Group #20241242 — subgroups

Subgroups — #20241242

Order 6016.

226 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 61 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	#15688986	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#407	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
5	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
6	C₂×C₄×Dic₄₇	C₄	1	2	yes	complemented
7	C₄₇⋊#95	—	2	1	no	—
8	C₄₇⋊#95	—	2	1	no	—
9	C₄₇⋊#95	C₄	1	2	yes	complemented
10	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₄₇	—	2	1	no	—
12	C₂²⋊C₈×C₄₇	—	2	1	no	—
13	C₂²⋊C₈×C₄₇	C₄	1	2	yes	complemented
14	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	—	4	2	no	—
16	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	Q₈	1	1	yes	characteristic
22	C₂²×Dic₄₇	—	2	2	no	—
23	C₂²×Dic₄₇	—	2	2	no	—
24	M₁₆×C₄₇	—	4	2	no	—
25	C₂×C₈×C₄₇	—	2	2	no	—
26	C₂²⋊Dic₄₇	—	2	4	no	—
27	C₂²⋊Dic₄₇	—	4	2	no	—
28	C₄×Dic₄₇	—	4	2	no	—
29	C₂³×C₄₇	—	2	2	no	—
30	C₂³×C₄₇	—	2	2	no	—
31	C₂³×C₄₇	—	4	1	no	—
32	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂³×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	—	2	2	no	—
36	C₂×C₄×C₄₇	—	2	2	no	—
37	C₂×C₄×C₄₇	—	4	1	no	—
38	C₂×C₄×C₄₇	—	4	1	no	—
39	C₂×C₄×C₄₇	—	4	1	no	—
40	C₂×C₄×C₄₇	—	4	1	no	—
41	C₂×C₄×C₄₇	C₄²	1	1	yes	characteristic
42	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×Dic₄₇	—	2	4	no	—
46	C₂×Dic₄₇	—	4	2	no	—
47	C₂×Dic₄₇	—	4	2	no	—
48	C₂×Dic₄₇	—	4	2	no	—
49	C₂×Dic₄₇	—	4	2	no	—
50	C₂×Dic₄₇	—	4	2	no	—
51	C₈×C₄₇	—	4	2	no	—
52	C₂²×C₄₇	—	2	1	no	—
53	C₂²×C₄₇	—	2	1	no	—
54	C₂²×C₄₇	—	2	1	no	—
55	C₂²×C₄₇	—	2	1	no	—
56	C₂²×C₄₇	—	2	1	no	—
57	C₂²×C₄₇	—	2	1	no	—
58	C₂²×C₄₇	—	2	1	no	—
59	C₂²×C₄₇	—	4	4	no	—
60	C₂²×C₄₇	#95	1	1	yes	characteristic
61	C₂²×C₄₇	#99	1	2	yes	—
62	C₂²×C₄₇	#104	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	Dic₄₇	—	4	4	no	—
68	Dic₄₇	—	8	2	no	—
69	#1284	—	47	1	no	maximal, Sylow
70	C₂×C₄₇	—	2	2	no	—
71	C₂×C₄₇	—	2	2	no	—
72	C₂×C₄₇	—	4	1	no	—
73	C₂×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
74	C₂×C₄₇	#339	1	1	yes	characteristic
75	C₂×C₄₇	#342	1	1	yes	characteristic
76	#407	D₄₇	1	1	yes	characteristic
77	#381	—	47	1	no	—
78	C₄×C₂²⋊C₄	—	47	1	no	—
79	C₄₇	#1284	1	1	yes	characteristic, complemented, Sylow
80	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
81	C₂×M₁₆	—	2	1	no	—
82	C₂×C₂²⋊C₄	—	94	1	no	—
83	C₂×C₂²⋊C₄	—	94	1	no	—
84	C₂×C₄²	—	47	2	no	—
85	C₂²⋊C₈	—	2	1	no	—
86	C₂²⋊C₈	Dic₄₇	1	2	yes	complemented
87	#95	—	47	2	no	—
88	#95	—	94	1	no	—
89	#95	—	94	1	no	—
90	C₂⁴	D₁₈₈	1	1	yes	characteristic
91	C₂²×C₄	—	4	2	no	—
92	C₂²×C₄	—	94	2	no	—
93	C₂²×C₄	—	94	2	no	—
94	C₂²×C₄	Dic₉₄	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
96	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
100	M₁₆	—	4	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂²⋊C₄	—	94	4	no	—
103	C₂²⋊C₄	—	188	2	no	—
104	C₄²	—	188	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	—	2	2	no	—
107	C₂³	—	4	1	no	—
108	C₂³	C₄⋊Dic₄₇	1	1	yes	characteristic
109	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
110	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	94	4	no	—
118	C₂×C₄	—	188	2	no	—
119	C₂×C₄	—	188	2	no	—
120	C₂×C₄	—	188	2	no	—
121	C₂×C₄	—	188	2	no	—
122	C₂×C₄	—	188	2	no	—
123	C₂×C₄	C₄×Dic₄₇	1	1	yes	characteristic
124	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
125	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
126	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	4	4	no	—
136	C₂²	C₄₇⋊#95	1	1	yes	characteristic
137	C₂²	#13093938	1	2	yes	—
138	C₂²	#13093950	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	—	376	2	no	—
144	C₄	—	188	4	no	—
145	C₂	—	2	2	no	—
146	C₂	—	2	2	no	—
147	C₂	—	4	1	no	—
148	C₂	#15688880	1	1	yes	characteristic
149	C₂	#15688884	1	1	yes	characteristic
150	C₂	#15688889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15688986

C₂

yes

maximal, characteristic

C₄×C₂²⋊Dic₄₇

C₂

yes

maximal, characteristic

C₄₇×#407

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₄₇

—

C₂×C₂²⋊Dic₄₇

—

C₂×C₄×Dic₄₇

C₄

yes

complemented

C₄₇⋊#95

—

C₄₇⋊#95

—

C₄₇⋊#95

C₄

yes

complemented

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

C₄

yes

complemented

C₂⁴×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

Q₈

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#95

yes

characteristic

C₂²×C₄₇

#99

yes

—

C₂²×C₄₇

#104

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

#1284

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

characteristic

#381

—

C₄×C₂²⋊C₄

—

C₄₇

#1284

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂×M₁₆

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₄₇

yes

—

—

—

C₂⁴

D₁₈₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₉₄

yes

characteristic

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

100

M₁₆

—

101

C₂×C₈

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

188

—

104

C₄²

—

188

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₄⋊Dic₄₇

yes

characteristic

109

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

110

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

188

—

119

C₂×C₄

—

188

—

120

C₂×C₄

—

188

—

121

C₂×C₄

—

188

—

122

C₂×C₄

—

188

—

123

C₂×C₄

C₄×Dic₄₇

yes

characteristic

124

C₂×C₄

Dic₄₇⋊C₄

yes

characteristic

125

C₂×C₄

Dic₄₇⋊C₄

yes

characteristic

126

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₄₇⋊#95

yes

characteristic

137

C₂²

#13093938

yes

—

138

C₂²

#13093950

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

376

—

144

C₄

—

188

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#15688880

yes

characteristic

149

C₂

#15688884

yes

characteristic

150

C₂

#15688889

yes

characteristic

	subgroup	G/H
center	C₂²	C₄₇⋊#95
commutator	C₂×C₄×C₄₇	C₄²
Frattini	C₂³×C₄	C₂×D₄₇
Fitting	C₄₇×#407	C₂
socle	C₂²×C₄₇	#95
hypercenter	#407	D₄₇
hypocenter	C₄₇	#1284
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₄₇⋊#95

commutator

C₂×C₄×C₄₇

C₄²

Frattini

C₂³×C₄

C₂×D₄₇

Fitting

C₄₇×#407

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
6. C₂×C₄×Dic₄₇	C₄	semidirect
9. C₄₇⋊#95	C₄	semidirect
13. C₂²⋊C₈×C₄₇	C₄	semidirect
79. C₄₇	#1284	semidirect
86. C₂²⋊C₈	Dic₄₇	semidirect

H ≅ G/N

type

6. C₂×C₄×Dic₄₇

C₄

semidirect

9. C₄₇⋊#95

C₄

semidirect

13. C₂²⋊C₈×C₄₇

C₄

semidirect

79. C₄₇

#1284

semidirect

86. C₂²⋊C₈

Dic₄₇

semidirect

Subgroups — #20241242

Conjugacy classes of subgroups

Complements