Group #20256489 — subgroups

Subgroups — #20256489

Order 6016.

360 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 209 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	#131×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#15689541	C₂	1	3	yes	maximal, complemented
3	#15689540	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689539	C₂	1	3	yes	maximal, complemented
5	C₂×#13093963	C₂	1	3	yes	maximal, complemented
6	M₃₂×D₄₇	C₂	1	3	yes	maximal, complemented
7	C₄₇×#504	C₂	1	1	yes	maximal, characteristic, complemented
8	#41×D₄₇	C₄	1	1	yes	characteristic
9	#13094032	C₄	1	3	yes	—
10	#13094031	C₄	1	1	yes	characteristic
11	#13094030	C₂²	1	3	yes	—
12	M₁₆×D₄₇	C₄	1	3	yes	—
13	C₂×C₈×D₄₇	C₂²	1	3	yes	—
14	#13093964	C₂²	1	3	yes	complemented
15	#13093963	C₂²	1	3	yes	—
16	#13093963	C₂²	1	1	yes	characteristic
17	#13093963	C₂²	1	3	yes	complemented
18	#13093963	C₂²	1	3	yes	complemented
19	C₂×C₄₇⋊C₁₆	C₂²	1	3	yes	complemented
20	C₁₆×D₄₇	C₂²	1	6	yes	complemented
21	#131×C₄₇	C₂²	1	1	yes	characteristic
22	M₃₂×C₄₇	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₄₇	C₂²	1	3	yes	complemented
24	#41×C₄₇	C₂×C₄	1	1	yes	characteristic
25	#10625354	C₈	1	1	yes	characteristic
26	Q₈×D₄₇	C₈	1	1	yes	characteristic
27	#10625352	C₈	1	3	yes	—
28	D₄×D₄₇	C₈	1	3	yes	—
29	#10625350	C₂×C₄	1	3	yes	—
30	C₂×C₄×D₄₇	C₂×C₄	1	3	yes	—
31	M₁₆×C₄₇	C₂×C₄	1	3	yes	—
32	C₂×C₈×C₄₇	C₂³	1	3	yes	—
33	#10625329	C₂×C₄	1	3	yes	—
34	C₂×C₄₇⋊C₈	C₂×C₄	1	3	yes	—
35	#10625324	C₂×C₄	1	6	yes	—
36	C₈×D₄₇	C₂×C₄	1	3	yes	—
37	C₈×D₄₇	C₂×C₄	1	3	yes	—
38	C₈×D₄₇	C₂³	1	3	yes	—
39	C₈×D₄₇	C₂³	1	1	yes	characteristic
40	C₁₆×C₄₇	C₂³	1	1	yes	characteristic
41	C₁₆×C₄₇	C₂³	1	3	yes	—
42	C₄₇⋊C₁₆	C₂³	1	3	yes	—
43	C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
44	C₂²×D₄₇	C₂×C₈	1	3	yes	—
45	Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
46	D₄×C₄₇	C₂×C₈	1	3	yes	—
47	C₂×C₄×C₄₇	C₂²×C₄	1	3	yes	—
48	C₂²⋊D₄₇	C₂×C₈	1	6	yes	—
49	C₂×Dic₄₇	C₂×C₈	1	3	yes	—
50	D₁₈₈	C₂×C₈	1	3	yes	—
51	C₄×D₄₇	C₂×C₈	1	3	yes	—
52	C₄×D₄₇	C₂×C₈	1	3	yes	—
53	C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₄×D₄₇	C₂²×C₄	1	3	yes	—
55	Dic₉₄	C₂×C₈	1	3	yes	—
56	C₈×C₄₇	C₂²×C₄	1	3	yes	—
57	C₈×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
59	C₄₇⋊C₈	C₂²×C₄	1	3	yes	—
60	C₂²×C₄₇	C₂²×C₈	1	3	yes	—
61	C₂×D₄₇	—	4	3	no	—
62	C₂×D₄₇	C₂²×C₈	1	1	yes	characteristic
63	C₂×D₄₇	C₂²×C₈	1	3	yes	—
64	C₄×C₄₇	C₂²×C₈	1	3	yes	—
65	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
66	Dic₄₇	C₂²×C₈	1	3	yes	—
67	Dic₄₇	C₂²×C₈	1	1	yes	characteristic
68	#3407	—	47	1	no	maximal, Sylow
69	C₂×C₄₇	—	2	3	no	—
70	C₂×C₄₇	C₂³×C₈	1	1	yes	characteristic
71	D₄₇	—	2	1	no	—
72	D₄₇	—	2	3	no	—
73	C₂×#131	—	47	1	no	—
74	#504	—	47	1	no	—
75	#504	—	47	3	no	—
76	#504	—	47	3	no	—
77	#504	D₄₇	1	1	yes	characteristic, complemented
78	C₂×M₃₂	—	47	3	no	—
79	C₂×M₃₂	—	47	3	no	—
80	C₄₇	#3407	1	1	yes	characteristic, complemented, Sylow
81	C₂×#41	—	47	1	no	—
82	#131	—	47	1	no	—
83	#131	—	47	3	no	—
84	#131	—	47	3	no	—
85	#131	C₂×D₄₇	1	1	yes	characteristic
86	C₂×M₁₆	—	47	3	no	—
87	C₂²×C₈	—	47	3	no	—
88	M₃₂	—	47	3	no	—
89	M₃₂	—	47	3	no	—
90	M₃₂	—	47	1	no	—
91	M₃₂	—	47	3	no	—
92	M₃₂	—	47	3	no	—
93	M₃₂	C₂×D₄₇	1	3	yes	complemented
94	C₂×C₁₆	—	47	3	no	—
95	C₂×C₁₆	—	47	6	no	—
96	C₂×C₁₆	C₂×D₄₇	1	3	yes	complemented
97	#41	—	47	1	no	—
98	#41	—	47	3	no	—
99	#41	—	47	3	no	—
100	#41	C₄×D₄₇	1	1	yes	characteristic
101	C₂×Q₈	—	47	1	no	—
102	C₂×D₄	—	47	3	no	—
103	C₂²×C₄	—	47	3	no	—
104	M₁₆	—	47	3	no	—
105	M₁₆	—	47	6	no	—
106	M₁₆	C₄×D₄₇	1	3	yes	—
107	C₂×C₈	—	47	3	no	—
108	C₂×C₈	—	47	3	no	—
109	C₂×C₈	—	47	3	no	—
110	C₂×C₈	—	47	3	no	—
111	C₂×C₈	—	47	1	no	—
112	C₂×C₈	C₂²×D₄₇	1	3	yes	—
113	C₁₆	—	47	3	no	—
114	C₁₆	—	47	1	no	—
115	C₁₆	C₂²×D₄₇	1	3	yes	—
116	C₁₆	C₂²×D₄₇	1	1	yes	characteristic
117	C₂³	—	47	3	no	—
118	Q₈	—	47	3	no	—
119	Q₈	C₈×D₄₇	1	1	yes	characteristic
120	D₄	—	47	6	no	—
121	D₄	—	47	3	no	—
122	D₄	C₈×D₄₇	1	3	yes	—
123	C₂×C₄	—	47	3	no	—
124	C₂×C₄	—	47	3	no	—
125	C₂×C₄	—	47	3	no	—
126	C₂×C₄	—	47	1	no	—
127	C₂×C₄	—	47	3	no	—
128	C₂×C₄	C₂×C₄×D₄₇	1	3	yes	—
129	C₈	—	47	1	no	—
130	C₈	—	47	3	no	—
131	C₈	C₂×C₄×D₄₇	1	3	yes	—
132	C₈	C₂³×D₄₇	1	1	yes	characteristic
133	C₂²	—	47	1	no	—
134	C₂²	—	47	3	no	—
135	C₂²	—	188	3	no	—
136	C₂²	C₂×C₈×D₄₇	1	3	yes	—
137	C₄	—	47	1	no	—
138	C₄	—	47	3	no	—
139	C₄	C₂×C₈×D₄₇	1	3	yes	—
140	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
141	C₂	—	2	3	no	—
142	C₂	—	94	1	no	—
143	C₂	—	94	3	no	—
144	C₂	C₂²×C₈×D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#131×D₄₇

C₂

yes

maximal, characteristic

#15689541

C₂

yes

maximal, complemented

#15689540

C₂

yes

maximal, characteristic, complemented

#15689539

C₂

yes

maximal, complemented

C₂×#13093963

C₂

yes

maximal, complemented

M₃₂×D₄₇

C₂

yes

maximal, complemented

C₄₇×#504

C₂

yes

maximal, characteristic, complemented

#41×D₄₇

C₄

yes

characteristic

#13094032

C₄

yes

—

#13094031

C₄

yes

characteristic

#13094030

C₂²

yes

—

M₁₆×D₄₇

C₄

yes

—

C₂×C₈×D₄₇

C₂²

yes

—

#13093964

C₂²

yes

complemented

#13093963

C₂²

yes

—

#13093963

C₂²

yes

characteristic

#13093963

C₂²

yes

complemented

#13093963

C₂²

yes

complemented

C₂×C₄₇⋊C₁₆

C₂²

yes

complemented

C₁₆×D₄₇

C₂²

yes

complemented

#131×C₄₇

C₂²

yes

characteristic

M₃₂×C₄₇

C₂²

yes

complemented

C₂×C₁₆×C₄₇

C₂²

yes

complemented

#41×C₄₇

C₂×C₄

yes

characteristic

#10625354

C₈

yes

characteristic

Q₈×D₄₇

C₈

yes

characteristic

#10625352

C₈

yes

—

D₄×D₄₇

C₈

yes

—

#10625350

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

M₁₆×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

#10625329

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

#10625324

C₂×C₄

yes

—

C₈×D₄₇

C₂×C₄

yes

—

C₈×D₄₇

C₂×C₄

yes

—

C₈×D₄₇

C₂³

yes

—

C₈×D₄₇

C₂³

yes

characteristic

C₁₆×C₄₇

C₂³

yes

characteristic

C₁₆×C₄₇

C₂³

yes

—

C₄₇⋊C₁₆

C₂³

yes

—

C₄₇⋊C₁₆

C₂³

yes

characteristic

C₂²×D₄₇

C₂×C₈

yes

—

Q₈×C₄₇

C₂×C₈

yes

characteristic

D₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂²⋊D₄₇

C₂×C₈

yes

—

C₂×Dic₄₇

C₂×C₈

yes

—

D₁₈₈

C₂×C₈

yes

—

C₄×D₄₇

C₂×C₈

yes

—

C₄×D₄₇

C₂×C₈

yes

—

C₄×D₄₇

C₂²×C₄

yes

characteristic

C₄×D₄₇

C₂²×C₄

yes

—

Dic₉₄

C₂×C₈

yes

—

C₈×C₄₇

C₂²×C₄

yes

—

C₈×C₄₇

C₂⁴

yes

characteristic

C₄₇⋊C₈

C₂²×C₄

yes

characteristic

C₄₇⋊C₈

C₂²×C₄

yes

—

C₂²×C₄₇

C₂²×C₈

yes

—

C₂×D₄₇

—

C₂×D₄₇

C₂²×C₈

yes

characteristic

C₂×D₄₇

C₂²×C₈

yes

—

C₄×C₄₇

C₂²×C₈

yes

—

C₄×C₄₇

C₂³×C₄

yes

characteristic

Dic₄₇

C₂²×C₈

yes

—

Dic₄₇

C₂²×C₈

yes

characteristic

#3407

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂³×C₈

yes

characteristic

D₄₇

—

D₄₇

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₂×M₃₂

—

C₂×M₃₂

—

C₄₇

#3407

yes

characteristic, complemented, Sylow

C₂×#41

—

—

—

—

C₂×D₄₇

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

C₂×D₄₇

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄₇

yes

—

—

—

100

#41

C₄×D₄₇

yes

characteristic

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

C₄×D₄₇

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₄₇

yes

—

113

C₁₆

—

114

C₁₆

—

115

C₁₆

C₂²×D₄₇

yes

—

116

C₁₆

C₂²×D₄₇

yes

characteristic

117

C₂³

—

118

Q₈

—

119

Q₈

C₈×D₄₇

yes

characteristic

120

D₄

—

121

D₄

—

122

D₄

C₈×D₄₇

yes

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₄×D₄₇

yes

—

129

C₈

—

130

C₈

—

131

C₈

C₂×C₄×D₄₇

yes

—

132

C₈

C₂³×D₄₇

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

188

—

136

C₂²

C₂×C₈×D₄₇

yes

—

137

C₄

—

138

C₄

—

139

C₄

C₂×C₈×D₄₇

yes

—

140

C₄

C₂²×C₄×D₄₇

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

C₂²×C₈×D₄₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×D₄₇
commutator, socle	C₂×C₄₇	C₂³×C₈
Fitting	C₄₇×#504	C₂
hypercenter	#504	D₄₇
hypocenter	C₄₇	#3407
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ #504

subgroup

G/H

center, Frattini

C₈

C₂³×D₄₇

commutator, socle

C₂×C₄₇

C₂³×C₈

Fitting

C₄₇×#504

C₂

D₄₇

C₄₇

G ▷ C₂×C₄₇ ▷ C₄₇

C₁ ◁ C₈ ◁ #504

N	H ≅ G/N	type
2. #15689541	C₂	semidirect
3. #15689540	C₂	semidirect
4. #15689539	C₂	semidirect
5. C₂×#13093963	C₂	semidirect
6. M₃₂×D₄₇	C₂	semidirect
7. C₄₇×#504	C₂	semidirect
14. #13093964	C₂²	semidirect
17. #13093963	C₂²	semidirect
18. #13093963	C₂²	semidirect
19. C₂×C₄₇⋊C₁₆	C₂²	semidirect
20. C₁₆×D₄₇	C₂²	semidirect
22. M₃₂×C₄₇	C₂²	semidirect
23. C₂×C₁₆×C₄₇	C₂²	semidirect
77. #504	D₄₇	semidirect
80. C₄₇	#3407	semidirect
93. M₃₂	C₂×D₄₇	semidirect
96. C₂×C₁₆	C₂×D₄₇	semidirect

H ≅ G/N

type

2. #15689541

C₂

semidirect

3. #15689540

C₂

semidirect

4. #15689539

C₂

semidirect

5. C₂×#13093963

C₂

semidirect

6. M₃₂×D₄₇

C₂

semidirect

7. C₄₇×#504

C₂

semidirect

14. #13093964

C₂²

semidirect

17. #13093963

C₂²

semidirect

18. #13093963

C₂²

semidirect

19. C₂×C₄₇⋊C₁₆

C₂²

semidirect

20. C₁₆×D₄₇

C₂²

semidirect

22. M₃₂×C₄₇

C₂²

semidirect

23. C₂×C₁₆×C₄₇

C₂²

semidirect

77. #504

D₄₇

semidirect

80. C₄₇

#3407

semidirect

93. M₃₂

C₂×D₄₇

semidirect

96. C₂×C₁₆

C₂×D₄₇

semidirect

Subgroups — #20256489

Conjugacy classes of subgroups

Complements