Group #21423610 — subgroups

Subgroups — #21423610

Order 7552.

166 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16837807	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837608	C₂	1	2	yes	maximal
3	#16837481	C₂	1	1	yes	maximal, characteristic
4	#16837149	C₂	1	2	yes	maximal, complemented
5	C₅₉×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745660	—	2	2	no	—
7	C₂×#11745659	—	2	1	no	—
8	#14273727	C₂²	1	2	yes	—
9	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
10	#14273661	—	2	1	no	—
11	#14273661	—	2	1	no	—
12	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	#14273652	—	2	2	no	—
14	(C₂×D₅₉)⋊C₈	—	2	1	no	—
15	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
16	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×D₂₃₆	D₄	1	1	yes	characteristic
18	C₂×C₄×D₅₉	—	2	1	no	—
19	C₂×Dic₁₁₈	—	2	2	no	—
20	C₂×C₈×C₅₉	—	2	2	no	—
21	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₄²×C₅₉	C₂³	1	1	yes	characteristic
24	(C₂×D₅₉)⋊C₄	—	2	1	no	—
25	C₄⋊Dic₅₉	—	2	2	no	—
26	C₄⋊Dic₅₉	—	2	2	no	—
27	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
28	C₂×C₅₉⋊C₈	—	2	1	no	—
29	#11745660	—	4	4	no	—
30	#11745659	—	4	2	no	—
31	C₂²×D₅₉	—	2	1	no	—
32	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
35	C₂×Dic₅₉	—	2	1	no	—
36	C₂×Dic₅₉	—	4	2	no	—
37	D₂₃₆	—	2	2	no	—
38	C₄×D₅₉	—	4	1	no	—
39	C₄×D₅₉	—	4	1	no	—
40	Dic₁₁₈	—	4	2	no	—
41	Dic₁₁₈	—	4	2	no	—
42	C₈×C₅₉	—	2	1	no	—
43	C₈×C₅₉	—	4	2	no	—
44	C₈×C₅₉	SD₁₆	1	2	yes	complemented
45	C₅₉⋊C₈	—	4	1	no	—
46	C₂²×C₅₉	#121	1	1	yes	characteristic
47	C₂×D₅₉	—	4	1	no	—
48	C₂×D₅₉	—	4	1	no	—
49	C₂×D₅₉	—	4	1	no	—
50	C₄×C₅₉	—	2	1	no	—
51	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
52	C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
53	C₄×C₅₉	#137	1	1	yes	characteristic
54	Dic₅₉	—	4	1	no	—
55	Dic₅₉	—	8	2	no	—
56	#1693	—	59	1	no	maximal, Sylow
57	C₂×C₅₉	#461	1	1	yes	characteristic
58	C₂×C₅₉	#469	1	1	yes	characteristic
59	C₂×C₅₉	#473	1	1	yes	characteristic
60	D₅₉	—	8	1	no	—
61	#500	—	59	1	no	—
62	#461	—	59	2	no	—
63	#436	—	59	1	no	—
64	#334	D₅₉	1	1	yes	characteristic, complemented
65	#332	—	59	2	no	—
66	C₅₉	#1693	1	1	yes	characteristic, complemented, Sylow
67	C₂×SD₁₆	—	118	2	no	—
68	C₂×M₁₆	—	118	1	no	—
69	C₄⋊Q₈	—	59	2	no	—
70	C₄×D₄	—	59	1	no	—
71	#107	—	118	1	no	—
72	#107	—	118	1	no	—
73	C₄⋊C₈	—	59	1	no	—
74	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
75	#103	—	118	2	no	—
76	C₂²⋊C₈	—	118	1	no	—
77	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
78	C₂×Q₈	—	118	2	no	—
79	C₂×D₄	—	59	1	no	—
80	C₂²×C₄	—	118	1	no	—
81	SD₁₆	—	236	4	no	—
82	M₁₆	—	236	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	118	1	no	—
85	C₂×C₈	D₂₃₆	1	1	yes	characteristic
86	C₂×C₈	D₂₃₆	1	1	yes	characteristic
87	C₄⋊C₄	—	59	1	no	—
88	C₄⋊C₄	—	118	2	no	—
89	C₄⋊C₄	—	118	2	no	—
90	C₂²⋊C₄	—	118	1	no	—
91	C₄²	C₂²×D₅₉	1	1	yes	characteristic
92	C₂³	—	118	1	no	—
93	Q₈	—	236	2	no	—
94	Q₈	—	236	2	no	—
95	D₄	—	118	2	no	—
96	C₂×C₄	—	118	1	no	—
97	C₂×C₄	—	236	2	no	—
98	C₂×C₄	—	236	1	no	—
99	C₂×C₄	—	236	1	no	—
100	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
101	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
102	C₂×C₄	#11745689	1	1	yes	characteristic
103	C₈	—	2	1	no	—
104	C₈	—	4	2	no	—
105	C₈	—	236	1	no	—
106	C₈	#11745660	1	2	yes	complemented
107	C₂²	—	236	1	no	—
108	C₂²	—	236	1	no	—
109	C₂²	—	236	1	no	—
110	C₂²	#14273702	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	472	2	no	—
113	C₄	—	236	1	no	—
114	C₄	C₂×#11745660	1	1	yes	characteristic
115	C₄	#14273754	1	2	yes	—
116	C₄	#14273760	1	1	yes	characteristic
117	C₂	—	472	1	no	—
118	C₂	#16837608	1	1	yes	characteristic
119	C₂	#16837656	1	1	yes	characteristic
120	C₂	#16837678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837807

C₂

yes

maximal, characteristic, complemented

#16837608

C₂

yes

maximal

#16837481

C₂

yes

maximal, characteristic

#16837149

C₂

yes

maximal, complemented

C₅₉×#334

C₂

yes

maximal, characteristic, complemented

C₂×#11745660

—

C₂×#11745659

—

#14273727

C₂²

yes

—

C₄×D₂₃₆

C₂²

yes

characteristic

#14273661

—

#14273661

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273652

—

(C₂×D₅₉)⋊C₈

—

C₄⋊C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₂₃₆

D₄

yes

characteristic

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

—

#11745660

—

#11745659

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

SD₁₆

yes

complemented

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₄×C₅₉

#137

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

—

—

—

D₅₉

yes

characteristic, complemented

#332

—

C₅₉

#1693

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

118

—

C₂×M₁₆

—

118

—

C₄⋊Q₈

—

C₄×D₄

—

#107

—

118

—

#107

—

118

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₉

yes

—

#103

—

118

—

C₂²⋊C₈

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

C₂×D₄

—

C₂²×C₄

—

118

—

SD₁₆

—

236

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

D₂₃₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

118

—

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

—

118

—

Q₈

—

236

—

Q₈

—

236

—

D₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

100

C₂×C₄

C₂×D₂₃₆

yes

characteristic

101

C₂×C₄

D₄×D₅₉

yes

characteristic

102

C₂×C₄

#11745689

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

—

236

—

106

C₈

#11745660

yes

complemented

107

C₂²

—

236

—

108

C₂²

—

236

—

109

C₂²

—

236

—

110

C₂²

#14273702

yes

characteristic

111

C₄

—

112

C₄

—

472

—

113

C₄

—

236

—

114

C₄

C₂×#11745660

yes

characteristic

115

C₄

#14273754

yes

—

116

C₄

#14273760

yes

characteristic

117

C₂

—

472

—

118

C₂

#16837608

yes

characteristic

119

C₂

#16837656

yes

characteristic

120

C₂

#16837678

yes

characteristic

	subgroup	G/H
center	C₂²	#14273702
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#334	C₂
socle	C₂²×C₅₉	#121
hypercenter	#334	D₅₉
hypocenter	C₅₉	#1693
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#14273702

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#334

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
1. #16837807	C₂	semidirect
4. #16837149	C₂	semidirect
5. C₅₉×#334	C₂	semidirect
44. C₈×C₅₉	SD₁₆	semidirect
64. #334	D₅₉	semidirect
66. C₅₉	#1693	semidirect
106. C₈	#11745660	semidirect

H ≅ G/N

type

1. #16837807

C₂

semidirect

4. #16837149

C₂

semidirect

5. C₅₉×#334

C₂

semidirect

44. C₈×C₅₉

SD₁₆

semidirect

64. #334

D₅₉

semidirect

66. C₅₉

#1693

semidirect

106. C₈

#11745660

semidirect

Subgroups — #21423610

Conjugacy classes of subgroups

Complements