Group #21461641 — subgroups

Subgroups — #21461641

Order 7808.

852 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 158 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	C₂×C₂²⋊#92420	C₂	1	6	yes	maximal, complemented
2	C₂⁴×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#92420	C₂²	1	4	yes	complemented
4	C₂²⋊#92420	—	2	12	no	—
5	C₂³×Dic₆₁	—	2	12	no	—
6	C₂³×Dic₆₁	C₄	1	3	yes	—
7	C₂³×Dic₆₁	C₂²	1	3	yes	—
8	C₂⁵×C₆₁	C₄	1	1	yes	characteristic
9	C₂⁴×C₆₁	—	2	6	no	—
10	C₂⁴×C₆₁	—	2	6	no	—
11	C₂⁴×C₆₁	C₈	1	4	yes	complemented
12	C₂⁴×C₆₁	C₂×C₄	1	3	yes	—
13	C₂²×Dic₆₁	—	2	32	no	—
14	C₂²×Dic₆₁	—	2	6	no	—
15	C₂²×Dic₆₁	—	2	12	no	—
16	C₂²×Dic₆₁	—	2	12	no	—
17	C₂²×Dic₆₁	C₂×C₄	1	3	yes	—
18	C₂²×Dic₆₁	D₄	1	12	yes	—
19	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
20	C₂×#92420	—	2	12	no	—
21	C₂³×C₆₁	—	2	8	no	—
22	C₂³×C₆₁	—	2	24	no	—
23	C₂³×C₆₁	—	2	24	no	—
24	C₂³×C₆₁	—	2	6	no	—
25	C₂³×C₆₁	—	2	6	no	—
26	C₂³×C₆₁	C₂²⋊C₄	1	6	yes	—
27	C₂³×C₆₁	C₂×C₈	1	6	yes	—
28	C₂³×C₆₁	M₁₆	1	6	yes	—
29	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₆₁	—	2	24	no	—
31	C₂×Dic₆₁	—	2	6	no	—
32	C₂×Dic₆₁	—	2	24	no	—
33	C₂×Dic₆₁	C₂²⋊C₄	1	6	yes	—
34	C₂×Dic₆₁	C₂×D₄	1	6	yes	—
35	#92420	—	4	4	no	—
36	C₂²×C₆₁	—	2	32	no	—
37	C₂²×C₆₁	—	2	6	no	—
38	C₂²×C₆₁	—	2	12	no	—
39	C₂²×C₆₁	—	2	6	no	—
40	C₂²×C₆₁	—	2	12	no	—
41	C₂²×C₆₁	C₂²⋊C₈	1	12	yes	complemented
42	C₂²×C₆₁	C₂×C₂²⋊C₄	1	3	yes	—
43	C₂²×C₆₁	C₂²×C₈	1	1	yes	characteristic
44	C₂²×C₆₁	C₂×M₁₆	1	3	yes	—
45	Dic₆₁	—	2	6	no	—
46	Dic₆₁	#120	1	4	yes	—
47	#1779	—	61	1	no	maximal, Sylow
48	C₂×C₆₁	—	2	12	no	—
49	C₂×C₆₁	#379	1	1	yes	characteristic
50	C₂×C₆₁	C₂×C₂²⋊C₈	1	3	yes	—
51	C₂×C₆₁	#407	1	3	yes	—
52	C₂⁴×C₄	—	61	1	no	—
53	C₂×C₂²⋊C₈	—	61	6	no	—
54	C₆₁	#1779	1	1	yes	characteristic, complemented, Sylow
55	C₂⁵	#6923	1	1	yes	characteristic
56	C₂³×C₄	—	61	3	no	—
57	C₂³×C₄	—	61	3	no	—
58	C₂³×C₄	—	122	12	no	—
59	C₂²×C₈	—	61	4	no	—
60	C₂²⋊C₈	—	122	12	no	—
61	C₂⁴	—	2	6	no	—
62	C₂⁴	—	2	6	no	—
63	C₂⁴	#92420	1	4	yes	complemented
64	C₂⁴	C₂×#6923	1	3	yes	—
65	C₂²×C₄	—	61	1	no	—
66	C₂²×C₄	—	61	12	no	—
67	C₂²×C₄	—	61	3	no	—
68	C₂²×C₄	—	122	12	no	—
69	C₂²×C₄	—	122	12	no	—
70	C₂²×C₄	—	122	32	no	—
71	C₂²×C₄	—	122	6	no	—
72	C₂×C₈	—	122	12	no	—
73	C₂³	—	2	8	no	—
74	C₂³	—	2	24	no	—
75	C₂³	—	2	24	no	—
76	C₂³	—	2	6	no	—
77	C₂³	—	2	6	no	—
78	C₂³	C₂×#92420	1	6	yes	—
79	C₂³	#11758217	1	6	yes	—
80	C₂³	C₂²⋊#6923	1	6	yes	—
81	C₂³	C₂²×#6923	1	1	yes	characteristic
82	C₂×C₄	—	61	6	no	—
83	C₂×C₄	—	61	6	no	—
84	C₂×C₄	—	122	24	no	—
85	C₂×C₄	—	122	24	no	—
86	C₂×C₄	—	122	6	no	—
87	C₈	—	244	4	no	—
88	C₂²	—	2	32	no	—
89	C₂²	—	2	12	no	—
90	C₂²	—	2	12	no	—
91	C₂²	—	2	6	no	—
92	C₂²	—	2	6	no	—
93	C₂²	C₂²⋊#92420	1	12	yes	complemented
94	C₂²	C₂×C₂²⋊#6923	1	3	yes	—
95	C₂²	C₂²×#92420	1	1	yes	characteristic
96	C₂²	C₂×#11758217	1	3	yes	—
97	C₄	—	61	4	no	—
98	C₄	—	122	6	no	—
99	C₂	—	2	12	no	—
100	C₂	#16840703	1	1	yes	characteristic
101	C₂	C₂×C₂²⋊#92420	1	3	yes	—
102	C₂	#16840732	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊#92420

C₂

yes

maximal, complemented

C₂⁴×Dic₆₁

C₂

yes

maximal, characteristic

C₂²×#92420

C₂²

yes

complemented

C₂²⋊#92420

—

C₂³×Dic₆₁

—

C₂³×Dic₆₁

C₄

yes

—

C₂³×Dic₆₁

C₂²

yes

—

C₂⁵×C₆₁

C₄

yes

characteristic

C₂⁴×C₆₁

—

C₂⁴×C₆₁

—

C₂⁴×C₆₁

C₈

yes

complemented

C₂⁴×C₆₁

C₂×C₄

yes

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

C₂×C₄

yes

—

C₂²×Dic₆₁

D₄

yes

—

C₂²×Dic₆₁

C₂³

yes

characteristic

C₂×#92420

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂²⋊C₄

yes

—

C₂³×C₆₁

C₂×C₈

yes

—

C₂³×C₆₁

M₁₆

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

—

#92420

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²⋊C₈

yes

complemented

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₆₁

C₂²×C₈

yes

characteristic

C₂²×C₆₁

C₂×M₁₆

yes

—

Dic₆₁

—

Dic₆₁

#120

yes

—

#1779

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#379

yes

characteristic

C₂×C₆₁

C₂×C₂²⋊C₈

yes

—

C₂×C₆₁

#407

yes

—

C₂⁴×C₄

—

C₂×C₂²⋊C₈

—

C₆₁

#1779

yes

characteristic, complemented, Sylow

C₂⁵

#6923

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

—

122

—

C₂²×C₈

—

C₂²⋊C₈

—

122

—

C₂⁴

—

C₂⁴

—

C₂⁴

#92420

yes

complemented

C₂⁴

C₂×#6923

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂×C₈

—

122

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×#92420

yes

—

C₂³

#11758217

yes

—

C₂³

C₂²⋊#6923

yes

—

C₂³

C₂²×#6923

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₈

—

244

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²⋊#92420

yes

complemented

C₂²

C₂×C₂²⋊#6923

yes

—

C₂²

C₂²×#92420

yes

characteristic

C₂²

C₂×#11758217

yes

—

C₄

—

C₄

—

122

—

C₂

—

100

C₂

#16840703

yes

characteristic

101

C₂

C₂×C₂²⋊#92420

yes

—

102

C₂

#16840732

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#6923
commutator	C₂²×C₆₁	C₂²×C₈
Fitting	C₂⁵×C₆₁	C₄
socle	C₂³×C₆₁	C₂²×C₄
hypercenter	C₂⁵	#6923
hypocenter	C₆₁	#1779
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵

subgroup

G/H

center, Frattini

C₂³

C₂²×#6923

commutator

C₂²×C₆₁

C₂²×C₈

Fitting

C₂⁵×C₆₁

C₄

socle

C₂³×C₆₁

C₂²×C₄

C₂⁵

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ C₂⁵

N	H ≅ G/N	type
1. C₂×C₂²⋊#92420	C₂	semidirect
3. C₂²×#92420	C₂²	semidirect
11. C₂⁴×C₆₁	C₈	semidirect
41. C₂²×C₆₁	C₂²⋊C₈	semidirect
54. C₆₁	#1779	semidirect
63. C₂⁴	#92420	semidirect
93. C₂²	C₂²⋊#92420	semidirect

H ≅ G/N

type

1. C₂×C₂²⋊#92420

C₂

semidirect

3. C₂²×#92420

C₂²

semidirect

11. C₂⁴×C₆₁

C₈

semidirect

41. C₂²×C₆₁

C₂²⋊C₈

semidirect

54. C₆₁

#1779

semidirect

63. C₂⁴

#92420

semidirect

93. C₂²

C₂²⋊#92420

semidirect

Subgroups — #21461641

Conjugacy classes of subgroups

Complements