Group #20256019 — subgroups

Subgroups — #20256019

Order 6016.

426 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 139 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×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689469	C₂	1	3	yes	maximal, complemented
3	#15689403	C₂	1	3	yes	maximal, complemented
4	#15689315	C₂	1	3	yes	maximal, complemented
5	Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#10625334	C₂	1	3	yes	maximal, complemented
7	C₄₇×#550	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₄₇	—	2	1	no	—
9	C₂×#10625334	C₂²	1	3	yes	—
10	#13094024	C₂²	1	3	yes	—
11	#13094004	—	2	3	no	—
12	C₄×D₁₈₈	C₂²	1	3	yes	—
13	#13093957	C₂²	1	3	yes	—
14	C₄⋊(C₄₇⋊C₈)	C₂²	1	3	yes	—
15	#13093945	C₂²	1	6	yes	—
16	#13093945	C₂²	1	3	yes	—
17	#13093944	C₂²	1	1	yes	characteristic
18	C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
19	C₂×#41×C₄₇	—	2	1	no	—
20	#127×C₄₇	C₂²	1	3	yes	—
21	#121×C₄₇	—	2	3	no	—
22	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₄₇	C₂²	1	3	yes	—
24	#41×C₄₇	—	4	4	no	—
25	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
26	C₂×D₄×C₄₇	—	2	3	no	—
27	C₂×D₄×C₄₇	—	2	3	no	—
28	C₂×D₄×C₄₇	C₂³	1	3	yes	—
29	C₂²×C₄×C₄₇	—	2	3	no	—
30	Q₈×D₄₇	—	4	4	no	—
31	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
32	C₂×C₄×D₄₇	—	2	3	no	—
33	C₂×Dic₉₄	—	2	3	no	—
34	C₄⋊C₄×C₄₇	D₄	1	3	yes	—
35	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²⋊C₄×C₄₇	—	2	3	no	—
37	C₄²×C₄₇	C₂³	1	3	yes	—
38	#10625334	—	2	6	no	—
39	(C₂×D₄₇)⋊C₄	—	2	3	no	—
40	C₄⋊Dic₄₇	—	2	3	no	—
41	C₄⋊Dic₄₇	C₂³	1	3	yes	—
42	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
43	C₂×C₄₇⋊C₈	C₂³	1	3	yes	—
44	C₂³×C₄₇	—	2	3	no	—
45	C₂²×D₄₇	—	2	1	no	—
46	Q₈×C₄₇	D₈	1	4	yes	complemented
47	D₄×C₄₇	—	2	3	no	—
48	D₄×C₄₇	—	2	3	no	—
49	D₄×C₄₇	—	4	6	no	—
50	D₄×C₄₇	—	8	3	no	—
51	C₂×C₄×C₄₇	—	2	1	no	—
52	C₂×C₄×C₄₇	—	4	6	no	—
53	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
55	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₄₇	—	2	3	no	—
57	D₁₈₈	—	2	1	no	—
58	D₁₈₈	#41	1	2	yes	complemented
59	C₄×D₄₇	—	4	6	no	—
60	Dic₉₄	—	4	6	no	—
61	C₄₇⋊C₈	—	2	3	no	—
62	C₄₇⋊C₈	—	2	1	no	—
63	C₂²×C₄₇	—	4	3	no	—
64	C₂²×C₄₇	—	4	3	no	—
65	C₂²×C₄₇	—	4	3	no	—
66	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₄₇	—	2	2	no	—
68	C₂×D₄₇	—	4	1	no	—
69	C₂×D₄₇	—	4	1	no	—
70	C₄×C₄₇	—	2	3	no	—
71	C₄×C₄₇	—	4	1	no	—
72	C₄×C₄₇	C₂×D₈	1	6	yes	—
73	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
74	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
75	Dic₄₇	—	4	3	no	—
76	#3358	—	47	1	no	maximal, Sylow
77	C₂×C₄₇	—	8	3	no	—
78	C₂×C₄₇	#548	1	1	yes	characteristic
79	C₂×C₄₇	C₂²×D₈	1	1	yes	characteristic
80	C₂×C₄₇	#577	1	1	yes	characteristic
81	D₄₇	—	4	2	no	—
82	#550	D₄₇	1	1	yes	characteristic, complemented
83	D₄×Q₈	—	47	1	no	—
84	#486	—	47	3	no	—
85	#474	—	47	3	no	—
86	#459	—	47	3	no	—
87	C₈×Q₈	—	47	1	no	—
88	C₄×D₈	—	47	3	no	—
89	C₄₇	#3358	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	2	1	no	—
91	C₂²×Q₈	—	94	1	no	—
92	C₂×D₈	—	47	3	no	—
93	C₄⋊Q₈	—	47	3	no	—
94	#127	C₂×D₄₇	1	3	yes	—
95	C₂²⋊Q₈	—	94	3	no	—
96	#121	—	2	3	no	—
97	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
98	C₄×D₄	—	47	3	no	—
99	C₄×D₄	C₂×D₄₇	1	3	yes	—
100	#107	—	47	3	no	—
101	C₄⋊C₈	—	47	3	no	—
102	#102	—	47	3	no	—
103	#102	—	47	6	no	—
104	#102	—	47	1	no	—
105	C₄×C₈	—	47	3	no	—
106	#41	—	4	4	no	—
107	C₂×Q₈	—	94	3	no	—
108	C₂×Q₈	—	188	4	no	—
109	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
110	C₂×D₄	—	2	3	no	—
111	C₂×D₄	—	2	3	no	—
112	C₂×D₄	—	47	1	no	—
113	C₂×D₄	C₂²×D₄₇	1	3	yes	—
114	C₂²×C₄	—	2	3	no	—
115	C₂²×C₄	—	94	3	no	—
116	D₈	—	94	6	no	—
117	C₂×C₈	—	47	1	no	—
118	C₂×C₈	—	47	3	no	—
119	C₄⋊C₄	—	47	3	no	—
120	C₄⋊C₄	—	94	3	no	—
121	C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
122	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
123	C₂²⋊C₄	—	2	3	no	—
124	C₂²⋊C₄	—	94	3	no	—
125	C₄²	C₂²×D₄₇	1	3	yes	—
126	C₂³	—	2	3	no	—
127	C₂³	—	94	1	no	—
128	Q₈	—	188	6	no	—
129	Q₈	#10625334	1	4	yes	complemented
130	D₄	—	2	3	no	—
131	D₄	—	2	3	no	—
132	D₄	—	4	6	no	—
133	D₄	—	8	3	no	—
134	D₄	—	47	2	no	—
135	D₄	—	94	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	6	no	—
138	C₂×C₄	—	94	3	no	—
139	C₂×C₄	—	188	6	no	—
140	C₂×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
141	C₂×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
142	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
143	C₈	—	94	3	no	—
144	C₈	—	94	1	no	—
145	C₂²	—	4	3	no	—
146	C₂²	—	4	3	no	—
147	C₂²	—	4	3	no	—
148	C₂²	—	94	2	no	—
149	C₂²	—	188	1	no	—
150	C₂²	—	188	1	no	—
151	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
152	C₄	—	2	3	no	—
153	C₄	—	4	1	no	—
154	C₄	—	188	3	no	—
155	C₄	C₂×#10625334	1	6	yes	—
156	C₄	#41×D₄₇	1	1	yes	characteristic
157	C₄	#13094071	1	1	yes	characteristic
158	C₂	—	8	3	no	—
159	C₂	—	188	2	no	—
160	C₂	#15689780	1	1	yes	characteristic
161	C₂	C₂²×#10625334	1	1	yes	characteristic
162	C₂	#15689934	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₁₈₈

C₂

yes

maximal, characteristic, complemented

#15689469

C₂

yes

maximal, complemented

#15689403

C₂

yes

maximal, complemented

#15689315

C₂

yes

maximal, complemented

Q₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#10625334

C₂

yes

maximal, complemented

C₄₇×#550

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₄₇

—

C₂×#10625334

C₂²

yes

—

#13094024

C₂²

yes

—

#13094004

—

C₄×D₁₈₈

C₂²

yes

—

#13093957

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093945

C₂²

yes

—

#13093945

C₂²

yes

—

#13093944

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

—

C₂×#41×C₄₇

—

#127×C₄₇

C₂²

yes

—

#121×C₄₇

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄×D₄×C₄₇

C₂²

yes

—

#41×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

—

Q₈×D₄₇

—

C₂×D₁₈₈

C₂³

yes

characteristic

C₂×C₄×D₄₇

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

C₂³

yes

characteristic

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

—

#10625334

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

D₈

yes

complemented

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

#41

yes

complemented

C₄×D₄₇

—

Dic₉₄

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

—

C₄×C₄₇

C₂×#41

yes

characteristic

C₄×C₄₇

2₋¹⁺⁴

yes

characteristic

Dic₄₇

—

#3358

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#548

yes

characteristic

C₂×C₄₇

C₂²×D₈

yes

characteristic

C₂×C₄₇

#577

yes

characteristic

D₄₇

—

#550

D₄₇

yes

characteristic, complemented

D₄×Q₈

—

#486

—

#474

—

#459

—

C₈×Q₈

—

C₄×D₈

—

C₄₇

#3358

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

C₂×D₄₇

yes

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

C₄×D₄

—

C₄×D₄

C₂×D₄₇

yes

—

100

#107

—

101

C₄⋊C₈

—

102

#102

—

103

#102

—

104

#102

—

105

C₄×C₈

—

106

#41

—

107

C₂×Q₈

—

108

C₂×Q₈

—

188

—

109

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

C₂²×D₄₇

yes

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

D₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

C₂²⋊D₄₇

yes

—

122

C₄⋊C₄

C₂²×D₄₇

yes

characteristic

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

C₂²×D₄₇

yes

—

126

C₂³

—

127

C₂³

—

128

Q₈

—

188

—

129

Q₈

#10625334

yes

complemented

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

188

—

140

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

141

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

142

C₂×C₄

C₂³×D₄₇

yes

characteristic

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

188

—

150

C₂²

—

188

—

151

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

188

—

155

C₄

C₂×#10625334

yes

—

156

C₄

#41×D₄₇

yes

characteristic

157

C₄

#13094071

yes

characteristic

158

C₂

—

159

C₂

—

188

—

160

C₂

#15689780

yes

characteristic

161

C₂

C₂²×#10625334

yes

characteristic

162

C₂

#15689934

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#550	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#550	D₄₇
hypocenter	C₄₇	#3358
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#550

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×D₁₈₈	C₂	semidirect
2. #15689469	C₂	semidirect
3. #15689403	C₂	semidirect
4. #15689315	C₂	semidirect
5. Q₈×C₄₇⋊C₈	C₂	semidirect
6. C₄×#10625334	C₂	semidirect
7. C₄₇×#550	C₂	semidirect
46. Q₈×C₄₇	D₈	semidirect
58. D₁₈₈	#41	semidirect
82. #550	D₄₇	semidirect
89. C₄₇	#3358	semidirect
129. Q₈	#10625334	semidirect

H ≅ G/N

type

1. Q₈×D₁₈₈

C₂

semidirect

2. #15689469

C₂

semidirect

3. #15689403

C₂

semidirect

4. #15689315

C₂

semidirect

5. Q₈×C₄₇⋊C₈

C₂

semidirect

6. C₄×#10625334

C₂

semidirect

7. C₄₇×#550

C₂

semidirect

46. Q₈×C₄₇

D₈

semidirect

58. D₁₈₈

#41

semidirect

82. #550

D₄₇

semidirect

89. C₄₇

#3358

semidirect

129. Q₈

#10625334

semidirect

Subgroups — #20256019

Conjugacy classes of subgroups

Complements