Group #20242813 — subgroups

Subgroups — #20242813

Order 6016.

166 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 55 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	#15689475	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689403	C₂	1	1	yes	maximal, characteristic
3	D₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#15688873	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688869	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688868	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#461	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄₇⋊C₈	—	2	1	no	—
9	#13094024	C₂²	1	1	yes	characteristic
10	#13093957	—	2	1	no	—
11	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#13093947	—	2	1	no	—
14	#13093947	—	2	1	no	—
15	#13093945	—	2	1	no	—
16	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
17	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₄₇	—	2	1	no	—
19	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
22	#103×C₄₇	—	2	1	no	—
23	C₂×Q₈×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	C₂×Dic₉₄	—	2	1	no	—
27	SD₁₆×C₄₇	—	4	2	no	—
28	C₂×C₈×C₄₇	—	2	1	no	—
29	C₄⋊C₄×C₄₇	—	2	1	no	—
30	C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
32	C₂²⋊C₄×C₄₇	—	2	1	no	—
33	C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	C₄⋊Dic₄₇	—	2	1	no	—
35	C₄⋊Dic₄₇	—	2	1	no	—
36	C₂×C₄₇⋊C₈	—	2	1	no	—
37	C₂×C₄₇⋊C₈	—	2	1	no	—
38	C₂×C₄₇⋊C₈	—	4	2	no	—
39	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
41	C₂³×C₄₇	—	2	1	no	—
42	Q₈×C₄₇	—	4	1	no	—
43	Q₈×C₄₇	—	4	1	no	—
44	D₄×C₄₇	—	2	1	no	—
45	D₄×C₄₇	SD₁₆	1	2	yes	—
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	4	1	no	—
48	C₂×C₄×C₄₇	—	4	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	—	4	1	no	—
54	Dic₉₄	—	4	1	no	—
55	Dic₉₄	—	4	1	no	—
56	C₈×C₄₇	—	4	1	no	—
57	C₄₇⋊C₈	—	2	1	no	—
58	C₄₇⋊C₈	—	2	1	no	—
59	C₄₇⋊C₈	—	4	1	no	—
60	C₄₇⋊C₈	—	4	1	no	—
61	C₂²×C₄₇	—	2	2	no	—
62	C₂²×C₄₇	—	4	1	no	—
63	C₂²×C₄₇	—	4	1	no	—
64	C₂²×C₄₇	#121	1	1	yes	characteristic
65	C₄×C₄₇	—	2	1	no	—
66	C₄×C₄₇	—	4	1	no	—
67	C₄×C₄₇	—	8	1	no	—
68	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₄₇	#135	1	1	yes	characteristic
70	C₄×C₄₇	#135	1	1	yes	characteristic
71	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
72	Dic₄₇	—	8	1	no	—
73	#1679	—	47	1	no	maximal, Sylow
74	C₂×C₄₇	—	4	2	no	—
75	C₂×C₄₇	#463	1	1	yes	characteristic
76	C₂×C₄₇	#465	1	1	yes	characteristic
77	C₂×C₄₇	#473	1	1	yes	characteristic
78	#487	—	47	1	no	—
79	#474	—	47	1	no	—
80	#461	D₄₇	1	1	yes	characteristic
81	C₈×D₄	—	47	1	no	—
82	#334	—	47	1	no	—
83	#332	—	47	1	no	—
84	#332	—	47	1	no	—
85	C₄₇	#1679	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	2	1	no	—
87	C₂²×C₈	—	94	1	no	—
88	C₄⋊Q₈	—	47	1	no	—
89	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
90	C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
91	#107	—	94	1	no	—
92	C₄⋊C₈	—	47	1	no	—
93	C₄⋊C₈	—	47	1	no	—
94	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
95	#103	—	2	1	no	—
96	#103	—	94	1	no	—
97	#103	—	94	1	no	—
98	#102	—	94	1	no	—
99	C₂²⋊C₈	—	94	1	no	—
100	C₄×C₈	—	47	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	94	1	no	—
103	C₂×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
104	C₂²×C₄	—	2	1	no	—
105	SD₁₆	—	4	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	47	1	no	—
108	C₂×C₈	—	47	1	no	—
109	C₂×C₈	—	94	1	no	—
110	C₂×C₈	—	94	1	no	—
111	C₂×C₈	—	188	2	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	94	1	no	—
115	C₄⋊C₄	—	94	1	no	—
116	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
117	C₂²⋊C₄	—	2	1	no	—
118	C₄²	C₂²×D₄₇	1	1	yes	characteristic
119	C₂³	—	2	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	4	1	no	—
122	Q₈	—	188	1	no	—
123	Q₈	—	188	1	no	—
124	D₄	—	2	1	no	—
125	D₄	#10625335	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	188	1	no	—
131	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
132	C₂×C₄	#10625352	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
134	C₈	—	4	1	no	—
135	C₈	—	94	1	no	—
136	C₈	—	94	1	no	—
137	C₈	—	188	1	no	—
138	C₈	—	188	1	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	#13093996	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	8	1	no	—
146	C₄	—	376	1	no	—
147	C₄	C₂×#10625335	1	1	yes	characteristic
148	C₄	#13094044	1	1	yes	characteristic
149	C₄	#13094045	1	1	yes	characteristic
150	C₄	#13094051	1	1	yes	characteristic
151	C₂	—	4	2	no	—
152	C₂	#15689340	1	1	yes	characteristic
153	C₂	#15689350	1	1	yes	characteristic
154	C₂	#15689396	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689475

C₂

yes

maximal, characteristic, complemented

#15689403

C₂

yes

maximal, characteristic

D₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic

#15688873

C₂

yes

maximal, characteristic, complemented

#15688869

C₂

yes

maximal, characteristic, complemented

#15688868

C₂

yes

maximal, characteristic, complemented

C₄₇×#461

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

—

#13094024

C₂²

yes

characteristic

#13093957

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

—

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄×D₄×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

#103×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂×Dic₉₄

—

SD₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂³×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×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

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1679

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

D₄₇

yes

characteristic

C₈×D₄

—

#334

—

#332

—

#332

—

C₄₇

#1679

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

C₄×D₄

C₂×D₄₇

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

—

—

—

—

C₂²⋊C₈

—

100

C₄×C₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×D₄

C₂²⋊D₄₇

yes

characteristic

104

C₂²×C₄

—

105

SD₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

188

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

117

C₂²⋊C₄

—

118

C₄²

C₂²×D₄₇

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

188

—

123

Q₈

—

188

—

124

D₄

—

125

D₄

#10625335

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

188

—

131

C₂×C₄

D₄×D₄₇

yes

characteristic

132

C₂×C₄

#10625352

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

188

—

138

C₈

—

188

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

#13093996

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

376

—

147

C₄

C₂×#10625335

yes

characteristic

148

C₄

#13094044

yes

characteristic

149

C₄

#13094045

yes

characteristic

150

C₄

#13094051

yes

characteristic

151

C₂

—

152

C₂

#15689340

yes

characteristic

153

C₂

#15689350

yes

characteristic

154

C₂

#15689396

yes

characteristic

	subgroup	G/H
center	C₂²	#13093996
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#461	C₂
socle	C₂²×C₄₇	#121
hypercenter	#461	D₄₇
hypocenter	C₄₇	#1679
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

subgroup

G/H

center

C₂²

#13093996

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#461

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #461

N	H ≅ G/N	type
1. #15689475	C₂	semidirect
4. #15688873	C₂	semidirect
5. #15688869	C₂	semidirect
6. #15688868	C₂	semidirect
85. C₄₇	#1679	semidirect

H ≅ G/N

type

1. #15689475

C₂

semidirect

4. #15688873

C₂

semidirect

5. #15688869

C₂

semidirect

6. #15688868

C₂

semidirect

85. C₄₇

#1679

semidirect

Subgroups — #20242813

Conjugacy classes of subgroups

Complements