Finite Group Databasebeta

← #28250849

Subgroups — #28250849

Order 13568.

4732 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 4732 conjugacy classes of subgroups — too many to draw. Only its 807 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#20288363C213yesmaximal, complemented
2#20284121C2112yesmaximal, complemented
3#20284107C2112yesmaximal, complemented
4#20283179C213yesmaximal, complemented
5C53×2−1+6C211yesmaximal, characteristic, complemented
6C22×Q8×D53C2211yescharacteristic, complemented
7Q8×C22⋊D53—212no—
8#16811280—212no—
9#16811194C22118yescomplemented
10#16811130C22118yescomplemented
11#16811101C2213yescomplemented
12#16810813C22148yescomplemented
13#16810810C22116yescomplemented
14#16810597C22136yescomplemented
15C53×#585—212no—
16C53×#585C22112yes—
17C2×2−1+4×C53—212no—
18C2×2−1+4×C53C2213yes—
19C22×Dic106C2313yes—
20C2×Q8×D53—212no—
21C22×C4×D53C2313yes—
22#13133215—218no—
23#13133204—218no—
24#13133198C23136yes—
25#13133196—236no—
26#13133193—218no—
27#13133188—218no—
28#13133185C23112yes—
29Q8×Dic53—212no—
30C4×C22⋊D53—236no—
31#13133176C23118yes—
32#13133132C23148yes—
332−1+4×C53—248no—
342−1+4×C53—236no—
352−1+4×C53—224no—
36H(5,𝔽2)×C53—216no—
37H(5,𝔽2)×C53—236no—
38H(5,𝔽2)×C53C23116yes—
39C2×#41×C53—248no—
40C2×#41×C53—272no—
41C2×#41×C53D4112yescomplemented
42C2×#41×C53C23118yes—
43C22×Q8×C53—216no—
44C22×Q8×C53D4112yescomplemented
45C22×Q8×C53C2311yescharacteristic
46C23×D53C2411yescharacteristic
47#41×C53—2144no—
48#41×C53—248no—
49#41×C53—248no—
50#41×C53—248no—
51#41×C53—272no—
52C2×Q8×C53—272no—
53C2×Q8×C53—248no—
54C2×Q8×C53C2×D4118yes—
55C2×Q8×C53C2×D4112yes—
56C2×D4×C53—272no—
57C2×D4×C53—248no—
58C2×D4×C53C2×D4118yes—
59C2×D4×C53C24112yes—
60C22×C4×C53—248no—
61C22×C4×C53C2×D4136yes—
62C22×C4×C53C2413yes—
63C2×C22⋊D53—212no—
64C22×Dic53C2413yes—
65Q8×D53—416no—
66C2×C4×D53—218no—
67C2×Dic106—218no—
68C22⋊Dic53—236no—
69C22⋊Dic53C24112yes—
70#11719921—212no—
71C4⋊Dic53—218no—
72#11719919—236no—
73C4×Dic53—218no—
74C23×C53—216no—
75C23×C53C22×D4112yes—
76C23×C53C2511yescharacteristic
77C22×D53—23no—
78C22×D53—42no—
79Q8×C53—216no—
80Q8×C53—236no—
81Q8×C53#120116yes—
82D4×C53—248no—
83D4×C53—224no—
84D4×C53—236no—
85C2×C4×C53—248no—
86C2×C4×C53—272no—
87C2×C4×C53C22×D4112yes—
88C2×C4×C53C22×D4118yes—
89C22⋊D53—424no—
90C2×Dic53—29no—
91C2×Dic53—212no—
92C4×D53—412no—
93Dic106—412no—
94#60400—531nomaximal, Sylow
95C22×C53—212no—
96C22×C53—41no—
97C22×C53—412no—
98C22×C53—816no—
99C22×C53C23×D413yes—
100C2×D53—26no—
101C2×D53—43no—
102C2×D53—41no—
103C4×C53—212no—
104C4×C53C2×#120112yes—
105Dic53—412no—
106Dic53—43no—
1072−1+6D5311yescharacteristic, complemented
108#3486—533no—
109#3028—5312no—
110#3025—5312no—
111#2884—533no—
112C2×C53—23no—
113C2×C53—412no—
114C2×C53C22×#12011yescharacteristic
115D53—42no—
116#585—212no—
117#585C2×D53112yes—
118C2×2−1+4—212no—
119C2×2−1+4C2×D5313yes—
120C23×Q8—531no—
121D4×Q8—10612no—
122#548—10612no—
123#536—5318no—
124#525—5318no—
125#519—533no—
126#458—5348no—
127B(4,𝔽2)—5316no—
128#410—5336no—
129C53#6040011yescharacteristic, complemented, Sylow
1302−1+4—248no—
1312−1+4—236no—
1322−1+4—224no—
133H(5,𝔽2)—216no—
134H(5,𝔽2)—236no—
135H(5,𝔽2)C22×D53116yes—
136C2×#41—248no—
137C2×#41—272no—
138C2×#41C22⋊D53112yescomplemented
139C2×#41C22×D53118yes—
140C22×Q8—216no—
141C22×Q8—533no—
142C22×Q8—10612no—
143C22×Q8C22⋊D53112yescomplemented
144C22×Q8C22×D5311yescharacteristic
145C23×C4—533no—
146C4⋊Q8—10618no—
147#124—10618no—
148#123—5336no—
149C22⋊Q8—10618no—
150C22⋊Q8—10636no—
151#121—10618no—
152#120—5312no—
153C4×Q8—10612no—
154C4×D4—10636no—
155#117—5318no—
156#99—5348no—
157C24—531no—
158#41—2144no—
159#41—248no—
160#41—248no—
161#41—248no—
162#41—272no—
163C2×Q8—272no—
164C2×Q8—248no—
165C2×Q8—10618no—
166C2×Q8—21216no—
167C2×Q8C2×C22⋊D53118yes—
168C2×Q8C2×C22⋊D53112yes—
169C2×D4—272no—
170C2×D4—248no—
171C2×D4—10612no—
172C2×D4C2×C22⋊D53118yes—
173C2×D4C23×D53112yes—
174C22×C4—248no—
175C22×C4—533no—
176C22×C4—10618no—
177C22×C4C2×C22⋊D53136yes—
178C22×C4C23×D5313yes—
179C4⋊C4—10618no—
180C4⋊C4—10636no—
181C22⋊C4—5312no—
182C22⋊C4—10636no—
183C22⋊C4—10612no—
184C42—10618no—
185C23—216no—
186C23—1063no—
187C23—2122no—
188C23C22×C22⋊D53112yes—
189C23C24×D5311yescharacteristic
190Q8—216no—
191Q8—236no—
192Q8—21212no—
193Q8#13133186116yes—
194D4—248no—
195D4—224no—
196D4—236no—
197D4—21224no—
198C2×C4—248no—
199C2×C4—272no—
200C2×C4—10612no—
201C2×C4—1069no—
202C2×C4—21212no—
203C2×C4C22×C22⋊D53112yes—
204C2×C4C22×C22⋊D53118yes—
205C22—212no—
206C22—41no—
207C22—412no—
208C22—816no—
209C22—1066no—
210C22—2123no—
211C22—2121no—
212C22C23×C22⋊D5313yes—
213C4—212no—
214C4—21212no—
215C4—2123no—
216C4C2×#13133186112yes—
217C2—23no—
218C2—412no—
219C2—2122no—
220C2C22×#1313318611yescharacteristic
subgroupG/H
centerC2C22×#13133186
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC53×2−1+6C2
socleC2×C53C22×#120
hypercenter2−1+6D53
hypocenterC53#60400
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×Q8 ◁ 2−1+6

Complements

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

NH ≅ G/Ntype
1. #20288363C2semidirect
2. #20284121C2semidirect
3. #20284107C2semidirect
4. #20283179C2semidirect
5. C53×2−1+6C2semidirect
6. C22×Q8×D53C22semidirect
9. #16811194C22semidirect
10. #16811130C22semidirect
11. #16811101C22semidirect
12. #16810813C22semidirect
13. #16810810C22semidirect
14. #16810597C22semidirect
41. C2×#41×C53D4semidirect
44. C22×Q8×C53D4semidirect
107. 2−1+6D53semidirect
129. C53#60400semidirect
138. C2×#41C22⋊D53semidirect
143. C22×Q8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202883632
#202841212
#202841072
#202831792
C53×2−1+62
#6040053