Group #21422864 — subgroups

Subgroups — #21422864

Order 7552.

166 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16837426	C₂	1	1	yes	maximal, characteristic
2	C₄×#11745664	C₂	1	1	yes	maximal, characteristic
3	#16837154	C₂	1	2	yes	maximal, complemented
4	#16837152	C₂	1	2	yes	maximal, complemented
5	C₅₉×#432	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	—	2	1	no	—
7	C₂×#11745664	C₄	1	2	yes	—
8	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
9	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
10	#14273638	C₄	1	1	yes	characteristic
11	#14273638	C₄	1	1	yes	characteristic
12	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂²⋊C₈×C₅₉	—	2	1	no	—
17	#97×C₅₉	—	2	1	no	—
18	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₅₉	—	2	1	no	—
22	C₂×C₈×C₅₉	—	2	1	no	—
23	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₅₉	D₄	1	1	yes	characteristic
25	C₄²×C₅₉	Q₈	1	1	yes	characteristic
26	C₄²×C₅₉	C₂³	1	1	yes	characteristic
27	#11745664	—	2	4	no	—
28	#11745664	—	4	2	no	—
29	C₂×C₅₉⋊C₈	—	2	2	no	—
30	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
32	C₂³×C₅₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	—	2	2	no	—
34	C₂×C₄×C₅₉	—	2	1	no	—
35	C₂×C₄×C₅₉	—	2	1	no	—
36	C₂×C₄×C₅₉	—	4	1	no	—
37	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₅₉	—	4	1	no	—
44	C₈×C₅₉	—	4	1	no	—
45	C₅₉⋊C₈	—	2	2	no	—
46	C₅₉⋊C₈	—	4	2	no	—
47	C₅₉⋊C₈	M₁₆	1	4	yes	complemented
48	C₂²×C₅₉	—	2	1	no	—
49	C₂²×C₅₉	—	2	1	no	—
50	C₂²×C₅₉	—	2	1	no	—
51	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₅₉	—	2	1	no	—
53	C₄×C₅₉	—	2	1	no	—
54	C₄×C₅₉	—	2	1	no	—
55	C₄×C₅₉	—	2	1	no	—
56	C₄×C₅₉	C₂×M₁₆	1	2	yes	—
57	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
58	C₄×C₅₉	#137	1	1	yes	characteristic
59	#1569	—	59	1	no	maximal, Sylow
60	C₂×C₅₉	—	4	1	no	—
61	C₂×C₅₉	#423	1	1	yes	characteristic
62	C₂×C₅₉	#428	1	1	yes	characteristic
63	C₂×C₅₉	#430	1	1	yes	characteristic
64	#432	D₅₉	1	1	yes	characteristic
65	#423	—	59	1	no	—
66	C₄×M₁₆	—	59	1	no	—
67	#335	—	59	2	no	—
68	#334	—	59	2	no	—
69	C₅₉	#1569	1	1	yes	characteristic, complemented, Sylow
70	C₂×M₁₆	—	59	2	no	—
71	C₂×M₁₆	—	118	1	no	—
72	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
73	C₄⋊C₈	—	59	2	no	—
74	C₄⋊C₈	—	118	1	no	—
75	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
76	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
77	C₂²⋊C₈	—	2	1	no	—
78	#97	—	2	1	no	—
79	#97	—	59	1	no	—
80	#97	—	59	1	no	—
81	C₄×C₈	—	59	2	no	—
82	C₂²×C₄	Dic₁₁₈	1	1	yes	characteristic
83	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
84	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
85	M₁₆	—	118	4	no	—
86	M₁₆	—	236	2	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	59	2	no	—
90	C₂×C₈	—	59	2	no	—
91	C₂×C₈	—	118	2	no	—
92	C₄²	Dic₁₁₈	1	1	yes	characteristic
93	C₄²	C₄×D₅₉	1	1	yes	characteristic
94	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
95	C₄²	C₂²×D₅₉	1	1	yes	characteristic
96	C₂³	Dic₅₉⋊C₄	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
102	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
103	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
104	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
107	C₈	—	4	1	no	—
108	C₈	—	4	1	no	—
109	C₈	—	59	4	no	—
110	C₈	—	118	2	no	—
111	C₈	—	236	2	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	M₁₆×D₅₉	1	2	yes	—
121	C₄	#14273752	1	1	yes	characteristic
122	C₄	#14273757	1	1	yes	characteristic
123	C₂	—	4	1	no	—
124	C₂	#16837425	1	1	yes	characteristic
125	C₂	#16837445	1	1	yes	characteristic
126	C₂	#16837450	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837426

C₂

yes

maximal, characteristic

C₄×#11745664

C₂

yes

maximal, characteristic

#16837154

C₂

yes

maximal, complemented

#16837152

C₂

yes

maximal, complemented

C₅₉×#432

C₂

yes

maximal, characteristic

C₂×#11745664

—

C₂×#11745664

C₄

yes

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273638

C₄

yes

characteristic

#14273638

C₄

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₂²⋊C₈×C₅₉

—

#97×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

Q₈

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

Q₈

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂³×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

M₁₆

yes

complemented

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×M₁₆

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

characteristic

#423

—

C₄×M₁₆

—

#335

—

#334

—

C₅₉

#1569

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₈

C₂×D₅₉

yes

characteristic

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₂²×C₄

Dic₁₁₈

yes

characteristic

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

M₁₆

—

118

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₄²

Dic₁₁₈

yes

characteristic

C₄²

C₄×D₅₉

yes

characteristic

C₄²

C₂²⋊D₅₉

yes

characteristic

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

Dic₅₉⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

102

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

103

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

104

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

105

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

118

—

111

C₈

—

236

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×Dic₅₉⋊C₄

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

M₁₆×D₅₉

yes

—

121

C₄

#14273752

yes

characteristic

122

C₄

#14273757

yes

characteristic

123

C₂

—

124

C₂

#16837425

yes

characteristic

125

C₂

#16837445

yes

characteristic

126

C₂

#16837450

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#432	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	#432	D₅₉
hypocenter	C₅₉	#1569
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#432

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #16837154	C₂	semidirect
4. #16837152	C₂	semidirect
47. C₅₉⋊C₈	M₁₆	semidirect
69. C₅₉	#1569	semidirect

H ≅ G/N

type

3. #16837154

C₂

semidirect

4. #16837152

C₂

semidirect

47. C₅₉⋊C₈

M₁₆

semidirect

69. C₅₉

#1569

semidirect

Subgroups — #21422864

Conjugacy classes of subgroups

Complements