Finite Group Databasebeta

← #27393355

Subgroups — #27393355

Order 13568.

1538 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1538 conjugacy classes of subgroups — too many to draw. Only its 235 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
1C2×#16811182C211yesmaximal, characteristic
2#20277311C212yesmaximal, complemented
3#20275549C218yesmaximal, complemented
4#20274630C211yesmaximal, characteristic, complemented
5#20274629C212yesmaximal, complemented
6C53×#1781C211yesmaximal, characteristic, complemented
7C23×C22⋊D53C2211yescharacteristic
8#16811182—24no—
9#16811182—24no—
10C2×#13133200C412yescomplemented
11C2×#13133198C412yescomplemented
12C2×#13133189C2214yes—
13C2×#13133186C412yescomplemented
14C2×#13133185C412yescomplemented
15C22×C22⋊Dic53C2211yescharacteristic
16C2×#13133132C2214yescomplemented
17C2×#13133131C2218yescomplemented
18#16810517—24no—
19#16810511—24no—
20#16810481C2214yescomplemented
21#16810479C2212yes—
22C22⋊C4×Dic53C2214yescomplemented
23C23×D4×C53C2211yescharacteristic
24C2×#99×C53C2214yescomplemented
25C53×#379C2212yes—
26C24×D53C2×C411yescharacteristic
27C22×C22⋊D53—22no—
28C22×C22⋊D53—22no—
29C22×C22⋊D53—44no—
30C22×C22⋊D53C2×C412yes—
31C22×C22⋊D53C2×C414yes—
32C22×C22⋊D53C2312yes—
33C23×Dic53C2×C411yescharacteristic
34C23×Dic53C2312yes—
35#13133200—28no—
36#13133198—24no—
37#13133198—24no—
38#13133189—28no—
39#13133189—28no—
40#13133186—28no—
41#13133185—24no—
42#13133185—24no—
43C2×#11719919—24no—
44C2×#11719919C2×C414yes—
45C2×C22⋊Dic53—22no—
46C2×C22⋊Dic53—22no—
47C2×C22⋊Dic53C2×C414yes—
48C2×C22⋊Dic53C2×C412yes—
49C2×C22⋊Dic53C2×C412yes—
50C2×C22⋊Dic53D414yes—
51C2×C22⋊Dic53D414yes—
52C2×C22⋊Dic53C2314yes—
53C2×#11719921C2×C414yes—
54C2×C4×Dic53—24no—
55#13133132—44no—
56#13133131—48no—
57#13133124—24no—
58#13133124—24no—
59C25×C53C2×C411yescharacteristic
60C25×C53C2311yescharacteristic
61C22×D4×C53—22no—
62C22×D4×C53—22no—
63C22×D4×C53—44no—
64C22×D4×C53C2×C412yes—
65C22×D4×C53C2312yes—
66C23×C4×C53C2×C411yescharacteristic
67C2×C22⋊C4×C53—22no—
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53C2314yes—
70#99×C53—44no—
71C24×C53—24no—
72C24×C53—22no—
73C24×C53—42no—
74C24×C53—82no—
75C24×C53—82no—
76C24×C53C22×C412yes—
77C24×C53C22×C411yescharacteristic
78C24×C53C22×C411yescharacteristic
79C24×C53C2×D412yes—
80C24×C53C2×D412yes—
81C24×C53C2411yescharacteristic
82C23×D53—21no—
83C23×D53—21no—
84C23×D53—42no—
85C23×D53—82no—
86C23×D53C22×C412yes—
87C2×D4×C53—22no—
88C2×D4×C53—22no—
89C2×D4×C53—24no—
90C2×D4×C53—24no—
91C2×D4×C53—24no—
92C2×D4×C53—42no—
93C2×D4×C53—42no—
94C2×D4×C53—48no—
95C2×D4×C53—48no—
96C22×C4×C53—22no—
97C22×C4×C53—24no—
98C22×C4×C53—42no—
99C22×C4×C53C22×C412yes—
100C2×C22⋊D53—24no—
101C2×C22⋊D53—24no—
102C2×C22⋊D53—28no—
103C2×C22⋊D53—28no—
104C2×C22⋊D53—44no—
105C2×C22⋊D53—48no—
106C2×C22⋊D53—48no—
107C2×C22⋊D53—84no—
108C22×Dic53—21no—
109C22×Dic53—21no—
110C22×Dic53—24no—
111C22×Dic53—42no—
112C22×Dic53—82no—
113C22×Dic53C22×C412yes—
114C22×Dic53C22×C414yes—
115C22×Dic53C2×D414yes—
116C22×Dic53C2×D414yes—
117C22⋊C4×C53—28no—
118C22⋊C4×C53—82no—
119C22⋊C4×C53—82no—
120C22⋊Dic53—28no—
121C22⋊Dic53—28no—
122C22⋊Dic53—82no—
123C22⋊Dic53—82no—
124C22⋊Dic53C22⋊C414yes—
125C22⋊Dic53C22⋊C414yes—
126C22⋊Dic53C22⋊C418yes—
127#11719921—28no—
128#11719919—28no—
129#11719919—84no—
130C4×Dic53—84no—
131C23×C53—22no—
132C23×C53—22no—
133C23×C53—24no—
134C23×C53—24no—
135C23×C53—22no—
136C23×C53—22no—
137C23×C53—22no—
138C23×C53—42no—
139C23×C53—42no—
140C23×C53—42no—
141C23×C53—42no—
142C23×C53—42no—
143C23×C53—42no—
144C23×C53—42no—
145C23×C53—44no—
146C23×C53—44no—
147C23×C53—82no—
148C23×C53—82no—
149C23×C53—88no—
150C23×C53—88no—
151C23×C53C2×C22⋊C411yescharacteristic
152C23×C53C2×C22⋊C411yescharacteristic
153C23×C53C2×C22⋊C412yes—
154C23×C53C23×C411yescharacteristic
155C23×C53C22×D412yes—
156C23×C53H(5,𝔽2)14yes—
157C22×D53—22no—
158C22×D53—24no—
159C22×D53—42no—
160C22×D53—42no—
161C22×D53—44no—
162C22×D53—82no—
163C22×D53—82no—
164C22×D53—88no—
165D4×C53—44no—
166D4×C53—44no—
167D4×C53—48no—
168C2×C4×C53—22no—
169C2×C4×C53—22no—
170C2×C4×C53—22no—
171C2×C4×C53—44no—
172C2×C4×C53—44no—
173C2×C4×C53—44no—
174C2×C4×C53—44no—
175C22⋊D53—48no—
176C22⋊D53—48no—
177C22⋊D53—88no—
178C2×Dic53—22no—
179C2×Dic53—24no—
180C2×Dic53—44no—
181C2×Dic53—44no—
182C2×Dic53—44no—
183C2×Dic53—42no—
184C2×Dic53—42no—
185C2×Dic53—44no—
186C2×Dic53—44no—
187C2×Dic53—44no—
188C2×Dic53C2×C22⋊C418yes—
189#21795—531nomaximal, Sylow
190C22×C53—22no—
191C22×C53—22no—
192C22×C53—22no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—22no—
196C22×C53—22no—
197C22×C53—22no—
198C22×C53—22no—
199C22×C53—22no—
200C22×C53—44no—
201C22×C53—44no—
202C22×C53—44no—
203C22×C53—42no—
204C22×C53—44no—
205C22×C53—84no—
206C22×C53—84no—
207C22×C53—82no—
208C22×C53—82no—
209C22×C53—88no—
210C22×C53C22×C22⋊C411yescharacteristic
211C22×C53#51812yes—
212C22×C53#53514yes—
213C2×D53—42no—
214C2×D53—44no—
215C2×D53—84no—
216C2×D53—84no—
217C2×D53—84no—
218C4×C53—42no—
219C4×C53—84no—
220Dic53—28no—
221Dic53—42no—
222Dic53—84no—
223C2×#534—531no—
224#2312—532no—
225#1964—538no—
226#1781—532no—
227#1781—531no—
228#1781D5311yescharacteristic, complemented
229C2×C53—24no—
230C2×C53—22no—
231C2×C53—41no—
232C2×C53—41no—
233C2×C53—42no—
234C2×C53—82no—
235C2×C53#230911yescharacteristic
236C2×C53#288312yes—
237D53—82no—
238C23×D4—531no—
239C23×D4C2×D5311yescharacteristic
240#534—1064no—
241#534—1064no—
242C2×#123—532no—
243C2×#123—532no—
244C2×#121—534no—
245C2×#120—532no—
246C2×#120—532no—
247C22×C22⋊C4—531no—
248C2×#99—534no—
249C2×#99—538no—
250C2×#99C2×D5314yescomplemented
251#388—1064no—
252#388—1064no—
253#380—534no—
254#379—532no—
255#379C2×D5312yes—
256C4×C22⋊C4—534no—
257C53#2179511yescharacteristic, complemented, Sylow
258C25—531no—
259C25C4×D5311yescharacteristic
260C25C22×D5311yescharacteristic
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—44no—
264C22×D4—532no—
265C22×D4—534no—
266C22×D4—532no—
267C22×D4—1062no—
268C22×D4—1062no—
269C22×D4—2124no—
270C22×D4C4×D5312yes—
271C22×D4C22×D5312yes—
272C23×C4—532no—
273C23×C4—531no—
274C23×C4C4×D5311yescharacteristic
275#123—1068no—
276#123—1064no—
277#123—1064no—
278#121—1068no—
279#121—1068no—
280#120—1064no—
281#120—1064no—
282#120—1068no—
283C2×C4⋊C4—534no—
284C2×C4⋊C4—1064no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—532no—
288C2×C22⋊C4—532no—
289C2×C22⋊C4—534no—
290C2×C22⋊C4—534no—
291C2×C22⋊C4—534no—
292C2×C22⋊C4—534no—
293C2×C22⋊C4—534no—
294C2×C22⋊C4—1062no—
295C2×C22⋊C4—1062no—
296C2×C22⋊C4C22×D5314yes—
297C2×C42—1064no—
298#99—44no—
299#99—2128no—
300#99—2124no—
301#95—1064no—
302#95—1064no—
303C24—24no—
304C24—22no—
305C24—42no—
306C24—82no—
307C24—82no—
308C24—532no—
309C24—1061no—
310C24—1061no—
311C24—4242no—
312C24—2122no—
313C24C2×C4×D5312yes—
314C24C2×C4×D5311yescharacteristic
315C24C2×C4×D5311yescharacteristic
316C24D4×D5312yes—
317C24D4×D5312yes—
318C24C23×D5311yescharacteristic
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—1068no—
329C2×D4—1064no—
330C2×D4—1064no—
331C2×D4—1068no—
332C2×D4—4244no—
333C2×D4—2124no—
334C2×D4—2128no—
335C2×D4—2128no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—42no—
339C22×C4—534no—
340C22×C4—534no—
341C22×C4—534no—
342C22×C4—532no—
343C22×C4—1061no—
344C22×C4—1061no—
345C22×C4—1064no—
346C22×C4—4242no—
347C22×C4—2122no—
348C22×C4C2×C4×D5312yes—
349C4⋊C4—1068no—
350C4⋊C4—4244no—
351C22⋊C4—28no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—538no—
355C22⋊C4—534no—
356C22⋊C4—534no—
357C22⋊C4—1068no—
358C22⋊C4—1068no—
359C22⋊C4—1068no—
360C22⋊C4—4242no—
361C22⋊C4—4242no—
362C42—4244no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—24no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—44no—
378C23—44no—
379C23—82no—
380C23—82no—
381C23—88no—
382C23—88no—
383C23—1064no—
384C23—1062no—
385C23—4242no—
386C23—4242no—
387C23—4248no—
388C23—2124no—
389C23—2122no—
390C23—2122no—
391C23C22⋊C4×D5312yes—
392C23C22⋊C4×D5311yescharacteristic
393C23C22⋊C4×D5311yescharacteristic
394C23C22×C4×D5311yescharacteristic
395C23C2×D4×D5312yes—
396C23#1313326214yes—
397D4—44no—
398D4—44no—
399D4—48no—
400D4—4248no—
401D4—2128no—
402D4—2128no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—538no—
411C2×C4—1062no—
412C2×C4—1064no—
413C2×C4—2124no—
414C2×C4—2124no—
415C2×C4—2124no—
416C2×C4—2124no—
417C2×C4—2124no—
418C2×C4—2124no—
419C2×C4—2122no—
420C2×C4—2122no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—21no—
430C22—21no—
431C22—44no—
432C22—44no—
433C22—44no—
434C22—44no—
435C22—42no—
436C22—84no—
437C22—84no—
438C22—82no—
439C22—82no—
440C22—88no—
441C22—4244no—
442C22—4244no—
443C22—4244no—
444C22—2124no—
445C22—2122no—
446C22C2×C22⋊C4×D5311yescharacteristic
447C22#1681109612yes—
448C22#1681118814yes—
449C4—42no—
450C4—84no—
451C4—1068no—
452C4—4244no—
453C4—2122no—
454C2—24no—
455C2—22no—
456C2—42no—
457C2—41no—
458C2—41no—
459C2—82no—
460C2—4242no—
461C2#2027729611yescharacteristic
462C2#2028317312yes—
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1781C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#16811182C2
cosocleC24×C53C24
hypercenter#1781D53
hypocenterC53#21795
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
2. #20277311C2semidirect
3. #20275549C2semidirect
4. #20274630C2semidirect
5. #20274629C2semidirect
6. C53×#1781C2semidirect
10. C2×#13133200C4semidirect
11. C2×#13133198C4semidirect
13. C2×#13133186C4semidirect
14. C2×#13133185C4semidirect
16. C2×#13133132C22semidirect
17. C2×#13133131C22semidirect
20. #16810481C22semidirect
22. C22⋊C4×Dic53C22semidirect
24. C2×#99×C53C22semidirect
228. #1781D53semidirect
250. C2×#99C2×D53semidirect
257. C53#21795semidirect

Maximal subgroups

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

subgroupindex
C2×#168111822
#202773112
#202755492
#202746302
#202746292
C53×#17812
#2179553