Group #21425612 — subgroups

Subgroups — #21425612

Order 7552.

222 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16837451	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837450	C₂	1	2	yes	maximal, complemented
3	#16837366	C₂	1	2	yes	maximal
4	C₅₉⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#407	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	2	yes	—
7	C₂×#11745664	C₂²	1	2	yes	—
8	#14273663	—	2	2	no	—
9	#14273662	—	2	4	no	—
10	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
11	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×C₅₉	—	2	2	no	—
15	C₂⁴×C₅₉	Q₈	1	1	yes	characteristic
16	C₂²×C₄×C₅₉	—	2	1	no	—
17	C₂²×C₄×C₅₉	—	2	2	no	—
18	C₂²×C₄×C₅₉	—	2	1	no	—
19	C₂²×C₄×C₅₉	—	2	2	no	—
20	C₂²×C₄×C₅₉	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
22	M₁₆×C₅₉	—	2	2	no	—
23	M₁₆×C₅₉	—	2	2	no	—
24	C₂×C₈×C₅₉	—	2	1	no	—
25	C₂×C₈×C₅₉	D₄	1	2	yes	—
26	#11745664	—	2	2	no	—
27	#11745664	—	2	2	no	—
28	#11745664	—	2	2	no	—
29	#11745664	—	2	2	no	—
30	C₂×C₅₉⋊C₈	—	2	2	no	—
31	C₂×C₅₉⋊C₈	D₄	1	4	yes	—
32	C₂³×C₅₉	—	2	2	no	—
33	C₂³×C₅₉	—	2	1	no	—
34	C₂³×C₅₉	—	4	2	no	—
35	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₅₉	—	2	2	no	—
37	C₂×C₄×C₅₉	—	2	1	no	—
38	C₂×C₄×C₅₉	—	4	1	no	—
39	C₂×C₄×C₅₉	—	4	2	no	—
40	C₂×C₄×C₅₉	—	4	1	no	—
41	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
44	C₂×C₄×C₅₉	#41	1	2	yes	—
45	C₈×C₅₉	—	2	2	no	—
46	C₈×C₅₉	—	4	1	no	—
47	C₅₉⋊C₈	—	2	4	no	—
48	C₅₉⋊C₈	—	4	2	no	—
49	C₂²×C₅₉	—	2	4	no	—
50	C₂²×C₅₉	—	2	2	no	—
51	C₂²×C₅₉	—	4	1	no	—
52	C₂²×C₅₉	—	4	2	no	—
53	C₂²×C₅₉	—	4	1	no	—
54	C₂²×C₅₉	—	4	1	no	—
55	C₂²×C₅₉	C₂²⋊Q₈	1	2	yes	—
56	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
57	C₄×C₅₉	—	4	1	no	—
58	C₄×C₅₉	#123	1	1	yes	characteristic
59	C₄×C₅₉	#123	1	2	yes	—
60	C₄×C₅₉	#127	1	1	yes	characteristic
61	#2091	—	59	1	no	maximal, Sylow
62	C₂×C₅₉	—	2	2	no	—
63	C₂×C₅₉	—	2	1	no	—
64	C₂×C₅₉	—	4	2	no	—
65	C₂×C₅₉	#399	1	1	yes	characteristic
66	#430	—	59	2	no	—
67	#430	—	59	1	no	—
68	#407	—	59	2	no	—
69	#407	D₅₉	1	1	yes	characteristic
70	#338	—	59	1	no	—
71	C₅₉	#2091	1	1	yes	characteristic, complemented, Sylow
72	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
73	C₂×M₁₆	—	59	2	no	—
74	C₂×M₁₆	—	59	2	no	—
75	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
76	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
77	#108	—	118	4	no	—
78	#108	—	118	2	no	—
79	C₂²⋊C₈	—	2	2	no	—
80	C₂²⋊C₈	—	118	4	no	—
81	C₂⁴	Dic₁₁₈	1	1	yes	characteristic
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	Dic₁₁₈	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	118	2	no	—
91	M₁₆	—	118	2	no	—
92	M₁₆	—	118	2	no	—
93	M₁₆	—	118	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	59	4	no	—
96	C₂×C₈	—	118	2	no	—
97	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
98	C₂³	—	2	2	no	—
99	C₂³	—	2	1	no	—
100	C₂³	—	4	2	no	—
101	C₂³	C₂×Dic₁₁₈	1	1	yes	characteristic
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	#11745685	1	1	yes	characteristic
108	C₂×C₄	D₄×D₅₉	1	2	yes	—
109	C₂×C₄	#11745687	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	118	4	no	—
114	C₈	—	236	2	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	4	1	no	—
118	C₂²	—	4	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	1	no	—
121	C₂²	#14273704	1	2	yes	—
122	C₂²	#14273706	1	1	yes	characteristic
123	C₄	—	4	1	no	—
124	C₄	#14273708	1	1	yes	characteristic
125	C₄	#14273709	1	2	yes	—
126	C₄	#14273723	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#16837338	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837451

C₂

yes

maximal, characteristic, complemented

#16837450

C₂

yes

maximal, complemented

#16837366

C₂

yes

maximal

C₅₉⋊#338

C₂

yes

maximal, characteristic, complemented

C₅₉×#407

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₂×#11745664

C₂²

yes

—

#14273663

—

#14273662

—

C₂²⋊(C₅₉⋊C₈)

—

C₂³×C₄×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂²⋊C₈×C₅₉

—

C₂⁴×C₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

—

—

—

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²⋊Q₈

yes

—

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#123

yes

characteristic

C₄×C₅₉

#123

yes

—

C₄×C₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

—

—

—

D₅₉

yes

characteristic

#338

—

C₅₉

#2091

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

C₂×M₁₆

C₂×D₅₉

yes

characteristic

#108

—

118

—

#108

—

118

—

C₂²⋊C₈

—

C₂²⋊C₈

—

118

—

C₂⁴

Dic₁₁₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₁₈

yes

characteristic

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₂²⋊D₅₉

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×Dic₁₁₈

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#11745685

yes

characteristic

108

C₂×C₄

D₄×D₅₉

yes

—

109

C₂×C₄

#11745687

yes

—

110

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

118

—

114

C₈

—

236

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#14273704

yes

—

122

C₂²

#14273706

yes

characteristic

123

C₄

—

124

C₄

#14273708

yes

characteristic

125

C₄

#14273709

yes

—

126

C₄

#14273723

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#16837338

yes

characteristic

	subgroup	G/H
center	C₄	#14273723
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#407	C₂
socle	C₂×C₅₉	#399
hypercenter	#407	D₅₉
hypocenter	C₅₉	#2091
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #407

subgroup

G/H

center

C₄

#14273723

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#407

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #407

N	H ≅ G/N	type
1. #16837451	C₂	semidirect
2. #16837450	C₂	semidirect
4. C₅₉⋊#338	C₂	semidirect
71. C₅₉	#2091	semidirect

H ≅ G/N

type

1. #16837451

C₂

semidirect

2. #16837450

C₂

semidirect

4. C₅₉⋊#338

C₂

semidirect

71. C₅₉

#2091

semidirect

Subgroups — #21425612

Conjugacy classes of subgroups

Complements