Group #20244193 — subgroups

Subgroups — #20244193

Order 6016.

436 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13094021	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093944	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₈₈	—	2	2	no	—
6	C₂²×D₁₈₈	C₂²	1	2	yes	—
7	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#13094021	—	2	2	no	—
9	#13094021	C₄	1	4	yes	complemented
10	#13093944	—	2	8	no	—
11	C₄×C₄₇⋊C₈	—	2	2	no	—
12	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
13	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₂³×D₄₇	—	2	2	no	—
16	C₂²×C₄×C₄₇	—	2	2	no	—
17	C₂²×C₄×C₄₇	D₄	1	2	yes	—
18	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂×D₁₈₈	—	2	4	no	—
20	C₂×D₁₈₈	—	2	4	no	—
21	C₂×D₁₈₈	—	2	8	no	—
22	C₂×D₁₈₈	—	2	2	no	—
23	C₂×D₁₈₈	—	4	4	no	—
24	C₂×D₁₈₈	—	4	4	no	—
25	C₂×D₁₈₈	C₂×C₄	1	4	yes	—
26	C₄⋊C₄×C₄₇	—	2	4	no	—
27	C₄⋊C₄×C₄₇	—	4	4	no	—
28	C₄²×C₄₇	C₂×C₄	1	2	yes	—
29	C₄²×C₄₇	D₄	1	2	yes	—
30	C₂×C₄₇⋊C₈	—	2	2	no	—
31	C₂×C₄₇⋊C₈	—	2	2	no	—
32	C₂×C₄₇⋊C₈	D₄	1	4	yes	complemented
33	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×D₄₇	—	2	4	no	—
35	C₂²×D₄₇	—	4	2	no	—
36	C₂²×D₄₇	—	4	2	no	—
37	C₂²×D₄₇	—	4	4	no	—
38	C₂²×D₄₇	—	4	4	no	—
39	C₂×C₄×C₄₇	—	4	2	no	—
40	C₂×C₄×C₄₇	—	4	4	no	—
41	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×C₄₇	D₈	1	4	yes	—
43	C₂×C₄×C₄₇	SD₁₆	1	4	yes	—
44	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₄₇	#41	1	2	yes	—
48	D₁₈₈	—	2	8	no	—
49	D₁₈₈	—	4	16	no	—
50	D₁₈₈	—	4	4	no	—
51	C₄₇⋊C₈	—	2	4	no	—
52	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄₇	#124	1	1	yes	characteristic
54	C₂²×C₄₇	#127	1	1	yes	characteristic
55	C₂²×C₄₇	C₂×D₈	1	2	yes	—
56	C₂²×C₄₇	C₂×SD₁₆	1	2	yes	—
57	C₂×D₄₇	—	4	4	no	—
58	C₂×D₄₇	—	4	8	no	—
59	C₂×D₄₇	—	8	4	no	—
60	C₂×D₄₇	—	8	2	no	—
61	C₂×D₄₇	—	8	2	no	—
62	C₄×C₄₇	—	8	2	no	—
63	C₄×C₄₇	#102	1	8	yes	complemented
64	C₄×C₄₇	C₄×D₄	1	2	yes	—
65	C₄×C₄₇	#121	1	2	yes	—
66	#1957	—	47	1	no	maximal, Sylow
67	C₂×C₄₇	#390	1	1	yes	characteristic
68	C₂×C₄₇	C₂×#102	1	2	yes	—
69	C₂×C₄₇	#486	1	2	yes	—
70	C₂×C₄₇	#492	1	1	yes	characteristic
71	C₂×C₄₇	#493	1	1	yes	characteristic
72	D₄₇	—	8	4	no	—
73	C₂×#127	—	47	1	no	—
74	C₂×#102	—	47	4	no	—
75	C₂×C₄×C₈	—	47	1	no	—
76	#384	D₄₇	1	1	yes	characteristic, complemented
77	C₄₇	#1957	1	1	yes	characteristic, complemented, Sylow
78	C₂²×D₄	—	47	2	no	—
79	C₂²×D₄	—	94	2	no	—
80	C₂²×C₈	—	47	2	no	—
81	#127	—	47	4	no	—
82	#127	—	94	2	no	—
83	C₂×C₄⋊C₄	—	2	2	no	—
84	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
85	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
86	#102	—	94	8	no	—
87	C₄×C₈	—	94	2	no	—
88	C₂⁴	—	94	2	no	—
89	C₂×D₄	—	47	4	no	—
90	C₂×D₄	—	94	8	no	—
91	C₂×D₄	—	94	2	no	—
92	C₂×D₄	—	94	4	no	—
93	C₂×D₄	—	94	4	no	—
94	C₂×D₄	—	188	4	no	—
95	C₂×D₄	—	188	4	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
98	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
99	C₂×C₈	—	47	4	no	—
100	C₂×C₈	—	94	2	no	—
101	C₂×C₈	—	94	2	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	4	4	no	—
104	C₄²	C₄×D₄₇	1	2	yes	—
105	C₄²	D₁₈₈	1	2	yes	—
106	C₂³	—	94	4	no	—
107	C₂³	—	188	2	no	—
108	C₂³	—	188	2	no	—
109	C₂³	—	188	4	no	—
110	C₂³	—	188	4	no	—
111	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
112	D₄	—	94	8	no	—
113	D₄	—	188	16	no	—
114	D₄	—	188	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
118	C₂×C₄	#10625334	1	4	yes	—
119	C₂×C₄	#10625336	1	4	yes	—
120	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
121	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
122	C₂×C₄	D₄×D₄₇	1	2	yes	—
123	C₂×C₄	#10625354	1	2	yes	—
124	C₈	—	94	4	no	—
125	C₂²	—	376	2	no	—
126	C₂²	—	376	2	no	—
127	C₂²	—	376	4	no	—
128	C₂²	—	188	4	no	—
129	C₂²	—	188	8	no	—
130	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
131	C₂²	#13094013	1	1	yes	characteristic
132	C₂²	#13094020	1	1	yes	characteristic
133	C₂²	C₂×#10625334	1	2	yes	—
134	C₂²	C₂×#10625336	1	2	yes	—
135	C₄	—	8	2	no	—
136	C₄	#13093944	1	8	yes	complemented
137	C₄	#13093986	1	2	yes	—
138	C₄	#13093999	1	2	yes	—
139	C₂	—	376	4	no	—
140	C₂	#15689024	1	1	yes	characteristic
141	C₂	C₂×#13093944	1	2	yes	—
142	C₂	#15689471	1	2	yes	—
143	C₂	#15689497	1	1	yes	characteristic
144	C₂	#15689499	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094021

C₂

yes

maximal, characteristic

C₂×#13093944

C₂

yes

maximal, complemented

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄₇×#384

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

—

C₂²×D₁₈₈

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

#13094021

—

#13094021

C₄

yes

complemented

#13093944

—

C₄×C₄₇⋊C₈

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂×C₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

complemented

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

D₈

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

C₂×D₈

yes

—

C₂²×C₄₇

C₂×SD₁₆

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#102

yes

complemented

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

#121

yes

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#486

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

C₂×#127

—

C₂×#102

—

C₂×C₄×C₈

—

#384

D₄₇

yes

characteristic, complemented

C₄₇

#1957

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#102

—

C₄×C₈

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₄×D₄₇

yes

—

105

C₄²

D₁₈₈

yes

—

106

C₂³

—

107

C₂³

—

188

—

108

C₂³

—

188

—

109

C₂³

—

188

—

110

C₂³

—

188

—

111

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

112

D₄

—

113

D₄

—

188

—

114

D₄

—

188

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

118

C₂×C₄

#10625334

yes

—

119

C₂×C₄

#10625336

yes

—

120

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

121

C₂×C₄

C₂×D₁₈₈

yes

characteristic

122

C₂×C₄

D₄×D₄₇

yes

—

123

C₂×C₄

#10625354

yes

—

124

C₈

—

125

C₂²

—

376

—

126

C₂²

—

376

—

127

C₂²

—

376

—

128

C₂²

—

188

—

129

C₂²

—

188

—

130

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

131

C₂²

#13094013

yes

characteristic

132

C₂²

#13094020

yes

characteristic

133

C₂²

C₂×#10625334

yes

—

134

C₂²

C₂×#10625336

yes

—

135

C₄

—

136

C₄

#13093944

yes

complemented

137

C₄

#13093986

yes

—

138

C₄

#13093999

yes

—

139

C₂

—

376

—

140

C₂

#15689024

yes

characteristic

141

C₂

C₂×#13093944

yes

—

142

C₂

#15689471

yes

—

143

C₂

#15689497

yes

characteristic

144

C₂

#15689499

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#384	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	#384	D₄₇
hypocenter	C₄₇	#1957
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#384

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
2. C₂×#13093944	C₂	semidirect
3. C₂×C₄×C₄₇⋊C₈	C₂	semidirect
4. C₄₇×#384	C₂	semidirect
9. #13094021	C₄	semidirect
32. C₂×C₄₇⋊C₈	D₄	semidirect
63. C₄×C₄₇	#102	semidirect
76. #384	D₄₇	semidirect
77. C₄₇	#1957	semidirect
136. C₄	#13093944	semidirect

H ≅ G/N

type

2. C₂×#13093944

C₂

semidirect

3. C₂×C₄×C₄₇⋊C₈

C₂

semidirect

4. C₄₇×#384

C₂

semidirect

9. #13094021

C₄

semidirect

32. C₂×C₄₇⋊C₈

D₄

semidirect

63. C₄×C₄₇

#102

semidirect

76. #384

D₄₇

semidirect

77. C₄₇

#1957

semidirect

136. C₄

#13093944

semidirect

Subgroups — #20244193

Conjugacy classes of subgroups

Complements