Group #20243048 — subgroups

Subgroups — #20243048

Order 6016.

186 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 61 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	#15689521	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689503	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689474	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689470	C₂	1	1	yes	maximal, characteristic
5	#15689470	C₂	1	1	yes	maximal, characteristic
6	#15689470	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₁₈₈	—	2	2	no	—
9	#13094024	C₂²	1	1	yes	characteristic
10	#13094024	C₂²	1	1	yes	characteristic
11	#13094024	C₂²	1	1	yes	characteristic
12	#13094021	C₂²	1	1	yes	characteristic
13	#13093956	—	2	2	no	—
14	#13093949	—	2	2	no	—
15	#13093946	—	2	2	no	—
16	#13093946	—	2	2	no	—
17	#13093946	—	2	2	no	—
18	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×D₁₈₈	—	2	1	no	—
22	C₂×D₁₈₈	—	2	1	no	—
23	C₂×D₁₈₈	—	2	1	no	—
24	C₂×Dic₉₄	—	2	1	no	—
25	C₂×Dic₉₄	—	2	1	no	—
26	C₂×Dic₉₄	—	2	1	no	—
27	C₂×C₈×C₄₇	D₄	1	2	yes	—
28	C₂×C₈×C₄₇	D₄	1	2	yes	—
29	C₂×C₈×C₄₇	D₄	1	2	yes	—
30	C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₄₇	—	2	1	no	—
32	C₄⋊Dic₄₇	—	2	1	no	—
33	C₄⋊Dic₄₇	—	2	1	no	—
34	C₄⋊Dic₄₇	—	2	1	no	—
35	C₄⋊Dic₄₇	—	2	1	no	—
36	C₄⋊Dic₄₇	—	2	1	no	—
37	Dic₁₈₈	—	4	4	no	—
38	C₂²×D₄₇	—	4	1	no	—
39	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₄₇	—	4	1	no	—
43	C₂×Dic₄₇	—	4	1	no	—
44	C₂×Dic₄₇	—	4	1	no	—
45	D₁₈₈	—	4	2	no	—
46	D₁₈₈	—	4	2	no	—
47	D₁₈₈	—	4	2	no	—
48	Dic₉₄	—	4	2	no	—
49	Dic₉₄	—	4	2	no	—
50	Dic₉₄	—	4	2	no	—
51	C₈×C₄₇	—	2	2	no	—
52	C₈×C₄₇	—	2	2	no	—
53	C₈×C₄₇	—	2	2	no	—
54	C₂²×C₄₇	#127	1	1	yes	characteristic
55	C₂×D₄₇	—	8	1	no	—
56	C₂×D₄₇	—	8	1	no	—
57	C₂×D₄₇	—	8	1	no	—
58	C₄×C₄₇	#135	1	2	yes	—
59	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
60	C₄×C₄₇	#137	1	2	yes	—
61	Dic₄₇	—	8	1	no	—
62	Dic₄₇	—	8	1	no	—
63	Dic₄₇	—	8	1	no	—
64	#1720	—	47	1	no	maximal, Sylow
65	C₂×C₄₇	#495	1	1	yes	characteristic
66	C₂×C₄₇	#496	1	1	yes	characteristic
67	C₂×C₄₇	#497	1	1	yes	characteristic
68	D₄₇	—	16	1	no	—
69	#498	—	47	1	no	—
70	#494	—	47	1	no	—
71	#487	—	47	1	no	—
72	#486	—	47	1	no	—
73	#486	—	47	1	no	—
74	#486	—	47	1	no	—
75	#322	D₄₇	1	1	yes	characteristic, complemented
76	C₄₇	#1720	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₁₆	—	94	2	no	—
78	C₄⋊Q₈	—	47	1	no	—
79	C₄⋊Q₈	—	47	1	no	—
80	C₄⋊Q₈	—	47	1	no	—
81	#127	—	47	1	no	—
82	#106	—	94	2	no	—
83	#103	—	94	2	no	—
84	#102	—	94	2	no	—
85	#102	—	94	2	no	—
86	#102	—	94	2	no	—
87	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
88	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
89	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
90	C₂×Q₈	—	94	1	no	—
91	C₂×Q₈	—	94	1	no	—
92	C₂×Q₈	—	94	1	no	—
93	C₂×D₄	—	94	1	no	—
94	C₂×D₄	—	94	1	no	—
95	C₂×D₄	—	94	1	no	—
96	Q₁₆	—	188	4	no	—
97	C₂×C₈	D₁₈₈	1	2	yes	—
98	C₂×C₈	D₁₈₈	1	2	yes	—
99	C₂×C₈	D₁₈₈	1	2	yes	—
100	C₄⋊C₄	—	94	1	no	—
101	C₄⋊C₄	—	94	1	no	—
102	C₄⋊C₄	—	94	1	no	—
103	C₄⋊C₄	—	94	1	no	—
104	C₄⋊C₄	—	94	1	no	—
105	C₄⋊C₄	—	94	1	no	—
106	C₄²	C₂²×D₄₇	1	1	yes	characteristic
107	C₂³	—	188	1	no	—
108	Q₈	—	188	2	no	—
109	Q₈	—	188	2	no	—
110	Q₈	—	188	2	no	—
111	D₄	—	188	2	no	—
112	D₄	—	188	2	no	—
113	D₄	—	188	2	no	—
114	C₂×C₄	—	188	1	no	—
115	C₂×C₄	—	188	1	no	—
116	C₂×C₄	—	188	1	no	—
117	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
118	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
119	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	2	2	no	—
122	C₈	—	2	2	no	—
123	C₂²	—	376	1	no	—
124	C₂²	—	376	1	no	—
125	C₂²	—	376	1	no	—
126	C₂²	#13094021	1	1	yes	characteristic
127	C₄	—	376	1	no	—
128	C₄	—	376	1	no	—
129	C₄	—	376	1	no	—
130	C₄	#13094046	1	2	yes	—
131	C₄	#13094052	1	2	yes	—
132	C₄	#13094057	1	2	yes	—
133	C₂	—	752	1	no	—
134	C₂	#15689507	1	1	yes	characteristic
135	C₂	#15689512	1	1	yes	characteristic
136	C₂	#15689517	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689521

C₂

yes

maximal, characteristic, complemented

#15689503

C₂

yes

maximal, characteristic, complemented

#15689474

C₂

yes

maximal, characteristic, complemented

#15689470

C₂

yes

maximal, characteristic

#15689470

C₂

yes

maximal, characteristic

#15689470

C₂

yes

maximal, characteristic

C₄₇×#322

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₈₈

—

#13094024

C₂²

yes

characteristic

#13094024

C₂²

yes

characteristic

#13094024

C₂²

yes

characteristic

#13094021

C₂²

yes

—

—

—

—

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

D₄

yes

—

C₂×C₈×C₄₇

D₄

yes

—

C₂×C₈×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₂²×C₄₇

#127

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

#135

yes

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₄×C₄₇

#137

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

—

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄₇

#1720

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

—

—

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₄×C₈

C₂×D₄₇

yes

characteristic

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

188

—

C₂×C₈

D₁₈₈

yes

—

C₂×C₈

D₁₈₈

yes

—

C₂×C₈

D₁₈₈

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

C₂²×D₄₇

yes

characteristic

107

C₂³

—

188

—

108

Q₈

—

188

—

109

Q₈

—

188

—

110

Q₈

—

188

—

111

D₄

—

188

—

112

D₄

—

188

—

113

D₄

—

188

—

114

C₂×C₄

—

188

—

115

C₂×C₄

—

188

—

116

C₂×C₄

—

188

—

117

C₂×C₄

C₂×D₁₈₈

yes

characteristic

118

C₂×C₄

C₂×D₁₈₈

yes

characteristic

119

C₂×C₄

C₂×D₁₈₈

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

376

—

124

C₂²

—

376

—

125

C₂²

—

376

—

126

C₂²

#13094021

yes

characteristic

127

C₄

—

376

—

128

C₄

—

376

—

129

C₄

—

376

—

130

C₄

#13094046

yes

—

131

C₄

#13094052

yes

—

132

C₄

#13094057

yes

—

133

C₂

—

752

—

134

C₂

#15689507

yes

characteristic

135

C₂

#15689512

yes

characteristic

136

C₂

#15689517

yes

characteristic

	subgroup	G/H
center	C₂²	#13094021
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#322	C₂
socle	C₂²×C₄₇	#127
hypercenter	#322	D₄₇
hypocenter	C₄₇	#1720
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#13094021

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#322

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #15689521	C₂	semidirect
2. #15689503	C₂	semidirect
3. #15689474	C₂	semidirect
7. C₄₇×#322	C₂	semidirect
75. #322	D₄₇	semidirect
76. C₄₇	#1720	semidirect

H ≅ G/N

type

1. #15689521

C₂

semidirect

2. #15689503

C₂

semidirect

3. #15689474

C₂

semidirect

7. C₄₇×#322

C₂

semidirect

75. #322

D₄₇

semidirect

76. C₄₇

#1720

semidirect

Subgroups — #20243048

Conjugacy classes of subgroups

Complements