Group #20241953 — subgroups

Subgroups — #20241953

Order 6016.

220 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 85 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	#15689680	C₂	1	1	yes	maximal, characteristic
2	#15688989	C₂	1	1	yes	maximal, characteristic
3	#15688861	C₂	1	4	yes	maximal
4	C₄₇×#431	C₂	1	1	yes	maximal, characteristic
5	#13094024	C₄	1	1	yes	characteristic, complemented
6	#13094024	C₄	1	1	yes	characteristic, complemented
7	#13094016	—	2	1	no	—
8	#13094003	—	2	1	no	—
9	#13094003	C₂²	1	2	yes	—
10	C₄×Dic₉₄	—	2	1	no	—
11	C₄×Dic₉₄	C₄	1	2	yes	complemented
12	#13093983	—	2	1	no	—
13	C₂×C₄⋊Dic₄₇	—	2	1	no	—
14	C₄₇⋊#95	—	2	1	no	—
15	C₄₇⋊#95	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₄₇	—	2	1	no	—
18	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
19	C₄×C₈×C₄₇	—	2	1	no	—
20	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₄₇	—	2	2	no	—
24	C₂×Dic₉₄	—	2	1	no	—
25	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₄₇	—	2	2	no	—
27	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₄₇	D₄	1	1	yes	characteristic
30	C₄²×C₄₇	D₄	1	1	yes	characteristic
31	C₂²⋊Dic₄₇	—	2	2	no	—
32	C₄⋊Dic₄₇	—	2	1	no	—
33	C₄⋊Dic₄₇	—	2	1	no	—
34	C₄⋊Dic₄₇	—	2	1	no	—
35	C₄⋊Dic₄₇	—	4	2	no	—
36	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
37	Dic₄₇⋊C₄	—	2	2	no	—
38	Dic₄₇⋊C₄	—	2	2	no	—
39	C₄×Dic₄₇	—	2	2	no	—
40	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	—	4	1	no	—
42	C₂×C₄×C₄₇	—	4	1	no	—
43	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
48	C₂×C₄×C₄₇	Q₁₆	1	2	yes	—
49	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₄₇	—	2	2	no	—
52	C₂×Dic₄₇	—	2	2	no	—
53	C₂×Dic₄₇	—	4	2	no	—
54	C₂×Dic₄₇	—	4	2	no	—
55	C₂×Dic₄₇	—	4	2	no	—
56	Dic₉₄	—	4	2	no	—
57	Dic₉₄	—	4	2	no	—
58	C₈×C₄₇	—	4	2	no	—
59	C₂²×C₄₇	—	2	1	no	—
60	C₂²×C₄₇	—	2	1	no	—
61	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂²×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₄₇	—	2	1	no	—
65	C₄×C₄₇	—	2	1	no	—
66	C₄×C₄₇	—	2	1	no	—
67	C₄×C₄₇	—	2	1	no	—
68	C₄×C₄₇	#103	1	2	yes	—
69	C₄×C₄₇	#103	1	2	yes	—
70	Dic₄₇	—	4	2	no	—
71	Dic₄₇	—	4	2	no	—
72	Dic₄₇	—	8	2	no	—
73	#1518	—	47	1	no	maximal, Sylow
74	C₂×C₄₇	—	2	1	no	—
75	C₂×C₄₇	—	2	1	no	—
76	C₂×C₄₇	#410	1	1	yes	characteristic
77	C₂×C₄₇	C₂×#103	1	1	yes	characteristic
78	C₂×C₄₇	#421	1	1	yes	characteristic
79	#533	—	47	1	no	—
80	#431	D₄₇	1	1	yes	characteristic
81	#382	—	47	1	no	—
82	#328	—	47	4	no	—
83	C₄₇	#1518	1	1	yes	characteristic, complemented, Sylow
84	C₄⋊Q₈	—	47	1	no	—
85	C₄⋊Q₈	—	47	1	no	—
86	#125	—	94	1	no	—
87	C₂²⋊Q₈	—	47	2	no	—
88	C₂²⋊Q₈	—	94	1	no	—
89	C₄×Q₈	—	47	2	no	—
90	C₄×Q₈	—	94	1	no	—
91	#117	—	94	1	no	—
92	C₂×C₄⋊C₄	—	94	1	no	—
93	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
94	C₄⋊C₈	—	2	1	no	—
95	C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
96	C₄×C₈	—	2	1	no	—
97	#95	—	47	2	no	—
98	#95	—	94	1	no	—
99	C₂×Q₈	—	47	2	no	—
100	C₂×Q₈	—	94	1	no	—
101	C₂²×C₄	—	94	2	no	—
102	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
103	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
105	C₂×C₈	—	2	2	no	—
106	C₄⋊C₄	—	47	2	no	—
107	C₄⋊C₄	—	94	2	no	—
108	C₄⋊C₄	—	94	1	no	—
109	C₄⋊C₄	—	94	1	no	—
110	C₄⋊C₄	—	94	1	no	—
111	C₄⋊C₄	—	94	2	no	—
112	C₄⋊C₄	—	188	2	no	—
113	C₂²⋊C₄	—	94	2	no	—
114	C₄²	—	94	2	no	—
115	C₄²	C₄×D₄₇	1	1	yes	characteristic
116	C₄²	C₄×D₄₇	1	1	yes	characteristic
117	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
118	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
119	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
120	Q₈	—	188	2	no	—
121	Q₈	—	188	2	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	94	2	no	—
125	C₂×C₄	—	94	2	no	—
126	C₂×C₄	—	188	2	no	—
127	C₂×C₄	—	188	2	no	—
128	C₂×C₄	—	188	2	no	—
129	C₂×C₄	#10625325	1	2	yes	—
130	C₂×C₄	Dic₁₈₈	1	2	yes	—
131	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
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×C₂²⋊D₄₇	1	1	yes	characteristic
137	C₈	—	4	2	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
141	C₂²	C₂×#10625325	1	1	yes	characteristic
142	C₂²	C₂×Dic₁₈₈	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	376	2	no	—
148	C₄	—	188	2	no	—
149	C₄	—	188	2	no	—
150	C₄	#13093949	1	2	yes	—
151	C₄	#13093949	1	2	yes	—
152	C₂	—	2	1	no	—
153	C₂	—	2	1	no	—
154	C₂	#15689096	1	1	yes	characteristic
155	C₂	C₂×#13093949	1	1	yes	characteristic
156	C₂	#15689139	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689680

C₂

yes

maximal, characteristic

#15688989

C₂

yes

maximal, characteristic

#15688861

C₂

yes

maximal

C₄₇×#431

C₂

yes

maximal, characteristic

#13094024

C₄

yes

characteristic, complemented

#13094024

C₄

yes

characteristic, complemented

#13094016

—

#13094003

—

#13094003

C₂²

yes

—

C₄×Dic₉₄

—

C₄×Dic₉₄

C₄

yes

complemented

#13093983

—

C₂×C₄⋊Dic₄₇

—

C₄₇⋊#95

—

C₄₇⋊#95

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂×C₄

yes

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

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

SD₁₆

yes

—

C₂×C₄×C₄₇

Q₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×SD₁₆

yes

characteristic

C₂²×C₄₇

C₂×Q₁₆

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#103

yes

—

C₄×C₄₇

#103

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#1518

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

yes

characteristic

#382

—

#328

—

C₄₇

#1518

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

C₂×D₄₇

yes

—

C₄×C₈

—

#95

—

#95

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

D₁₈₈

yes

characteristic

103

C₂²×C₄

D₁₈₈

yes

characteristic

104

C₂²×C₄

C₂²×D₄₇

yes

characteristic

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

188

—

113

C₂²⋊C₄

—

114

C₄²

—

115

C₄²

C₄×D₄₇

yes

characteristic

116

C₄²

C₄×D₄₇

yes

characteristic

117

C₄²

C₂²⋊D₄₇

yes

characteristic

118

C₄²

C₂²⋊D₄₇

yes

characteristic

119

C₂³

C₂×D₁₈₈

yes

characteristic

120

Q₈

—

188

—

121

Q₈

—

188

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

188

—

127

C₂×C₄

—

188

—

128

C₂×C₄

—

188

—

129

C₂×C₄

#10625325

yes

—

130

C₂×C₄

Dic₁₈₈

yes

—

131

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

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×C₂²⋊D₄₇

yes

characteristic

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

141

C₂²

C₂×#10625325

yes

characteristic

142

C₂²

C₂×Dic₁₈₈

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

376

—

148

C₄

—

188

—

149

C₄

—

188

—

150

C₄

#13093949

yes

—

151

C₄

#13093949

yes

—

152

C₂

—

153

C₂

—

154

C₂

#15689096

yes

characteristic

155

C₂

C₂×#13093949

yes

characteristic

156

C₂

#15689139

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#431	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#431	D₄₇
hypocenter	C₄₇	#1518
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×(C₂×D₄₇)⋊C₄

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#431

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
5. #13094024	C₄	semidirect
6. #13094024	C₄	semidirect
11. C₄×Dic₉₄	C₄	semidirect
83. C₄₇	#1518	semidirect

H ≅ G/N

type

5. #13094024

C₄

semidirect

6. #13094024

C₄

semidirect

11. C₄×Dic₉₄

C₄

semidirect

83. C₄₇

#1518

semidirect

Subgroups — #20241953

Conjugacy classes of subgroups

Complements