Group #21451538 — subgroups

Subgroups — #21451538

Order 7808.

782 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14519760	C₂	1	3	yes	maximal
2	#16839681	C₂	1	4	yes	maximal, complemented
3	#16839673	C₂	1	4	yes	maximal, complemented
4	#16839625	C₂	1	3	yes	maximal, complemented
5	C₂³×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂²⋊D₆₁	C₂²	1	1	yes	characteristic
7	#14519760	—	2	12	no	—
8	C₂×#11758196	C₂²	1	6	yes	—
9	C₂×C₂²⋊Dic₆₁	C₂²	1	3	yes	—
10	C₂×#11758198	C₂²	1	6	yes	—
11	#14519684	C₂²	1	12	yes	—
12	C₂²×Q₈×C₆₁	—	2	12	no	—
13	C₂²×Q₈×C₆₁	C₂²	1	4	yes	—
14	C₂³×C₄×C₆₁	C₂²	1	3	yes	—
15	C₂⁴×C₆₁	C₂³	1	1	yes	characteristic
16	C₂³×D₆₁	C₂³	1	1	yes	characteristic
17	C₂×Q₈×C₆₁	—	2	32	no	—
18	C₂×Q₈×C₆₁	—	2	24	no	—
19	C₂²×C₄×C₆₁	—	2	12	no	—
20	C₂²×C₄×C₆₁	D₄	1	12	yes	—
21	C₂²×C₄×C₆₁	C₂³	1	6	yes	—
22	C₂×C₂²⋊D₆₁	—	2	6	no	—
23	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₆₁	C₂³	1	6	yes	—
25	C₂²⋊Dic₆₁	—	2	6	no	—
26	#11758198	—	2	12	no	—
27	#11758196	—	2	12	no	—
28	C₂³×C₆₁	—	2	4	no	—
29	C₂³×C₆₁	C₂×D₄	1	6	yes	—
30	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
31	C₂²×D₆₁	—	2	3	no	—
32	C₂²×D₆₁	—	2	4	no	—
33	Q₈×C₆₁	—	2	32	no	—
34	C₂×C₄×C₆₁	—	2	24	no	—
35	C₂×C₄×C₆₁	—	2	12	no	—
36	C₂×C₄×C₆₁	C₂×D₄	1	12	yes	—
37	C₂²⋊D₆₁	—	4	4	no	—
38	C₂×Dic₆₁	—	2	3	no	—
39	C₂×Dic₆₁	—	2	6	no	—
40	C₂×Dic₆₁	—	2	12	no	—
41	C₂²×C₆₁	—	2	12	no	—
42	C₂²×C₆₁	#120	1	4	yes	—
43	C₂²×C₆₁	C₂²×D₄	1	3	yes	—
44	C₂²×C₆₁	2₋¹⁺⁴	1	4	yes	—
45	C₂×D₆₁	—	4	1	no	—
46	C₂×D₆₁	—	4	6	no	—
47	C₄×C₆₁	—	2	12	no	—
48	Dic₆₁	—	4	1	no	—
49	Dic₆₁	—	4	6	no	—
50	#2614	—	61	1	no	maximal, Sylow
51	C₂×C₆₁	—	2	4	no	—
52	C₂×C₆₁	C₂×#120	1	1	yes	characteristic
53	C₂×C₆₁	#536	1	6	yes	—
54	D₆₁	—	8	1	no	—
55	C₂³×Q₈	D₆₁	1	1	yes	characteristic, complemented
56	C₂×#123	—	61	3	no	—
57	#395	—	61	4	no	—
58	#393	—	61	4	no	—
59	#381	—	61	3	no	—
60	C₆₁	#2614	1	1	yes	characteristic, complemented, Sylow
61	C₂²×Q₈	—	2	12	no	—
62	C₂²×Q₈	C₂×D₆₁	1	4	yes	—
63	C₂²×D₄	—	61	1	no	—
64	C₂³×C₄	C₂×D₆₁	1	3	yes	—
65	#123	—	122	12	no	—
66	C₂×C₄⋊C₄	—	61	6	no	—
67	C₂×C₂²⋊C₄	—	61	3	no	—
68	C₂×C₂²⋊C₄	—	61	6	no	—
69	#95	—	61	12	no	—
70	C₂⁴	—	61	1	no	—
71	C₂⁴	C₂²×D₆₁	1	1	yes	characteristic
72	C₂×Q₈	—	2	32	no	—
73	C₂×Q₈	—	2	24	no	—
74	C₂×D₄	—	122	6	no	—
75	C₂²×C₄	—	2	12	no	—
76	C₂²×C₄	—	61	6	no	—
77	C₂²×C₄	—	61	1	no	—
78	C₂²×C₄	C₂²⋊D₆₁	1	12	yes	—
79	C₂²×C₄	C₂²×D₆₁	1	6	yes	—
80	C₄⋊C₄	—	122	12	no	—
81	C₂²⋊C₄	—	122	6	no	—
82	C₂²⋊C₄	—	122	12	no	—
83	C₂³	—	2	4	no	—
84	C₂³	—	122	3	no	—
85	C₂³	—	122	4	no	—
86	C₂³	C₂×C₂²⋊D₆₁	1	6	yes	—
87	C₂³	C₂³×D₆₁	1	1	yes	characteristic
88	Q₈	—	2	32	no	—
89	D₄	—	244	4	no	—
90	C₂×C₄	—	2	12	no	—
91	C₂×C₄	—	2	24	no	—
92	C₂×C₄	—	122	6	no	—
93	C₂×C₄	—	122	3	no	—
94	C₂×C₄	—	122	12	no	—
95	C₂×C₄	C₂×C₂²⋊D₆₁	1	12	yes	—
96	C₂²	—	2	12	no	—
97	C₂²	—	244	1	no	—
98	C₂²	—	244	6	no	—
99	C₂²	#14519746	1	4	yes	—
100	C₂²	C₂²×C₂²⋊D₆₁	1	3	yes	—
101	C₂²	#14519825	1	4	yes	—
102	C₄	—	2	12	no	—
103	C₄	—	244	1	no	—
104	C₄	—	244	6	no	—
105	C₂	—	2	4	no	—
106	C₂	—	488	1	no	—
107	C₂	C₂×#14519746	1	1	yes	characteristic
108	C₂	#16840336	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519760

C₂

yes

maximal

#16839681

C₂

yes

maximal, complemented

#16839673

C₂

yes

maximal, complemented

#16839625

C₂

yes

maximal, complemented

C₂³×Q₈×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₆₁

C₂²

yes

characteristic

#14519760

—

C₂×#11758196

C₂²

yes

—

C₂×C₂²⋊Dic₆₁

C₂²

yes

—

C₂×#11758198

C₂²

yes

—

#14519684

C₂²

yes

—

C₂²×Q₈×C₆₁

—

C₂²×Q₈×C₆₁

C₂²

yes

—

C₂³×C₄×C₆₁

C₂²

yes

—

C₂⁴×C₆₁

C₂³

yes

characteristic

C₂³×D₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂×C₂²⋊D₆₁

—

C₂²×Dic₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

C₂³

yes

—

C₂²⋊Dic₆₁

—

#11758198

—

#11758196

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂×D₄

yes

—

C₂³×C₆₁

C₂⁴

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#120

yes

—

C₂²×C₆₁

C₂²×D₄

yes

—

C₂²×C₆₁

2₋¹⁺⁴

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

Dic₆₁

—

Dic₆₁

—

#2614

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×#120

yes

characteristic

C₂×C₆₁

#536

yes

—

D₆₁

—

C₂³×Q₈

D₆₁

yes

characteristic, complemented

—

—

—

—

C₆₁

#2614

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₆₁

yes

—

C₂²×D₄

—

C₂³×C₄

C₂×D₆₁

yes

—

#123

—

122

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

C₂⁴

—

C₂⁴

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₄⋊C₄

—

122

—

C₂²⋊C₄

—

122

—

C₂²⋊C₄

—

122

—

C₂³

—

C₂³

—

122

—

C₂³

—

122

—

C₂³

C₂×C₂²⋊D₆₁

yes

—

C₂³

C₂³×D₆₁

yes

characteristic

Q₈

—

D₄

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

C₂²

—

C₂²

—

244

—

C₂²

—

244

—

C₂²

#14519746

yes

—

100

C₂²

C₂²×C₂²⋊D₆₁

yes

—

101

C₂²

#14519825

yes

—

102

C₄

—

103

C₄

—

244

—

104

C₄

—

244

—

105

C₂

—

106

C₂

—

488

—

107

C₂

C₂×#14519746

yes

characteristic

108

C₂

#16840336

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂⁴
Fitting	C₂³×Q₈×C₆₁	C₂
hypercenter	C₂³×Q₈	D₆₁
hypocenter	C₆₁	#2614
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator, socle

C₂³×C₆₁

C₂⁴

Fitting

C₂³×Q₈×C₆₁

C₂

hypercenter

C₂³×Q₈

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. #16839681	C₂	semidirect
3. #16839673	C₂	semidirect
4. #16839625	C₂	semidirect
5. C₂³×Q₈×C₆₁	C₂	semidirect
55. C₂³×Q₈	D₆₁	semidirect
60. C₆₁	#2614	semidirect

H ≅ G/N

type

2. #16839681

C₂

semidirect

3. #16839673

C₂

semidirect

4. #16839625

C₂

semidirect

5. C₂³×Q₈×C₆₁

C₂

semidirect

55. C₂³×Q₈

D₆₁

semidirect

60. C₆₁

#2614

semidirect

Subgroups — #21451538

Conjugacy classes of subgroups

Complements