Group #21425020 — subgroups

Subgroups — #21425020

Order 7552.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273651	C₂	1	4	yes	maximal
3	C₄×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
7	C₂×C₄×Dic₅₉	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
9	#14273651	—	2	8	no	—
10	C₅₉⋊#95	—	2	1	no	—
11	C₂²×Q₈×C₅₉	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₅₉	—	2	2	no	—
13	C₄⋊Q₈×C₅₉	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
15	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
16	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₉	—	2	2	no	—
18	C₂×Q₈×C₅₉	—	2	4	no	—
19	C₂×Q₈×C₅₉	—	2	4	no	—
20	C₂×Q₈×C₅₉	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₅₉	—	2	2	no	—
22	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₉	—	2	2	no	—
26	C₄⋊C₄×C₅₉	—	2	4	no	—
27	C₄⋊C₄×C₅₉	—	2	4	no	—
28	C₄²×C₅₉	C₂×C₄	1	2	yes	—
29	C₄²×C₅₉	D₄	1	2	yes	—
30	C₄⋊Dic₅₉	—	2	2	no	—
31	C₄⋊Dic₅₉	D₄	1	4	yes	—
32	C₄×Dic₅₉	—	4	2	no	—
33	C₂×C₅₉⋊C₈	—	2	4	no	—
34	C₂×C₅₉⋊C₈	—	2	2	no	—
35	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₅₉	—	2	8	no	—
37	Q₈×C₅₉	—	4	2	no	—
38	Q₈×C₅₉	—	4	2	no	—
39	C₂×C₄×C₅₉	—	2	2	no	—
40	C₂×C₄×C₅₉	—	2	4	no	—
41	C₂×C₄×C₅₉	—	4	4	no	—
42	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
45	C₂×C₄×C₅₉	Q₁₆	1	2	yes	—
46	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	#41	1	2	yes	—
50	C₂×Dic₅₉	—	2	4	no	—
51	C₂×Dic₅₉	—	4	2	no	—
52	C₂×Dic₅₉	—	4	2	no	—
53	C₅₉⋊C₈	—	4	2	no	—
54	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₅₉	#121	1	2	yes	—
56	C₂²×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₅₉	#137	1	2	yes	—
59	C₄×C₅₉	—	2	2	no	—
60	C₄×C₅₉	—	4	4	no	—
61	C₄×C₅₉	#103	1	4	yes	complemented
62	C₄×C₅₉	C₄×D₄	1	2	yes	—
63	C₄×C₅₉	#124	1	1	yes	characteristic
64	C₄×C₅₉	#127	1	1	yes	characteristic
65	Dic₅₉	—	4	4	no	—
66	#1981	—	59	1	no	maximal, Sylow
67	C₂×C₅₉	#390	1	1	yes	characteristic
68	C₂×C₅₉	C₂×#103	1	1	yes	characteristic
69	C₂×C₅₉	#419	1	1	yes	characteristic
70	C₂×C₅₉	#461	1	2	yes	—
71	C₂×C₅₉	#462	1	2	yes	—
72	C₂×C₄⋊Q₈	D₅₉	1	1	yes	characteristic
73	C₂×C₄⋊C₈	—	59	1	no	—
74	C₂×#103	—	59	4	no	—
75	C₄×C₄⋊C₄	—	59	1	no	—
76	C₅₉	#1981	1	1	yes	characteristic, complemented, Sylow
77	C₂²×Q₈	C₂×D₅₉	1	2	yes	—
78	C₂²×C₈	—	59	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	Dic₅₉	1	4	yes	complemented
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	59	2	no	—
84	C₂×C₄²	—	118	1	no	—
85	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
86	C₄⋊C₈	—	118	2	no	—
87	#103	—	118	8	no	—
88	#95	—	118	1	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	2	4	no	—
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	C₂×Dic₅₉	1	4	yes	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	118	2	no	—
95	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
98	C₂×C₈	—	118	4	no	—
99	C₂×C₈	—	118	2	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	59	4	no	—
103	C₄⋊C₄	—	118	2	no	—
104	C₄²	—	236	2	no	—
105	C₄²	C₂×Dic₅₉	1	2	yes	—
106	C₄²	C₂²⋊D₅₉	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
108	Q₈	—	2	8	no	—
109	Q₈	—	4	2	no	—
110	Q₈	—	4	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	4	no	—
114	C₂×C₄	—	118	4	no	—
115	C₂×C₄	—	236	2	no	—
116	C₂×C₄	—	236	2	no	—
117	C₂×C₄	#11745671	1	2	yes	—
118	C₂×C₄	#11745672	1	2	yes	—
119	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
120	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
121	C₂×C₄	D₄×D₅₉	1	2	yes	—
122	C₂×C₄	#11745687	1	2	yes	—
123	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
124	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
125	C₈	—	236	2	no	—
126	C₂²	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
127	C₂²	#14273698	1	2	yes	—
128	C₂²	C₂×#11745671	1	1	yes	characteristic
129	C₂²	C₂×#11745672	1	1	yes	characteristic
130	C₂²	#14273758	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	236	4	no	—
134	C₄	#14273651	1	4	yes	complemented
135	C₄	D₄×Dic₅₉	1	2	yes	—
136	C₄	#14273714	1	1	yes	characteristic
137	C₄	#14273723	1	1	yes	characteristic
138	C₂	#16837305	1	1	yes	characteristic
139	C₂	C₂×#14273651	1	1	yes	characteristic
140	C₂	#16837411	1	1	yes	characteristic
141	C₂	#16837604	1	2	yes	—
142	C₂	#16837609	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14273651

C₂

yes

maximal

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

—

C₄⋊(C₅₉⋊C₈)

—

#14273651

—

C₅₉⋊#95

—

C₂²×Q₈×C₅₉

C₂²

yes

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

Q₁₆

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

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₉

#121

yes

—

C₂²×C₅₉

C₂×SD₁₆

yes

characteristic

C₂²×C₅₉

C₂×Q₁₆

yes

characteristic

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#103

yes

complemented

C₄×C₅₉

C₄×D₄

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#462

yes

—

C₂×C₄⋊Q₈

D₅₉

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#103

—

C₄×C₄⋊C₄

—

C₅₉

#1981

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₉

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₅₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

118

—

#103

—

118

—

#95

—

118

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

118

—

C₂×C₈

—

118

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

118

—

104

C₄²

—

236

—

105

C₄²

C₂×Dic₅₉

yes

—

106

C₄²

C₂²⋊D₅₉

yes

—

107

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

118

—

115

C₂×C₄

—

236

—

116

C₂×C₄

—

236

—

117

C₂×C₄

#11745671

yes

—

118

C₂×C₄

#11745672

yes

—

119

C₂×C₄

C₂²⋊Dic₅₉

yes

—

120

C₂×C₄

C₂²⋊Dic₅₉

yes

—

121

C₂×C₄

D₄×D₅₉

yes

—

122

C₂×C₄

#11745687

yes

—

123

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

124

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

125

C₈

—

236

—

126

C₂²

C₂×C₂²⋊Dic₅₉

yes

characteristic

127

C₂²

#14273698

yes

—

128

C₂²

C₂×#11745671

yes

characteristic

129

C₂²

C₂×#11745672

yes

characteristic

130

C₂²

#14273758

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

236

—

134

C₄

#14273651

yes

complemented

135

C₄

D₄×Dic₅₉

yes

—

136

C₄

#14273714

yes

characteristic

137

C₄

#14273723

yes

characteristic

138

C₂

#16837305

yes

characteristic

139

C₂

C₂×#14273651

yes

characteristic

140

C₂

#16837411

yes

characteristic

141

C₂

#16837604

yes

—

142

C₂

#16837609

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×C₄⋊Q₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₅₉
hypocenter	C₅₉	#1981
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×C₄⋊Q₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₅₉

hypocenter

C₅₉

#1981

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
13. C₄⋊Q₈×C₅₉	C₄	semidirect
61. C₄×C₅₉	#103	semidirect
76. C₅₉	#1981	semidirect
80. C₄⋊Q₈	Dic₅₉	semidirect
134. C₄	#14273651	semidirect

H ≅ G/N

type

13. C₄⋊Q₈×C₅₉

C₄

semidirect

61. C₄×C₅₉

#103

semidirect

76. C₅₉

#1981

semidirect

80. C₄⋊Q₈

Dic₅₉

semidirect

134. C₄

#14273651

semidirect

Subgroups — #21425020

Conjugacy classes of subgroups

Complements