Group #21421998 — subgroups

Subgroups — #21421998

Order 7552.

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

normal subgroups only

This group has 176 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×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#431	C₂	1	1	yes	maximal, characteristic
4	C₂×#11745664	—	2	1	no	—
5	C₂×C₄⋊Dic₅₉	—	2	1	no	—
6	C₂×C₄×Dic₅₉	—	2	1	no	—
7	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₈)	C₄	1	2	yes	complemented
9	C₅₉⋊#95	C₄	1	2	yes	—
10	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₅₉	C₄	1	2	yes	complemented
12	C₂²⋊C₈×C₅₉	—	2	1	no	—
13	C₄×C₈×C₅₉	—	2	1	no	—
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	C₂×C₈×C₅₉	—	2	2	no	—
19	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₅₉	D₄	1	1	yes	characteristic
22	C₄²×C₅₉	Q₈	1	1	yes	characteristic
23	C₄⋊Dic₅₉	—	2	4	no	—
24	C₄×Dic₅₉	—	4	1	no	—
25	C₄×Dic₅₉	—	4	1	no	—
26	#11745664	—	4	1	no	—
27	#11745664	—	4	1	no	—
28	C₂×C₅₉⋊C₈	—	2	2	no	—
29	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₉	—	2	2	no	—
31	C₂×C₄×C₅₉	—	4	1	no	—
32	C₂×C₄×C₅₉	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
39	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
41	C₂×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
42	C₂×Dic₅₉	—	2	4	no	—
43	C₂×Dic₅₉	—	4	2	no	—
44	C₂×Dic₅₉	—	4	2	no	—
45	C₈×C₅₉	—	4	2	no	—
46	C₅₉⋊C₈	—	4	2	no	—
47	C₂²×C₅₉	—	2	1	no	—
48	C₂²×C₅₉	—	2	1	no	—
49	C₂²×C₅₉	#95	1	1	yes	characteristic
50	C₂²×C₅₉	#102	1	1	yes	characteristic
51	C₂²×C₅₉	#103	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	#102	1	1	yes	characteristic
56	C₄×C₅₉	#103	1	1	yes	characteristic
57	C₄×C₅₉	#104	1	2	yes	—
58	C₄×C₅₉	#106	1	1	yes	characteristic
59	C₄×C₅₉	#107	1	1	yes	characteristic
60	Dic₅₉	—	4	4	no	—
61	#1303	—	59	1	no	maximal, Sylow
62	C₂×C₅₉	—	2	1	no	—
63	C₂×C₅₉	—	2	1	no	—
64	C₂×C₅₉	#339	1	1	yes	characteristic
65	C₂×C₅₉	#340	1	1	yes	characteristic
66	C₂×C₅₉	#344	1	1	yes	characteristic
67	#431	D₅₉	1	1	yes	characteristic
68	#423	—	59	1	no	—
69	C₄×C₄⋊C₄	—	59	1	no	—
70	C₅₉	#1303	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	—	118	1	no	—
72	C₂×C₄⋊C₄	—	118	1	no	—
73	C₂×C₄²	—	118	1	no	—
74	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
75	C₄⋊C₈	—	59	2	no	—
76	C₄⋊C₈	—	118	1	no	—
77	C₄⋊C₈	Dic₅₉	1	2	yes	complemented
78	C₂²⋊C₈	—	2	1	no	—
79	C₄×C₈	—	2	1	no	—
80	#95	—	59	2	no	—
81	C₂²×C₄	—	118	2	no	—
82	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
83	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
84	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
85	M₁₆	—	236	1	no	—
86	M₁₆	—	236	1	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	118	2	no	—
89	C₄⋊C₄	—	118	4	no	—
90	C₄²	—	236	1	no	—
91	C₄²	—	236	1	no	—
92	C₄²	Dic₁₁₈	1	1	yes	characteristic
93	C₄²	C₄×D₅₉	1	1	yes	characteristic
94	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
95	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
96	C₂³	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	118	4	no	—
100	C₂×C₄	—	236	2	no	—
101	C₂×C₄	—	236	2	no	—
102	C₂×C₄	#11745660	1	1	yes	characteristic
103	C₂×C₄	#11745660	1	1	yes	characteristic
104	C₂×C₄	D₄₇₂	1	1	yes	characteristic
105	C₂×C₄	Dic₂₃₆	1	1	yes	characteristic
106	C₂×C₄	C₄×Dic₅₉	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	C₄⋊Dic₅₉	1	1	yes	characteristic
110	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
111	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
112	C₈	—	4	2	no	—
113	C₈	—	236	2	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	C₅₉⋊#95	1	1	yes	characteristic
117	C₂²	#14273649	1	1	yes	characteristic
118	C₂²	#14273652	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	—	236	4	no	—
123	C₄	#14273649	1	1	yes	characteristic
124	C₄	#14273652	1	1	yes	characteristic
125	C₄	#14273654	1	2	yes	—
126	C₄	#14273659	1	1	yes	characteristic
127	C₄	#14273661	1	1	yes	characteristic
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#16837161	1	1	yes	characteristic
131	C₂	#16837165	1	1	yes	characteristic
132	C₂	#16837174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837426

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic

C₅₉×#431

C₂

yes

maximal, characteristic

C₂×#11745664

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄×Dic₅₉

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₄

yes

complemented

C₅₉⋊#95

C₄

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₄

yes

complemented

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×Dic₅₉

—

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

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

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

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1303

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

characteristic

#423

—

C₄×C₄⋊C₄

—

C₅₉

#1303

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂×C₄⋊C₄

—

118

—

C₂×C₄²

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

Dic₅₉

yes

complemented

C₂²⋊C₈

—

C₄×C₈

—

#95

—

C₂²×C₄

—

118

—

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

D₂₃₆

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

M₁₆

—

236

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

118

—

C₄⋊C₄

—

118

—

C₄²

—

236

—

C₄²

—

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

—

118

—

100

C₂×C₄

—

236

—

101

C₂×C₄

—

236

—

102

C₂×C₄

#11745660

yes

characteristic

103

C₂×C₄

#11745660

yes

characteristic

104

C₂×C₄

D₄₇₂

yes

characteristic

105

C₂×C₄

Dic₂₃₆

yes

characteristic

106

C₂×C₄

C₄×Dic₅₉

yes

characteristic

107

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

108

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

109

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

110

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

111

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

112

C₈

—

113

C₈

—

236

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₅₉⋊#95

yes

characteristic

117

C₂²

#14273649

yes

characteristic

118

C₂²

#14273652

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

236

—

123

C₄

#14273649

yes

characteristic

124

C₄

#14273652

yes

characteristic

125

C₄

#14273654

yes

—

126

C₄

#14273659

yes

characteristic

127

C₄

#14273661

yes

characteristic

128

C₂

—

129

C₂

—

130

C₂

#16837161

yes

characteristic

131

C₂

#16837165

yes

characteristic

132

C₂

#16837174

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₅₉)⋊C₄
commutator	C₂×C₄×C₅₉	C₄²
Frattini	C₂×C₄²	C₂×D₅₉
Fitting	C₅₉×#431	C₂
socle	C₂²×C₅₉	#95
hypercenter	#431	D₅₉
hypocenter	C₅₉	#1303
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

(C₂×D₅₉)⋊C₄

commutator

C₂×C₄×C₅₉

C₄²

Frattini

C₂×C₄²

C₂×D₅₉

Fitting

C₅₉×#431

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
8. C₄⋊(C₅₉⋊C₈)	C₄	semidirect
11. C₄⋊C₈×C₅₉	C₄	semidirect
70. C₅₉	#1303	semidirect
77. C₄⋊C₈	Dic₅₉	semidirect

H ≅ G/N

type

8. C₄⋊(C₅₉⋊C₈)

C₄

semidirect

11. C₄⋊C₈×C₅₉

C₄

semidirect

70. C₅₉

#1303

semidirect

77. C₄⋊C₈

Dic₅₉

semidirect

Subgroups — #21421998

Conjugacy classes of subgroups

Complements