Finite Group Databasebeta

← #27302760

Subgroups — #27302760

Order 13568.

556 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 93 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#20279666C211yesmaximal, characteristic
2#20274701C211yesmaximal, characteristic
3#99×Dic53C211yesmaximal, characteristic
4#20272840C211yesmaximal, characteristic
5#20272839C211yesmaximal, characteristic
6#20272839C211yesmaximal, characteristic
7C53×#1393C211yesmaximal, characteristic
8C2×#13133216C411yescharacteristic, complemented
9C2×#13133194—21no—
10C2×#13133194—21no—
11C2×D4×Dic53C2211yescharacteristic
12C2×#13133132C2211yescharacteristic
13C2×#13133132C2211yescharacteristic
14#16810559—21no—
15#16810559—21no—
16#16810526C2211yescharacteristic
17#16810493C411yescharacteristic, complemented
18#16810492C411yescharacteristic, complemented
19#16810481—21no—
20C4×#11719919C411yescharacteristic, complemented
21C22⋊C4×Dic53—21no—
22#16810389—21no—
23#16810389—21no—
24C2×#99×C53C2211yescharacteristic
25C2×#99×C53C2211yescharacteristic
26C53×#390C2211yescharacteristic
27C22×Dic106—21no—
28C23×Dic53D411yescharacteristic
29C23×Dic53D411yescharacteristic
30#13133216—41no—
31#13133216—41no—
32#13133194—44no—
33#13133194—44no—
34D4×Dic53—41no—
35D4×Dic53—41no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×#11719919—21no—
44C2×#11719919—41no—
45C2×C22⋊Dic53—21no—
46C2×C22⋊Dic53—21no—
47C2×C22⋊Dic53—21no—
48C2×C22⋊Dic53—21no—
49C2×C22⋊Dic53D411yescharacteristic
50C2×C22⋊Dic53D411yescharacteristic
51C2×C4×Dic53—41no—
52C2×C4×Dic53C2×C411yescharacteristic
53C2×C4×Dic53C2×C411yescharacteristic
54#13133132—21no—
55#13133132—21no—
56#13133132—21no—
57#13133132Q812yescomplemented
58#13133124—21no—
59#13133124—41no—
60#13133124—41no—
61C22×D4×C53C2311yescharacteristic
62C2×C4⋊C4×C53C2×C411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C2×C42×C53C2×C411yescharacteristic
68#99×C53—21no—
69#99×C53—21no—
70#99×C53—21no—
71#99×C53Q812yescomplemented
72C24×C53C2×D411yescharacteristic
73C24×C53C2×D411yescharacteristic
74C2×D4×C53—41no—
75C2×D4×C53—41no—
76C2×D4×C53C2×Q811yescharacteristic
77C2×D4×C53C2×Q811yescharacteristic
78C2×D4×C53#4111yescharacteristic
79C2×D4×C53#4111yescharacteristic
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53C22×C411yescharacteristic
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C22×Dic53—21no—
90C22×Dic53—21no—
91C22×Dic53—42no—
92C22×Dic53—42no—
93C22×Dic53—41no—
94C22×Dic53—41no—
95C22×Dic53C22⋊C411yescharacteristic
96C22×Dic53C22⋊C411yescharacteristic
97C22×Dic53C22⋊C411yescharacteristic
98C22×Dic53C22⋊C411yescharacteristic
99C2×Dic106—42no—
100C2×Dic106—42no—
101C2×Dic106—41no—
102C2×Dic106—41no—
103C4⋊C4×C53—41no—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—42no—
106C22⋊C4×C53—42no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109C22⋊C4×C53—41no—
110C42×C53—41no—
111C22⋊Dic53—22no—
112C22⋊Dic53—41no—
113C22⋊Dic53—41no—
114C22⋊Dic53—42no—
115C22⋊Dic53—42no—
116C22⋊Dic53—41no—
117C22⋊Dic53—41no—
118C22⋊Dic53—41no—
119C4⋊Dic53—41no—
120C4⋊Dic53—42no—
121C4⋊Dic53—42no—
122C4⋊Dic53—41no—
123#11719919—24no—
124#11719919—42no—
125#11719919—42no—
126#11719919—42no—
127#11719919—82no—
128C4×Dic53—21no—
129C4×Dic53—21no—
130C4×Dic53—41no—
131C4×Dic53—82no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—41no—
140C23×C53C2×C22⋊C411yescharacteristic
141C23×C53C22⋊Q811yescharacteristic
142C23×C53C22⋊Q811yescharacteristic
143C23×C53#12411yescharacteristic
144C23×C53C4⋊Q811yescharacteristic
145D4×C53—41no—
146D4×C53—41no—
147D4×C53—41no—
148D4×C53—41no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—22no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—42no—
157C2×C4×C53—41no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53—81no—
161C2×C4×C53—81no—
162C2×C4×C53—81no—
163C2×C4×C53C4×Q811yescharacteristic
164C2×C4×C53C4×Q811yescharacteristic
165C2×Dic53—22no—
166C2×Dic53—22no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—42no—
173C2×Dic53—42no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53—42no—
180C2×Dic53—41no—
181C2×Dic53—81no—
182C2×Dic53—81no—
183C2×Dic53—81no—
184C2×Dic53—81no—
185C2×Dic53—81no—
186C2×Dic53—81no—
187C2×Dic53—81no—
188C2×Dic53#9911yescharacteristic
189C2×Dic53#9911yescharacteristic
190Dic106—82no—
191Dic106—82no—
192#13298—531nomaximal, Sylow
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—81no—
208C22×C53—81no—
209C22×C53#39111yescharacteristic
210C22×C53C2×#9911yescharacteristic
211C22×C53#41011yescharacteristic
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—42no—
215C4×C53—82no—
216C4×C53—81no—
217C4×C53—161no—
218Dic53—21no—
219Dic53—42no—
220Dic53—41no—
221Dic53—41no—
222Dic53—41no—
223Dic53—82no—
224Dic53—81no—
225Dic53—161no—
226Dic53#35312yes—
227#2542—531no—
228#1799—531no—
229C4×#99—531no—
230#1393—531no—
231#1393—531no—
232#1393—531no—
233#1393D5311yescharacteristic
234C2×C53—21no—
235C2×C53—21no—
236C2×C53—41no—
237C2×C53—41no—
238C2×C53—41no—
239C2×C53—41no—
240C2×C53#199611yescharacteristic
241C2×C53C2×#35311yescharacteristic
242C2×C53#212911yescharacteristic
243C2×C4⋊Q8—531no—
244C2×C22⋊Q8—1061no—
245C2×C22⋊Q8—1061no—
246C2×C4×D4—531no—
247C2×#99—531no—
248C2×#99—531no—
249C2×#99C2×D5311yescharacteristic
250C2×#99C2×D5311yescharacteristic
251#399—1061no—
252#399—1061no—
253#390—531no—
254#390C2×D5311yescharacteristic
255#384—531no—
256#384—531no—
257#380—1061no—
258C4×C4⋊C4—531no—
259C4×C22⋊C4—1061no—
260#342—1061no—
261#342—1061no—
262C53#1329811yescharacteristic, complemented, Sylow
263C22×Q8—1061no—
264C22×D4C22×D5311yescharacteristic
265C23×C4—531no—
266C23×C4—531no—
267C4⋊Q8—2121no—
268C4⋊Q8—2121no—
269C22⋊Q8—2124no—
270C22⋊Q8—2124no—
271C4×D4—2121no—
272C4×D4—2121no—
273C2×C4⋊C4—531no—
274C2×C4⋊C4—531no—
275C2×C4⋊C4—1061no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4—1061no—
278C2×C4⋊C4—1061no—
279C2×C4⋊C4—1061no—
280C2×C4⋊C4—1061no—
281C2×C4⋊C4—2121no—
282C2×C4⋊C4C4×D5311yescharacteristic
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—531no—
288C2×C22⋊C4—531no—
289C2×C22⋊C4—1061no—
290C2×C22⋊C4—1061no—
291C2×C22⋊C4—1061no—
292C2×C22⋊C4—1061no—
293C2×C42—531no—
294C2×C42—531no—
295C2×C42—2121no—
296C2×C42C4×D5311yescharacteristic
297#99—21no—
298#99—21no—
299#99—21no—
300#99—532no—
301#99—1061no—
302#99—1061no—
303#99—1061no—
304#99Dic10612yescomplemented
305#95—1061no—
306#95—2121no—
307#95—2121no—
308C24D4×D5311yescharacteristic
309C24D4×D5311yescharacteristic
310C2×Q8—2121no—
311C2×Q8—2121no—
312C2×Q8—2122no—
313C2×Q8—2122no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4C2×Dic10611yescharacteristic
317C2×D4#1171994511yescharacteristic
318C2×D4#1171994711yescharacteristic
319C2×D4Q8×D5311yescharacteristic
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—531no—
325C22×C4—531no—
326C22×C4—531no—
327C22×C4—531no—
328C22×C4—1061no—
329C22×C4—1061no—
330C22×C4—1061no—
331C22×C4—1061no—
332C22×C4—1061no—
333C22×C4—1061no—
334C22×C4—2122no—
335C22×C4—2122no—
336C22×C4—2121no—
337C22×C4—2121no—
338C22×C4C2×C4×D5311yescharacteristic
339C4⋊C4—41no—
340C4⋊C4—1064no—
341C4⋊C4—4242no—
342C4⋊C4—2121no—
343C4⋊C4—2121no—
344C4⋊C4—2122no—
345C4⋊C4—2122no—
346C4⋊C4—2122no—
347C4⋊C4—2122no—
348C4⋊C4—2122no—
349C22⋊C4—22no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—1062no—
356C22⋊C4—2121no—
357C22⋊C4—2121no—
358C22⋊C4—2122no—
359C22⋊C4—2122no—
360C22⋊C4—2121no—
361C22⋊C4—2121no—
362C22⋊C4—2121no—
363C42—41no—
364C42—1061no—
365C42—1061no—
366C42—4242no—
367C42—2121no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23C22⋊C4×D5311yescharacteristic
377C23#1313319411yescharacteristic
378C23#1313319511yescharacteristic
379C23#1313320311yescharacteristic
380C23#1313321611yescharacteristic
381Q8—4242no—
382Q8—4242no—
383D4—41no—
384D4—41no—
385D4—41no—
386D4—41no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—531no—
402C2×C4—531no—
403C2×C4—1062no—
404C2×C4—1062no—
405C2×C4—1061no—
406C2×C4—1061no—
407C2×C4—1061no—
408C2×C4—4241no—
409C2×C4—4241no—
410C2×C4—4241no—
411C2×C4—4241no—
412C2×C4—4241no—
413C2×C4—4241no—
414C2×C4—4241no—
415C2×C4—2122no—
416C2×C4—2122no—
417C2×C4—2122no—
418C2×C4—2121no—
419C2×C4—2121no—
420C2×C4—2121no—
421C2×C4—2121no—
422C2×C4—2121no—
423C2×C4—2121no—
424C2×C4—2121no—
425C2×C4—2121no—
426C2×C4#1313318211yescharacteristic
427C2×C4C4×Dic10611yescharacteristic
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—81no—
443C22—81no—
444C22#1681052711yescharacteristic
445C22#99×D5311yescharacteristic
446C22#1681059511yescharacteristic
447C4—21no—
448C4—21no—
449C4—42no—
450C4—82no—
451C4—81no—
452C4—161no—
453C4—532no—
454C4—8481no—
455C4—1061no—
456C4—4242no—
457C4—4241no—
458C4—2122no—
459C4—2121no—
460C4—2121no—
461C4—2121no—
462C2—21no—
463C2—21no—
464C2—41no—
465C2—41no—
466C2—41no—
467C2—41no—
468C2#2027569811yescharacteristic
469C2#353×D5311yescharacteristic
470C2#2027638811yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1393C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#1393D53
hypocenterC53#13298
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
8. C2×#13133216C4semidirect
17. #16810493C4semidirect
18. #16810492C4semidirect
20. C4×#11719919C4semidirect
57. #13133132Q8semidirect
71. #99×C53Q8semidirect
262. C53#13298semidirect
304. #99Dic106semidirect

Maximal subgroups

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

subgroupindex
#202796662
#202747012
#99×Dic532
#202728402
#202728392
C53×#13932
#1329853