Group #20241731 — subgroups

Subgroups — #20241731

Order 6016.

236 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 77 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	#15689180	C₂	1	1	yes	maximal, characteristic
3	#15688848	C₂	1	4	yes	maximal
4	C₄₇×#432	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₉₄	C₄	1	1	yes	characteristic
6	C₄×Dic₉₄	—	2	1	no	—
7	C₄×Dic₉₄	—	2	1	no	—
8	C₄×Dic₉₄	—	2	1	no	—
9	C₄×Dic₉₄	—	2	1	no	—
10	C₂×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
11	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄×Dic₄₇	C₄	1	2	yes	—
13	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
14	C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	#13093935	—	2	1	no	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₄₇	—	2	1	no	—
18	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
19	#97×C₄₇	—	2	1	no	—
20	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
24	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
25	C₂×Dic₉₄	—	2	1	no	—
26	C₂×Dic₉₄	—	2	1	no	—
27	C₂×Dic₉₄	—	2	4	no	—
28	C₂×C₈×C₄₇	—	2	2	no	—
29	C₄²×C₄₇	D₄	1	1	yes	characteristic
30	C₄²×C₄₇	D₄	1	1	yes	characteristic
31	C₄²×C₄₇	D₄	1	1	yes	characteristic
32	C₄²×C₄₇	D₄	1	1	yes	characteristic
33	C₄⋊Dic₄₇	—	2	1	no	—
34	C₄⋊Dic₄₇	—	2	1	no	—
35	Dic₄₇⋊C₄	—	2	2	no	—
36	Dic₄₇⋊C₄	—	2	2	no	—
37	C₄×Dic₄₇	—	2	2	no	—
38	C₄×Dic₄₇	—	2	2	no	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	4	1	no	—
44	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₄₇	—	2	2	no	—
51	C₂×Dic₄₇	—	2	2	no	—
52	C₂×Dic₄₇	—	2	2	no	—
53	C₂×Dic₄₇	—	4	2	no	—
54	C₂×Dic₄₇	M₁₆	1	4	yes	—
55	Dic₉₄	—	2	4	no	—
56	Dic₉₄	—	2	4	no	—
57	C₈×C₄₇	—	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×M₁₆	1	2	yes	—
63	C₄×C₄₇	—	2	1	no	—
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	#101	1	2	yes	—
69	Dic₄₇	—	2	4	no	—
70	Dic₄₇	—	4	2	no	—
71	#1467	—	47	1	no	maximal, Sylow
72	C₂×C₄₇	—	2	2	no	—
73	C₂×C₄₇	#407	1	1	yes	characteristic
74	C₂×C₄₇	#410	1	1	yes	characteristic
75	C₂×C₄₇	C₂×#101	1	1	yes	characteristic
76	C₂×C₄×Q₈	—	47	1	no	—
77	#432	—	47	1	no	—
78	#432	D₄₇	1	1	yes	characteristic
79	#324	—	47	4	no	—
80	C₄₇	#1467	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	—	47	1	no	—
82	C₄×Q₈	—	94	1	no	—
83	C₄×Q₈	—	94	1	no	—
84	C₄×Q₈	—	94	1	no	—
85	C₄×Q₈	—	94	1	no	—
86	C₂×C₄⋊C₄	—	47	1	no	—
87	C₂×C₄⋊C₄	—	47	2	no	—
88	C₂×C₄²	—	47	2	no	—
89	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
90	C₄⋊C₈	—	2	1	no	—
91	C₄⋊C₈	—	94	1	no	—
92	C₂²⋊C₈	—	47	2	no	—
93	C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
94	#97	—	2	1	no	—
95	#97	—	94	1	no	—
96	C₂×Q₈	—	94	4	no	—
97	C₂×Q₈	—	94	1	no	—
98	C₂×Q₈	—	94	1	no	—
99	C₂²×C₄	—	47	2	no	—
100	C₂²×C₄	—	47	2	no	—
101	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
102	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	94	2	no	—
106	C₄⋊C₄	—	94	1	no	—
107	C₄⋊C₄	—	94	1	no	—
108	C₄⋊C₄	—	94	2	no	—
109	C₄⋊C₄	—	94	2	no	—
110	C₄²	—	94	2	no	—
111	C₄²	—	94	2	no	—
112	C₄²	D₁₈₈	1	1	yes	characteristic
113	C₄²	D₁₈₈	1	1	yes	characteristic
114	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
115	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
116	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
117	Q₈	—	94	4	no	—
118	Q₈	—	94	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	47	4	no	—
123	C₂×C₄	—	94	2	no	—
124	C₂×C₄	—	94	2	no	—
125	C₂×C₄	—	94	2	no	—
126	C₂×C₄	—	188	2	no	—
127	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
128	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
129	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
130	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
131	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
133	C₈	—	4	2	no	—
134	C₈	—	188	2	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
138	C₂²	M₁₆×D₄₇	1	2	yes	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	94	4	no	—
145	C₄	—	188	2	no	—
146	C₄	#13093942	1	2	yes	—
147	C₂	—	2	2	no	—
148	C₂	#15689086	1	1	yes	characteristic
149	C₂	#15689096	1	1	yes	characteristic
150	C₂	C₂×#13093942	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₉₄

C₂

yes

maximal, characteristic

#15689180

C₂

yes

maximal, characteristic

#15688848

C₂

yes

maximal

C₄₇×#432

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₄

yes

characteristic

C₄×Dic₉₄

—

C₄×Dic₉₄

—

C₄×Dic₉₄

—

C₄×Dic₉₄

—

C₂×C₄⋊Dic₄₇

C₄

yes

characteristic

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093935

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

C₂²

yes

—

#97×C₄₇

—

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×Dic₉₄

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

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

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

M₁₆

yes

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#101

yes

—

Dic₄₇

—

Dic₄₇

—

#1467

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄×Q₈

—

#432

—

#432

D₄₇

yes

characteristic

#324

—

C₄₇

#1467

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

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₄₇

yes

—

#97

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₄×D₄₇

yes

characteristic

102

C₂²×C₄

C₄×D₄₇

yes

characteristic

103

C₂²×C₄

C₂²×D₄₇

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₄²

D₁₈₈

yes

characteristic

113

C₄²

D₁₈₈

yes

characteristic

114

C₄²

C₂²⋊D₄₇

yes

characteristic

115

C₄²

C₂²⋊D₄₇

yes

characteristic

116

C₂³

C₂×C₄×D₄₇

yes

characteristic

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

188

—

127

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

128

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

129

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

130

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

131

C₂×C₄

C₂×D₁₈₈

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

133

C₈

—

134

C₈

—

188

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

138

C₂²

M₁₆×D₄₇

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

188

—

146

C₄

#13093942

yes

—

147

C₂

—

148

C₂

#15689086

yes

characteristic

149

C₂

#15689096

yes

characteristic

150

C₂

C₂×#13093942

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂³×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#432	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#432	D₄₇
hypocenter	C₄₇	#1467
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×(C₂×D₄₇)⋊C₄

commutator

C₂³×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#432

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
80. C₄₇	#1467	semidirect

H ≅ G/N

type

80. C₄₇

#1467

semidirect

Subgroups — #20241731

Conjugacy classes of subgroups

Complements