Group #21422842 — subgroups

Subgroups — #21422842

Order 7552.

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

normal subgroups only

This group has 180 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	#16837455	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#16837154	C₂	1	2	yes	maximal, complemented
4	#16837152	C₂	1	2	yes	maximal, complemented
5	C₅₉×#431	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	C₄	1	1	yes	characteristic
7	C₂²×C₅₉⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
10	C₄×C₅₉⋊C₈	—	2	1	no	—
11	C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
12	C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
13	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₅₉	—	2	1	no	—
18	C₄×C₈×C₅₉	—	2	1	no	—
19	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₅₉	—	2	2	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	C₂×C₅₉⋊C₈	—	2	2	no	—
28	C₂×C₅₉⋊C₈	—	2	2	no	—
29	C₂×C₅₉⋊C₈	C₈	1	4	yes	complemented
30	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₅₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	—	2	2	no	—
36	C₂×C₄×C₅₉	—	2	2	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	C₂×C₈	1	2	yes	—
41	C₂×C₄×C₅₉	M₁₆	1	2	yes	—
42	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₅₉	—	4	2	no	—
46	C₅₉⋊C₈	—	2	1	no	—
47	C₅₉⋊C₈	—	2	1	no	—
48	C₅₉⋊C₈	—	4	2	no	—
49	C₅₉⋊C₈	C₂×C₈	1	2	yes	—
50	C₅₉⋊C₈	C₂×C₈	1	2	yes	—
51	C₂²×C₅₉	—	2	1	no	—
52	C₂²×C₅₉	—	2	1	no	—
53	C₂²×C₅₉	C₄⋊C₈	1	2	yes	—
54	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₅₉	—	2	1	no	—
56	C₄×C₅₉	—	2	1	no	—
57	C₄×C₅₉	C₄⋊C₈	1	2	yes	—
58	C₄×C₅₉	#108	1	2	yes	—
59	C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
60	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
61	C₄×C₅₉	#135	1	2	yes	—
62	#1564	—	59	1	no	maximal, Sylow
63	C₂×C₅₉	—	2	2	no	—
64	C₂×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₅₉	#427	1	1	yes	characteristic
66	C₂×C₅₉	C₂×#108	1	1	yes	characteristic
67	#431	—	59	1	no	—
68	#431	D₅₉	1	1	yes	characteristic
69	C₂×C₄×C₈	—	59	1	no	—
70	#335	—	59	2	no	—
71	#334	—	59	2	no	—
72	C₅₉	#1564	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	59	1	no	—
74	C₂²×C₈	—	59	1	no	—
75	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
76	C₄⋊C₈	—	59	2	no	—
77	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
78	C₂²⋊C₈	—	2	1	no	—
79	C₂²⋊C₈	—	118	1	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	—	59	1	no	—
82	C₄×C₈	—	59	1	no	—
83	C₄×C₈	—	59	1	no	—
84	C₄×C₈	—	59	1	no	—
85	C₄×C₈	—	118	1	no	—
86	C₂²×C₄	Dic₁₁₈	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	59	4	no	—
91	C₂×C₈	—	59	1	no	—
92	C₂×C₈	—	59	1	no	—
93	C₂×C₈	—	59	1	no	—
94	C₂×C₈	—	59	1	no	—
95	C₂×C₈	—	118	2	no	—
96	C₂×C₈	—	118	2	no	—
97	C₄²	Dic₁₁₈	1	1	yes	characteristic
98	C₄²	C₄×D₅₉	1	1	yes	characteristic
99	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
100	C₄²	C₂²×D₅₉	1	1	yes	characteristic
101	C₂³	Dic₅₉⋊C₄	1	1	yes	characteristic
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	C₈×D₅₉	1	2	yes	—
105	C₂×C₄	#11745659	1	2	yes	—
106	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
107	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
108	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
112	C₈	—	4	2	no	—
113	C₈	—	59	2	no	—
114	C₈	—	59	2	no	—
115	C₈	—	118	1	no	—
116	C₈	—	118	1	no	—
117	C₈	—	236	2	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	Dic₅₉⋊C₈	1	2	yes	—
121	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	Dic₅₉⋊C₈	1	2	yes	—
125	C₄	#14273662	1	2	yes	—
126	C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
127	C₄	C₂×#11745659	1	1	yes	characteristic
128	C₄	#14273747	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
131	C₂	#16837438	1	1	yes	characteristic
132	C₂	C₂×#14273662	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837455

C₂

yes

maximal, characteristic

C₂×C₄×C₅₉⋊C₈

C₂

yes

maximal, characteristic

#16837154

C₂

yes

maximal, complemented

#16837152

C₂

yes

maximal, complemented

C₅₉×#431

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₄

yes

characteristic

C₂²×C₅₉⋊C₈

C₄

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

C₄

yes

characteristic

C₄×C₅₉⋊C₈

C₄

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

Q₈

yes

characteristic

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

Q₈

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₈

yes

complemented

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂³×C₅₉

C₄⋊C₄

yes

characteristic

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

C₂×C₈

yes

—

C₂×C₄×C₅₉

M₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₂×C₈

yes

—

C₅₉⋊C₈

C₂×C₈

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄⋊C₈

yes

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄⋊C₈

yes

—

C₄×C₅₉

#108

yes

—

C₄×C₅₉

C₂²×C₈

yes

characteristic

C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₄×C₅₉

#135

yes

—

#1564

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×C₄⋊C₈

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

characteristic

C₂×C₄×C₈

—

#335

—

#334

—

C₅₉

#1564

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₉

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

118

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

118

—

C₂²×C₄

Dic₁₁₈

yes

characteristic

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₄²

Dic₁₁₈

yes

characteristic

C₄²

C₄×D₅₉

yes

characteristic

C₄²

C₂²⋊D₅₉

yes

characteristic

100

C₄²

C₂²×D₅₉

yes

characteristic

101

C₂³

Dic₅₉⋊C₄

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₈×D₅₉

yes

—

105

C₂×C₄

#11745659

yes

—

106

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

107

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

108

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

109

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

110

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

118

—

116

C₈

—

118

—

117

C₈

—

236

—

118

C₂²

—

119

C₂²

—

120

C₂²

Dic₅₉⋊C₈

yes

—

121

C₂²

C₂×Dic₅₉⋊C₄

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

Dic₅₉⋊C₈

yes

—

125

C₄

#14273662

yes

—

126

C₄

C₂×C₈×D₅₉

yes

characteristic

127

C₄

C₂×#11745659

yes

characteristic

128

C₄

#14273747

yes

—

129

C₂

—

130

C₂

C₂×Dic₅₉⋊C₈

yes

characteristic

131

C₂

#16837438

yes

characteristic

132

C₂

C₂×#14273662

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊D₅₉
commutator	C₄×C₅₉	C₂²×C₈
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#431	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	#431	D₅₉
hypocenter	C₅₉	#1564
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₄²

C₂²⋊D₅₉

commutator

C₄×C₅₉

C₂²×C₈

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#431

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

D₅₉

C₅₉

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16837154	C₂	semidirect
4. #16837152	C₂	semidirect
29. C₂×C₅₉⋊C₈	C₈	semidirect
72. C₅₉	#1564	semidirect

H ≅ G/N

type

3. #16837154

C₂

semidirect

4. #16837152

C₂

semidirect

29. C₂×C₅₉⋊C₈

C₈

semidirect

72. C₅₉

#1564

semidirect

Subgroups — #21422842

Conjugacy classes of subgroups

Complements