Group #21423479 — subgroups

Subgroups — #21423479

Order 7552.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16837800	C₂	1	1	yes	maximal, characteristic
2	#16837698	C₂	1	1	yes	maximal, characteristic
3	#16837613	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
5	#16837150	C₂	1	1	yes	maximal, characteristic
6	#16837149	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#334	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₃₆	—	2	1	no	—
9	#14273727	C₂²	1	1	yes	characteristic
10	#14273727	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
12	#14273661	—	2	1	no	—
13	#14273659	—	2	1	no	—
14	#14273659	—	2	1	no	—
15	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
16	Dic₅₉⋊C₈	—	2	1	no	—
17	#14273652	—	2	1	no	—
18	#14273652	—	2	1	no	—
19	C₈×Dic₅₉	—	2	1	no	—
20	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₁₈	—	2	1	no	—
24	C₂×Dic₁₁₈	—	2	1	no	—
25	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
26	C₂×C₈×C₅₉	—	2	1	no	—
27	C₂×C₈×C₅₉	—	2	1	no	—
28	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
29	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
30	C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₅₉	—	2	1	no	—
32	C₄⋊Dic₅₉	—	2	1	no	—
33	C₄⋊Dic₅₉	—	2	1	no	—
34	C₄⋊Dic₅₉	—	2	1	no	—
35	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
36	Dic₅₉⋊C₄	—	2	1	no	—
37	C₄×Dic₅₉	—	2	1	no	—
38	C₂×C₅₉⋊C₈	—	2	1	no	—
39	Dic₂₃₆	—	4	2	no	—
40	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
43	C₂×Dic₅₉	—	2	1	no	—
44	C₂×Dic₅₉	—	2	1	no	—
45	C₂×Dic₅₉	—	4	1	no	—
46	C₂×Dic₅₉	—	4	1	no	—
47	Dic₁₁₈	—	2	1	no	—
48	Dic₁₁₈	—	4	1	no	—
49	Dic₁₁₈	—	4	1	no	—
50	Dic₁₁₈	—	8	1	no	—
51	Dic₁₁₈	SD₁₆	1	2	yes	—
52	C₈×C₅₉	—	2	1	no	—
53	C₈×C₅₉	—	4	1	no	—
54	C₈×C₅₉	—	4	1	no	—
55	C₈×C₅₉	Q₁₆	1	2	yes	complemented
56	C₅₉⋊C₈	—	4	1	no	—
57	C₂²×C₅₉	#121	1	1	yes	characteristic
58	C₄×C₅₉	—	2	1	no	—
59	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
60	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₅₉	#135	1	1	yes	characteristic
62	C₄×C₅₉	#137	1	1	yes	characteristic
63	Dic₅₉	—	2	2	no	—
64	Dic₅₉	—	4	1	no	—
65	Dic₅₉	—	8	1	no	—
66	Dic₅₉	—	8	1	no	—
67	#1672	—	59	1	no	maximal, Sylow
68	C₂×C₅₉	#462	1	1	yes	characteristic
69	C₂×C₅₉	#465	1	1	yes	characteristic
70	C₂×C₅₉	#471	1	1	yes	characteristic
71	#498	—	59	1	no	—
72	#477	—	59	1	no	—
73	#462	—	59	1	no	—
74	C₈×Q₈	—	59	1	no	—
75	#334	D₅₉	1	1	yes	characteristic
76	#333	—	59	1	no	—
77	#332	—	59	1	no	—
78	C₅₉	#1672	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	—	118	1	no	—
80	C₄⋊Q₈	—	59	1	no	—
81	C₄⋊Q₈	—	59	1	no	—
82	C₄×Q₈	—	59	1	no	—
83	#107	—	118	1	no	—
84	#106	—	118	1	no	—
85	#106	—	118	1	no	—
86	C₄⋊C₈	—	59	1	no	—
87	C₄⋊C₈	—	118	1	no	—
88	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
89	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
90	#103	—	118	1	no	—
91	#103	—	118	1	no	—
92	C₄×C₈	—	118	1	no	—
93	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
94	C₂×Q₈	—	59	1	no	—
95	C₂×Q₈	—	118	1	no	—
96	C₂×Q₈	—	118	1	no	—
97	Q₁₆	—	236	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	118	1	no	—
101	C₂×C₈	D₂₃₆	1	1	yes	characteristic
102	C₂×C₈	D₂₃₆	1	1	yes	characteristic
103	C₄⋊C₄	—	59	1	no	—
104	C₄⋊C₄	—	118	1	no	—
105	C₄⋊C₄	—	118	1	no	—
106	C₄⋊C₄	—	118	1	no	—
107	C₄⋊C₄	—	118	1	no	—
108	C₄⋊C₄	—	118	1	no	—
109	C₄²	—	118	1	no	—
110	C₄²	C₂²×D₅₉	1	1	yes	characteristic
111	Q₈	—	59	2	no	—
112	Q₈	—	118	1	no	—
113	Q₈	—	472	1	no	—
114	Q₈	—	236	1	no	—
115	Q₈	—	236	1	no	—
116	C₂×C₄	—	118	1	no	—
117	C₂×C₄	—	118	1	no	—
118	C₂×C₄	—	236	1	no	—
119	C₂×C₄	—	236	1	no	—
120	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
121	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
122	C₂×C₄	#11745689	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	4	1	no	—
126	C₈	—	236	1	no	—
127	C₈	Dic₂₃₆	1	2	yes	complemented
128	C₂²	#14273702	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	118	2	no	—
131	C₄	—	472	1	no	—
132	C₄	—	472	1	no	—
133	C₄	—	236	1	no	—
134	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
135	C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
136	C₄	#14273748	1	1	yes	characteristic
137	C₄	#14273760	1	1	yes	characteristic
138	C₂	#16837613	1	1	yes	characteristic
139	C₂	#16837632	1	1	yes	characteristic
140	C₂	#16837666	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837800

C₂

yes

maximal, characteristic

#16837698

C₂

yes

maximal, characteristic

#16837613

C₂

yes

maximal, characteristic

C₈×Dic₁₁₈

C₂

yes

maximal, characteristic

#16837150

C₂

yes

maximal, characteristic

#16837149

C₂

yes

maximal, characteristic

C₅₉×#334

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

—

#14273727

C₂²

yes

characteristic

#14273727

C₂²

yes

characteristic

C₄×Dic₁₁₈

C₂²

yes

—

—

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

#14273652

—

#14273652

—

C₈×Dic₅₉

—

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

SD₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

Q₁₆

yes

complemented

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

C₈×Q₈

—

#334

D₅₉

yes

characteristic

#333

—

#332

—

C₅₉

#1672

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

118

—

#106

—

118

—

#106

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

118

—

#103

—

118

—

C₄×C₈

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

118

—

C₂×Q₈

—

118

—

Q₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

118

—

101

C₂×C₈

D₂₃₆

yes

characteristic

102

C₂×C₈

D₂₃₆

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

118

—

105

C₄⋊C₄

—

118

—

106

C₄⋊C₄

—

118

—

107

C₄⋊C₄

—

118

—

108

C₄⋊C₄

—

118

—

109

C₄²

—

118

—

110

C₄²

C₂²×D₅₉

yes

characteristic

111

Q₈

—

112

Q₈

—

118

—

113

Q₈

—

472

—

114

Q₈

—

236

—

115

Q₈

—

236

—

116

C₂×C₄

—

118

—

117

C₂×C₄

—

118

—

118

C₂×C₄

—

236

—

119

C₂×C₄

—

236

—

120

C₂×C₄

C₂×D₂₃₆

yes

characteristic

121

C₂×C₄

D₄×D₅₉

yes

characteristic

122

C₂×C₄

#11745689

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

236

—

127

C₈

Dic₂₃₆

yes

complemented

128

C₂²

#14273702

yes

characteristic

129

C₄

—

130

C₄

—

118

—

131

C₄

—

472

—

132

C₄

—

472

—

133

C₄

—

236

—

134

C₄

SD₁₆×D₅₉

yes

characteristic

135

C₄

C₂×Dic₂₃₆

yes

characteristic

136

C₄

#14273748

yes

characteristic

137

C₄

#14273760

yes

characteristic

138

C₂

#16837613

yes

characteristic

139

C₂

#16837632

yes

characteristic

140

C₂

#16837666

yes

characteristic

	subgroup	G/H
center	C₂²	#14273702
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#334	C₂
socle	C₂²×C₅₉	#121
hypercenter	#334	D₅₉
hypocenter	C₅₉	#1672
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#14273702

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#334

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
55. C₈×C₅₉	Q₁₆	semidirect
78. C₅₉	#1672	semidirect
127. C₈	Dic₂₃₆	semidirect

H ≅ G/N

type

55. C₈×C₅₉

Q₁₆

semidirect

78. C₅₉

#1672

semidirect

127. C₈

Dic₂₃₆

semidirect

Subgroups — #21423479

Conjugacy classes of subgroups

Complements