Group #21423945 — subgroups

Subgroups — C₄×C₄⋊(C₅₉⋊C₈) #21423945

Order 7552.

316 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 213 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×C₄⋊(C₅₉⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄×C₅₉⋊C₈	C₂	1	2	yes	maximal
3	#16837157	C₂	1	2	yes	maximal
4	C₄³×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₈	C₂²	1	4	yes	—
6	C₄⋊(C₅₉⋊C₈)	C₄	1	8	yes	direct product factor
7	C₄×C₅₉⋊C₈	—	2	4	no	—
8	C₂×C₄²×C₅₉	C₄	1	2	yes	—
9	C₂×C₄²×C₅₉	C₄	1	2	yes	—
10	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
18	C₄²×C₅₉	—	2	4	no	—
19	C₄²×C₅₉	—	2	2	no	—
20	C₄²×C₅₉	C₈	1	8	yes	complemented
21	C₄²×C₅₉	C₂×C₄	1	4	yes	—
22	C₄²×C₅₉	D₄	1	2	yes	—
23	C₄²×C₅₉	Q₈	1	2	yes	—
24	C₂×C₅₉⋊C₈	—	2	4	no	—
25	C₂×C₅₉⋊C₈	—	2	4	no	—
26	C₂×C₅₉⋊C₈	C₂×C₄	1	8	yes	—
27	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₉	—	2	2	no	—
29	C₂×C₄×C₅₉	—	2	2	no	—
30	C₂×C₄×C₅₉	—	2	2	no	—
31	C₂×C₄×C₅₉	C₄²	1	2	yes	—
32	C₂×C₄×C₅₉	C₄²	1	2	yes	—
33	C₂×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
34	C₂×C₄×C₅₉	C₂×C₈	1	4	yes	—
35	C₂×C₄×C₅₉	C₂×C₈	1	4	yes	—
36	C₂×C₄×C₅₉	C₂×C₈	1	4	yes	—
37	C₂×C₄×C₅₉	M₁₆	1	4	yes	—
38	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₉	#41	1	2	yes	—
42	C₅₉⋊C₈	—	2	8	no	—
43	C₂²×C₅₉	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₅₉	#117	1	1	yes	characteristic
46	C₂²×C₅₉	C₂²×C₈	1	2	yes	—
47	C₂²×C₅₉	C₂×M₁₆	1	2	yes	—
48	C₄×C₅₉	—	2	2	no	—
49	C₄×C₅₉	—	2	4	no	—
50	C₄×C₅₉	C₄×C₈	1	4	yes	complemented
51	C₄×C₅₉	C₄⋊C₈	1	8	yes	complemented
52	C₄×C₅₉	C₄×D₄	1	2	yes	—
53	C₄×C₅₉	C₄×Q₈	1	2	yes	—
54	C₄×C₄⋊C₈	—	59	1	no	maximal, Sylow
55	C₂×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
57	C₂×C₅₉	C₄×M₁₆	1	1	yes	characteristic
58	C₂×C₅₉	C₂×C₄⋊C₈	1	2	yes	—
59	C₂×C₅₉	#431	1	2	yes	—
60	C₂×C₄⋊C₈	—	59	2	no	—
61	C₂×C₄×C₈	—	59	2	no	—
62	C₄³	D₅₉	1	1	yes	characteristic
63	#336	—	59	2	no	—
64	C₅₉	C₄×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₈	—	59	4	no	—
66	C₂×C₄²	Dic₅₉	1	2	yes	—
67	C₂×C₄²	Dic₅₉	1	2	yes	—
68	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
69	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
71	C₄⋊C₈	—	59	8	no	—
72	C₄×C₈	—	118	4	no	—
73	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
74	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
75	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₅₉	1	2	yes	—
77	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
78	C₂×C₈	—	59	8	no	—
79	C₂×C₈	—	118	4	no	—
80	C₂×C₈	—	118	4	no	—
81	C₄²	—	2	4	no	—
82	C₄²	—	2	2	no	—
83	C₄²	C₅₉⋊C₈	1	8	yes	complemented
84	C₄²	Dic₁₁₈	1	2	yes	—
85	C₄²	C₄×D₅₉	1	4	yes	—
86	C₄²	D₂₃₆	1	2	yes	—
87	C₂³	C₂²×Dic₅₉	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	C₂×C₅₉⋊C₈	1	4	yes	—
92	C₂×C₄	C₂×C₅₉⋊C₈	1	4	yes	—
93	C₂×C₄	C₂×C₅₉⋊C₈	1	4	yes	—
94	C₂×C₄	#11745664	1	4	yes	—
95	C₂×C₄	C₄×Dic₅₉	1	2	yes	—
96	C₂×C₄	C₄×Dic₅₉	1	2	yes	—
97	C₂×C₄	C₄⋊Dic₅₉	1	4	yes	—
98	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
99	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
100	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
101	C₂×C₄	#11745685	1	2	yes	—
102	C₈	—	118	8	no	—
103	C₂²	C₂×C₄×Dic₅₉	1	1	yes	characteristic
104	C₂²	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
105	C₂²	#14273686	1	1	yes	characteristic
106	C₂²	C₂²×C₅₉⋊C₈	1	2	yes	—
107	C₂²	C₂×#11745664	1	2	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	2	2	no	—
110	C₄	C₄×C₅₉⋊C₈	1	4	yes	complemented
111	C₄	C₄⋊(C₅₉⋊C₈)	1	8	yes	direct product factor
112	C₄	C₄×D₂₃₆	1	2	yes	—
113	C₄	C₄×Dic₁₁₈	1	2	yes	—
114	C₂	C₄×C₄⋊Dic₅₉	1	1	yes	characteristic
115	C₂	C₂×C₄×C₅₉⋊C₈	1	1	yes	characteristic
116	C₂	C₄×#11745664	1	1	yes	characteristic
117	C₂	C₂×C₄⋊(C₅₉⋊C₈)	1	2	yes	—
118	C₂	#16837455	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₅₉⋊C₈)

C₂

yes

maximal

C₂×C₄×C₅₉⋊C₈

C₂

yes

maximal

#16837157

C₂

yes

maximal

C₄³×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₄

yes

direct product factor

C₄×C₅₉⋊C₈

—

C₂×C₄²×C₅₉

C₄

yes

—

C₂×C₄²×C₅₉

C₄

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₈

yes

complemented

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

D₄

yes

—

C₄²×C₅₉

Q₈

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄²

yes

—

C₂×C₄×C₅₉

C₄²

yes

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂×C₈

yes

—

C₂×C₄×C₅₉

C₂×C₈

yes

—

C₂×C₄×C₅₉

C₂×C₈

yes

—

C₂×C₄×C₅₉

M₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₄²

yes

characteristic

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

#117

yes

characteristic

C₂²×C₅₉

C₂²×C₈

yes

—

C₂²×C₅₉

C₂×M₁₆

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×C₈

yes

complemented

C₄×C₅₉

C₄⋊C₈

yes

complemented

C₄×C₅₉

C₄×D₄

yes

—

C₄×C₅₉

C₄×Q₈

yes

—

C₄×C₄⋊C₈

—

maximal, Sylow

C₂×C₅₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₉

C₂×C₄×C₈

yes

characteristic

C₂×C₅₉

C₄×M₁₆

yes

characteristic

C₂×C₅₉

C₂×C₄⋊C₈

yes

—

C₂×C₅₉

#431

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₄³

D₅₉

yes

characteristic

#336

—

C₅₉

C₄×C₄⋊C₈

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

Dic₅₉

yes

—

C₂×C₄²

Dic₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

118

—

C₂²×C₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

C₂²×C₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₄²

—

C₄²

—

C₄²

C₅₉⋊C₈

yes

complemented

C₄²

Dic₁₁₈

yes

—

C₄²

C₄×D₅₉

yes

—

C₄²

D₂₃₆

yes

—

C₂³

C₂²×Dic₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₅₉⋊C₈

yes

—

C₂×C₄

C₂×C₅₉⋊C₈

yes

—

C₂×C₄

C₂×C₅₉⋊C₈

yes

—

C₂×C₄

#11745664

yes

—

C₂×C₄

C₄×Dic₅₉

yes

—

C₂×C₄

C₄×Dic₅₉

yes

—

C₂×C₄

C₄⋊Dic₅₉

yes

—

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

C₂×C₄

C₂×C₄×D₅₉

yes

—

100

C₂×C₄

C₂×D₂₃₆

yes

characteristic

101

C₂×C₄

#11745685

yes

—

102

C₈

—

118

—

103

C₂²

C₂×C₄×Dic₅₉

yes

characteristic

104

C₂²

C₂×C₄⋊Dic₅₉

yes

characteristic

105

C₂²

#14273686

yes

characteristic

106

C₂²

C₂²×C₅₉⋊C₈

yes

—

107

C₂²

C₂×#11745664

yes

—

108

C₄

—

109

C₄

—

110

C₄

C₄×C₅₉⋊C₈

yes

complemented

111

C₄

C₄⋊(C₅₉⋊C₈)

yes

direct product factor

112

C₄

C₄×D₂₃₆

yes

—

113

C₄

C₄×Dic₁₁₈

yes

—

114

C₂

C₄×C₄⋊Dic₅₉

yes

characteristic

115

C₂

C₂×C₄×C₅₉⋊C₈

yes

characteristic

116

C₂

C₄×#11745664

yes

characteristic

117

C₂

C₂×C₄⋊(C₅₉⋊C₈)

yes

—

118

C₂

#16837455

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₅₉
commutator	C₂×C₅₉	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₄³×C₅₉	C₂
socle	C₂³×C₅₉	C₂²×C₄
hypercenter	C₄³	D₅₉
hypocenter	C₅₉	C₄×C₄⋊C₈
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄²

C₂×D₅₉

commutator

C₂×C₅₉

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₄³×C₅₉

C₂

socle

C₂³×C₅₉

C₂²×C₄

hypercenter

C₄³

D₅₉

hypocenter

C₅₉

C₄×C₄⋊C₈

LCS

G ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
6. C₄⋊(C₅₉⋊C₈)	C₄	direct (N × H)
20. C₄²×C₅₉	C₈	semidirect
50. C₄×C₅₉	C₄×C₈	semidirect
51. C₄×C₅₉	C₄⋊C₈	semidirect
64. C₅₉	C₄×C₄⋊C₈	semidirect
83. C₄²	C₅₉⋊C₈	semidirect
110. C₄	C₄×C₅₉⋊C₈	semidirect
111. C₄	C₄⋊(C₅₉⋊C₈)	direct (N × H)

H ≅ G/N

type

6. C₄⋊(C₅₉⋊C₈)

C₄

direct (N × H)

20. C₄²×C₅₉

C₈

semidirect

50. C₄×C₅₉

C₄×C₈

semidirect

51. C₄×C₅₉

C₄⋊C₈

semidirect

64. C₅₉

C₄×C₄⋊C₈

semidirect

83. C₄²

C₅₉⋊C₈

semidirect

110. C₄

C₄×C₅₉⋊C₈

semidirect

111. C₄

C₄⋊(C₅₉⋊C₈)

direct (N × H)

Subgroups — C₄×C₄⋊(C₅₉⋊C₈) #21423945

Conjugacy classes of subgroups

Complements

Subgroups — C4×C4⋊(C59⋊C8) #21423945

Conjugacy classes of subgroups

Complements

Subgroups — C₄×C₄⋊(C₅₉⋊C₈) #21423945