Group #20256257 — subgroups

Subgroups — #20256257

Order 6016.

352 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 125 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	#15689834	C₂	1	2	yes	maximal, complemented
2	#15689524	C₂	1	2	yes	maximal, complemented
3	#15689426	C₂	1	4	yes	maximal, complemented
4	#15689410	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic
7	#15689205	C₂	1	2	yes	maximal
8	C₄₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₃₇₆	C₂²	1	1	yes	characteristic
10	#13094015	C₂²	1	4	yes	—
11	#13094004	—	2	2	no	—
12	#13094000	—	2	4	no	—
13	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
14	C₄×D₁₈₈	C₂²	1	2	yes	—
15	#13093986	C₂²	1	4	yes	—
16	C₄⋊C₄×D₄₇	—	2	2	no	—
17	#13093958	C₂²	1	1	yes	characteristic
18	#13093955	C₂²	1	4	yes	—
19	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₄₇⋊C₈	C₂²	1	2	yes	—
21	#13093946	C₂²	1	2	yes	—
22	#13093944	C₂²	1	4	yes	—
23	C₈×Dic₄₇	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
25	#107×C₄₇	C₂²	1	2	yes	—
26	#107×C₄₇	C₂²	1	2	yes	—
27	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₄₇	—	2	2	no	—
29	C₂×D₁₈₈	C₂³	1	2	yes	—
30	C₂×C₄×D₄₇	—	2	2	no	—
31	C₂×C₄×D₄₇	—	2	4	no	—
32	C₂×Dic₉₄	D₄	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
37	C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	(C₂×D₄₇)⋊C₄	—	2	2	no	—
39	(C₂×D₄₇)⋊C₄	—	2	4	no	—
40	C₄⋊Dic₄₇	—	2	4	no	—
41	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
42	C₄⋊Dic₄₇	C₂³	1	2	yes	—
43	Dic₄₇⋊C₄	—	2	4	no	—
44	Dic₄₇⋊C₄	D₄	1	2	yes	—
45	C₄×Dic₄₇	C₂³	1	2	yes	—
46	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
47	D₃₇₆	Q₈	1	4	yes	complemented
48	C₂²×D₄₇	—	2	2	no	—
49	Q₈×C₄₇	—	8	2	no	—
50	C₂×C₄×C₄₇	—	2	4	no	—
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₄₇	—	2	2	no	—
55	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
56	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
57	D₁₈₈	—	2	2	no	—
58	D₁₈₈	C₂×Q₈	1	4	yes	—
59	C₄×D₄₇	—	4	4	no	—
60	C₄×D₄₇	—	8	4	no	—
61	Dic₉₄	—	2	2	no	—
62	C₈×C₄₇	—	2	1	no	—
63	C₈×C₄₇	C₂×Q₈	1	2	yes	—
64	C₄₇⋊C₈	—	2	2	no	—
65	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₄₇	—	2	4	no	—
67	C₂×D₄₇	—	4	2	no	—
68	C₂×D₄₇	—	4	2	no	—
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	4	4	no	—
71	C₄×C₄₇	#135	1	2	yes	—
72	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
73	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
74	Dic₄₇	—	2	2	no	—
75	Dic₄₇	—	2	2	no	—
76	Dic₄₇	—	4	2	no	—
77	#3380	—	47	1	no	maximal, Sylow
78	C₂×C₄₇	D₄×Q₈	1	1	yes	characteristic
79	C₂×C₄₇	C₂×#135	1	1	yes	characteristic
80	C₂×C₄₇	#576	1	1	yes	characteristic
81	D₄₇	—	4	4	no	—
82	#554	—	47	2	no	—
83	#500	D₄₇	1	1	yes	characteristic, complemented
84	#499	—	47	2	no	—
85	#478	—	47	4	no	—
86	#476	—	47	2	no	—
87	C₈×Q₈	—	47	1	no	—
88	C₄×D₈	—	47	2	no	—
89	C₄×D₈	—	47	1	no	—
90	C₄₇	#3380	1	1	yes	characteristic, complemented, Sylow
91	C₂×D₈	—	47	1	no	—
92	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
93	#125	—	47	4	no	—
94	C₂²⋊Q₈	—	94	2	no	—
95	C₂²⋊Q₈	—	94	4	no	—
96	C₄×Q₈	—	47	1	no	—
97	C₄×D₄	—	47	4	no	—
98	C₄×D₄	—	47	2	no	—
99	C₂×C₄⋊C₄	—	94	2	no	—
100	#107	—	47	1	no	—
101	#107	C₂×D₄₇	1	2	yes	—
102	#107	C₂×D₄₇	1	2	yes	—
103	#106	—	47	4	no	—
104	C₄⋊C₈	—	47	2	no	—
105	C₄⋊C₈	—	47	1	no	—
106	#102	—	47	2	no	—
107	#102	—	47	4	no	—
108	C₄×C₈	—	47	2	no	—
109	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	47	1	no	—
112	C₂×D₄	—	47	2	no	—
113	C₂²×C₄	—	94	4	no	—
114	C₂²×C₄	—	94	2	no	—
115	D₈	—	47	4	no	—
116	C₂×C₈	—	47	2	no	—
117	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	47	2	no	—
121	C₄⋊C₄	—	47	1	no	—
122	C₄⋊C₄	—	47	2	no	—
123	C₄⋊C₄	—	94	4	no	—
124	C₄⋊C₄	—	94	4	no	—
125	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
126	C₂²⋊C₄	—	94	4	no	—
127	C₂²⋊C₄	—	94	2	no	—
128	C₄²	—	47	2	no	—
129	C₄²	C₂²×D₄₇	1	1	yes	characteristic
130	C₂³	—	94	2	no	—
131	Q₈	—	8	2	no	—
132	Q₈	—	94	2	no	—
133	D₄	—	47	4	no	—
134	D₄	—	94	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	47	2	no	—
137	C₂×C₄	—	47	2	no	—
138	C₂×C₄	—	94	2	no	—
139	C₂×C₄	—	376	4	no	—
140	C₂×C₄	—	188	4	no	—
141	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
142	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	94	2	no	—
146	C₈	Q₈×D₄₇	1	2	yes	—
147	C₂²	—	94	4	no	—
148	C₂²	—	188	2	no	—
149	C₂²	—	188	2	no	—
150	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	4	4	no	—
153	C₄	—	94	2	no	—
154	C₄	—	94	2	no	—
155	C₄	—	188	2	no	—
156	C₄	#13094047	1	2	yes	—
157	C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
158	C₄	#13094072	1	1	yes	characteristic
159	C₂	—	188	4	no	—
160	C₂	#15689788	1	1	yes	characteristic
161	C₂	C₂×#13094047	1	1	yes	characteristic
162	C₂	#15689924	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689834

C₂

yes

maximal, complemented

#15689524

C₂

yes

maximal, complemented

#15689426

C₂

yes

maximal, complemented

#15689410

C₂

yes

maximal, complemented

C₈×Dic₉₄

C₂

yes

maximal, characteristic, complemented

C₄×D₃₇₆

C₂

yes

maximal, characteristic

#15689205

C₂

yes

maximal

C₄₇×#500

C₂

yes

maximal, characteristic, complemented

C₂×D₃₇₆

C₂²

yes

characteristic

#13094015

C₂²

yes

—

#13094004

—

#13094000

—

C₄×Dic₉₄

C₂²

yes

characteristic

C₄×D₁₈₈

C₂²

yes

—

#13093986

C₂²

yes

—

C₄⋊C₄×D₄₇

—

#13093958

C₂²

yes

characteristic

#13093955

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

Dic₄₇⋊C₈

C₂²

yes

—

#13093946

C₂²

yes

—

#13093944

C₂²

yes

—

C₈×Dic₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

#107×C₄₇

C₂²

yes

—

#107×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

D₄

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

characteristic

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

D₄

yes

—

C₄×Dic₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

D₃₇₆

Q₈

yes

complemented

C₂²×D₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×D₄

yes

—

D₁₈₈

—

D₁₈₈

C₂×Q₈

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂×Q₈

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#135

yes

—

C₄×C₄₇

C₂²×Q₈

yes

characteristic

C₄×C₄₇

2₋¹⁺⁴

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#3380

—

maximal, Sylow

C₂×C₄₇

D₄×Q₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#554

—

#500

D₄₇

yes

characteristic, complemented

#499

—

#478

—

#476

—

C₈×Q₈

—

C₄×D₈

—

C₄×D₈

—

C₄₇

#3380

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₄⋊Q₈

C₂×D₄₇

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

100

#107

—

101

#107

C₂×D₄₇

yes

—

102

#107

C₂×D₄₇

yes

—

103

#106

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

#102

—

107

#102

—

108

C₄×C₈

—

109

C₄×C₈

C₂×D₄₇

yes

characteristic

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×D₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

D₈

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₄₇

yes

characteristic

118

C₂×C₈

C₂²×D₄₇

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

C₂²×D₄₇

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

C₂²×D₄₇

yes

characteristic

130

C₂³

—

131

Q₈

—

132

Q₈

—

133

D₄

—

134

D₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

376

—

140

C₂×C₄

—

188

—

141

C₂×C₄

D₄×D₄₇

yes

characteristic

142

C₂×C₄

D₄×D₄₇

yes

characteristic

143

C₂×C₄

C₂³×D₄₇

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

Q₈×D₄₇

yes

—

147

C₂²

—

148

C₂²

—

188

—

149

C₂²

—

188

—

150

C₂²

C₂×D₄×D₄₇

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

188

—

156

C₄

#13094047

yes

—

157

C₄

C₂×Q₈×D₄₇

yes

characteristic

158

C₄

#13094072

yes

characteristic

159

C₂

—

188

—

160

C₂

#15689788

yes

characteristic

161

C₂

C₂×#13094047

yes

characteristic

162

C₂

#15689924

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#500	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#500	D₄₇
hypocenter	C₄₇	#3380
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#500

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #15689834	C₂	semidirect
2. #15689524	C₂	semidirect
3. #15689426	C₂	semidirect
4. #15689410	C₂	semidirect
5. C₈×Dic₉₄	C₂	semidirect
8. C₄₇×#500	C₂	semidirect
47. D₃₇₆	Q₈	semidirect
83. #500	D₄₇	semidirect
90. C₄₇	#3380	semidirect

H ≅ G/N

type

1. #15689834

C₂

semidirect

2. #15689524

C₂

semidirect

3. #15689426

C₂

semidirect

4. #15689410

C₂

semidirect

5. C₈×Dic₉₄

C₂

semidirect

8. C₄₇×#500

C₂

semidirect

47. D₃₇₆

Q₈

semidirect

83. #500

D₄₇

semidirect

90. C₄₇

#3380

semidirect

Subgroups — #20256257

Conjugacy classes of subgroups

Complements