Group #21459093 — subgroups

Subgroups — C₂²×#14519775 #21459093

Order 7808.

1624 conjugacy classes of subgroups; only the 725 normal subgroups are shown.

normal subgroups only

This group has 1624 conjugacy classes of subgroups — too many to draw. Only its 725 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	C₂³×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519775	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#11758196	C₂	1	4	yes	maximal
4	C₂²×C₄×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	—	2	8	no	—
7	C₂²×Dic₁₂₂	C₂²	1	12	yes	—
8	C₂³×Dic₆₁	C₂²	1	2	yes	—
9	C₂³×Dic₆₁	C₂²	1	2	yes	—
10	#14519775	C₂²	1	64	yes	direct product factor
11	C₂×#11758196	C₂²	1	48	yes	—
12	C₂×C₄×Dic₆₁	C₂²	1	12	yes	—
13	C₂²×Q₈×C₆₁	—	2	8	no	—
14	C₂²×Q₈×C₆₁	C₂²	1	12	yes	—
15	C₂³×C₄×C₆₁	C₂²	1	2	yes	—
16	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
17	C₂⁴×C₆₁	C₂³	1	1	yes	characteristic
18	C₂×Q₈×C₆₁	—	2	48	no	—
19	C₂×Q₈×C₆₁	C₂³	1	16	yes	—
20	C₂²×C₄×C₆₁	—	2	8	no	—
21	C₂²×C₄×C₆₁	D₄	1	8	yes	—
22	C₂²×C₄×C₆₁	C₂³	1	12	yes	—
23	C₂²×C₄×C₆₁	C₂³	1	6	yes	—
24	C₂²×Dic₆₁	—	2	8	no	—
25	C₂²×Dic₆₁	Q₈	1	16	yes	complemented
26	C₂²×Dic₆₁	C₂³	1	12	yes	—
27	C₂²×Dic₆₁	C₂³	1	12	yes	—
28	C₂×Dic₁₂₂	—	2	48	no	—
29	C₂×Dic₁₂₂	C₂³	1	16	yes	—
30	#11758196	C₂³	1	64	yes	—
31	C₄×Dic₆₁	C₂³	1	16	yes	—
32	C₂³×C₆₁	C₂×D₄	1	4	yes	—
33	C₂³×C₆₁	C₂×Q₈	1	8	yes	—
34	C₂³×C₆₁	C₂⁴	1	3	yes	—
35	Q₈×C₆₁	—	2	32	no	—
36	C₂×C₄×C₆₁	—	2	24	no	—
37	C₂×C₄×C₆₁	C₂×D₄	1	24	yes	—
38	C₂×C₄×C₆₁	C₂⁴	1	8	yes	—
39	C₂×C₄×C₆₁	C₂⁴	1	4	yes	—
40	C₂×Dic₆₁	—	2	24	no	—
41	C₂×Dic₆₁	C₂×Q₈	1	48	yes	complemented
42	C₂×Dic₆₁	C₂⁴	1	8	yes	—
43	C₂×Dic₆₁	C₂⁴	1	8	yes	—
44	Dic₁₂₂	—	2	32	no	—
45	C₂²×C₆₁	C₄⋊Q₈	1	16	yes	complemented
46	C₂²×C₆₁	C₂²×D₄	1	6	yes	—
47	C₂²×C₆₁	C₂²×Q₈	1	12	yes	—
48	C₂²×C₆₁	C₂⁵	1	1	yes	characteristic
49	C₄×C₆₁	—	2	8	no	—
50	C₄×C₆₁	C₂²×D₄	1	8	yes	—
51	Dic₆₁	—	2	8	no	—
52	Dic₆₁	C₂²×Q₈	1	16	yes	complemented
53	C₂²×C₄⋊Q₈	—	61	1	no	maximal, Sylow
54	C₂×C₆₁	C₂×C₄⋊Q₈	1	12	yes	complemented
55	C₂×C₆₁	C₂³×D₄	1	1	yes	characteristic
56	C₂×C₆₁	C₂³×Q₈	1	2	yes	—
57	C₂³×Q₈	—	61	1	no	—
58	C₂³×Q₈	D₆₁	1	1	yes	characteristic
59	C₂×C₄⋊Q₈	—	61	24	no	—
60	C₂²×C₄⋊C₄	—	61	4	no	—
61	C₂²×C₄²	—	61	1	no	—
62	C₆₁	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
63	C₂²×Q₈	—	2	8	no	—
64	C₂²×Q₈	—	61	12	no	—
65	C₂²×Q₈	—	122	8	no	—
66	C₂²×Q₈	C₂×D₆₁	1	12	yes	—
67	C₂³×C₄	—	61	2	no	—
68	C₂³×C₄	—	61	2	no	—
69	C₂³×C₄	C₂×D₆₁	1	2	yes	—
70	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
71	C₄⋊Q₈	—	61	64	no	—
72	C₂×C₄⋊C₄	—	61	48	no	—
73	C₂×C₄²	—	61	12	no	—
74	C₂⁴	C₂²×D₆₁	1	1	yes	characteristic
75	C₂×Q₈	—	2	48	no	—
76	C₂×Q₈	—	61	16	no	—
77	C₂×Q₈	—	122	48	no	—
78	C₂×Q₈	C₂²×D₆₁	1	16	yes	—
79	C₂²×C₄	—	2	8	no	—
80	C₂²×C₄	—	61	16	no	—
81	C₂²×C₄	—	61	12	no	—
82	C₂²×C₄	—	61	12	no	—
83	C₂²×C₄	—	122	8	no	—
84	C₂²×C₄	C₂²⋊D₆₁	1	8	yes	—
85	C₂²×C₄	C₂²×D₆₁	1	12	yes	—
86	C₂²×C₄	C₂²×D₆₁	1	6	yes	—
87	C₄⋊C₄	—	61	64	no	—
88	C₄²	—	61	16	no	—
89	C₂³	Q₈×D₆₁	1	8	yes	—
90	C₂³	C₂×C₂²⋊D₆₁	1	4	yes	—
91	C₂³	C₂³×D₆₁	1	3	yes	—
92	Q₈	—	2	32	no	—
93	Q₈	—	122	32	no	—
94	C₂×C₄	—	2	24	no	—
95	C₂×C₄	—	61	48	no	—
96	C₂×C₄	—	61	8	no	—
97	C₂×C₄	—	61	8	no	—
98	C₂×C₄	—	122	24	no	—
99	C₂×C₄	C₂×C₂²⋊D₆₁	1	24	yes	—
100	C₂×C₄	C₂³×D₆₁	1	8	yes	—
101	C₂×C₄	C₂³×D₆₁	1	4	yes	—
102	C₂²	#14519775	1	16	yes	direct product factor
103	C₂²	C₂²×C₂²⋊D₆₁	1	6	yes	—
104	C₂²	C₂×Q₈×D₆₁	1	12	yes	—
105	C₂²	C₂⁴×D₆₁	1	1	yes	characteristic
106	C₄	—	2	8	no	—
107	C₄	—	61	16	no	—
108	C₄	—	122	8	no	—
109	C₄	C₂²×C₂²⋊D₆₁	1	8	yes	—
110	C₂	C₂×#14519775	1	12	yes	direct product factor
111	C₂	C₂³×C₂²⋊D₆₁	1	1	yes	characteristic
112	C₂	C₂²×Q₈×D₆₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₁₂₂

C₂

yes

maximal, characteristic

C₂×#14519775

C₂

yes

maximal, direct product factor

C₂²×#11758196

C₂

yes

maximal

C₂²×C₄×Dic₆₁

C₂

yes

maximal, characteristic

C₂³×Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

—

C₂²×Dic₁₂₂

C₂²

yes

—

C₂³×Dic₆₁

C₂²

yes

—

C₂³×Dic₆₁

C₂²

yes

—

#14519775

C₂²

yes

direct product factor

C₂×#11758196

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

C₂²×Q₈×C₆₁

—

C₂²×Q₈×C₆₁

C₂²

yes

—

C₂³×C₄×C₆₁

C₂²

yes

—

C₂³×C₄×C₆₁

C₂²

yes

characteristic

C₂⁴×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

Q₈

yes

complemented

C₂²×Dic₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂³

yes

—

#11758196

C₂³

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

—

C₂³×C₆₁

C₂×Q₈

yes

—

C₂³×C₆₁

C₂⁴

yes

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×Q₈

yes

complemented

C₂×Dic₆₁

C₂⁴

yes

—

C₂×Dic₆₁

C₂⁴

yes

—

Dic₁₂₂

—

C₂²×C₆₁

C₄⋊Q₈

yes

complemented

C₂²×C₆₁

C₂²×D₄

yes

—

C₂²×C₆₁

C₂²×Q₈

yes

—

C₂²×C₆₁

C₂⁵

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂²×D₄

yes

—

Dic₆₁

—

Dic₆₁

C₂²×Q₈

yes

complemented

C₂²×C₄⋊Q₈

—

maximal, Sylow

C₂×C₆₁

C₂×C₄⋊Q₈

yes

complemented

C₂×C₆₁

C₂³×D₄

yes

characteristic

C₂×C₆₁

C₂³×Q₈

yes

—

C₂³×Q₈

—

C₂³×Q₈

D₆₁

yes

characteristic

C₂×C₄⋊Q₈

—

C₂²×C₄⋊C₄

—

C₂²×C₄²

—

C₆₁

C₂²×C₄⋊Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

122

—

C₂²×Q₈

C₂×D₆₁

yes

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×D₆₁

yes

—

C₂³×C₄

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂⁴

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

C₂²×D₆₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄²

—

C₂³

Q₈×D₆₁

yes

—

C₂³

C₂×C₂²⋊D₆₁

yes

—

C₂³

C₂³×D₆₁

yes

—

Q₈

—

Q₈

—

122

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

100

C₂×C₄

C₂³×D₆₁

yes

—

101

C₂×C₄

C₂³×D₆₁

yes

—

102

C₂²

#14519775

yes

direct product factor

103

C₂²

C₂²×C₂²⋊D₆₁

yes

—

104

C₂²

C₂×Q₈×D₆₁

yes

—

105

C₂²

C₂⁴×D₆₁

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

122

—

109

C₄

C₂²×C₂²⋊D₆₁

yes

—

110

C₂

C₂×#14519775

yes

direct product factor

111

C₂

C₂³×C₂²⋊D₆₁

yes

characteristic

112

C₂

C₂²×Q₈×D₆₁

yes

—

	subgroup	G/H
center	C₂⁴	C₂²×D₆₁
commutator	C₂²×C₆₁	C₂⁵
Frattini	C₂²	C₂⁴×D₆₁
Fitting	C₂³×Q₈×C₆₁	C₂
socle	C₂⁴×C₆₁	C₂³
hypercenter	C₂³×Q₈	D₆₁
hypocenter	C₆₁	C₂²×C₄⋊Q₈
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

subgroup

G/H

center

C₂⁴

C₂²×D₆₁

commutator

C₂²×C₆₁

C₂⁵

Frattini

C₂²

C₂⁴×D₆₁

Fitting

C₂³×Q₈×C₆₁

C₂

socle

C₂⁴×C₆₁

C₂³

hypercenter

C₂³×Q₈

D₆₁

hypocenter

C₆₁

C₂²×C₄⋊Q₈

LCS

G ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂⁴ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. C₂×#14519775	C₂	direct (N × H)
10. #14519775	C₂²	direct (N × H)
25. C₂²×Dic₆₁	Q₈	semidirect
41. C₂×Dic₆₁	C₂×Q₈	semidirect
45. C₂²×C₆₁	C₄⋊Q₈	semidirect
52. Dic₆₁	C₂²×Q₈	semidirect
54. C₂×C₆₁	C₂×C₄⋊Q₈	semidirect
62. C₆₁	C₂²×C₄⋊Q₈	semidirect
102. C₂²	#14519775	direct (N × H)
110. C₂	C₂×#14519775	direct (N × H)

H ≅ G/N

type

2. C₂×#14519775

C₂

direct (N × H)

10. #14519775

C₂²

direct (N × H)

25. C₂²×Dic₆₁

Q₈

semidirect

41. C₂×Dic₆₁

C₂×Q₈

semidirect

45. C₂²×C₆₁

C₄⋊Q₈

semidirect

52. Dic₆₁

C₂²×Q₈

semidirect

54. C₂×C₆₁

C₂×C₄⋊Q₈

semidirect

62. C₆₁

C₂²×C₄⋊Q₈

semidirect

102. C₂²

#14519775

direct (N × H)

110. C₂

C₂×#14519775

direct (N × H)

Subgroups — C₂²×#14519775 #21459093

Conjugacy classes of subgroups

Complements

Subgroups — C22×#14519775 #21459093

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#14519775 #21459093