Group #29596403 — subgroups

Subgroups — #29596403

Order 15616.

282 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 155 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	#21445991	C₂	1	2	yes	maximal
2	#21443913	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519713	—	2	2	no	—
5	C₂×#14519713	C₂²	1	4	yes	—
6	#16839786	—	2	2	no	—
7	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₆₁	C₄	1	2	yes	—
10	C₄×C₁₆×C₆₁	C₄	1	2	yes	—
11	C₂×#11758194	—	2	4	no	—
12	#14519713	—	2	8	no	—
13	#14519713	—	4	4	no	—
14	C₄⋊#92418	—	4	4	no	—
15	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₆₁	—	2	4	no	—
19	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
23	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
24	C₂²×C₄×C₆₁	Q₁₆	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
29	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₈×C₆₁	#41	1	4	yes	—
32	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
33	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
34	C₄²×C₆₁	D₈	1	2	yes	—
35	#11758194	—	4	4	no	—
36	#11758194	—	4	4	no	—
37	C₂×#92418	—	4	4	no	—
38	C₁₆×C₆₁	—	2	4	no	—
39	C₂³×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
40	C₂×C₄×C₆₁	—	2	4	no	—
41	C₂×C₄×C₆₁	#107	1	2	yes	—
42	C₂×C₄×C₆₁	#107	1	2	yes	—
43	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	#124	1	2	yes	—
45	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
47	C₈×C₆₁	#106	1	2	yes	—
48	C₈×C₆₁	#106	1	2	yes	—
49	C₈×C₆₁	#117	1	2	yes	—
50	C₈×C₆₁	#125	1	2	yes	—
51	#92418	—	8	4	no	—
52	#12891	—	61	1	no	maximal, Sylow
53	C₂²×C₆₁	—	2	2	no	—
54	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
55	C₂²×C₆₁	#487	1	2	yes	—
56	C₄×C₆₁	—	2	2	no	—
57	C₄×C₆₁	#368	1	4	yes	—
58	C₄×C₆₁	#382	1	1	yes	characteristic
59	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
60	C₄×C₆₁	#486	1	2	yes	—
61	C₂×C₄×C₁₆	D₆₁	1	1	yes	characteristic
62	#1848	—	61	2	no	—
63	#1381	—	61	4	no	—
64	C₂×C₆₁	—	2	2	no	—
65	C₂×C₆₁	#1834	1	1	yes	characteristic
66	C₂×C₆₁	C₂×#368	1	2	yes	—
67	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
68	C₂×#108	—	61	4	no	—
69	C₂×#108	—	122	2	no	—
70	#423	—	122	2	no	—
71	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
72	C₄×C₁₆	Dic₆₁	1	2	yes	—
73	C₄×C₁₆	Dic₆₁	1	2	yes	—
74	C₆₁	#12891	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	—	122	4	no	—
76	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
77	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
78	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
79	C₂×C₁₆	—	2	4	no	—
80	C₂×C₁₆	C₂×Dic₆₁	1	4	yes	—
81	#108	—	122	8	no	—
82	#108	—	244	4	no	—
83	C₄⋊C₈	—	244	4	no	—
84	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
85	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
87	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
88	C₂²×C₄	Dic₂₄₄	1	2	yes	—
89	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
90	M₁₆	—	244	4	no	—
91	M₁₆	—	244	4	no	—
92	C₂×C₈	—	244	4	no	—
93	C₂×C₈	#11758190	1	4	yes	—
94	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
95	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
96	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
97	C₂×C₈	#11758222	1	4	yes	—
98	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
99	C₄²	D₄₈₈	1	2	yes	—
100	C₄²	C₄⋊Dic₆₁	1	1	yes	characteristic
101	C₄²	C₄⋊Dic₆₁	1	1	yes	characteristic
102	C₁₆	—	2	4	no	—
103	C₂³	C₂×Dic₂₄₄	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	#14519711	1	2	yes	—
106	C₂×C₄	#14519711	1	2	yes	—
107	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
108	C₂×C₄	#14519765	1	2	yes	—
109	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
110	C₂×C₄	C₂×#11758190	1	2	yes	—
111	C₈	—	488	4	no	—
112	C₈	#14519709	1	2	yes	—
113	C₈	#14519709	1	2	yes	—
114	C₈	#14519736	1	2	yes	—
115	C₈	#14519769	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×#14519711	1	1	yes	characteristic
118	C₂²	#16840113	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	#16839592	1	4	yes	—
121	C₄	#16839628	1	1	yes	characteristic
122	C₄	C₂×#14519709	1	1	yes	characteristic
123	C₄	#16840109	1	2	yes	—
124	C₂	—	2	2	no	—
125	C₂	#21445947	1	1	yes	characteristic
126	C₂	C₂×#16839592	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445991

C₂

yes

maximal

#21443913

C₂

yes

maximal

C₂×C₄×C₁₆×C₆₁

C₂

yes

maximal, characteristic

C₂×#14519713

—

C₂×#14519713

C₂²

yes

—

#16839786

—

C₂²×C₁₆×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₄×C₁₆×C₆₁

C₄

yes

—

C₄×C₁₆×C₆₁

C₄

yes

—

—

—

—

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

Q₈

yes

characteristic

C₄×C₈×C₆₁

Q₈

yes

characteristic

C₂²×C₄×C₆₁

Q₁₆

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₄²×C₆₁

C₄⋊C₄

yes

characteristic

C₄²×C₆₁

C₄⋊C₄

yes

characteristic

C₄²×C₆₁

D₈

yes

—

#11758194

—

#11758194

—

C₂×#92418

—

C₁₆×C₆₁

—

C₂³×C₆₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#107

yes

—

C₂×C₄×C₆₁

#107

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#124

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₈×C₆₁

#106

yes

—

C₈×C₆₁

#106

yes

—

C₈×C₆₁

#117

yes

—

C₈×C₆₁

#125

yes

—

#92418

—

#12891

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

C₂×#107

yes

characteristic

C₂²×C₆₁

#487

yes

—

C₄×C₆₁

—

C₄×C₆₁

#368

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₂×C₄×C₁₆

D₆₁

yes

characteristic

#1848

—

#1381

—

C₂×C₆₁

—

C₂×C₆₁

#1834

yes

characteristic

C₂×C₆₁

C₂×#368

yes

—

C₂²×C₁₆

C₂×D₆₁

yes

—

C₂×#108

—

C₂×#108

—

122

—

#423

—

122

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄×C₁₆

Dic₆₁

yes

—

C₄×C₁₆

Dic₆₁

yes

—

C₆₁

#12891

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂²×C₈

D₂₄₄

yes

characteristic

C₂²×C₈

C₂²×D₆₁

yes

characteristic

C₂×C₄²

D₂₄₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₆₁

yes

—

#108

—

122

—

#108

—

244

—

C₄⋊C₈

—

244

—

C₄×C₈

Dic₁₂₂

yes

characteristic

C₄×C₈

Dic₁₂₂

yes

characteristic

C₄×C₈

C₂×Dic₆₁

yes

characteristic

C₄×C₈

C₂×Dic₆₁

yes

characteristic

C₂²×C₄

Dic₂₄₄

yes

—

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

M₁₆

—

244

—

M₁₆

—

244

—

C₂×C₈

—

244

—

C₂×C₈

#11758190

yes

—

C₂×C₈

C₄⋊Dic₆₁

yes

characteristic

C₂×C₈

C₄⋊Dic₆₁

yes

characteristic

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

C₂×C₈

#11758222

yes

—

C₂×C₈

C₂²×Dic₆₁

yes

characteristic

C₄²

D₄₈₈

yes

—

100

C₄²

C₄⋊Dic₆₁

yes

characteristic

101

C₄²

C₄⋊Dic₆₁

yes

characteristic

102

C₁₆

—

103

C₂³

C₂×Dic₂₄₄

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#14519711

yes

—

106

C₂×C₄

#14519711

yes

—

107

C₂×C₄

C₂×C₄⋊Dic₆₁

yes

characteristic

108

C₂×C₄

#14519765

yes

—

109

C₂×C₄

C₂×D₄₈₈

yes

characteristic

110

C₂×C₄

C₂×#11758190

yes

—

111

C₈

—

488

—

112

C₈

#14519709

yes

—

113

C₈

#14519709

yes

—

114

C₈

#14519736

yes

—

115

C₈

#14519769

yes

—

116

C₂²

—

117

C₂²

C₂×#14519711

yes

characteristic

118

C₂²

#16840113

yes

—

119

C₄

—

120

C₄

#16839592

yes

—

121

C₄

#16839628

yes

characteristic

122

C₄

C₂×#14519709

yes

characteristic

123

C₄

#16840109

yes

—

124

C₂

—

125

C₂

#21445947

yes

characteristic

126

C₂

C₂×#16839592

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₈₈
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₄×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂×#107
prime radical	C₂³×C₆₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₁
hypocenter	C₆₁	#12891
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₄₈₈

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₂×C₄×C₁₆×C₆₁

C₂

socle

C₂²×C₆₁

C₂×#107

prime radical

C₂³×C₆₁

C₂×Q₁₆

cosocle

C₂²×C₈×C₆₁

C₂³

hypercenter

C₂×C₄×C₁₆

D₆₁

hypocenter

C₆₁

#12891

LCS

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
74. C₆₁	#12891	semidirect

H ≅ G/N

type

74. C₆₁

#12891

semidirect

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

subgroup	index
#21445991	2
#21443913	2
C₂×C₄×C₁₆×C₆₁	2
#12891	61

subgroup

index

#21445991

#21443913

C₂×C₄×C₁₆×C₆₁

#12891

Subgroups — #29596403

Conjugacy classes of subgroups

Complements

Maximal subgroups