Group #20244220 — subgroups

Subgroups — #20244220

Order 6016.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#15689673	C₂	1	1	yes	maximal, characteristic
2	#15689147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13093952	C₂	1	4	yes	maximal, complemented
4	C₄₇×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10625350	C₂²	1	2	yes	—
6	C₂×#10625329	C₂²	1	2	yes	—
7	#13094024	C₄	1	1	yes	characteristic, complemented
8	#13094021	C₄	1	1	yes	characteristic, complemented
9	#13094012	C₄	1	2	yes	complemented
10	#13093998	—	2	2	no	—
11	C₄×D₁₈₈	—	2	2	no	—
12	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
13	#13093952	—	2	8	no	—
14	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	#95×C₄₇	—	2	2	no	—
17	C₂²×C₄×C₄₇	—	2	2	no	—
18	C₂²×C₄×C₄₇	D₄	1	2	yes	—
19	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
20	C₂×C₂²⋊D₄₇	—	2	2	no	—
21	#10625350	—	2	4	no	—
22	#10625350	—	2	4	no	—
23	C₂×D₁₈₈	—	2	2	no	—
24	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₄₇	—	2	2	no	—
26	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
27	C₄²×C₄₇	—	2	2	no	—
28	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₄₇	D₄	1	2	yes	—
31	C₄²×C₄₇	D₄	1	4	yes	complemented
32	(C₂×D₄₇)⋊C₄	—	2	4	no	—
33	C₄⋊Dic₄₇	—	2	2	no	—
34	#10625329	—	2	4	no	—
35	C₂×C₄₇⋊C₈	—	2	2	no	—
36	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×D₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	—	2	4	no	—
39	C₂×C₄×C₄₇	—	4	2	no	—
40	C₂×C₄×C₄₇	—	4	4	no	—
41	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	#41	1	2	yes	—
47	C₂²⋊D₄₇	—	4	4	no	—
48	C₂×Dic₄₇	—	2	2	no	—
49	D₁₈₈	—	2	4	no	—
50	D₁₈₈	—	4	4	no	—
51	C₄×D₄₇	—	4	2	no	—
52	C₄×D₄₇	—	4	2	no	—
53	Dic₉₄	—	2	4	no	—
54	C₄₇⋊C₈	—	4	2	no	—
55	C₂²×C₄₇	—	2	2	no	—
56	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄₇	#121	1	2	yes	—
58	C₂×D₄₇	—	4	2	no	—
59	C₂×D₄₇	—	4	4	no	—
60	C₄×C₄₇	—	2	2	no	—
61	C₄×C₄₇	—	2	2	no	—
62	C₄×C₄₇	—	4	4	no	—
63	C₄×C₄₇	C₄×D₄	1	2	yes	—
64	C₄×C₄₇	#124	1	1	yes	characteristic
65	C₄×C₄₇	#127	1	1	yes	characteristic
66	Dic₄₇	—	4	2	no	—
67	#1963	—	47	1	no	maximal, Sylow
68	C₂×C₄₇	—	2	2	no	—
69	C₂×C₄₇	#390	1	1	yes	characteristic
70	C₂×C₄₇	#421	1	2	yes	—
71	D₄₇	—	8	2	no	—
72	#532	—	47	1	no	—
73	#423	—	47	1	no	—
74	C₂×#104	—	47	4	no	—
75	#376	D₄₇	1	1	yes	characteristic, complemented
76	C₄₇	#1963	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	—	47	2	no	—
78	C₂×M₁₆	—	47	2	no	—
79	C₄⋊Q₈	—	47	1	no	—
80	#127	—	47	1	no	—
81	#124	—	47	2	no	—
82	#121	—	94	2	no	—
83	C₄×D₄	—	94	2	no	—
84	C₂×C₄²	C₂×D₄₇	1	2	yes	—
85	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
86	C₄⋊C₈	—	94	2	no	—
87	#104	—	94	8	no	—
88	#95	—	2	2	no	—
89	#41	—	94	4	no	—
90	#41	—	94	4	no	—
91	C₂×Q₈	—	47	2	no	—
92	C₂×D₄	—	47	2	no	—
93	C₂×D₄	—	94	2	no	—
94	C₂×D₄	—	94	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	94	2	no	—
97	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
98	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
99	M₁₆	—	94	4	no	—
100	C₂×C₈	—	94	2	no	—
101	C₄⋊C₄	—	94	2	no	—
102	C₂²⋊C₄	—	94	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₄×D₄₇	1	1	yes	characteristic
105	C₄²	C₄×D₄₇	1	1	yes	characteristic
106	C₄²	D₁₈₈	1	2	yes	—
107	C₄²	D₁₈₈	1	4	yes	complemented
108	C₂³	—	94	2	no	—
109	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
110	Q₈	—	94	4	no	—
111	D₄	—	94	4	no	—
112	D₄	—	188	4	no	—
113	D₄	—	188	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	94	2	no	—
118	C₂×C₄	—	188	2	no	—
119	C₂×C₄	—	188	2	no	—
120	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
121	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
123	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
124	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
125	C₂×C₄	#10625350	1	2	yes	—
126	C₈	—	188	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	188	2	no	—
129	C₂²	—	188	4	no	—
130	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
131	C₂²	#13093998	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	4	no	—
135	C₄	—	188	2	no	—
136	C₄	C₄×D₁₈₈	1	2	yes	—
137	C₄	#13094012	1	1	yes	characteristic
138	C₄	#13094021	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	376	2	no	—
141	C₂	#15689025	1	1	yes	characteristic
142	C₂	#15689141	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689673

C₂

yes

maximal, characteristic

#15689147

C₂

yes

maximal, characteristic, complemented

C₂×#13093952

C₂

yes

maximal, complemented

C₄₇×#376

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₂²

yes

—

C₂×#10625329

C₂²

yes

—

#13094024

C₄

yes

characteristic, complemented

#13094021

C₄

yes

characteristic, complemented

#13094012

C₄

yes

complemented

#13093998

—

C₄×D₁₈₈

—

C₄⋊(C₄₇⋊C₈)

—

#13093952

—

C₂×C₄²×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

#10625350

—

#10625350

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂×C₄

yes

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

C₂×C₄

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

D₄

yes

complemented

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#121

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1963

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#390

yes

characteristic

C₂×C₄₇

#421

yes

—

D₄₇

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄₇

#1963

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

C₄×D₄₇

yes

characteristic

105

C₄²

C₄×D₄₇

yes

characteristic

106

C₄²

D₁₈₈

yes

—

107

C₄²

D₁₈₈

yes

complemented

108

C₂³

—

109

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

188

—

113

D₄

—

188

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

188

—

119

C₂×C₄

—

188

—

120

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

121

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

122

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

123

C₂×C₄

C₂×D₁₈₈

yes

—

124

C₂×C₄

C₂×D₁₈₈

yes

characteristic

125

C₂×C₄

#10625350

yes

—

126

C₈

—

188

—

127

C₂²

—

128

C₂²

—

188

—

129

C₂²

—

188

—

130

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

131

C₂²

#13093998

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

188

—

136

C₄

C₄×D₁₈₈

yes

—

137

C₄

#13094012

yes

characteristic

138

C₄

#13094021

yes

characteristic

139

C₂

—

140

C₂

—

376

—

141

C₂

#15689025

yes

characteristic

142

C₂

#15689141

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#376	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#376	D₄₇
hypocenter	C₄₇	#1963
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#376

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
2. #15689147	C₂	semidirect
3. C₂×#13093952	C₂	semidirect
4. C₄₇×#376	C₂	semidirect
7. #13094024	C₄	semidirect
8. #13094021	C₄	semidirect
9. #13094012	C₄	semidirect
31. C₄²×C₄₇	D₄	semidirect
75. #376	D₄₇	semidirect
76. C₄₇	#1963	semidirect
107. C₄²	D₁₈₈	semidirect

H ≅ G/N

type

2. #15689147

C₂

semidirect

3. C₂×#13093952

C₂

semidirect

4. C₄₇×#376

C₂

semidirect

7. #13094024

C₄

semidirect

8. #13094021

C₄

semidirect

9. #13094012

C₄

semidirect

31. C₄²×C₄₇

D₄

semidirect

75. #376

D₄₇

semidirect

76. C₄₇

#1963

semidirect

107. C₄²

D₁₈₈

semidirect

Subgroups — #20244220

Conjugacy classes of subgroups

Complements