Group #20241857 — subgroups

Subgroups — #20241857

Order 6016.

246 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 81 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	#15689180	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689176	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₄₇⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₄₇×#532	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	#13093935	—	2	1	no	—
10	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
11	C₂×#41×C₄₇	—	2	1	no	—
12	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
13	#127×C₄₇	C₄	1	1	yes	characteristic
14	#124×C₄₇	—	2	1	no	—
15	#121×C₄₇	—	2	1	no	—
16	#121×C₄₇	C₄	1	2	yes	—
17	C₄×D₄×C₄₇	—	2	1	no	—
18	C₄×D₄×C₄₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	#41×C₄₇	—	4	4	no	—
21	C₂×Q₈×C₄₇	—	2	1	no	—
22	C₂×D₄×C₄₇	—	2	2	no	—
23	C₂×D₄×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	—	2	1	no	—
25	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₄₇	—	2	2	no	—
27	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
30	C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂²⋊C₄×C₄₇	—	2	2	no	—
33	C₂²⋊C₄×C₄₇	—	2	2	no	—
34	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₄₇	D₄	1	1	yes	characteristic
36	C₄²×C₄₇	D₄	1	1	yes	characteristic
37	C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄₇⋊C₈	—	2	2	no	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	C₂³×C₄₇	—	2	2	no	—
41	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₄₇	—	4	2	no	—
43	D₄×C₄₇	—	2	4	no	—
44	D₄×C₄₇	—	4	2	no	—
45	D₄×C₄₇	—	4	2	no	—
46	D₄×C₄₇	—	4	4	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	4	1	no	—
51	C₂×C₄×C₄₇	—	4	4	no	—
52	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₄₇⋊C₈	—	4	2	no	—
59	C₄₇⋊C₈	—	4	2	no	—
60	C₂²×C₄₇	—	2	1	no	—
61	C₂²×C₄₇	—	2	1	no	—
62	C₂²×C₄₇	—	2	1	no	—
63	C₂²×C₄₇	—	4	2	no	—
64	C₂²×C₄₇	—	4	2	no	—
65	C₂²×C₄₇	—	4	2	no	—
66	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₄₇	—	2	1	no	—
68	C₄×C₄₇	—	2	1	no	—
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	—	4	2	no	—
72	C₄×C₄₇	#131	1	2	yes	—
73	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
74	#1496	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	—	4	1	no	—
76	C₂×C₄₇	—	8	2	no	—
77	C₂×C₄₇	#408	1	1	yes	characteristic
78	C₂×C₄₇	#418	1	1	yes	characteristic
79	C₂×C₄₇	#421	1	1	yes	characteristic
80	#532	D₄₇	1	1	yes	characteristic
81	#432	—	47	1	no	—
82	#431	—	47	1	no	—
83	D₄⋊C₈	—	47	4	no	—
84	C₄₇	#1496	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	—	2	1	no	—
86	C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
87	#127	Dic₄₇	1	1	yes	characteristic
88	#124	—	2	1	no	—
89	#121	—	2	1	no	—
90	#121	Dic₄₇	1	2	yes	—
91	C₄×D₄	—	2	1	no	—
92	C₄×D₄	C₂×D₄₇	1	2	yes	—
93	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
94	C₄⋊C₈	—	47	2	no	—
95	C₄⋊C₈	—	94	1	no	—
96	C₂²⋊C₈	—	94	1	no	—
97	C₂²⋊C₈	—	94	1	no	—
98	#97	—	94	1	no	—
99	C₄×C₈	—	47	2	no	—
100	#41	—	4	4	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	C₂×Dic₄₇	1	2	yes	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
108	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
110	C₂×C₈	—	94	2	no	—
111	C₂×C₈	—	94	2	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	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	—	2	2	no	—
121	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
122	Q₈	—	4	2	no	—
123	D₄	—	2	4	no	—
124	D₄	—	4	2	no	—
125	D₄	—	4	2	no	—
126	D₄	—	4	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	4	no	—
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	—	2	1	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	2	no	—
146	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	2	no	—
152	C₄	#13094031	1	2	yes	—
153	C₄	#13094050	1	2	yes	—
154	C₂	—	4	1	no	—
155	C₂	—	8	2	no	—
156	C₂	#15689090	1	1	yes	characteristic
157	C₂	#15689128	1	1	yes	characteristic
158	C₂	#15689140	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689180

C₂

yes

maximal, characteristic, complemented

#15689176

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, complemented

C₄₇×#532

C₂

yes

maximal, characteristic

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₂×#41×C₄₇

—

C₄⋊Q₈×C₄₇

C₄

yes

characteristic

#127×C₄₇

C₄

yes

characteristic

#124×C₄₇

—

#121×C₄₇

—

#121×C₄₇

C₄

yes

—

C₄×D₄×C₄₇

—

C₄×D₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#41×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

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

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×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

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

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×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#131

yes

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

#1496

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

characteristic

#432

—

#431

—

D₄⋊C₈

—

C₄₇

#1496

yes

characteristic, complemented, Sylow

C₂×#41

—

C₄⋊Q₈

Dic₄₇

yes

characteristic

#127

Dic₄₇

yes

—

—

Dic₄₇

yes

—

C₄×D₄

—

C₄×D₄

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

100

#41

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

C₂×Dic₄₇

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

108

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

109

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₂×Dic₄₇

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

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

—

121

C₂³

C₂²⋊Dic₄₇

yes

characteristic

122

Q₈

—

123

D₄

—

124

D₄

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

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

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#13094031

yes

—

153

C₄

#13094050

yes

—

154

C₂

—

155

C₂

—

156

C₂

#15689090

yes

characteristic

157

C₂

#15689128

yes

characteristic

158

C₂

#15689140

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#532	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#532	D₄₇
hypocenter	C₄₇	#1496
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#532

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #15689180	C₂	semidirect
2. #15689176	C₂	semidirect
3. D₄⋊(C₄₇⋊C₈)	C₂	semidirect
84. C₄₇	#1496	semidirect

H ≅ G/N

type

1. #15689180

C₂

semidirect

2. #15689176

C₂

semidirect

3. D₄⋊(C₄₇⋊C₈)

C₂

semidirect

84. C₄₇

#1496

semidirect

Subgroups — #20241857

Conjugacy classes of subgroups

Complements