Group #20242137 — subgroups

Subgroups — #20242137

Order 6016.

180 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 83 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	#15689175	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#10625324	C₂	1	1	yes	maximal, characteristic
4	#15688876	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688875	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688873	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625324	C₄	1	1	yes	characteristic
9	C₂×#10625324	C₄	1	1	yes	characteristic
10	C₂×C₈×D₄₇	—	2	1	no	—
11	C₄²×D₄₇	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₄₇⋊C₈	—	2	1	no	—
15	(C₂×D₄₇)⋊C₈	—	2	1	no	—
16	#13093934	C₄	1	1	yes	characteristic
17	#13093934	C₄	1	1	yes	characteristic
18	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₄₇	—	2	1	no	—
20	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
25	C₂×C₄×D₄₇	Q₈	1	1	yes	characteristic
26	C₂×C₈×C₄₇	—	2	1	no	—
27	C₂×C₈×C₄₇	—	2	1	no	—
28	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₄₇	D₄	1	1	yes	characteristic
33	C₄×Dic₄₇	Q₈	1	1	yes	characteristic
34	C₂×C₄₇⋊C₈	—	2	1	no	—
35	C₂×C₄₇⋊C₈	—	2	1	no	—
36	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
38	#10625324	—	2	2	no	—
39	#10625324	C₈	1	4	yes	complemented
40	C₈×D₄₇	—	4	2	no	—
41	C₂²×D₄₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄×D₄₇	—	2	1	no	—
49	C₄×D₄₇	—	2	1	no	—
50	C₄×D₄₇	—	4	1	no	—
51	C₄×D₄₇	C₂×C₈	1	2	yes	—
52	C₄×D₄₇	M₁₆	1	2	yes	—
53	C₈×C₄₇	—	2	1	no	—
54	C₈×C₄₇	—	4	1	no	—
55	C₈×C₄₇	—	4	1	no	—
56	C₈×C₄₇	C₂×C₈	1	2	yes	complemented
57	C₄₇⋊C₈	—	2	1	no	—
58	C₄₇⋊C₈	—	4	1	no	—
59	C₄₇⋊C₈	—	4	1	no	—
60	C₄₇⋊C₈	C₂×C₈	1	2	yes	complemented
61	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₄₇	—	2	1	no	—
63	C₂×D₄₇	—	2	1	no	—
64	C₂×D₄₇	C₄⋊C₈	1	2	yes	—
65	C₄×C₄₇	—	2	1	no	—
66	C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
67	C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
68	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₄₇	#137	1	1	yes	characteristic
70	Dic₄₇	—	2	1	no	—
71	Dic₄₇	—	2	1	no	—
72	Dic₄₇	C₄⋊C₈	1	2	yes	—
73	#1565	—	47	1	no	maximal, Sylow
74	C₂×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
75	C₂×C₄₇	#428	1	1	yes	characteristic
76	C₂×C₄₇	#430	1	1	yes	characteristic
77	D₄₇	—	2	2	no	—
78	#431	—	47	1	no	—
79	C₂×C₄⋊C₈	—	47	1	no	—
80	C₄×M₁₆	—	47	1	no	—
81	#335	—	47	1	no	—
82	#335	—	47	1	no	—
83	#334	—	47	1	no	—
84	#334	D₄₇	1	1	yes	characteristic, complemented
85	C₄₇	#1565	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₁₆	—	47	1	no	—
87	C₂×M₁₆	—	47	1	no	—
88	C₂²×C₈	—	94	1	no	—
89	C₂×C₄²	—	47	1	no	—
90	C₄⋊C₈	—	47	1	no	—
91	C₄⋊C₈	—	47	1	no	—
92	C₄⋊C₈	—	94	1	no	—
93	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
94	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
95	C₂²⋊C₈	—	94	1	no	—
96	#97	—	47	1	no	—
97	#97	—	47	1	no	—
98	C₄×C₈	—	47	1	no	—
99	C₄×C₈	—	94	1	no	—
100	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
101	C₂²×C₄	—	47	1	no	—
102	C₂²×C₄	—	47	1	no	—
103	C₂²×C₄	—	47	1	no	—
104	M₁₆	—	47	4	no	—
105	M₁₆	—	94	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	47	1	no	—
109	C₂×C₈	—	47	1	no	—
110	C₂×C₈	—	94	1	no	—
111	C₂×C₈	—	94	1	no	—
112	C₂×C₈	—	188	2	no	—
113	C₂×C₈	C₄×D₄₇	1	1	yes	characteristic
114	C₂×C₈	C₄×D₄₇	1	1	yes	characteristic
115	C₄²	—	47	1	no	—
116	C₄²	—	47	1	no	—
117	C₄²	—	47	1	no	—
118	C₄²	C₂²×D₄₇	1	1	yes	characteristic
119	C₂³	—	47	1	no	—
120	C₂×C₄	—	47	1	no	—
121	C₂×C₄	—	47	2	no	—
122	C₂×C₄	—	47	2	no	—
123	C₂×C₄	—	47	1	no	—
124	C₂×C₄	—	47	1	no	—
125	C₂×C₄	—	94	1	no	—
126	C₂×C₄	—	94	1	no	—
127	C₂×C₄	—	188	1	no	—
128	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
129	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
130	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	47	2	no	—
135	C₈	—	94	1	no	—
136	C₈	—	188	1	no	—
137	C₈	—	188	1	no	—
138	C₈	C₈×D₄₇	1	2	yes	complemented
139	C₂²	—	47	2	no	—
140	C₂²	—	94	1	no	—
141	C₂²	—	94	1	no	—
142	C₂²	C₄⋊C₄×D₄₇	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	47	2	no	—
145	C₄	—	94	1	no	—
146	C₄	—	94	1	no	—
147	C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
148	C₄	M₁₆×D₄₇	1	1	yes	characteristic
149	C₄	#13094051	1	1	yes	characteristic
150	C₄	#13094053	1	1	yes	characteristic
151	C₂	—	94	2	no	—
152	C₂	C₄⋊C₈×D₄₇	1	1	yes	characteristic
153	C₂	#15689162	1	1	yes	characteristic
154	C₂	#15689170	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689175

C₂

yes

maximal, characteristic

C₄⋊C₈×D₄₇

C₂

yes

maximal, characteristic

C₄×#10625324

C₂

yes

maximal, characteristic

#15688876

C₂

yes

maximal, characteristic, complemented

#15688875

C₂

yes

maximal, characteristic, complemented

#15688873

C₂

yes

maximal, characteristic, complemented

C₄₇×#334

C₂

yes

maximal, characteristic, complemented

C₂×#10625324

C₄

yes

characteristic

C₂×#10625324

C₄

yes

characteristic

C₂×C₈×D₄₇

—

C₄²×D₄₇

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

Dic₄₇⋊C₈

—

(C₂×D₄₇)⋊C₈

—

#13093934

C₄

yes

characteristic

#13093934

C₄

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₄₇

—

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄×D₄₇

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

D₄

yes

characteristic

C₂×C₄×D₄₇

Q₈

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

characteristic

C₄×Dic₄₇

Q₈

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

#10625324

—

#10625324

C₈

yes

complemented

C₈×D₄₇

—

C₂²×D₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×Dic₄₇

C₄⋊C₄

yes

characteristic

C₂×Dic₄₇

C₄⋊C₄

yes

characteristic

C₂×Dic₄₇

C₄⋊C₄

yes

characteristic

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

C₂×C₈

yes

—

C₄×D₄₇

M₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₈

yes

complemented

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂×C₈

yes

complemented

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₄⋊C₈

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂²×C₈

yes

characteristic

C₄×C₄₇

C₂×M₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₄⋊C₈

yes

—

#1565

—

maximal, Sylow

C₂×C₄₇

C₂×C₄⋊C₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#431

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄₇

#1565

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₈

C₂×D₄₇

yes

characteristic

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₄₇

yes

characteristic

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

M₁₆

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

188

—

113

C₂×C₈

C₄×D₄₇

yes

characteristic

114

C₂×C₈

C₄×D₄₇

yes

characteristic

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₂²×D₄₇

yes

characteristic

119

C₂³

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

188

—

128

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

129

C₂×C₄

D₄×D₄₇

yes

characteristic

130

C₂×C₄

Q₈×D₄₇

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

188

—

137

C₈

—

188

—

138

C₈

C₈×D₄₇

yes

complemented

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₄⋊C₄×D₄₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

C₂×C₈×D₄₇

yes

characteristic

148

C₄

M₁₆×D₄₇

yes

characteristic

149

C₄

#13094051

yes

characteristic

150

C₄

#13094053

yes

characteristic

151

C₂

—

152

C₂

C₄⋊C₈×D₄₇

yes

characteristic

153

C₂

#15689162

yes

characteristic

154

C₂

#15689170

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₈
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#334	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₄
hypercenter	#334	D₄₇
hypocenter	C₄₇	#1565
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₄₇

commutator

C₄×C₄₇

C₂²×C₈

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#334

C₂

socle

C₂²×C₄₇

C₂×C₄⋊C₄

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
4. #15688876	C₂	semidirect
5. #15688875	C₂	semidirect
6. #15688873	C₂	semidirect
7. C₄₇×#334	C₂	semidirect
39. #10625324	C₈	semidirect
56. C₈×C₄₇	C₂×C₈	semidirect
60. C₄₇⋊C₈	C₂×C₈	semidirect
84. #334	D₄₇	semidirect
85. C₄₇	#1565	semidirect
138. C₈	C₈×D₄₇	semidirect

H ≅ G/N

type

4. #15688876

C₂

semidirect

5. #15688875

C₂

semidirect

6. #15688873

C₂

semidirect

7. C₄₇×#334

C₂

semidirect

39. #10625324

C₈

semidirect

56. C₈×C₄₇

C₂×C₈

semidirect

60. C₄₇⋊C₈

C₂×C₈

semidirect

84. #334

D₄₇

semidirect

85. C₄₇

#1565

semidirect

138. C₈

C₈×D₄₇

semidirect

Subgroups — #20242137

Conjugacy classes of subgroups

Complements