Group #28500681 — subgroups

Subgroups — #28500681

Order 15104.

210 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₄×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic
2	#21422994	C₂	1	2	yes	maximal
3	#21422168	C₂	1	1	yes	maximal, characteristic
4	#21422061	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₇₂	C₄	1	1	yes	characteristic
7	#16837532	—	2	2	no	—
8	C₄×Dic₂₃₆	C₂²	1	2	yes	—
9	#16837228	C₄	1	1	yes	characteristic
10	#16837207	C₄	1	2	yes	—
11	#16837155	C₂²	1	2	yes	—
12	#16837134	—	2	2	no	—
13	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₈²×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₁₈	—	2	2	no	—
18	Dic₄₇₂	—	2	2	no	—
19	#14273661	C₂×C₄	1	2	yes	—
20	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
21	Dic₅₉⋊C₈	—	4	2	no	—
22	#14273652	—	2	2	no	—
23	#14273637	—	4	2	no	—
24	C₂×C₁₆×C₅₉	—	2	1	no	—
25	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₁₈	—	2	2	no	—
31	C₂×C₈×C₅₉	—	2	1	no	—
32	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
33	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
34	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
36	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₅₉	—	2	2	no	—
38	Dic₅₉⋊C₄	—	4	2	no	—
39	C₄×Dic₅₉	—	4	2	no	—
40	C₂×C₅₉⋊C₈	—	4	2	no	—
41	Dic₂₃₆	—	2	2	no	—
42	Dic₂₃₆	—	2	2	no	—
43	C₁₆×C₅₉	—	2	2	no	—
44	C₁₆×C₅₉	—	2	1	no	—
45	C₁₆×C₅₉	M₁₆	1	2	yes	—
46	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
49	C₂×Dic₅₉	—	4	2	no	—
50	C₂×Dic₅₉	—	4	2	no	—
51	Dic₁₁₈	—	4	2	no	—
52	Dic₁₁₈	—	4	2	no	—
53	C₈×C₅₉	—	2	2	no	—
54	C₈×C₅₉	—	2	1	no	—
55	C₈×C₅₉	—	2	1	no	—
56	C₈×C₅₉	Q₃₂	1	2	yes	complemented
57	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
58	C₈×C₅₉	#131	1	1	yes	characteristic
59	C₅₉⋊C₈	—	8	2	no	—
60	#12083	—	59	1	no	maximal, Sylow
61	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
62	C₄×C₅₉	—	2	1	no	—
63	C₄×C₅₉	#436	1	1	yes	characteristic
64	C₄×C₅₉	#443	1	1	yes	characteristic
65	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
66	C₄×C₅₉	#508	1	1	yes	characteristic
67	Dic₅₉	—	8	2	no	—
68	Dic₅₉	—	8	2	no	—
69	C₄×Q₃₂	—	59	1	no	—
70	#1591	—	59	2	no	—
71	#1371	—	59	1	no	—
72	#1332	—	59	2	no	—
73	C₈×C₁₆	D₅₉	1	1	yes	characteristic
74	C₂×C₅₉	#1589	1	1	yes	characteristic
75	C₂×C₅₉	C₄×Q₃₂	1	1	yes	characteristic
76	C₂×C₅₉	#2178	1	1	yes	characteristic
77	C₂×Q₃₂	—	59	1	no	—
78	#446	—	118	2	no	—
79	C₄×Q₁₆	—	59	2	no	—
80	#366	—	59	1	no	—
81	#358	—	59	2	no	—
82	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
83	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
84	#335	—	59	2	no	—
85	Q₈⋊C₈	—	118	2	no	—
86	C₈²	C₂×D₅₉	1	1	yes	characteristic
87	C₅₉	#12083	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	—	59	2	no	—
89	C₄×Q₈	—	118	2	no	—
90	Q₃₂	—	118	2	no	—
91	C₂×C₁₆	—	2	1	no	—
92	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
93	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
94	#107	—	59	2	no	—
95	C₄⋊C₈	—	118	2	no	—
96	C₄⋊C₈	—	236	2	no	—
97	#103	—	118	2	no	—
98	#97	—	236	2	no	—
99	C₄×C₈	D₂₃₆	1	1	yes	characteristic
100	C₄×C₈	D₂₃₆	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
102	C₂×Q₈	—	118	2	no	—
103	Q₁₆	—	118	2	no	—
104	Q₁₆	—	118	2	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	236	2	no	—
107	C₂×C₈	D₄₇₂	1	1	yes	characteristic
108	C₂×C₈	D₄₇₂	1	1	yes	characteristic
109	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
110	C₂×C₈	#11745685	1	1	yes	characteristic
111	C₄⋊C₄	—	118	2	no	—
112	C₄⋊C₄	—	236	2	no	—
113	C₄²	—	236	2	no	—
114	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
115	C₁₆	—	2	2	no	—
116	C₁₆	—	2	1	no	—
117	C₁₆	#11745659	1	2	yes	—
118	Q₈	—	236	2	no	—
119	Q₈	—	236	2	no	—
120	C₂×C₄	—	236	2	no	—
121	C₂×C₄	—	236	2	no	—
122	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
123	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
124	C₂×C₄	#14273749	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	472	2	no	—
129	C₈	Dic₄₇₂	1	2	yes	complemented
130	C₈	C₂×#11745659	1	1	yes	characteristic
131	C₈	#14273733	1	1	yes	characteristic
132	C₂²	C₄×D₄₇₂	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	472	2	no	—
135	C₄	—	472	2	no	—
136	C₄	#16837481	1	1	yes	characteristic
137	C₄	#16837520	1	1	yes	characteristic
138	C₄	C₂×Dic₄₇₂	1	1	yes	characteristic
139	C₄	#16837834	1	1	yes	characteristic
140	C₂	#21422982	1	1	yes	characteristic
141	C₂	C₄×Dic₄₇₂	1	1	yes	characteristic
142	C₂	#21425978	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₄₇₂

C₂

yes

maximal, characteristic

#21422994

C₂

yes

maximal

#21422168

C₂

yes

maximal, characteristic

#21422061

C₂

yes

maximal

C₈×C₁₆×C₅₉

C₂

yes

maximal, characteristic

C₂×Dic₄₇₂

C₄

yes

characteristic

#16837532

—

C₄×Dic₂₃₆

C₂²

yes

—

#16837228

C₄

yes

characteristic

#16837207

C₄

yes

—

#16837155

C₂²

yes

—

#16837134

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₈²×C₅₉

C₂²

yes

characteristic

C₂×Dic₂₃₆

C₂×C₄

yes

—

C₄×Dic₁₁₈

—

Dic₄₇₂

—

#14273661

C₂×C₄

yes

—

C₄⋊(C₅₉⋊C₈)

—

Dic₅₉⋊C₈

—

#14273652

—

#14273637

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₈

yes

characteristic

C₂×C₈×C₅₉

D₈

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

Dic₂₃₆

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

M₁₆

yes

—

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

Q₃₂

yes

complemented

C₈×C₅₉

C₂×M₁₆

yes

characteristic

C₈×C₅₉

#131

yes

characteristic

C₅₉⋊C₈

—

#12083

—

maximal, Sylow

C₂²×C₅₉

C₄×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂×Q₃₂

yes

characteristic

C₄×C₅₉

#508

yes

characteristic

Dic₅₉

—

Dic₅₉

—

C₄×Q₃₂

—

#1591

—

#1371

—

#1332

—

C₈×C₁₆

D₅₉

yes

characteristic

C₂×C₅₉

#1589

yes

characteristic

C₂×C₅₉

C₄×Q₃₂

yes

characteristic

C₂×C₅₉

#2178

yes

characteristic

C₂×Q₃₂

—

#446

—

118

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₄×C₁₆

C₂×D₅₉

yes

characteristic

#335

—

Q₈⋊C₈

—

118

—

C₈²

C₂×D₅₉

yes

characteristic

C₅₉

#12083

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

118

—

Q₃₂

—

118

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅₉

yes

characteristic

C₂×C₁₆

C₄×D₅₉

yes

characteristic

#107

—

C₄⋊C₈

—

118

—

C₄⋊C₈

—

236

—

#103

—

118

—

#97

—

236

—

C₄×C₈

D₂₃₆

yes

characteristic

100

C₄×C₈

D₂₃₆

yes

characteristic

101

C₄×C₈

C₂²×D₅₉

yes

characteristic

102

C₂×Q₈

—

118

—

103

Q₁₆

—

118

—

104

Q₁₆

—

118

—

105

C₂×C₈

—

106

C₂×C₈

—

236

—

107

C₂×C₈

D₄₇₂

yes

characteristic

108

C₂×C₈

D₄₇₂

yes

characteristic

109

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

110

C₂×C₈

#11745685

yes

characteristic

111

C₄⋊C₄

—

118

—

112

C₄⋊C₄

—

236

—

113

C₄²

—

236

—

114

C₄²

C₂×D₂₃₆

yes

characteristic

115

C₁₆

—

116

C₁₆

—

117

C₁₆

#11745659

yes

—

118

Q₈

—

236

—

119

Q₈

—

236

—

120

C₂×C₄

—

236

—

121

C₂×C₄

—

236

—

122

C₂×C₄

C₄×D₂₃₆

yes

characteristic

123

C₂×C₄

C₂×D₄₇₂

yes

characteristic

124

C₂×C₄

#14273749

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

472

—

129

C₈

Dic₄₇₂

yes

complemented

130

C₈

C₂×#11745659

yes

characteristic

131

C₈

#14273733

yes

characteristic

132

C₂²

C₄×D₄₇₂

yes

characteristic

133

C₄

—

134

C₄

—

472

—

135

C₄

—

472

—

136

C₄

#16837481

yes

characteristic

137

C₄

#16837520

yes

characteristic

138

C₄

C₂×Dic₄₇₂

yes

characteristic

139

C₄

#16837834

yes

characteristic

140

C₂

#21422982

yes

characteristic

141

C₂

C₄×Dic₄₇₂

yes

characteristic

142

C₂

#21425978

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₈×C₁₆×C₅₉	C₂
socle, prime radical	C₂²×C₅₉	C₄×D₈
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₈×C₁₆	D₅₉
hypocenter	C₅₉	#12083
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₄₇₂

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₈×C₁₆×C₅₉

C₂

socle, prime radical

C₂²×C₅₉

C₄×D₈

cosocle

C₄×C₈×C₅₉

C₂³

hypercenter

C₈×C₁₆

D₅₉

hypocenter

C₅₉

#12083

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
56. C₈×C₅₉	Q₃₂	semidirect
87. C₅₉	#12083	semidirect
129. C₈	Dic₄₇₂	semidirect

H ≅ G/N

type

56. C₈×C₅₉

Q₃₂

semidirect

87. C₅₉

#12083

semidirect

129. C₈

Dic₄₇₂

semidirect

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

subgroup	index
C₄×Dic₄₇₂	2
#21422994	2
#21422168	2
#21422061	2
C₈×C₁₆×C₅₉	2
#12083	59

subgroup

index

C₄×Dic₄₇₂

C₈×C₁₆×C₅₉

Subgroups — #28500681

Conjugacy classes of subgroups

Complements

Maximal subgroups