Group #21458970 — subgroups

Subgroups — #21458970

Order 7808.

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

normal subgroups only

This group has 350 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	#16840579	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840203	C₂	1	3	yes	maximal, complemented
3	#16840202	C₂	1	3	yes	maximal, complemented
4	#16840192	C₂	1	3	yes	maximal, complemented
5	C₂×#14519723	C₂	1	3	yes	maximal, complemented
6	#16840179	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#578	C₂	1	1	yes	maximal, characteristic, complemented
8	#14519824	—	2	1	no	—
9	#14519810	—	2	3	no	—
10	#14519799	C₂²	1	3	yes	—
11	C₂×Dic₂₄₄	C₂²	1	3	yes	—
12	#14519723	C₂²	1	3	yes	—
13	#14519723	C₂²	1	6	yes	—
14	#14519721	C₂²	1	3	yes	—
15	#14519720	C₂²	1	3	yes	—
16	#14519718	C₂²	1	1	yes	characteristic
17	#14519717	C₂²	1	3	yes	—
18	C₂×#11758185	C₂²	1	3	yes	—
19	2₋¹⁺⁴×C₆₁	—	2	1	no	—
20	#137×C₆₁	—	2	3	no	—
21	#135×C₆₁	C₂²	1	3	yes	—
22	C₂×Q₁₆×C₆₁	C₂²	1	3	yes	—
23	#131×C₆₁	C₂²	1	1	yes	characteristic
24	#41×C₆₁	—	2	3	no	—
25	#41×C₆₁	—	4	3	no	—
26	#41×C₆₁	D₄	1	1	yes	characteristic
27	C₂×Q₈×C₆₁	—	2	3	no	—
28	C₂×Q₈×C₆₁	—	4	1	no	—
29	Q₈×D₆₁	—	4	1	no	—
30	#11758224	—	4	3	no	—
31	#11758222	—	2	3	no	—
32	C₂×Dic₁₂₂	—	2	3	no	—
33	Q₁₆×C₆₁	—	2	3	no	—
34	Q₁₆×C₆₁	C₂³	1	3	yes	—
35	SD₁₆×C₆₁	—	2	3	no	—
36	D₈×C₆₁	C₂³	1	1	yes	characteristic
37	M₁₆×C₆₁	D₄	1	3	yes	—
38	C₂×C₈×C₆₁	C₂³	1	3	yes	—
39	Dic₂₄₄	—	2	3	no	—
40	Dic₂₄₄	C₂³	1	3	yes	—
41	D₄₈₈	C₂³	1	1	yes	characteristic
42	#11758190	—	2	3	no	—
43	#11758185	C₂³	1	1	yes	characteristic
44	#11758185	C₂³	1	3	yes	—
45	Q₈×C₆₁	—	2	3	no	—
46	Q₈×C₆₁	—	4	3	no	—
47	Q₈×C₆₁	D₈	1	1	yes	characteristic
48	D₄×C₆₁	—	2	1	no	—
49	D₄×C₆₁	—	4	3	no	—
50	D₄×C₆₁	D₈	1	3	yes	—
51	C₂×C₄×C₆₁	—	4	3	no	—
52	C₂×C₄×C₆₁	—	4	3	no	—
53	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
54	C₂²⋊D₆₁	—	4	3	no	—
55	C₂×Dic₆₁	—	4	3	no	—
56	D₂₄₄	—	2	1	no	—
57	C₄×D₆₁	—	4	3	no	—
58	Dic₁₂₂	—	2	3	no	—
59	Dic₁₂₂	—	4	3	no	—
60	C₈×C₆₁	C₂×D₄	1	3	yes	—
61	C₈×C₆₁	C₂⁴	1	1	yes	characteristic
62	C₂²×C₆₁	—	4	1	no	—
63	C₂²×C₆₁	C₂×D₈	1	3	yes	—
64	C₂×D₆₁	—	4	1	no	—
65	C₄×C₆₁	—	4	3	no	—
66	C₄×C₆₁	C₂×D₈	1	3	yes	—
67	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
68	Dic₆₁	—	4	3	no	—
69	#3417	—	61	1	no	maximal, Sylow
70	C₂×C₆₁	—	2	3	no	—
71	C₂×C₆₁	—	8	1	no	—
72	C₂×C₆₁	C₂²×D₈	1	1	yes	characteristic
73	D₆₁	—	8	1	no	—
74	#578	—	61	1	no	—
75	#578	D₆₁	1	1	yes	characteristic, complemented
76	#510	—	61	3	no	—
77	#510	—	61	3	no	—
78	#508	—	61	3	no	—
79	C₂×Q₃₂	—	61	3	no	—
80	#504	—	61	1	no	—
81	C₆₁	#3417	1	1	yes	characteristic, complemented, Sylow
82	2₋¹⁺⁴	—	2	1	no	—
83	2₋¹⁺⁴	—	122	1	no	—
84	#137	—	2	3	no	—
85	#137	—	122	3	no	—
86	#135	—	61	3	no	—
87	#135	C₂×D₆₁	1	3	yes	—
88	C₂×Q₁₆	—	61	3	no	—
89	C₂×Q₁₆	C₂×D₆₁	1	3	yes	—
90	#131	C₂×D₆₁	1	1	yes	characteristic
91	Q₃₂	—	61	3	no	—
92	Q₃₂	—	61	6	no	—
93	SD₃₂	—	61	3	no	—
94	SD₃₂	—	61	3	no	—
95	D₁₆	—	61	1	no	—
96	M₃₂	—	61	3	no	—
97	C₂×C₁₆	—	61	3	no	—
98	#41	—	2	3	no	—
99	#41	—	4	3	no	—
100	#41	—	122	3	no	—
101	#41	—	244	3	no	—
102	#41	C₂²⋊D₆₁	1	1	yes	characteristic
103	C₂×Q₈	—	2	3	no	—
104	C₂×Q₈	—	4	1	no	—
105	C₂×Q₈	—	122	3	no	—
106	C₂×Q₈	—	244	1	no	—
107	Q₁₆	—	2	3	no	—
108	Q₁₆	—	61	3	no	—
109	Q₁₆	—	122	3	no	—
110	Q₁₆	C₂²×D₆₁	1	3	yes	—
111	SD₁₆	—	2	3	no	—
112	SD₁₆	—	122	3	no	—
113	D₈	—	61	1	no	—
114	D₈	C₂²×D₆₁	1	1	yes	characteristic
115	M₁₆	C₂²⋊D₆₁	1	3	yes	—
116	C₂×C₈	C₂²×D₆₁	1	3	yes	—
117	C₁₆	—	61	1	no	—
118	C₁₆	—	61	3	no	—
119	Q₈	—	2	3	no	—
120	Q₈	—	4	3	no	—
121	Q₈	—	122	3	no	—
122	Q₈	—	244	3	no	—
123	Q₈	#11758199	1	1	yes	characteristic
124	D₄	—	2	1	no	—
125	D₄	—	4	3	no	—
126	D₄	—	122	1	no	—
127	D₄	—	244	3	no	—
128	D₄	#11758199	1	3	yes	—
129	C₂×C₄	—	4	3	no	—
130	C₂×C₄	—	4	3	no	—
131	C₂×C₄	—	244	3	no	—
132	C₂×C₄	—	244	3	no	—
133	C₂×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
134	C₈	C₂×C₂²⋊D₆₁	1	3	yes	—
135	C₈	C₂³×D₆₁	1	1	yes	characteristic
136	C₂²	—	4	1	no	—
137	C₂²	—	244	1	no	—
138	C₂²	C₂×#11758199	1	3	yes	—
139	C₄	—	4	3	no	—
140	C₄	—	244	3	no	—
141	C₄	C₂×#11758199	1	3	yes	—
142	C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
143	C₂	—	2	3	no	—
144	C₂	—	8	1	no	—
145	C₂	—	488	1	no	—
146	C₂	C₂²×#11758199	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840579

C₂

yes

maximal, characteristic, complemented

#16840203

C₂

yes

maximal, complemented

#16840202

C₂

yes

maximal, complemented

#16840192

C₂

yes

maximal, complemented

C₂×#14519723

C₂

yes

maximal, complemented

#16840179

C₂

yes

maximal, characteristic, complemented

C₆₁×#578

C₂

yes

maximal, characteristic, complemented

#14519824

—

#14519810

—

#14519799

C₂²

yes

—

C₂×Dic₂₄₄

C₂²

yes

—

#14519723

C₂²

yes

—

#14519723

C₂²

yes

—

#14519721

C₂²

yes

—

#14519720

C₂²

yes

—

#14519718

C₂²

yes

characteristic

#14519717

C₂²

yes

—

C₂×#11758185

C₂²

yes

—

2₋¹⁺⁴×C₆₁

—

#137×C₆₁

—

#135×C₆₁

C₂²

yes

—

C₂×Q₁₆×C₆₁

C₂²

yes

—

#131×C₆₁

C₂²

yes

characteristic

#41×C₆₁

—

#41×C₆₁

—

#41×C₆₁

D₄

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

Q₈×D₆₁

—

#11758224

—

#11758222

—

C₂×Dic₁₂₂

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

C₂³

yes

—

SD₁₆×C₆₁

—

D₈×C₆₁

C₂³

yes

characteristic

M₁₆×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

Dic₂₄₄

—

Dic₂₄₄

C₂³

yes

—

D₄₈₈

C₂³

yes

characteristic

#11758190

—

#11758185

C₂³

yes

characteristic

#11758185

C₂³

yes

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

D₈

yes

characteristic

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

D₈

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

C₂×D₄

yes

—

C₈×C₆₁

C₂⁴

yes

characteristic

C₂²×C₆₁

—

C₂²×C₆₁

C₂×D₈

yes

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

—

C₄×C₆₁

C₂²×D₄

yes

characteristic

Dic₆₁

—

#3417

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₂²×D₈

yes

characteristic

D₆₁

—

#578

—

#578

D₆₁

yes

characteristic, complemented

#510

—

#510

—

#508

—

C₂×Q₃₂

—

#504

—

C₆₁

#3417

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

2₋¹⁺⁴

—

122

—

#137

—

#137

—

122

—

#135

—

#135

C₂×D₆₁

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₆₁

yes

—

#131

C₂×D₆₁

yes

characteristic

Q₃₂

—

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

#41

—

#41

—

100

#41

—

122

—

101

#41

—

244

—

102

#41

C₂²⋊D₆₁

yes

characteristic

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

122

—

106

C₂×Q₈

—

244

—

107

Q₁₆

—

108

Q₁₆

—

109

Q₁₆

—

122

—

110

Q₁₆

C₂²×D₆₁

yes

—

111

SD₁₆

—

112

SD₁₆

—

122

—

113

D₈

—

114

D₈

C₂²×D₆₁

yes

characteristic

115

M₁₆

C₂²⋊D₆₁

yes

—

116

C₂×C₈

C₂²×D₆₁

yes

—

117

C₁₆

—

118

C₁₆

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

—

122

Q₈

—

244

—

123

Q₈

#11758199

yes

characteristic

124

D₄

—

125

D₄

—

126

D₄

—

122

—

127

D₄

—

244

—

128

D₄

#11758199

yes

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

244

—

132

C₂×C₄

—

244

—

133

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

134

C₈

C₂×C₂²⋊D₆₁

yes

—

135

C₈

C₂³×D₆₁

yes

characteristic

136

C₂²

—

137

C₂²

—

244

—

138

C₂²

C₂×#11758199

yes

—

139

C₄

—

140

C₄

—

244

—

141

C₄

C₂×#11758199

yes

—

142

C₄

C₂²×C₂²⋊D₆₁

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

488

—

146

C₂

C₂²×#11758199

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#11758199
commutator	C₈×C₆₁	C₂⁴
Frattini	C₈	C₂³×D₆₁
Fitting	C₆₁×#578	C₂
socle	C₂×C₆₁	C₂²×D₈
hypercenter	#578	D₆₁
hypocenter	C₆₁	#3417
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #578

subgroup

G/H

center

C₂

C₂²×#11758199

commutator

C₈×C₆₁

C₂⁴

Frattini

C₈

C₂³×D₆₁

Fitting

C₆₁×#578

C₂

socle

C₂×C₆₁

C₂²×D₈

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #578

N	H ≅ G/N	type
1. #16840579	C₂	semidirect
2. #16840203	C₂	semidirect
3. #16840202	C₂	semidirect
4. #16840192	C₂	semidirect
5. C₂×#14519723	C₂	semidirect
6. #16840179	C₂	semidirect
7. C₆₁×#578	C₂	semidirect
75. #578	D₆₁	semidirect
81. C₆₁	#3417	semidirect

H ≅ G/N

type

1. #16840579

C₂

semidirect

2. #16840203

C₂

semidirect

3. #16840202

C₂

semidirect

4. #16840192

C₂

semidirect

5. C₂×#14519723

C₂

semidirect

6. #16840179

C₂

semidirect

7. C₆₁×#578

C₂

semidirect

75. #578

D₆₁

semidirect

81. C₆₁

#3417

semidirect

Subgroups — #21458970

Conjugacy classes of subgroups

Complements