Group #20242527 — subgroups

Subgroups — #20242527

Order 6016.

206 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	#15689502	C₂	1	1	yes	maximal, characteristic
3	#15689415	C₂	1	1	yes	maximal, characteristic
4	#15688871	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#15688853	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#477	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625337	—	2	2	no	—
9	#13094024	—	2	1	no	—
10	#13094024	C₂²	1	1	yes	characteristic
11	#13094023	—	2	2	no	—
12	#13094023	—	2	1	no	—
13	Q₈×Dic₄₇	—	2	1	no	—
14	C₄×Dic₉₄	—	2	1	no	—
15	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
16	#13093957	—	2	1	no	—
17	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
18	#13093947	—	2	1	no	—
19	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
22	#107×C₄₇	—	2	1	no	—
23	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
24	#103×C₄₇	—	2	1	no	—
25	C₂×Q₈×C₄₇	—	2	1	no	—
26	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
27	C₂×Dic₉₄	—	2	2	no	—
28	C₂×Dic₉₄	D₄	1	1	yes	characteristic
29	C₂×Dic₉₄	D₄	1	2	yes	—
30	C₂×C₈×C₄₇	—	2	1	no	—
31	C₄⋊C₄×C₄₇	—	2	1	no	—
32	C₄⋊C₄×C₄₇	—	2	1	no	—
33	C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
35	C₄²×C₄₇	—	2	1	no	—
36	C₄²×C₄₇	C₂³	1	1	yes	characteristic
37	#10625337	—	4	4	no	—
38	C₄⋊Dic₄₇	—	2	2	no	—
39	C₄⋊Dic₄₇	—	2	1	no	—
40	C₄⋊Dic₄₇	—	2	1	no	—
41	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
42	Dic₄₇⋊C₄	—	2	2	no	—
43	Dic₄₇⋊C₄	—	2	1	no	—
44	C₄×Dic₄₇	—	2	2	no	—
45	C₄×Dic₄₇	—	2	1	no	—
46	C₂×C₄₇⋊C₈	—	2	1	no	—
47	C₂×C₄₇⋊C₈	—	2	1	no	—
48	Q₈×C₄₇	—	2	1	no	—
49	Q₈×C₄₇	—	4	2	no	—
50	Q₈×C₄₇	Q₁₆	1	2	yes	complemented
51	C₂×C₄×C₄₇	—	2	1	no	—
52	C₂×C₄×C₄₇	—	2	1	no	—
53	C₂×C₄×C₄₇	—	4	1	no	—
54	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₄₇	—	2	1	no	—
58	C₂×Dic₄₇	—	2	2	no	—
59	C₂×Dic₄₇	—	2	1	no	—
60	Dic₉₄	—	2	1	no	—
61	Dic₉₄	—	4	2	no	—
62	Dic₉₄	—	8	2	no	—
63	Dic₉₄	Q₁₆	1	2	yes	complemented
64	C₈×C₄₇	—	4	1	no	—
65	C₄₇⋊C₈	—	2	2	no	—
66	C₄₇⋊C₈	—	4	1	no	—
67	C₂²×C₄₇	#120	1	1	yes	characteristic
68	C₄×C₄₇	—	2	2	no	—
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	4	1	no	—
71	C₄×C₄₇	—	8	1	no	—
72	C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
73	C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₄₇	#137	1	1	yes	characteristic
75	C₄×C₄₇	#137	1	1	yes	characteristic
76	Dic₄₇	—	2	2	no	—
77	Dic₄₇	—	4	2	no	—
78	Dic₄₇	—	4	1	no	—
79	#1633	—	47	1	no	maximal, Sylow
80	C₂×C₄₇	#451	1	1	yes	characteristic
81	C₂×C₄₇	#451	1	1	yes	characteristic
82	C₂×C₄₇	#453	1	1	yes	characteristic
83	Q₈²	—	47	1	no	—
84	#494	—	47	1	no	—
85	#477	—	47	1	no	—
86	#477	D₄₇	1	1	yes	characteristic
87	#333	—	47	1	no	—
88	Q₈⋊C₈	—	47	1	no	—
89	Q₈⋊C₈	—	47	1	no	—
90	C₄₇	#1633	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₁₆	—	94	2	no	—
92	C₄⋊Q₈	—	47	1	no	—
93	C₄⋊Q₈	—	94	1	no	—
94	C₄⋊Q₈	—	94	2	no	—
95	C₄⋊Q₈	—	94	1	no	—
96	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
97	C₄×Q₈	—	47	1	no	—
98	C₄×Q₈	—	94	1	no	—
99	C₄×Q₈	—	94	1	no	—
100	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	#107	—	94	1	no	—
103	C₄⋊C₈	—	47	1	no	—
104	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
105	#103	—	2	1	no	—
106	#103	—	94	1	no	—
107	C₄×C₈	—	47	1	no	—
108	C₂×Q₈	—	2	1	no	—
109	C₂×Q₈	—	47	1	no	—
110	C₂×Q₈	—	47	2	no	—
111	C₂×Q₈	—	94	2	no	—
112	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
113	Q₁₆	—	188	4	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	94	1	no	—
116	C₂×C₈	—	94	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	47	1	no	—
121	C₄⋊C₄	—	94	2	no	—
122	C₄⋊C₄	—	94	2	no	—
123	C₄⋊C₄	—	94	1	no	—
124	C₄⋊C₄	—	94	1	no	—
125	C₄⋊C₄	—	94	1	no	—
126	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
127	C₄²	—	2	1	no	—
128	C₄²	—	94	2	no	—
129	C₄²	—	94	1	no	—
130	C₄²	C₂²×D₄₇	1	1	yes	characteristic
131	Q₈	—	2	1	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	47	2	no	—
134	Q₈	—	94	1	no	—
135	Q₈	—	376	2	no	—
136	Q₈	—	188	2	no	—
137	Q₈	#10625337	1	2	yes	complemented
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	94	1	no	—
142	C₂×C₄	—	94	1	no	—
143	C₂×C₄	—	94	2	no	—
144	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
145	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
147	C₈	—	4	1	no	—
148	C₈	—	94	2	no	—
149	C₈	—	188	1	no	—
150	C₂²	#13093992	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	1	no	—
153	C₄	—	4	1	no	—
154	C₄	—	8	1	no	—
155	C₄	—	94	2	no	—
156	C₄	—	188	1	no	—
157	C₄	—	188	2	no	—
158	C₄	Q₁₆×D₄₇	1	1	yes	characteristic
159	C₄	C₂×#10625337	1	1	yes	characteristic
160	C₄	#13094054	1	1	yes	characteristic
161	C₄	#13094056	1	1	yes	characteristic
162	C₂	#15689277	1	1	yes	characteristic
163	C₂	#15689278	1	1	yes	characteristic
164	C₂	#15689289	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₉₄

C₂

yes

maximal, characteristic

#15689502

C₂

yes

maximal, characteristic

#15689415

C₂

yes

maximal, characteristic

#15688871

C₂

yes

maximal, characteristic

Q₈⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

#15688853

C₂

yes

maximal, characteristic

C₄₇×#477

C₂

yes

maximal, characteristic

C₂×#10625337

—

#13094024

—

#13094024

C₂²

yes

characteristic

#13094023

—

#13094023

—

Q₈×Dic₄₇

—

C₄×Dic₉₄

—

C₄×Dic₉₄

C₂²

yes

characteristic

#13093957

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093947

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

C₂²

yes

characteristic

#107×C₄₇

—

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

#103×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

D₄

yes

characteristic

C₂×Dic₉₄

D₄

yes

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625337

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

characteristic

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

Q₁₆

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

Q₁₆

yes

complemented

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

#120

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₄×C₄₇

C₂×Q₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

Q₈²

—

#494

—

#477

—

#477

D₄₇

yes

characteristic

#333

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₄₇

#1633

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

C₂×D₄₇

yes

characteristic

101

#107

—

102

#107

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₄₇

yes

characteristic

105

#103

—

106

#103

—

107

C₄×C₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

113

Q₁₆

—

188

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₂²×D₄₇

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

376

—

136

Q₈

—

188

—

137

Q₈

#10625337

yes

complemented

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

D₄×D₄₇

yes

characteristic

145

C₂×C₄

D₄×D₄₇

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

188

—

150

C₂²

#13093992

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

188

—

157

C₄

—

188

—

158

C₄

Q₁₆×D₄₇

yes

characteristic

159

C₄

C₂×#10625337

yes

characteristic

160

C₄

#13094054

yes

characteristic

161

C₄

#13094056

yes

characteristic

162

C₂

#15689277

yes

characteristic

163

C₂

#15689278

yes

characteristic

164

C₂

#15689289

yes

characteristic

	subgroup	G/H
center	C₂²	#13093992
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#477	C₂
socle	C₂²×C₄₇	#120
hypercenter	#477	D₄₇
hypocenter	C₄₇	#1633
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#13093992

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#477

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
50. Q₈×C₄₇	Q₁₆	semidirect
63. Dic₉₄	Q₁₆	semidirect
90. C₄₇	#1633	semidirect
137. Q₈	#10625337	semidirect

H ≅ G/N

type

50. Q₈×C₄₇

Q₁₆

semidirect

63. Dic₉₄

Q₁₆

semidirect

90. C₄₇

#1633

semidirect

137. Q₈

#10625337

semidirect

Subgroups — #20242527

Conjugacy classes of subgroups

Complements