Group #29594707 — subgroups

Subgroups — #29594707

Order 15616.

260 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 83 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	#21448047	C₂	1	2	yes	maximal, complemented
2	#21448043	C₂	1	2	yes	maximal
3	#21445429	C₂	1	1	yes	maximal, characteristic, complemented
4	#21445425	C₂	1	1	yes	maximal, characteristic
5	C₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840167	C₂²	1	2	yes	—
7	#16840142	C₂²	1	1	yes	characteristic
8	#16840139	C₂²	1	1	yes	characteristic
9	#16840113	—	2	2	no	—
10	#16840109	—	2	2	no	—
11	#16839567	—	2	4	no	—
12	#16839564	—	2	4	no	—
13	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
14	C₈²×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₄₄	—	2	2	no	—
16	C₂×D₄₈₈	—	2	2	no	—
17	#14519777	—	2	1	no	—
18	#14519777	—	2	2	no	—
19	#14519774	—	2	1	no	—
20	#14519711	—	2	2	no	—
21	#14519711	—	2	2	no	—
22	#14519702	—	4	4	no	—
23	#14519699	—	4	4	no	—
24	C₂×C₁₆×C₆₁	D₄	1	4	yes	—
25	C₄×C₈×C₆₁	D₄	1	2	yes	—
26	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
27	C₂×D₂₄₄	—	4	1	no	—
28	C₂×D₂₄₄	—	4	2	no	—
29	C₂×Dic₁₂₂	—	4	1	no	—
30	C₂×Dic₁₂₂	—	4	2	no	—
31	C₂×C₈×C₆₁	D₈	1	4	yes	—
32	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
33	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₄⋊Dic₆₁	—	4	2	no	—
35	C₄⋊Dic₆₁	—	4	2	no	—
36	C₄⋊Dic₆₁	—	4	2	no	—
37	Dic₂₄₄	—	4	4	no	—
38	D₄₈₈	—	4	4	no	—
39	C₁₆×C₆₁	—	2	4	no	—
40	C₂²×D₆₁	—	8	1	no	—
41	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
42	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
43	C₂×Dic₆₁	—	8	1	no	—
44	C₂×Dic₆₁	—	8	2	no	—
45	D₂₄₄	—	8	2	no	—
46	D₂₄₄	—	8	4	no	—
47	Dic₁₂₂	—	8	2	no	—
48	Dic₁₂₂	—	8	4	no	—
49	C₈×C₆₁	—	2	4	no	—
50	C₈×C₆₁	#135	1	4	yes	—
51	#12511	—	61	1	no	maximal, Sylow
52	C₂²×C₆₁	#493	1	1	yes	characteristic
53	C₂×D₆₁	—	16	1	no	—
54	C₂×D₆₁	—	16	2	no	—
55	C₄×C₆₁	#495	1	2	yes	—
56	C₄×C₆₁	#508	1	4	yes	—
57	Dic₆₁	—	16	1	no	—
58	Dic₆₁	—	16	2	no	—
59	#2241	—	61	2	no	—
60	#2240	—	61	2	no	—
61	#1710	—	61	1	no	—
62	#1709	—	61	1	no	—
63	C₈×C₁₆	D₆₁	1	1	yes	characteristic, complemented
64	C₂×C₆₁	#1709	1	1	yes	characteristic
65	C₂×C₆₁	#2249	1	2	yes	—
66	D₆₁	—	32	1	no	—
67	#500	—	61	2	no	—
68	#494	—	61	1	no	—
69	#493	—	61	1	no	—
70	#487	—	122	2	no	—
71	#486	—	122	2	no	—
72	#358	—	122	4	no	—
73	#357	—	122	4	no	—
74	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
75	C₈²	C₂×D₆₁	1	1	yes	characteristic
76	C₆₁	#12511	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₁₆	—	122	2	no	—
78	C₂×D₈	—	122	2	no	—
79	C₄⋊Q₈	—	122	1	no	—
80	C₄⋊Q₈	—	122	2	no	—
81	#127	—	122	1	no	—
82	C₂×C₁₆	D₂₄₄	1	4	yes	—
83	#107	—	122	2	no	—
84	#107	—	122	2	no	—
85	#103	—	244	4	no	—
86	#102	—	244	4	no	—
87	C₄×C₈	D₂₄₄	1	2	yes	—
88	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
89	C₂×Q₈	—	244	1	no	—
90	C₂×Q₈	—	244	2	no	—
91	C₂×D₄	—	244	1	no	—
92	C₂×D₄	—	244	2	no	—
93	Q₁₆	—	244	4	no	—
94	D₈	—	244	4	no	—
95	C₂×C₈	D₄₈₈	1	4	yes	—
96	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
97	C₄⋊C₄	—	244	2	no	—
98	C₄⋊C₄	—	244	2	no	—
99	C₄⋊C₄	—	244	2	no	—
100	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
101	C₁₆	—	2	4	no	—
102	C₂³	—	488	1	no	—
103	Q₈	—	488	4	no	—
104	Q₈	—	488	2	no	—
105	D₄	—	488	4	no	—
106	D₄	—	488	2	no	—
107	C₂×C₄	—	488	1	no	—
108	C₂×C₄	—	488	2	no	—
109	C₂×C₄	#14519774	1	1	yes	characteristic
110	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#14519799	1	4	yes	—
113	C₂²	—	976	1	no	—
114	C₂²	—	976	2	no	—
115	C₂²	#16840139	1	1	yes	characteristic
116	C₄	—	976	1	no	—
117	C₄	—	976	2	no	—
118	C₄	#16840146	1	2	yes	—
119	C₄	#16840194	1	4	yes	—
120	C₂	—	1952	1	no	—
121	C₂	#21445425	1	1	yes	characteristic
122	C₂	#21448080	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448047

C₂

yes

maximal, complemented

#21448043

C₂

yes

maximal

#21445429

C₂

yes

maximal, characteristic, complemented

#21445425

C₂

yes

maximal, characteristic

C₈×C₁₆×C₆₁

C₂

yes

maximal, characteristic, complemented

#16840167

C₂²

yes

—

#16840142

C₂²

yes

characteristic

#16840139

C₂²

yes

—

—

—

—

C₄×C₁₆×C₆₁

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

C₂×D₄₈₈

—

—

—

—

—

—

—

—

C₂×C₁₆×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

Dic₂₄₄

—

D₄₈₈

—

C₁₆×C₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

#135

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

#495

yes

—

C₄×C₆₁

#508

yes

—

Dic₆₁

—

Dic₆₁

—

—

—

—

—

C₈×C₁₆

D₆₁

yes

characteristic, complemented

C₂×C₆₁

#1709

yes

characteristic

C₂×C₆₁

#2249

yes

—

D₆₁

—

—

—

—

—

122

—

#486

—

122

—

#358

—

122

—

#357

—

122

—

C₄×C₁₆

C₂×D₆₁

yes

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12511

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×D₈

—

122

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

122

—

#127

—

122

—

C₂×C₁₆

D₂₄₄

yes

—

#107

—

122

—

#107

—

122

—

#103

—

244

—

#102

—

244

—

C₄×C₈

D₂₄₄

yes

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

244

—

C₂×Q₈

—

244

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

Q₁₆

—

244

—

D₈

—

244

—

C₂×C₈

D₄₈₈

yes

—

C₂×C₈

C₂×D₂₄₄

yes

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

100

C₄²

C₂×D₂₄₄

yes

characteristic

101

C₁₆

—

102

C₂³

—

488

—

103

Q₈

—

488

—

104

Q₈

—

488

—

105

D₄

—

488

—

106

D₄

—

488

—

107

C₂×C₄

—

488

—

108

C₂×C₄

—

488

—

109

C₂×C₄

#14519774

yes

characteristic

110

C₂×C₄

C₂×D₄₈₈

yes

—

111

C₈

—

112

C₈

#14519799

yes

—

113

C₂²

—

976

—

114

C₂²

—

976

—

115

C₂²

#16840139

yes

characteristic

116

C₄

—

976

—

117

C₄

—

976

—

118

C₄

#16840146

yes

—

119

C₄

#16840194

yes

—

120

C₂

—

1952

—

121

C₂

#21445425

yes

characteristic

122

C₂

#21448080

yes

—

	subgroup	G/H
center	C₂²	#16840139
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₈×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	#493
prime radical	#16840139	C₂²
hypercenter	C₈×C₁₆	D₆₁
hypocenter	C₆₁	#12511
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#16840139

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₈×C₁₆×C₆₁

C₂

C₂²×C₆₁

C₂²

C₈×C₁₆

D₆₁

hypocenter

C₆₁

#12511

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #21448047	C₂	semidirect
3. #21445429	C₂	semidirect
5. C₈×C₁₆×C₆₁	C₂	semidirect
63. C₈×C₁₆	D₆₁	semidirect
76. C₆₁	#12511	semidirect

H ≅ G/N

type

1. #21448047

C₂

semidirect

3. #21445429

C₂

semidirect

5. C₈×C₁₆×C₆₁

C₂

semidirect

63. C₈×C₁₆

D₆₁

semidirect

76. C₆₁

#12511

semidirect

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

subgroup	index
#21448047	2
#21448043	2
#21445429	2
#21445425	2
C₈×C₁₆×C₆₁	2
#12511	61

subgroup

index

C₈×C₁₆×C₆₁

Subgroups — #29594707

Conjugacy classes of subgroups

Complements

Maximal subgroups