Group #29583090 — subgroups

Subgroups — #29583090

Order 15616.

476 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 93 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	#21446642	C₂	1	3	yes	maximal
2	#21443700	C₂	1	3	yes	maximal, complemented
3	C₆₁×#2845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14519774	C₂²	1	1	yes	characteristic
5	C₂×#14519697	—	2	6	no	—
6	C₂×#14519688	C₂²	1	3	yes	—
7	C₆₁×#398	—	2	6	no	—
8	C₆₁×#382	C₂²	1	3	yes	—
9	C₂²×D₂₄₄	—	2	3	no	—
10	C₂²×#92418	—	2	3	no	—
11	#14519774	—	2	3	no	—
12	#14519774	Q₈	1	2	yes	—
13	#14519697	—	4	12	no	—
14	#14519688	—	2	6	no	—
15	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
16	C₂×C₄⋊C₄×C₆₁	D₄	1	6	yes	—
17	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
18	#95×C₆₁	—	2	2	no	—
19	#95×C₆₁	—	2	6	no	—
20	C₂³×D₆₁	—	4	1	no	—
21	C₂²×C₄×C₆₁	—	2	6	no	—
22	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
23	C₂×D₂₄₄	—	2	6	no	—
24	C₂×D₂₄₄	—	4	6	no	—
25	C₂×D₂₄₄	—	4	6	no	—
26	C₂×D₂₄₄	—	4	3	no	—
27	C₄⋊C₄×C₆₁	—	4	6	no	—
28	C₄⋊C₄×C₆₁	—	8	6	no	—
29	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₆₁	#41	1	3	yes	—
31	C₂×#92418	—	2	6	no	—
32	C₂×#92418	—	4	3	no	—
33	C₂×#92418	—	4	3	no	—
34	C₂³×C₆₁	#120	1	1	yes	characteristic
35	C₂²×D₆₁	—	4	2	no	—
36	C₂²×D₆₁	—	8	3	no	—
37	C₂²×D₆₁	—	8	3	no	—
38	C₂×C₄×C₆₁	—	2	3	no	—
39	C₂×C₄×C₆₁	—	2	3	no	—
40	C₂×C₄×C₆₁	—	4	6	no	—
41	C₂×C₄×C₆₁	—	4	6	no	—
42	C₂×C₄×C₆₁	—	4	6	no	—
43	C₂×C₄×C₆₁	C₂²⋊Q₈	1	3	yes	—
44	C₂×C₄×C₆₁	#124	1	3	yes	—
45	D₂₄₄	—	4	12	no	—
46	D₂₄₄	—	8	6	no	—
47	#92418	—	4	6	no	—
48	#10360	—	61	1	no	maximal, Sylow
49	C₂²×C₆₁	#393	1	1	yes	characteristic
50	C₂²×C₆₁	#453	1	3	yes	—
51	C₂²×C₆₁	#454	1	3	yes	—
52	C₂×D₆₁	—	8	6	no	—
53	C₂×D₆₁	—	16	3	no	—
54	C₂×D₆₁	—	16	1	no	—
55	C₄×C₆₁	—	2	3	no	—
56	C₄×C₆₁	—	2	3	no	—
57	C₄×C₆₁	—	8	6	no	—
58	#2845	D₆₁	1	1	yes	characteristic, complemented
59	#1961	—	61	3	no	—
60	#1295	—	61	3	no	—
61	C₂×C₆₁	#1656	1	3	yes	—
62	C₂×C₆₁	#1658	1	1	yes	characteristic
63	C₂×C₆₁	#2027	1	3	yes	—
64	D₆₁	—	16	2	no	—
65	C₂×#127	—	61	1	no	—
66	C₂×#102	—	122	6	no	—
67	C₂×#97	—	61	3	no	—
68	#398	—	2	6	no	—
69	#382	C₂×D₆₁	1	3	yes	—
70	C₆₁	#10360	1	1	yes	characteristic, complemented, Sylow
71	C₂²×D₄	—	122	3	no	—
72	C₂²×C₈	—	122	3	no	—
73	#127	—	61	2	no	—
74	#127	—	122	3	no	—
75	C₂×C₄⋊C₄	—	2	6	no	—
76	C₂×C₄⋊C₄	C₂²⋊D₆₁	1	6	yes	—
77	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
78	#102	—	244	12	no	—
79	#97	—	122	6	no	—
80	#95	—	2	2	no	—
81	#95	—	2	6	no	—
82	C₂⁴	—	244	1	no	—
83	C₂×D₄	—	122	6	no	—
84	C₂×D₄	—	244	6	no	—
85	C₂×D₄	—	244	6	no	—
86	C₂×D₄	—	244	3	no	—
87	C₂²×C₄	—	2	6	no	—
88	C₂²×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
89	C₂×C₈	—	122	6	no	—
90	C₂×C₈	—	244	3	no	—
91	C₂×C₈	—	244	3	no	—
92	C₄⋊C₄	—	4	6	no	—
93	C₄⋊C₄	—	8	6	no	—
94	C₄²	Q₈×D₆₁	1	1	yes	characteristic
95	C₄²	#11758226	1	3	yes	—
96	C₂³	—	488	3	no	—
97	C₂³	—	488	3	no	—
98	C₂³	—	244	2	no	—
99	C₂³	#14519746	1	1	yes	characteristic
100	D₄	—	488	6	no	—
101	D₄	—	244	12	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	#14519753	1	3	yes	—
108	C₂×C₄	#14519766	1	3	yes	—
109	C₈	—	244	6	no	—
110	C₂²	—	976	1	no	—
111	C₂²	—	976	3	no	—
112	C₂²	—	488	6	no	—
113	C₂²	#16839673	1	1	yes	characteristic
114	C₂²	#16839930	1	3	yes	—
115	C₂²	#16839935	1	3	yes	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	8	6	no	—
119	C₂	—	976	2	no	—
120	C₂	#21445112	1	3	yes	—
121	C₂	#21445121	1	1	yes	characteristic
122	C₂	#21446972	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21446642

C₂

yes

maximal

#21443700

C₂

yes

maximal, complemented

C₆₁×#2845

C₂

yes

maximal, characteristic, complemented

C₂×#14519774

C₂²

yes

characteristic

C₂×#14519697

—

C₂×#14519688

C₂²

yes

—

C₆₁×#398

—

C₆₁×#382

C₂²

yes

—

C₂²×D₂₄₄

—

C₂²×#92418

—

#14519774

—

#14519774

Q₈

yes

—

#14519697

—

#14519688

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#95×C₆₁

—

#95×C₆₁

—

C₂³×D₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

#120

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

#124

yes

—

D₂₄₄

—

D₂₄₄

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#454

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

#2845

D₆₁

yes

characteristic, complemented

#1961

—

#1295

—

C₂×C₆₁

#1656

yes

—

C₂×C₆₁

#1658

yes

characteristic

C₂×C₆₁

#2027

yes

—

D₆₁

—

C₂×#127

—

C₂×#102

—

122

—

C₂×#97

—

#398

—

#382

C₂×D₆₁

yes

—

C₆₁

#10360

yes

characteristic, complemented, Sylow

C₂²×D₄

—

122

—

C₂²×C₈

—

122

—

#127

—

#127

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₆₁

yes

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

#102

—

244

—

#97

—

122

—

#95

—

#95

—

C₂⁴

—

244

—

C₂×D₄

—

122

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

C₂×D₄

—

244

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

—

244

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×D₆₁

yes

characteristic

C₄²

#11758226

yes

—

C₂³

—

488

—

C₂³

—

488

—

C₂³

—

244

—

C₂³

#14519746

yes

characteristic

100

D₄

—

488

—

101

D₄

—

244

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#14519753

yes

—

108

C₂×C₄

#14519766

yes

—

109

C₈

—

244

—

110

C₂²

—

976

—

111

C₂²

—

976

—

112

C₂²

—

488

—

113

C₂²

#16839673

yes

characteristic

114

C₂²

#16839930

yes

—

115

C₂²

#16839935

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₂

—

976

—

120

C₂

#21445112

yes

—

121

C₂

#21445121

yes

characteristic

122

C₂

#21446972

yes

—

	subgroup	G/H
center	C₂³	#14519746
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#2845	C₂
socle	C₂³×C₆₁	#120
prime radical	C₂×#14519774	C₂²
hypercenter	#2845	D₆₁
hypocenter	C₆₁	#10360
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#14519746

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#2845

C₂

C₂³×C₆₁

C₂²

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
2. #21443700	C₂	semidirect
3. C₆₁×#2845	C₂	semidirect
58. #2845	D₆₁	semidirect
70. C₆₁	#10360	semidirect

H ≅ G/N

type

2. #21443700

C₂

semidirect

3. C₆₁×#2845

C₂

semidirect

58. #2845

D₆₁

semidirect

70. C₆₁

#10360

semidirect

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

subgroup	index
#21446642	2
#21443700	2
C₆₁×#2845	2
#10360	61

subgroup

index

#21446642

#21443700

C₆₁×#2845

#10360

Subgroups — #29583090

Conjugacy classes of subgroups

Complements

Maximal subgroups