Group #21447969 — subgroups

Subgroups — #21447969

Order 7808.

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

normal subgroups only

This group has 130 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	#16840166	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#16839588	C₂	1	1	yes	maximal, characteristic
4	#16839587	C₂	1	1	yes	maximal, characteristic
5	#16839580	C₂	1	1	yes	maximal, characteristic
6	#16839566	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#358	C₂	1	1	yes	maximal, characteristic
8	#14519776	—	2	1	no	—
9	Q₈×Dic₆₁	—	2	1	no	—
10	C₂×#11758185	C₂²	1	1	yes	characteristic
11	#14519711	C₂²	1	1	yes	characteristic
12	#14519711	C₂²	1	1	yes	characteristic
13	#14519710	—	2	1	no	—
14	#14519701	—	2	1	no	—
15	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	#107×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₆₁	—	2	1	no	—
20	C₂×Dic₁₂₂	—	4	1	no	—
21	Q₁₆×C₆₁	—	2	1	no	—
22	Q₁₆×C₆₁	Q₈	1	2	yes	complemented
23	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₆₁	—	2	1	no	—
25	C₄⋊Dic₆₁	—	2	1	no	—
26	C₄⋊Dic₆₁	—	2	1	no	—
27	C₄⋊Dic₆₁	—	4	1	no	—
28	#11758196	—	4	1	no	—
29	C₄×Dic₆₁	—	4	1	no	—
30	C₄×Dic₆₁	D₄	1	1	yes	characteristic
31	C₂×#92418	D₄	1	1	yes	characteristic
32	C₁₆×C₆₁	—	2	1	no	—
33	#11758185	—	2	1	no	—
34	Q₈×C₆₁	—	2	2	no	—
35	Q₈×C₆₁	—	4	1	no	—
36	C₂×C₄×C₆₁	—	4	1	no	—
37	C₂×C₄×C₆₁	—	4	1	no	—
38	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₆₁	—	4	1	no	—
40	C₂×Dic₆₁	—	4	1	no	—
41	C₂×Dic₆₁	D₈	1	1	yes	characteristic
42	C₂×Dic₆₁	D₈	1	1	yes	characteristic
43	Dic₁₂₂	—	8	2	no	—
44	C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₆₁	#41	1	1	yes	characteristic
46	#92418	—	2	1	no	—
47	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
48	C₄×C₆₁	—	4	2	no	—
49	C₄×C₆₁	—	8	1	no	—
50	C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
51	C₄×C₆₁	#137	1	1	yes	characteristic
52	Dic₆₁	—	2	1	no	—
53	Dic₆₁	—	8	1	no	—
54	Dic₆₁	—	8	1	no	—
55	Dic₆₁	Q₃₂	1	2	yes	—
56	#2227	—	61	1	no	maximal, Sylow
57	C₂×C₆₁	#474	1	1	yes	characteristic
58	C₂×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
59	C₂×C₆₁	#509	1	1	yes	characteristic
60	#500	—	61	1	no	—
61	C₄×Q₁₆	—	61	1	no	—
62	#366	—	61	1	no	—
63	#366	—	61	1	no	—
64	C₄⋊C₁₆	—	61	1	no	—
65	#358	—	61	1	no	—
66	#358	D₆₁	1	1	yes	characteristic
67	C₆₁	#2227	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
69	C₄⋊Q₈	—	122	1	no	—
70	C₄×Q₈	—	122	1	no	—
71	C₂×C₁₆	—	61	1	no	—
72	C₂×C₁₆	C₂×D₆₁	1	1	yes	characteristic
73	#107	—	61	1	no	—
74	#107	—	61	1	no	—
75	#107	—	122	1	no	—
76	#107	C₂×D₆₁	1	1	yes	characteristic
77	#103	—	122	1	no	—
78	C₄×C₈	—	61	1	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	244	1	no	—
81	Q₁₆	—	2	1	no	—
82	Q₁₆	Dic₁₂₂	1	2	yes	complemented
83	C₂×C₈	—	61	1	no	—
84	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	122	1	no	—
87	C₄⋊C₄	—	122	1	no	—
88	C₄⋊C₄	—	244	1	no	—
89	C₄⋊C₄	—	244	1	no	—
90	C₄²	—	61	1	no	—
91	C₄²	—	244	1	no	—
92	C₁₆	—	2	1	no	—
93	C₁₆	—	122	1	no	—
94	Q₈	—	2	2	no	—
95	Q₈	—	4	1	no	—
96	Q₈	—	488	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	61	1	no	—
100	C₂×C₄	—	61	1	no	—
101	C₂×C₄	—	244	1	no	—
102	C₂×C₄	—	244	1	no	—
103	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
104	C₈	—	122	1	no	—
105	C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
106	C₈	#11758224	1	1	yes	characteristic
107	C₂²	D₈×D₆₁	1	1	yes	characteristic
108	C₄	—	4	2	no	—
109	C₄	—	8	1	no	—
110	C₄	—	61	2	no	—
111	C₄	—	122	1	no	—
112	C₄	—	488	1	no	—
113	C₄	—	488	1	no	—
114	C₄	#14519754	1	1	yes	characteristic
115	C₄	#14519806	1	1	yes	characteristic
116	C₂	#16840040	1	1	yes	characteristic
117	C₂	Q₃₂×D₆₁	1	1	yes	characteristic
118	C₂	#16840199	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840166

C₂

yes

maximal, characteristic

Q₁₆×Dic₆₁

C₂

yes

maximal, characteristic

#16839588

C₂

yes

maximal, characteristic

#16839587

C₂

yes

maximal, characteristic

#16839580

C₂

yes

maximal, characteristic

#16839566

C₂

yes

maximal, characteristic

C₆₁×#358

C₂

yes

maximal, characteristic

#14519776

—

Q₈×Dic₆₁

—

C₂×#11758185

C₂²

yes

characteristic

#14519711

C₂²

yes

characteristic

#14519711

C₂²

yes

characteristic

#14519710

—

#14519701

—

C₈×Dic₆₁

C₂²

yes

characteristic

C₂×Q₁₆×C₆₁

C₂²

yes

characteristic

C₂×C₁₆×C₆₁

C₂²

yes

characteristic

#107×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Dic₁₂₂

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

Q₈

yes

complemented

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

characteristic

C₂×#92418

D₄

yes

characteristic

C₁₆×C₆₁

—

#11758185

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

D₈

yes

characteristic

C₂×Dic₆₁

D₈

yes

characteristic

Dic₁₂₂

—

C₈×C₆₁

C₂×Q₈

yes

C₈×C₆₁

yes

—

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₄×C₆₁

#137

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

Q₃₂

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×Q₃₂

yes

C₂×C₆₁

yes

—

C₄×Q₁₆

—

#366

—

#366

—

C₄⋊C₁₆

—

#358

—

#358

D₆₁

yes

characteristic

C₆₁

#2227

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

122

—

C₄×Q₈

—

122

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₁

yes

—

—

—

122

—

#107

C₂×D₆₁

yes

characteristic

#103

—

122

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

244

—

Q₁₆

—

Q₁₆

Dic₁₂₂

yes

complemented

C₂×C₈

—

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₄²

—

C₄²

—

244

—

C₁₆

—

C₁₆

—

122

—

Q₈

—

Q₈

—

Q₈

—

488

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

244

—

102

C₂×C₄

—

244

—

103

C₂×C₄

D₄×D₆₁

yes

characteristic

104

C₈

—

122

—

105

C₈

C₂×Dic₁₂₂

yes

characteristic

106

C₈

#11758224

yes

characteristic

107

C₂²

D₈×D₆₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

122

—

112

C₄

—

488

—

113

C₄

—

488

—

114

C₄

#14519754

yes

characteristic

115

C₄

#14519806

yes

characteristic

116

C₂

#16840040

yes

characteristic

117

C₂

Q₃₂×D₆₁

yes

characteristic

118

C₂

#16840199

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₆₁
commutator	C₂×C₈×C₆₁	C₂³
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×#358	C₂
socle	C₂²×C₆₁	C₂×D₈
hypercenter	#358	D₆₁
hypocenter	C₆₁	#2227
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₆₁

commutator

C₂×C₈×C₆₁

C₂³

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₆₁×#358

C₂

socle

C₂²×C₆₁

C₂×D₈

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
22. Q₁₆×C₆₁	Q₈	semidirect
67. C₆₁	#2227	semidirect
82. Q₁₆	Dic₁₂₂	semidirect

H ≅ G/N

type

22. Q₁₆×C₆₁

Q₈

semidirect

67. C₆₁

#2227

semidirect

82. Q₁₆

Dic₁₂₂

semidirect

Subgroups — #21447969

Conjugacy classes of subgroups

Complements