Group #21422516 — subgroups

Subgroups — #21422516

Order 7552.

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

normal subgroups only

This group has 218 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	#16837959	C₂	1	1	yes	maximal, characteristic
2	C₄×#11745664	C₂	1	1	yes	maximal, characteristic
3	#16837132	C₂	1	4	yes	maximal, complemented
4	C₅₉×#431	C₂	1	1	yes	maximal, characteristic
5	C₂×#11745664	—	2	1	no	—
6	#14273727	C₄	1	1	yes	characteristic
7	#14273727	C₄	1	1	yes	characteristic
8	#14273719	—	2	1	no	—
9	#14273706	—	2	1	no	—
10	#14273706	C₄	1	2	yes	—
11	C₄×Dic₁₁₈	—	2	1	no	—
12	C₄×Dic₁₁₈	C₂²	1	2	yes	—
13	#14273686	—	2	1	no	—
14	#14273638	—	2	1	no	—
15	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
18	C₂²⋊C₈×C₅₉	—	2	1	no	—
19	C₄×C₈×C₅₉	—	2	1	no	—
20	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
23	C₂×Dic₁₁₈	—	2	1	no	—
24	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₅₉	—	2	2	no	—
26	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₅₉	D₄	1	1	yes	characteristic
28	C₄²×C₅₉	D₄	1	1	yes	characteristic
29	C₄²×C₅₉	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₅₉	—	2	2	no	—
31	C₄⋊Dic₅₉	—	2	1	no	—
32	C₄⋊Dic₅₉	—	2	1	no	—
33	C₄⋊Dic₅₉	—	2	1	no	—
34	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
35	Dic₅₉⋊C₄	—	2	2	no	—
36	Dic₅₉⋊C₄	—	2	2	no	—
37	C₄×Dic₅₉	—	2	2	no	—
38	#11745664	—	2	4	no	—
39	C₂×C₅₉⋊C₈	—	2	2	no	—
40	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	—	2	2	no	—
42	C₂×C₄×C₅₉	—	2	2	no	—
43	C₂×C₄×C₅₉	—	2	2	no	—
44	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₅₉	—	2	2	no	—
51	C₂×Dic₅₉	—	2	2	no	—
52	Dic₁₁₈	—	2	4	no	—
53	Dic₁₁₈	—	4	2	no	—
54	C₈×C₅₉	—	4	2	no	—
55	C₅₉⋊C₈	—	2	4	no	—
56	C₂²×C₅₉	—	2	1	no	—
57	C₂²×C₅₉	—	2	1	no	—
58	C₂²×C₅₉	—	2	1	no	—
59	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₅₉	—	2	1	no	—
61	C₄×C₅₉	—	2	1	no	—
62	C₄×C₅₉	—	2	1	no	—
63	C₄×C₅₉	#104	1	2	yes	—
64	C₄×C₅₉	#131	1	2	yes	—
65	C₄×C₅₉	#137	1	2	yes	—
66	Dic₅₉	—	4	2	no	—
67	Dic₅₉	—	4	2	no	—
68	#1485	—	59	1	no	maximal, Sylow
69	C₂×C₅₉	—	4	1	no	—
70	C₂×C₅₉	#408	1	1	yes	characteristic
71	C₂×C₅₉	#419	1	1	yes	characteristic
72	C₂×C₅₉	C₂×#104	1	1	yes	characteristic
73	#533	—	59	1	no	—
74	#431	D₅₉	1	1	yes	characteristic
75	C₄×M₁₆	—	59	1	no	—
76	Q₈⋊C₈	—	59	4	no	—
77	C₅₉	#1485	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₁₆	—	118	1	no	—
79	C₄⋊Q₈	—	59	1	no	—
80	C₄⋊Q₈	—	59	1	no	—
81	#125	—	118	1	no	—
82	C₂²⋊Q₈	—	59	2	no	—
83	C₂²⋊Q₈	—	118	1	no	—
84	C₄×Q₈	—	59	2	no	—
85	C₄×Q₈	—	118	1	no	—
86	#117	—	118	1	no	—
87	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
88	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
89	C₂²⋊C₈	—	2	1	no	—
90	#97	—	118	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	59	2	no	—
93	C₂×Q₈	—	59	2	no	—
94	C₂×Q₈	—	118	1	no	—
95	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
96	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
98	M₁₆	—	118	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	118	2	no	—
101	C₄⋊C₄	—	59	2	no	—
102	C₄⋊C₄	—	118	1	no	—
103	C₄⋊C₄	—	118	1	no	—
104	C₄⋊C₄	—	118	2	no	—
105	C₄⋊C₄	—	118	1	no	—
106	C₄⋊C₄	—	118	2	no	—
107	C₂²⋊C₄	—	118	2	no	—
108	C₄²	—	118	2	no	—
109	C₄²	C₄×D₅₉	1	1	yes	characteristic
110	C₄²	D₂₃₆	1	1	yes	characteristic
111	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
112	C₄²	C₂²×D₅₉	1	1	yes	characteristic
113	C₂³	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
114	Q₈	—	118	4	no	—
115	Q₈	—	236	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	118	2	no	—
120	C₂×C₄	—	118	2	no	—
121	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
125	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₈	—	118	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	236	2	no	—
137	C₄	—	236	2	no	—
138	C₄	#14273653	1	2	yes	—
139	C₄	#14273733	1	2	yes	—
140	C₄	#14273757	1	2	yes	—
141	C₂	—	4	1	no	—
142	C₂	#16837368	1	1	yes	characteristic
143	C₂	#16837410	1	1	yes	characteristic
144	C₂	C₂×#14273653	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal, characteristic

C₄×#11745664

C₂

yes

maximal, characteristic

#16837132

C₂

yes

maximal, complemented

C₅₉×#431

C₂

yes

maximal, characteristic

C₂×#11745664

—

#14273727

C₄

yes

characteristic

#14273727

C₄

yes

—

—

C₄

yes

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₂²

yes

—

#14273686

—

#14273638

—

C₄×C₅₉⋊C₈

C₂²

yes

—

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

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#104

yes

—

C₄×C₅₉

#131

yes

—

C₄×C₅₉

#137

yes

—

Dic₅₉

—

Dic₅₉

—

#1485

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

characteristic

C₄×M₁₆

—

Q₈⋊C₈

—

C₅₉

#1485

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

118

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

C₄×Q₈

—

118

—

#117

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

C₂×D₅₉

yes

—

C₂²⋊C₈

—

#97

—

118

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂²×C₄

C₄×D₅₉

yes

characteristic

C₂²×C₄

D₂₃₆

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

M₁₆

—

118

—

C₂×C₈

—

100

C₂×C₈

—

118

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

118

—

103

C₄⋊C₄

—

118

—

104

C₄⋊C₄

—

118

—

105

C₄⋊C₄

—

118

—

106

C₄⋊C₄

—

118

—

107

C₂²⋊C₄

—

118

—

108

C₄²

—

118

—

109

C₄²

C₄×D₅₉

yes

characteristic

110

C₄²

D₂₃₆

yes

characteristic

111

C₄²

C₂²⋊D₅₉

yes

characteristic

112

C₄²

C₂²×D₅₉

yes

characteristic

113

C₂³

(C₂×D₅₉)⋊C₄

yes

characteristic

114

Q₈

—

118

—

115

Q₈

—

236

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

118

—

120

C₂×C₄

—

118

—

121

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

122

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

123

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

124

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

125

C₂×C₄

C₂×D₂₃₆

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

127

C₈

—

128

C₈

—

118

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

236

—

137

C₄

—

236

—

138

C₄

#14273653

yes

—

139

C₄

#14273733

yes

—

140

C₄

#14273757

yes

—

141

C₂

—

142

C₂

#16837368

yes

characteristic

143

C₂

#16837410

yes

characteristic

144

C₂

C₂×#14273653

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#431	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#431	D₅₉
hypocenter	C₅₉	#1485
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#431

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16837132	C₂	semidirect
77. C₅₉	#1485	semidirect

H ≅ G/N

type

3. #16837132

C₂

semidirect

77. C₅₉

#1485

semidirect

Subgroups — #21422516

Conjugacy classes of subgroups

Complements