Group #20252471 — subgroups

Subgroups — #20252471

Order 6016.

356 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 177 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×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	#15689253	C₂	1	4	yes	maximal, complemented
3	#15689250	C₂	1	4	yes	maximal, complemented
4	#15689177	C₂	1	4	yes	maximal
5	#15689176	C₂	1	1	yes	maximal, characteristic
6	C₄₇×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₉₄	C₄	1	1	yes	characteristic
8	C₄×Dic₉₄	C₄	1	2	yes	—
9	C₄×Dic₉₄	C₄	1	2	yes	—
10	C₄×Dic₉₄	C₂²	1	2	yes	—
11	C₄×Dic₉₄	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
13	C₂×Dic₄₇⋊C₄	C₄	1	2	yes	—
14	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
15	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	Dic₄₇⋊C₈	C₂²	1	8	yes	—
17	C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
18	#13093934	C₂²	1	8	yes	—
19	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
20	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
22	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
23	C₄×C₈×C₄₇	C₂²	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₉₄	—	2	4	no	—
30	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
31	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₄₇	C₂³	1	4	yes	—
33	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	C₄²×C₄₇	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
39	Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
40	Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
41	C₄×Dic₄₇	C₂×C₄	1	4	yes	—
42	C₄×Dic₄₇	C₂³	1	4	yes	—
43	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
44	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	4	1	no	—
47	C₂×C₄×C₄₇	M₁₆	1	4	yes	—
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	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₄₇	—	2	4	no	—
55	C₂×Dic₄₇	—	4	2	no	—
56	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
57	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
58	Dic₉₄	—	2	4	no	—
59	Dic₉₄	—	2	4	no	—
60	C₈×C₄₇	—	2	4	no	—
61	C₄₇⋊C₈	—	2	4	no	—
62	C₂²×C₄₇	—	2	1	no	—
63	C₂²×C₄₇	—	2	1	no	—
64	C₂²×C₄₇	C₂×M₁₆	1	2	yes	—
65	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₄₇	—	2	1	no	—
67	C₄×C₄₇	—	2	1	no	—
68	C₄×C₄₇	—	2	1	no	—
69	C₄×C₄₇	C₂×M₁₆	1	2	yes	—
70	C₄×C₄₇	C₂×M₁₆	1	2	yes	—
71	C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
72	Dic₄₇	—	2	4	no	—
73	Dic₄₇	—	2	4	no	—
74	#2983	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	—	2	2	no	—
76	C₂×C₄₇	#519	1	1	yes	characteristic
77	C₂×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
78	C₂×C₄₇	#568	1	1	yes	characteristic
79	C₂×C₄×Q₈	—	47	1	no	—
80	#446	—	47	4	no	—
81	#446	—	47	4	no	—
82	#432	—	47	4	no	—
83	#431	—	47	1	no	—
84	#431	D₄₇	1	1	yes	characteristic
85	C₄₇	#2983	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	47	1	no	—
87	C₄×Q₈	—	47	2	no	—
88	C₄×Q₈	—	47	2	no	—
89	C₄×Q₈	—	47	2	no	—
90	C₄×Q₈	—	47	2	no	—
91	C₂×C₄⋊C₄	—	47	2	no	—
92	C₂×C₄⋊C₄	—	47	1	no	—
93	C₂×C₄²	—	47	2	no	—
94	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
95	C₄⋊C₈	—	47	8	no	—
96	C₄⋊C₈	—	47	2	no	—
97	C₄⋊C₈	C₂×D₄₇	1	2	yes	—
98	C₂²⋊C₈	—	47	2	no	—
99	C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
100	#97	—	47	8	no	—
101	C₄×C₈	—	47	2	no	—
102	C₄×C₈	C₂×D₄₇	1	2	yes	—
103	C₂×Q₈	—	47	2	no	—
104	C₂×Q₈	—	47	2	no	—
105	C₂×Q₈	—	94	4	no	—
106	C₂²×C₄	—	47	2	no	—
107	C₂²×C₄	—	47	2	no	—
108	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
109	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
111	C₂×C₈	—	47	4	no	—
112	C₂×C₈	C₂²×D₄₇	1	4	yes	—
113	C₄⋊C₄	—	47	4	no	—
114	C₄⋊C₄	—	47	4	no	—
115	C₄⋊C₄	—	47	2	no	—
116	C₄⋊C₄	—	47	2	no	—
117	C₄²	—	47	4	no	—
118	C₄²	—	47	4	no	—
119	C₄²	C₄×D₄₇	1	1	yes	characteristic
120	C₄²	C₄×D₄₇	1	1	yes	characteristic
121	C₄²	C₂²×D₄₇	1	1	yes	characteristic
122	C₄²	C₂²×D₄₇	1	1	yes	characteristic
123	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
124	Q₈	—	94	4	no	—
125	Q₈	—	94	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	47	4	no	—
129	C₂×C₄	—	47	4	no	—
130	C₂×C₄	—	94	4	no	—
131	C₂×C₄	—	188	2	no	—
132	C₂×C₄	#10625324	1	4	yes	—
133	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	94	4	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#10625324	1	2	yes	—
144	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	94	4	no	—
149	C₄	—	94	4	no	—
150	C₄	C₂×#10625324	1	2	yes	—
151	C₄	C₂×#10625324	1	2	yes	—
152	C₄	#13094071	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#15689600	1	1	yes	characteristic
155	C₂	C₂²×#10625324	1	1	yes	characteristic
156	C₂	#15689889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₉₄

C₂

yes

maximal, characteristic

#15689253

C₂

yes

maximal, complemented

#15689250

C₂

yes

maximal, complemented

#15689177

C₂

yes

maximal

#15689176

C₂

yes

maximal, characteristic

C₄₇×#431

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₄

yes

characteristic

C₄×Dic₉₄

C₄

yes

—

C₄×Dic₉₄

C₄

yes

—

C₄×Dic₉₄

C₂²

yes

—

C₄×Dic₉₄

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₄

yes

characteristic

C₂×Dic₄₇⋊C₄

C₄

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

Dic₄₇⋊C₈

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093934

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₂²⋊C₈×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂×C₄

yes

—

C₂×Dic₉₄

C₂×C₄

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

Dic₄₇⋊C₄

C₂×C₄

yes

—

Dic₄₇⋊C₄

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

M₁₆

yes

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×M₁₆

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×M₁₆

yes

—

C₄×C₄₇

C₂×M₁₆

yes

—

C₄×C₄₇

2₋¹⁺⁴

yes

—

Dic₄₇

—

Dic₄₇

—

#2983

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#519

yes

characteristic

C₂×C₄₇

C₂²×M₁₆

yes

characteristic

C₂×C₄₇

#568

yes

characteristic

C₂×C₄×Q₈

—

—

—

—

—

D₄₇

yes

characteristic

C₄₇

#2983

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₄₇

yes

—

100

#97

—

101

C₄×C₈

—

102

C₄×C₈

C₂×D₄₇

yes

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₄₇

yes

characteristic

109

C₂²×C₄

C₄×D₄₇

yes

characteristic

110

C₂²×C₄

C₂²×D₄₇

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₄₇

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₄×D₄₇

yes

characteristic

120

C₄²

C₄×D₄₇

yes

characteristic

121

C₄²

C₂²×D₄₇

yes

characteristic

122

C₄²

C₂²×D₄₇

yes

characteristic

123

C₂³

C₂×C₄×D₄₇

yes

characteristic

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

188

—

132

C₂×C₄

#10625324

yes

—

133

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

134

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

135

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

136

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

137

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

138

C₂×C₄

C₂³×D₄₇

yes

characteristic

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×#10625324

yes

—

144

C₂²

C₂²×C₄×D₄₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

C₂×#10625324

yes

—

151

C₄

C₂×#10625324

yes

—

152

C₄

#13094071

yes

—

153

C₂

—

154

C₂

#15689600

yes

characteristic

155

C₂

C₂²×#10625324

yes

characteristic

156

C₂

#15689889

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#431	C₂
hypercenter	#431	D₄₇
hypocenter	C₄₇	#2983
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₇

commutator, socle

C₂²×C₄₇

C₂³×C₄

Fitting

C₄₇×#431

C₂

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #15689253	C₂	semidirect
3. #15689250	C₂	semidirect
85. C₄₇	#2983	semidirect

H ≅ G/N

type

2. #15689253

C₂

semidirect

3. #15689250

C₂

semidirect

85. C₄₇

#2983

semidirect

Subgroups — #20252471

Conjugacy classes of subgroups

Complements