Finite Group Databasebeta

← #28250843

Subgroups — #28250843

Order 13568.

5128 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 5128 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#20288346C213yesmaximal, complemented
2#20284117C214yesmaximal, complemented
3#20284112C218yesmaximal, complemented
4#20284107C214yesmaximal, complemented
5C2×#16810810C218yesmaximal, complemented
6#20283176C213yesmaximal, complemented
7C53×H(7,𝔽2)C211yesmaximal, characteristic, complemented
8C22×D4×D53C2211yescharacteristic, complemented
9#16811261—212no—
10D4×C22⋊D53—212no—
11#16811187C2216yescomplemented
12#16811182C22112yescomplemented
13#16811130C2216yescomplemented
14C2×#13133185C22112yescomplemented
15#16811098C2213yescomplemented
16#16810813C22116yescomplemented
17#16810810C22116yescomplemented
18#16810810C22132yescomplemented
19#16810597C22112yescomplemented
20C2×#13133132C22124yescomplemented
21C53×#585—212no—
22C53×#585C2214yescomplemented
23C2×H(5,𝔽2)×C53—212no—
24C2×H(5,𝔽2)×C53C2213yes—
25C2×H(5,𝔽2)×C53C2218yescomplemented
26C24×D53C2312yescomplemented
27C22×D212C2311yescharacteristic, complemented
28C22×C22⋊D53—212no—
29C22×C22⋊D53C2312yescomplemented
30C2×D4×D53—212no—
31C22×C4×D53C2311yescharacteristic, complemented
32#13133213—26no—
33#13133206—26no—
34#13133200—212no—
35#13133198C23112yescomplemented
36#13133193—26no—
37#13133191—212no—
38#13133189—224no—
39#13133188—26no—
40#13133186—212no—
41#13133185—224no—
42#13133185C23112yescomplemented
43#13133185C23124yescomplemented
44D4×Dic53—212no—
45C4×C22⋊D53—212no—
46#13133176C2316yescomplemented
47C2×C22⋊Dic53C23112yescomplemented
48C2×#11719921—212no—
49#13133132C23148yescomplemented
502−1+4×C53—216no—
512−1+4×C53—212no—
52H(5,𝔽2)×C53—248no—
53H(5,𝔽2)×C53—224no—
54H(5,𝔽2)×C53—212no—
55H(5,𝔽2)×C53—248no—
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×D4112yescomplemented
69C24×C53C2412yes—
70C23×D53—212no—
71C23×D53—84no—
72C23×D53C2411yescharacteristic
73C23×D53C2412yes—
74#41×C53—296no—
75#41×C53—248no—
76#41×C53—216no—
77#41×C53—248no—
78#41×C53—248no—
79#41×C53—224no—
80C2×Q8×C53—224no—
81C2×Q8×C53—28no—
82C2×Q8×C53C2×D416yescomplemented
83C2×D4×C53—296no—
84C2×D4×C53—224no—
85C2×D4×C53—224no—
86C2×D4×C53—248no—
87C2×D4×C53—248no—
88C2×D4×C53—248no—
89C2×D4×C53C2×D4112yes—
90C2×D4×C53C2×D416yescomplemented
91C2×D4×C53C2×D4124yescomplemented
92C2×D4×C53C24112yes—
93C22×C4×C53—224no—
94C22×C4×C53—216no—
95C22×C4×C53C2×D4112yescomplemented
96C22×C4×C53C2×D4112yescomplemented
97C22×C4×C53C2411yescharacteristic
98C2×C22⋊D53—212no—
99C2×C22⋊D53—212no—
100C2×C22⋊D53—224no—
101C2×C22⋊D53—824no—
102C22×Dic53—212no—
103C22×Dic53C2411yescharacteristic
104D4×D53—416no—
105C2×D212—26no—
106C2×C4×D53—26no—
107C22⋊Dic53—212no—
108C22⋊Dic53—212no—
109C22⋊Dic53C24112yes—
110#11719921—212no—
111#11719921—224no—
112C4⋊Dic53—26no—
113#11719919—212no—
114C4×Dic53—26no—
115C23×C53—216no—
116C23×C53—224no—
117C23×C53—82no—
118C23×C53—812no—
119C23×C53—168no—
120C23×C53C22×D4112yes—
121C23×C53C22×D4112yes—
122C23×C53C2511yescharacteristic
123C22×D53—23no—
124C22×D53—26no—
125C22×D53—42no—
126C22×D53—424no—
127C22×D53—48no—
128C22×D53—82no—
129C22×D53—812no—
130Q8×C53—216no—
131Q8×C53—212no—
132D4×C53—248no—
133D4×C53—248no—
134D4×C53—224no—
135D4×C53—212no—
136D4×C53#120116yes—
137C2×C4×C53—248no—
138C2×C4×C53—224no—
139C2×C4×C53—224no—
140C2×C4×C53C22×D4112yes—
141C2×C4×C53C22×D416yes—
142C22⋊D53—424no—
143C22⋊D53—48no—
144C22⋊D53—824no—
145C2×Dic53—23no—
146C2×Dic53—212no—
147C2×Dic53—812no—
148D212—44no—
149C4×D53—44no—
150#60399—531nomaximal, Sylow
151C22×C53—212no—
152C22×C53—412no—
153C22×C53—41no—
154C22×C53—412no—
155C22×C53—824no—
156C22×C53—816no—
157C22×C53C2×#12018yes—
158C22×C53C23×D413yes—
159C2×D53—26no—
160C2×D53—43no—
161C2×D53—41no—
162C2×D53—412no—
163C2×D53—42no—
164C2×D53—816no—
165C4×C53—212no—
166C4×C53C2×#12014yes—
167Dic53—412no—
168Dic53—41no—
169H(7,𝔽2)D5311yescharacteristic, complemented
170#3484—533no—
171#3027—534no—
172#3026—538no—
173#3025—534no—
174C2×B(4,𝔽2)—538no—
175#2883—533no—
176C2×C53—23no—
177C2×C53—28no—
178C2×C53—412no—
179C2×C53C22×#12011yescharacteristic
180D53—42no—
181D53—82no—
182#585—212no—
183#585C2×D5314yescomplemented
184C2×H(5,𝔽2)—212no—
185C2×H(5,𝔽2)C2×D5313yes—
186C2×H(5,𝔽2)C2×D5318yescomplemented
187C23×D4—531no—
188#546—10612no—
189D42—10612no—
190#535—536no—
191#534—5312no—
192#525—536no—
193C2×#120—5312no—
194#518—533no—
195#458—5316no—
196B(4,𝔽2)—5316no—
197B(4,𝔽2)—5332no—
198#410—5312no—
199C2×#99—5324no—
200C53#6039911yescharacteristic, complemented, Sylow
201C25—532no—
2022−1+4—216no—
2032−1+4—212no—
204H(5,𝔽2)—248no—
205H(5,𝔽2)—224no—
206H(5,𝔽2)—212no—
207H(5,𝔽2)—248no—
208H(5,𝔽2)C22×D53116yes—
209C2×#41—248no—
210C2×#41—224no—
211C2×#41—224no—
212C2×#41C22⋊D53112yescomplemented
213C2×#41C22×D5316yes—
214C22×D4—216no—
215C22×D4—224no—
216C22×D4—532no—
217C22×D4—531no—
218C22×D4—10612no—
219C22×D4—10612no—
220C22×D4C22⋊D53112yescomplemented
221C22×D4C22×D53112yes—
222C22×D4C22×D5311yescharacteristic
223C23×C4—531no—
224#127—1066no—
225#124—1066no—
226#123—5312no—
227#123—10612no—
228C22⋊Q8—1066no—
229#121—1066no—
230#121—10624no—
231#121—10612no—
232#120—5312no—
233#120—5324no—
234#120—10612no—
235#120—10624no—
236C4×D4—10612no—
237C4×D4—10612no—
238#117—536no—
239C2×C22⋊C4—5312no—
240C2×C22⋊C4—10612no—
241#99—5348no—
242C24—28no—
243C24—532no—
244C24—531no—
245C24—10612no—
246C24—4244no—
247C24C2×C22⋊D53112yescomplemented
248C24C23×D5312yes—
249#41—296no—
250#41—248no—
251#41—216no—
252#41—248no—
253#41—248no—
254#41—224no—
255C2×Q8—224no—
256C2×Q8—28no—
257C2×Q8C2×C22⋊D5316yescomplemented
258C2×D4—296no—
259C2×D4—224no—
260C2×D4—224no—
261C2×D4—248no—
262C2×D4—248no—
263C2×D4—248no—
264C2×D4—10624no—
265C2×D4—10612no—
266C2×D4—10612no—
267C2×D4—1066no—
268C2×D4—42424no—
269C2×D4—21216no—
270C2×D4C2×C22⋊D53112yes—
271C2×D4C2×C22⋊D53124yescomplemented
272C2×D4C2×C22⋊D5316yescomplemented
273C2×D4C23×D53112yes—
274C22×C4—224no—
275C22×C4—216no—
276C22×C4—531no—
277C22×C4—10612no—
278C22×C4—1066no—
279C22×C4C2×C22⋊D53112yescomplemented
280C22×C4C2×C22⋊D53112yescomplemented
281C22×C4C23×D5311yescharacteristic
282C4⋊C4—1066no—
283C4⋊C4—10612no—
284C22⋊C4—5312no—
285C22⋊C4—10612no—
286C22⋊C4—10612no—
287C22⋊C4—10612no—
288C22⋊C4—10624no—
289C42—1066no—
290C23—216no—
291C23—224no—
292C23—82no—
293C23—812no—
294C23—168no—
295C23—1063no—
296C23—1066no—
297C23—42412no—
298C23—4242no—
299C23—2122no—
300C23—21224no—
301C23—2128no—
302C23C22×C22⋊D53112yes—
303C23C22×C22⋊D53112yes—
304C23C24×D5311yescharacteristic
305Q8—216no—
306Q8—212no—
307D4—248no—
308D4—248no—
309D4—224no—
310D4—212no—
311D4—42424no—
312D4—21224no—
313D4—2128no—
314D4—2124no—
315D4#13133186116yes—
316C2×C4—248no—
317C2×C4—224no—
318C2×C4—224no—
319C2×C4—10612no—
320C2×C4—1063no—
321C2×C4—42412no—
322C2×C4—2124no—
323C2×C4C22×C22⋊D53112yes—
324C2×C4C22×C22⋊D5316yes—
325C22—212no—
326C22—412no—
327C22—41no—
328C22—412no—
329C22—824no—
330C22—816no—
331C22—1066no—
332C22—42416no—
333C22—21212no—
334C22—2123no—
335C22—2122no—
336C22—2121no—
337C22C2×#1313318618yes—
338C22C23×C22⋊D5313yes—
339C4—212no—
340C4—21212no—
341C4—2121no—
342C4C2×#1313318614yes—
343C2—23no—
344C2—28no—
345C2—412no—
346C2—4242no—
347C2—2122no—
348C2C22×#1313318611yescharacteristic
subgroupG/H
centerC2C22×#13133186
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC53×H(7,𝔽2)C2
socleC2×C53C22×#120
hypercenterH(7,𝔽2)D53
hypocenterC53#60399
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. #20288346C2semidirect
2. #20284117C2semidirect
3. #20284112C2semidirect
4. #20284107C2semidirect
5. C2×#16810810C2semidirect
6. #20283176C2semidirect
7. C53×H(7,𝔽2)C2semidirect
8. C22×D4×D53C22semidirect
11. #16811187C22semidirect
12. #16811182C22semidirect
13. #16811130C22semidirect
14. C2×#13133185C22semidirect
15. #16811098C22semidirect
16. #16810813C22semidirect
17. #16810810C22semidirect
18. #16810810C22semidirect
19. #16810597C22semidirect
20. C2×#13133132C22semidirect
22. C53×#585C22semidirect
25. C2×H(5,𝔽2)×C53C22semidirect
26. C24×D53C23semidirect
27. C22×D212C23semidirect
29. C22×C22⋊D53C23semidirect
31. C22×C4×D53C23semidirect
35. #13133198C23semidirect
42. #13133185C23semidirect
43. #13133185C23semidirect
46. #13133176C23semidirect
47. C2×C22⋊Dic53C23semidirect
49. #13133132C23semidirect
60. C2×#41×C53D4semidirect
64. C22×D4×C53D4semidirect
68. C24×C53C2×D4semidirect
82. C2×Q8×C53C2×D4semidirect
90. C2×D4×C53C2×D4semidirect
91. C2×D4×C53C2×D4semidirect
95. C22×C4×C53C2×D4semidirect
96. C22×C4×C53C2×D4semidirect
169. H(7,𝔽2)D53semidirect
183. #585C2×D53semidirect
186. C2×H(5,𝔽2)C2×D53semidirect
200. C53#60399semidirect
212. C2×#41C22⋊D53semidirect
220. C22×D4C22⋊D53semidirect
247. C24C2×C22⋊D53semidirect
257. C2×Q8C2×C22⋊D53semidirect
271. C2×D4C2×C22⋊D53semidirect
272. C2×D4C2×C22⋊D53semidirect
279. C22×C4C2×C22⋊D53semidirect
280. C22×C4C2×C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202883462
#202841172
#202841122
#202841072
C2×#168108102
#202831762
C53×H(7,𝔽2)2
#6039953