Group #20242609 — subgroups

Subgroups — #20242609

Order 6016.

200 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 65 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	#15689473	C₂	1	1	yes	maximal, characteristic
2	#15689408	C₂	1	1	yes	maximal, characteristic
3	#15689333	C₂	1	1	yes	maximal, characteristic
4	#15688872	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	Q₈⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	Q₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625337	—	2	1	no	—
9	#13094024	C₂²	1	1	yes	characteristic
10	#13093955	—	2	1	no	—
11	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#13093948	—	2	2	no	—
14	#13093948	—	2	1	no	—
15	#13093947	—	2	1	no	—
16	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₄₇	—	2	2	no	—
18	C₄⋊Q₈×C₄₇	—	2	1	no	—
19	C₄⋊Q₈×C₄₇	—	2	1	no	—
20	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₄₇	—	2	1	no	—
22	C₄×Q₈×C₄₇	—	2	1	no	—
23	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₄₇	—	2	2	no	—
26	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
27	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂×Q₈×C₄₇	D₄	1	2	yes	—
29	C₂×Dic₉₄	—	2	1	no	—
30	C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₄⋊C₄×C₄₇	—	2	1	no	—
33	C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₄⋊C₄×C₄₇	—	2	1	no	—
35	C₄⋊C₄×C₄₇	—	2	1	no	—
36	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₄²×C₄₇	—	2	2	no	—
39	C₄²×C₄₇	—	2	1	no	—
40	C₄²×C₄₇	—	2	1	no	—
41	C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	#10625337	—	4	2	no	—
43	C₄⋊Dic₄₇	—	2	1	no	—
44	C₄⋊Dic₄₇	—	2	1	no	—
45	C₂×C₄₇⋊C₈	—	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	—	2	1	no	—
50	Q₈×C₄₇	—	4	2	no	—
51	Q₈×C₄₇	—	8	2	no	—
52	Q₈×C₄₇	SD₁₆	1	2	yes	—
53	Q₈×C₄₇	Q₁₆	1	2	yes	complemented
54	C₂×C₄×C₄₇	—	2	1	no	—
55	C₂×C₄×C₄₇	—	2	1	no	—
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	2	1	no	—
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×Dic₄₇	—	4	1	no	—
63	Dic₉₄	—	4	1	no	—
64	Dic₉₄	—	4	1	no	—
65	C₄₇⋊C₈	—	4	1	no	—
66	C₄₇⋊C₈	—	4	1	no	—
67	C₄₇⋊C₈	—	4	1	no	—
68	C₂²×C₄₇	#120	1	1	yes	characteristic
69	C₄×C₄₇	—	2	2	no	—
70	C₄×C₄₇	—	2	2	no	—
71	C₄×C₄₇	—	2	1	no	—
72	C₄×C₄₇	—	4	2	no	—
73	C₄×C₄₇	—	4	1	no	—
74	C₄×C₄₇	—	4	1	no	—
75	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
77	C₄×C₄₇	#137	1	1	yes	characteristic
78	C₄×C₄₇	#137	1	1	yes	characteristic
79	Dic₄₇	—	8	1	no	—
80	#1645	—	47	1	no	maximal, Sylow
81	C₂×C₄₇	#448	1	1	yes	characteristic
82	C₂×C₄₇	#451	1	1	yes	characteristic
83	C₂×C₄₇	#455	1	1	yes	characteristic
84	Q₈²	D₄₇	1	1	yes	characteristic
85	#487	—	47	1	no	—
86	#475	—	47	1	no	—
87	#462	—	47	1	no	—
88	#333	—	47	1	no	—
89	Q₈⋊C₈	—	47	1	no	—
90	Q₈⋊C₈	—	47	1	no	—
91	C₄₇	#1645	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	94	1	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	C₄⋊Q₈	—	2	1	no	—
95	C₄⋊Q₈	—	2	1	no	—
96	C₄⋊Q₈	—	47	1	no	—
97	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
98	C₄×Q₈	—	2	1	no	—
99	C₄×Q₈	—	2	1	no	—
100	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
101	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
102	#106	—	94	1	no	—
103	C₄⋊C₈	—	47	1	no	—
104	C₄⋊C₈	—	47	1	no	—
105	#103	—	94	1	no	—
106	#103	—	94	2	no	—
107	#103	—	94	1	no	—
108	C₄×C₈	—	47	1	no	—
109	C₂×Q₈	—	2	2	no	—
110	C₂×Q₈	—	94	1	no	—
111	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
112	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
113	C₂×Q₈	C₂²⋊D₄₇	1	2	yes	—
114	Q₁₆	—	188	2	no	—
115	C₂×C₈	—	94	1	no	—
116	C₂×C₈	—	94	1	no	—
117	C₂×C₈	—	94	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	94	1	no	—
125	C₄⋊C₄	—	94	1	no	—
126	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
127	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
128	C₄²	—	2	2	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	C₂²×D₄₇	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	2	1	no	—
134	Q₈	—	4	2	no	—
135	Q₈	—	8	2	no	—
136	Q₈	—	188	1	no	—
137	Q₈	—	188	1	no	—
138	Q₈	#10625336	1	2	yes	—
139	Q₈	#10625337	1	2	yes	complemented
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	188	1	no	—
146	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
149	C₈	—	188	1	no	—
150	C₈	—	188	1	no	—
151	C₈	—	188	1	no	—
152	C₂²	#13093993	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	2	1	no	—
156	C₄	—	4	2	no	—
157	C₄	—	4	1	no	—
158	C₄	—	4	1	no	—
159	C₄	—	376	1	no	—
160	C₄	C₂×#10625336	1	1	yes	characteristic
161	C₄	C₂×#10625337	1	1	yes	characteristic
162	C₄	#13094055	1	1	yes	characteristic
163	C₄	#13094055	1	1	yes	characteristic
164	C₂	#15689262	1	1	yes	characteristic
165	C₂	#15689279	1	1	yes	characteristic
166	C₂	#15689301	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689473

C₂

yes

maximal, characteristic

#15689408

C₂

yes

maximal, characteristic

#15689333

C₂

yes

maximal, characteristic

#15688872

C₂

yes

maximal, characteristic

Q₈⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

Q₈⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

Q₈²×C₄₇

C₂

yes

maximal, characteristic

C₂×#10625337

—

#13094024

C₂²

yes

characteristic

#13093955

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

—

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×Q₈×C₄₇

D₄

yes

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625337

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

SD₁₆

yes

—

Q₈×C₄₇

Q₁₆

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

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

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

#120

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₄×C₄₇

C₂×Q₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

Q₈²

D₄₇

yes

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₄₇

#1645

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

C₂×D₄₇

yes

characteristic

101

C₄×Q₈

C₂×D₄₇

yes

characteristic

102

#106

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#103

—

106

#103

—

107

#103

—

108

C₄×C₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

112

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

113

C₂×Q₈

C₂²⋊D₄₇

yes

—

114

Q₁₆

—

188

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

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

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×D₄₇

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

188

—

137

Q₈

—

188

—

138

Q₈

#10625336

yes

—

139

Q₈

#10625337

yes

complemented

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

188

—

146

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

147

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

149

C₈

—

188

—

150

C₈

—

188

—

151

C₈

—

188

—

152

C₂²

#13093993

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

376

—

160

C₄

C₂×#10625336

yes

characteristic

161

C₄

C₂×#10625337

yes

characteristic

162

C₄

#13094055

yes

characteristic

163

C₄

#13094055

yes

characteristic

164

C₂

#15689262

yes

characteristic

165

C₂

#15689279

yes

characteristic

166

C₂

#15689301

yes

characteristic

	subgroup	G/H
center	C₂²	#13093993
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	Q₈²×C₄₇	C₂
socle	C₂²×C₄₇	#120
hypercenter	Q₈²	D₄₇
hypocenter	C₄₇	#1645
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#13093993

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

Q₈²×C₄₇

C₂

socle

C₂²×C₄₇

#120

hypercenter

Q₈²

D₄₇

hypocenter

C₄₇

#1645

LCS

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
53. Q₈×C₄₇	Q₁₆	semidirect
91. C₄₇	#1645	semidirect
139. Q₈	#10625337	semidirect

H ≅ G/N

type

53. Q₈×C₄₇

Q₁₆

semidirect

91. C₄₇

#1645

semidirect

139. Q₈

#10625337

semidirect

Subgroups — #20242609

Conjugacy classes of subgroups

Complements