Group #21447697 — subgroups

Subgroups — #21447697

Order 7808.

146 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16839879	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839878	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839591	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839570	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#359	C₂	1	1	yes	maximal, characteristic, complemented
8	#14519799	C₂²	1	1	yes	characteristic
9	#14519785	—	2	1	no	—
10	#14519784	—	2	1	no	—
11	#14519723	C₄	1	1	yes	characteristic, complemented
12	#14519721	C₄	1	1	yes	characteristic, complemented
13	#14519720	C₄	1	1	yes	characteristic, complemented
14	#14519718	C₄	1	1	yes	characteristic, complemented
15	C₂×#11758185	C₂²	1	1	yes	characteristic
16	#14519713	C₂²	1	1	yes	characteristic
17	#14519704	—	2	1	no	—
18	#14519703	—	2	1	no	—
19	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
20	#135×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
22	#108×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	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₄×Dic₆₁	D₄	1	1	yes	characteristic
31	#11758194	—	2	1	no	—
32	#11758194	—	4	1	no	—
33	C₂×#92418	—	4	1	no	—
34	C₂×#92418	D₄	1	1	yes	characteristic
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	C₁₆×C₆₁	—	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	—	4	1	no	—
47	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₆₁	—	4	1	no	—
49	C₂×Dic₆₁	—	2	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	C₂²×C₄	1	1	yes	characteristic
55	C₈×C₆₁	#41	1	1	yes	characteristic
56	#92418	—	4	1	no	—
57	#92418	D₈	1	1	yes	characteristic
58	#92418	D₈	1	1	yes	characteristic
59	C₂²×C₆₁	—	4	1	no	—
60	C₂²×C₆₁	#135	1	1	yes	characteristic
61	C₂×D₆₁	—	4	1	no	—
62	C₄×C₆₁	—	4	1	no	—
63	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
64	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
65	Dic₆₁	—	2	2	no	—
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	—	61	1	no	—
75	#368	—	61	1	no	—
76	#359	—	61	1	no	—
77	#359	D₆₁	1	1	yes	characteristic, complemented
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	—	122	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	C₂×D₆₁	1	1	yes	characteristic
90	#108	—	61	1	no	—
91	#108	C₂×D₆₁	1	1	yes	characteristic
92	#104	—	122	1	no	—
93	#104	—	122	1	no	—
94	C₄×C₈	—	61	1	no	—
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	—	122	1	no	—
105	M₁₆	—	244	1	no	—
106	M₁₆	—	244	1	no	—
107	C₂×C₈	—	61	1	no	—
108	C₂×C₈	—	244	1	no	—
109	C₂×C₈	—	244	1	no	—
110	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
111	C₄²	—	61	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	61	1	no	—
114	C₁₆	—	61	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	—	4	1	no	—
122	C₂×C₄	—	122	1	no	—
123	C₂×C₄	—	244	1	no	—
124	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	61	1	no	—
127	C₈	—	61	1	no	—
128	C₈	—	244	1	no	—
129	C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
130	C₈	#11758224	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	244	1	no	—
133	C₂²	#14519798	1	1	yes	characteristic
134	C₄	—	4	1	no	—
135	C₄	—	122	2	no	—
136	C₄	—	244	1	no	—
137	C₄	#14519737	1	1	yes	characteristic
138	C₄	D₈×D₆₁	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	8	1	no	—
141	C₂	—	488	1	no	—
142	C₂	#16839842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840192

C₂

yes

maximal, characteristic

#16839879

C₂

yes

maximal, characteristic, complemented

#16839878

C₂

yes

maximal, characteristic, complemented

#16839591

C₂

yes

maximal, characteristic, complemented

#16839570

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₆₁

C₂

yes

maximal, characteristic, complemented

C₆₁×#359

C₂

yes

maximal, characteristic, complemented

#14519799

C₂²

yes

—

—

C₄

yes

characteristic, complemented

#14519721

C₄

yes

characteristic, complemented

#14519720

C₄

yes

characteristic, complemented

#14519718

C₄

yes

characteristic, complemented

C₂×#11758185

C₂²

yes

characteristic

#14519713

C₂²

yes

characteristic

#14519704

—

#14519703

—

C₈×Dic₆₁

C₂²

yes

characteristic

#135×C₆₁

C₂²

yes

characteristic

C₂×C₁₆×C₆₁

C₂²

yes

characteristic

#108×C₆₁

C₂²

yes

characteristic

#41×C₆₁

—

#11758222

—

Q₁₆×C₆₁

C₂×C₄

yes

characteristic

SD₁₆×C₆₁

—

D₈×C₆₁

C₂×C₄

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄×Dic₆₁

D₄

yes

—

—

—

D₄

yes

characteristic

Dic₂₄₄

C₂×C₄

yes

characteristic

D₄₈₈

C₂×C₄

yes

characteristic

#11758190

—

#11758189

—

C₈×D₆₁

—

C₁₆×C₆₁

—

#11758185

C₂×C₄

yes

characteristic

#11758185

C₂×C₄

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂²×C₄

yes

C₈×C₆₁

yes

—

D₈

yes

characteristic

#92418

D₈

yes

characteristic

C₂²×C₆₁

—

C₂²×C₆₁

#135

yes

characteristic

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

characteristic

C₄×C₆₁

C₂×D₈

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#2178

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₄×D₈

yes

characteristic

D₆₁

—

—

—

—

—

—

D₆₁

yes

characteristic, complemented

C₄×C₁₆

—

C₆₁

#2178

yes

characteristic, complemented, Sylow

#135

—

#135

C₂×D₆₁

yes

characteristic

#131

—

122

—

#131

—

122

—

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₁

yes

characteristic

#108

—

#108

C₂×D₆₁

yes

characteristic

#104

—

122

—

#104

—

122

—

C₄×C₈

—

#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

—

122

—

105

M₁₆

—

244

—

106

M₁₆

—

244

—

107

C₂×C₈

—

108

C₂×C₈

—

244

—

109

C₂×C₈

—

244

—

110

C₂×C₈

C₂²×D₆₁

yes

characteristic

111

C₄²

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

Q₈

—

116

Q₈

—

122

—

117

D₄

—

118

D₄

—

119

D₄

—

122

—

120

D₄

—

244

—

121

C₂×C₄

—

122

C₂×C₄

—

122

—

123

C₂×C₄

—

244

—

124

C₂×C₄

D₄×D₆₁

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

244

—

129

C₈

C₂×C₄×D₆₁

yes

characteristic

130

C₈

#11758224

yes

characteristic

131

C₂²

—

132

C₂²

—

244

—

133

C₂²

#14519798

yes

characteristic

134

C₄

—

135

C₄

—

122

—

136

C₄

—

244

—

137

C₄

#14519737

yes

characteristic

138

C₄

D₈×D₆₁

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

—

488

—

142

C₂

#16839842

yes

characteristic

	subgroup	G/H
center	C₄	D₈×D₆₁
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×#359	C₂
socle	C₂×C₆₁	C₄×D₈
hypercenter	#359	D₆₁
hypocenter	C₆₁	#2178
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

subgroup

G/H

center

C₄

D₈×D₆₁

commutator

C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₆₁×#359

C₂

socle

C₂×C₆₁

C₄×D₈

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

N	H ≅ G/N	type
2. #16839879	C₂	semidirect
3. #16839878	C₂	semidirect
4. #16839591	C₂	semidirect
5. #16839570	C₂	semidirect
6. C₁₆×Dic₆₁	C₂	semidirect
7. C₆₁×#359	C₂	semidirect
11. #14519723	C₄	semidirect
12. #14519721	C₄	semidirect
13. #14519720	C₄	semidirect
14. #14519718	C₄	semidirect
77. #359	D₆₁	semidirect
79. C₆₁	#2178	semidirect

H ≅ G/N

type

2. #16839879

C₂

semidirect

3. #16839878

C₂

semidirect

4. #16839591

C₂

semidirect

5. #16839570

C₂

semidirect

6. C₁₆×Dic₆₁

C₂

semidirect

7. C₆₁×#359

C₂

semidirect

11. #14519723

C₄

semidirect

12. #14519721

C₄

semidirect

13. #14519720

C₄

semidirect

14. #14519718

C₄

semidirect

77. #359

D₆₁

semidirect

79. C₆₁

#2178

semidirect

Subgroups — #21447697

Conjugacy classes of subgroups

Complements