Group #21445314 — subgroups

Subgroups — #21445314

Order 7808.

152 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16840142	C₂	1	1	yes	maximal, characteristic
2	#16840048	C₂	1	2	yes	maximal
3	#16839892	C₂	1	1	yes	maximal, characteristic
4	#16839507	C₂	1	2	yes	maximal
5	C₆₁×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₄₄	—	2	1	no	—
7	C₂×Dic₂₄₄	—	2	1	no	—
8	#14519777	C₂²	1	2	yes	—
9	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
10	#14519709	—	2	2	no	—
11	C₄⋊#92418	C₂²	1	1	yes	characteristic
12	#14519706	—	2	1	no	—
13	#14519702	—	2	2	no	—
14	#14519687	—	2	1	no	—
15	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
16	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₂₂	—	2	2	no	—
18	C₂×Dic₁₂₂	D₄	1	1	yes	characteristic
19	C₂×C₈×C₆₁	—	2	2	no	—
20	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
21	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
22	C₄²×C₆₁	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₆₁	—	2	2	no	—
24	C₄⋊Dic₆₁	—	2	2	no	—
25	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
26	#11758196	—	2	1	no	—
27	C₄×Dic₆₁	—	2	1	no	—
28	C₂×#92418	—	2	1	no	—
29	Dic₂₄₄	—	4	2	no	—
30	Dic₂₄₄	—	4	2	no	—
31	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
34	C₂×Dic₆₁	—	2	1	no	—
35	C₂×Dic₆₁	—	2	1	no	—
36	C₂×Dic₆₁	—	4	2	no	—
37	Dic₁₂₂	—	2	2	no	—
38	Dic₁₂₂	—	4	2	no	—
39	Dic₁₂₂	—	4	2	no	—
40	C₈×C₆₁	—	2	1	no	—
41	C₈×C₆₁	—	4	2	no	—
42	C₈×C₆₁	SD₁₆	1	2	yes	—
43	#92418	—	4	1	no	—
44	C₂²×C₆₁	#121	1	1	yes	characteristic
45	C₄×C₆₁	—	2	1	no	—
46	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
47	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
48	C₄×C₆₁	#137	1	2	yes	—
49	Dic₆₁	—	4	1	no	—
50	Dic₆₁	—	4	1	no	—
51	Dic₆₁	—	8	2	no	—
52	#1690	—	61	1	no	maximal, Sylow
53	C₂×C₆₁	#460	1	1	yes	characteristic
54	C₂×C₆₁	#470	1	1	yes	characteristic
55	C₂×C₆₁	#473	1	1	yes	characteristic
56	#494	—	61	1	no	—
57	#475	—	61	2	no	—
58	#446	—	61	1	no	—
59	#334	D₆₁	1	1	yes	characteristic
60	#332	—	61	2	no	—
61	C₆₁	#1690	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₁₆	—	122	1	no	—
63	C₂×Q₁₆	—	122	1	no	—
64	C₄⋊Q₈	—	61	2	no	—
65	C₄×Q₈	—	61	1	no	—
66	#106	—	122	2	no	—
67	C₄⋊C₈	—	61	1	no	—
68	C₄⋊C₈	—	122	1	no	—
69	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
70	#103	—	122	2	no	—
71	#97	—	122	1	no	—
72	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
73	C₂×Q₈	—	61	1	no	—
74	C₂×Q₈	—	122	2	no	—
75	Q₁₆	—	244	2	no	—
76	Q₁₆	—	244	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	122	1	no	—
79	C₂×C₈	D₂₄₄	1	1	yes	characteristic
80	C₂×C₈	D₂₄₄	1	1	yes	characteristic
81	C₄⋊C₄	—	61	1	no	—
82	C₄⋊C₄	—	122	1	no	—
83	C₄⋊C₄	—	122	2	no	—
84	C₄⋊C₄	—	122	2	no	—
85	C₄²	—	122	1	no	—
86	C₄²	C₂²×D₆₁	1	1	yes	characteristic
87	Q₈	—	122	2	no	—
88	Q₈	—	244	2	no	—
89	Q₈	—	244	2	no	—
90	C₂×C₄	—	122	1	no	—
91	C₂×C₄	—	122	1	no	—
92	C₂×C₄	—	244	2	no	—
93	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
94	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
95	C₂×C₄	#11758226	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	4	2	no	—
98	C₈	—	244	1	no	—
99	C₈	#11758190	1	2	yes	—
100	C₂²	#14519752	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	488	2	no	—
103	C₄	—	244	1	no	—
104	C₄	—	244	1	no	—
105	C₄	C₂×#11758190	1	1	yes	characteristic
106	C₄	#14519805	1	1	yes	characteristic
107	C₄	#14519806	1	2	yes	—
108	C₂	#16839963	1	1	yes	characteristic
109	C₂	#16840021	1	1	yes	characteristic
110	C₂	#16840038	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840142

C₂

yes

maximal, characteristic

#16840048

C₂

yes

maximal

#16839892

C₂

yes

maximal, characteristic

#16839507

C₂

yes

maximal

C₆₁×#334

C₂

yes

maximal, characteristic

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

#14519777

C₂²

yes

—

C₄×Dic₁₂₂

C₂²

yes

characteristic

#14519709

—

C₄⋊#92418

C₂²

yes

—

—

—

C₄⋊C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

D₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

characteristic

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

Dic₂₄₄

—

Dic₂₄₄

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

SD₁₆

yes

—

#92418

—

C₂²×C₆₁

#121

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

—

D₆₁

yes

characteristic

#332

—

C₆₁

#1690

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

122

—

C₄⋊Q₈

—

C₄×Q₈

—

#106

—

122

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄⋊C₈

C₂×D₆₁

yes

—

#103

—

122

—

#97

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

122

—

Q₁₆

—

244

—

Q₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

D₂₄₄

yes

characteristic

C₂×C₈

D₂₄₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄²

—

122

—

C₄²

C₂²×D₆₁

yes

characteristic

Q₈

—

122

—

Q₈

—

244

—

Q₈

—

244

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

C₂×D₂₄₄

yes

characteristic

C₂×C₄

D₄×D₆₁

yes

characteristic

C₂×C₄

#11758226

yes

characteristic

C₈

—

C₈

—

C₈

—

244

—

C₈

#11758190

yes

—

100

C₂²

#14519752

yes

characteristic

101

C₄

—

102

C₄

—

488

—

103

C₄

—

244

—

104

C₄

—

244

—

105

C₄

C₂×#11758190

yes

characteristic

106

C₄

#14519805

yes

characteristic

107

C₄

#14519806

yes

—

108

C₂

#16839963

yes

characteristic

109

C₂

#16840021

yes

characteristic

110

C₂

#16840038

yes

characteristic

	subgroup	G/H
center	C₂²	#14519752
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#334	C₂
socle	C₂²×C₆₁	#121
hypercenter	#334	D₆₁
hypocenter	C₆₁	#1690
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#14519752

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#334

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
61. C₆₁	#1690	semidirect

H ≅ G/N

type

61. C₆₁

#1690

semidirect

Subgroups — #21445314

Conjugacy classes of subgroups

Complements