Group #21424132 — subgroups

Subgroups — #21424132

Order 7552.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂³×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273662	C₂	1	2	yes	maximal, complemented
3	#16837366	C₂	1	1	yes	maximal, characteristic
4	#16837166	C₂	1	2	yes	maximal, complemented
5	C₅₉×#407	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	2	yes	—
7	C₂²×C₅₉⋊C₈	—	2	4	no	—
8	C₂²×C₅₉⋊C₈	C₄	1	4	yes	—
9	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	#14273662	—	2	4	no	—
12	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
13	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₅₉	—	2	2	no	—
16	C₂⁴×C₅₉	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₅₉	—	2	2	no	—
18	C₂²×C₄×C₅₉	—	2	2	no	—
19	C₂²×C₄×C₅₉	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₅₉	D₄	1	2	yes	—
21	C₂²×C₄×C₅₉	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
23	M₁₆×C₅₉	—	2	2	no	—
24	M₁₆×C₅₉	—	2	2	no	—
25	C₂×C₈×C₅₉	—	2	2	no	—
26	#11745664	—	2	2	no	—
27	#11745664	—	2	2	no	—
28	C₂×C₅₉⋊C₈	—	2	2	no	—
29	C₂×C₅₉⋊C₈	—	2	8	no	—
30	C₂×C₅₉⋊C₈	—	2	2	no	—
31	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₅₉⋊C₈	D₄	1	4	yes	—
34	C₂³×C₅₉	—	2	4	no	—
35	C₂³×C₅₉	—	2	2	no	—
36	C₂³×C₅₉	C₄⋊C₄	1	2	yes	—
37	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₅₉	—	2	2	no	—
39	C₂×C₄×C₅₉	—	2	2	no	—
40	C₂×C₄×C₅₉	—	2	2	no	—
41	C₂×C₄×C₅₉	—	2	4	no	—
42	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	#41	1	2	yes	—
47	C₈×C₅₉	—	4	2	no	—
48	C₅₉⋊C₈	—	2	2	no	—
49	C₅₉⋊C₈	—	4	2	no	—
50	C₅₉⋊C₈	C₂²⋊C₄	1	4	yes	—
51	C₂²×C₅₉	—	2	8	no	—
52	C₂²×C₅₉	—	2	2	no	—
53	C₂²×C₅₉	—	2	2	no	—
54	C₂²×C₅₉	—	2	2	no	—
55	C₂²×C₅₉	#108	1	4	yes	complemented
56	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₅₉	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₅₉	—	2	2	no	—
59	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₅₉	#117	1	1	yes	characteristic
61	C₄×C₅₉	#121	1	2	yes	—
62	#1810	—	59	1	no	maximal, Sylow
63	C₂×C₅₉	—	2	4	no	—
64	C₂×C₅₉	—	2	2	no	—
65	C₂×C₅₉	#380	1	1	yes	characteristic
66	C₂×C₅₉	C₂×#108	1	2	yes	—
67	C₂³×C₈	—	59	1	no	—
68	C₂×#108	—	59	2	no	—
69	#407	—	59	1	no	—
70	#407	D₅₉	1	1	yes	characteristic
71	#341	—	59	2	no	—
72	C₅₉	#1810	1	1	yes	characteristic, complemented, Sylow
73	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
74	C₂×M₁₆	—	59	2	no	—
75	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
76	C₂²×C₈	—	59	1	no	—
77	C₂²×C₈	—	59	4	no	—
78	C₂²×C₈	—	59	1	no	—
79	C₂²×C₈	—	118	4	no	—
80	#108	—	118	4	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	C₂²⋊C₈	—	118	2	no	—
83	C₂⁴	Dic₁₁₈	1	1	yes	characteristic
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	Dic₁₁₈	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₅₉	1	2	yes	—
88	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
89	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	118	2	no	—
93	M₁₆	—	118	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	59	2	no	—
96	C₂×C₈	—	59	4	no	—
97	C₂×C₈	—	59	2	no	—
98	C₂×C₈	—	118	8	no	—
99	C₂×C₈	—	118	2	no	—
100	C₂×C₈	—	118	2	no	—
101	C₂³	—	2	4	no	—
102	C₂³	—	2	2	no	—
103	C₂³	Dic₅₉⋊C₄	1	2	yes	—
104	C₂³	C₂×Dic₁₁₈	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
111	C₂×C₄	D₄×D₅₉	1	2	yes	—
112	C₂×C₄	#11745687	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₈	—	59	4	no	—
116	C₈	—	118	2	no	—
117	C₈	—	236	2	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#14273662	1	4	yes	complemented
123	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
124	C₂²	#14273704	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
127	C₄	#14273683	1	1	yes	characteristic
128	C₄	#14273699	1	2	yes	—
129	C₂	—	2	4	no	—
130	C₂	—	2	2	no	—
131	C₂	#16837260	1	1	yes	characteristic
132	C₂	C₂×#14273662	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₅₉⋊C₈

C₂

yes

maximal, characteristic

C₂×#14273662

C₂

yes

maximal, complemented

#16837366

C₂

yes

maximal, characteristic

#16837166

C₂

yes

maximal, complemented

C₅₉×#407

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₂²×C₅₉⋊C₈

—

C₂²×C₅₉⋊C₈

C₄

yes

—

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273662

—

C₂²⋊(C₅₉⋊C₈)

—

C₂³×C₄×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂²⋊C₈×C₅₉

—

C₂⁴×C₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

Q₈

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₄⋊C₄

yes

—

C₂³×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#108

yes

complemented

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

#1810

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#380

yes

characteristic

C₂×C₅₉

C₂×#108

yes

—

C₂³×C₈

—

C₂×#108

—

#407

—

#407

D₅₉

yes

characteristic

#341

—

C₅₉

#1810

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

118

—

#108

—

118

—

C₂²⋊C₈

—

C₂²⋊C₈

—

118

—

C₂⁴

Dic₁₁₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₁₈

yes

characteristic

C₂²×C₄

C₄×D₅₉

yes

—

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

100

C₂×C₈

—

118

—

101

C₂³

—

102

C₂³

—

103

C₂³

Dic₅₉⋊C₄

yes

—

104

C₂³

C₂×Dic₁₁₈

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Dic₅₉⋊C₄

yes

—

110

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

111

C₂×C₄

D₄×D₅₉

yes

—

112

C₂×C₄

#11745687

yes

—

113

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

118

—

117

C₈

—

236

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#14273662

yes

complemented

123

C₂²

C₂×Dic₅₉⋊C₄

yes

characteristic

124

C₂²

#14273704

yes

—

125

C₄

—

126

C₄

C₂²⋊C₄×D₅₉

yes

characteristic

127

C₄

#14273683

yes

characteristic

128

C₄

#14273699

yes

—

129

C₂

—

130

C₂

—

131

C₂

#16837260

yes

characteristic

132

C₂

C₂×#14273662

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#407	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	#407	D₅₉
hypocenter	C₅₉	#1810
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#407

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#14273662	C₂	semidirect
4. #16837166	C₂	semidirect
55. C₂²×C₅₉	#108	semidirect
72. C₅₉	#1810	semidirect
122. C₂²	#14273662	semidirect

H ≅ G/N

type

2. C₂×#14273662

C₂

semidirect

4. #16837166

C₂

semidirect

55. C₂²×C₅₉

#108

semidirect

72. C₅₉

#1810

semidirect

122. C₂²

#14273662

semidirect

Subgroups — #21424132

Conjugacy classes of subgroups

Complements