Finite Group Databasebeta

← #27300058

Subgroups — #27300058

Order 13568.

540 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 103 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#123×Dic53C211yesmaximal, characteristic
2C2×#16810413C211yesmaximal, characteristic
3#20274705C211yesmaximal, characteristic
4#20274701C211yesmaximal, characteristic
5#20272834C211yesmaximal, characteristic
6#20272832C211yesmaximal, characteristic
7C2×C53×#352C211yesmaximal, characteristic
8C2×D4×Dic53C2211yescharacteristic
9C22×C4×Dic53C411yescharacteristic, complemented
10#16810646—21no—
11C2×#13133134C2211yescharacteristic
12C2×#13133132C2211yescharacteristic
13#16810516—21no—
14#16810509—21no—
15#16810503C2211yescharacteristic
16#16810498C411yescharacteristic, complemented
17#16810486C411yescharacteristic, complemented
18#16810481—21no—
19C4⋊C4×Dic53—21no—
20C22⋊C4×Dic53—21no—
21C22⋊C4×Dic53C411yescharacteristic, complemented
22#16810413—22no—
23#16810391—21no—
24#16810389—21no—
25C2×#123×C53C2211yescharacteristic
26C2×#100×C53C2211yescharacteristic
27C2×#99×C53C2211yescharacteristic
28C53×#352—22no—
29C23×Dic53C2×C411yescharacteristic
30C23×Dic53C2×C411yescharacteristic
31C23×Dic53D411yescharacteristic
32C2×#11719917—21no—
33D4×Dic53—41no—
34D4×Dic53—41no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719919—41no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53C2×C411yescharacteristic
41C2×C22⋊Dic53C2×C411yescharacteristic
42C2×C22⋊Dic53D411yescharacteristic
43C2×C4×Dic53—21no—
44C2×C4×Dic53—21no—
45C2×C4×Dic53—21no—
46C2×C4×Dic53—41no—
47C2×C4×Dic53—41no—
48C2×C4×Dic53D411yescharacteristic
49#13133146—41no—
50#13133146—41no—
51#13133134—21no—
52#13133134—21no—
53#13133132—21no—
54#13133132—21no—
55#13133124—21no—
56#13133124—21no—
57#13133124—21no—
58#13133124—21no—
59#13133124—21no—
60#13133124—41no—
61C22×D4×C53C2311yescharacteristic
62C23×C4×C53C2×C411yescharacteristic
63C2×M16×C53—21no—
64#123×C53—21no—
65#123×C53—21no—
66#123×C53D412yes—
67#123×C53Q812yes—
68C2×C4⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53C2×C411yescharacteristic
72#100×C53—21no—
73#100×C53—21no—
74#99×C53—21no—
75#99×C53—21no—
76C24×C53C22×C411yescharacteristic
77C24×C53C2×D411yescharacteristic
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53C2×D411yescharacteristic
81C2×D4×C53C2×Q811yescharacteristic
82C2×D4×C53#4111yescharacteristic
83C2×D4×C53#4111yescharacteristic
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53C4⋊C412yescomplemented
91C22×C4×C53C2×D411yescharacteristic
92C22×Dic53—21no—
93C22×Dic53—21no—
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—42no—
99C22×Dic53—42no—
100C22×Dic53—42no—
101C22×Dic53—41no—
102C22×Dic53—41no—
103C22×Dic53C22⋊C411yescharacteristic
104C22×Dic53C22⋊C411yescharacteristic
105C22×Dic53C22⋊C411yescharacteristic
106C22×Dic53C22⋊C411yescharacteristic
107M16×C53—41no—
108M16×C53—41no—
109C2×C8×C53—41no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—22no—
112C22⋊C4×C53—21no—
113C22⋊C4×C53—22no—
114C22⋊C4×C53—22no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C22⋊C4×C53C4⋊C412yescomplemented
118C22⋊Dic53—41no—
119C22⋊Dic53—41no—
120C22⋊Dic53—42no—
121C22⋊Dic53—41no—
122C22⋊Dic53—41no—
123C22⋊Dic53—41no—
124C4⋊Dic53—41no—
125C4⋊Dic53—42no—
126#11719919—82no—
127C4×Dic53—21no—
128C4×Dic53—21no—
129C4×Dic53—42no—
130C4×Dic53—42no—
131C4×Dic53—41no—
132C4×Dic53—41no—
133#11719917—41no—
134#11719917—41no—
135C2×#88322—41no—
136C23×C53—21no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53C2×C22⋊C411yescharacteristic
145C23×C53C2×C4⋊C411yescharacteristic
146C23×C53#11711yescharacteristic
147C23×C53#12111yescharacteristic
148C23×C53C22⋊Q811yescharacteristic
149D4×C53—41no—
150D4×C53—41no—
151D4×C53—41no—
152D4×C53—41no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—42no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—42no—
163C2×C4×C53—81no—
164C2×C4×C53—81no—
165C2×C4×C53—81no—
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53C22⋊Q811yescharacteristic
168C2×Dic53—22no—
169C2×Dic53—22no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—21no—
173C2×Dic53—42no—
174C2×Dic53—41no—
175C2×Dic53—42no—
176C2×Dic53—41no—
177C2×Dic53—42no—
178C2×Dic53—42no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—41no—
182C2×Dic53—42no—
183C2×Dic53—41no—
184C2×Dic53—41no—
185C2×Dic53—41no—
186C2×Dic53—81no—
187C2×Dic53—81no—
188C2×Dic53—81no—
189C2×Dic53—81no—
190C2×Dic53#9911yescharacteristic
191C2×Dic53#9911yescharacteristic
192C8×C53—81no—
193#88322—81no—
194#12794—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—42no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53—81no—
211C22×C53#38011yescharacteristic
212C22×C53C2×#9911yescharacteristic
213C22×C53#41011yescharacteristic
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—42no—
217C4×C53—42no—
218C4×C53—161no—
219Dic53—21no—
220Dic53—41no—
221Dic53—41no—
222Dic53—41no—
223Dic53—42no—
224Dic53—81no—
225Dic53—161no—
226Dic53#35212yescomplemented
227C4×#123—531no—
228C2×#352—531no—
229C2×#352D5311yescharacteristic
230#1800—531no—
231#1799—531no—
232#1390—531no—
233#1390—531no—
234C2×C53—21no—
235C2×C53—21no—
236C2×C53—41no—
237C2×C53—41no—
238C2×C53—41no—
239C2×C53—41no—
240C2×C53#179911yescharacteristic
241C2×C53C2×#35211yescharacteristic
242C2×C53#212011yescharacteristic
243C2×#123C2×D5311yescharacteristic
244C2×C4×D4—531no—
245C22×C42—531no—
246#423—1061no—
247C2×#100—531no—
248C2×#100C2×D5311yescharacteristic
249C2×#99—531no—
250C2×#99C2×D5311yescharacteristic
251#388—1061no—
252#387—1061no—
253#386—531no—
254#385—531no—
255#381—531no—
256#380—1061no—
257C4×C4⋊C4—1061no—
258C4×C22⋊C4—531no—
259C4×C22⋊C4—1061no—
260#352—22no—
261#352—1062no—
262#343—1061no—
263#342—1061no—
264C53#1279411yescharacteristic, complemented, Sylow
265C22×D4C22×D5311yescharacteristic
266C23×C4—531no—
267C23×C4—531no—
268C23×C4—531no—
269C23×C4C4×D5311yescharacteristic
270C2×M16—21no—
271C2×M16—1061no—
272#123—21no—
273#123—21no—
274#123Dic10612yes—
275#123C22⋊D5312yes—
276C4×D4—2121no—
277C4×D4—2121no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—531no—
280C2×C4⋊C4—1061no—
281C2×C4⋊C4—2121no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—531no—
285C2×C22⋊C4—531no—
286C2×C22⋊C4—531no—
287C2×C22⋊C4—1061no—
288C2×C22⋊C4—1061no—
289C2×C22⋊C4C4×D5311yescharacteristic
290C2×C42—531no—
291C2×C42—1061no—
292C2×C42—1061no—
293C2×C42—1061no—
294C2×C42—2121no—
295C2×C42—2121no—
296C4⋊C8—2121no—
297C4⋊C8—2121no—
298#100—21no—
299#100—21no—
300#100—1061no—
301#100—1061no—
302#99—21no—
303#99—21no—
304#99—1061no—
305#99—1061no—
306#95—1061no—
307#95—1061no—
308#95—1061no—
309#95—1061no—
310#95—1061no—
311#95—2121no—
312C24C2×C4×D5311yescharacteristic
313C24D4×D5311yescharacteristic
314C2×D4—41no—
315C2×D4—41no—
316C2×D4C2×Dic10611yescharacteristic
317C2×D4#1171994711yescharacteristic
318C2×D4#1171994711yescharacteristic
319C2×D4C2×C22⋊D5311yescharacteristic
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—531no—
327C22×C4—531no—
328C22×C4—531no—
329C22×C4—531no—
330C22×C4—1061no—
331C22×C4—1061no—
332C22×C4—1061no—
333C22×C4—1061no—
334C22×C4—1061no—
335C22×C4—1061no—
336C22×C4—2122no—
337C22×C4—2122no—
338C22×C4—2122no—
339C22×C4—2121no—
340C22×C4—2121no—
341C22×C4#1171991912yescomplemented
342C22×C4D4×D5311yescharacteristic
343M16—41no—
344M16—41no—
345M16—2121no—
346M16—2121no—
347C2×C8—41no—
348C2×C8—2121no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—4242no—
352C4⋊C4—2121no—
353C4⋊C4—2122no—
354C22⋊C4—21no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—2121no—
360C22⋊C4—2121no—
361C22⋊C4—2122no—
362C22⋊C4—2121no—
363C22⋊C4—2121no—
364C22⋊C4—2121no—
365C22⋊C4#1171991912yescomplemented
366C42—1061no—
367C42—1061no—
368C42—2121no—
369C42—2121no—
370C42—2122no—
371C42—2122no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23C22⋊C4×D5311yescharacteristic
381C23C2×#1171991911yescharacteristic
382C23#1313317311yescharacteristic
383C23#1313318911yescharacteristic
384C23#1313319411yescharacteristic
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—531no—
403C2×C4—531no—
404C2×C4—1062no—
405C2×C4—1062no—
406C2×C4—1061no—
407C2×C4—1061no—
408C2×C4—1061no—
409C2×C4—4241no—
410C2×C4—4241no—
411C2×C4—4241no—
412C2×C4—4241no—
413C2×C4—2122no—
414C2×C4—2122no—
415C2×C4—2122no—
416C2×C4—2121no—
417C2×C4—2121no—
418C2×C4—2121no—
419C2×C4—2121no—
420C2×C4—2122no—
421C2×C4—2121no—
422C2×C4—2121no—
423C2×C4—2121no—
424C2×C4—2121no—
425C2×C4—2122no—
426C2×C4#1313318911yescharacteristic
427C2×C4#1313319411yescharacteristic
428C8—81no—
429C8—4241no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—42no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—81no—
446C22#1681048111yescharacteristic
447C22#99×D5311yescharacteristic
448C22#1681059511yescharacteristic
449C4—21no—
450C4—21no—
451C4—42no—
452C4—42no—
453C4—161no—
454C4—532no—
455C4—8481no—
456C4—1061no—
457C4—4241no—
458C4—2121no—
459C4—2121no—
460C4—2122no—
461C4—2121no—
462C2—21no—
463C2—21no—
464C2—41no—
465C2—41no—
466C2—41no—
467C2—41no—
468C2#2027470111yescharacteristic
469C2#352×D5311yescharacteristic
470C2#2027635511yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC2×C53×#352C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenterC2×#352D53
hypocenterC53#12794
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#352

Complements

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

NH ≅ G/Ntype
9. C22×C4×Dic53C4semidirect
16. #16810498C4semidirect
17. #16810486C4semidirect
21. C22⋊C4×Dic53C4semidirect
90. C22×C4×C53C4⋊C4semidirect
117. C22⋊C4×C53C4⋊C4semidirect
226. Dic53#352semidirect
264. C53#12794semidirect
341. C22×C4#11719919semidirect
365. C22⋊C4#11719919semidirect

Maximal subgroups

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

subgroupindex
#123×Dic532
C2×#168104132
#202747052
#202747012
#202728342
#202728322
C2×C53×#3522
#1279453