Group #29594724 — subgroups

Subgroups — #29594724

Order 15616.

398 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 103 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	#21448068	C₂	1	4	yes	maximal, complemented
2	#21445434	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14519718	—	2	8	no	—
6	#16840139	—	2	2	no	—
7	#16840139	C₂²	1	2	yes	—
8	C₄×#11758185	C₂²	1	2	yes	—
9	C₆₁×#493	—	2	2	no	—
10	C₆₁×#493	C₂²	1	2	yes	—
11	C₈²×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×D₄₈₈	—	2	4	no	—
13	C₂×D₄₈₈	—	4	4	no	—
14	#14519774	—	2	2	no	—
15	#14519718	—	4	16	no	—
16	C₂×#11758185	D₄	1	4	yes	—
17	C₂×D₈×C₆₁	—	2	4	no	—
18	C₂×D₈×C₆₁	—	4	4	no	—
19	#127×C₆₁	—	2	2	no	—
20	C₄×C₈×C₆₁	D₄	1	2	yes	—
21	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
22	C₂×D₄×C₆₁	—	4	2	no	—
23	C₂×D₄×C₆₁	—	4	4	no	—
24	C₂×D₂₄₄	—	4	2	no	—
25	C₂×D₂₄₄	—	4	4	no	—
26	D₈×C₆₁	—	4	8	no	—
27	D₈×C₆₁	—	8	8	no	—
28	C₂×C₈×C₆₁	D₈	1	4	yes	—
29	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
30	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
31	D₄₈₈	—	4	8	no	—
32	D₄₈₈	—	8	8	no	—
33	#11758185	D₈	1	8	yes	complemented
34	C₂³×C₆₁	—	8	2	no	—
35	C₂²×D₆₁	—	8	2	no	—
36	D₄×C₆₁	—	8	4	no	—
37	D₄×C₆₁	—	8	8	no	—
38	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
40	D₂₄₄	—	8	4	no	—
41	D₂₄₄	—	8	8	no	—
42	C₈×C₆₁	D₁₆	1	8	yes	complemented
43	C₈×C₆₁	C₂×D₈	1	4	yes	—
44	#12515	—	61	1	no	maximal, Sylow
45	C₂²×C₆₁	—	16	2	no	—
46	C₂²×C₆₁	—	16	4	no	—
47	C₂²×C₆₁	#493	1	1	yes	characteristic
48	C₂×D₆₁	—	16	2	no	—
49	C₂×D₆₁	—	16	4	no	—
50	C₄×C₆₁	#493	1	2	yes	—
51	C₄×C₆₁	C₂×D₁₆	1	4	yes	—
52	#2246	—	61	4	no	—
53	#1712	—	61	1	no	—
54	#1712	D₆₁	1	1	yes	characteristic, complemented
55	C₈×C₁₆	—	61	1	no	—
56	C₂×C₆₁	—	32	2	no	—
57	C₂×C₆₁	#1712	1	1	yes	characteristic
58	C₂×C₆₁	#2246	1	2	yes	—
59	D₆₁	—	32	2	no	—
60	C₂×D₁₆	—	122	8	no	—
61	#493	—	2	2	no	—
62	#493	—	61	2	no	—
63	#493	—	122	2	no	—
64	#493	C₂×D₆₁	1	2	yes	—
65	C₄×C₁₆	—	61	2	no	—
66	C₈²	C₂×D₆₁	1	1	yes	characteristic
67	C₆₁	#12515	1	1	yes	characteristic, complemented, Sylow
68	C₂×D₈	—	2	4	no	—
69	C₂×D₈	—	4	4	no	—
70	C₂×D₈	—	122	4	no	—
71	C₂×D₈	—	244	4	no	—
72	#127	—	2	2	no	—
73	#127	—	122	2	no	—
74	D₁₆	—	244	16	no	—
75	C₂×C₁₆	—	61	4	no	—
76	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
77	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
78	C₂×D₄	—	4	2	no	—
79	C₂×D₄	—	4	4	no	—
80	C₂×D₄	—	244	2	no	—
81	C₂×D₄	—	244	4	no	—
82	D₈	—	4	8	no	—
83	D₈	—	8	8	no	—
84	D₈	—	488	8	no	—
85	D₈	—	244	8	no	—
86	C₂×C₈	#11758199	1	4	yes	—
87	C₂×C₈	D₄×D₆₁	1	2	yes	—
88	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
89	C₁₆	—	61	8	no	—
90	C₂³	—	8	2	no	—
91	C₂³	—	488	2	no	—
92	D₄	—	8	4	no	—
93	D₄	—	8	8	no	—
94	D₄	—	488	4	no	—
95	D₄	—	488	8	no	—
96	C₂×C₄	#14519773	1	1	yes	characteristic
97	C₂×C₄	C₂×#11758199	1	2	yes	—
98	C₈	#14519718	1	8	yes	complemented
99	C₈	D₈×D₆₁	1	4	yes	—
100	C₂²	—	16	2	no	—
101	C₂²	—	16	4	no	—
102	C₂²	—	976	2	no	—
103	C₂²	—	976	4	no	—
104	C₂²	#16840138	1	1	yes	characteristic
105	C₄	#16840137	1	2	yes	—
106	C₄	C₂×#14519718	1	4	yes	—
107	C₂	—	32	2	no	—
108	C₂	—	1952	2	no	—
109	C₂	#21445433	1	1	yes	characteristic
110	C₂	#21448068	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448068

C₂

yes

maximal, complemented

#21445434

C₂

yes

maximal, characteristic, complemented

C₈×#11758185

C₂

yes

maximal, characteristic, complemented

C₆₁×#1712

C₂

yes

maximal, characteristic, complemented

C₂×#14519718

—

#16840139

—

#16840139

C₂²

yes

—

C₄×#11758185

C₂²

yes

—

C₆₁×#493

—

C₆₁×#493

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×D₄₈₈

—

C₂×D₄₈₈

—

#14519774

—

#14519718

—

C₂×#11758185

D₄

yes

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

#127×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

D₈×C₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

D₄₈₈

—

D₄₈₈

—

#11758185

D₈

yes

complemented

C₂³×C₆₁

—

C₂²×D₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

D₁₆

yes

complemented

C₈×C₆₁

C₂×D₈

yes

—

#12515

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#493

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

#493

yes

—

C₄×C₆₁

C₂×D₁₆

yes

—

#2246

—

#1712

—

#1712

D₆₁

yes

characteristic, complemented

C₈×C₁₆

—

C₂×C₆₁

—

C₂×C₆₁

#1712

yes

characteristic

C₂×C₆₁

#2246

yes

—

D₆₁

—

C₂×D₁₆

—

122

—

#493

—

#493

—

#493

—

122

—

#493

C₂×D₆₁

yes

—

C₄×C₁₆

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12515

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

122

—

C₂×D₈

—

244

—

#127

—

#127

—

122

—

D₁₆

—

244

—

C₂×C₁₆

—

C₄×C₈

C₂²⋊D₆₁

yes

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

D₈

—

D₈

—

D₈

—

488

—

D₈

—

244

—

C₂×C₈

#11758199

yes

—

C₂×C₈

D₄×D₆₁

yes

—

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

C₁₆

—

C₂³

—

C₂³

—

488

—

D₄

—

D₄

—

D₄

—

488

—

D₄

—

488

—

C₂×C₄

#14519773

yes

characteristic

C₂×C₄

C₂×#11758199

yes

—

C₈

#14519718

yes

complemented

C₈

D₈×D₆₁

yes

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

976

—

103

C₂²

—

976

—

104

C₂²

#16840138

yes

characteristic

105

C₄

#16840137

yes

—

106

C₄

C₂×#14519718

yes

—

107

C₂

—

108

C₂

—

1952

—

109

C₂

#21445433

yes

characteristic

110

C₂

#21448068

yes

—

	subgroup	G/H
center	C₂²	#16840138
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1712	C₂
socle	C₂²×C₆₁	#493
hypercenter	#1712	D₆₁
hypocenter	C₆₁	#12515
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#16840138

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1712

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #21448068	C₂	semidirect
2. #21445434	C₂	semidirect
3. C₈×#11758185	C₂	semidirect
4. C₆₁×#1712	C₂	semidirect
33. #11758185	D₈	semidirect
42. C₈×C₆₁	D₁₆	semidirect
54. #1712	D₆₁	semidirect
67. C₆₁	#12515	semidirect
98. C₈	#14519718	semidirect

H ≅ G/N

type

1. #21448068

C₂

semidirect

2. #21445434

C₂

semidirect

3. C₈×#11758185

C₂

semidirect

4. C₆₁×#1712

C₂

semidirect

33. #11758185

D₈

semidirect

42. C₈×C₆₁

D₁₆

semidirect

54. #1712

D₆₁

semidirect

67. C₆₁

#12515

semidirect

98. C₈

#14519718

semidirect

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

subgroup	index
#21448068	2
#21445434	2
C₈×#11758185	2
C₆₁×#1712	2
#12515	61

subgroup

index

C₆₁×#1712

Subgroups — #29594724

Conjugacy classes of subgroups

Complements

Maximal subgroups