Group #20254827 — subgroups

Subgroups — #20254827

Order 6016.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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	#15689728	C₂	1	2	yes	maximal, complemented
2	#15689424	C₂	1	4	yes	maximal, complemented
3	#15689376	C₂	1	2	yes	maximal, complemented
4	#15689359	C₂	1	2	yes	maximal, complemented
5	#15689319	C₂	1	4	yes	maximal, complemented
6	C₄₇×#424	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₃₇₆	C₂²	1	2	yes	—
8	C₂×D₃₇₆	C₂²	1	2	yes	—
9	#13094021	C₂²	1	2	yes	—
10	#13094016	C₂²	1	2	yes	—
11	#13094009	—	2	4	no	—
12	#13093998	C₂²	1	4	yes	—
13	#13093997	—	2	4	no	—
14	C₄×D₁₈₈	C₂²	1	4	yes	—
15	#13093958	C₂²	1	2	yes	—
16	#13093956	C₂²	1	2	yes	—
17	#13093946	C₂²	1	4	yes	—
18	#13093946	C₂²	1	4	yes	—
19	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	#117×C₄₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
23	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂×C₂²⋊D₄₇	—	2	4	no	—
25	C₂×D₁₈₈	—	2	4	no	—
26	C₂×D₁₈₈	C₂³	1	4	yes	—
27	C₂×C₄×D₄₇	—	2	4	no	—
28	M₁₆×C₄₇	—	4	1	no	—
29	C₂×C₈×C₄₇	—	4	1	no	—
30	C₂×C₈×C₄₇	C₂³	1	2	yes	—
31	C₂×C₈×C₄₇	C₂³	1	2	yes	—
32	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
33	C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
34	C₄²×C₄₇	C₂³	1	2	yes	—
35	(C₂×D₄₇)⋊C₄	—	2	4	no	—
36	(C₂×D₄₇)⋊C₄	—	2	4	no	—
37	C₄⋊Dic₄₇	C₂³	1	4	yes	—
38	Dic₄₇⋊C₄	—	2	4	no	—
39	D₃₇₆	—	4	2	no	—
40	D₃₇₆	—	4	2	no	—
41	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₄₇	—	2	4	no	—
43	C₂×C₄×C₄₇	—	2	1	no	—
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
46	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₂²⋊D₄₇	—	8	4	no	—
49	C₂×Dic₄₇	—	2	4	no	—
50	D₁₈₈	—	2	4	no	—
51	D₁₈₈	—	2	4	no	—
52	D₁₈₈	—	8	4	no	—
53	C₄×D₄₇	—	8	4	no	—
54	C₈×C₄₇	—	2	2	no	—
55	C₈×C₄₇	—	2	2	no	—
56	C₂²×C₄₇	—	2	1	no	—
57	C₂²×C₄₇	—	2	2	no	—
58	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
59	C₂×D₄₇	—	4	4	no	—
60	C₂×D₄₇	—	4	4	no	—
61	C₂×D₄₇	—	4	4	no	—
62	C₄×C₄₇	—	2	1	no	—
63	C₄×C₄₇	—	2	4	no	—
64	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
65	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
66	Dic₄₇	—	4	4	no	—
67	#3259	—	47	1	no	maximal, Sylow
68	C₂×C₄₇	—	4	1	no	—
69	C₂×C₄₇	#537	1	1	yes	characteristic
70	C₂×C₄₇	#576	1	2	yes	—
71	D₄₇	—	8	4	no	—
72	#540	—	47	2	no	—
73	#478	—	47	4	no	—
74	#469	—	47	2	no	—
75	#466	—	47	2	no	—
76	#459	—	47	4	no	—
77	#424	D₄₇	1	1	yes	characteristic, complemented
78	C₄₇	#3259	1	1	yes	characteristic, complemented, Sylow
79	C₂×D₈	—	47	2	no	—
80	C₂×D₈	—	47	2	no	—
81	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
83	#127	—	47	2	no	—
84	#125	—	47	2	no	—
85	#123	—	94	4	no	—
86	#121	—	47	4	no	—
87	#121	—	94	4	no	—
88	C₄×D₄	—	47	4	no	—
89	#117	C₂×D₄₇	1	1	yes	characteristic
90	#107	—	47	2	no	—
91	#106	—	47	2	no	—
92	C₄⋊C₈	C₂×D₄₇	1	4	yes	—
93	#102	—	47	4	no	—
94	#102	—	47	4	no	—
95	C₂×D₄	—	47	4	no	—
96	C₂×D₄	—	94	4	no	—
97	C₂×D₄	—	94	4	no	—
98	C₂²×C₄	—	94	4	no	—
99	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
100	D₈	—	188	2	no	—
101	D₈	—	188	2	no	—
102	M₁₆	—	4	1	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	C₂²×D₄₇	1	2	yes	—
105	C₂×C₈	C₂²×D₄₇	1	2	yes	—
106	C₄⋊C₄	—	47	4	no	—
107	C₄⋊C₄	—	94	4	no	—
108	C₄⋊C₄	D₁₈₈	1	2	yes	—
109	C₂²⋊C₄	—	94	4	no	—
110	C₂²⋊C₄	—	94	4	no	—
111	C₂²⋊C₄	D₁₈₈	1	2	yes	—
112	C₄²	C₂²×D₄₇	1	2	yes	—
113	C₂³	—	94	4	no	—
114	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
115	D₄	—	94	4	no	—
116	D₄	—	94	4	no	—
117	D₄	—	376	4	no	—
118	D₄	—	376	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	94	4	no	—
122	C₂×C₄	—	376	4	no	—
123	C₂×C₄	C₂×D₁₈₈	1	4	yes	—
124	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
125	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	188	4	no	—
131	C₂²	—	188	4	no	—
132	C₂²	—	188	4	no	—
133	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
134	C₄	—	2	4	no	—
135	C₄	—	2	1	no	—
136	C₄	—	188	4	no	—
137	C₄	#13094069	1	1	yes	characteristic
138	C₄	#13094072	1	1	yes	characteristic
139	C₂	—	4	1	no	—
140	C₂	—	376	4	no	—
141	C₂	#15689704	1	1	yes	characteristic
142	C₂	#15689928	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689728

C₂

yes

maximal, complemented

#15689424

C₂

yes

maximal, complemented

#15689376

C₂

yes

maximal, complemented

#15689359

C₂

yes

maximal, complemented

#15689319

C₂

yes

maximal, complemented

C₄₇×#424

C₂

yes

maximal, characteristic, complemented

C₂×D₃₇₆

C₂²

yes

—

C₂×D₃₇₆

C₂²

yes

—

#13094021

C₂²

yes

—

#13094016

C₂²

yes

—

#13094009

—

#13093998

C₂²

yes

—

#13093997

—

C₄×D₁₈₈

C₂²

yes

—

#13093958

C₂²

yes

—

#13093956

C₂²

yes

—

#13093946

C₂²

yes

—

#13093946

C₂²

yes

—

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

#117×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

D₄

yes

—

C₂²⋊C₄×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

D₃₇₆

—

D₃₇₆

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

H(5,𝔽₂)

yes

characteristic

C₄×C₄₇

2₋¹⁺⁴

yes

characteristic

Dic₄₇

—

#3259

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#537

yes

characteristic

C₂×C₄₇

#576

yes

—

D₄₇

—

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄₇

#3259

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₂²×C₈

C₂×D₄₇

yes

—

—

—

—

—

C₄×D₄

—

#117

C₂×D₄₇

yes

characteristic

#107

—

#106

—

C₄⋊C₈

C₂×D₄₇

yes

—

#102

—

#102

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

100

D₈

—

188

—

101

D₈

—

188

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

C₂²×D₄₇

yes

—

105

C₂×C₈

C₂²×D₄₇

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

D₁₈₈

yes

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

D₁₈₈

yes

—

112

C₄²

C₂²×D₄₇

yes

—

113

C₂³

—

114

C₂³

C₂×D₁₈₈

yes

characteristic

115

D₄

—

116

D₄

—

117

D₄

—

376

—

118

D₄

—

376

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

376

—

123

C₂×C₄

C₂×D₁₈₈

yes

—

124

C₂×C₄

C₂×D₁₈₈

yes

characteristic

125

C₂×C₄

C₂³×D₄₇

yes

characteristic

126

C₈

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

188

—

131

C₂²

—

188

—

132

C₂²

—

188

—

133

C₂²

C₂²×D₁₈₈

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

188

—

137

C₄

#13094069

yes

characteristic

138

C₄

#13094072

yes

characteristic

139

C₂

—

140

C₂

—

376

—

141

C₂

#15689704

yes

characteristic

142

C₂

#15689928

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#424	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#424	D₄₇
hypocenter	C₄₇	#3259
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#424

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
1. #15689728	C₂	semidirect
2. #15689424	C₂	semidirect
3. #15689376	C₂	semidirect
4. #15689359	C₂	semidirect
5. #15689319	C₂	semidirect
6. C₄₇×#424	C₂	semidirect
77. #424	D₄₇	semidirect
78. C₄₇	#3259	semidirect

H ≅ G/N

type

1. #15689728

C₂

semidirect

2. #15689424

C₂

semidirect

3. #15689376

C₂

semidirect

4. #15689359

C₂

semidirect

5. #15689319

C₂

semidirect

6. C₄₇×#424

C₂

semidirect

77. #424

D₄₇

semidirect

78. C₄₇

#3259

semidirect

Subgroups — #20254827

Conjugacy classes of subgroups

Complements