Group #21445224 — subgroups

Subgroups — #21445224

Order 7808.

144 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16840167	C₂	1	1	yes	maximal, characteristic
2	#16840058	C₂	1	2	yes	maximal
3	C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
4	#16839510	C₂	1	2	yes	maximal
5	C₆₁×#335	C₂	1	1	yes	maximal, characteristic
6	#14519777	C₂²	1	2	yes	—
7	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
8	#14519711	—	2	1	no	—
9	#14519711	—	2	1	no	—
10	#14519711	—	2	2	no	—
11	C₄⋊#92418	C₂²	1	1	yes	characteristic
12	#14519706	—	2	1	no	—
13	#14519702	—	2	2	no	—
14	C₈×Dic₆₁	—	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	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
32	C₂×Dic₆₁	—	2	1	no	—
33	C₂×Dic₆₁	—	2	1	no	—
34	C₂×Dic₆₁	—	4	2	no	—
35	Dic₁₂₂	—	2	1	no	—
36	Dic₁₂₂	—	8	2	no	—
37	Dic₁₂₂	Q₁₆	1	2	yes	—
38	C₈×C₆₁	—	2	1	no	—
39	C₈×C₆₁	—	4	2	no	—
40	C₈×C₆₁	Q₁₆	1	2	yes	—
41	#92418	—	4	1	no	—
42	C₂²×C₆₁	#121	1	1	yes	characteristic
43	C₄×C₆₁	—	2	1	no	—
44	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
45	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₆₁	#135	1	1	yes	characteristic
47	C₄×C₆₁	#137	1	1	yes	characteristic
48	Dic₆₁	—	2	2	no	—
49	Dic₆₁	—	4	1	no	—
50	Dic₆₁	—	8	2	no	—
51	#1676	—	61	1	no	maximal, Sylow
52	C₂×C₆₁	#462	1	1	yes	characteristic
53	C₂×C₆₁	#467	1	1	yes	characteristic
54	C₂×C₆₁	#472	1	1	yes	characteristic
55	#500	—	61	1	no	—
56	#477	—	61	2	no	—
57	C₈×Q₈	—	61	1	no	—
58	#335	D₆₁	1	1	yes	characteristic
59	#333	—	61	2	no	—
60	C₆₁	#1676	1	1	yes	characteristic, complemented, Sylow
61	C₄⋊Q₈	—	61	2	no	—
62	C₄×Q₈	—	61	1	no	—
63	#107	—	122	2	no	—
64	#107	—	122	1	no	—
65	#107	—	122	1	no	—
66	C₄⋊C₈	—	61	1	no	—
67	C₄⋊C₈	—	122	1	no	—
68	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
69	#103	—	122	2	no	—
70	C₄×C₈	—	122	1	no	—
71	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
72	C₂×Q₈	—	61	1	no	—
73	C₂×Q₈	—	122	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	122	1	no	—
76	C₂×C₈	D₂₄₄	1	1	yes	characteristic
77	C₂×C₈	D₂₄₄	1	1	yes	characteristic
78	C₄⋊C₄	—	61	1	no	—
79	C₄⋊C₄	—	122	1	no	—
80	C₄⋊C₄	—	122	2	no	—
81	C₄⋊C₄	—	122	2	no	—
82	C₄²	—	122	1	no	—
83	C₄²	C₂²×D₆₁	1	1	yes	characteristic
84	Q₈	—	61	2	no	—
85	Q₈	—	122	1	no	—
86	Q₈	—	488	2	no	—
87	C₂×C₄	—	122	1	no	—
88	C₂×C₄	—	122	1	no	—
89	C₂×C₄	—	244	2	no	—
90	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
91	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
92	C₂×C₄	#11758226	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	4	2	no	—
95	C₈	—	244	1	no	—
96	C₈	Dic₂₄₄	1	2	yes	—
97	C₂²	#14519752	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	122	2	no	—
100	C₄	—	488	2	no	—
101	C₄	—	244	1	no	—
102	C₄	Q₁₆×D₆₁	1	1	yes	characteristic
103	C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
104	C₄	#14519796	1	1	yes	characteristic
105	C₄	#14519810	1	1	yes	characteristic
106	C₂	#16839973	1	1	yes	characteristic
107	C₂	#16840004	1	1	yes	characteristic
108	C₂	#16840033	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840167

C₂

yes

maximal, characteristic

#16840058

C₂

yes

maximal

C₈×Dic₁₂₂

C₂

yes

maximal, characteristic

#16839510

C₂

yes

maximal

C₆₁×#335

C₂

yes

maximal, characteristic

#14519777

C₂²

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

—

—

C₂²

yes

characteristic

#14519706

—

#14519702

—

C₈×Dic₆₁

—

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

—

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

Q₁₆

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

Q₁₆

yes

—

#92418

—

C₂²×C₆₁

#121

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂×Q₁₆

yes

characteristic

C₄×C₆₁

C₂×Q₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

C₈×Q₈

—

#335

D₆₁

yes

characteristic

#333

—

C₆₁

#1676

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

122

—

#107

—

122

—

#107

—

122

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄⋊C₈

C₂×D₆₁

yes

—

#103

—

122

—

C₄×C₈

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

122

—

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

—

Q₈

—

122

—

Q₈

—

488

—

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

Dic₂₄₄

yes

—

C₂²

#14519752

yes

characteristic

C₄

—

C₄

—

122

—

100

C₄

—

488

—

101

C₄

—

244

—

102

C₄

Q₁₆×D₆₁

yes

characteristic

103

C₄

C₂×Dic₂₄₄

yes

characteristic

104

C₄

#14519796

yes

characteristic

105

C₄

#14519810

yes

characteristic

106

C₂

#16839973

yes

characteristic

107

C₂

#16840004

yes

characteristic

108

C₂

#16840033

yes

characteristic

	subgroup	G/H
center	C₂²	#14519752
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#335	C₂
socle	C₂²×C₆₁	#121
hypercenter	#335	D₆₁
hypocenter	C₆₁	#1676
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#14519752

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#335

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
60. C₆₁	#1676	semidirect

H ≅ G/N

type

60. C₆₁

#1676

semidirect

Subgroups — #21445224

Conjugacy classes of subgroups

Complements