Group #21421696 — subgroups

Subgroups — Q₈×D₈×C₅₉ #21421696

Order 7552.

404 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 208 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	D₄×Q₈×C₅₉	C₂	1	2	yes	maximal, complemented
2	C₅₉×#500	C₂	1	3	yes	maximal, complemented
3	C₅₉×#474	C₂	1	6	yes	maximal, complemented
4	C₈×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈×C₅₉	C₂	1	3	yes	maximal
6	C₂²×Q₈×C₅₉	—	2	2	no	—
7	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
8	C₄⋊Q₈×C₅₉	C₂²	1	6	yes	—
9	C₂²⋊Q₈×C₅₉	—	2	6	no	—
10	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₄×D₄×C₅₉	C₂²	1	6	yes	—
12	#107×C₅₉	C₂²	1	3	yes	—
13	#107×C₅₉	C₂²	1	6	yes	—
14	C₄⋊C₈×C₅₉	C₂²	1	3	yes	—
15	#102×C₅₉	C₂²	1	6	yes	—
16	C₄×C₈×C₅₉	C₂²	1	3	yes	—
17	C₂×Q₈×C₅₉	—	2	6	no	—
18	C₂×Q₈×C₅₉	—	4	8	no	—
19	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
20	C₂×D₄×C₅₉	C₂³	1	2	yes	—
21	C₂²×C₄×C₅₉	—	2	6	no	—
22	D₈×C₅₉	Q₈	1	4	yes	direct product factor
23	C₂×C₈×C₅₉	C₂³	1	3	yes	—
24	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₅₉	—	2	6	no	—
26	C₄⋊C₄×C₅₉	D₄	1	3	yes	—
27	C₄⋊C₄×C₅₉	C₂³	1	6	yes	—
28	C₂²⋊C₄×C₅₉	—	2	6	no	—
29	C₄²×C₅₉	C₂³	1	3	yes	—
30	C₂³×C₅₉	—	2	2	no	—
31	Q₈×C₅₉	—	4	12	no	—
32	Q₈×C₅₉	D₈	1	4	yes	direct product factor
33	D₄×C₅₉	—	2	2	no	—
34	D₄×C₅₉	C₂×Q₈	1	4	yes	complemented
35	C₂×C₄×C₅₉	—	2	6	no	—
36	C₂×C₄×C₅₉	—	4	12	no	—
37	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
38	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
39	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
40	C₈×C₅₉	—	2	3	no	—
41	C₈×C₅₉	C₂×Q₈	1	2	yes	complemented
42	C₂²×C₅₉	—	2	4	no	—
43	C₂²×C₅₉	—	4	2	no	—
44	C₂²×C₅₉	—	4	2	no	—
45	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
46	C₄×C₅₉	—	2	3	no	—
47	C₄×C₅₉	—	4	6	no	—
48	C₄×C₅₉	C₂×D₈	1	6	yes	—
49	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
50	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
51	Q₈×D₈	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×C₅₉	—	4	4	no	—
53	C₂×C₅₉	D₄×Q₈	1	1	yes	characteristic
54	C₂×C₅₉	C₂²×D₈	1	1	yes	characteristic
55	C₂×C₅₉	#578	1	1	yes	characteristic
56	D₄×Q₈	C₂×C₅₉	1	2	yes	complemented
57	#500	C₂×C₅₉	1	3	yes	complemented
58	#474	C₂×C₅₉	1	6	yes	complemented
59	C₈×Q₈	C₂×C₅₉	1	1	yes	characteristic, complemented
60	C₄×D₈	C₂×C₅₉	1	3	yes	—
61	C₅₉	Q₈×D₈	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²×Q₈	—	2	2	no	—
63	C₂×D₈	C₂²×C₅₉	1	1	yes	characteristic
64	C₄⋊Q₈	C₂²×C₅₉	1	6	yes	—
65	C₂²⋊Q₈	—	2	6	no	—
66	C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
67	C₄×D₄	C₂²×C₅₉	1	6	yes	—
68	#107	C₂²×C₅₉	1	3	yes	—
69	#107	C₂²×C₅₉	1	6	yes	—
70	C₄⋊C₈	C₂²×C₅₉	1	3	yes	—
71	#102	C₂²×C₅₉	1	6	yes	—
72	C₄×C₈	C₂²×C₅₉	1	3	yes	—
73	C₂×Q₈	—	2	6	no	—
74	C₂×Q₈	—	4	8	no	—
75	C₂×Q₈	D₄×C₅₉	1	1	yes	characteristic
76	C₂×D₄	C₂³×C₅₉	1	2	yes	—
77	C₂²×C₄	—	2	6	no	—
78	D₈	Q₈×C₅₉	1	4	yes	direct product factor
79	C₂×C₈	C₂³×C₅₉	1	3	yes	—
80	C₂×C₈	C₂³×C₅₉	1	1	yes	characteristic
81	C₄⋊C₄	—	2	6	no	—
82	C₄⋊C₄	D₄×C₅₉	1	3	yes	—
83	C₄⋊C₄	C₂³×C₅₉	1	6	yes	—
84	C₂²⋊C₄	—	2	6	no	—
85	C₄²	C₂³×C₅₉	1	3	yes	—
86	C₂³	—	2	2	no	—
87	Q₈	—	4	12	no	—
88	Q₈	D₈×C₅₉	1	4	yes	direct product factor
89	D₄	—	2	2	no	—
90	D₄	C₂×Q₈×C₅₉	1	4	yes	complemented
91	C₂×C₄	—	2	6	no	—
92	C₂×C₄	—	4	12	no	—
93	C₂×C₄	C₂×D₄×C₅₉	1	3	yes	—
94	C₂×C₄	C₂×D₄×C₅₉	1	3	yes	—
95	C₂×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
96	C₈	—	2	3	no	—
97	C₈	C₂×Q₈×C₅₉	1	2	yes	complemented
98	C₂²	—	2	4	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	C₂²×D₄×C₅₉	1	1	yes	characteristic
102	C₄	—	2	3	no	—
103	C₄	—	4	6	no	—
104	C₄	C₂×D₈×C₅₉	1	6	yes	—
105	C₄	C₂²×Q₈×C₅₉	1	1	yes	characteristic
106	C₄	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
107	C₂	—	4	4	no	—
108	C₂	D₄×Q₈×C₅₉	1	1	yes	characteristic
109	C₂	C₂²×D₈×C₅₉	1	1	yes	characteristic
110	C₂	C₅₉×#578	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×Q₈×C₅₉

C₂

yes

maximal, complemented

C₅₉×#500

C₂

yes

maximal, complemented

C₅₉×#474

C₂

yes

maximal, complemented

C₈×Q₈×C₅₉

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₅₉

C₂

yes

maximal

C₂²×Q₈×C₅₉

—

C₂×D₈×C₅₉

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

—

C₂²⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄×D₄×C₅₉

C₂²

yes

—

#107×C₅₉

C₂²

yes

—

#107×C₅₉

C₂²

yes

—

C₄⋊C₈×C₅₉

C₂²

yes

—

#102×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×D₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

—

D₈×C₅₉

Q₈

yes

direct product factor

C₂×C₈×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

—

C₂³×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

D₈

yes

direct product factor

D₄×C₅₉

—

D₄×C₅₉

C₂×Q₈

yes

complemented

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₈

yes

complemented

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

—

C₄×C₅₉

C₂²×Q₈

yes

characteristic

C₄×C₅₉

2₋¹⁺⁴

yes

characteristic

Q₈×D₈

C₅₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₉

—

C₂×C₅₉

D₄×Q₈

yes

characteristic

C₂×C₅₉

C₂²×D₈

yes

characteristic

C₂×C₅₉

#578

yes

characteristic

D₄×Q₈

C₂×C₅₉

yes

complemented

#500

C₂×C₅₉

yes

complemented

#474

C₂×C₅₉

yes

complemented

C₈×Q₈

C₂×C₅₉

yes

characteristic, complemented

C₄×D₈

C₂×C₅₉

yes

—

C₅₉

Q₈×D₈

yes

characteristic, direct product factor, Sylow

C₂²×Q₈

—

C₂×D₈

C₂²×C₅₉

yes

characteristic

C₄⋊Q₈

C₂²×C₅₉

yes

—

C₂²⋊Q₈

—

C₄×Q₈

C₂²×C₅₉

yes

characteristic

C₄×D₄

C₂²×C₅₉

yes

—

#107

C₂²×C₅₉

yes

—

#107

C₂²×C₅₉

yes

—

C₄⋊C₈

C₂²×C₅₉

yes

—

#102

C₂²×C₅₉

yes

—

C₄×C₈

C₂²×C₅₉

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄×C₅₉

yes

characteristic

C₂×D₄

C₂³×C₅₉

yes

—

C₂²×C₄

—

D₈

Q₈×C₅₉

yes

direct product factor

C₂×C₈

C₂³×C₅₉

yes

—

C₂×C₈

C₂³×C₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₄×C₅₉

yes

—

C₄⋊C₄

C₂³×C₅₉

yes

—

C₂²⋊C₄

—

C₄²

C₂³×C₅₉

yes

—

C₂³

—

Q₈

—

Q₈

D₈×C₅₉

yes

direct product factor

D₄

—

D₄

C₂×Q₈×C₅₉

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₅₉

yes

—

C₂×C₄

C₂×D₄×C₅₉

yes

—

C₂×C₄

C₂⁴×C₅₉

yes

characteristic

C₈

—

C₈

C₂×Q₈×C₅₉

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

C₂²×D₄×C₅₉

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₂×D₈×C₅₉

yes

—

105

C₄

C₂²×Q₈×C₅₉

yes

characteristic

106

C₄

2₋¹⁺⁴×C₅₉

yes

characteristic

107

C₂

—

108

C₂

D₄×Q₈×C₅₉

yes

characteristic

109

C₂

C₂²×D₈×C₅₉

yes

characteristic

110

C₂

C₅₉×#578

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅₉
prime radical	C₂×D₈×C₅₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₄×Q₈×C₅₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₉

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₅₉

prime radical

C₂×D₈×C₅₉

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₉ ◁ C₄×Q₈×C₅₉ ◁ G

N	H ≅ G/N	type
1. D₄×Q₈×C₅₉	C₂	semidirect
2. C₅₉×#500	C₂	semidirect
3. C₅₉×#474	C₂	semidirect
4. C₈×Q₈×C₅₉	C₂	semidirect
22. D₈×C₅₉	Q₈	direct (N × H)
32. Q₈×C₅₉	D₈	direct (N × H)
34. D₄×C₅₉	C₂×Q₈	semidirect
41. C₈×C₅₉	C₂×Q₈	semidirect
51. Q₈×D₈	C₅₉	direct (N × H)
56. D₄×Q₈	C₂×C₅₉	semidirect
57. #500	C₂×C₅₉	semidirect
58. #474	C₂×C₅₉	semidirect
59. C₈×Q₈	C₂×C₅₉	semidirect
61. C₅₉	Q₈×D₈	direct (N × H)
78. D₈	Q₈×C₅₉	direct (N × H)
88. Q₈	D₈×C₅₉	direct (N × H)
90. D₄	C₂×Q₈×C₅₉	semidirect
97. C₈	C₂×Q₈×C₅₉	semidirect

H ≅ G/N

type

1. D₄×Q₈×C₅₉

C₂

semidirect

2. C₅₉×#500

C₂

semidirect

3. C₅₉×#474

C₂

semidirect

4. C₈×Q₈×C₅₉

C₂

semidirect

22. D₈×C₅₉

Q₈

direct (N × H)

32. Q₈×C₅₉

D₈

direct (N × H)

34. D₄×C₅₉

C₂×Q₈

semidirect

41. C₈×C₅₉

C₂×Q₈

semidirect

51. Q₈×D₈

C₅₉

direct (N × H)

56. D₄×Q₈

C₂×C₅₉

semidirect

57. #500

C₂×C₅₉

semidirect

58. #474

C₂×C₅₉

semidirect

59. C₈×Q₈

C₂×C₅₉

semidirect

61. C₅₉

Q₈×D₈

direct (N × H)

78. D₈

Q₈×C₅₉

direct (N × H)

88. Q₈

D₈×C₅₉

direct (N × H)

90. D₄

C₂×Q₈×C₅₉

semidirect

97. C₈

C₂×Q₈×C₅₉

semidirect

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

subgroup	index
D₄×Q₈×C₅₉	2
C₅₉×#500	2
C₅₉×#474	2
C₈×Q₈×C₅₉	2
C₄×D₈×C₅₉	2
Q₈×D₈	59

subgroup

index

D₄×Q₈×C₅₉

C₅₉×#500

C₅₉×#474

C₈×Q₈×C₅₉

C₄×D₈×C₅₉

Q₈×D₈

Subgroups — Q₈×D₈×C₅₉ #21421696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×D8×C59 #21421696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₈×C₅₉ #21421696