	94	4	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	4	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	8	2	no	—
152	C₂²	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
153	C₂²	#13093992	1	1	yes	characteristic
154	C₂²	#13094005	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	8	1	no	—
157	C₄	—	188	2	no	—
158	C₄	—	188	2	no	—
159	C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
160	C₄	#13094001	1	2	yes	—
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	4	4	no	—
164	C₂	#15688995	1	1	yes	characteristic
165	C₂	#15689290	1	1	yes	characteristic
166	C₂	#15689300	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689598

C₂

yes

maximal, characteristic

#15689163

C₂

yes

maximal, complemented

C₂×#13093941

C₂

yes

maximal, characteristic

#15688883

C₂

yes

maximal, complemented

C₄₇×#390

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

—

D₄×Dic₄₇

—

D₄×Dic₄₇

—

#13093983

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

—

#13093957

—

#13093955

—

#13093941

C₄

yes

complemented

C₂²×D₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

D₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

—

C₄⋊Dic₄₇

D₄

yes

—

C₄×Dic₄₇

—

#10625329

—

#10625329

C₂×C₄

yes

complemented

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₄⋊C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

#1856

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

D₄₇

yes

characteristic

#339

—

C₄₇

#1856

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×D₄₇

yes

characteristic

C₂×M₁₆

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#107

—

100

#106

—

101

#100

—

102

C₂⁴

C₂²⋊D₄₇

yes

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

Dic₉₄

yes

—

106

C₂×D₄

C₄×D₄₇

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²×D₄₇

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

Dic₄₇⋊C₄

yes

—

128

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

376

—

139

C₂×C₄

—

376

—

140

C₂×C₄

C₂×Dic₉₄

yes

characteristic

141

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

142

C₂×C₄

D₄×D₄₇

yes

—

143

C₂×C₄

#10625352

yes

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×Dic₄₇⋊C₄

yes

characteristic

153

C₂²

#13093992

yes

characteristic

154

C₂²

#13094005

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

188

—

158

C₄

—

188

—

159

C₄

(C₂²⋊D₄₇)⋊C₄

yes

—

160

C₄

#13094001

yes

—

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#15688995

yes

characteristic

165

C₂

#15689290

yes

characteristic

166

C₂

#15689300

yes

characteristic

	subgroup	G/H
center	C₂²	#13093992
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#390	C₂
socle	C₂²×C₄₇	#120
hypercenter	#390	D₄₇
hypocenter	C₄₇	#1856
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #390

subgroup

G/H

center

C₂²

#13093992

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#390

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #390

N	H ≅ G/N	type
2. #15689163	C₂	semidirect
4. #15688883	C₂	semidirect
13. #13093941	C₄	semidirect
37. #10625329	C₂×C₄	semidirect
89. C₄₇	#1856	semidirect

H ≅ G/N

type

2. #15689163

C₂

semidirect

4. #15688883

C₂

semidirect

13. #13093941

C₄

semidirect

37. #10625329

C₂×C₄

semidirect

89. C₄₇

#1856

semidirect

Subgroups — #20243597

Conjugacy classes of subgroups

Complements