Group #29567122 — subgroups

Subgroups — #29567122

Order 15616.

200 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 91 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	#21444578	C₂	1	2	yes	maximal
2	C₆₁×#2150	C₂	1	1	yes	maximal, characteristic
3	#16839786	—	2	2	no	—
4	#16839515	—	2	2	no	—
5	#16839515	C₄	1	4	yes	—
6	C₂×M₃₂×C₆₁	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×C₄⋊C₁₆	—	2	2	no	—
9	C₂×#11758194	—	4	2	no	—
10	C₄⋊#92418	—	2	4	no	—
11	C₄⋊#92418	—	4	2	no	—
12	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
15	M₃₂×C₆₁	—	4	2	no	—
16	C₂×C₁₆×C₆₁	—	2	2	no	—
17	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₆₁	D₄	1	2	yes	—
19	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
21	C₂×C₈×C₆₁	—	2	1	no	—
22	C₂×C₈×C₆₁	—	2	1	no	—
23	C₂×C₈×C₆₁	—	4	1	no	—
24	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
25	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
27	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
28	C₄²×C₆₁	D₈	1	1	yes	characteristic
29	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
30	#11758194	—	8	4	no	—
31	C₂×#92418	—	4	4	no	—
32	C₁₆×C₆₁	—	4	2	no	—
33	C₂³×C₆₁	#106	1	1	yes	characteristic
34	C₂×C₄×C₆₁	—	2	1	no	—
35	C₂×C₄×C₆₁	—	2	1	no	—
36	C₂×C₄×C₆₁	—	2	2	no	—
37	C₂×C₄×C₆₁	—	2	2	no	—
38	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
39	C₂×C₄×C₆₁	#102	1	2	yes	—
40	C₂×C₄×C₆₁	#103	1	2	yes	—
41	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
42	C₈×C₆₁	—	2	2	no	—
43	C₈×C₆₁	—	2	1	no	—
44	C₈×C₆₁	#100	1	1	yes	characteristic
45	C₈×C₆₁	#101	1	1	yes	characteristic
46	#92418	—	8	4	no	—
47	#7245	—	61	1	no	maximal, Sylow
48	C₂²×C₆₁	—	2	1	no	—
49	C₂²×C₆₁	—	2	2	no	—
50	C₂²×C₆₁	#340	1	1	yes	characteristic
51	C₄×C₆₁	—	2	1	no	—
52	C₄×C₆₁	—	2	2	no	—
53	C₄×C₆₁	#338	1	1	yes	characteristic
54	C₄×C₆₁	#343	1	1	yes	characteristic
55	C₄×C₆₁	#361	1	2	yes	—
56	C₄×C₆₁	#362	1	2	yes	—
57	#2150	D₆₁	1	1	yes	characteristic
58	#1567	—	61	2	no	—
59	C₂×C₆₁	—	4	1	no	—
60	C₂×C₆₁	#1292	1	1	yes	characteristic
61	C₂×C₆₁	#1387	1	2	yes	—
62	C₂×M₃₂	Dic₆₁	1	2	yes	—
63	#423	—	122	2	no	—
64	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
65	C₄⋊C₁₆	—	2	2	no	—
66	#335	—	61	4	no	—
67	#335	—	122	2	no	—
68	C₆₁	#7245	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₁₆	—	244	2	no	—
70	C₂²×C₈	Dic₁₂₂	1	1	yes	characteristic
71	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
72	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
73	M₃₂	—	4	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₄⋊C₈	—	122	4	no	—
76	C₄⋊C₈	—	244	2	no	—
77	C₄×C₈	C₄×D₆₁	1	2	yes	—
78	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
79	C₂²×C₄	#11758190	1	2	yes	—
80	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
81	M₁₆	—	488	4	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	244	4	no	—
86	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
87	C₂×C₈	#11758196	1	2	yes	—
88	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
89	C₄²	D₄₈₈	1	1	yes	characteristic
90	C₄²	Dic₂₄₄	1	1	yes	characteristic
91	C₄²	#11758198	1	2	yes	—
92	C₁₆	—	4	2	no	—
93	C₂³	#14519709	1	1	yes	characteristic
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	#14519684	1	1	yes	characteristic
99	C₂×C₄	#14519699	1	2	yes	—
100	C₂×C₄	#14519702	1	2	yes	—
101	C₂×C₄	#14519711	1	1	yes	characteristic
102	C₈	—	2	2	no	—
103	C₈	—	2	1	no	—
104	C₈	—	488	4	no	—
105	C₈	#14519694	1	1	yes	characteristic
106	C₈	#14519696	1	1	yes	characteristic
107	C₂²	—	2	2	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#16839525	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	2	1	no	—
112	C₄	#16839520	1	1	yes	characteristic
113	C₄	#16839531	1	1	yes	characteristic
114	C₄	#16839576	1	2	yes	—
115	C₄	#16839579	1	2	yes	—
116	C₂	—	4	1	no	—
117	C₂	#21443694	1	1	yes	characteristic
118	C₂	#21443928	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21444578

C₂

yes

maximal

C₆₁×#2150

C₂

yes

maximal, characteristic

#16839786

—

#16839515

—

#16839515

C₄

yes

—

C₂×M₃₂×C₆₁

C₄

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×C₄⋊C₁₆

—

C₂×#11758194

—

C₄⋊#92418

—

C₄⋊#92418

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

Q₈

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂²⋊C₄

yes

—

C₄²×C₆₁

D₈

yes

characteristic

C₄²×C₆₁

Q₁₆

yes

characteristic

#11758194

—

C₂×#92418

—

C₁₆×C₆₁

—

C₂³×C₆₁

#106

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#95

yes

characteristic

C₂×C₄×C₆₁

#102

yes

—

C₂×C₄×C₆₁

#103

yes

—

C₂×C₄×C₆₁

#107

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#340

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₄×C₆₁

#362

yes

—

#2150

D₆₁

yes

characteristic

#1567

—

C₂×C₆₁

—

C₂×C₆₁

#1292

yes

characteristic

C₂×C₆₁

#1387

yes

—

C₂×M₃₂

Dic₆₁

yes

—

#423

—

122

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₁₆

—

#335

—

#335

—

122

—

C₆₁

#7245

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

244

—

C₂²×C₈

Dic₁₂₂

yes

characteristic

C₂²×C₈

C₂×Dic₆₁

yes

characteristic

C₂×C₄²

D₂₄₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

244

—

C₄×C₈

C₄×D₆₁

yes

—

C₄×C₈

C₂²⋊D₆₁

yes

—

C₂²×C₄

#11758190

yes

—

C₂²×C₄

C₄⋊Dic₆₁

yes

characteristic

M₁₆

—

488

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

C₄×Dic₆₁

yes

characteristic

C₂×C₈

#11758196

yes

—

C₂×C₈

C₂²⋊Dic₆₁

yes

characteristic

C₄²

D₄₈₈

yes

characteristic

C₄²

Dic₂₄₄

yes

characteristic

C₄²

#11758198

yes

—

C₁₆

—

C₂³

#14519709

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#14519684

yes

characteristic

C₂×C₄

#14519699

yes

—

100

C₂×C₄

#14519702

yes

—

101

C₂×C₄

#14519711

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

488

—

105

C₈

#14519694

yes

characteristic

106

C₈

#14519696

yes

characteristic

107

C₂²

—

108

C₂²

—

109

C₂²

#16839525

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

#16839520

yes

characteristic

113

C₄

#16839531

yes

characteristic

114

C₄

#16839576

yes

—

115

C₄

#16839579

yes

—

116

C₂

—

117

C₂

#21443694

yes

characteristic

118

C₂

#21443928

yes

—

	subgroup	G/H
center	C₂²	#16839525
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₆₁×#2150	C₂
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#106
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	#2150	D₆₁
hypocenter	C₆₁	#7245
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂²

#16839525

commutator

C₂×C₈×C₆₁

C₄²

Frattini

C₂×C₄×C₈

C₂×D₆₁

Fitting

C₆₁×#2150

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄×C₈×C₆₁

C₂²

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
68. C₆₁	#7245	semidirect

H ≅ G/N

type

68. C₆₁

#7245

semidirect

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

subgroup	index
#21444578	2
C₆₁×#2150	2
#7245	61

subgroup

index

#21444578

C₆₁×#2150

#7245

Subgroups — #29567122

Conjugacy classes of subgroups

Complements

Maximal subgroups