Group #21421956 — subgroups

Subgroups — #21421956

Order 7552.

156 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16837268	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#423	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₅₉	—	2	1	no	—
5	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
6	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
7	C₄×C₅₉⋊C₈	C₄	1	2	yes	complemented
8	C₅₉⋊#95	—	2	1	no	—
9	C₅₉⋊#95	C₄	1	2	yes	—
10	C₂×M₁₆×C₅₉	—	2	1	no	—
11	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₅₉	—	2	1	no	—
13	C₄⋊C₈×C₅₉	C₄	1	2	yes	complemented
14	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₅₉	—	2	2	no	—
18	M₁₆×C₅₉	—	4	1	no	—
19	M₁₆×C₅₉	—	4	1	no	—
20	C₂×C₈×C₅₉	—	2	2	no	—
21	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₉	D₄	1	1	yes	characteristic
24	C₄²×C₅₉	Q₈	1	1	yes	characteristic
25	C₄⋊Dic₅₉	—	4	1	no	—
26	C₄⋊Dic₅₉	—	4	1	no	—
27	C₂×C₅₉⋊C₈	—	2	2	no	—
28	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₉	—	4	1	no	—
30	C₂×C₄×C₅₉	—	4	1	no	—
31	C₂×C₄×C₅₉	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
38	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
41	C₂×Dic₅₉	—	4	2	no	—
42	C₂×Dic₅₉	—	4	2	no	—
43	C₂×Dic₅₉	—	4	2	no	—
44	C₈×C₅₉	—	4	2	no	—
45	C₅₉⋊C₈	—	4	2	no	—
46	C₂²×C₅₉	—	2	1	no	—
47	C₂²×C₅₉	—	2	1	no	—
48	C₂²×C₅₉	#95	1	1	yes	characteristic
49	C₂²×C₅₉	#102	1	1	yes	characteristic
50	C₂²×C₅₉	#103	1	1	yes	characteristic
51	C₄×C₅₉	—	2	1	no	—
52	C₄×C₅₉	—	2	1	no	—
53	C₄×C₅₉	—	2	1	no	—
54	C₄×C₅₉	—	2	1	no	—
55	C₄×C₅₉	#102	1	1	yes	characteristic
56	C₄×C₅₉	#103	1	1	yes	characteristic
57	C₄×C₅₉	#106	1	1	yes	characteristic
58	C₄×C₅₉	#107	1	1	yes	characteristic
59	Dic₅₉	—	8	2	no	—
60	#1286	—	59	1	no	maximal, Sylow
61	C₂×C₅₉	—	2	1	no	—
62	C₂×C₅₉	—	2	1	no	—
63	C₂×C₅₉	H(3,ℤ/4)	1	1	yes	characteristic
64	C₂×C₅₉	#340	1	1	yes	characteristic
65	C₂×C₅₉	#344	1	1	yes	characteristic
66	#431	—	59	1	no	—
67	#423	D₅₉	1	1	yes	characteristic
68	#382	—	59	1	no	—
69	C₅₉	#1286	1	1	yes	characteristic, complemented, Sylow
70	C₂×M₁₆	—	2	1	no	—
71	C₂×C₄⋊C₄	—	118	1	no	—
72	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
73	C₄⋊C₈	—	2	1	no	—
74	C₄⋊C₈	—	118	1	no	—
75	C₄⋊C₈	Dic₅₉	1	2	yes	complemented
76	C₂²⋊C₈	—	118	1	no	—
77	C₄×C₈	—	59	2	no	—
78	#95	—	59	2	no	—
79	#95	—	118	1	no	—
80	C₂²×C₄	—	118	2	no	—
81	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
82	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
83	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
84	M₁₆	—	4	1	no	—
85	M₁₆	—	4	1	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	118	2	no	—
88	C₄⋊C₄	—	236	1	no	—
89	C₄⋊C₄	—	236	1	no	—
90	C₄²	Dic₁₁₈	1	1	yes	characteristic
91	C₄²	C₄×D₅₉	1	1	yes	characteristic
92	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
93	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
94	C₂³	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	236	2	no	—
98	C₂×C₄	—	236	2	no	—
99	C₂×C₄	—	236	2	no	—
100	C₂×C₄	C₄×Dic₅₉	1	1	yes	characteristic
101	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
102	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
103	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
104	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
105	C₂×C₄	#11745669	1	1	yes	characteristic
106	C₂×C₄	#11745670	1	1	yes	characteristic
107	C₂×C₄	#11745671	1	1	yes	characteristic
108	C₂×C₄	#11745672	1	1	yes	characteristic
109	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
110	C₈	—	4	2	no	—
111	C₈	—	236	2	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	C₅₉⋊#95	1	1	yes	characteristic
115	C₂²	#14273647	1	1	yes	characteristic
116	C₂²	#14273650	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	—	472	2	no	—
122	C₄	#14273648	1	1	yes	characteristic
123	C₄	#14273651	1	1	yes	characteristic
124	C₄	#14273658	1	1	yes	characteristic
125	C₄	#14273660	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	2	1	no	—
128	C₂	#16837159	1	1	yes	characteristic
129	C₂	#16837164	1	1	yes	characteristic
130	C₂	#16837174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837455

C₂

yes

maximal, characteristic

#16837268

C₂

yes

maximal, characteristic

C₅₉×#423

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₉

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

C₄

yes

complemented

C₅₉⋊#95

—

C₅₉⋊#95

C₄

yes

—

C₂×M₁₆×C₅₉

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

C₄

yes

complemented

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×Dic₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

Q₈

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄²

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

characteristic

C₂×C₄×C₅₉

Q₁₆

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1286

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

characteristic

#382

—

C₅₉

#1286

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

Dic₅₉

yes

complemented

C₂²⋊C₈

—

118

—

C₄×C₈

—

#95

—

#95

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

D₂₃₆

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

118

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄²

Dic₁₁₈

yes

characteristic

C₄²

C₄×D₅₉

yes

characteristic

C₄²

C₂×Dic₅₉

yes

characteristic

C₄²

C₂²⋊D₅₉

yes

characteristic

C₂³

(C₂×D₅₉)⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

100

C₂×C₄

C₄×Dic₅₉

yes

characteristic

101

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

102

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

103

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

104

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

105

C₂×C₄

#11745669

yes

characteristic

106

C₂×C₄

#11745670

yes

characteristic

107

C₂×C₄

#11745671

yes

characteristic

108

C₂×C₄

#11745672

yes

characteristic

109

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

110

C₈

—

111

C₈

—

236

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₅₉⋊#95

yes

characteristic

115

C₂²

#14273647

yes

characteristic

116

C₂²

#14273650

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

472

—

122

C₄

#14273648

yes

characteristic

123

C₄

#14273651

yes

characteristic

124

C₄

#14273658

yes

characteristic

125

C₄

#14273660

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#16837159

yes

characteristic

129

C₂

#16837164

yes

characteristic

130

C₂

#16837174

yes

characteristic

	subgroup	G/H
center	C₂²	C₅₉⋊#95
commutator	C₂×C₄×C₅₉	C₄²
Frattini	C₂×C₄²	C₂×D₅₉
Fitting	C₅₉×#423	C₂
socle	C₂²×C₅₉	#95
hypercenter	#423	D₅₉
hypocenter	C₅₉	#1286
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₅₉⋊#95

commutator

C₂×C₄×C₅₉

C₄²

Frattini

C₂×C₄²

C₂×D₅₉

Fitting

C₅₉×#423

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
7. C₄×C₅₉⋊C₈	C₄	semidirect
13. C₄⋊C₈×C₅₉	C₄	semidirect
69. C₅₉	#1286	semidirect
75. C₄⋊C₈	Dic₅₉	semidirect

H ≅ G/N

type

7. C₄×C₅₉⋊C₈

C₄

semidirect

13. C₄⋊C₈×C₅₉

C₄

semidirect

69. C₅₉

#1286

semidirect

75. C₄⋊C₈

Dic₅₉

semidirect

Subgroups — #21421956

Conjugacy classes of subgroups

Complements