Group #20242829 — subgroups

Subgroups — #20242829

Order 6016.

166 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 53 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	#15689330	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689320	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688876	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688869	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688867	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#486	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625337	—	2	1	no	—
9	C₂×#10625335	—	2	1	no	—
10	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
11	C₄⋊(C₄₇⋊C₈)	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	#13093948	—	2	1	no	—
16	#13093945	—	2	1	no	—
17	C₈×Dic₄₇	—	2	1	no	—
18	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	#127×C₄₇	C₂²	1	1	yes	characteristic
20	#102×C₄₇	—	2	1	no	—
21	#102×C₄₇	—	2	1	no	—
22	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	—	2	1	no	—
25	C₂×D₄×C₄₇	—	2	1	no	—
26	C₂×D₄×C₄₇	—	2	1	no	—
27	C₂×Dic₉₄	D₄	1	1	yes	characteristic
28	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₄⋊C₄×C₄₇	—	2	1	no	—
32	C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	#10625337	—	4	2	no	—
34	#10625335	—	4	2	no	—
35	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
36	Dic₄₇⋊C₄	—	2	1	no	—
37	C₄×Dic₄₇	—	2	1	no	—
38	C₂×C₄₇⋊C₈	—	2	1	no	—
39	C₂×C₄₇⋊C₈	—	2	1	no	—
40	C₂×C₄₇⋊C₈	—	2	1	no	—
41	C₂³×C₄₇	—	4	1	no	—
42	Q₈×C₄₇	—	4	1	no	—
43	Q₈×C₄₇	—	4	1	no	—
44	D₄×C₄₇	—	4	1	no	—
45	D₄×C₄₇	—	4	1	no	—
46	D₄×C₄₇	—	4	1	no	—
47	D₄×C₄₇	—	4	1	no	—
48	D₄×C₄₇	—	8	1	no	—
49	C₂×C₄×C₄₇	—	4	1	no	—
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
53	C₂×Dic₄₇	—	2	1	no	—
54	C₂×Dic₄₇	—	2	1	no	—
55	Dic₉₄	—	2	1	no	—
56	Dic₉₄	D₈	1	2	yes	—
57	C₈×C₄₇	—	2	1	no	—
58	C₈×C₄₇	—	2	1	no	—
59	C₄₇⋊C₈	—	4	1	no	—
60	C₄₇⋊C₈	—	4	1	no	—
61	C₄₇⋊C₈	—	4	1	no	—
62	C₂²×C₄₇	—	8	1	no	—
63	C₂²×C₄₇	—	8	1	no	—
64	C₂²×C₄₇	—	8	1	no	—
65	C₂²×C₄₇	#121	1	1	yes	characteristic
66	C₄×C₄₇	—	2	1	no	—
67	C₄×C₄₇	—	8	1	no	—
68	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
69	C₄×C₄₇	#135	1	1	yes	characteristic
70	C₄×C₄₇	#135	1	1	yes	characteristic
71	C₄×C₄₇	#137	1	1	yes	characteristic
72	Dic₄₇	—	2	2	no	—
73	Dic₄₇	—	4	1	no	—
74	#1682	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	—	16	1	no	—
76	C₂×C₄₇	#464	1	1	yes	characteristic
77	C₂×C₄₇	#466	1	1	yes	characteristic
78	C₂×C₄₇	#471	1	1	yes	characteristic
79	#486	D₄₇	1	1	yes	characteristic
80	#462	—	47	1	no	—
81	#460	—	47	1	no	—
82	C₈×Q₈	—	47	1	no	—
83	#335	—	47	1	no	—
84	#332	—	47	1	no	—
85	#331	—	47	1	no	—
86	C₄₇	#1682	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₁₆	—	94	1	no	—
88	C₂×SD₁₆	—	94	1	no	—
89	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
90	#127	C₂×D₄₇	1	1	yes	characteristic
91	C₄×Q₈	—	47	1	no	—
92	C₄⋊C₈	—	47	1	no	—
93	C₄⋊C₈	—	47	1	no	—
94	C₄⋊C₈	—	47	1	no	—
95	C₄⋊C₈	—	94	1	no	—
96	#103	—	94	1	no	—
97	#102	—	2	1	no	—
98	#102	—	2	1	no	—
99	#102	—	94	1	no	—
100	C₄×C₈	—	94	1	no	—
101	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	47	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	Q₁₆	—	188	2	no	—
108	SD₁₆	—	188	2	no	—
109	C₂×C₈	—	94	1	no	—
110	C₂×C₈	—	94	1	no	—
111	C₂×C₈	—	94	1	no	—
112	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
113	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	47	1	no	—
117	C₄⋊C₄	—	94	1	no	—
118	C₄²	—	94	1	no	—
119	C₄²	C₂²×D₄₇	1	1	yes	characteristic
120	C₂³	—	4	1	no	—
121	Q₈	—	4	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	47	2	no	—
124	Q₈	—	94	1	no	—
125	D₄	—	4	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	8	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	94	1	no	—
132	C₂×C₄	—	94	1	no	—
133	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
134	C₂×C₄	#10625352	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	188	1	no	—
139	C₈	—	188	1	no	—
140	C₈	—	188	1	no	—
141	C₂²	—	8	1	no	—
142	C₂²	—	8	1	no	—
143	C₂²	—	8	1	no	—
144	C₂²	#13093995	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	8	1	no	—
147	C₄	—	94	2	no	—
148	C₄	—	188	1	no	—
149	C₄	D₈×D₄₇	1	1	yes	characteristic
150	C₄	#13094043	1	1	yes	characteristic
151	C₄	#13094044	1	1	yes	characteristic
152	C₄	#13094054	1	1	yes	characteristic
153	C₂	—	16	1	no	—
154	C₂	#15689342	1	1	yes	characteristic
155	C₂	#15689356	1	1	yes	characteristic
156	C₂	#15689383	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689330

C₂

yes

maximal, characteristic, complemented

#15689320

C₂

yes

maximal, characteristic

C₈×Dic₉₄

C₂

yes

maximal, characteristic, complemented

#15688876

C₂

yes

maximal, characteristic, complemented

#15688869

C₂

yes

maximal, characteristic, complemented

#15688867

C₂

yes

maximal, characteristic

C₄₇×#486

C₂

yes

maximal, characteristic

C₂×#10625337

—

C₂×#10625335

—

C₄×Dic₉₄

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

Dic₄₇⋊C₈

—

#13093948

—

#13093945

—

C₈×Dic₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

#127×C₄₇

C₂²

yes

characteristic

#102×C₄₇

—

#102×C₄₇

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×Dic₉₄

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625337

—

#10625335

—

C₄⋊Dic₄₇

D₄

yes

characteristic

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

D₈

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

#1682

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

characteristic

#462

—

#460

—

C₈×Q₈

—

#335

—

#332

—

#331

—

C₄₇

#1682

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

#127

C₂×D₄₇

yes

characteristic

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

—

—

—

—

100

C₄×C₈

—

101

C₄×C₈

C₂×D₄₇

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

Q₁₆

—

188

—

108

SD₁₆

—

188

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

113

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

—

119

C₄²

C₂²×D₄₇

yes

characteristic

120

C₂³

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

D₄×D₄₇

yes

characteristic

134

C₂×C₄

#10625352

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

188

—

139

C₈

—

188

—

140

C₈

—

188

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#13093995

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

188

—

149

C₄

D₈×D₄₇

yes

characteristic

150

C₄

#13094043

yes

characteristic

151

C₄

#13094044

yes

characteristic

152

C₄

#13094054

yes

characteristic

153

C₂

—

154

C₂

#15689342

yes

characteristic

155

C₂

#15689356

yes

characteristic

156

C₂

#15689383

yes

characteristic

	subgroup	G/H
center	C₂²	#13093995
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#486	C₂
socle	C₂²×C₄₇	#121
hypercenter	#486	D₄₇
hypocenter	C₄₇	#1682
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#13093995

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#486

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #15689330	C₂	semidirect
3. C₈×Dic₉₄	C₂	semidirect
4. #15688876	C₂	semidirect
5. #15688869	C₂	semidirect
86. C₄₇	#1682	semidirect

H ≅ G/N

type

1. #15689330

C₂

semidirect

3. C₈×Dic₉₄

C₂

semidirect

4. #15688876

C₂

semidirect

5. #15688869

C₂

semidirect

86. C₄₇

#1682

semidirect

Subgroups — #20242829

Conjugacy classes of subgroups

Complements