Group #21423604 — subgroups

Subgroups — #21423604

Order 7552.

146 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16837798	C₂	1	1	yes	maximal, characteristic
2	#16837612	C₂	1	1	yes	maximal, characteristic
3	#16837533	C₂	1	1	yes	maximal, characteristic
4	#16837154	C₂	1	1	yes	maximal, characteristic
5	#16837151	C₂	1	1	yes	maximal, characteristic
6	#16837149	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#477	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745672	—	2	1	no	—
9	#14273727	C₂²	1	1	yes	characteristic
10	#14273658	—	2	1	no	—
11	#14273658	—	2	1	no	—
12	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
13	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	#14273650	—	2	1	no	—
16	#14273638	—	2	1	no	—
17	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
20	#107×C₅₉	—	2	1	no	—
21	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	#103×C₅₉	—	2	1	no	—
23	C₂×Q₈×C₅₉	—	2	1	no	—
24	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
25	C₂×Dic₁₁₈	—	2	1	no	—
26	C₂×C₈×C₅₉	—	2	1	no	—
27	C₄⋊C₄×C₅₉	—	2	1	no	—
28	C₄⋊C₄×C₅₉	—	2	1	no	—
29	C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
31	C₄²×C₅₉	—	2	1	no	—
32	C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	#11745672	—	4	2	no	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	C₄⋊Dic₅₉	—	2	1	no	—
36	C₂×C₅₉⋊C₈	—	2	1	no	—
37	C₂×C₅₉⋊C₈	—	2	1	no	—
38	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
39	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
40	Q₈×C₅₉	—	2	1	no	—
41	Q₈×C₅₉	—	2	1	no	—
42	Q₈×C₅₉	—	8	1	no	—
43	C₂×C₄×C₅₉	—	2	1	no	—
44	C₂×C₄×C₅₉	—	2	1	no	—
45	C₂×C₄×C₅₉	—	4	1	no	—
46	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
49	C₂×Dic₅₉	—	4	1	no	—
50	Dic₁₁₈	—	4	1	no	—
51	Dic₁₁₈	—	4	1	no	—
52	C₈×C₅₉	—	4	1	no	—
53	C₅₉⋊C₈	—	2	1	no	—
54	C₅₉⋊C₈	—	4	1	no	—
55	C₅₉⋊C₈	—	4	1	no	—
56	C₅₉⋊C₈	Q₁₆	1	2	yes	complemented
57	C₂²×C₅₉	#121	1	1	yes	characteristic
58	C₄×C₅₉	—	2	1	no	—
59	C₄×C₅₉	—	4	1	no	—
60	C₄×C₅₉	—	4	1	no	—
61	C₄×C₅₉	—	8	1	no	—
62	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₄×C₅₉	#137	1	1	yes	characteristic
65	C₄×C₅₉	#137	1	1	yes	characteristic
66	Dic₅₉	—	8	1	no	—
67	#1692	—	59	1	no	maximal, Sylow
68	C₂×C₅₉	#462	1	1	yes	characteristic
69	C₂×C₅₉	#468	1	1	yes	characteristic
70	C₂×C₅₉	#471	1	1	yes	characteristic
71	#498	—	59	1	no	—
72	#477	D₅₉	1	1	yes	characteristic
73	#462	—	59	1	no	—
74	#446	—	59	1	no	—
75	#335	—	59	1	no	—
76	#333	—	59	1	no	—
77	#332	—	59	1	no	—
78	C₅₉	#1692	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	—	118	1	no	—
80	C₄⋊Q₈	—	59	1	no	—
81	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
82	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
83	#107	—	2	1	no	—
84	#106	—	118	1	no	—
85	#106	—	118	1	no	—
86	C₄⋊C₈	—	59	1	no	—
87	C₄⋊C₈	—	59	1	no	—
88	C₄⋊C₈	—	118	1	no	—
89	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
90	#103	—	2	1	no	—
91	#103	—	118	1	no	—
92	#97	—	118	1	no	—
93	C₄×C₈	—	59	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	118	1	no	—
96	C₂×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
97	Q₁₆	—	236	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	59	1	no	—
100	C₂×C₈	—	59	1	no	—
101	C₂×C₈	—	118	1	no	—
102	C₂×C₈	—	118	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	118	1	no	—
107	C₄⋊C₄	—	118	1	no	—
108	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	C₂²×D₅₉	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	2	1	no	—
113	Q₈	—	8	1	no	—
114	Q₈	—	236	1	no	—
115	Q₈	—	236	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	236	1	no	—
120	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
121	C₂×C₄	#11745687	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	59	2	no	—
125	C₈	—	118	1	no	—
126	C₈	—	236	1	no	—
127	C₈	—	236	1	no	—
128	C₂²	#14273699	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	4	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	8	1	no	—
133	C₄	—	472	1	no	—
134	C₄	Q₁₆×D₅₉	1	1	yes	characteristic
135	C₄	#14273752	1	1	yes	characteristic
136	C₄	#14273758	1	1	yes	characteristic
137	C₄	#14273759	1	1	yes	characteristic
138	C₂	#16837610	1	1	yes	characteristic
139	C₂	#16837650	1	1	yes	characteristic
140	C₂	#16837663	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837798

C₂

yes

maximal, characteristic

#16837612

C₂

yes

maximal, characteristic

#16837533

C₂

yes

maximal, characteristic

#16837154

C₂

yes

maximal, characteristic

#16837151

C₂

yes

maximal, characteristic

#16837149

C₂

yes

maximal, characteristic

C₅₉×#477

C₂

yes

maximal, characteristic

C₂×#11745672

—

#14273727

C₂²

yes

characteristic

#14273658

—

#14273658

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273650

—

#14273638

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄×Q₈×C₅₉

C₂²

yes

characteristic

#107×C₅₉

—

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

#103×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745672

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

Q₁₆

yes

complemented

C₂²×C₅₉

#121

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

yes

—

—

—

—

—

C₅₉

#1692

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

C₄×Q₈

C₂×D₅₉

yes

characteristic

#107

—

#106

—

118

—

#106

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

#103

—

118

—

#97

—

118

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

C₂²⋊D₅₉

yes

characteristic

Q₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

118

—

102

C₂×C₈

—

118

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

118

—

107

C₄⋊C₄

—

118

—

108

C₄⋊C₄

C₂²⋊D₅₉

yes

characteristic

109

C₄²

—

110

C₄²

C₂²×D₅₉

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

236

—

115

Q₈

—

236

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

236

—

120

C₂×C₄

D₄×D₅₉

yes

characteristic

121

C₂×C₄

#11745687

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

118

—

126

C₈

—

236

—

127

C₈

—

236

—

128

C₂²

#14273699

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

472

—

134

C₄

Q₁₆×D₅₉

yes

characteristic

135

C₄

#14273752

yes

characteristic

136

C₄

#14273758

yes

characteristic

137

C₄

#14273759

yes

characteristic

138

C₂

#16837610

yes

characteristic

139

C₂

#16837650

yes

characteristic

140

C₂

#16837663

yes

characteristic

	subgroup	G/H
center	C₂²	#14273699
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#477	C₂
socle	C₂²×C₅₉	#121
hypercenter	#477	D₅₉
hypocenter	C₅₉	#1692
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#14273699

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#477

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
56. C₅₉⋊C₈	Q₁₆	semidirect
78. C₅₉	#1692	semidirect

H ≅ G/N

type

56. C₅₉⋊C₈

Q₁₆

semidirect

78. C₅₉

#1692

semidirect

Subgroups — #21423604

Conjugacy classes of subgroups

Complements