Group #20252794 — subgroups

Subgroups — #20252794

Order 6016.

716 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 161 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	#15689694	C₂	1	3	yes	maximal, complemented
2	#15689313	C₂	1	8	yes	maximal, complemented
3	#15689097	C₂	1	3	yes	maximal, complemented
4	C₄₇×#585	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic, complemented
6	#13094022	—	2	3	no	—
7	#13094011	—	2	3	no	—
8	#13094005	C₂²	1	12	yes	complemented
9	#13094003	—	2	6	no	—
10	#13093983	C₂²	1	3	yes	complemented
11	#13093939	C₂²	1	12	yes	complemented
12	2₋¹⁺⁴×C₄₇	—	2	3	no	—
13	H(5,𝔽₂)×C₄₇	—	2	3	no	—
14	H(5,𝔽₂)×C₄₇	C₂²	1	4	yes	—
15	C₂×#41×C₄₇	—	2	6	no	—
16	C₂×#41×C₄₇	C₂²	1	3	yes	—
17	#41×C₄₇	—	2	4	no	—
18	#41×C₄₇	—	2	12	no	—
19	#41×C₄₇	—	2	6	no	—
20	#41×C₄₇	—	2	12	no	—
21	#41×C₄₇	—	2	6	no	—
22	C₂×Q₈×C₄₇	—	2	6	no	—
23	C₂×Q₈×C₄₇	D₄	1	3	yes	—
24	C₂×D₄×C₄₇	—	2	6	no	—
25	C₂×D₄×C₄₇	—	2	12	no	—
26	C₂×D₄×C₄₇	D₄	1	3	yes	—
27	C₂×D₄×C₄₇	C₂³	1	6	yes	—
28	C₂²×C₄×C₄₇	—	2	4	no	—
29	C₂²×C₄×C₄₇	D₄	1	6	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×Dic₄₇	C₂³	1	2	yes	—
32	C₂×Dic₉₄	—	2	3	no	—
33	C₂×Dic₉₄	—	2	3	no	—
34	C₂²⋊Dic₄₇	—	2	12	no	—
35	C₂²⋊Dic₄₇	C₂³	1	6	yes	—
36	C₄⋊Dic₄₇	—	2	3	no	—
37	Dic₄₇⋊C₄	—	2	12	no	—
38	C₄×Dic₄₇	—	2	3	no	—
39	C₂³×C₄₇	—	2	4	no	—
40	C₂³×C₄₇	C₂×D₄	1	6	yes	—
41	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
42	Q₈×C₄₇	—	2	4	no	—
43	Q₈×C₄₇	—	2	6	no	—
44	D₄×C₄₇	—	2	12	no	—
45	D₄×C₄₇	—	2	12	no	—
46	D₄×C₄₇	—	2	6	no	—
47	C₂×C₄×C₄₇	—	2	12	no	—
48	C₂×C₄×C₄₇	—	2	6	no	—
49	C₂×C₄×C₄₇	—	2	6	no	—
50	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
51	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
53	C₂×Dic₄₇	—	2	6	no	—
54	C₂×Dic₄₇	—	2	6	no	—
55	Dic₉₄	—	4	1	no	—
56	Dic₉₄	—	4	3	no	—
57	C₂²×C₄₇	—	2	6	no	—
58	C₂²×C₄₇	—	4	1	no	—
59	C₂²×C₄₇	—	4	6	no	—
60	C₂²×C₄₇	—	8	4	no	—
61	C₂²×C₄₇	C₂²×D₄	1	3	yes	—
62	C₄×C₄₇	—	2	6	no	—
63	C₄×C₄₇	#120	1	1	yes	characteristic
64	C₄×C₄₇	#120	1	3	yes	—
65	Dic₄₇	—	4	6	no	—
66	Dic₄₇	—	4	2	no	—
67	#3028	—	47	1	no	maximal, Sylow
68	C₂×C₄₇	—	2	3	no	—
69	C₂×C₄₇	—	4	6	no	—
70	C₂×C₄₇	C₂×#120	1	1	yes	characteristic
71	#585	D₄₇	1	1	yes	characteristic
72	#536	—	47	3	no	—
73	#458	—	47	8	no	—
74	#410	—	47	3	no	—
75	C₄₇	#3028	1	1	yes	characteristic, complemented, Sylow
76	2₋¹⁺⁴	—	2	3	no	—
77	H(5,𝔽₂)	—	2	3	no	—
78	H(5,𝔽₂)	C₂×D₄₇	1	4	yes	—
79	C₂×#41	—	2	6	no	—
80	C₂×#41	C₂×D₄₇	1	3	yes	—
81	C₂²×Q₈	—	47	1	no	—
82	C₄⋊Q₈	—	94	3	no	—
83	#124	—	94	3	no	—
84	#123	—	47	12	no	—
85	C₂²⋊Q₈	—	94	6	no	—
86	#117	—	47	3	no	—
87	#99	—	47	12	no	—
88	#41	—	2	4	no	—
89	#41	—	2	12	no	—
90	#41	—	2	6	no	—
91	#41	—	2	12	no	—
92	#41	—	2	6	no	—
93	C₂×Q₈	—	2	6	no	—
94	C₂×Q₈	—	94	3	no	—
95	C₂×Q₈	—	94	3	no	—
96	C₂×Q₈	C₂²⋊D₄₇	1	3	yes	—
97	C₂×D₄	—	2	6	no	—
98	C₂×D₄	—	2	12	no	—
99	C₂×D₄	C₂²⋊D₄₇	1	3	yes	—
100	C₂×D₄	C₂²×D₄₇	1	6	yes	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	47	2	no	—
103	C₂²×C₄	C₂²⋊D₄₇	1	6	yes	—
104	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
105	C₄⋊C₄	—	94	12	no	—
106	C₄⋊C₄	—	94	3	no	—
107	C₂²⋊C₄	—	47	6	no	—
108	C₂²⋊C₄	—	94	12	no	—
109	C₄²	—	94	3	no	—
110	C₂³	—	2	4	no	—
111	C₂³	C₂×C₂²⋊D₄₇	1	6	yes	—
112	C₂³	C₂³×D₄₇	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	6	no	—
115	Q₈	—	188	1	no	—
116	Q₈	—	188	3	no	—
117	D₄	—	2	12	no	—
118	D₄	—	2	12	no	—
119	D₄	—	2	6	no	—
120	C₂×C₄	—	2	12	no	—
121	C₂×C₄	—	2	6	no	—
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	—	94	6	no	—
124	C₂×C₄	—	94	6	no	—
125	C₂×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
127	C₂×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
128	C₂²	—	2	6	no	—
129	C₂²	—	4	1	no	—
130	C₂²	—	4	6	no	—
131	C₂²	—	8	4	no	—
132	C₂²	C₂²×C₂²⋊D₄₇	1	3	yes	—
133	C₄	—	2	6	no	—
134	C₄	—	188	6	no	—
135	C₄	—	188	2	no	—
136	C₄	#13093993	1	1	yes	characteristic
137	C₄	#13093993	1	3	yes	—
138	C₂	—	2	3	no	—
139	C₂	—	4	6	no	—
140	C₂	C₂×#13093993	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689694

C₂

yes

maximal, complemented

#15689313

C₂

yes

maximal, complemented

#15689097

C₂

yes

maximal, complemented

C₄₇×#585

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

characteristic, complemented

#13094022

—

#13094011

—

#13094005

C₂²

yes

complemented

#13094003

—

#13093983

C₂²

yes

complemented

#13093939

C₂²

yes

complemented

2₋¹⁺⁴×C₄₇

—

H(5,𝔽₂)×C₄₇

—

H(5,𝔽₂)×C₄₇

C₂²

yes

—

C₂×#41×C₄₇

—

C₂×#41×C₄₇

C₂²

yes

—

#41×C₄₇

—

#41×C₄₇

—

#41×C₄₇

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

—

C₂×D₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

C₂³

yes

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

—

C₄×C₄₇

—

C₄×C₄₇

#120

yes

characteristic

C₄×C₄₇

#120

yes

—

Dic₄₇

—

Dic₄₇

—

#3028

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#120

yes

characteristic

#585

D₄₇

yes

—

—

—

C₄₇

#3028

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₄₇

yes

—

C₂×#41

—

C₂×#41

C₂×D₄₇

yes

—

C₂²×Q₈

—

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

—

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₄₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₄₇

yes

—

100

C₂×D₄

C₂²×D₄₇

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊D₄₇

yes

—

104

C₂²×C₄

C₂²×D₄₇

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₂³

—

111

C₂³

C₂×C₂²⋊D₄₇

yes

—

112

C₂³

C₂³×D₄₇

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

188

—

116

Q₈

—

188

—

117

D₄

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

126

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

127

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂²×C₂²⋊D₄₇

yes

—

133

C₄

—

134

C₄

—

188

—

135

C₄

—

188

—

136

C₄

#13093993

yes

characteristic

137

C₄

#13093993

yes

—

138

C₂

—

139

C₂

—

140

C₂

C₂×#13093993

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#13093993
commutator	C₂³×C₄₇	C₂⁴
Frattini	C₂³	C₂³×D₄₇
Fitting	C₄₇×#585	C₂
socle	C₂×C₄₇	C₂×#120
hypercenter	#585	D₄₇
hypocenter	C₄₇	#3028
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₂

C₂×#13093993

commutator

C₂³×C₄₇

C₂⁴

Frattini

C₂³

C₂³×D₄₇

Fitting

C₄₇×#585

C₂

C₂×C₄₇

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #15689694	C₂	semidirect
2. #15689313	C₂	semidirect
3. #15689097	C₂	semidirect
5. C₂²×Dic₉₄	C₂²	semidirect
8. #13094005	C₂²	semidirect
10. #13093983	C₂²	semidirect
11. #13093939	C₂²	semidirect
75. C₄₇	#3028	semidirect

H ≅ G/N

type

1. #15689694

C₂

semidirect

2. #15689313

C₂

semidirect

3. #15689097

C₂

semidirect

5. C₂²×Dic₉₄

C₂²

semidirect

8. #13094005

C₂²

semidirect

10. #13093983

C₂²

semidirect

11. #13093939

C₂²

semidirect

75. C₄₇

#3028

semidirect

Subgroups — #20252794

Conjugacy classes of subgroups

Complements