Group #21458958 — subgroups

Subgroups — #21458958

Order 7808.

370 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 129 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	#16840567	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840198	C₂	1	3	yes	maximal, complemented
3	#16840197	C₂	1	3	yes	maximal, complemented
4	#16840192	C₂	1	3	yes	maximal, complemented
5	C₂×#14519718	C₂	1	3	yes	maximal, complemented
6	#16840179	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#576	C₂	1	1	yes	maximal, characteristic, complemented
8	#14519823	—	2	1	no	—
9	#14519805	—	2	3	no	—
10	#14519799	C₂²	1	3	yes	complemented
11	C₂×D₄₈₈	C₂²	1	3	yes	complemented
12	#14519723	C₂²	1	1	yes	characteristic, complemented
13	#14519721	C₂²	1	3	yes	complemented
14	#14519720	C₂²	1	3	yes	complemented
15	#14519718	C₂²	1	3	yes	complemented
16	#14519718	C₂²	1	6	yes	complemented
17	#14519717	C₂²	1	3	yes	complemented
18	C₂×#11758185	C₂²	1	3	yes	complemented
19	H(5,𝔽₂)×C₆₁	—	2	1	no	—
20	AGL(1,ℤ/8)×C₆₁	—	2	3	no	—
21	#135×C₆₁	C₂²	1	3	yes	complemented
22	C₂×D₈×C₆₁	C₂²	1	3	yes	complemented
23	#131×C₆₁	C₂²	1	1	yes	characteristic
24	#41×C₆₁	—	2	3	no	—
25	#41×C₆₁	—	4	1	no	—
26	#41×C₆₁	D₄	1	1	yes	characteristic
27	C₂×D₄×C₆₁	—	2	3	no	—
28	C₂×D₄×C₆₁	—	4	3	no	—
29	#11758226	—	4	1	no	—
30	D₄×D₆₁	—	4	3	no	—
31	#11758222	—	2	3	no	—
32	C₂×D₂₄₄	—	2	3	no	—
33	Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
34	SD₁₆×C₆₁	—	2	3	no	—
35	D₈×C₆₁	—	2	3	no	—
36	D₈×C₆₁	C₂³	1	3	yes	—
37	M₁₆×C₆₁	D₄	1	3	yes	—
38	C₂×C₈×C₆₁	C₂³	1	3	yes	—
39	Dic₂₄₄	C₂³	1	1	yes	characteristic
40	D₄₈₈	—	2	3	no	—
41	D₄₈₈	C₂³	1	3	yes	—
42	#11758190	—	2	3	no	—
43	#11758185	C₂³	1	1	yes	characteristic
44	#11758185	C₂³	1	3	yes	—
45	C₂³×C₆₁	—	4	3	no	—
46	C₂²×D₆₁	—	4	3	no	—
47	Q₈×C₆₁	—	2	1	no	—
48	Q₈×C₆₁	D₈	1	1	yes	characteristic
49	D₄×C₆₁	—	2	3	no	—
50	D₄×C₆₁	—	4	3	no	—
51	D₄×C₆₁	—	4	3	no	—
52	D₄×C₆₁	D₈	1	3	yes	—
53	C₂×C₄×C₆₁	—	4	3	no	—
54	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
55	C₂²⋊D₆₁	—	4	3	no	—
56	D₂₄₄	—	2	3	no	—
57	D₂₄₄	—	4	3	no	—
58	C₄×D₆₁	—	4	3	no	—
59	Dic₁₂₂	—	2	1	no	—
60	C₈×C₆₁	C₂×D₄	1	3	yes	—
61	C₈×C₆₁	C₂⁴	1	1	yes	characteristic
62	C₂²×C₆₁	—	4	3	no	—
63	C₂²×C₆₁	—	16	3	no	—
64	C₂²×C₆₁	C₂×D₈	1	3	yes	—
65	C₂×D₆₁	—	4	3	no	—
66	C₂×D₆₁	—	16	3	no	—
67	C₄×C₆₁	—	4	1	no	—
68	C₄×C₆₁	C₂×D₈	1	3	yes	—
69	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
70	Dic₆₁	—	4	1	no	—
71	#3415	—	61	1	no	maximal, Sylow
72	C₂×C₆₁	—	2	3	no	—
73	C₂×C₆₁	—	8	3	no	—
74	C₂×C₆₁	C₂²×D₈	1	1	yes	characteristic
75	D₆₁	—	8	3	no	—
76	#576	—	61	1	no	—
77	#576	D₆₁	1	1	yes	characteristic, complemented
78	#509	—	61	3	no	—
79	#509	—	61	3	no	—
80	#508	—	61	3	no	—
81	C₂×D₁₆	—	61	3	no	—
82	#504	—	61	1	no	—
83	C₆₁	#3415	1	1	yes	characteristic, complemented, Sylow
84	H(5,𝔽₂)	—	2	1	no	—
85	H(5,𝔽₂)	—	122	1	no	—
86	AGL(1,ℤ/8)	—	2	3	no	—
87	AGL(1,ℤ/8)	—	122	3	no	—
88	#135	—	61	3	no	—
89	#135	C₂×D₆₁	1	3	yes	complemented
90	C₂×D₈	—	61	3	no	—
91	C₂×D₈	C₂×D₆₁	1	3	yes	complemented
92	#131	C₂×D₆₁	1	1	yes	characteristic
93	Q₃₂	—	61	1	no	—
94	SD₃₂	—	61	3	no	—
95	SD₃₂	—	61	3	no	—
96	D₁₆	—	61	3	no	—
97	D₁₆	—	61	6	no	—
98	M₃₂	—	61	3	no	—
99	C₂×C₁₆	—	61	3	no	—
100	#41	—	2	3	no	—
101	#41	—	4	1	no	—
102	#41	—	122	3	no	—
103	#41	—	244	1	no	—
104	#41	C₂²⋊D₆₁	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	4	3	no	—
107	C₂×D₄	—	122	3	no	—
108	C₂×D₄	—	244	3	no	—
109	Q₁₆	—	61	1	no	—
110	Q₁₆	C₂²×D₆₁	1	1	yes	characteristic
111	SD₁₆	—	2	3	no	—
112	SD₁₆	—	122	3	no	—
113	D₈	—	2	3	no	—
114	D₈	—	61	3	no	—
115	D₈	—	122	3	no	—
116	D₈	C₂²×D₆₁	1	3	yes	—
117	M₁₆	C₂²⋊D₆₁	1	3	yes	—
118	C₂×C₈	C₂²×D₆₁	1	3	yes	—
119	C₁₆	—	61	1	no	—
120	C₁₆	—	61	3	no	—
121	C₂³	—	4	3	no	—
122	C₂³	—	244	3	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	122	1	no	—
125	Q₈	#11758199	1	1	yes	characteristic
126	D₄	—	2	3	no	—
127	D₄	—	4	3	no	—
128	D₄	—	4	3	no	—
129	D₄	—	122	3	no	—
130	D₄	—	244	3	no	—
131	D₄	—	244	3	no	—
132	D₄	#11758199	1	3	yes	—
133	C₂×C₄	—	4	3	no	—
134	C₂×C₄	—	244	3	no	—
135	C₂×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
136	C₈	C₂×C₂²⋊D₆₁	1	3	yes	—
137	C₈	C₂³×D₆₁	1	1	yes	characteristic
138	C₂²	—	4	3	no	—
139	C₂²	—	16	3	no	—
140	C₂²	—	976	3	no	—
141	C₂²	—	244	3	no	—
142	C₂²	C₂×#11758199	1	3	yes	—
143	C₄	—	4	1	no	—
144	C₄	—	244	1	no	—
145	C₄	C₂×#11758199	1	3	yes	—
146	C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
147	C₂	—	2	3	no	—
148	C₂	—	8	3	no	—
149	C₂	—	488	3	no	—
150	C₂	C₂²×#11758199	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840567

C₂

yes

maximal, characteristic, complemented

#16840198

C₂

yes

maximal, complemented

#16840197

C₂

yes

maximal, complemented

#16840192

C₂

yes

maximal, complemented

C₂×#14519718

C₂

yes

maximal, complemented

#16840179

C₂

yes

maximal, characteristic, complemented

C₆₁×#576

C₂

yes

maximal, characteristic, complemented

#14519823

—

#14519805

—

#14519799

C₂²

yes

complemented

C₂×D₄₈₈

C₂²

yes

complemented

#14519723

C₂²

yes

characteristic, complemented

#14519721

C₂²

yes

complemented

#14519720

C₂²

yes

complemented

#14519718

C₂²

yes

complemented

#14519718

C₂²

yes

complemented

#14519717

C₂²

yes

complemented

C₂×#11758185

C₂²

yes

complemented

H(5,𝔽₂)×C₆₁

—

AGL(1,ℤ/8)×C₆₁

—

#135×C₆₁

C₂²

yes

complemented

C₂×D₈×C₆₁

C₂²

yes

complemented

#131×C₆₁

C₂²

yes

characteristic

#41×C₆₁

—

#41×C₆₁

—

#41×C₆₁

D₄

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

#11758226

—

D₄×D₆₁

—

#11758222

—

C₂×D₂₄₄

—

Q₁₆×C₆₁

C₂³

yes

characteristic

SD₁₆×C₆₁

—

D₈×C₆₁

—

D₈×C₆₁

C₂³

yes

—

M₁₆×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

Dic₂₄₄

C₂³

yes

characteristic

D₄₈₈

—

D₄₈₈

C₂³

yes

—

#11758190

—

#11758185

C₂³

yes

characteristic

#11758185

C₂³

yes

—

C₂³×C₆₁

—

C₂²×D₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

D₈

yes

characteristic

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

D₈

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂²⋊D₆₁

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

C₂×D₄

yes

—

C₈×C₆₁

C₂⁴

yes

characteristic

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×D₈

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

—

C₄×C₆₁

C₂²×D₄

yes

characteristic

Dic₆₁

—

#3415

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₂²×D₈

yes

characteristic

D₆₁

—

#576

—

#576

D₆₁

yes

characteristic, complemented

#509

—

#509

—

#508

—

C₂×D₁₆

—

#504

—

C₆₁

#3415

yes

characteristic, complemented, Sylow

H(5,𝔽₂)

—

H(5,𝔽₂)

—

122

—

AGL(1,ℤ/8)

—

AGL(1,ℤ/8)

—

122

—

#135

—

#135

C₂×D₆₁

yes

complemented

C₂×D₈

—

C₂×D₈

C₂×D₆₁

yes

complemented

#131

C₂×D₆₁

yes

characteristic

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

100

#41

—

101

#41

—

102

#41

—

122

—

103

#41

—

244

—

104

#41

C₂²⋊D₆₁

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

122

—

108

C₂×D₄

—

244

—

109

Q₁₆

—

110

Q₁₆

C₂²×D₆₁

yes

characteristic

111

SD₁₆

—

112

SD₁₆

—

122

—

113

D₈

—

114

D₈

—

115

D₈

—

122

—

116

D₈

C₂²×D₆₁

yes

—

117

M₁₆

C₂²⋊D₆₁

yes

—

118

C₂×C₈

C₂²×D₆₁

yes

—

119

C₁₆

—

120

C₁₆

—

121

C₂³

—

122

C₂³

—

244

—

123

Q₈

—

124

Q₈

—

122

—

125

Q₈

#11758199

yes

characteristic

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

122

—

130

D₄

—

244

—

131

D₄

—

244

—

132

D₄

#11758199

yes

—

133

C₂×C₄

—

134

C₂×C₄

—

244

—

135

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

136

C₈

C₂×C₂²⋊D₆₁

yes

—

137

C₈

C₂³×D₆₁

yes

characteristic

138

C₂²

—

139

C₂²

—

140

C₂²

—

976

—

141

C₂²

—

244

—

142

C₂²

C₂×#11758199

yes

—

143

C₄

—

144

C₄

—

244

—

145

C₄

C₂×#11758199

yes

—

146

C₄

C₂²×C₂²⋊D₆₁

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

—

488

—

150

C₂

C₂²×#11758199

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#11758199
commutator	C₈×C₆₁	C₂⁴
Frattini	C₈	C₂³×D₆₁
Fitting	C₆₁×#576	C₂
socle	C₂×C₆₁	C₂²×D₈
hypercenter	#576	D₆₁
hypocenter	C₆₁	#3415
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

subgroup

G/H

center

C₂

C₂²×#11758199

commutator

C₈×C₆₁

C₂⁴

Frattini

C₈

C₂³×D₆₁

Fitting

C₆₁×#576

C₂

socle

C₂×C₆₁

C₂²×D₈

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

N	H ≅ G/N	type
1. #16840567	C₂	semidirect
2. #16840198	C₂	semidirect
3. #16840197	C₂	semidirect
4. #16840192	C₂	semidirect
5. C₂×#14519718	C₂	semidirect
6. #16840179	C₂	semidirect
7. C₆₁×#576	C₂	semidirect
10. #14519799	C₂²	semidirect
11. C₂×D₄₈₈	C₂²	semidirect
12. #14519723	C₂²	semidirect
13. #14519721	C₂²	semidirect
14. #14519720	C₂²	semidirect
15. #14519718	C₂²	semidirect
16. #14519718	C₂²	semidirect
17. #14519717	C₂²	semidirect
18. C₂×#11758185	C₂²	semidirect
21. #135×C₆₁	C₂²	semidirect
22. C₂×D₈×C₆₁	C₂²	semidirect
77. #576	D₆₁	semidirect
83. C₆₁	#3415	semidirect
89. #135	C₂×D₆₁	semidirect
91. C₂×D₈	C₂×D₆₁	semidirect

H ≅ G/N

type

1. #16840567

C₂

semidirect

2. #16840198

C₂

semidirect

3. #16840197

C₂

semidirect

4. #16840192

C₂

semidirect

5. C₂×#14519718

C₂

semidirect

6. #16840179

C₂

semidirect

7. C₆₁×#576

C₂

semidirect

10. #14519799

C₂²

semidirect

11. C₂×D₄₈₈

C₂²

semidirect

12. #14519723

C₂²

semidirect

13. #14519721

C₂²

semidirect

14. #14519720

C₂²

semidirect

15. #14519718

C₂²

semidirect

16. #14519718

C₂²

semidirect

17. #14519717

C₂²

semidirect

18. C₂×#11758185

C₂²

semidirect

21. #135×C₆₁

C₂²

semidirect

22. C₂×D₈×C₆₁

C₂²

semidirect

77. #576

D₆₁

semidirect

83. C₆₁

#3415

semidirect

89. #135

C₂×D₆₁

semidirect

91. C₂×D₈

C₂×D₆₁

semidirect

Subgroups — #21458958

Conjugacy classes of subgroups

Complements