Group #20241952 — subgroups

Subgroups — #20241952

Order 6016.

220 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 93 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	#15689176	C₂	1	1	yes	maximal, characteristic
2	#15688989	C₂	1	1	yes	maximal, characteristic
3	#15688860	C₂	1	4	yes	maximal
4	C₄₇×#533	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₄₇	—	2	1	no	—
6	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₄×C₄₇⋊C₈	—	2	1	no	—
9	C₄₇⋊#95	—	2	1	no	—
10	C₄₇⋊#95	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic, complemented
12	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic, complemented
13	#125×C₄₇	—	2	1	no	—
14	C₂²⋊Q₈×C₄₇	—	2	1	no	—
15	C₂²⋊Q₈×C₄₇	C₂²	1	2	yes	—
16	C₄×Q₈×C₄₇	—	2	1	no	—
17	C₄×Q₈×C₄₇	C₄	1	2	yes	complemented
18	#117×C₄₇	—	2	1	no	—
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₄₇	—	2	1	no	—
21	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₇	—	2	2	no	—
26	C₄⋊C₄×C₄₇	—	2	1	no	—
27	C₄⋊C₄×C₄₇	—	2	1	no	—
28	C₄⋊C₄×C₄₇	—	2	1	no	—
29	C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂²⋊C₄×C₄₇	—	2	2	no	—
33	C₄²×C₄₇	—	2	2	no	—
34	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₄₇	D₄	1	1	yes	characteristic
37	C₄²×C₄₇	D₄	1	1	yes	characteristic
38	C₄⋊Dic₄₇	—	4	2	no	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₄₇	—	4	2	no	—
42	Q₈×C₄₇	—	4	2	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	—	4	1	no	—
46	C₂×C₄×C₄₇	—	4	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
52	C₂×C₄×C₄₇	Q₁₆	1	2	yes	—
53	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₄₇	—	4	2	no	—
56	C₂×Dic₄₇	—	4	2	no	—
57	C₂×Dic₄₇	—	4	2	no	—
58	C₄₇⋊C₈	—	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	—	4	2	no	—
69	C₄×C₄₇	—	4	2	no	—
70	C₄×C₄₇	#103	1	2	yes	—
71	C₄×C₄₇	#103	1	2	yes	—
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	D₄₇	1	1	yes	characteristic
80	#431	—	47	1	no	—
81	#382	—	47	1	no	—
82	#328	—	47	4	no	—
83	C₄₇	#1518	1	1	yes	characteristic, complemented, Sylow
84	C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic, complemented
85	C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic, complemented
86	#125	—	2	1	no	—
87	C₂²⋊Q₈	—	2	1	no	—
88	C₂²⋊Q₈	C₂×D₄₇	1	2	yes	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	Dic₄₇	1	2	yes	complemented
91	#117	—	2	1	no	—
92	C₂×C₄⋊C₄	—	94	1	no	—
93	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
94	C₄⋊C₈	—	94	1	no	—
95	C₂²⋊C₈	—	47	2	no	—
96	C₄×C₈	—	94	1	no	—
97	#95	—	47	2	no	—
98	#95	—	94	1	no	—
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	C₂×Dic₄₇	1	2	yes	—
101	C₂²×C₄	—	94	2	no	—
102	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
103	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
105	C₂×C₈	—	94	2	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	188	2	no	—
112	C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
113	C₂²⋊C₄	—	2	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
116	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
117	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
118	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
119	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	188	2	no	—
127	C₂×C₄	—	188	2	no	—
128	C₂×C₄	—	188	2	no	—
129	C₂×C₄	#10625336	1	2	yes	—
130	C₂×C₄	#10625337	1	2	yes	—
131	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
132	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
133	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
135	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
137	C₈	—	188	2	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
141	C₂²	C₂×#10625336	1	1	yes	characteristic
142	C₂²	C₂×#10625337	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	—	4	2	no	—
148	C₄	—	4	2	no	—
149	C₄	—	376	2	no	—
150	C₄	#13093948	1	2	yes	—
151	C₄	#13093948	1	2	yes	—
152	C₂	—	2	1	no	—
153	C₂	—	2	1	no	—
154	C₂	#15689097	1	1	yes	characteristic
155	C₂	C₂×#13093948	1	1	yes	characteristic
156	C₂	#15689140	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689176

C₂

yes

maximal, characteristic

#15688989

C₂

yes

maximal, characteristic

#15688860

C₂

yes

maximal

C₄₇×#533

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₄₇

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄×C₄₇⋊C₈

—

C₄₇⋊#95

—

C₄₇⋊#95

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₄

yes

characteristic, complemented

C₄⋊Q₈×C₄₇

C₄

yes

characteristic, complemented

#125×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₂²⋊Q₈×C₄₇

C₂²

yes

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

C₄

yes

complemented

#117×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

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

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

—

C₄₇⋊C₈

—

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

—

C₄×C₄₇

—

C₄×C₄₇

#103

yes

—

C₄×C₄₇

#103

yes

—

Dic₄₇

—

#1518

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

—

—

—

C₄₇

#1518

yes

characteristic, complemented, Sylow

C₄⋊Q₈

Dic₄₇

yes

characteristic, complemented

C₄⋊Q₈

Dic₄₇

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×D₄₇

yes

—

C₄×Q₈

—

C₄×Q₈

Dic₄₇

yes

complemented

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#95

—

#95

—

C₂×Q₈

—

100

C₂×Q₈

C₂×Dic₄₇

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

103

C₂²×C₄

C₂²⋊D₄₇

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

—

188

—

112

C₄⋊C₄

C₂×Dic₄₇

yes

—

113

C₂²⋊C₄

—

114

C₄²

—

115

C₄²

C₂×Dic₄₇

yes

characteristic

116

C₄²

C₂×Dic₄₇

yes

characteristic

117

C₄²

C₂²⋊D₄₇

yes

characteristic

118

C₄²

C₂²⋊D₄₇

yes

characteristic

119

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

120

Q₈

—

121

Q₈

—

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

#10625336

yes

—

130

C₂×C₄

#10625337

yes

—

131

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

132

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

133

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

134

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

135

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

137

C₈

—

188

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

141

C₂²

C₂×#10625336

yes

characteristic

142

C₂²

C₂×#10625337

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

376

—

150

C₄

#13093948

yes

—

151

C₄

#13093948

yes

—

152

C₂

—

153

C₂

—

154

C₂

#15689097

yes

characteristic

155

C₂

C₂×#13093948

yes

characteristic

156

C₂

#15689140

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#533	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#533	D₄₇
hypocenter	C₄₇	#1518
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#533

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
11. C₄⋊Q₈×C₄₇	C₄	semidirect
12. C₄⋊Q₈×C₄₇	C₄	semidirect
17. C₄×Q₈×C₄₇	C₄	semidirect
83. C₄₇	#1518	semidirect
84. C₄⋊Q₈	Dic₄₇	semidirect
85. C₄⋊Q₈	Dic₄₇	semidirect
90. C₄×Q₈	Dic₄₇	semidirect

H ≅ G/N

type

11. C₄⋊Q₈×C₄₇

C₄

semidirect

12. C₄⋊Q₈×C₄₇

C₄

semidirect

17. C₄×Q₈×C₄₇

C₄

semidirect

83. C₄₇

#1518

semidirect

84. C₄⋊Q₈

Dic₄₇

semidirect

85. C₄⋊Q₈

Dic₄₇

semidirect

90. C₄×Q₈

Dic₄₇

semidirect

Subgroups — #20241952

Conjugacy classes of subgroups

Complements