Group #20244112 — subgroups

Subgroups — #20244112

Order 6016.

248 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 121 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×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093935	C₂	1	1	yes	maximal, characteristic
4	#15688878	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
10	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
11	#13093935	—	2	2	no	—
12	C₂×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
13	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	#95×C₄₇	C₄	1	1	yes	characteristic
16	#95×C₄₇	C₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₄₇	—	2	4	no	—
23	C₄²×C₄₇	—	2	2	no	—
24	C₄²×C₄₇	—	2	2	no	—
25	C₄²×C₄₇	D₄	1	2	yes	—
26	C₄²×C₄₇	Q₈	1	2	yes	—
27	C₂×C₄₇⋊C₈	—	2	1	no	—
28	C₂×C₄₇⋊C₈	—	2	1	no	—
29	C₂×C₄₇⋊C₈	—	2	1	no	—
30	C₂×C₄₇⋊C₈	—	2	1	no	—
31	C₂×C₄₇⋊C₈	—	2	2	no	—
32	C₂×C₄₇⋊C₈	—	2	2	no	—
33	C₂×C₄₇⋊C₈	—	2	2	no	—
34	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
35	C₂×C₄₇⋊C₈	Q₈	1	2	yes	—
36	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	—	2	2	no	—
39	C₂×C₄×C₄₇	—	2	2	no	—
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₄₇	M₁₆	1	4	yes	—
47	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
53	C₄₇⋊C₈	—	2	2	no	—
54	C₄₇⋊C₈	—	2	2	no	—
55	C₄₇⋊C₈	—	4	2	no	—
56	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₄₇	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₄₇	C₂×M₁₆	1	2	yes	—
60	C₂²×C₄₇	#131	1	2	yes	—
61	C₄×C₄₇	—	2	2	no	—
62	C₄×C₄₇	—	2	2	no	—
63	C₄×C₄₇	—	2	4	no	—
64	C₄×C₄₇	—	4	2	no	—
65	C₄×C₄₇	#121	1	1	yes	characteristic
66	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
67	C₄×C₄₇	#125	1	1	yes	characteristic
68	C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
69	#1940	—	47	1	no	maximal, Sylow
70	C₂×C₄₇	#389	1	1	yes	characteristic
71	C₂×C₄₇	#423	1	1	yes	characteristic
72	C₂×C₄₇	#424	1	1	yes	characteristic
73	C₂×C₄₇	#435	1	2	yes	—
74	C₂×C₄₇	#446	1	2	yes	—
75	C₂×C₄⋊C₈	—	47	2	no	—
76	C₂×C₄⋊C₈	—	47	1	no	—
77	C₂×#97	—	47	1	no	—
78	C₄×C₄⋊C₄	D₄₇	1	1	yes	characteristic
79	#336	—	47	2	no	—
80	C₄₇	#1940	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	47	1	no	—
82	C₂²×C₈	—	47	1	no	—
83	C₂²×C₈	—	47	2	no	—
84	C₂×C₄⋊C₄	Dic₄₇	1	2	yes	—
85	C₂×C₄²	C₂×D₄₇	1	2	yes	—
86	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
87	C₄⋊C₈	—	94	4	no	—
88	C₄⋊C₈	—	94	2	no	—
89	#97	—	94	2	no	—
90	#95	Dic₄₇	1	1	yes	characteristic
91	#95	Dic₄₇	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
93	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
94	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
95	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
97	C₂×C₈	—	47	2	no	—
98	C₂×C₈	—	47	2	no	—
99	C₂×C₈	—	94	1	no	—
100	C₂×C₈	—	94	1	no	—
101	C₂×C₈	—	94	1	no	—
102	C₂×C₈	—	94	1	no	—
103	C₂×C₈	—	94	2	no	—
104	C₂×C₈	—	94	2	no	—
105	C₂×C₈	—	94	2	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	Dic₉₄	1	2	yes	—
110	C₄²	D₁₈₈	1	2	yes	—
111	C₂³	C₂²×Dic₄₇	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	#10625329	1	4	yes	—
120	C₂×C₄	C₄⋊Dic₄₇	1	2	yes	—
121	C₂×C₄	C₄⋊Dic₄₇	1	2	yes	—
122	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
123	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
124	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
125	C₂×C₄	#10625352	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
127	C₂×C₄	#10625354	1	1	yes	characteristic
128	C₈	—	94	2	no	—
129	C₈	—	94	2	no	—
130	C₈	—	188	2	no	—
131	C₂²	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
132	C₂²	D₄×Dic₄₇	1	1	yes	characteristic
133	C₂²	Q₈×Dic₄₇	1	1	yes	characteristic
134	C₂²	C₂×#10625329	1	2	yes	—
135	C₂²	#13094031	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	4	2	no	—
140	C₄	#13093999	1	1	yes	characteristic
141	C₄	#13094004	1	1	yes	characteristic
142	C₄	#13094015	1	1	yes	characteristic
143	C₄	#13094023	1	1	yes	characteristic
144	C₂	#15689022	1	1	yes	characteristic
145	C₂	#15689147	1	1	yes	characteristic
146	C₂	#15689150	1	1	yes	characteristic
147	C₂	#15689195	1	2	yes	—
148	C₂	#15689254	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×#13093935

C₂

yes

maximal, characteristic

#15688878

C₂

yes

maximal

C₄×C₄⋊C₄×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

#13093935

—

C₂×C₄⋊C₄×C₄₇

C₄

yes

—

C₂×C₄²×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

C₄

yes

characteristic

#95×C₄₇

C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

Q₈

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂×C₄₇⋊C₈

Q₈

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

M₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

C₄×D₄

yes

characteristic

C₂²×C₄₇

C₄×Q₈

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

—

C₂²×C₄₇

#131

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#121

yes

characteristic

C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₄×C₄₇

#125

yes

characteristic

C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#446

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₄×C₄⋊C₄

D₄₇

yes

characteristic

#336

—

C₄₇

#1940

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

Dic₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#95

Dic₄₇

yes

characteristic

#95

Dic₄₇

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

Dic₉₄

yes

—

110

C₄²

D₁₈₈

yes

—

111

C₂³

C₂²×Dic₄₇

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#10625329

yes

—

120

C₂×C₄

C₄⋊Dic₄₇

yes

—

121

C₂×C₄

C₄⋊Dic₄₇

yes

—

122

C₂×C₄

C₂×Dic₉₄

yes

characteristic

123

C₂×C₄

C₂×D₁₈₈

yes

characteristic

124

C₂×C₄

D₄×D₄₇

yes

characteristic

125

C₂×C₄

#10625352

yes

characteristic

126

C₂×C₄

Q₈×D₄₇

yes

characteristic

127

C₂×C₄

#10625354

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

188

—

131

C₂²

C₂×C₄⋊Dic₄₇

yes

characteristic

132

C₂²

D₄×Dic₄₇

yes

characteristic

133

C₂²

Q₈×Dic₄₇

yes

characteristic

134

C₂²

C₂×#10625329

yes

—

135

C₂²

#13094031

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#13093999

yes

characteristic

141

C₄

#13094004

yes

characteristic

142

C₄

#13094015

yes

characteristic

143

C₄

#13094023

yes

characteristic

144

C₂

#15689022

yes

characteristic

145

C₂

#15689147

yes

characteristic

146

C₂

#15689150

yes

characteristic

147

C₂

#15689195

yes

—

148

C₂

#15689254

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₇
commutator, socle	C₂³×C₄₇	C₂²×C₄
Fitting	C₄×C₄⋊C₄×C₄₇	C₂
hypercenter	C₄×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#1940
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₇

commutator, socle

C₂³×C₄₇

C₂²×C₄

Fitting

C₄×C₄⋊C₄×C₄₇

C₂

hypercenter

C₄×C₄⋊C₄

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
80. C₄₇	#1940	semidirect

H ≅ G/N

type

80. C₄₇

#1940

semidirect

Subgroups — #20244112

Conjugacy classes of subgroups

Complements