Group #20250420 — subgroups

Subgroups — #20250420

Order 6016.

456 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13094022	C₂	1	2	yes	maximal
2	#15689053	C₂	1	2	yes	maximal
3	#15689044	C₂	1	4	yes	maximal
4	#15689021	C₂	1	2	yes	maximal
5	#15689008	C₂	1	4	yes	maximal
6	C₄₇×#391	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₉₄	C₂²	1	2	yes	—
8	#13094022	—	2	8	no	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
14	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
15	C₄₇⋊#95	C₂²	1	2	yes	—
16	C₄₇⋊#95	C₂²	1	4	yes	—
17	C₄₇⋊#95	C₂²	1	4	yes	—
18	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	#95×C₄₇	C₂²	1	4	yes	—
22	C₂×Q₈×C₄₇	—	4	1	no	—
23	C₂×Q₈×C₄₇	—	4	1	no	—
24	C₂×Q₈×C₄₇	D₄	1	4	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
28	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
29	C₂²×Dic₄₇	C₂³	1	4	yes	—
30	C₂²×Dic₄₇	C₂³	1	4	yes	—
31	C₂×Dic₉₄	—	2	4	no	—
32	C₂×Dic₉₄	—	4	2	no	—
33	C₂×Dic₉₄	—	4	2	no	—
34	C₄⋊C₄×C₄₇	—	4	1	no	—
35	C₄²×C₄₇	—	4	1	no	—
36	C₄⋊Dic₄₇	—	4	2	no	—
37	C₄⋊Dic₄₇	—	4	2	no	—
38	C₄⋊Dic₄₇	Q₈	1	8	yes	complemented
39	Dic₄₇⋊C₄	—	2	8	no	—
40	Dic₄₇⋊C₄	—	2	8	no	—
41	C₄×Dic₄₇	—	2	4	no	—
42	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
43	Q₈×C₄₇	—	4	4	no	—
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	—	2	1	no	—
46	C₂×C₄×C₄₇	—	2	1	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	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	4	no	—
52	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
53	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
55	C₂×Dic₄₇	—	2	4	no	—
56	C₂×Dic₄₇	—	2	4	no	—
57	C₂×Dic₄₇	—	2	4	no	—
58	C₂×Dic₄₇	—	2	4	no	—
59	C₂×Dic₄₇	—	2	4	no	—
60	C₂×Dic₄₇	C₂×Q₈	1	8	yes	—
61	Dic₉₄	—	4	8	no	—
62	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
63	C₂²×C₄₇	C₂²×Q₈	1	2	yes	—
64	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
66	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
67	C₄×C₄₇	—	2	4	no	—
68	C₄×C₄₇	—	2	2	no	—
69	C₄×C₄₇	—	2	4	no	—
70	C₄×C₄₇	—	4	2	no	—
71	Dic₄₇	—	2	8	no	—
72	Dic₄₇	—	4	4	no	—
73	#2734	—	47	1	no	maximal, Sylow
74	C₂×C₄₇	#534	1	1	yes	characteristic
75	C₂×C₄₇	#544	1	2	yes	—
76	C₂×C₄₇	D₄×Q₈	1	2	yes	—
77	C₂×C₄₇	Q₈²	1	1	yes	characteristic
78	C₂×C₄₇	#563	1	1	yes	characteristic
79	C₂×C₄⋊Q₈	—	47	2	no	—
80	#398	—	47	2	no	—
81	#395	—	47	4	no	—
82	#391	D₄₇	1	1	yes	characteristic
83	#389	—	47	2	no	—
84	#387	—	47	4	no	—
85	C₄₇	#2734	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	47	2	no	—
87	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
88	C₄⋊Q₈	—	94	8	no	—
89	C₂×C₄⋊C₄	—	47	2	no	—
90	C₂×C₄⋊C₄	—	47	4	no	—
91	C₂×C₄⋊C₄	—	47	4	no	—
92	C₂×C₄⋊C₄	—	47	2	no	—
93	C₂×C₄⋊C₄	—	47	2	no	—
94	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
95	C₂×C₄²	—	47	2	no	—
96	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
97	#95	—	47	2	no	—
98	#95	—	47	4	no	—
99	#95	—	47	4	no	—
100	#95	C₂×D₄₇	1	4	yes	—
101	C₂×Q₈	—	4	1	no	—
102	C₂×Q₈	—	4	1	no	—
103	C₂×Q₈	—	94	4	no	—
104	C₂×Q₈	—	188	2	no	—
105	C₂×Q₈	—	188	2	no	—
106	C₂×Q₈	D₁₈₈	1	4	yes	—
107	C₂²×C₄	—	47	4	no	—
108	C₂²×C₄	—	47	4	no	—
109	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
110	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
112	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	47	8	no	—
115	C₄⋊C₄	—	94	8	no	—
116	C₄⋊C₄	—	94	8	no	—
117	C₄⋊C₄	—	188	2	no	—
118	C₄⋊C₄	—	188	2	no	—
119	C₄²	—	4	1	no	—
120	C₄²	—	94	4	no	—
121	C₂³	C₂³×D₄₇	1	1	yes	characteristic
122	Q₈	—	4	4	no	—
123	Q₈	—	188	8	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	47	8	no	—
133	C₂×C₄	—	94	4	no	—
134	C₂×C₄	—	94	4	no	—
135	C₂×C₄	—	94	4	no	—
136	C₂×C₄	—	94	4	no	—
137	C₂×C₄	—	94	4	no	—
138	C₂×C₄	C₂×D₁₈₈	1	4	yes	—
139	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
140	C₂×C₄	Q₈×D₄₇	1	4	yes	—
141	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
142	C₂²	C₂×Q₈×D₄₇	1	2	yes	—
143	C₂²	#13094069	1	1	yes	characteristic
144	C₂²	#13094069	1	1	yes	characteristic
145	C₂²	#13094071	1	2	yes	—
146	C₄	—	2	4	no	—
147	C₄	—	2	4	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	94	8	no	—
151	C₄	—	188	4	no	—
152	C₂	#15689685	1	1	yes	characteristic
153	C₂	#15689752	1	2	yes	—
154	C₂	Q₈×D₁₈₈	1	2	yes	—
155	C₂	#15689853	1	1	yes	characteristic
156	C₂	#15689873	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094022

C₂

yes

maximal

#15689053

C₂

yes

maximal

#15689044

C₂

yes

maximal

#15689021

C₂

yes

maximal

#15689008

C₂

yes

maximal

C₄₇×#391

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

#13094022

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

Q₈

yes

complemented

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×Q₈

yes

—

Dic₉₄

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

D₄×Q₈

yes

—

C₂×C₄₇

Q₈²

yes

characteristic

C₂×C₄₇

#563

yes

characteristic

C₂×C₄⋊Q₈

—

#398

—

#395

—

#391

D₄₇

yes

characteristic

#389

—

#387

—

C₄₇

#2734

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

—

—

—

100

#95

C₂×D₄₇

yes

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

188

—

105

C₂×Q₈

—

188

—

106

C₂×Q₈

D₁₈₈

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂²×D₄₇

yes

—

110

C₂²×C₄

C₂²×D₄₇

yes

—

111

C₂²×C₄

C₂²×D₄₇

yes

characteristic

112

C₂²×C₄

C₂²×D₄₇

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

188

—

118

C₄⋊C₄

—

188

—

119

C₄²

—

120

C₄²

—

121

C₂³

C₂³×D₄₇

yes

characteristic

122

Q₈

—

123

Q₈

—

188

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂×D₁₈₈

yes

—

139

C₂×C₄

C₂×D₁₈₈

yes

—

140

C₂×C₄

Q₈×D₄₇

yes

—

141

C₂²

C₂²×D₁₈₈

yes

characteristic

142

C₂²

C₂×Q₈×D₄₇

yes

—

143

C₂²

#13094069

yes

characteristic

144

C₂²

#13094069

yes

characteristic

145

C₂²

#13094071

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

188

—

152

C₂

#15689685

yes

characteristic

153

C₂

#15689752

yes

—

154

C₂

Q₈×D₁₈₈

yes

—

155

C₂

#15689853

yes

characteristic

156

C₂

#15689873

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#391	C₂
hypercenter	#391	D₄₇
hypocenter	C₄₇	#2734
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #391

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#391

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #391

N	H ≅ G/N	type
38. C₄⋊Dic₄₇	Q₈	semidirect
85. C₄₇	#2734	semidirect

H ≅ G/N

type

38. C₄⋊Dic₄₇

Q₈

semidirect

85. C₄₇

#2734

semidirect

Subgroups — #20250420

Conjugacy classes of subgroups

Complements