Group #28479859 — subgroups

Subgroups — #28479859

Order 15104.

366 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 179 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	#21422844	C₂	1	2	yes	maximal
2	#21422388	C₂	1	2	yes	maximal
3	#21421929	C₂	1	2	yes	maximal
4	C₄×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
5	#16837455	C₂²	1	4	yes	—
6	#16837155	C₄	1	4	yes	complemented
7	#16837153	C₄	1	4	yes	complemented
8	#16837124	—	2	4	no	—
9	C₂×#97×C₅₉	C₄	1	2	yes	—
10	C₂×#97×C₅₉	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₄³×C₅₉	C₂²	1	1	yes	characteristic
14	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	8	yes	—
15	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
16	C₄×C₅₉⋊C₈	—	2	4	no	—
17	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₉	—	2	2	no	—
21	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
23	#97×C₅₉	—	2	4	no	—
24	#97×C₅₉	—	2	4	no	—
25	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₅₉	D₄	1	2	yes	—
28	C₄×C₈×C₅₉	Q₈	1	2	yes	—
29	C₂²×C₄×C₅₉	—	2	2	no	—
30	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	#41	1	2	yes	—
32	C₂×C₈×C₅₉	—	2	2	no	—
33	C₂×C₈×C₅₉	—	2	2	no	—
34	C₂×C₈×C₅₉	—	2	2	no	—
35	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₄²	1	2	yes	—
38	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₅₉	M₁₆	1	4	yes	—
40	C₄²×C₅₉	—	2	2	no	—
41	C₄²×C₅₉	—	4	4	no	—
42	C₄²×C₅₉	M₁₆	1	4	yes	—
43	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
44	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₅₉⋊C₈	—	2	8	no	—
47	C₂³×C₅₉	#117	1	1	yes	characteristic
48	C₂×C₄×C₅₉	—	2	4	no	—
49	C₂×C₄×C₅₉	—	2	4	no	—
50	C₂×C₄×C₅₉	—	4	4	no	—
51	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
54	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
55	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
57	C₈×C₅₉	—	2	1	no	—
58	C₈×C₅₉	—	2	1	no	—
59	C₈×C₅₉	—	2	4	no	—
60	C₈×C₅₉	#97	1	4	yes	—
61	C₅₉⋊C₈	—	4	8	no	—
62	#8269	—	59	1	no	maximal, Sylow
63	C₂²×C₅₉	—	2	2	no	—
64	C₂²×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₅₉	#432	1	2	yes	—
66	C₄×C₅₉	—	2	2	no	—
67	C₄×C₅₉	—	4	4	no	—
68	C₄×C₅₉	C₂×#97	1	1	yes	characteristic
69	C₄×C₅₉	C₄×M₁₆	1	1	yes	characteristic
70	C₄×C₅₉	#423	1	2	yes	—
71	C₄×C₅₉	#444	1	4	yes	—
72	C₄×#97	D₅₉	1	1	yes	characteristic
73	#1564	—	59	2	no	—
74	#1452	—	59	2	no	—
75	#1276	—	59	2	no	—
76	C₂×C₅₉	—	2	2	no	—
77	C₂×C₅₉	#1767	1	1	yes	characteristic
78	C₂×C₅₉	#1773	1	2	yes	—
79	#431	—	59	4	no	—
80	C₂×#97	Dic₅₉	1	2	yes	—
81	C₂×#97	Dic₅₉	1	2	yes	—
82	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
84	C₄³	C₂×D₅₉	1	1	yes	characteristic
85	#335	—	59	4	no	—
86	#334	—	59	4	no	—
87	#322	—	118	4	no	—
88	C₅₉	#8269	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
91	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
92	C₂×C₄²	—	2	2	no	—
93	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
94	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
95	C₄⋊C₈	—	59	8	no	—
96	C₂²⋊C₈	—	118	4	no	—
97	#97	—	2	4	no	—
98	#97	—	2	4	no	—
99	C₄×C₈	—	118	4	no	—
100	C₄×C₈	Dic₁₁₈	1	2	yes	—
101	C₄×C₈	C₄×D₅₉	1	2	yes	—
102	C₄×C₈	C₄×D₅₉	1	2	yes	—
103	C₄×C₈	D₂₃₆	1	2	yes	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	#11745685	1	2	yes	—
106	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	118	8	no	—
111	C₂×C₈	#11745664	1	4	yes	—
112	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
113	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
114	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
115	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
116	C₄²	—	2	2	no	—
117	C₄²	—	4	4	no	—
118	C₄²	#11745664	1	4	yes	—
119	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
120	C₄²	C₂×C₄×D₅₉	1	2	yes	—
121	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
122	C₂³	#14273686	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
128	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
129	C₂×C₄	C₄×Dic₁₁₈	1	2	yes	—
130	C₂×C₄	C₂×#11745664	1	2	yes	—
131	C₂×C₄	C₂×#11745664	1	2	yes	—
132	C₈	—	2	4	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	236	8	no	—
136	C₈	#14273638	1	4	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	C₄×C₄⋊Dic₅₉	1	1	yes	characteristic
139	C₂²	#16837459	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	C₂×#14273638	1	1	yes	characteristic
143	C₄	C₄×#11745664	1	1	yes	characteristic
144	C₄	#16837426	1	2	yes	—
145	C₄	#16837525	1	4	yes	—
146	C₂	—	2	2	no	—
147	C₂	#21423949	1	1	yes	characteristic
148	C₂	#21423980	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21422844

C₂

yes

maximal

#21422388

C₂

yes

maximal

#21421929

C₂

yes

maximal

C₄×#97×C₅₉

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

#16837155

C₄

yes

complemented

#16837153

C₄

yes

complemented

#16837124

—

C₂×#97×C₅₉

C₄

yes

—

C₂×#97×C₅₉

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄³×C₅₉

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#97×C₅₉

—

#97×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄²

yes

characteristic

C₂×C₈×C₅₉

C₄²

yes

characteristic

C₂×C₈×C₅₉

C₄²

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

M₁₆

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

M₁₆

yes

—

C₄²×C₅₉

C₂²×C₄

yes

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#117

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

C₄×Q₈

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#97

yes

—

C₅₉⋊C₈

—

#8269

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

#432

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#97

yes

characteristic

C₄×C₅₉

C₄×M₁₆

yes

characteristic

C₄×C₅₉

#423

yes

—

C₄×C₅₉

#444

yes

—

C₄×#97

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

#1767

yes

characteristic

C₂×C₅₉

#1773

yes

—

#431

—

C₂×#97

Dic₅₉

yes

—

C₂×#97

Dic₅₉

yes

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₄³

C₂×D₅₉

yes

—

—

—

118

—

C₅₉

#8269

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

C₂²×C₈

C₂×Dic₅₉

yes

—

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₅₉

yes

—

C₂×C₄²

C₂²×D₅₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

118

—

#97

—

#97

—

C₄×C₈

—

118

—

100

C₄×C₈

Dic₁₁₈

yes

—

101

C₄×C₈

C₄×D₅₉

yes

—

102

C₄×C₈

C₄×D₅₉

yes

—

103

C₄×C₈

D₂₃₆

yes

—

104

C₂²×C₄

—

105

C₂²×C₄

#11745685

yes

—

106

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

118

—

111

C₂×C₈

#11745664

yes

—

112

C₂×C₈

C₄×Dic₅₉

yes

characteristic

113

C₂×C₈

C₄×Dic₅₉

yes

characteristic

114

C₂×C₈

C₄×Dic₅₉

yes

—

115

C₂×C₈

C₄⋊Dic₅₉

yes

—

116

C₄²

—

117

C₄²

—

118

C₄²

#11745664

yes

—

119

C₄²

C₂×Dic₁₁₈

yes

characteristic

120

C₄²

C₂×C₄×D₅₉

yes

—

121

C₄²

C₂×D₂₃₆

yes

characteristic

122

C₂³

#14273686

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂×C₄×Dic₅₉

yes

characteristic

127

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

128

C₂×C₄

C₄×D₂₃₆

yes

—

129

C₂×C₄

C₄×Dic₁₁₈

yes

—

130

C₂×C₄

C₂×#11745664

yes

—

131

C₂×C₄

C₂×#11745664

yes

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

236

—

136

C₈

#14273638

yes

—

137

C₂²

—

138

C₂²

C₄×C₄⋊Dic₅₉

yes

characteristic

139

C₂²

#16837459

yes

—

140

C₄

—

141

C₄

—

142

C₄

C₂×#14273638

yes

characteristic

143

C₄

C₄×#11745664

yes

characteristic

144

C₄

#16837426

yes

—

145

C₄

#16837525

yes

—

146

C₂

—

147

C₂

#21423949

yes

characteristic

148

C₂

#21423980

yes

—

	subgroup	G/H
center	C₄²	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄×#97×C₅₉	C₂
socle	C₂²×C₅₉	C₄×C₄⋊C₄
prime radical	C₂³×C₅₉	#117
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄×#97	D₅₉
hypocenter	C₅₉	#8269
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×#97

subgroup

G/H

center

C₄²

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₄×#97×C₅₉

C₂

socle

C₂²×C₅₉

C₄×C₄⋊C₄

prime radical

C₂³×C₅₉

#117

cosocle

C₂×C₄²×C₅₉

C₂³

hypercenter

C₄×#97

D₅₉

hypocenter

C₅₉

#8269

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×#97

N	H ≅ G/N	type
6. #16837155	C₄	semidirect
7. #16837153	C₄	semidirect
88. C₅₉	#8269	semidirect

H ≅ G/N

type

6. #16837155

C₄

semidirect

7. #16837153

C₄

semidirect

88. C₅₉

#8269

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21422844	2
#21422388	2
#21421929	2
C₄×#97×C₅₉	2
#8269	59

subgroup

index

C₄×#97×C₅₉

Subgroups — #28479859

Conjugacy classes of subgroups

Complements

Maximal subgroups