Finite Group Databasebeta

← #27590515

Subgroups — #27590515

Order 13568.

1778 conjugacy classes of subgroups; only the 539 normal subgroups are shown.

normal subgroups only

This group has 1778 conjugacy classes of subgroups — too many to draw. Only its 539 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20282979C213yesmaximal
2#20282975C2112yesmaximal, complemented
3#20277982C2112yesmaximal, complemented
4#20277093C213yesmaximal
5C53×#2840C211yesmaximal, characteristic
6C24×Dic53C2211yescharacteristic
7C2×Q8×Dic53C2214yescomplemented
8C2×#13133173C2216yes—
9#16810563C22112yescomplemented
10#16810559C22112yes—
11#16810557C22124yescomplemented
12#16810546C22124yes—
13#16810542C22112yescomplemented
14#16810484C2216yes—
15#16810481C22112yes—
16#16810479C2213yes—
17#16810474C22112yescomplemented
18C22⋊C4×Dic53C22112yes—
19C53×#393C22112yes—
20C53×#379C2213yes—
21C23×Dic53—44no—
22C23×Dic53Q818yescomplemented
23C23×Dic53C2313yes—
24C23×Dic53C2313yes—
25Q8×Dic53—48no—
26#13133173—412no—
27C2×C4⋊Dic53C23112yes—
28C2×#11719919C23124yes—
29C2×#11719919C23112yes—
30C2×C22⋊Dic53C2316yes—
31C2×C22⋊Dic53C23112yes—
32C2×C4×Dic53C23112yes—
33#13133124C23112yes—
34#13133124C23124yes—
35C25×C53C2311yescharacteristic
36C22×Q8×C53C2314yes—
37C2×C22⋊C4×C53C23112yes—
38C2×C22⋊C4×C53C2316yes—
39#95×C53C23112yes—
40C24×C53—44no—
41C24×C53—42no—
42C24×C53C2×Q814yes—
43C24×C53C2413yes—
44C2×Q8×C53—48no—
45C2×Q8×C53—44no—
46C22×C4×C53C24112yes—
47C22×Dic53—212no—
48C22×Dic53—416no—
49C22×Dic53—46no—
50C22×Dic53C2×Q8112yes—
51C22×Dic53C2×Q8112yes—
52C22×Dic53C24112yes—
53C22×Dic53C2413yes—
54C22×Dic53C2411yescharacteristic
55C22⋊C4×C53—412no—
56C22⋊C4×C53—46no—
57C22⋊Dic53—412no—
58C22⋊Dic53—46no—
59C4⋊Dic53—412no—
60#11719919—424no—
61#11719919—412no—
62C4×Dic53—224no—
63C23×C53—26no—
64C23×C53—212no—
65C23×C53—44no—
66C23×C53—48no—
67C23×C53—416no—
68C23×C53C22×Q816yes—
69C23×C53C2511yescharacteristic
70Q8×C53—416no—
71C2×C4×C53—212no—
72C2×C4×C53—224no—
73C2×Dic53—26no—
74C2×Dic53—224no—
75C2×Dic53—212no—
76C2×Dic53—224no—
77C2×Dic53—412no—
78C2×Dic53C22×Q816yes—
79C2×Dic53C22×Q812yes—
80C2×Dic53H(5,𝔽2)14yes—
81#36674—531nomaximal, Sylow
82C22×C53—26no—
83C22×C53—212no—
84C22×C53—416no—
85C22×C53—412no—
86C22×C53C23×Q811yescharacteristic
87C22×C53C2×H(5,𝔽2)12yes—
88C22×C53#58514yes—
89C4×C53—412no—
90Dic53—26no—
91Dic53—412no—
92Dic53#54314yes—
93#2842—533no—
94#2841—5312no—
95#2840D5311yescharacteristic
96#2396—5312no—
97#2280—533no—
98C2×C53—46no—
99C2×C53C2×#54311yescharacteristic
100C2×C53#345812yes—
101C2×C53#352914yes—
102C24×C4—531no—
103C2×C4×Q8—534no—
104C2×#117—536no—
105#400—5312no—
106#399—5312no—
107#398—5324no—
108#396—5324no—
109#395—5312no—
110#393C2×D53112yes—
111#381—536no—
112#380—5312no—
113#379—533no—
114#379C2×D5313yes—
115C4×C4⋊C4—5312no—
116C4×C22⋊C4—5312no—
117C53#3667411yescharacteristic, complemented, Sylow
118C25C22×D5311yescharacteristic
119C22×Q8C22×D5314yes—
120C23×C4—533no—
121C23×C4—538no—
122C23×C4—533no—
123C23×C4—2124no—
124C4×Q8—2128no—
125#117—21212no—
126C2×C4⋊C4—5312no—
127C2×C4⋊C4—5312no—
128C2×C4⋊C4—5324no—
129C2×C22⋊C4—536no—
130C2×C22⋊C4—5312no—
131C2×C22⋊C4C22×D5316yes—
132C2×C22⋊C4C22×D53112yes—
133C2×C42—5312no—
134#95—5324no—
135#95—5312no—
136#95C22×D53112yes—
137C24—42no—
138C24—44no—
139C24Q8×D5314yes—
140C24C23×D5313yes—
141C2×Q8—44no—
142C2×Q8—48no—
143C22×C4—531no—
144C22×C4—5312no—
145C22×C4—533no—
146C22×C4—5312no—
147C22×C4—5312no—
148C22×C4—10612no—
149C22×C4—21216no—
150C22×C4—2126no—
151C22×C4C23×D53112yes—
152C4⋊C4—21212no—
153C4⋊C4—21212no—
154C4⋊C4—21224no—
155C22⋊C4—46no—
156C22⋊C4—412no—
157C22⋊C4—2126no—
158C22⋊C4—21212no—
159C42—10624no—
160C23—26no—
161C23—212no—
162C23—44no—
163C23—48no—
164C23—416no—
165C23C2×Q8×D5316yes—
166C23C24×D5311yescharacteristic
167Q8—416no—
168C2×C4—212no—
169C2×C4—224no—
170C2×C4—532no—
171C2×C4—534no—
172C2×C4—536no—
173C2×C4—10624no—
174C2×C4—1066no—
175C2×C4—10612no—
176C2×C4—10624no—
177C2×C4—21212no—
178C22—26no—
179C22—212no—
180C22—416no—
181C22—412no—
182C22C22×Q8×D5311yescharacteristic
183C22H(5,𝔽2)×D5312yes—
184C22#1681145814yes—
185C4—412no—
186C4—534no—
187C4—1066no—
188C4—21212no—
189C2—46no—
190C2#543×D5311yescharacteristic
191C2#2028811712yes—
192C2#2028885514yes—
subgroupG/H
center, FrattiniC23C24×D53
commutator, socle, cosocleC23×C53C25
FittingC53×#2840C2
prime radicalC25×C53C23
hypercenter#2840D53
hypocenterC53#36674
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #2840

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20282975C2semidirect
3. #20277982C2semidirect
7. C2×Q8×Dic53C22semidirect
9. #16810563C22semidirect
11. #16810557C22semidirect
13. #16810542C22semidirect
17. #16810474C22semidirect
22. C23×Dic53Q8semidirect
117. C53#36674semidirect

Maximal subgroups

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

subgroupindex
#202829792
#202829752
#202779822
#202770932
C53×#28402
#3667453