Group #20242101 — subgroups

Subgroups — #20242101

Order 6016.

332 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₂×#13094021	C₂	1	1	yes	maximal, characteristic
2	#15689180	C₂	1	1	yes	maximal, characteristic, complemented
3	#15688866	C₂	1	4	yes	maximal, complemented
4	C₄₇×#432	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₈₈	—	2	1	no	—
6	C₂²×D₁₈₈	—	2	1	no	—
7	C₂²×D₁₈₈	C₄	1	2	yes	—
8	#13094021	—	2	1	no	—
9	#13094021	—	2	1	no	—
10	#13094021	C₄	1	2	yes	—
11	#13094021	C₂²	1	2	yes	—
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	complemented
13	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
14	#13093935	—	2	1	no	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₄₇	C₂²	1	2	yes	complemented
17	C₂²⋊C₈×C₄₇	—	2	1	no	—
18	#97×C₄₇	—	2	1	no	—
19	C₂³×D₄₇	—	2	2	no	—
20	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂×D₁₈₈	—	2	2	no	—
24	C₂×D₁₈₈	—	2	2	no	—
25	C₂×D₁₈₈	—	2	2	no	—
26	C₂×D₁₈₈	—	2	2	no	—
27	C₂×D₁₈₈	—	2	2	no	—
28	C₂×D₁₈₈	—	4	4	no	—
29	C₂×D₁₈₈	—	4	4	no	—
30	C₂×D₁₈₈	—	4	4	no	—
31	C₂×D₁₈₈	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₄₇	D₄	1	1	yes	characteristic
35	C₄²×C₄₇	D₄	1	1	yes	characteristic
36	C₄²×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×C₄₇⋊C₈	—	2	2	no	—
38	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×D₄₇	—	2	4	no	—
40	C₂²×D₄₇	—	4	4	no	—
41	C₂²×D₄₇	—	4	4	no	—
42	C₂²×D₄₇	—	4	4	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	D₁₈₈	—	4	4	no	—
53	D₁₈₈	—	4	4	no	—
54	D₁₈₈	—	4	4	no	—
55	D₁₈₈	—	4	4	no	—
56	D₁₈₈	—	4	4	no	—
57	D₁₈₈	—	4	4	no	—
58	C₈×C₄₇	—	4	2	no	—
59	C₄₇⋊C₈	—	4	2	no	—
60	C₂²×C₄₇	—	2	1	no	—
61	C₂²×C₄₇	—	2	1	no	—
62	C₂²×C₄₇	#100	1	2	yes	—
63	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×D₄₇	—	4	4	no	—
65	C₂×D₄₇	—	4	4	no	—
66	C₂×D₄₇	—	4	4	no	—
67	C₂×D₄₇	—	8	8	no	—
68	C₄×C₄₇	—	2	1	no	—
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	—	2	1	no	—
72	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
73	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
74	#1555	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	—	2	2	no	—
76	C₂×C₄₇	C₂×#100	1	1	yes	characteristic
77	C₂×C₄₇	#418	1	1	yes	characteristic
78	C₂×C₄₇	#418	1	1	yes	characteristic
79	D₄₇	—	8	4	no	—
80	C₂×#127	—	47	1	no	—
81	#432	—	47	1	no	—
82	#432	D₄₇	1	1	yes	characteristic, complemented
83	#331	—	47	4	no	—
84	C₄₇	#1555	1	1	yes	characteristic, complemented, Sylow
85	C₂²×D₄	—	47	2	no	—
86	C₂²×D₄	—	94	1	no	—
87	C₂²×D₄	—	94	1	no	—
88	#127	—	47	2	no	—
89	#127	—	47	2	no	—
90	#127	—	94	1	no	—
91	#127	—	94	1	no	—
92	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
93	C₄⋊C₈	—	47	2	no	—
94	C₄⋊C₈	C₂×D₄₇	1	2	yes	complemented
95	C₂²⋊C₈	—	2	1	no	—
96	C₂²⋊C₈	—	94	1	no	—
97	#97	—	2	1	no	—
98	#97	—	94	1	no	—
99	C₂⁴	—	94	2	no	—
100	C₂×D₄	—	47	4	no	—
101	C₂×D₄	—	94	2	no	—
102	C₂×D₄	—	94	2	no	—
103	C₂×D₄	—	94	2	no	—
104	C₂×D₄	—	94	2	no	—
105	C₂×D₄	—	94	2	no	—
106	C₂×D₄	—	188	4	no	—
107	C₂×D₄	—	188	4	no	—
108	C₂×D₄	—	188	4	no	—
109	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
110	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
111	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	94	2	no	—
114	C₄²	C₄×D₄₇	1	1	yes	characteristic
115	C₄²	D₁₈₈	1	1	yes	characteristic
116	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
117	C₄²	C₂²×D₄₇	1	1	yes	characteristic
118	C₂³	—	94	4	no	—
119	C₂³	—	188	4	no	—
120	C₂³	—	188	4	no	—
121	C₂³	—	188	4	no	—
122	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
123	D₄	—	188	4	no	—
124	D₄	—	188	4	no	—
125	D₄	—	188	4	no	—
126	D₄	—	188	4	no	—
127	D₄	—	188	4	no	—
128	D₄	—	188	4	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
133	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
136	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	188	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	376	8	no	—
143	C₂²	—	188	4	no	—
144	C₂²	—	188	4	no	—
145	C₂²	—	188	4	no	—
146	C₂²	#13093940	1	2	yes	—
147	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	#13094049	1	2	yes	—
153	C₄	#13094052	1	2	yes	—
154	C₂	—	2	2	no	—
155	C₂	—	376	4	no	—
156	C₂	C₂×#13093940	1	1	yes	characteristic
157	C₂	#15689127	1	1	yes	characteristic
158	C₂	#15689129	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094021

C₂

yes

maximal, characteristic

#15689180

C₂

yes

maximal, characteristic, complemented

#15688866

C₂

yes

maximal, complemented

C₄₇×#432

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

—

C₂²×D₁₈₈

—

C₂²×D₁₈₈

C₄

yes

—

#13094021

—

#13094021

—

#13094021

C₄

yes

—

#13094021

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

complemented

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

complemented

C₂²⋊C₈×C₄₇

—

#97×C₄₇

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂×C₄

yes

—

C₂×C₈×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

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

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

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#100

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

#1555

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

C₂×#127

—

#432

—

#432

D₄₇

yes

characteristic, complemented

#331

—

C₄₇

#1555

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

—

—

—

—

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

complemented

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₂⁴

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

188

—

107

C₂×D₄

—

188

—

108

C₂×D₄

—

188

—

109

C₂²×C₄

C₄×D₄₇

yes

characteristic

110

C₂²×C₄

D₁₈₈

yes

characteristic

111

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄²

C₄×D₄₇

yes

characteristic

115

C₄²

D₁₈₈

yes

characteristic

116

C₄²

C₂²⋊D₄₇

yes

characteristic

117

C₄²

C₂²×D₄₇

yes

characteristic

118

C₂³

—

119

C₂³

—

188

—

120

C₂³

—

188

—

121

C₂³

—

188

—

122

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

123

D₄

—

188

—

124

D₄

—

188

—

125

D₄

—

188

—

126

D₄

—

188

—

127

D₄

—

188

—

128

D₄

—

188

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

133

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

134

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

135

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

136

C₂×C₄

C₂×D₁₈₈

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

138

C₈

—

139

C₈

—

188

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

376

—

143

C₂²

—

188

—

144

C₂²

—

188

—

145

C₂²

—

188

—

146

C₂²

#13093940

yes

—

147

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#13094049

yes

—

153

C₄

#13094052

yes

—

154

C₂

—

155

C₂

—

376

—

156

C₂

C₂×#13093940

yes

characteristic

157

C₂

#15689127

yes

characteristic

158

C₂

#15689129

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#432	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#432	D₄₇
hypocenter	C₄₇	#1555
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×(C₂×D₄₇)⋊C₄

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#432

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
2. #15689180	C₂	semidirect
3. #15688866	C₂	semidirect
4. C₄₇×#432	C₂	semidirect
12. C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
16. C₄⋊C₈×C₄₇	C₂²	semidirect
82. #432	D₄₇	semidirect
84. C₄₇	#1555	semidirect
94. C₄⋊C₈	C₂×D₄₇	semidirect

H ≅ G/N

type

2. #15689180

C₂

semidirect

3. #15688866

C₂

semidirect

4. C₄₇×#432

C₂

semidirect

12. C₄⋊(C₄₇⋊C₈)

C₂²

semidirect

16. C₄⋊C₈×C₄₇

C₂²

semidirect

82. #432

D₄₇

semidirect

84. C₄₇

#1555

semidirect

94. C₄⋊C₈

C₂×D₄₇

semidirect

Subgroups — #20242101

Conjugacy classes of subgroups

Complements