Group #21447698 — subgroups

Subgroups — #21447698

Order 7808.

146 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16840192	C₂	1	1	yes	maximal, characteristic
2	#16839880	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839591	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839569	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839568	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#443	C₂	1	1	yes	maximal, characteristic, complemented
8	#14519799	C₂²	1	1	yes	characteristic
9	#14519783	—	2	1	no	—
10	#14519723	C₄	1	1	yes	characteristic, complemented
11	#14519721	C₄	1	1	yes	characteristic, complemented
12	#14519720	C₄	1	1	yes	characteristic, complemented
13	#14519718	C₄	1	1	yes	characteristic, complemented
14	C₂×#11758185	C₂²	1	1	yes	characteristic
15	C₂×#11758185	C₂²	1	1	yes	characteristic
16	#14519713	C₂²	1	1	yes	characteristic
17	#14519705	—	2	1	no	—
18	#135×C₆₁	C₂²	1	1	yes	characteristic
19	#131×C₆₁	—	2	1	no	—
20	#108×C₆₁	C₂²	1	1	yes	characteristic
21	#104×C₆₁	—	2	1	no	—
22	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	#41×C₆₁	—	2	1	no	—
24	#11758222	—	2	1	no	—
25	Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
26	SD₁₆×C₆₁	—	2	1	no	—
27	D₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	M₁₆×C₆₁	—	2	1	no	—
29	M₁₆×C₆₁	—	4	1	no	—
30	C₂×C₈×C₆₁	—	4	1	no	—
31	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
33	C₄²×C₆₁	D₄	1	1	yes	characteristic
34	#11758194	—	2	1	no	—
35	Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
36	D₄₈₈	C₂×C₄	1	1	yes	characteristic
37	#11758190	—	2	1	no	—
38	#11758189	—	4	1	no	—
39	C₈×D₆₁	—	4	1	no	—
40	#11758185	—	2	1	no	—
41	#11758185	C₂×C₄	1	1	yes	characteristic
42	#11758185	C₂×C₄	1	1	yes	characteristic
43	Q₈×C₆₁	—	2	1	no	—
44	D₄×C₆₁	—	2	1	no	—
45	D₄×C₆₁	—	4	1	no	—
46	C₂×C₄×C₆₁	—	2	1	no	—
47	C₂×C₄×C₆₁	—	4	1	no	—
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂²⋊D₆₁	—	4	1	no	—
50	D₂₄₄	—	2	1	no	—
51	C₄×D₆₁	—	4	1	no	—
52	Dic₁₂₂	—	2	1	no	—
53	C₈×C₆₁	—	4	1	no	—
54	C₈×C₆₁	D₈	1	1	yes	characteristic
55	C₈×C₆₁	D₈	1	1	yes	characteristic
56	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₈×C₆₁	#41	1	1	yes	characteristic
58	#92418	—	4	1	no	—
59	C₂²×C₆₁	—	4	1	no	—
60	C₂²×C₆₁	#135	1	1	yes	characteristic
61	C₂×D₆₁	—	4	1	no	—
62	C₄×C₆₁	—	2	2	no	—
63	C₄×C₆₁	—	4	1	no	—
64	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
65	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
66	Dic₆₁	—	4	1	no	—
67	#2178	—	61	1	no	maximal, Sylow
68	C₂×C₆₁	—	2	1	no	—
69	C₂×C₆₁	—	8	1	no	—
70	C₂×C₆₁	C₄×D₈	1	1	yes	characteristic
71	D₆₁	—	8	1	no	—
72	#508	—	61	1	no	—
73	#443	—	61	1	no	—
74	#443	D₆₁	1	1	yes	characteristic, complemented
75	#368	—	61	1	no	—
76	#359	—	61	1	no	—
77	#359	—	61	1	no	—
78	C₄×C₁₆	—	61	1	no	—
79	C₆₁	#2178	1	1	yes	characteristic, complemented, Sylow
80	#135	—	61	1	no	—
81	#135	C₂×D₆₁	1	1	yes	characteristic
82	#131	—	2	1	no	—
83	#131	—	122	1	no	—
84	Q₃₂	—	61	1	no	—
85	SD₃₂	—	61	1	no	—
86	SD₃₂	—	61	1	no	—
87	D₁₆	—	61	1	no	—
88	C₂×C₁₆	—	61	1	no	—
89	C₂×C₁₆	—	61	1	no	—
90	#108	—	61	1	no	—
91	#108	C₂×D₆₁	1	1	yes	characteristic
92	#104	—	2	1	no	—
93	#104	—	122	1	no	—
94	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
95	#41	—	2	1	no	—
96	#41	—	122	1	no	—
97	Q₁₆	—	61	1	no	—
98	Q₁₆	C₄×D₆₁	1	1	yes	characteristic
99	SD₁₆	—	2	1	no	—
100	SD₁₆	—	122	1	no	—
101	D₈	—	61	1	no	—
102	D₈	C₄×D₆₁	1	1	yes	characteristic
103	M₁₆	—	2	1	no	—
104	M₁₆	—	4	1	no	—
105	M₁₆	—	122	1	no	—
106	M₁₆	—	244	1	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	—	244	1	no	—
109	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
110	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
111	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
112	C₁₆	—	61	1	no	—
113	C₁₆	—	61	1	no	—
114	C₁₆	—	122	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	122	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	4	1	no	—
119	D₄	—	122	1	no	—
120	D₄	—	244	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	244	1	no	—
124	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	244	1	no	—
127	C₈	#11758199	1	1	yes	characteristic
128	C₈	#11758199	1	1	yes	characteristic
129	C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
130	C₈	#11758222	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	244	1	no	—
133	C₂²	#14519797	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	4	1	no	—
136	C₄	—	244	1	no	—
137	C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
138	C₄	C₂×#11758199	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	8	1	no	—
141	C₂	—	488	1	no	—
142	C₂	C₄×#11758199	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840192

C₂

yes

maximal, characteristic

#16839880

C₂

yes

maximal, characteristic, complemented

#16839591

C₂

yes

maximal, characteristic, complemented

#16839569

C₂

yes

maximal, characteristic, complemented

#16839568

C₂

yes

maximal, characteristic, complemented

C₄×#11758185

C₂

yes

maximal, characteristic, complemented

C₆₁×#443

C₂

yes

maximal, characteristic, complemented

#14519799

C₂²

yes

characteristic

#14519783

—

#14519723

C₄

yes

characteristic, complemented

#14519721

C₄

yes

characteristic, complemented

#14519720

C₄

yes

characteristic, complemented

#14519718

C₄

yes

characteristic, complemented

C₂×#11758185

C₂²

yes

characteristic

C₂×#11758185

C₂²

yes

characteristic

#14519713

C₂²

yes

characteristic

#14519705

—

#135×C₆₁

C₂²

yes

characteristic

#131×C₆₁

—

#108×C₆₁

C₂²

yes

characteristic

#104×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

#41×C₆₁

—

#11758222

—

Q₁₆×C₆₁

C₂×C₄

yes

characteristic

SD₁₆×C₆₁

—

D₈×C₆₁

C₂×C₄

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

#11758194

—

Dic₂₄₄

C₂×C₄

yes

characteristic

D₄₈₈

C₂×C₄

yes

characteristic

#11758190

—

#11758189

—

C₈×D₆₁

—

#11758185

—

#11758185

C₂×C₄

yes

characteristic

#11758185

C₂×C₄

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²⋊D₆₁

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

D₈

yes

characteristic

C₈×C₆₁

D₈

yes

characteristic

C₈×C₆₁

C₂²×C₄

yes

C₈×C₆₁

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

#135

yes

characteristic

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

characteristic

C₄×C₆₁

C₂×D₈

yes

characteristic

Dic₆₁

—

#2178

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₄×D₈

yes

characteristic

D₆₁

—

#508

—

#443

—

#443

D₆₁

yes

characteristic, complemented

#368

—

#359

—

#359

—

C₄×C₁₆

—

C₆₁

#2178

yes

characteristic, complemented, Sylow

#135

—

#135

C₂×D₆₁

yes

characteristic

#131

—

#131

—

122

—

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#108

C₂×D₆₁

yes

characteristic

#104

—

#104

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

#41

—

#41

—

122

—

Q₁₆

—

Q₁₆

C₄×D₆₁

yes

characteristic

SD₁₆

—

100

SD₁₆

—

122

—

101

D₈

—

102

D₈

C₄×D₆₁

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

122

—

106

M₁₆

—

244

—

107

C₂×C₈

—

108

C₂×C₈

—

244

—

109

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

110

C₂×C₈

C₂²×D₆₁

yes

characteristic

111

C₄²

C₂²⋊D₆₁

yes

characteristic

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

122

—

115

Q₈

—

116

Q₈

—

122

—

117

D₄

—

118

D₄

—

119

D₄

—

122

—

120

D₄

—

244

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

244

—

124

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

125

C₈

—

126

C₈

—

244

—

127

C₈

#11758199

yes

characteristic

128

C₈

#11758199

yes

characteristic

129

C₈

C₂×C₄×D₆₁

yes

characteristic

130

C₈

#11758222

yes

characteristic

131

C₂²

—

132

C₂²

—

244

—

133

C₂²

#14519797

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

244

—

137

C₄

C₄×C₂²⋊D₆₁

yes

characteristic

138

C₄

C₂×#11758199

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

—

488

—

142

C₂

C₄×#11758199

yes

characteristic

	subgroup	G/H
center	C₈	#11758199
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×#443	C₂
socle	C₂×C₆₁	C₄×D₈
hypercenter	#443	D₆₁
hypocenter	C₆₁	#2178
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #443

subgroup

G/H

center

C₈

#11758199

commutator

C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₆₁×#443

C₂

socle

C₂×C₆₁

C₄×D₈

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #443

N	H ≅ G/N	type
2. #16839880	C₂	semidirect
3. #16839591	C₂	semidirect
4. #16839569	C₂	semidirect
5. #16839568	C₂	semidirect
6. C₄×#11758185	C₂	semidirect
7. C₆₁×#443	C₂	semidirect
10. #14519723	C₄	semidirect
11. #14519721	C₄	semidirect
12. #14519720	C₄	semidirect
13. #14519718	C₄	semidirect
74. #443	D₆₁	semidirect
79. C₆₁	#2178	semidirect

H ≅ G/N

type

2. #16839880

C₂

semidirect

3. #16839591

C₂

semidirect

4. #16839569

C₂

semidirect

5. #16839568

C₂

semidirect

6. C₄×#11758185

C₂

semidirect

7. C₆₁×#443

C₂

semidirect

10. #14519723

C₄

semidirect

11. #14519721

C₄

semidirect

12. #14519720

C₄

semidirect

13. #14519718

C₄

semidirect

74. #443

D₆₁

semidirect

79. C₆₁

#2178

semidirect

Subgroups — #21447698

Conjugacy classes of subgroups

Complements