Group #21423586 — subgroups

Subgroups — #21423586

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	#16837782	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837683	C₂	1	2	yes	maximal
3	#16837481	C₂	1	1	yes	maximal, characteristic
4	#16837147	C₂	1	2	yes	maximal, complemented
5	C₅₉×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₃₆	—	2	1	no	—
7	C₂×Dic₂₃₆	—	2	1	no	—
8	C₂×#11745659	—	2	1	no	—
9	#14273727	C₂²	1	2	yes	—
10	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
11	#14273661	—	2	2	no	—
12	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	#14273649	—	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	Dic₂₃₆	—	4	2	no	—
30	Dic₂₃₆	—	4	2	no	—
31	#11745659	—	4	2	no	—
32	C₂²×D₅₉	—	2	1	no	—
33	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
36	C₂×Dic₅₉	—	2	1	no	—
37	C₂×Dic₅₉	—	4	2	no	—
38	D₂₃₆	—	2	2	no	—
39	C₄×D₅₉	—	4	1	no	—
40	C₄×D₅₉	—	4	1	no	—
41	Dic₁₁₈	—	4	2	no	—
42	Dic₁₁₈	—	4	2	no	—
43	C₈×C₅₉	—	2	1	no	—
44	C₈×C₅₉	—	4	2	no	—
45	C₈×C₅₉	D₈	1	2	yes	—
46	C₅₉⋊C₈	—	4	1	no	—
47	C₂²×C₅₉	#121	1	1	yes	characteristic
48	C₂×D₅₉	—	4	1	no	—
49	C₂×D₅₉	—	4	1	no	—
50	C₂×D₅₉	—	4	1	no	—
51	C₄×C₅₉	—	2	1	no	—
52	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
53	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₄×C₅₉	#137	1	2	yes	—
55	Dic₅₉	—	4	1	no	—
56	Dic₅₉	—	8	2	no	—
57	#1689	—	59	1	no	maximal, Sylow
58	C₂×C₅₉	#459	1	1	yes	characteristic
59	C₂×C₅₉	#470	1	1	yes	characteristic
60	C₂×C₅₉	#473	1	1	yes	characteristic
61	D₅₉	—	8	1	no	—
62	#494	—	59	1	no	—
63	#474	—	59	2	no	—
64	#436	—	59	1	no	—
65	#334	D₅₉	1	1	yes	characteristic, complemented
66	#332	—	59	2	no	—
67	C₅₉	#1689	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₁₆	—	118	1	no	—
69	C₂×Q₁₆	—	118	1	no	—
70	C₂×M₁₆	—	118	1	no	—
71	C₄⋊Q₈	—	59	2	no	—
72	C₄×D₄	—	59	1	no	—
73	#107	—	118	2	no	—
74	C₄⋊C₈	—	59	1	no	—
75	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
76	#102	—	118	2	no	—
77	C₂²⋊C₈	—	118	1	no	—
78	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
79	C₂×Q₈	—	118	2	no	—
80	C₂×D₄	—	59	1	no	—
81	C₂²×C₄	—	118	1	no	—
82	Q₁₆	—	236	2	no	—
83	Q₁₆	—	236	2	no	—
84	M₁₆	—	236	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	118	1	no	—
87	C₂×C₈	D₂₃₆	1	1	yes	characteristic
88	C₂×C₈	D₂₃₆	1	1	yes	characteristic
89	C₄⋊C₄	—	59	1	no	—
90	C₄⋊C₄	—	118	2	no	—
91	C₄⋊C₄	—	118	2	no	—
92	C₂²⋊C₄	—	118	1	no	—
93	C₄²	C₂²×D₅₉	1	1	yes	characteristic
94	C₂³	—	118	1	no	—
95	Q₈	—	236	2	no	—
96	Q₈	—	236	2	no	—
97	D₄	—	118	2	no	—
98	C₂×C₄	—	118	1	no	—
99	C₂×C₄	—	236	2	no	—
100	C₂×C₄	—	236	1	no	—
101	C₂×C₄	—	236	1	no	—
102	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
103	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
104	C₂×C₄	#11745689	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	4	2	no	—
107	C₈	—	236	1	no	—
108	C₈	D₄₇₂	1	2	yes	—
109	C₂²	—	236	1	no	—
110	C₂²	—	236	1	no	—
111	C₂²	—	236	1	no	—
112	C₂²	#14273702	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	472	2	no	—
115	C₄	—	236	1	no	—
116	C₄	C₂×D₄₇₂	1	1	yes	characteristic
117	C₄	#14273755	1	1	yes	characteristic
118	C₄	#14273756	1	2	yes	—
119	C₂	—	472	1	no	—
120	C₂	#16837598	1	1	yes	characteristic
121	C₂	#16837661	1	1	yes	characteristic
122	C₂	#16837678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837782

C₂

yes

maximal, characteristic, complemented

#16837683

C₂

yes

maximal

#16837481

C₂

yes

maximal, characteristic

#16837147

C₂

yes

maximal, complemented

C₅₉×#334

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×#11745659

—

#14273727

C₂²

yes

—

C₄×D₂₃₆

C₂²

yes

characteristic

#14273661

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273649

—

(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

—

Dic₂₃₆

—

Dic₂₃₆

—

#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

D₈

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

—

—

—

D₅₉

yes

characteristic, complemented

#332

—

C₅₉

#1689

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₂×M₁₆

—

118

—

C₄⋊Q₈

—

C₄×D₄

—

#107

—

118

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₉

yes

—

#102

—

118

—

C₂²⋊C₈

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

C₂×D₄

—

C₂²×C₄

—

118

—

Q₁₆

—

236

—

Q₁₆

—

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

—

100

C₂×C₄

—

236

—

101

C₂×C₄

—

236

—

102

C₂×C₄

C₂×D₂₃₆

yes

characteristic

103

C₂×C₄

D₄×D₅₉

yes

characteristic

104

C₂×C₄

#11745689

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

236

—

108

C₈

D₄₇₂

yes

—

109

C₂²

—

236

—

110

C₂²

—

236

—

111

C₂²

—

236

—

112

C₂²

#14273702

yes

characteristic

113

C₄

—

114

C₄

—

472

—

115

C₄

—

236

—

116

C₄

C₂×D₄₇₂

yes

characteristic

117

C₄

#14273755

yes

characteristic

118

C₄

#14273756

yes

—

119

C₂

—

472

—

120

C₂

#16837598

yes

characteristic

121

C₂

#16837661

yes

characteristic

122

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	#1689
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. #16837782	C₂	semidirect
4. #16837147	C₂	semidirect
5. C₅₉×#334	C₂	semidirect
65. #334	D₅₉	semidirect
67. C₅₉	#1689	semidirect

H ≅ G/N

type

1. #16837782

C₂

semidirect

4. #16837147

C₂

semidirect

5. C₅₉×#334

C₂

semidirect

65. #334

D₅₉

semidirect

67. C₅₉

#1689

semidirect

Subgroups — #21423586

Conjugacy classes of subgroups

Complements