Group #21447326 — subgroups

Subgroups — #21447326

Order 7808.

176 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 71 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	#16839804	C₂	1	2	yes	maximal
2	C₂×#14519711	C₂	1	1	yes	maximal, characteristic
3	#16839788	C₂	1	1	yes	maximal, characteristic
4	#16839530	C₂	1	2	yes	maximal
5	C₆₁×#341	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758194	C₂²	1	1	yes	characteristic
7	#14519736	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
9	#14519711	—	2	2	no	—
10	#14519711	—	2	2	no	—
11	#14519709	—	2	2	no	—
12	#14519706	—	2	2	no	—
13	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
16	C₂²×Dic₆₁	—	2	2	no	—
17	M₁₆×C₆₁	—	2	2	no	—
18	M₁₆×C₆₁	Q₈	1	4	yes	—
19	C₂×C₈×C₆₁	—	2	2	no	—
20	C₂×C₈×C₆₁	—	2	2	no	—
21	C₂×C₈×C₆₁	D₄	1	2	yes	—
22	C₂²⋊Dic₆₁	—	2	1	no	—
23	C₄⋊Dic₆₁	—	2	2	no	—
24	C₄⋊Dic₆₁	—	2	2	no	—
25	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
26	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
27	C₄×Dic₆₁	—	2	1	no	—
28	#11758194	—	2	2	no	—
29	C₂×#92418	—	2	1	no	—
30	C₂³×C₆₁	#41	1	1	yes	characteristic
31	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₆₁	#41	1	2	yes	—
35	C₂×Dic₆₁	—	2	1	no	—
36	C₂×Dic₆₁	—	2	1	no	—
37	C₂×Dic₆₁	—	4	2	no	—
38	C₂×Dic₆₁	—	8	2	no	—
39	C₈×C₆₁	—	2	4	no	—
40	C₈×C₆₁	—	2	2	no	—
41	#92418	—	4	1	no	—
42	C₂²×C₆₁	—	2	1	no	—
43	C₂²×C₆₁	—	2	1	no	—
44	C₂²×C₆₁	#121	1	1	yes	characteristic
45	C₂²×C₆₁	#125	1	2	yes	—
46	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₆₁	#123	1	1	yes	characteristic
48	C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
49	Dic₆₁	—	4	1	no	—
50	Dic₆₁	—	4	1	no	—
51	Dic₆₁	—	8	2	no	—
52	#2089	—	61	1	no	maximal, Sylow
53	C₂×C₆₁	—	2	2	no	—
54	C₂×C₆₁	#398	1	1	yes	characteristic
55	C₂×C₆₁	#472	1	1	yes	characteristic
56	C₂×C₆₁	#473	1	1	yes	characteristic
57	#428	—	61	2	no	—
58	C₂×#107	—	61	1	no	—
59	#424	—	61	1	no	—
60	#343	—	61	2	no	—
61	#341	D₆₁	1	1	yes	characteristic
62	C₆₁	#2089	1	1	yes	characteristic, complemented, Sylow
63	C₂×M₁₆	—	61	1	no	—
64	C₂×M₁₆	C₂×D₆₁	1	2	yes	—
65	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
66	#117	—	61	1	no	—
67	C₂×C₄⋊C₄	—	61	2	no	—
68	#107	—	122	2	no	—
69	#107	—	122	2	no	—
70	#106	—	122	2	no	—
71	C₄⋊C₈	—	122	2	no	—
72	C₂²×C₄	—	122	2	no	—
73	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	—	122	2	no	—
76	M₁₆	Dic₁₂₂	1	4	yes	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	122	1	no	—
80	C₂×C₈	D₂₄₄	1	2	yes	—
81	C₄⋊C₄	—	61	1	no	—
82	C₄⋊C₄	—	61	1	no	—
83	C₄⋊C₄	—	122	2	no	—
84	C₄⋊C₄	—	122	2	no	—
85	C₂²⋊C₄	—	122	1	no	—
86	C₄²	—	122	1	no	—
87	C₂³	#11758226	1	1	yes	characteristic
88	C₂×C₄	—	122	1	no	—
89	C₂×C₄	—	122	1	no	—
90	C₂×C₄	—	488	2	no	—
91	C₂×C₄	—	244	2	no	—
92	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
93	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
94	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
95	C₂×C₄	#11758224	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	2	2	no	—
98	C₈	—	244	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#14519752	1	1	yes	characteristic
102	C₂²	#14519768	1	2	yes	—
103	C₄	—	488	2	no	—
104	C₄	—	244	1	no	—
105	C₄	—	244	1	no	—
106	C₄	#14519754	1	2	yes	—
107	C₄	#14519761	1	1	yes	characteristic
108	C₄	#14519777	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#16839694	1	1	yes	characteristic
111	C₂	#16840033	1	1	yes	characteristic
112	C₂	#16840038	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839804

C₂

yes

maximal

C₂×#14519711

C₂

yes

maximal, characteristic

#16839788

C₂

yes

maximal, characteristic

#16839530

C₂

yes

maximal

C₆₁×#341

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

characteristic

#14519736

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₆₁

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

Q₈

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

characteristic

C₄⋊Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

—

#11758194

—

C₂×#92418

—

C₂³×C₆₁

#41

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#121

yes

characteristic

C₂²×C₆₁

#125

yes

—

C₄×C₆₁

C₂²⋊Q₈

yes

—

C₄×C₆₁

#123

yes

characteristic

C₄×C₆₁

C₄⋊Q₈

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#2089

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

—

—

D₆₁

yes

characteristic

C₆₁

#2089

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₁

yes

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

#107

—

122

—

#107

—

122

—

#106

—

122

—

C₄⋊C₈

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

M₁₆

—

122

—

M₁₆

Dic₁₂₂

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

D₂₄₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₂²⋊C₄

—

122

—

C₄²

—

122

—

C₂³

#11758226

yes

characteristic

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

488

—

C₂×C₄

—

244

—

C₂×C₄

C₂×Dic₁₂₂

yes

—

C₂×C₄

C₂×D₂₄₄

yes

characteristic

C₂×C₄

D₄×D₆₁

yes

characteristic

C₂×C₄

#11758224

yes

—

C₈

—

C₈

—

C₈

—

244

—

C₂²

—

100

C₂²

—

101

C₂²

#14519752

yes

characteristic

102

C₂²

#14519768

yes

—

103

C₄

—

488

—

104

C₄

—

244

—

105

C₄

—

244

—

106

C₄

#14519754

yes

—

107

C₄

#14519761

yes

characteristic

108

C₄

#14519777

yes

characteristic

109

C₂

—

110

C₂

#16839694

yes

characteristic

111

C₂

#16840033

yes

characteristic

112

C₂

#16840038

yes

characteristic

	subgroup	G/H
center	C₂²	#14519752
commutator	C₂²×C₄×C₆₁	C₂³
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#341	C₂
socle	C₂²×C₆₁	#121
hypercenter	#341	D₆₁
hypocenter	C₆₁	#2089
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#14519752

commutator

C₂²×C₄×C₆₁

C₂³

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#341

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
62. C₆₁	#2089	semidirect

H ≅ G/N

type

62. C₆₁

#2089

semidirect

Subgroups — #21447326

Conjugacy classes of subgroups

Complements