Group #28503071 — subgroups

Subgroups — #28503071

Order 15104.

188 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 67 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	#21426371	C₂	1	1	yes	maximal, characteristic
2	#21426004	C₂	1	1	yes	maximal, characteristic
3	#21423501	C₂	1	2	yes	maximal
4	#21422155	C₂	1	2	yes	maximal
5	C₅₉×#1372	C₂	1	1	yes	maximal, characteristic
6	#16837807	C₂²	1	2	yes	—
7	#16837698	—	2	2	no	—
8	C₈×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	#16837228	—	2	1	no	—
10	#16837228	—	2	1	no	—
11	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
12	Dic₅₉⋊C₁₆	—	2	1	no	—
13	#16837177	—	2	1	no	—
14	#16837150	—	2	2	no	—
15	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#335	C₂²	1	2	yes	—
17	#14273727	—	2	2	no	—
18	C₄×Dic₁₁₈	D₄	1	1	yes	characteristic
19	C₂×C₅₉⋊C₁₆	—	2	1	no	—
20	#14273661	—	2	2	no	—
21	#14273661	—	2	2	no	—
22	#14273661	—	4	2	no	—
23	C₄⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
24	Dic₅₉⋊C₈	—	2	1	no	—
25	#14273652	—	4	2	no	—
26	C₈×Dic₅₉	—	2	1	no	—
27	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₅₉	—	2	2	no	—
30	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₁₈	—	4	2	no	—
32	C₂×Dic₁₁₈	D₈	1	1	yes	characteristic
33	C₂×C₈×C₅₉	—	4	2	no	—
34	C₂×C₈×C₅₉	C₂×D₄	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	—	4	2	no	—
38	C₄⋊Dic₅₉	—	4	2	no	—
39	C₄⋊Dic₅₉	D₈	1	1	yes	characteristic
40	Dic₅₉⋊C₄	—	2	1	no	—
41	C₄×Dic₅₉	—	2	1	no	—
42	C₂×C₅₉⋊C₈	—	2	1	no	—
43	C₁₆×C₅₉	—	2	1	no	—
44	C₁₆×C₅₉	Q₁₆	1	2	yes	—
45	C₅₉⋊C₁₆	—	4	1	no	—
46	C₂×C₄×C₅₉	#121	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	—	2	1	no	—
50	C₂×Dic₅₉	—	2	1	no	—
51	C₂×Dic₅₉	—	8	2	no	—
52	Dic₁₁₈	—	2	2	no	—
53	Dic₁₁₈	—	16	2	no	—
54	C₈×C₅₉	—	2	1	no	—
55	C₈×C₅₉	—	8	2	no	—
56	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
57	C₈×C₅₉	#137	1	1	yes	characteristic
58	C₅₉⋊C₈	—	4	1	no	—
59	#12461	—	59	1	no	maximal, Sylow
60	C₂²×C₅₉	#466	1	1	yes	characteristic
61	C₄×C₅₉	—	2	1	no	—
62	C₄×C₅₉	#462	1	1	yes	characteristic
63	C₄×C₅₉	#473	1	1	yes	characteristic
64	C₄×C₅₉	#510	1	2	yes	—
65	Dic₅₉	—	4	1	no	—
66	Dic₅₉	—	4	1	no	—
67	Dic₅₉	—	16	2	no	—
68	#2252	—	59	1	no	—
69	#2183	—	59	1	no	—
70	#1676	—	59	2	no	—
71	#1372	D₅₉	1	1	yes	characteristic
72	#1365	—	59	2	no	—
73	C₂×C₅₉	#1674	1	1	yes	characteristic
74	C₂×C₅₉	#2219	1	1	yes	characteristic
75	C₂×C₅₉	#2221	1	1	yes	characteristic
76	#500	—	59	2	no	—
77	#477	—	118	2	no	—
78	C₈×Q₈	—	59	1	no	—
79	#366	—	118	1	no	—
80	#366	—	118	1	no	—
81	C₄⋊C₁₆	—	59	1	no	—
82	C₄⋊C₁₆	—	118	1	no	—
83	#346	—	118	1	no	—
84	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
85	#335	C₂×D₅₉	1	2	yes	—
86	#333	—	118	2	no	—
87	C₅₉	#12461	1	1	yes	characteristic, complemented, Sylow
88	C₄⋊Q₈	—	118	2	no	—
89	C₄×Q₈	—	59	1	no	—
90	C₂×C₁₆	—	118	1	no	—
91	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
92	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
93	#107	—	118	2	no	—
94	#107	—	118	2	no	—
95	#107	—	236	2	no	—
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	59	1	no	—
98	C₄⋊C₈	—	118	1	no	—
99	#103	—	236	2	no	—
100	C₄×C₈	—	118	1	no	—
101	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
102	C₂×Q₈	—	59	1	no	—
103	C₂×Q₈	—	236	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	118	1	no	—
106	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
107	C₂×C₈	#11745689	1	1	yes	characteristic
108	C₄⋊C₄	—	59	1	no	—
109	C₄⋊C₄	—	118	1	no	—
110	C₄⋊C₄	—	236	2	no	—
111	C₄⋊C₄	—	236	2	no	—
112	C₄²	—	118	1	no	—
113	C₄²	D₄×D₅₉	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	236	1	no	—
116	C₁₆	Dic₂₃₆	1	2	yes	—
117	Q₈	—	118	2	no	—
118	Q₈	—	944	2	no	—
119	C₂×C₄	—	118	1	no	—
120	C₂×C₄	—	118	1	no	—
121	C₂×C₄	—	472	2	no	—
122	C₂×C₄	#14273702	1	1	yes	characteristic
123	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
124	C₂×C₄	#14273750	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	8	2	no	—
127	C₈	—	236	1	no	—
128	C₈	C₂×Dic₂₃₆	1	1	yes	characteristic
129	C₈	#14273760	1	1	yes	characteristic
130	C₂²	#16837639	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	944	2	no	—
133	C₄	—	236	1	no	—
134	C₄	—	236	1	no	—
135	C₄	#16837613	1	1	yes	characteristic
136	C₄	#16837678	1	1	yes	characteristic
137	C₄	#16837841	1	2	yes	—
138	C₂	#21423491	1	1	yes	characteristic
139	C₂	#21426205	1	1	yes	characteristic
140	C₂	#21426217	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426371

C₂

yes

maximal, characteristic

#21426004

C₂

yes

maximal, characteristic

#21423501

C₂

yes

maximal

#21422155

C₂

yes

maximal

C₅₉×#1372

C₂

yes

maximal, characteristic

#16837807

C₂²

yes

—

#16837698

—

C₈×Dic₁₁₈

C₂²

yes

characteristic

#16837228

—

#16837228

—

C₄⋊(C₅₉⋊C₁₆)

C₂²

yes

characteristic

Dic₅₉⋊C₁₆

—

#16837177

—

#16837150

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₅₉×#335

C₂²

yes

—

#14273727

—

C₄×Dic₁₁₈

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

#14273661

—

#14273661

—

#14273661

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

characteristic

Dic₅₉⋊C₈

—

#14273652

—

C₈×Dic₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

D₈

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₈

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₁₆×C₅₉

Q₁₆

yes

—

C₅₉⋊C₁₆

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₁₆

yes

characteristic

C₈×C₅₉

#137

yes

characteristic

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

—

—

D₅₉

yes

characteristic

#1365

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

118

—

C₈×Q₈

—

#366

—

118

—

#366

—

118

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

118

—

#346

—

118

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

#335

C₂×D₅₉

yes

—

#333

—

118

—

C₅₉

#12461

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

118

—

C₄×Q₈

—

C₂×C₁₆

—

118

—

C₂×C₁₆

D₂₃₆

yes

characteristic

C₂×C₁₆

D₂₃₆

yes

characteristic

#107

—

118

—

#107

—

118

—

#107

—

236

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

#103

—

236

—

100

C₄×C₈

—

118

—

101

C₄×C₈

C₂²×D₅₉

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

236

—

104

C₂×C₈

—

105

C₂×C₈

—

118

—

106

C₂×C₈

C₂×D₂₃₆

yes

characteristic

107

C₂×C₈

#11745689

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

118

—

110

C₄⋊C₄

—

236

—

111

C₄⋊C₄

—

236

—

112

C₄²

—

118

—

113

C₄²

D₄×D₅₉

yes

characteristic

114

C₁₆

—

115

C₁₆

—

236

—

116

C₁₆

Dic₂₃₆

yes

—

117

Q₈

—

118

—

118

Q₈

—

944

—

119

C₂×C₄

—

118

—

120

C₂×C₄

—

118

—

121

C₂×C₄

—

472

—

122

C₂×C₄

#14273702

yes

characteristic

123

C₂×C₄

D₈×D₅₉

yes

characteristic

124

C₂×C₄

#14273750

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

236

—

128

C₈

C₂×Dic₂₃₆

yes

characteristic

129

C₈

#14273760

yes

characteristic

130

C₂²

#16837639

yes

characteristic

131

C₄

—

132

C₄

—

944

—

133

C₄

—

236

—

134

C₄

—

236

—

135

C₄

#16837613

yes

characteristic

136

C₄

#16837678

yes

characteristic

137

C₄

#16837841

yes

—

138

C₂

#21423491

yes

characteristic

139

C₂

#21426205

yes

characteristic

140

C₂

#21426217

yes

characteristic

	subgroup	G/H
center	C₂²	#16837639
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1372	C₂
socle, prime radical	C₂²×C₅₉	#466
hypercenter	#1372	D₅₉
hypocenter	C₅₉	#12461
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂²

#16837639

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1372

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
87. C₅₉	#12461	semidirect

H ≅ G/N

type

87. C₅₉

#12461

semidirect

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

subgroup	index
#21426371	2
#21426004	2
#21423501	2
#21422155	2
C₅₉×#1372	2
#12461	59

subgroup

index

C₅₉×#1372

Subgroups — #28503071

Conjugacy classes of subgroups

Complements

Maximal subgroups