Group #21421953 — subgroups

Subgroups — #21421953

Order 7552.

164 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16837459	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#432	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₅₉	—	2	1	no	—
5	C₂×C₄×Dic₅₉	C₄	1	2	yes	—
6	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
7	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
8	#14273638	C₄	1	2	yes	complemented
9	C₅₉⋊#95	—	2	1	no	—
10	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₅₉	C₄	1	2	yes	complemented
12	C₂²⋊C₈×C₅₉	—	2	1	no	—
13	#97×C₅₉	—	2	1	no	—
14	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₅₉	—	2	2	no	—
18	C₂×C₈×C₅₉	—	2	2	no	—
19	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₅₉	D₄	1	1	yes	characteristic
22	C₄²×C₅₉	Q₈	1	1	yes	characteristic
23	C₄⋊Dic₅₉	—	2	4	no	—
24	C₄×Dic₅₉	—	4	2	no	—
25	C₂×C₅₉⋊C₈	—	2	2	no	—
26	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₅₉	—	2	1	no	—
28	C₂×C₄×C₅₉	—	2	1	no	—
29	C₂×C₄×C₅₉	—	2	2	no	—
30	C₂×C₄×C₅₉	—	4	1	no	—
31	C₂×C₄×C₅₉	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₅₉	—	2	4	no	—
38	C₂×Dic₅₉	—	4	2	no	—
39	C₂×Dic₅₉	—	4	2	no	—
40	C₈×C₅₉	—	4	2	no	—
41	C₅₉⋊C₈	—	4	2	no	—
42	C₂²×C₅₉	—	2	1	no	—
43	C₂²×C₅₉	—	2	1	no	—
44	C₂²×C₅₉	#95	1	1	yes	characteristic
45	C₂²×C₅₉	#100	1	1	yes	characteristic
46	C₂²×C₅₉	#101	1	1	yes	characteristic
47	C₄×C₅₉	—	2	1	no	—
48	C₄×C₅₉	—	2	1	no	—
49	C₄×C₅₉	—	2	1	no	—
50	C₄×C₅₉	—	2	1	no	—
51	C₄×C₅₉	—	2	1	no	—
52	C₄×C₅₉	#104	1	2	yes	—
53	Dic₅₉	—	4	4	no	—
54	#1285	—	59	1	no	maximal, Sylow
55	C₂×C₅₉	—	2	1	no	—
56	C₂×C₅₉	—	2	1	no	—
57	C₂×C₅₉	H(3,ℤ/4)	1	1	yes	characteristic
58	C₂×C₅₉	#339	1	1	yes	characteristic
59	C₂×C₅₉	#343	1	1	yes	characteristic
60	#432	—	59	1	no	—
61	#432	D₅₉	1	1	yes	characteristic
62	C₄×C₄⋊C₄	—	59	1	no	—
63	C₅₉	#1285	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊C₄	—	118	1	no	—
65	C₂×C₄²	—	59	2	no	—
66	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
67	C₄⋊C₈	—	118	1	no	—
68	C₄⋊C₈	Dic₅₉	1	2	yes	complemented
69	C₂²⋊C₈	—	2	1	no	—
70	C₂²⋊C₈	—	118	1	no	—
71	#97	—	2	1	no	—
72	#97	—	59	2	no	—
73	#95	—	118	1	no	—
74	C₂²×C₄	—	118	2	no	—
75	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
76	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	118	2	no	—
80	C₄⋊C₄	—	118	4	no	—
81	C₄²	—	236	2	no	—
82	C₄²	Dic₁₁₈	1	1	yes	characteristic
83	C₄²	C₄×D₅₉	1	1	yes	characteristic
84	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
85	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
86	C₂³	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	—	118	4	no	—
92	C₂×C₄	—	236	2	no	—
93	C₂×C₄	—	236	2	no	—
94	C₂×C₄	C₄×Dic₅₉	1	1	yes	characteristic
95	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
96	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
97	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
98	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
99	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
100	C₈	—	4	2	no	—
101	C₈	—	236	2	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	C₅₉⋊#95	1	1	yes	characteristic
105	C₂²	#14273643	1	1	yes	characteristic
106	C₂²	#14273645	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	236	4	no	—
113	C₄	#14273654	1	2	yes	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#16837159	1	1	yes	characteristic
117	C₂	#16837161	1	1	yes	characteristic
118	C₂	#16837170	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837459

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic

C₅₉×#432

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₉

—

C₂×C₄×Dic₅₉

C₄

yes

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

C₄

yes

complemented

C₅₉⋊#95

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₄

yes

complemented

C₂²⋊C₈×C₅₉

—

#97×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

Q₈

yes

characteristic

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄²

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#104

yes

—

Dic₅₉

—

#1285

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

characteristic

C₄×C₄⋊C₄

—

C₅₉

#1285

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

118

—

C₄⋊C₈

Dic₅₉

yes

complemented

C₂²⋊C₈

—

C₂²⋊C₈

—

118

—

#97

—

#97

—

#95

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

D₂₃₆

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

118

—

C₄⋊C₄

—

118

—

C₄²

—

236

—

C₄²

Dic₁₁₈

yes

characteristic

C₄²

C₄×D₅₉

yes

characteristic

C₄²

C₂×Dic₅₉

yes

characteristic

C₄²

C₂²⋊D₅₉

yes

characteristic

C₂³

(C₂×D₅₉)⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

C₄×Dic₅₉

yes

characteristic

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

100

C₈

—

101

C₈

—

236

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₅₉⋊#95

yes

characteristic

105

C₂²

#14273643

yes

characteristic

106

C₂²

#14273645

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

236

—

113

C₄

#14273654

yes

—

114

C₂

—

115

C₂

—

116

C₂

#16837159

yes

characteristic

117

C₂

#16837161

yes

characteristic

118

C₂

#16837170

yes

characteristic

	subgroup	G/H
center	C₂²	C₅₉⋊#95
commutator	C₂×C₄×C₅₉	C₄²
Frattini	C₂×C₄²	C₂×D₅₉
Fitting	C₅₉×#432	C₂
socle	C₂²×C₅₉	#95
hypercenter	#432	D₅₉
hypocenter	C₅₉	#1285
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₅₉⋊#95

commutator

C₂×C₄×C₅₉

C₄²

Frattini

C₂×C₄²

C₂×D₅₉

Fitting

C₅₉×#432

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
8. #14273638	C₄	semidirect
11. C₄⋊C₈×C₅₉	C₄	semidirect
63. C₅₉	#1285	semidirect
68. C₄⋊C₈	Dic₅₉	semidirect

H ≅ G/N

type

8. #14273638

C₄

semidirect

11. C₄⋊C₈×C₅₉

C₄

semidirect

63. C₅₉

#1285

semidirect

68. C₄⋊C₈

Dic₅₉

semidirect

Subgroups — #21421953

Conjugacy classes of subgroups

Complements