Group #20241513 — subgroups

Subgroups — #20241513

Order 6016.

232 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 57 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×#13093939	C₂	1	2	yes	maximal
2	C₄₇×#390	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
4	#13093939	—	2	2	no	—
5	#13093939	C₄	1	4	yes	complemented
6	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic, complemented
8	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
9	C₂×C₄²×C₄₇	C₄	1	1	yes	characteristic, complemented
10	C₂⁴×C₄₇	D₄	1	2	yes	—
11	C₂×D₄×C₄₇	—	4	1	no	—
12	C₂×D₄×C₄₇	—	4	1	no	—
13	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
14	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
15	C₂×D₄×C₄₇	Q₈	1	1	yes	characteristic
16	C₂²×C₄×C₄₇	—	2	1	no	—
17	C₂²×C₄×C₄₇	—	2	1	no	—
18	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₄₇	—	4	2	no	—
20	C₄⋊C₄×C₄₇	—	4	1	no	—
21	C₂²⋊C₄×C₄₇	—	4	4	no	—
22	C₄²×C₄₇	—	4	1	no	—
23	C₂²⋊Dic₄₇	—	2	4	no	—
24	C₂²⋊Dic₄₇	—	4	2	no	—
25	C₂³×C₄₇	—	2	2	no	—
26	C₂³×C₄₇	—	2	2	no	—
27	C₂³×C₄₇	—	4	2	no	—
28	C₂³×C₄₇	—	4	2	no	—
29	C₂³×C₄₇	C₂²⋊C₄	1	2	yes	—
30	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂³×C₄₇	C₄⋊C₄	1	2	yes	—
32	D₄×C₄₇	—	4	2	no	—
33	D₄×C₄₇	—	4	1	no	—
34	D₄×C₄₇	—	4	1	no	—
35	C₂×C₄×C₄₇	—	2	1	no	—
36	C₂×C₄×C₄₇	—	2	1	no	—
37	C₂×C₄×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	—	4	1	no	—
39	C₂×C₄×C₄₇	—	4	1	no	—
40	C₂×C₄×C₄₇	—	4	2	no	—
41	C₂×C₄×C₄₇	—	4	1	no	—
42	C₂×C₄×C₄₇	C₄²	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₄₇	—	4	4	no	—
45	C₂×Dic₄₇	—	8	2	no	—
46	C₂×Dic₄₇	—	8	2	no	—
47	C₂²×C₄₇	—	2	1	no	—
48	C₂²×C₄₇	—	2	1	no	—
49	C₂²×C₄₇	—	2	2	no	—
50	C₂²×C₄₇	—	2	2	no	—
51	C₂²×C₄₇	—	4	2	no	—
52	C₂²×C₄₇	—	4	2	no	—
53	C₂²×C₄₇	—	4	2	no	—
54	C₂²×C₄₇	—	4	2	no	—
55	C₂²×C₄₇	—	8	2	no	—
56	C₂²×C₄₇	#95	1	1	yes	characteristic
57	C₂²×C₄₇	#99	1	1	yes	characteristic
58	C₂²×C₄₇	#99	1	1	yes	characteristic
59	C₄×C₄₇	—	2	1	no	—
60	C₄×C₄₇	—	2	1	no	—
61	C₄×C₄₇	—	4	2	no	—
62	C₄×C₄₇	—	8	1	no	—
63	Dic₄₇	—	8	4	no	—
64	#1393	—	47	1	no	maximal, Sylow
65	C₂×C₄₇	—	2	1	no	—
66	C₂×C₄₇	—	2	1	no	—
67	C₂×C₄₇	—	4	2	no	—
68	C₂×C₄₇	—	4	2	no	—
69	C₂×C₄₇	#342	1	1	yes	characteristic
70	C₂×C₄₇	#353	1	1	yes	characteristic
71	C₂×C₄₇	#354	1	1	yes	characteristic
72	C₂×#99	—	47	2	no	—
73	#390	D₄₇	1	1	yes	characteristic
74	C₄₇	#1393	1	1	yes	characteristic, complemented, Sylow
75	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
76	C₂×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic, complemented
77	C₂×C₂²⋊C₄	—	2	2	no	—
78	C₂×C₂²⋊C₄	—	94	2	no	—
79	C₂×C₄²	Dic₄₇	1	1	yes	characteristic, complemented
80	#99	—	47	4	no	—
81	#99	—	94	2	no	—
82	C₂⁴	C₂²⋊D₄₇	1	2	yes	—
83	C₂×D₄	—	4	1	no	—
84	C₂×D₄	—	4	1	no	—
85	C₂×D₄	Dic₉₄	1	1	yes	characteristic
86	C₂×D₄	C₄×D₄₇	1	2	yes	—
87	C₂×D₄	D₁₈₈	1	1	yes	characteristic
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	188	2	no	—
91	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
92	C₄⋊C₄	—	4	1	no	—
93	C₂²⋊C₄	—	4	4	no	—
94	C₂²⋊C₄	—	94	4	no	—
95	C₂²⋊C₄	—	188	2	no	—
96	C₄²	—	4	1	no	—
97	C₂³	—	2	2	no	—
98	C₂³	—	2	2	no	—
99	C₂³	—	4	2	no	—
100	C₂³	—	4	2	no	—
101	C₂³	Dic₄₇⋊C₄	1	2	yes	—
102	C₂³	(C₂×D₄₇)⋊C₄	1	2	yes	—
103	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
104	D₄	—	4	1	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	376	2	no	—
115	C₂×C₄	—	376	2	no	—
116	C₂×C₄	—	188	4	no	—
117	C₂×C₄	C₄×Dic₄₇	1	1	yes	characteristic
118	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
119	C₂²	—	2	1	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	4	2	no	—
124	C₂²	—	4	2	no	—
125	C₂²	—	4	2	no	—
126	C₂²	—	4	2	no	—
127	C₂²	—	8	2	no	—
128	C₂²	C₄₇⋊#95	1	1	yes	characteristic
129	C₂²	#13093939	1	1	yes	characteristic
130	C₂²	#13093939	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	2	no	—
134	C₄	—	8	1	no	—
135	C₄	—	376	4	no	—
136	C₂	—	2	1	no	—
137	C₂	—	2	1	no	—
138	C₂	—	4	2	no	—
139	C₂	—	4	2	no	—
140	C₂	#15688890	1	1	yes	characteristic
141	C₂	#15688916	1	1	yes	characteristic
142	C₂	#15688918	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093939

C₂

yes

maximal

C₄₇×#390

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₄₇

—

#13093939

—

#13093939

C₄

yes

complemented

C₂²×D₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₄

yes

characteristic, complemented

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₄

yes

characteristic, complemented

C₂⁴×C₄₇

D₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

D₄

yes

characteristic

C₂×D₄×C₄₇

Q₈

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

C₄⋊C₄

yes

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

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

C₄⋊C₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#1393

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×#99

—

#390

D₄₇

yes

characteristic

C₄₇

#1393

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×D₄₇

yes

characteristic

C₂×C₄⋊C₄

Dic₄₇

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

Dic₄₇

yes

characteristic, complemented

#99

—

#99

—

C₂⁴

C₂²⋊D₄₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₉₄

yes

characteristic

C₂×D₄

C₄×D₄₇

yes

—

C₂×D₄

D₁₈₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

188

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

188

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

Dic₄₇⋊C₄

yes

—

102

C₂³

(C₂×D₄₇)⋊C₄

yes

—

103

C₂³

C₂²⋊Dic₄₇

yes

characteristic

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

376

—

115

C₂×C₄

—

376

—

116

C₂×C₄

—

188

—

117

C₂×C₄

C₄×Dic₄₇

yes

characteristic

118

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

C₄₇⋊#95

yes

characteristic

129

C₂²

#13093939

yes

characteristic

130

C₂²

#13093939

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

376

—

136

C₂

—

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

#15688890

yes

characteristic

141

C₂

#15688916

yes

characteristic

142

C₂

#15688918

yes

characteristic

	subgroup	G/H
center	C₂²	#13093939
commutator	C₂×C₄×C₄₇	C₄²
Frattini	C₂²×D₄	C₂×D₄₇
Fitting	C₄₇×#390	C₂
socle	C₂²×C₄₇	#99
hypercenter	#390	D₄₇
hypocenter	C₄₇	#1393
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #390

subgroup

G/H

center

C₂²

#13093939

commutator

C₂×C₄×C₄₇

C₄²

Frattini

C₂²×D₄

C₂×D₄₇

Fitting

C₄₇×#390

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ #390

N	H ≅ G/N	type
5. #13093939	C₄	semidirect
7. C₂×C₄⋊C₄×C₄₇	C₄	semidirect
9. C₂×C₄²×C₄₇	C₄	semidirect
74. C₄₇	#1393	semidirect
76. C₂×C₄⋊C₄	Dic₄₇	semidirect
79. C₂×C₄²	Dic₄₇	semidirect

H ≅ G/N

type

5. #13093939

C₄

semidirect

7. C₂×C₄⋊C₄×C₄₇

C₄

semidirect

9. C₂×C₄²×C₄₇

C₄

semidirect

74. C₄₇

#1393

semidirect

76. C₂×C₄⋊C₄

Dic₄₇

semidirect

79. C₂×C₄²

Dic₄₇

semidirect

Subgroups — #20241513

Conjugacy classes of subgroups

Complements