Group #29607247 — subgroups

Subgroups — #29607247

Order 15616.

1118 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 411 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	#21444081	C₂	1	8	yes	maximal
2	#21444049	C₂	1	6	yes	maximal
3	C₂×C₆₁×#376	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₆₁	C₂²	1	4	yes	—
5	C₂×#14519684	C₂²	1	24	yes	—
6	C₂²×C₄²×C₆₁	C₂²	1	3	yes	—
7	C₆₁×#376	C₄	1	8	yes	complemented
8	C₂×#95×C₆₁	C₂²	1	4	yes	—
9	C₂³×Dic₆₁	C₂³	1	8	yes	—
10	C₂×C₄⋊Dic₆₁	—	2	24	no	—
11	#14519684	—	4	24	no	—
12	C₂³×C₄×C₆₁	C₂³	1	6	yes	—
13	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₆₁	—	2	12	no	—
15	C₂×C₄²×C₆₁	C₂×C₄	1	12	yes	—
16	#95×C₆₁	C₂×C₄	1	16	yes	—
17	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
18	C₂²×C₄×C₆₁	—	2	6	no	—
19	C₂²×C₄×C₆₁	—	2	6	no	—
20	C₂²×C₄×C₆₁	—	2	12	no	—
21	C₂²×C₄×C₆₁	—	2	12	no	—
22	C₂²×C₄×C₆₁	C₂²×C₄	1	12	yes	—
23	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
24	C₂²×C₄×C₆₁	#41	1	12	yes	—
25	C₂²×Dic₆₁	—	2	24	no	—
26	C₂²×Dic₆₁	—	2	24	no	—
27	C₂²×Dic₆₁	—	2	8	no	—
28	C₄²×C₆₁	—	2	24	no	—
29	C₄⋊Dic₆₁	—	4	16	no	—
30	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
31	C₂³×C₆₁	C₂×#41	1	6	yes	—
32	C₂³×C₆₁	H(5,𝔽₂)	1	4	yes	—
33	C₂³×C₆₁	2₋¹⁺⁴	1	4	yes	—
34	C₂×C₄×C₆₁	—	2	12	no	—
35	C₂×C₄×C₆₁	—	2	24	no	—
36	C₂×C₄×C₆₁	—	4	6	no	—
37	C₂×C₄×C₆₁	—	4	6	no	—
38	C₂×C₄×C₆₁	—	4	12	no	—
39	C₂×C₄×C₆₁	#117	1	12	yes	—
40	C₂×C₄×C₆₁	#126	1	16	yes	—
41	C₂×Dic₆₁	—	4	8	no	—
42	C₂×Dic₆₁	—	4	24	no	—
43	C₂×Dic₆₁	—	4	24	no	—
44	#14350	—	61	1	no	maximal, Sylow
45	C₂²×C₆₁	C₂×#117	1	3	yes	—
46	C₂²×C₆₁	#520	1	4	yes	—
47	C₂²×C₆₁	C₂×#126	1	4	yes	—
48	C₂²×C₆₁	#540	1	6	yes	—
49	C₂²×C₆₁	#541	1	6	yes	—
50	C₂²×C₆₁	#542	1	12	yes	—
51	C₄×C₆₁	—	4	12	no	—
52	C₄×C₆₁	#383	1	8	yes	—
53	Dic₆₁	—	8	8	no	—
54	C₂×#376	D₆₁	1	1	yes	characteristic
55	#1443	—	61	8	no	—
56	C₄⋊#95	—	61	6	no	—
57	C₂×C₆₁	C₂×#383	1	1	yes	characteristic
58	C₂×C₆₁	#2314	1	6	yes	—
59	C₂×C₆₁	#2644	1	4	yes	—
60	C₂×C₆₁	#2645	1	4	yes	—
61	C₂²×C₄⋊C₄	—	61	4	no	—
62	C₂²×C₄²	C₂×D₆₁	1	3	yes	—
63	#376	Dic₆₁	1	8	yes	complemented
64	C₂×#95	—	61	24	no	—
65	C₂×#95	C₂×D₆₁	1	4	yes	—
66	C₆₁	#14350	1	1	yes	characteristic, complemented, Sylow
67	C₂³×C₄	—	61	8	no	—
68	C₂³×C₄	C₂²×D₆₁	1	6	yes	—
69	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	122	24	no	—
71	C₂×C₄²	—	2	12	no	—
72	C₂×C₄²	C₂×Dic₆₁	1	12	yes	—
73	#95	—	244	24	no	—
74	#95	C₂×Dic₆₁	1	16	yes	—
75	C₂⁴	C₂³×D₆₁	1	1	yes	characteristic
76	C₂²×C₄	—	2	6	no	—
77	C₂²×C₄	—	2	6	no	—
78	C₂²×C₄	—	2	12	no	—
79	C₂²×C₄	—	2	12	no	—
80	C₂²×C₄	—	122	24	no	—
81	C₂²×C₄	—	122	24	no	—
82	C₂²×C₄	—	122	8	no	—
83	C₂²×C₄	#11758222	1	12	yes	—
84	C₂²×C₄	C₂²×Dic₆₁	1	12	yes	—
85	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
86	C₄⋊C₄	—	244	16	no	—
87	C₄²	—	2	24	no	—
88	C₂³	C₂³×Dic₆₁	1	1	yes	characteristic
89	C₂³	C₂×#11758222	1	6	yes	—
90	C₂³	#14519823	1	4	yes	—
91	C₂³	#14519824	1	4	yes	—
92	C₂×C₄	—	2	12	no	—
93	C₂×C₄	—	2	24	no	—
94	C₂×C₄	—	4	6	no	—
95	C₂×C₄	—	4	6	no	—
96	C₂×C₄	—	4	12	no	—
97	C₂×C₄	—	244	8	no	—
98	C₂×C₄	—	244	24	no	—
99	C₂×C₄	—	244	24	no	—
100	C₂×C₄	#14519736	1	12	yes	—
101	C₂×C₄	#14519771	1	16	yes	—
102	C₂²	C₂×#14519736	1	3	yes	—
103	C₂²	#16840245	1	4	yes	—
104	C₂²	C₂×#14519771	1	4	yes	—
105	C₂²	#16840367	1	6	yes	—
106	C₂²	#16840374	1	6	yes	—
107	C₂²	#16840383	1	12	yes	—
108	C₄	—	4	12	no	—
109	C₄	—	488	8	no	—
110	C₄	#16839630	1	8	yes	—
111	C₂	C₂×#16839630	1	1	yes	characteristic
112	C₂	#21448428	1	6	yes	—
113	C₂	#21451779	1	4	yes	—
114	C₂	#21451784	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21444081

C₂

yes

maximal

#21444049

C₂

yes

maximal

C₂×C₆₁×#376

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#14519684

C₂²

yes

—

C₂²×C₄²×C₆₁

C₂²

yes

—

C₆₁×#376

C₄

yes

complemented

C₂×#95×C₆₁

C₂²

yes

—

C₂³×Dic₆₁

C₂³

yes

—

C₂×C₄⋊Dic₆₁

—

#14519684

—

C₂³×C₄×C₆₁

C₂³

yes

—

C₂³×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

#95×C₆₁

C₂×C₄

yes

—

C₂⁴×C₆₁

C₂⁴

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

—

C₂²×C₄×C₆₁

#41

yes

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₄²×C₆₁

—

C₄⋊Dic₆₁

—

C₂³×C₆₁

C₂³×C₄

yes

characteristic

C₂³×C₆₁

C₂×#41

yes

—

C₂³×C₆₁

H(5,𝔽₂)

yes

—

C₂³×C₆₁

2₋¹⁺⁴

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

#126

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

#14350

—

maximal, Sylow

C₂²×C₆₁

C₂×#117

yes

—

C₂²×C₆₁

#520

yes

—

C₂²×C₆₁

C₂×#126

yes

—

C₂²×C₆₁

#540

yes

—

C₂²×C₆₁

#541

yes

—

C₂²×C₆₁

#542

yes

—

C₄×C₆₁

—

C₄×C₆₁

#383

yes

—

Dic₆₁

—

C₂×#376

D₆₁

yes

characteristic

#1443

—

C₄⋊#95

—

C₂×C₆₁

C₂×#383

yes

characteristic

C₂×C₆₁

#2314

yes

—

C₂×C₆₁

#2644

yes

—

C₂×C₆₁

#2645

yes

—

C₂²×C₄⋊C₄

—

C₂²×C₄²

C₂×D₆₁

yes

—

#376

Dic₆₁

yes

complemented

C₂×#95

—

C₂×#95

C₂×D₆₁

yes

—

C₆₁

#14350

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

C₂²×D₆₁

yes

—

C₂³×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₆₁

yes

—

#95

—

244

—

#95

C₂×Dic₆₁

yes

—

C₂⁴

C₂³×D₆₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

#11758222

yes

—

C₂²×C₄

C₂²×Dic₆₁

yes

—

C₂²×C₄

C₂²×Dic₆₁

yes

—

C₄⋊C₄

—

244

—

C₄²

—

C₂³

C₂³×Dic₆₁

yes

characteristic

C₂³

C₂×#11758222

yes

—

C₂³

#14519823

yes

—

C₂³

#14519824

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

244

—

C₂×C₄

—

244

—

C₂×C₄

—

244

—

100

C₂×C₄

#14519736

yes

—

101

C₂×C₄

#14519771

yes

—

102

C₂²

C₂×#14519736

yes

—

103

C₂²

#16840245

yes

—

104

C₂²

C₂×#14519771

yes

—

105

C₂²

#16840367

yes

—

106

C₂²

#16840374

yes

—

107

C₂²

#16840383

yes

—

108

C₄

—

109

C₄

—

488

—

110

C₄

#16839630

yes

—

111

C₂

C₂×#16839630

yes

characteristic

112

C₂

#21448428

yes

—

113

C₂

#21451779

yes

—

114

C₂

#21451784

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₆₁
commutator	C₂³×C₆₁	C₂³×C₄
Fitting	C₂×C₆₁×#376	C₂
socle, prime radical, cosocle	C₂⁴×C₆₁	C₂⁴
hypercenter	C₂×#376	D₆₁
hypocenter	C₆₁	#14350
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₆₁

commutator

C₂³×C₆₁

C₂³×C₄

Fitting

C₂×C₆₁×#376

C₂

socle, prime radical, cosocle

C₂⁴×C₆₁

C₂⁴

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂⁴ ◁ C₂×#376

N	H ≅ G/N	type
7. C₆₁×#376	C₄	semidirect
63. #376	Dic₆₁	semidirect
66. C₆₁	#14350	semidirect

H ≅ G/N

type

7. C₆₁×#376

C₄

semidirect

63. #376

Dic₆₁

semidirect

66. C₆₁

#14350

semidirect

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

subgroup	index
#21444081	2
#21444049	2
C₂×C₆₁×#376	2
#14350	61

subgroup

index

#21444081

#21444049

C₂×C₆₁×#376

#14350

Subgroups — #29607247

Conjugacy classes of subgroups

Complements

Maximal subgroups