Group #20241783 — subgroups

Subgroups — #20241783

Order 6016.

248 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 113 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	#15689176	C₂	1	1	yes	maximal, characteristic
2	#15689176	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₄₇⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
9	C₄×C₄₇⋊C₈	—	2	1	no	—
10	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
11	C₂²×Q₈×C₄₇	C₄	1	1	yes	characteristic
12	C₄×Q₈×C₄₇	—	2	1	no	—
13	C₄×Q₈×C₄₇	—	2	1	no	—
14	C₄×Q₈×C₄₇	C₄	1	2	yes	—
15	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
17	C₂×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
18	C₂×C₄²×C₄₇	—	2	1	no	—
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₄₇	—	2	1	no	—
21	C₂×Q₈×C₄₇	—	2	2	no	—
22	C₂×Q₈×C₄₇	C₈	1	4	yes	complemented
23	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₄₇	—	2	1	no	—
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₄²×C₄₇	—	2	2	no	—
34	C₄²×C₄₇	—	2	2	no	—
35	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₄₇	D₄	1	1	yes	characteristic
37	C₄²×C₄₇	D₄	1	1	yes	characteristic
38	C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	C₂×C₄₇⋊C₈	—	2	2	no	—
41	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₄₇	—	2	4	no	—
43	Q₈×C₄₇	—	2	2	no	—
44	Q₈×C₄₇	C₂×C₈	1	4	yes	—
45	C₂×C₄×C₄₇	—	2	4	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	4	1	no	—
50	C₂×C₄×C₄₇	—	4	2	no	—
51	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×C₈	1	2	yes	—
55	C₂×C₄×C₄₇	M₁₆	1	2	yes	—
56	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₄₇⋊C₈	—	4	2	no	—
60	C₄₇⋊C₈	—	4	2	no	—
61	C₂²×C₄₇	—	2	1	no	—
62	C₂²×C₄₇	—	2	1	no	—
63	C₂²×C₄₇	C₂²⋊C₈	1	2	yes	—
64	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₄₇	—	2	4	no	—
66	C₄×C₄₇	—	2	1	no	—
67	C₄×C₄₇	—	2	1	no	—
68	C₄×C₄₇	—	2	1	no	—
69	C₄×C₄₇	—	4	2	no	—
70	C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
71	C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
72	C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
73	C₄×C₄₇	#137	1	2	yes	—
74	#1479	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	—	2	2	no	—
76	C₂×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₄₇	#419	1	1	yes	characteristic
78	C₂×C₄₇	#421	1	1	yes	characteristic
79	C₂×C₄×Q₈	D₄₇	1	1	yes	characteristic
80	#431	—	47	1	no	—
81	#431	—	47	1	no	—
82	Q₈⋊C₈	—	47	4	no	—
83	C₄₇	#1479	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₈	Dic₄₇	1	1	yes	characteristic
85	C₄×Q₈	—	2	1	no	—
86	C₄×Q₈	—	2	1	no	—
87	C₄×Q₈	Dic₄₇	1	2	yes	—
88	C₄×Q₈	C₂×D₄₇	1	2	yes	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
91	C₂×C₄²	—	2	1	no	—
92	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
93	C₄⋊C₈	—	47	2	no	—
94	C₄⋊C₈	—	94	1	no	—
95	C₂²⋊C₈	—	94	1	no	—
96	C₂²⋊C₈	—	94	1	no	—
97	C₄×C₈	—	47	2	no	—
98	C₄×C₈	—	94	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	C₄₇⋊C₈	1	4	yes	complemented
102	C₂×Q₈	C₂×Dic₄₇	1	2	yes	—
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
106	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
107	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
108	C₂×C₈	—	94	2	no	—
109	C₂×C₈	—	94	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
114	C₄²	—	2	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
117	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
118	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
119	C₄²	C₂²×D₄₇	1	1	yes	characteristic
120	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	2	no	—
123	Q₈	C₂×C₄₇⋊C₈	1	4	yes	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
131	C₂×C₄	#10625329	1	2	yes	—
132	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
133	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
135	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
138	C₈	—	188	2	no	—
139	C₈	—	188	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
143	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
144	C₄	—	2	4	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	2	no	—
149	C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
150	C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
151	C₄	C₂×#10625329	1	1	yes	characteristic
152	C₄	#13094055	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
155	C₂	#15689132	1	1	yes	characteristic
156	C₂	#15689140	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689176

C₂

yes

maximal, characteristic

#15689176

C₂

yes

maximal, characteristic

Q₈⋊(C₄₇⋊C₈)

C₂

yes

maximal, complemented

C₂×C₄×Q₈×C₄₇

C₂

yes

maximal, characteristic

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₂²×Q₈×C₄₇

C₄

yes

characteristic

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

C₄

yes

—

C₄×Q₈×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₄

yes

characteristic

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₈

yes

complemented

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

C₂×C₈

yes

—

C₂×C₄×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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

M₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²⋊C₈

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊C₈

yes

—

C₄×C₄₇

C₂²×C₈

yes

characteristic

C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₄×C₄₇

#137

yes

—

#1479

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×C₂²⋊C₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄×Q₈

D₄₇

yes

characteristic

#431

—

#431

—

Q₈⋊C₈

—

C₄₇

#1479

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₄₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

Dic₄₇

yes

—

C₄×Q₈

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₄₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₄₇⋊C₈

yes

complemented

102

C₂×Q₈

C₂×Dic₄₇

yes

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

106

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

107

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₂×Dic₄₇

yes

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₂×Dic₄₇

yes

characteristic

117

C₄²

C₂²⋊D₄₇

yes

characteristic

118

C₄²

C₂²⋊D₄₇

yes

characteristic

119

C₄²

C₂²×D₄₇

yes

characteristic

120

C₂³

C₂²⋊Dic₄₇

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

C₂×C₄₇⋊C₈

yes

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×C₄₇⋊C₈

yes

—

131

C₂×C₄

#10625329

yes

—

132

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

133

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

134

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

135

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

138

C₈

—

188

—

139

C₈

—

188

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂²⋊(C₄₇⋊C₈)

yes

—

143

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

150

C₄

C₂²×C₄₇⋊C₈

yes

characteristic

151

C₄

C₂×#10625329

yes

characteristic

152

C₄

#13094055

yes

—

153

C₂

—

154

C₂

C₂×C₂²⋊(C₄₇⋊C₈)

yes

characteristic

155

C₂

#15689132

yes

characteristic

156

C₂

#15689140

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂²×C₈
Frattini	C₄²	C₂²×D₄₇
Fitting	C₂×C₄×Q₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	C₂×C₄×Q₈	D₄₇
hypocenter	C₄₇	#1479
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₄₇

commutator

C₄×C₄₇

C₂²×C₈

Frattini

C₄²

C₂²×D₄₇

Fitting

C₂×C₄×Q₈×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

hypercenter

C₂×C₄×Q₈

D₄₇

hypocenter

C₄₇

#1479

LCS

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
3. Q₈⋊(C₄₇⋊C₈)	C₂	semidirect
22. C₂×Q₈×C₄₇	C₈	semidirect
83. C₄₇	#1479	semidirect
101. C₂×Q₈	C₄₇⋊C₈	semidirect

H ≅ G/N

type

3. Q₈⋊(C₄₇⋊C₈)

C₂

semidirect

22. C₂×Q₈×C₄₇

C₈

semidirect

83. C₄₇

#1479

semidirect

101. C₂×Q₈

C₄₇⋊C₈

semidirect

Subgroups — #20241783

Conjugacy classes of subgroups

Complements