Group #20242502 — subgroups

Subgroups — #20242502

Order 6016.

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

normal subgroups only

This group has 314 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	#15689758	C₂	1	1	yes	maximal, characteristic
2	#15689410	C₂	1	2	yes	maximal, complemented
3	#15688867	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688852	C₂	1	2	yes	maximal, complemented
5	C₄₇×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×D₄₇	—	2	2	no	—
7	#13093995	—	2	2	no	—
8	#13093992	—	2	2	no	—
9	C₄×D₁₈₈	C₂²	1	2	yes	—
10	#13093955	—	2	2	no	—
11	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	complemented
12	#13093944	—	2	2	no	—
13	C₂×SD₁₆×C₄₇	—	2	2	no	—
14	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
15	#127×C₄₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₄₇	—	2	2	no	—
18	C₂×Q₈×C₄₇	—	2	1	no	—
19	C₂×D₄×C₄₇	—	2	2	no	—
20	C₂×D₄×C₄₇	D₄	1	2	yes	—
21	C₂×C₂²⋊D₄₇	—	2	4	no	—
22	D₄×D₄₇	—	4	4	no	—
23	D₄×D₄₇	—	4	4	no	—
24	C₂×D₁₈₈	D₄	1	2	yes	—
25	C₂×C₄×D₄₇	—	2	2	no	—
26	SD₁₆×C₄₇	—	4	4	no	—
27	C₂×C₈×C₄₇	—	2	1	no	—
28	C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₄₇	—	2	2	no	—
31	(C₂×D₄₇)⋊C₄	—	2	2	no	—
32	C₄⋊Dic₄₇	D₄	1	2	yes	—
33	C₂×C₄₇⋊C₈	—	2	2	no	—
34	C₂³×C₄₇	—	2	2	no	—
35	C₂²×D₄₇	—	2	2	no	—
36	C₂²×D₄₇	—	4	8	no	—
37	C₂²×D₄₇	—	8	2	no	—
38	Q₈×C₄₇	—	4	2	no	—
39	D₄×C₄₇	—	4	2	no	—
40	D₄×C₄₇	—	4	2	no	—
41	D₄×C₄₇	—	4	2	no	—
42	C₂×C₄×C₄₇	—	4	1	no	—
43	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂²⋊D₄₇	—	4	8	no	—
46	C₂×Dic₄₇	—	2	2	no	—
47	D₁₈₈	—	2	2	no	—
48	D₁₈₈	SD₁₆	1	4	yes	complemented
49	C₄×D₄₇	—	4	2	no	—
50	C₄×D₄₇	—	4	2	no	—
51	C₈×C₄₇	—	2	2	no	—
52	C₄₇⋊C₈	—	4	2	no	—
53	C₂²×C₄₇	—	4	4	no	—
54	C₂²×C₄₇	—	4	2	no	—
55	C₂²×C₄₇	#120	1	1	yes	characteristic
56	C₂×D₄₇	—	2	4	no	—
57	C₂×D₄₇	—	4	2	no	—
58	C₂×D₄₇	—	4	2	no	—
59	C₂×D₄₇	—	8	8	no	—
60	C₄×C₄₇	—	2	1	no	—
61	C₄×C₄₇	—	8	1	no	—
62	C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
63	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
64	Dic₄₇	—	4	2	no	—
65	#1629	—	47	1	no	maximal, Sylow
66	C₂×C₄₇	—	8	2	no	—
67	C₂×C₄₇	#450	1	2	yes	—
68	C₂×C₄₇	#453	1	1	yes	characteristic
69	D₄₇	—	4	4	no	—
70	D₄²	—	47	1	no	—
71	#492	D₄₇	1	1	yes	characteristic, complemented
72	#476	—	47	2	no	—
73	#331	—	47	1	no	—
74	D₄⋊C₈	—	47	2	no	—
75	C₄₇	#1629	1	1	yes	characteristic, complemented, Sylow
76	C₂²×D₄	—	94	2	no	—
77	C₂×SD₁₆	—	2	2	no	—
78	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic, complemented
79	#127	C₂×D₄₇	1	1	yes	characteristic
80	#121	—	94	2	no	—
81	#120	—	94	2	no	—
82	C₄×D₄	—	47	2	no	—
83	#106	—	94	2	no	—
84	C₄⋊C₈	—	47	2	no	—
85	#102	—	94	2	no	—
86	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic, complemented
87	C₂⁴	—	94	2	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×D₄	—	2	2	no	—
90	C₂×D₄	—	47	2	no	—
91	C₂×D₄	—	94	4	no	—
92	C₂×D₄	—	188	4	no	—
93	C₂×D₄	—	188	4	no	—
94	C₂×D₄	C₂²⋊D₄₇	1	2	yes	—
95	C₂²×C₄	—	94	2	no	—
96	SD₁₆	—	4	4	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	94	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	47	2	no	—
101	C₂²⋊C₄	—	94	2	no	—
102	C₂²⋊C₄	—	94	2	no	—
103	C₄²	C₂²×D₄₇	1	1	yes	characteristic
104	C₂³	—	2	2	no	—
105	C₂³	—	94	2	no	—
106	C₂³	—	376	2	no	—
107	C₂³	—	188	8	no	—
108	Q₈	—	4	2	no	—
109	D₄	—	4	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	2	no	—
112	D₄	—	47	4	no	—
113	D₄	—	94	2	no	—
114	D₄	—	188	8	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	94	2	no	—
117	C₂×C₄	—	188	2	no	—
118	C₂×C₄	—	188	2	no	—
119	C₂×C₄	D₄×D₄₇	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	188	2	no	—
123	C₂²	—	4	4	no	—
124	C₂²	—	4	2	no	—
125	C₂²	—	94	4	no	—
126	C₂²	—	376	8	no	—
127	C₂²	—	188	2	no	—
128	C₂²	—	188	2	no	—
129	C₂²	#13093992	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	8	1	no	—
132	C₄	—	188	2	no	—
133	C₄	SD₁₆×D₄₇	1	2	yes	—
134	C₄	#13094051	1	2	yes	—
135	C₂	—	8	2	no	—
136	C₂	—	188	4	no	—
137	C₂	#15689272	1	2	yes	—
138	C₂	#15689290	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689758

C₂

yes

maximal, characteristic

#15689410

C₂

yes

maximal, complemented

#15688867

C₂

yes

maximal, characteristic, complemented

#15688852

C₂

yes

maximal, complemented

C₄₇×#492

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₄₇

—

#13093995

—

#13093992

—

C₄×D₁₈₈

C₂²

yes

—

#13093955

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

complemented

#13093944

—

C₂×SD₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic, complemented

#127×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic, complemented

C₂³×D₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

—

C₂×C₂²⋊D₄₇

—

D₄×D₄₇

—

D₄×D₄₇

—

C₂×D₁₈₈

D₄

yes

—

C₂×C₄×D₄₇

—

SD₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

SD₁₆

yes

complemented

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#120

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

Dic₄₇

—

#1629

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#450

yes

—

C₂×C₄₇

#453

yes

characteristic

D₄₇

—

D₄²

—

#492

D₄₇

yes

characteristic, complemented

#476

—

#331

—

D₄⋊C₈

—

C₄₇

#1629

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic, complemented

#127

C₂×D₄₇

yes

characteristic

#121

—

#120

—

C₄×D₄

—

#106

—

C₄⋊C₈

—

#102

—

C₄×C₈

C₂×D₄₇

yes

characteristic, complemented

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

C₂×D₄

C₂²⋊D₄₇

yes

—

C₂²×C₄

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

C₂²×D₄₇

yes

characteristic

104

C₂³

—

105

C₂³

—

106

C₂³

—

376

—

107

C₂³

—

188

—

108

Q₈

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

114

D₄

—

188

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

188

—

118

C₂×C₄

—

188

—

119

C₂×C₄

D₄×D₄₇

yes

—

120

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

121

C₈

—

122

C₈

—

188

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

376

—

127

C₂²

—

188

—

128

C₂²

—

188

—

129

C₂²

#13093992

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

188

—

133

C₄

SD₁₆×D₄₇

yes

—

134

C₄

#13094051

yes

—

135

C₂

—

136

C₂

—

188

—

137

C₂

#15689272

yes

—

138

C₂

#15689290

yes

characteristic

	subgroup	G/H
center	C₂²	#13093992
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#492	C₂
socle	C₂²×C₄₇	#120
hypercenter	#492	D₄₇
hypocenter	C₄₇	#1629
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#13093992

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#492

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #15689410	C₂	semidirect
3. #15688867	C₂	semidirect
4. #15688852	C₂	semidirect
5. C₄₇×#492	C₂	semidirect
11. C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
14. C₄⋊Q₈×C₄₇	C₂²	semidirect
16. C₄×C₈×C₄₇	C₂²	semidirect
48. D₁₈₈	SD₁₆	semidirect
71. #492	D₄₇	semidirect
75. C₄₇	#1629	semidirect
78. C₄⋊Q₈	C₂×D₄₇	semidirect
86. C₄×C₈	C₂×D₄₇	semidirect

H ≅ G/N

type

2. #15689410

C₂

semidirect

3. #15688867

C₂

semidirect

4. #15688852

C₂

semidirect

5. C₄₇×#492

C₂

semidirect

11. C₄⋊(C₄₇⋊C₈)

C₂²

semidirect

14. C₄⋊Q₈×C₄₇

C₂²

semidirect

16. C₄×C₈×C₄₇

C₂²

semidirect

48. D₁₈₈

SD₁₆

semidirect

71. #492

D₄₇

semidirect

75. C₄₇

#1629

semidirect

78. C₄⋊Q₈

C₂×D₄₇

semidirect

86. C₄×C₈

C₂×D₄₇

semidirect

Subgroups — #20242502

Conjugacy classes of subgroups

Complements