Group #21447988 — subgroups

Subgroups — #21447988

Order 7808.

126 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16840159	C₂	1	1	yes	maximal, characteristic, complemented
2	#16839879	C₂	1	1	yes	maximal, characteristic
3	#16839585	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839584	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839582	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839572	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#360	C₂	1	1	yes	maximal, characteristic
8	#14519784	—	2	1	no	—
9	#14519776	—	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	#14519708	—	2	1	no	—
14	#14519704	—	2	1	no	—
15	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
16	#135×C₆₁	C₂²	1	1	yes	characteristic
17	M₃₂×C₆₁	C₂²	1	1	yes	characteristic
18	#106×C₆₁	C₂²	1	1	yes	characteristic
19	#41×C₆₁	—	2	1	no	—
20	C₂×Dic₁₂₂	—	4	1	no	—
21	Q₁₆×C₆₁	Q₈	1	1	yes	characteristic, complemented
22	SD₁₆×C₆₁	—	2	1	no	—
23	D₈×C₆₁	Q₈	1	1	yes	characteristic, complemented
24	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₆₁	—	2	1	no	—
26	C₄⋊Dic₆₁	—	2	1	no	—
27	#11758196	—	4	1	no	—
28	C₄×Dic₆₁	D₄	1	1	yes	characteristic
29	#11758194	—	2	1	no	—
30	#11758194	—	4	1	no	—
31	C₂×#92418	—	4	1	no	—
32	C₂×#92418	D₄	1	1	yes	characteristic
33	C₁₆×C₆₁	—	2	1	no	—
34	#11758185	—	2	1	no	—
35	Q₈×C₆₁	—	2	1	no	—
36	D₄×C₆₁	—	2	1	no	—
37	D₄×C₆₁	—	4	1	no	—
38	C₂×C₄×C₆₁	—	4	1	no	—
39	C₂×C₄×C₆₁	—	4	1	no	—
40	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₆₁	—	2	1	no	—
42	C₂×Dic₆₁	—	4	1	no	—
43	Dic₁₂₂	—	8	2	no	—
44	C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₆₁	#41	1	1	yes	characteristic
46	#92418	—	4	1	no	—
47	#92418	SD₁₆	1	1	yes	characteristic
48	#92418	SD₁₆	1	1	yes	characteristic
49	C₂²×C₆₁	—	4	1	no	—
50	C₂²×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₄×C₆₁	—	4	1	no	—
52	C₄×C₆₁	—	8	1	no	—
53	C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
55	Dic₆₁	—	2	2	no	—
56	Dic₆₁	—	8	1	no	—
57	#2230	—	61	1	no	maximal, Sylow
58	C₂×C₆₁	—	2	1	no	—
59	C₂×C₆₁	—	8	1	no	—
60	C₂×C₆₁	#476	1	1	yes	characteristic
61	#498	—	61	1	no	—
62	#443	—	61	1	no	—
63	#365	—	61	1	no	—
64	#365	—	61	1	no	—
65	#364	—	61	1	no	—
66	#360	—	61	1	no	—
67	#360	D₆₁	1	1	yes	characteristic
68	C₆₁	#2230	1	1	yes	characteristic, complemented, Sylow
69	#135	C₂×D₆₁	1	1	yes	characteristic
70	#131	—	122	1	no	—
71	C₄⋊Q₈	—	122	1	no	—
72	M₃₂	—	61	1	no	—
73	M₃₂	C₂×D₆₁	1	1	yes	characteristic
74	#108	—	61	1	no	—
75	#106	—	61	1	no	—
76	#106	—	122	1	no	—
77	#106	C₂×D₆₁	1	1	yes	characteristic
78	#104	—	122	1	no	—
79	C₄×C₈	—	61	1	no	—
80	#41	—	2	1	no	—
81	C₂×Q₈	—	244	1	no	—
82	Q₁₆	Dic₁₂₂	1	1	yes	characteristic, complemented
83	SD₁₆	—	2	1	no	—
84	D₈	Dic₁₂₂	1	1	yes	characteristic, complemented
85	M₁₆	—	122	1	no	—
86	M₁₆	—	244	1	no	—
87	C₂×C₈	—	61	1	no	—
88	C₂×C₈	—	244	1	no	—
89	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	122	1	no	—
92	C₄⋊C₄	—	244	1	no	—
93	C₄²	—	61	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	122	1	no	—
96	Q₈	—	2	1	no	—
97	Q₈	—	488	2	no	—
98	D₄	—	2	1	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	122	1	no	—
103	C₂×C₄	—	244	1	no	—
104	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
105	C₈	—	61	1	no	—
106	C₈	—	61	1	no	—
107	C₈	—	244	1	no	—
108	C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
109	C₈	#11758224	1	1	yes	characteristic
110	C₂²	—	4	1	no	—
111	C₂²	#14519801	1	1	yes	characteristic
112	C₄	—	4	1	no	—
113	C₄	—	8	1	no	—
114	C₄	—	122	2	no	—
115	C₄	—	488	1	no	—
116	C₄	#14519754	1	1	yes	characteristic
117	C₄	SD₁₆×D₆₁	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	8	1	no	—
120	C₂	#16840050	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840159

C₂

yes

maximal, characteristic, complemented

#16839879

C₂

yes

maximal, characteristic

#16839585

C₂

yes

maximal, characteristic, complemented

#16839584

C₂

yes

maximal, characteristic, complemented

#16839582

C₂

yes

maximal, characteristic, complemented

#16839572

C₂

yes

maximal, characteristic

C₆₁×#360

C₂

yes

maximal, characteristic

#14519784

—

#14519776

—

#14519717

C₂²

yes

characteristic

#14519713

C₂²

yes

characteristic

#14519709

C₂²

yes

characteristic

#14519708

—

#14519704

—

C₈×Dic₆₁

C₂²

yes

characteristic

#135×C₆₁

C₂²

yes

characteristic

M₃₂×C₆₁

C₂²

yes

characteristic

#106×C₆₁

C₂²

yes

characteristic

#41×C₆₁

—

C₂×Dic₁₂₂

—

Q₁₆×C₆₁

Q₈

yes

characteristic, complemented

SD₁₆×C₆₁

—

D₈×C₆₁

Q₈

yes

characteristic, complemented

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

D₄

yes

—

—

—

D₄

yes

characteristic

C₁₆×C₆₁

—

#11758185

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

C₂×Q₈

yes

C₈×C₆₁

yes

—

SD₁₆

yes

characteristic

#92418

SD₁₆

yes

characteristic

C₂²×C₆₁

—

C₂²×C₆₁

AGL(1,ℤ/8)

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#2230

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

—

—

—

—

—

—

D₆₁

yes

characteristic

C₆₁

#2230

yes

characteristic, complemented, Sylow

#135

C₂×D₆₁

yes

characteristic

#131

—

122

—

C₄⋊Q₈

—

122

—

M₃₂

—

M₃₂

C₂×D₆₁

yes

—

—

—

122

—

#106

C₂×D₆₁

yes

characteristic

#104

—

122

—

C₄×C₈

—

#41

—

C₂×Q₈

—

244

—

Q₁₆

Dic₁₂₂

yes

characteristic, complemented

SD₁₆

—

D₈

Dic₁₂₂

yes

characteristic, complemented

M₁₆

—

122

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₄²

—

C₁₆

—

C₁₆

—

122

—

Q₈

—

Q₈

—

488

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

122

—

103

C₂×C₄

—

244

—

104

C₂×C₄

D₄×D₆₁

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

244

—

108

C₈

C₂×Dic₁₂₂

yes

characteristic

109

C₈

#11758224

yes

characteristic

110

C₂²

—

111

C₂²

#14519801

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

122

—

115

C₄

—

488

—

116

C₄

#14519754

yes

characteristic

117

C₄

SD₁₆×D₆₁

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

#16840050

yes

characteristic

	subgroup	G/H
center	C₂	#16840050
commutator	C₂×C₈×C₆₁	C₂³
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×#360	C₂
socle	C₂×C₆₁	#476
hypercenter	#360	D₆₁
hypocenter	C₆₁	#2230
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

subgroup

G/H

center

C₂

#16840050

commutator

C₂×C₈×C₆₁

C₂³

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₆₁×#360

C₂

C₂×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

N	H ≅ G/N	type
1. #16840159	C₂	semidirect
3. #16839585	C₂	semidirect
4. #16839584	C₂	semidirect
5. #16839582	C₂	semidirect
21. Q₁₆×C₆₁	Q₈	semidirect
23. D₈×C₆₁	Q₈	semidirect
68. C₆₁	#2230	semidirect
82. Q₁₆	Dic₁₂₂	semidirect
84. D₈	Dic₁₂₂	semidirect

H ≅ G/N

type

1. #16840159

C₂

semidirect

3. #16839585

C₂

semidirect

4. #16839584

C₂

semidirect

5. #16839582

C₂

semidirect

21. Q₁₆×C₆₁

Q₈

semidirect

23. D₈×C₆₁

Q₈

semidirect

68. C₆₁

#2230

semidirect

82. Q₁₆

Dic₁₂₂

semidirect

84. D₈

Dic₁₂₂

semidirect

Subgroups — #21447988

Conjugacy classes of subgroups

Complements