Group #20251704 — subgroups

Subgroups — #20251704

Order 6016.

582 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13094018	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689015	C₂	1	3	yes	maximal, complemented
4	#15689008	C₂	1	3	yes	maximal, complemented
5	#15689001	C₂	1	3	yes	maximal
6	#15688997	C₂	1	3	yes	maximal, complemented
7	C₄₇×#383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄×D₄₇	C₂²	1	3	yes	—
10	#13094018	—	2	4	no	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	3	yes	—
12	C₂×Dic₄₇⋊C₄	C₂²	1	3	yes	—
13	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	3	yes	—
14	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	3	yes	—
15	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	3	yes	—
16	C₂×C₄×Dic₄₇	C₂²	1	3	yes	—
17	C₄²×D₄₇	—	2	4	no	—
18	C₄₇⋊#95	C₂²	1	3	yes	—
19	C₄₇⋊#95	C₂²	1	3	yes	—
20	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	#95×C₄₇	C₂²	1	3	yes	—
22	#95×C₄₇	C₂²	1	3	yes	—
23	C₂³×D₄₇	C₂³	1	1	yes	characteristic
24	C₂³×D₄₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
28	C₂²×Dic₄₇	C₂³	1	3	yes	—
29	C₂²×Dic₄₇	C₂³	1	3	yes	—
30	C₂×D₁₈₈	—	2	6	no	—
31	C₂×C₄×D₄₇	—	2	6	no	—
32	C₂×C₄×D₄₇	—	2	6	no	—
33	C₂×C₄×D₄₇	—	2	6	no	—
34	C₄²×C₄₇	D₄	1	4	yes	complemented
35	(C₂×D₄₇)⋊C₄	—	2	6	no	—
36	(C₂×D₄₇)⋊C₄	—	2	6	no	—
37	(C₂×D₄₇)⋊C₄	—	2	6	no	—
38	C₄⋊Dic₄₇	—	2	6	no	—
39	Dic₄₇⋊C₄	—	2	6	no	—
40	C₄×Dic₄₇	—	2	6	no	—
41	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
42	C₂²×D₄₇	—	2	1	no	—
43	C₂²×D₄₇	—	2	3	no	—
44	C₂²×D₄₇	—	2	1	no	—
45	C₂²×D₄₇	—	2	3	no	—
46	C₂²×D₄₇	—	2	3	no	—
47	C₂²×D₄₇	#41	1	6	yes	—
48	C₂×C₄×C₄₇	—	2	3	no	—
49	C₂×C₄×C₄₇	—	2	3	no	—
50	C₂×C₄×C₄₇	—	2	3	no	—
51	C₂×C₄×C₄₇	—	2	3	no	—
52	C₂×C₄×C₄₇	—	2	3	no	—
53	C₂×C₄×C₄₇	—	2	3	no	—
54	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
55	C₂×Dic₄₇	—	2	3	no	—
56	C₂×Dic₄₇	—	2	3	no	—
57	C₂×Dic₄₇	—	2	3	no	—
58	C₂×Dic₄₇	—	2	3	no	—
59	C₂×Dic₄₇	—	2	3	no	—
60	C₂×Dic₄₇	#41	1	6	yes	—
61	D₁₈₈	—	4	4	no	—
62	C₄×D₄₇	—	2	24	no	—
63	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
64	C₂²×C₄₇	C₂×#41	1	3	yes	—
65	C₂²×C₄₇	C₂×#41	1	3	yes	—
66	C₂×D₄₇	—	2	6	no	—
67	C₂×D₄₇	—	2	6	no	—
68	C₂×D₄₇	—	4	3	no	—
69	C₂×D₄₇	—	4	1	no	—
70	C₂×D₄₇	—	4	3	no	—
71	C₂×D₄₇	#126	1	4	yes	complemented
72	C₄×C₄₇	—	2	6	no	—
73	C₄×C₄₇	—	4	1	no	—
74	C₄×C₄₇	—	4	3	no	—
75	Dic₄₇	—	2	6	no	—
76	Dic₄₇	—	4	3	no	—
77	#2852	—	47	1	no	maximal, Sylow
78	C₂×C₄₇	#525	1	3	yes	—
79	C₂×C₄₇	C₂×#126	1	1	yes	characteristic
80	C₂×C₄₇	#529	1	3	yes	—
81	D₄₇	—	2	4	no	—
82	D₄₇	—	8	1	no	—
83	C₂×#126	—	47	1	no	—
84	C₂²×C₄²	—	47	1	no	—
85	#388	—	47	3	no	—
86	#387	—	47	3	no	—
87	#386	—	47	3	no	—
88	#385	—	47	3	no	—
89	#383	D₄₇	1	1	yes	characteristic, complemented
90	C₄₇	#2852	1	1	yes	characteristic, complemented, Sylow
91	C₂²×D₄	—	47	1	no	—
92	C₂³×C₄	—	47	3	no	—
93	#126	—	94	4	no	—
94	C₂×C₄⋊C₄	—	47	3	no	—
95	C₂×C₄⋊C₄	—	47	3	no	—
96	C₂×C₂²⋊C₄	—	47	3	no	—
97	C₂×C₂²⋊C₄	—	47	3	no	—
98	C₂×C₂²⋊C₄	—	47	3	no	—
99	C₂×C₄²	—	47	3	no	—
100	C₂×C₄²	—	94	4	no	—
101	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
102	#95	—	47	3	no	—
103	#95	—	47	3	no	—
104	#95	C₂×D₄₇	1	3	yes	—
105	#95	C₂×D₄₇	1	3	yes	—
106	C₂⁴	—	47	1	no	—
107	C₂⁴	—	47	1	no	—
108	C₂×D₄	—	94	6	no	—
109	C₂²×C₄	—	47	3	no	—
110	C₂²×C₄	—	47	3	no	—
111	C₂²×C₄	—	94	6	no	—
112	C₂²×C₄	—	94	6	no	—
113	C₂²×C₄	—	94	6	no	—
114	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
116	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
117	C₄⋊C₄	—	94	6	no	—
118	C₄⋊C₄	—	94	6	no	—
119	C₂²⋊C₄	—	94	6	no	—
120	C₂²⋊C₄	—	94	6	no	—
121	C₂²⋊C₄	—	94	6	no	—
122	C₄²	—	94	6	no	—
123	C₄²	D₁₈₈	1	4	yes	complemented
124	C₂³	—	47	6	no	—
125	C₂³	—	94	1	no	—
126	C₂³	—	94	3	no	—
127	C₂³	—	94	1	no	—
128	C₂³	—	94	3	no	—
129	C₂³	—	94	3	no	—
130	C₂³	C₂³×D₄₇	1	1	yes	characteristic
131	D₄	—	188	4	no	—
132	C₂×C₄	—	2	3	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	2	3	no	—
135	C₂×C₄	—	2	3	no	—
136	C₂×C₄	—	2	3	no	—
137	C₂×C₄	—	2	3	no	—
138	C₂×C₄	—	47	6	no	—
139	C₂×C₄	—	94	24	no	—
140	C₂×C₄	—	94	3	no	—
141	C₂×C₄	—	94	3	no	—
142	C₂×C₄	—	94	3	no	—
143	C₂×C₄	—	94	3	no	—
144	C₂×C₄	—	94	3	no	—
145	C₂×C₄	C₂×D₁₈₈	1	6	yes	—
146	C₂²	—	47	4	no	—
147	C₂²	—	94	6	no	—
148	C₂²	—	94	6	no	—
149	C₂²	—	188	3	no	—
150	C₂²	—	188	1	no	—
151	C₂²	—	188	3	no	—
152	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
153	C₂²	#41×D₄₇	1	3	yes	—
154	C₂²	#41×D₄₇	1	3	yes	—
155	C₄	—	2	6	no	—
156	C₄	—	4	1	no	—
157	C₄	—	4	3	no	—
158	C₄	—	94	6	no	—
159	C₄	—	188	3	no	—
160	C₂	—	94	4	no	—
161	C₂	—	376	1	no	—
162	C₂	#15689635	1	3	yes	—
163	C₂	#126×D₄₇	1	1	yes	characteristic
164	C₂	#15689656	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094018

C₂

yes

maximal, characteristic, complemented

C₂×C₄²×D₄₇

C₂

yes

maximal, characteristic, complemented

#15689015

C₂

yes

maximal, complemented

#15689008

C₂

yes

maximal, complemented

#15689001

C₂

yes

maximal

#15688997

C₂

yes

maximal, complemented

C₄₇×#383

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

characteristic

C₂²×C₄×D₄₇

C₂²

yes

—

#13094018

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄²×D₄₇

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

C₂²

yes

—

#95×C₄₇

C₂²

yes

—

C₂³×D₄₇

C₂³

yes

characteristic

C₂³×D₄₇

C₂³

yes

characteristic

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×D₁₈₈

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₄²×C₄₇

D₄

yes

complemented

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

#41

yes

—

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×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#41

yes

—

D₁₈₈

—

C₄×D₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

#126

yes

complemented

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

#2852

—

maximal, Sylow

C₂×C₄₇

#525

yes

—

C₂×C₄₇

C₂×#126

yes

characteristic

C₂×C₄₇

#529

yes

—

D₄₇

—

D₄₇

—

C₂×#126

—

C₂²×C₄²

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄₇

#2852

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

#126

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂×D₄₇

yes

characteristic

102

#95

—

103

#95

—

104

#95

C₂×D₄₇

yes

—

105

#95

C₂×D₄₇

yes

—

106

C₂⁴

—

107

C₂⁴

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂²×D₄₇

yes

characteristic

115

C₂²×C₄

C₂²×D₄₇

yes

—

116

C₂²×C₄

C₂²×D₄₇

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

—

123

C₄²

D₁₈₈

yes

complemented

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

C₂³×D₄₇

yes

characteristic

131

D₄

—

188

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

C₂×D₁₈₈

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

188

—

150

C₂²

—

188

—

151

C₂²

—

188

—

152

C₂²

C₂²×D₁₈₈

yes

characteristic

153

C₂²

#41×D₄₇

yes

—

154

C₂²

#41×D₄₇

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

188

—

160

C₂

—

161

C₂

—

376

—

162

C₂

#15689635

yes

—

163

C₂

#126×D₄₇

yes

characteristic

164

C₂

#15689656

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#383	C₂
hypercenter	#383	D₄₇
hypocenter	C₄₇	#2852
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#383

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
1. C₂×#13094018	C₂	semidirect
2. C₂×C₄²×D₄₇	C₂	semidirect
3. #15689015	C₂	semidirect
4. #15689008	C₂	semidirect
6. #15688997	C₂	semidirect
7. C₄₇×#383	C₂	semidirect
34. C₄²×C₄₇	D₄	semidirect
71. C₂×D₄₇	#126	semidirect
89. #383	D₄₇	semidirect
90. C₄₇	#2852	semidirect
123. C₄²	D₁₈₈	semidirect

H ≅ G/N

type

1. C₂×#13094018

C₂

semidirect

2. C₂×C₄²×D₄₇

C₂

semidirect

3. #15689015

C₂

semidirect

4. #15689008

C₂

semidirect

6. #15688997

C₂

semidirect

7. C₄₇×#383

C₂

semidirect

34. C₄²×C₄₇

D₄

semidirect

71. C₂×D₄₇

#126

semidirect

89. #383

D₄₇

semidirect

90. C₄₇

#2852

semidirect

123. C₄²

D₁₈₈

semidirect

Subgroups — #20251704

Conjugacy classes of subgroups

Complements