Group #21447709 — subgroups

Subgroups — #21447709

Order 7808.

142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16840205	C₂	1	1	yes	maximal, characteristic
2	#16839883	C₂	1	1	yes	maximal, characteristic
3	#16839868	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839577	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839571	C₂	1	1	yes	maximal, characteristic
6	#16839539	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#365	C₂	1	1	yes	maximal, characteristic, complemented
8	#14519799	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
10	#14519785	—	2	1	no	—
11	#14519742	—	2	1	no	—
12	Dic₄₈₈	C₄	1	2	yes	complemented
13	#14519722	C₄	1	2	yes	complemented
14	#14519717	C₂²	1	1	yes	characteristic
15	#14519703	—	2	1	no	—
16	#14519700	—	2	1	no	—
17	#14519687	C₂²	1	1	yes	characteristic
18	M₃₂×C₆₁	C₂²	1	1	yes	characteristic
19	#108×C₆₁	C₂²	1	1	yes	characteristic
20	#106×C₆₁	C₂²	1	1	yes	characteristic
21	#11758222	—	2	1	no	—
22	C₂×Dic₁₂₂	—	2	1	no	—
23	M₁₆×C₆₁	—	2	1	no	—
24	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₆₁	—	2	1	no	—
26	#11758196	—	4	1	no	—
27	C₄×Dic₆₁	—	4	1	no	—
28	C₄×Dic₆₁	D₄	1	1	yes	characteristic
29	C₂×#92418	D₄	1	1	yes	characteristic
30	Dic₂₄₄	—	2	1	no	—
31	Dic₂₄₄	C₂×C₄	1	2	yes	—
32	Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
33	D₄₈₈	C₂×C₄	1	1	yes	characteristic
34	#11758190	—	2	1	no	—
35	#11758189	—	4	1	no	—
36	C₈×D₆₁	—	4	1	no	—
37	C₁₆×C₆₁	C₂×C₄	1	2	yes	—
38	#11758185	—	2	1	no	—
39	C₂×C₄×C₆₁	—	4	1	no	—
40	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²⋊D₆₁	—	4	1	no	—
42	C₂×Dic₆₁	—	2	1	no	—
43	C₂×Dic₆₁	—	4	1	no	—
44	D₂₄₄	—	2	1	no	—
45	C₄×D₆₁	—	4	1	no	—
46	Dic₁₂₂	—	2	1	no	—
47	Dic₁₂₂	—	2	2	no	—
48	Dic₁₂₂	—	4	1	no	—
49	C₈×C₆₁	—	4	1	no	—
50	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₆₁	#41	1	1	yes	characteristic
52	#92418	—	2	1	no	—
53	C₂²×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₂×D₆₁	—	4	1	no	—
55	C₄×C₆₁	—	8	1	no	—
56	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
57	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
58	Dic₆₁	—	4	1	no	—
59	Dic₆₁	—	4	2	no	—
60	Dic₆₁	—	4	1	no	—
61	#2180	—	61	1	no	maximal, Sylow
62	C₂×C₆₁	—	2	1	no	—
63	C₂×C₆₁	#442	1	1	yes	characteristic
64	D₆₁	—	8	1	no	—
65	#510	—	61	1	no	—
66	#444	—	61	1	no	—
67	#441	—	61	1	no	—
68	#365	D₆₁	1	1	yes	characteristic, complemented
69	#362	—	61	1	no	—
70	#360	—	61	1	no	—
71	#347	—	61	1	no	—
72	C₆₁	#2180	1	1	yes	characteristic, complemented, Sylow
73	#135	—	61	1	no	—
74	C₂×Q₁₆	—	61	1	no	—
75	#131	—	122	1	no	—
76	C₄×Q₈	—	122	1	no	—
77	Q₃₂	—	61	2	no	—
78	SD₃₂	—	61	2	no	—
79	M₃₂	—	61	1	no	—
80	M₃₂	C₂×D₆₁	1	1	yes	characteristic
81	#108	C₂×D₆₁	1	1	yes	characteristic
82	#106	C₂×D₆₁	1	1	yes	characteristic
83	#104	—	122	1	no	—
84	#103	—	122	1	no	—
85	#97	—	61	1	no	—
86	#41	—	122	1	no	—
87	C₂×Q₈	—	122	1	no	—
88	Q₁₆	—	61	2	no	—
89	Q₁₆	—	61	1	no	—
90	Q₁₆	—	122	1	no	—
91	SD₁₆	—	122	1	no	—
92	D₈	—	61	1	no	—
93	M₁₆	—	2	1	no	—
94	M₁₆	—	244	1	no	—
95	C₂×C₈	—	61	1	no	—
96	C₂×C₈	—	244	1	no	—
97	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	244	1	no	—
100	C₄²	—	61	1	no	—
101	C₄²	—	244	1	no	—
102	C₁₆	—	122	1	no	—
103	C₁₆	C₄×D₆₁	1	2	yes	—
104	Q₈	—	122	1	no	—
105	Q₈	—	122	2	no	—
106	Q₈	—	244	1	no	—
107	D₄	—	122	1	no	—
108	D₄	—	244	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	122	1	no	—
111	C₂×C₄	—	244	1	no	—
112	C₂×C₄	—	244	1	no	—
113	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
114	C₈	—	4	1	no	—
115	C₈	—	122	1	no	—
116	C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
117	C₈	#11758226	1	1	yes	characteristic
118	C₂²	—	244	1	no	—
119	C₂²	#14519804	1	1	yes	characteristic
120	C₄	—	8	1	no	—
121	C₄	—	244	1	no	—
122	C₄	—	244	1	no	—
123	C₄	—	244	2	no	—
124	C₄	#14519739	1	1	yes	characteristic
125	C₄	#14519804	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	488	1	no	—
128	C₂	#16839873	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840205

C₂

yes

maximal, characteristic

#16839883

C₂

yes

maximal, characteristic

#16839868

C₂

yes

maximal, characteristic, complemented

#16839577

C₂

yes

maximal, characteristic, complemented

#16839571

C₂

yes

maximal, characteristic

#16839539

C₂

yes

maximal, characteristic, complemented

C₆₁×#365

C₂

yes

maximal, characteristic, complemented

#14519799

C₂²

yes

characteristic

C₂×Dic₂₄₄

C₂²

yes

characteristic

#14519785

—

#14519742

—

Dic₄₈₈

C₄

yes

complemented

#14519722

C₄

yes

complemented

#14519717

C₂²

yes

—

—

C₂²

yes

characteristic

M₃₂×C₆₁

C₂²

yes

characteristic

#108×C₆₁

C₂²

yes

characteristic

#106×C₆₁

C₂²

yes

characteristic

#11758222

—

C₂×Dic₁₂₂

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

characteristic

C₂×#92418

D₄

yes

characteristic

Dic₂₄₄

—

Dic₂₄₄

C₂×C₄

yes

—

Dic₂₄₄

C₂×C₄

yes

characteristic

D₄₈₈

C₂×C₄

yes

characteristic

#11758190

—

#11758189

—

C₈×D₆₁

—

C₁₆×C₆₁

C₂×C₄

yes

—

#11758185

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂²×C₄

yes

C₈×C₆₁

yes

—

C₂²×C₆₁

AGL(1,ℤ/8)

yes

characteristic

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

characteristic

C₄×C₆₁

AGL(1,ℤ/8)

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#2180

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#442

yes

characteristic

D₆₁

—

—

—

—

D₆₁

yes

characteristic, complemented

#362

—

#360

—

#347

—

C₆₁

#2180

yes

characteristic, complemented, Sylow

#135

—

C₂×Q₁₆

—

#131

—

122

—

C₄×Q₈

—

122

—

Q₃₂

—

SD₃₂

—

M₃₂

—

M₃₂

C₂×D₆₁

yes

characteristic

#108

C₂×D₆₁

yes

characteristic

#106

C₂×D₆₁

yes

characteristic

#104

—

122

—

#103

—

122

—

#97

—

#41

—

122

—

C₂×Q₈

—

122

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

122

—

SD₁₆

—

122

—

D₈

—

M₁₆

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

244

—

100

C₄²

—

101

C₄²

—

244

—

102

C₁₆

—

122

—

103

C₁₆

C₄×D₆₁

yes

—

104

Q₈

—

122

—

105

Q₈

—

122

—

106

Q₈

—

244

—

107

D₄

—

122

—

108

D₄

—

244

—

109

C₂×C₄

—

110

C₂×C₄

—

122

—

111

C₂×C₄

—

244

—

112

C₂×C₄

—

244

—

113

C₂×C₄

D₄×D₆₁

yes

characteristic

114

C₈

—

115

C₈

—

122

—

116

C₈

C₂×C₄×D₆₁

yes

characteristic

117

C₈

#11758226

yes

characteristic

118

C₂²

—

244

—

119

C₂²

#14519804

yes

characteristic

120

C₄

—

121

C₄

—

244

—

122

C₄

—

244

—

123

C₄

—

244

—

124

C₄

#14519739

yes

characteristic

125

C₄

#14519804

yes

characteristic

126

C₂

—

127

C₂

—

488

—

128

C₂

#16839873

yes

characteristic

	subgroup	G/H
center	C₂	#16839873
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×#365	C₂
socle	C₂×C₆₁	#442
hypercenter	#365	D₆₁
hypocenter	C₆₁	#2180
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#16839873

commutator

C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₆₁×#365

C₂

C₂×C₆₁

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
3. #16839868	C₂	semidirect
4. #16839577	C₂	semidirect
6. #16839539	C₂	semidirect
7. C₆₁×#365	C₂	semidirect
12. Dic₄₈₈	C₄	semidirect
13. #14519722	C₄	semidirect
68. #365	D₆₁	semidirect
72. C₆₁	#2180	semidirect

H ≅ G/N

type

3. #16839868

C₂

semidirect

4. #16839577

C₂

semidirect

6. #16839539

C₂

semidirect

7. C₆₁×#365

C₂

semidirect

12. Dic₄₈₈

C₄

semidirect

13. #14519722

C₄

semidirect

68. #365

D₆₁

semidirect

72. C₆₁

#2180

semidirect

Subgroups — #21447709

Conjugacy classes of subgroups

Complements