Group #21447711 — subgroups

Subgroups — #21447711

Order 7808.

142 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 59 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	#16839884	C₂	1	1	yes	maximal, characteristic
2	#16839869	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839585	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839578	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839572	C₂	1	1	yes	maximal, characteristic
6	#16839539	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#510	C₂	1	1	yes	maximal, characteristic
8	#14519784	—	2	1	no	—
9	Q₈×Dic₆₁	—	2	1	no	—
10	#14519717	C₂²	1	1	yes	characteristic
11	#14519713	C₂²	1	1	yes	characteristic
12	#14519709	C₂²	1	1	yes	characteristic
13	#14519704	—	2	1	no	—
14	#14519701	—	2	1	no	—
15	#14519687	C₂²	1	1	yes	characteristic
16	#135×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	Q₃₂×C₆₁	C₄	1	2	yes	complemented
19	SD₃₂×C₆₁	C₄	1	2	yes	complemented
20	M₃₂×C₆₁	C₂²	1	1	yes	characteristic
21	#41×C₆₁	—	2	1	no	—
22	C₂×Q₈×C₆₁	—	2	1	no	—
23	Q₁₆×C₆₁	—	2	1	no	—
24	Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
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	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₆₁	—	2	1	no	—
30	C₄⋊Dic₆₁	—	4	1	no	—
31	C₄×Dic₆₁	—	4	1	no	—
32	C₄×Dic₆₁	D₄	1	1	yes	characteristic
33	#11758194	—	2	1	no	—
34	#11758194	—	4	1	no	—
35	C₂×#92418	—	4	1	no	—
36	C₂×#92418	D₄	1	1	yes	characteristic
37	C₁₆×C₆₁	C₂×C₄	1	2	yes	—
38	#11758185	—	2	1	no	—
39	Q₈×C₆₁	—	2	1	no	—
40	Q₈×C₆₁	—	2	2	no	—
41	Q₈×C₆₁	—	4	1	no	—
42	D₄×C₆₁	—	2	1	no	—
43	D₄×C₆₁	—	4	1	no	—
44	C₂×C₄×C₆₁	—	4	1	no	—
45	C₂×C₄×C₆₁	—	4	1	no	—
46	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₆₁	—	2	1	no	—
48	C₂×Dic₆₁	—	4	1	no	—
49	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₈×C₆₁	#41	1	1	yes	characteristic
51	#92418	—	2	1	no	—
52	#92418	—	4	1	no	—
53	C₂²×C₆₁	—	4	1	no	—
54	C₂²×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₄×C₆₁	—	4	1	no	—
56	C₄×C₆₁	—	4	2	no	—
57	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
58	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
59	Dic₆₁	—	4	1	no	—
60	Dic₆₁	—	8	1	no	—
61	#2180	—	61	1	no	maximal, Sylow
62	C₂×C₆₁	—	2	1	no	—
63	C₂×C₆₁	—	8	1	no	—
64	C₂×C₆₁	#442	1	1	yes	characteristic
65	#510	D₆₁	1	1	yes	characteristic
66	#444	—	61	1	no	—
67	#441	—	61	1	no	—
68	#365	—	61	1	no	—
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	C₂×D₆₁	1	1	yes	characteristic
74	C₂×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
75	#131	—	122	1	no	—
76	C₄×Q₈	—	122	1	no	—
77	Q₃₂	Dic₆₁	1	2	yes	complemented
78	SD₃₂	Dic₆₁	1	2	yes	complemented
79	M₃₂	—	61	1	no	—
80	M₃₂	C₂×D₆₁	1	1	yes	characteristic
81	#108	—	61	1	no	—
82	#106	—	61	1	no	—
83	#104	—	122	1	no	—
84	#103	—	122	1	no	—
85	#97	—	61	1	no	—
86	#41	—	2	1	no	—
87	C₂×Q₈	—	2	1	no	—
88	Q₁₆	—	2	1	no	—
89	Q₁₆	C₂×Dic₆₁	1	2	yes	—
90	Q₁₆	C₂×Dic₆₁	1	1	yes	characteristic
91	SD₁₆	—	2	1	no	—
92	D₈	C₂×Dic₆₁	1	1	yes	characteristic
93	M₁₆	—	122	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	—	122	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	C₂×Dic₆₁	1	2	yes	—
104	Q₈	—	2	1	no	—
105	Q₈	—	2	2	no	—
106	Q₈	—	4	1	no	—
107	D₄	—	2	1	no	—
108	D₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	122	1	no	—
112	C₂×C₄	—	244	1	no	—
113	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
114	C₈	—	122	1	no	—
115	C₈	—	244	1	no	—
116	C₈	#11758224	1	1	yes	characteristic
117	C₈	C₂²×Dic₆₁	1	1	yes	characteristic
118	C₂²	—	4	1	no	—
119	C₂²	#14519801	1	1	yes	characteristic
120	C₄	—	4	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	488	1	no	—
123	C₄	—	244	1	no	—
124	C₄	D₄×Dic₆₁	1	1	yes	characteristic
125	C₄	#14519801	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	8	1	no	—
128	C₂	#16839874	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839884

C₂

yes

maximal, characteristic

#16839869

C₂

yes

maximal, characteristic, complemented

#16839585

C₂

yes

maximal, characteristic, complemented

#16839578

C₂

yes

maximal, characteristic, complemented

#16839572

C₂

yes

maximal, characteristic

#16839539

C₂

yes

maximal, characteristic, complemented

C₆₁×#510

C₂

yes

maximal, characteristic

#14519784

—

Q₈×Dic₆₁

—

#14519717

C₂²

yes

characteristic

#14519713

C₂²

yes

characteristic

#14519709

C₂²

yes

—

—

C₂²

yes

characteristic

#135×C₆₁

C₂²

yes

characteristic

C₂×Q₁₆×C₆₁

C₂²

yes

characteristic

Q₃₂×C₆₁

C₄

yes

complemented

SD₃₂×C₆₁

C₄

yes

complemented

M₃₂×C₆₁

C₂²

yes

characteristic

#41×C₆₁

—

C₂×Q₈×C₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

C₂×C₄

yes

—

Q₁₆×C₆₁

C₂×C₄

yes

characteristic

SD₁₆×C₆₁

—

D₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

—

—

—

D₄

yes

characteristic

C₁₆×C₆₁

C₂×C₄

yes

—

#11758185

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

C₂²×C₄

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

AGL(1,ℤ/8)

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

characteristic

C₄×C₆₁

AGL(1,ℤ/8)

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#2180

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#442

yes

characteristic

#510

D₆₁

yes

—

—

—

—

—

—

C₆₁

#2180

yes

characteristic, complemented, Sylow

#135

C₂×D₆₁

yes

characteristic

C₂×Q₁₆

C₂×D₆₁

yes

characteristic

#131

—

122

—

C₄×Q₈

—

122

—

Q₃₂

Dic₆₁

yes

complemented

SD₃₂

Dic₆₁

yes

complemented

M₃₂

—

M₃₂

C₂×D₆₁

yes

—

—

—

122

—

#103

—

122

—

#97

—

#41

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

C₂×Dic₆₁

yes

—

Q₁₆

C₂×Dic₆₁

yes

characteristic

SD₁₆

—

D₈

C₂×Dic₆₁

yes

characteristic

M₁₆

—

122

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

100

C₄²

—

101

C₄²

—

244

—

102

C₁₆

—

122

—

103

C₁₆

C₂×Dic₆₁

yes

—

104

Q₈

—

105

Q₈

—

106

Q₈

—

107

D₄

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

122

—

112

C₂×C₄

—

244

—

113

C₂×C₄

D₄×D₆₁

yes

characteristic

114

C₈

—

122

—

115

C₈

—

244

—

116

C₈

#11758224

yes

characteristic

117

C₈

C₂²×Dic₆₁

yes

characteristic

118

C₂²

—

119

C₂²

#14519801

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

488

—

123

C₄

—

244

—

124

C₄

D₄×Dic₆₁

yes

characteristic

125

C₄

#14519801

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#16839874

yes

characteristic

	subgroup	G/H
center	C₂	#16839874
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×#510	C₂
socle	C₂×C₆₁	#442
hypercenter	#510	D₆₁
hypocenter	C₆₁	#2180
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #510

subgroup

G/H

center

C₂

#16839874

commutator

C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₆₁×#510

C₂

C₂×C₆₁

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #510

N	H ≅ G/N	type
2. #16839869	C₂	semidirect
3. #16839585	C₂	semidirect
4. #16839578	C₂	semidirect
6. #16839539	C₂	semidirect
18. Q₃₂×C₆₁	C₄	semidirect
19. SD₃₂×C₆₁	C₄	semidirect
72. C₆₁	#2180	semidirect
77. Q₃₂	Dic₆₁	semidirect
78. SD₃₂	Dic₆₁	semidirect

H ≅ G/N

type

2. #16839869

C₂

semidirect

3. #16839585

C₂

semidirect

4. #16839578

C₂

semidirect

6. #16839539

C₂

semidirect

18. Q₃₂×C₆₁

C₄

semidirect

19. SD₃₂×C₆₁

C₄

semidirect

72. C₆₁

#2180

semidirect

77. Q₃₂

Dic₆₁

semidirect

78. SD₃₂

Dic₆₁

semidirect

Subgroups — #21447711

Conjugacy classes of subgroups

Complements