Finite Group Databasebeta

← #28250855

Subgroups — #28250855

Order 13568.

4854 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 4854 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#20288355C213yesmaximal, complemented
2#20284121C214yesmaximal, complemented
3#20284112C218yesmaximal, complemented
4#20284107C214yesmaximal, complemented
5C2×#16810813C218yesmaximal, complemented
6#20283176C213yesmaximal, complemented
7C53×H(7,𝔽2)C211yesmaximal, characteristic, complemented
8C22×#11719947C2211yescharacteristic, complemented
9#16811272—212no—
10#16811267—212no—
11#16811194C2216yescomplemented
12#16811182C22112yescomplemented
13#16811130C2216yescomplemented
14C2×#13133198C22112yescomplemented
15#16811098C2213yescomplemented
16#16810813C22116yescomplemented
17#16810813C22132yescomplemented
18#16810810C22116yescomplemented
19#16810597C22112yescomplemented
20C2×#13133132C22124yescomplemented
21C53×#585—212no—
22C53×#585C2214yes—
23C2×H(5,𝔽2)×C53—212no—
24C2×H(5,𝔽2)×C53C2218yes—
25C2×H(5,𝔽2)×C53C2213yes—
26C2×#11719947—212no—
27C22×Dic106C2311yescharacteristic, complemented
28C22×C22⋊D53C2312yescomplemented
29C23×Dic53C2312yescomplemented
30C22×C4×D53C2311yescharacteristic, complemented
31#13133215—26no—
32#13133204—26no—
33#13133200—212no—
34#13133198—224no—
35#13133198C23112yescomplemented
36#13133198C23124yescomplemented
37#13133196—212no—
38#13133193—26no—
39#13133189—224no—
40#13133188—26no—
41#13133186—212no—
42#13133185C23112yescomplemented
43D4×Dic53—212no—
44C4×C22⋊D53—212no—
45#13133176C2316yescomplemented
46C2×#11719919—212no—
47C2×C22⋊Dic53—212no—
48C2×C22⋊Dic53C23112yescomplemented
49#13133132C23148yescomplemented
502−1+4×C53—216no—
512−1+4×C53—212no—
52H(5,𝔽2)×C53—248no—
53H(5,𝔽2)×C53—212no—
54H(5,𝔽2)×C53—248no—
55H(5,𝔽2)×C53—224no—
56H(5,𝔽2)×C53C23116yes—
57C2×#41×C53—248no—
58C2×#41×C53—224no—
59C2×#41×C53—224no—
60C2×#41×C53D4112yescomplemented
61C2×#41×C53C2316yes—
62C22×D4×C53—216no—
63C22×D4×C53—224no—
64C22×D4×C53D4112yescomplemented
65C22×D4×C53C23112yes—
66C22×D4×C53C2311yescharacteristic
67C24×C53—28no—
68C24×C53C2×D4112yes—
69C24×C53C2412yes—
70C23×D53C2411yescharacteristic
71#41×C53—296no—
72#41×C53—248no—
73#41×C53—216no—
74#41×C53—248no—
75#41×C53—248no—
76#41×C53—224no—
77C2×Q8×C53—224no—
78C2×Q8×C53—28no—
79C2×Q8×C53C2×D416yes—
80C2×D4×C53—296no—
81C2×D4×C53—224no—
82C2×D4×C53—224no—
83C2×D4×C53—248no—
84C2×D4×C53—248no—
85C2×D4×C53—248no—
86C2×D4×C53C2×D416yes—
87C2×D4×C53C2×D4124yes—
88C2×D4×C53C2×D4112yes—
89C2×D4×C53C24112yes—
90C22×C4×C53—224no—
91C22×C4×C53—216no—
92C22×C4×C53C2×D4112yes—
93C22×C4×C53C2×D4112yes—
94C22×C4×C53C2411yescharacteristic
95C2×C22⋊D53—212no—
96C2×C22⋊D53—212no—
97C22×Dic53—212no—
98C22×Dic53—212no—
99C22×Dic53C2411yescharacteristic
100C22×Dic53C2412yes—
101#11719947—416no—
102C2×C4×D53—26no—
103C2×Dic106—26no—
104C22⋊Dic53—212no—
105C22⋊Dic53—224no—
106C22⋊Dic53—212no—
107C22⋊Dic53C24112yes—
108#11719921—212no—
109C4⋊Dic53—26no—
110#11719919—212no—
111#11719919—224no—
112C4×Dic53—26no—
113C23×C53—216no—
114C23×C53—224no—
115C23×C53—82no—
116C23×C53—812no—
117C23×C53—168no—
118C23×C53C22×D4112yes—
119C23×C53C22×D4112yes—
120C23×C53C2511yescharacteristic
121C22×D53—23no—
122C22×D53—81no—
123Q8×C53—216no—
124Q8×C53—212no—
125D4×C53—248no—
126D4×C53—248no—
127D4×C53—224no—
128D4×C53—212no—
129D4×C53#120116yes—
130C2×C4×C53—248no—
131C2×C4×C53—224no—
132C2×C4×C53—224no—
133C2×C4×C53C22×D4112yes—
134C2×C4×C53C22×D416yes—
135C22⋊D53—48no—
136C22⋊D53—812no—
137C2×Dic53—26no—
138C2×Dic53—23no—
139C2×Dic53—212no—
140C2×Dic53—48no—
141C2×Dic53—812no—
142C4×D53—44no—
143Dic106—44no—
144#60401—531nomaximal, Sylow
145C22×C53—212no—
146C22×C53—412no—
147C22×C53—41no—
148C22×C53—412no—
149C22×C53—824no—
150C22×C53—816no—
151C22×C53C2×#12018yes—
152C22×C53C23×D413yes—
153C2×D53—46no—
154C2×D53—41no—
155C4×C53—212no—
156C4×C53C2×#12014yes—
157Dic53—412no—
158Dic53—42no—
159Dic53—41no—
160H(7,𝔽2)D5311yescharacteristic, complemented
161#3485—533no—
162#3028—534no—
163#3026—538no—
164#3025—534no—
165C2×#458—538no—
166#2883—533no—
167C2×C53—23no—
168C2×C53—28no—
169C2×C53—412no—
170C2×C53C22×#12011yescharacteristic
171D53—81no—
172#585—212no—
173#585C2×D5314yes—
174C2×H(5,𝔽2)—212no—
175C2×H(5,𝔽2)C2×D5318yes—
176C2×H(5,𝔽2)C2×D5313yes—
177C22×#41—531no—
178#547—10612no—
179#546—10612no—
180#536—536no—
181#534—5312no—
182#525—536no—
183C2×#123—5312no—
184#518—533no—
185#458—5316no—
186#458—5332no—
187B(4,𝔽2)—5316no—
188#410—5312no—
189C2×#99—5324no—
190C53#6040111yescharacteristic, complemented, Sylow
1912−1+4—216no—
1922−1+4—212no—
193H(5,𝔽2)—248no—
194H(5,𝔽2)—212no—
195H(5,𝔽2)—248no—
196H(5,𝔽2)—224no—
197H(5,𝔽2)C22×D53116yes—
198C2×#41—248no—
199C2×#41—224no—
200C2×#41—224no—
201C2×#41—10612no—
202C2×#41C22⋊D53112yescomplemented
203C2×#41C22×D5316yes—
204C22×Q8—531no—
205C22×D4—216no—
206C22×D4—224no—
207C22×D4—532no—
208C22×D4C22⋊D53112yescomplemented
209C22×D4C22×D53112yes—
210C22×D4C22×D5311yescharacteristic
211C23×C4—532no—
212C23×C4—531no—
213C4⋊Q8—1066no—
214#124—1066no—
215#123—5312no—
216#123—5324no—
217#123—10624no—
218#123—10612no—
219C22⋊Q8—1066no—
220C22⋊Q8—10612no—
221#121—1066no—
222#121—10624no—
223#120—5312no—
224#120—10612no—
225C4×D4—10612no—
226C4×D4—10612no—
227#117—536no—
228C2×C4⋊C4—10612no—
229C2×C22⋊C4—5312no—
230C2×C22⋊C4—10612no—
231#99—5348no—
232C24—28no—
233C24—531no—
234C24C2×C22⋊D53112yes—
235C24C23×D5312yes—
236#41—296no—
237#41—248no—
238#41—216no—
239#41—248no—
240#41—248no—
241#41—224no—
242#41—21216no—
243C2×Q8—224no—
244C2×Q8—28no—
245C2×Q8—1066no—
246C2×Q8C2×C22⋊D5316yes—
247C2×D4—296no—
248C2×D4—224no—
249C2×D4—224no—
250C2×D4—248no—
251C2×D4—248no—
252C2×D4—248no—
253C2×D4—10612no—
254C2×D4—10612no—
255C2×D4C2×C22⋊D53124yes—
256C2×D4C2×C22⋊D5316yes—
257C2×D4C2×C22⋊D53112yes—
258C2×D4C23×D53112yes—
259C22×C4—224no—
260C22×C4—216no—
261C22×C4—531no—
262C22×C4—532no—
263C22×C4—10612no—
264C22×C4—10612no—
265C22×C4—1066no—
266C22×C4C2×C22⋊D53112yes—
267C22×C4C2×C22⋊D53112yes—
268C22×C4C23×D5311yescharacteristic
269C4⋊C4—10612no—
270C4⋊C4—10624no—
271C4⋊C4—1066no—
272C22⋊C4—5312no—
273C22⋊C4—10612no—
274C22⋊C4—10612no—
275C22⋊C4—10624no—
276C22⋊C4—10612no—
277C42—1066no—
278C23—216no—
279C23—224no—
280C23—82no—
281C23—812no—
282C23—168no—
283C23—1063no—
284C23—4241no—
285C23C22×C22⋊D53112yes—
286C23C22×C22⋊D53112yes—
287C23C24×D5311yescharacteristic
288Q8—216no—
289Q8—212no—
290Q8—2124no—
291D4—248no—
292D4—248no—
293D4—224no—
294D4—212no—
295D4—42412no—
296D4—2128no—
297D4#13133186116yes—
298C2×C4—248no—
299C2×C4—224no—
300C2×C4—224no—
301C2×C4—10612no—
302C2×C4—1063no—
303C2×C4—1066no—
304C2×C4—42412no—
305C2×C4—2128no—
306C2×C4—2124no—
307C2×C4C22×C22⋊D53112yes—
308C2×C4C22×C22⋊D5316yes—
309C22—212no—
310C22—412no—
311C22—41no—
312C22—412no—
313C22—824no—
314C22—816no—
315C22—2126no—
316C22—2121no—
317C22C2×#1313318618yes—
318C22C23×C22⋊D5313yes—
319C4—212no—
320C4—21212no—
321C4—2121no—
322C4—2122no—
323C4C2×#1313318614yes—
324C2—23no—
325C2—28no—
326C2—412no—
327C2—4241no—
328C2C22×#1313318611yescharacteristic
subgroupG/H
centerC2C22×#13133186
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC53×H(7,𝔽2)C2
socleC2×C53C22×#120
hypercenterH(7,𝔽2)D53
hypocenterC53#60401
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×D4 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20288355C2semidirect
2. #20284121C2semidirect
3. #20284112C2semidirect
4. #20284107C2semidirect
5. C2×#16810813C2semidirect
6. #20283176C2semidirect
7. C53×H(7,𝔽2)C2semidirect
8. C22×#11719947C22semidirect
11. #16811194C22semidirect
12. #16811182C22semidirect
13. #16811130C22semidirect
14. C2×#13133198C22semidirect
15. #16811098C22semidirect
16. #16810813C22semidirect
17. #16810813C22semidirect
18. #16810810C22semidirect
19. #16810597C22semidirect
20. C2×#13133132C22semidirect
27. C22×Dic106C23semidirect
28. C22×C22⋊D53C23semidirect
29. C23×Dic53C23semidirect
30. C22×C4×D53C23semidirect
35. #13133198C23semidirect
36. #13133198C23semidirect
42. #13133185C23semidirect
45. #13133176C23semidirect
48. C2×C22⋊Dic53C23semidirect
49. #13133132C23semidirect
60. C2×#41×C53D4semidirect
64. C22×D4×C53D4semidirect
160. H(7,𝔽2)D53semidirect
190. C53#60401semidirect
202. C2×#41C22⋊D53semidirect
208. C22×D4C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202883552
#202841212
#202841122
#202841072
C2×#168108132
#202831762
C53×H(7,𝔽2)2
#6040153