Finite Group Databasebeta

← #27358266

Subgroups — #27358266

Order 13568.

1256 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 183 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#20285512C212yesmaximal, complemented
2#20280195C211yesmaximal, characteristic, complemented
3#20273752C214yesmaximal, complemented
4#20273650C214yesmaximal, complemented
5#20273231C212yesmaximal, complemented
6#20273045C211yesmaximal, characteristic, complemented
7C2×C53×#537C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9#16810941—22no—
10#16810939C2214yescomplemented
11#16810874—24no—
12#16810757—22no—
13#16810756C2214yescomplemented
14#16810587C2212yes—
15#16810562C2212yescomplemented
16#16810540C2212yescomplemented
17#16810513—22no—
18#16810502—21no—
19#16810502—21no—
20C4×C22⋊Dic53C2211yescharacteristic
21#16810357C2218yescomplemented
22#16810349C2214yescomplemented
23C22×#41×C53—21no—
24C53×#537—24no—
25C53×#537—22no—
26C53×#537C2214yes—
27C2×C22⋊Q8×C53—22no—
28C2×#121×C53—22no—
29C2×#121×C53—21no—
30C2×#121×C53C2212yes—
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×D212C2311yescharacteristic
33C22×C22⋊D53—21no—
34C2×#11719922—24no—
35C2×#11719917—22no—
36#13133191—22no—
37#13133191—22no—
38C2×C4⋊Dic53C2311yescharacteristic
39C2×C22⋊Dic53—21no—
40C2×#11719921—22no—
41C2×#11719921—21no—
42C2×#11719921—22no—
43C2×C4×Dic53—21no—
44#13133148—24no—
45#13133138—24no—
46#13133137—24no—
47C22⋊#88322—22no—
48C22⋊#88322C2314yes—
49#13133124—22no—
50#13133124—22no—
51#13133124C2312yes—
52C2×#41×C53—24no—
53C2×#41×C53—24no—
54C2×#41×C53—48no—
55C22×Q8×C53—21no—
56C22×D4×C53—21no—
57C22×D4×C53—22no—
58C22×D4×C53—21no—
59C22×D4×C53D412yes—
60C23×C4×C53—22no—
61C23×C4×C53D411yescharacteristic
62C23×C4×C53D411yescharacteristic
63C23×C4×C53C2311yescharacteristic
64C22⋊Q8×C53—28no—
65C22⋊Q8×C53—28no—
66#121×C53—28no—
67#121×C53—24no—
68#121×C53—28no—
69#121×C53—24no—
70#121×C53—22no—
71#121×C53—22no—
72#121×C53D414yes—
73#121×C53C2314yes—
74C2×C4⋊C4×C53—21no—
75C2×C4⋊C4×C53—22no—
76C2×C4⋊C4×C53—22no—
77C2×C4⋊C4×C53D412yes—
78C2×C4⋊C4×C53D412yes—
79C2×C4⋊C4×C53C2312yes—
80C2×C22⋊C4×C53—22no—
81C2×C22⋊C4×C53—22no—
82C24×C53—22no—
83C24×C53C2×D411yescharacteristic
84C23×D53—21no—
85#41×C53—416no—
86#41×C53—48no—
87#41×C53—48no—
88C2×Q8×C53—24no—
89C2×Q8×C53—44no—
90C2×D4×C53—24no—
91C2×D4×C53—24no—
92C2×D4×C53—24no—
93C2×D4×C53—24no—
94C2×D4×C53—22no—
95C2×D4×C53—44no—
96C2×D4×C53—48no—
97C2×D4×C53—82no—
98C2×D4×C53—82no—
99C2×D4×C53C2×D414yes—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—24no—
103C22×C4×C53—28no—
104C22×C4×C53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—21no—
107C22×C4×C53—48no—
108C22×C4×C53—81no—
109C22×C4×C53—81no—
110C22×C4×C53C2×D412yes—
111C22×C4×C53C2×D412yes—
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D412yes—
114C22×C4×C53C2×D412yes—
115C22×C4×C53C2411yescharacteristic
116C2×C22⋊D53—44no—
117C2×C22⋊D53—41no—
118C2×C22⋊D53—41no—
119C22×Dic53—21no—
120C22×Dic53—22no—
121C2×D212—22no—
122C2×D212—81no—
123C4⋊C4×C53—24no—
124C4⋊C4×C53—24no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—24no—
127C4⋊C4×C53C2×D414yes—
128C22⋊C4×C53—28no—
129C22⋊C4×C53—24no—
130C22⋊C4×C53—24no—
131C22⋊Dic53—42no—
132#11719922—84no—
133#11719921—42no—
134#11719921—44no—
135#11719921—82no—
136C4⋊Dic53—22no—
137C4×Dic53—81no—
138#11719917—82no—
139C2×#88322—24no—
140C23×C53—22no—
141C23×C53—22no—
142C23×C53—24no—
143C23×C53—41no—
144C23×C53—44no—
145C23×C53—44no—
146C23×C53—44no—
147C23×C53#12011yescharacteristic
148C23×C53#12011yescharacteristic
149C23×C53C22×D411yescharacteristic
150C22×D53—42no—
151C22×D53—42no—
152C22×D53—41no—
153C22×D53—42no—
154Q8×C53—48no—
155D4×C53—44no—
156D4×C53—48no—
157D4×C53—48no—
158D4×C53—48no—
159D4×C53—82no—
160D4×C53—84no—
161C2×C4×C53—24no—
162C2×C4×C53—24no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53—24no—
166C2×C4×C53—22no—
167C2×C4×C53—41no—
168C2×C4×C53—42no—
169C2×C4×C53—42no—
170C2×C4×C53—41no—
171C2×C4×C53—416no—
172C2×C4×C53—44no—
173C2×C4×C53—42no—
174C2×C4×C53—42no—
175C2×C4×C53—81no—
176C2×C4×C53#12012yes—
177C2×C4×C53AGL(1,ℤ/8)14yes—
178C2×C4×C53C22×D412yes—
179C22⋊D53—82no—
180C22⋊D53—82no—
181C2×Dic53—42no—
182C2×Dic53—42no—
183C2×Dic53—42no—
184C2×Dic53—41no—
185C2×Dic53—42no—
186D212—44no—
187#88322—44no—
188#18324—531nomaximal, Sylow
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—44no—
197C22×C53—44no—
198C22×C53—44no—
199C22×C53—44no—
200C22×C53—88no—
201C22×C53#45412yes—
202C22×C53C2×#12011yescharacteristic
203C22×C53C2×AGL(1,ℤ/8)12yes—
204C2×D53—84no—
205C2×D53—81no—
206C2×D53—81no—
207C2×D53—81no—
208C4×C53—24no—
209C4×C53—22no—
210C4×C53—22no—
211C4×C53—44no—
212C4×C53—42no—
213Dic53—82no—
214Dic53—81no—
215C2×#537D5311yescharacteristic, complemented
216#3192—532no—
217#2591—531no—
218#1616—534no—
219#1598—534no—
220#1507—532no—
221#1463—531no—
222C2×C53—22no—
223C2×C53—22no—
224C2×C53—41no—
225C2×C53—84no—
226C2×C53#300311yescharacteristic
227C2×C53C2×#45411yescharacteristic
228C2×C53#302611yescharacteristic
229D53—161no—
230C22×#41—21no—
231#537—24no—
232#537—22no—
233#537C2×D5314yes—
234C2×C22⋊Q8—22no—
235C2×#121—22no—
236C2×#121—21no—
237C2×#121—531no—
238C2×#121C2×D5312yes—
239C22×C4⋊C4C2×D5311yescharacteristic
240#481—534no—
241#481—1062no—
242#469—1064no—
243#447—534no—
244#447—1062no—
245#407—532no—
246#400—532no—
247#394—532no—
248#388—1062no—
249#386—1061no—
250#386—1061no—
251C4×C22⋊C4—531no—
252#327—538no—
253#324—534no—
254C53#1832411yescharacteristic, complemented, Sylow
255C2×#41—24no—
256C2×#41—24no—
257C2×#41—48no—
258C22×Q8—21no—
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—21no—
262C22×D4—531no—
263C22×D4—1061no—
264C22×D4C22⋊D5312yes—
265C23×C4—22no—
266C23×C4C22⋊D5311yescharacteristic
267C23×C4C22⋊D5311yescharacteristic
268C23×C4C22×D5311yescharacteristic
269C2×D8—1064no—
270C2×M16—1062no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273#121—24no—
274#121—28no—
275#121—28no—
276#121—24no—
277#121—22no—
278#121—22no—
279#121—1062no—
280#121—1062no—
281#121C22⋊D5314yes—
282#121C22×D5314yes—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—531no—
287C2×C4⋊C4C22⋊D5312yes—
288C2×C4⋊C4C22⋊D5312yes—
289C2×C4⋊C4C22×D5312yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—1062no—
293C2×C22⋊C4—1061no—
294C2×C22⋊C4—1061no—
295C2×C22⋊C4—1062no—
296C2×C42—1061no—
297#106—1064no—
298#102—1064no—
299#102—1064no—
300C22⋊C8—534no—
301C22⋊C8—1062no—
302#95—532no—
303#95—1062no—
304#95—1062no—
305C24—22no—
306C24—1061no—
307C24C2×C22⋊D5311yescharacteristic
308#41—416no—
309#41—48no—
310#41—48no—
311C2×Q8—24no—
312C2×Q8—44no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—44no—
319C2×D4—48no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—1062no—
323C2×D4—4241no—
324C2×D4—2124no—
325C2×D4—2121no—
326C2×D4—2121no—
327C2×D4C2×C22⋊D5314yes—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—48no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—1062no—
339C22×C4—1061no—
340C22×C4C2×C22⋊D5312yes—
341C22×C4C2×C22⋊D5312yes—
342C22×C4C2×C22⋊D5311yescharacteristic
343C22×C4C2×C22⋊D5312yes—
344C22×C4C2×C22⋊D5312yes—
345C22×C4C23×D5311yescharacteristic
346D8—4244no—
347M16—4242no—
348C2×C8—1064no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—1062no—
354C4⋊C4C2×C22⋊D5314yes—
355C22⋊C4—28no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—4242no—
359C22⋊C4—2122no—
360C22⋊C4—2122no—
361C22⋊C4—2124no—
362C42—4241no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—41no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—2122no—
371C23—2122no—
372C23—2121no—
373C23—2122no—
374C23#1313318611yescharacteristic
375C23#1313318611yescharacteristic
376C23C22×C22⋊D5311yescharacteristic
377Q8—48no—
378D4—44no—
379D4—48no—
380D4—48no—
381D4—48no—
382D4—82no—
383D4—84no—
384D4—4242no—
385D4—4242no—
386D4—2124no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—416no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—81no—
402C2×C4—2122no—
403C2×C4—2122no—
404C2×C4—2122no—
405C2×C4—2121no—
406C2×C4—2122no—
407C2×C4#1313318612yes—
408C2×C4#1313324314yes—
409C2×C4C22×C22⋊D5312yes—
410C8—2124no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—44no—
419C22—44no—
420C22—44no—
421C22—44no—
422C22—88no—
423C22—4244no—
424C22—4241no—
425C22—4241no—
426C22—4241no—
427C22#1681079612yes—
428C22C2×#1313318611yescharacteristic
429C22C2×#1313324312yes—
430C4—24no—
431C4—22no—
432C4—22no—
433C4—44no—
434C4—42no—
435C4—4242no—
436C4—4241no—
437C2—22no—
438C2—22no—
439C2—41no—
440C2—84no—
441C2—8481no—
442C2#2028394211yescharacteristic
443C2C2×#1681079611yescharacteristic
444C2#2028411211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#537C2
socleC22×C53C2×#120
hypercenterC2×#537D53
hypocenterC53#18324
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #20285512C2semidirect
2. #20280195C2semidirect
3. #20273752C2semidirect
4. #20273650C2semidirect
5. #20273231C2semidirect
6. #20273045C2semidirect
7. C2×C53×#537C2semidirect
10. #16810939C22semidirect
13. #16810756C22semidirect
15. #16810562C22semidirect
16. #16810540C22semidirect
21. #16810357C22semidirect
22. #16810349C22semidirect
215. C2×#537D53semidirect
254. C53#18324semidirect

Maximal subgroups

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

subgroupindex
#202855122
#202801952
#202737522
#202736502
#202732312
#202730452
C2×C53×#5372
#1832453