Group #29584698 — subgroups

Subgroups — #29584698

Order 15616.

328 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 79 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	#21444610	C₂	1	3	yes	maximal, complemented
2	#21444488	C₂	1	3	yes	maximal
3	C₆₁×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14519774	C₂²	1	1	yes	characteristic
5	#16839815	C₂²	1	3	yes	—
6	#16839514	—	2	3	no	—
7	#16839512	—	2	3	no	—
8	#16839505	—	2	3	no	—
9	#16839505	—	2	3	no	—
10	C₆₁×#423	C₂²	1	3	yes	—
11	C₂²×D₂₄₄	—	2	3	no	—
12	#14519774	—	2	3	no	—
13	#14519774	Q₈	1	2	yes	—
14	C₄⋊#92418	D₄	1	3	yes	—
15	C₄⋊#92418	D₄	1	3	yes	—
16	C₂²⋊#92418	—	2	3	no	—
17	C₄×#92418	—	2	3	no	—
18	C₂×M₁₆×C₆₁	—	2	3	no	—
19	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₈×C₆₁	—	2	3	no	—
21	C₄⋊C₈×C₆₁	—	2	3	no	—
22	C₂³×D₆₁	—	4	1	no	—
23	C₂²×C₄×C₆₁	#41	1	3	yes	—
24	C₂×D₂₄₄	—	2	6	no	—
25	C₂×D₂₄₄	—	4	3	no	—
26	C₂×D₂₄₄	—	8	3	no	—
27	C₂×D₂₄₄	—	8	3	no	—
28	M₁₆×C₆₁	—	8	3	no	—
29	C₂×C₈×C₆₁	—	4	3	no	—
30	C₄²×C₆₁	C₂×D₄	1	3	yes	—
31	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×#92418	—	2	3	no	—
33	C₂×#92418	—	2	3	no	—
34	C₂³×C₆₁	#126	1	1	yes	characteristic
35	C₂²×D₆₁	—	4	2	no	—
36	C₂²×D₆₁	—	8	6	no	—
37	C₂×C₄×C₆₁	—	2	3	no	—
38	C₂×C₄×C₆₁	—	2	3	no	—
39	C₂×C₄×C₆₁	#121	1	3	yes	—
40	C₂×C₄×C₆₁	C₂²⋊Q₈	1	3	yes	—
41	D₂₄₄	—	8	6	no	—
42	D₂₄₄	—	8	3	no	—
43	D₂₄₄	—	8	3	no	—
44	C₈×C₆₁	—	8	3	no	—
45	#92418	—	4	3	no	—
46	#92418	—	4	3	no	—
47	#10612	—	61	1	no	maximal, Sylow
48	C₂²×C₆₁	—	2	3	no	—
49	C₂²×C₆₁	#396	1	1	yes	characteristic
50	C₂×D₆₁	—	8	6	no	—
51	C₂×D₆₁	—	16	4	no	—
52	C₄×C₆₁	—	2	3	no	—
53	C₄×C₆₁	#471	1	3	yes	—
54	C₄×C₆₁	#472	1	3	yes	—
55	#1574	—	61	3	no	—
56	#1545	—	61	3	no	—
57	#1309	D₆₁	1	1	yes	characteristic, complemented
58	C₂×C₆₁	—	4	1	no	—
59	C₂×C₆₁	#2063	1	3	yes	—
60	D₆₁	—	16	2	no	—
61	C₂×#127	—	61	1	no	—
62	#431	—	61	3	no	—
63	#423	C₂×D₆₁	1	3	yes	—
64	#335	—	122	3	no	—
65	#334	—	122	3	no	—
66	#331	—	122	3	no	—
67	#331	—	122	3	no	—
68	C₆₁	#10612	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₄	—	122	3	no	—
70	C₂×M₁₆	—	2	3	no	—
71	#127	—	61	2	no	—
72	#127	—	122	3	no	—
73	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
74	C₄⋊C₈	—	2	3	no	—
75	C₄⋊C₈	—	2	3	no	—
76	C₄⋊C₈	—	61	3	no	—
77	C₄⋊C₈	—	61	3	no	—
78	C₂²⋊C₈	—	122	3	no	—
79	C₄×C₈	—	122	3	no	—
80	C₂⁴	—	244	1	no	—
81	C₂×D₄	—	122	6	no	—
82	C₂×D₄	—	488	3	no	—
83	C₂×D₄	—	488	3	no	—
84	C₂×D₄	—	244	3	no	—
85	C₂²×C₄	#11758222	1	3	yes	—
86	M₁₆	—	8	3	no	—
87	C₂×C₈	—	4	3	no	—
88	C₂×C₈	—	122	3	no	—
89	C₂×C₈	—	122	3	no	—
90	C₄²	D₄×D₆₁	1	3	yes	—
91	C₄²	Q₈×D₆₁	1	1	yes	characteristic
92	C₂³	—	488	6	no	—
93	C₂³	—	244	2	no	—
94	C₂³	#14519771	1	1	yes	characteristic
95	D₄	—	488	6	no	—
96	D₄	—	488	3	no	—
97	D₄	—	488	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#14519750	1	3	yes	—
101	C₂×C₄	#14519755	1	3	yes	—
102	C₈	—	8	3	no	—
103	C₈	—	244	3	no	—
104	C₈	—	244	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	976	4	no	—
107	C₂²	—	488	6	no	—
108	C₂²	#16839686	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	#16840028	1	3	yes	—
111	C₄	#16840031	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	976	2	no	—
114	C₂	#21447180	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21444610

C₂

yes

maximal, complemented

#21444488

C₂

yes

maximal

C₆₁×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#14519774

C₂²

yes

characteristic

#16839815

C₂²

yes

—

—

—

—

—

C₆₁×#423

C₂²

yes

—

C₂²×D₂₄₄

—

#14519774

—

#14519774

Q₈

yes

—

C₄⋊#92418

D₄

yes

—

C₄⋊#92418

D₄

yes

—

C₂²⋊#92418

—

C₄×#92418

—

C₂×M₁₆×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

C₂³×D₆₁

—

C₂²×C₄×C₆₁

#41

yes

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

#126

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

#396

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#471

yes

—

C₄×C₆₁

#472

yes

—

#1574

—

#1545

—

#1309

D₆₁

yes

characteristic, complemented

C₂×C₆₁

—

C₂×C₆₁

#2063

yes

—

D₆₁

—

C₂×#127

—

#431

—

#423

C₂×D₆₁

yes

—

#335

—

122

—

#334

—

122

—

#331

—

122

—

#331

—

122

—

C₆₁

#10612

yes

characteristic, complemented, Sylow

C₂²×D₄

—

122

—

C₂×M₁₆

—

#127

—

#127

—

122

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

122

—

C₄×C₈

—

122

—

C₂⁴

—

244

—

C₂×D₄

—

122

—

C₂×D₄

—

488

—

C₂×D₄

—

488

—

C₂×D₄

—

244

—

C₂²×C₄

#11758222

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄²

D₄×D₆₁

yes

—

C₄²

Q₈×D₆₁

yes

characteristic

C₂³

—

488

—

C₂³

—

244

—

C₂³

#14519771

yes

characteristic

D₄

—

488

—

D₄

—

488

—

D₄

—

488

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#14519750

yes

—

101

C₂×C₄

#14519755

yes

—

102

C₈

—

103

C₈

—

244

—

104

C₈

—

244

—

105

C₂²

—

106

C₂²

—

976

—

107

C₂²

—

488

—

108

C₂²

#16839686

yes

characteristic

109

C₄

—

110

C₄

#16840028

yes

—

111

C₄

#16840031

yes

—

112

C₂

—

113

C₂

—

976

—

114

C₂

#21447180

yes

—

	subgroup	G/H
center	C₂²	#16839686
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1309	C₂
socle	C₂²×C₆₁	#396
prime radical	C₂×#14519774	C₂²
hypercenter	#1309	D₆₁
hypocenter	C₆₁	#10612
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#16839686

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1309

C₂

C₂²×C₆₁

C₂²

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
1. #21444610	C₂	semidirect
3. C₆₁×#1309	C₂	semidirect
57. #1309	D₆₁	semidirect
68. C₆₁	#10612	semidirect

H ≅ G/N

type

1. #21444610

C₂

semidirect

3. C₆₁×#1309

C₂

semidirect

57. #1309

D₆₁

semidirect

68. C₆₁

#10612

semidirect

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

subgroup	index
#21444610	2
#21444488	2
C₆₁×#1309	2
#10612	61

subgroup

index

#21444610

#21444488

C₆₁×#1309

#10612

Subgroups — #29584698

Conjugacy classes of subgroups

Complements

Maximal subgroups