Group #29573787 — subgroups

Subgroups — #29573787

Order 15616.

422 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 227 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	#21445976	C₂	1	2	yes	maximal
2	#21445947	C₂	1	4	yes	maximal
3	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519709	—	2	4	no	—
5	#16839632	C₂²	1	4	yes	—
6	#16839525	—	2	8	no	—
7	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₈²×C₆₁	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₆₁	—	2	8	no	—
11	C₂×C₄⋊Dic₆₁	—	2	4	no	—
12	#14519709	—	4	8	no	—
13	C₂²×C₈×C₆₁	D₄	1	4	yes	—
14	C₂²×C₈×C₆₁	D₄	1	2	yes	—
15	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₆₁	Q₈	1	4	yes	—
19	C₄×C₈×C₆₁	Q₈	1	2	yes	—
20	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
22	C₂²×Dic₆₁	—	4	4	no	—
23	C₂×C₈×C₆₁	—	2	8	no	—
24	C₂×C₈×C₆₁	C₄⋊C₄	1	8	yes	—
25	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
26	C₂×C₈×C₆₁	SD₁₆	1	8	yes	—
27	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
28	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
29	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
30	C₄⋊Dic₆₁	—	4	8	no	—
31	C₄⋊Dic₆₁	—	4	16	no	—
32	C₂³×C₆₁	#127	1	1	yes	characteristic
33	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
38	C₂×C₄×C₆₁	#135	1	8	yes	—
39	C₂×Dic₆₁	—	8	4	no	—
40	C₂×Dic₆₁	—	8	8	no	—
41	C₈×C₆₁	—	2	8	no	—
42	C₈×C₆₁	#106	1	8	yes	—
43	#8815	—	61	1	no	maximal, Sylow
44	C₂²×C₆₁	#384	1	1	yes	characteristic
45	C₂²×C₆₁	#492	1	2	yes	—
46	C₂²×C₆₁	#495	1	4	yes	—
47	C₄×C₆₁	C₂×#106	1	2	yes	—
48	C₄×C₆₁	#427	1	4	yes	—
49	C₄×C₆₁	#498	1	2	yes	—
50	C₄×C₆₁	#499	1	4	yes	—
51	Dic₆₁	—	16	4	no	—
52	#1844	—	61	2	no	—
53	#1834	—	61	4	no	—
54	C₂×C₈²	D₆₁	1	1	yes	characteristic
55	C₂×C₆₁	#1708	1	4	yes	—
56	C₂×C₆₁	#1844	1	1	yes	characteristic
57	C₂×C₆₁	#1846	1	2	yes	—
58	C₂×#106	—	122	4	no	—
59	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
60	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
61	#384	—	61	4	no	—
62	#340	—	122	8	no	—
63	C₈²	Dic₆₁	1	4	yes	complemented
64	C₆₁	#8815	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₈	D₂₄₄	1	4	yes	—
66	C₂²×C₈	D₂₄₄	1	2	yes	—
67	C₂×C₄⋊C₄	—	122	8	no	—
68	C₂×C₄⋊C₄	—	122	4	no	—
69	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
70	#106	—	244	8	no	—
71	C₄×C₈	Dic₁₂₂	1	4	yes	—
72	C₄×C₈	Dic₁₂₂	1	2	yes	—
73	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
74	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
75	C₂²×C₄	—	244	4	no	—
76	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₂₄₄	1	2	yes	—
78	C₂×C₈	—	2	8	no	—
79	C₂×C₈	#11758190	1	8	yes	—
80	C₂×C₈	C₄⋊Dic₆₁	1	8	yes	—
81	C₂×C₈	C₄⋊Dic₆₁	1	4	yes	—
82	C₄⋊C₄	—	244	8	no	—
83	C₄⋊C₄	—	244	16	no	—
84	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
85	C₄²	C₂×Dic₁₂₂	1	2	yes	—
86	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
87	C₂³	#14519774	1	1	yes	characteristic
88	C₂×C₄	—	488	4	no	—
89	C₂×C₄	—	488	8	no	—
90	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄⋊Dic₆₁	1	2	yes	—
92	C₂×C₄	#14519777	1	1	yes	characteristic
93	C₂×C₄	#14519777	1	2	yes	—
94	C₂×C₄	C₂×#11758190	1	4	yes	—
95	C₂×C₄	#14519799	1	8	yes	—
96	C₈	—	2	8	no	—
97	C₈	#14519709	1	8	yes	—
98	C₂²	#16839632	1	1	yes	characteristic
99	C₂²	#16840135	1	2	yes	—
100	C₂²	#16840146	1	4	yes	—
101	C₄	—	976	4	no	—
102	C₄	C₂×#14519709	1	2	yes	—
103	C₄	#16839800	1	4	yes	—
104	C₄	#16840160	1	2	yes	—
105	C₄	#16840164	1	4	yes	—
106	C₂	#21445420	1	4	yes	—
107	C₂	#21445976	1	1	yes	characteristic
108	C₂	#21445983	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445976

C₂

yes

maximal

#21445947

C₂

yes

maximal

C₂×C₈²×C₆₁

C₂

yes

maximal, characteristic

C₂×#14519709

—

#16839632

C₂²

yes

—

#16839525

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₈²×C₆₁

C₄

yes

complemented

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

#14519709

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂³×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

#135

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#106

yes

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#495

yes

—

C₄×C₆₁

C₂×#106

yes

—

C₄×C₆₁

#427

yes

—

C₄×C₆₁

#498

yes

—

C₄×C₆₁

#499

yes

—

Dic₆₁

—

#1844

—

#1834

—

C₂×C₈²

D₆₁

yes

characteristic

C₂×C₆₁

#1708

yes

—

C₂×C₆₁

#1844

yes

characteristic

C₂×C₆₁

#1846

yes

—

C₂×#106

—

122

—

C₂×C₄×C₈

C₂×D₆₁

yes

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

#384

—

#340

—

122

—

C₈²

Dic₆₁

yes

complemented

C₆₁

#8815

yes

characteristic, complemented, Sylow

C₂²×C₈

D₂₄₄

yes

—

C₂²×C₈

D₂₄₄

yes

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

#106

—

244

—

C₄×C₈

Dic₁₂₂

yes

—

C₄×C₈

Dic₁₂₂

yes

—

C₄×C₈

C₂×Dic₆₁

yes

—

C₄×C₈

C₂×Dic₆₁

yes

—

C₂²×C₄

—

244

—

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

C₂²×C₄

C₂×D₂₄₄

yes

—

C₂×C₈

—

C₂×C₈

#11758190

yes

—

C₂×C₈

C₄⋊Dic₆₁

yes

—

C₂×C₈

C₄⋊Dic₆₁

yes

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₄²

C₂×Dic₁₂₂

yes

characteristic

C₄²

C₂×Dic₁₂₂

yes

—

C₄²

C₂²×Dic₆₁

yes

characteristic

C₂³

#14519774

yes

characteristic

C₂×C₄

—

488

—

C₂×C₄

—

488

—

C₂×C₄

C₂×C₄⋊Dic₆₁

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₆₁

yes

—

C₂×C₄

#14519777

yes

characteristic

C₂×C₄

#14519777

yes

—

C₂×C₄

C₂×#11758190

yes

—

C₂×C₄

#14519799

yes

—

C₈

—

C₈

#14519709

yes

—

C₂²

#16839632

yes

characteristic

C₂²

#16840135

yes

—

100

C₂²

#16840146

yes

—

101

C₄

—

976

—

102

C₄

C₂×#14519709

yes

—

103

C₄

#16839800

yes

—

104

C₄

#16840160

yes

—

105

C₄

#16840164

yes

—

106

C₂

#21445420

yes

—

107

C₂

#21445976

yes

characteristic

108

C₂

#21445983

yes

—

	subgroup	G/H
center	C₂³	#14519774
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₈²×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₈²	D₆₁
hypocenter	C₆₁	#8815
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#14519774

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₂×C₈²×C₆₁

C₂

socle, prime radical

C₂³×C₆₁

#127

cosocle

C₂×C₄²×C₆₁

C₂³

hypercenter

C₂×C₈²

D₆₁

hypocenter

C₆₁

#8815

LCS

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
9. C₈²×C₆₁	C₄	semidirect
63. C₈²	Dic₆₁	semidirect
64. C₆₁	#8815	semidirect

H ≅ G/N

type

9. C₈²×C₆₁

C₄

semidirect

63. C₈²

Dic₆₁

semidirect

64. C₆₁

#8815

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21445976	2
#21445947	2
C₂×C₈²×C₆₁	2
#8815	61

subgroup

index

#21445976

#21445947

C₂×C₈²×C₆₁

#8815

Subgroups — #29573787

Conjugacy classes of subgroups

Complements

Maximal subgroups