Finite Group Databasebeta

← #27299233

Subgroups — #27299233

Order 13568.

548 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 113 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#20277389C211yesmaximal, characteristic
2C2×#16810414C211yesmaximal, characteristic
3C2×#16810412C211yesmaximal, characteristic
4#100×Dic53C211yesmaximal, characteristic
5#99×Dic53C211yesmaximal, characteristic
6#20272833C211yesmaximal, characteristic
7C53×#1390C211yesmaximal, characteristic
8C2×#13133198C2211yescharacteristic
9C2×D4×Dic53C2211yescharacteristic
10C22×#11719919C411yescharacteristic, complemented
11C2×#13133134C2211yescharacteristic
12C2×#13133132C2211yescharacteristic
13M16×Dic53—21no—
14#16810520—21no—
15#16810511—21no—
16#16810505—21no—
17#16810496C411yescharacteristic, complemented
18#16810495—21no—
19#16810484C411yescharacteristic, complemented
20C22⋊C4×Dic53—21no—
21C22⋊C4×Dic53C411yescharacteristic, complemented
22#16810414C414yescomplemented
23#16810412C414yescomplemented
24#16810391—21no—
25#16810389—21no—
26C2×#100×C53C2211yescharacteristic
27C2×#99×C53C2211yescharacteristic
28C53×#386C2211yescharacteristic
29C23×Dic53C2×C411yescharacteristic
30C23×Dic53C2×C411yescharacteristic
31C23×Dic53D411yescharacteristic
32C2×#11719917—21no—
33#13133198—21no—
34#13133198—21no—
35#13133198C2×C412yes—
36#13133198C2×C412yes—
37D4×Dic53—41no—
38D4×Dic53—41no—
39C2×C4⋊Dic53D411yescharacteristic
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×#11719919—21no—
44C2×#11719919—41no—
45C2×#11719919—41no—
46C2×C22⋊Dic53—21no—
47C2×C22⋊Dic53—21no—
48C2×C22⋊Dic53C2×C411yescharacteristic
49C2×C22⋊Dic53C2×C411yescharacteristic
50C2×C22⋊Dic53D411yescharacteristic
51C2×C4×Dic53—21no—
52C2×C4×Dic53—41no—
53C2×C4×Dic53D411yescharacteristic
54#13133134—21no—
55#13133134C2×C412yes—
56#13133132—21no—
57#13133132C2×C412yes—
58#13133127—41no—
59C8×Dic53—41no—
60#13133124—21no—
61#13133124—21no—
62#13133124—21no—
63#13133124—21no—
64#13133124—41no—
65C22×D4×C53C2311yescharacteristic
66C23×C4×C53C2×C411yescharacteristic
67C2×M16×C53—21no—
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53C2×C411yescharacteristic
71#100×C53—21no—
72#100×C53C2×C412yes—
73#99×C53—21no—
74#99×C53C2×C412yes—
75#95×C53—21no—
76C24×C53C22×C411yescharacteristic
77C24×C53C2×D411yescharacteristic
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53C22×C411yescharacteristic
81C2×D4×C53C22×C411yescharacteristic
82C2×D4×C53#4111yescharacteristic
83C2×D4×C53#4111yescharacteristic
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—81no—
89C22×C4×C53C2×D411yescharacteristic
90C22×Dic53—21no—
91C22×Dic53—21no—
92C22×Dic53—21no—
93C22×Dic53—21no—
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—41no—
98C22×Dic53—41no—
99C22×Dic53—41no—
100C22×Dic53—81no—
101C22×Dic53—81no—
102C22×Dic53C4212yescomplemented
103C22×Dic53C22⋊C411yescharacteristic
104C22×Dic53C22⋊C411yescharacteristic
105C22×Dic53C22⋊C411yescharacteristic
106C22×Dic53C22⋊C411yescharacteristic
107M16×C53—22no—
108M16×C53—41no—
109C2×C8×C53—41no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—81no—
114C22⋊Dic53—21no—
115C22⋊Dic53—22no—
116C22⋊Dic53—22no—
117C22⋊Dic53—22no—
118C22⋊Dic53—41no—
119C22⋊Dic53—41no—
120C22⋊Dic53—41no—
121C22⋊Dic53C4212yescomplemented
122C4⋊Dic53—41no—
123#11719919—22no—
124#11719919—22no—
125#11719919—42no—
126#11719919—41no—
127#11719919—41no—
128#11719919—81no—
129C4×Dic53—21no—
130C4×Dic53—21no—
131C4×Dic53—81no—
132C4×Dic53—82no—
133#11719917—22no—
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×C4211yescharacteristic
145C23×C53C2×C22⋊C411yescharacteristic
146C23×C53#11711yescharacteristic
147C23×C53C4×D411yescharacteristic
148C23×C53C4×D411yescharacteristic
149D4×C53—41no—
150D4×C53—41no—
151D4×C53—41no—
152D4×C53—41no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—81no—
163C2×C4×C53—81no—
164C2×C4×C53—82no—
165C2×C4×C53C4×D411yescharacteristic
166C2×C4×C53C4×D411yescharacteristic
167C2×Dic53—22no—
168C2×Dic53—22no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—41no—
173C2×Dic53—42no—
174C2×Dic53—41no—
175C2×Dic53—42no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53—42no—
180C2×Dic53—41no—
181C2×Dic53—41no—
182C2×Dic53—41no—
183C2×Dic53—42no—
184C2×Dic53—81no—
185C2×Dic53—81no—
186C2×Dic53—81no—
187C2×Dic53—81no—
188C2×Dic53—81no—
189C2×Dic53#9911yescharacteristic
190C2×Dic53#9911yescharacteristic
191C8×C53—42no—
192#88322—42no—
193#12621—531nomaximal, Sylow
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—42no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53—81no—
210C22×C53C4×C22⋊C411yescharacteristic
211C22×C53C2×#9911yescharacteristic
212C22×C53#41011yescharacteristic
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—81no—
216C4×C53—82no—
217C4×C53—81no—
218Dic53—21no—
219Dic53—21no—
220Dic53—42no—
221Dic53—41no—
222Dic53—41no—
223Dic53—41no—
224Dic53—42no—
225Dic53—82no—
226#2322—531no—
227C2×#352—531no—
228C2×#352—531no—
229C4×#100—531no—
230C4×#99—531no—
231#1390—531no—
232#1390D5311yescharacteristic
233C2×C53—21no—
234C2×C53—21no—
235C2×C53—41no—
236C2×C53—41no—
237C2×C53—41no—
238C2×C53—41no—
239C2×C53C4×#9911yescharacteristic
240C2×C53#212111yescharacteristic
241C2×C53#212311yescharacteristic
242C2×#123—531no—
243C2×C4×D4—531no—
244C22×C4⋊C4—531no—
245C2×#100—531no—
246C2×#100C2×D5311yescharacteristic
247C2×#99—531no—
248C2×#99C2×D5311yescharacteristic
249C4×M16—1061no—
250#389—1061no—
251#388—1061no—
252#387—1061no—
253#386C2×D5311yescharacteristic
254#385—531no—
255#385—1061no—
256#381—531no—
257C4×C22⋊C4—531no—
258C4×C22⋊C4—1061no—
259#352—534no—
260#352—534no—
261#343—1061no—
262#342—1061no—
263C53#1262111yescharacteristic, complemented, Sylow
264C22×D4C22×D5311yescharacteristic
265C23×C4—531no—
266C23×C4—531no—
267C23×C4—531no—
268C23×C4C4×D5311yescharacteristic
269C2×M16—21no—
270C2×M16—1061no—
271#123—532no—
272#123—532no—
273#123—1061no—
274#123—1061no—
275C4×D4—2121no—
276C4×D4—2121no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4—1061no—
279C2×C4⋊C4—1061no—
280C2×C4⋊C4—1061no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4—2121no—
283C2×C4⋊C4—2121no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—531no—
287C2×C22⋊C4—531no—
288C2×C22⋊C4—531no—
289C2×C22⋊C4—1061no—
290C2×C22⋊C4—1061no—
291C2×C22⋊C4C4×D5311yescharacteristic
292C2×C42—531no—
293C2×C42—1061no—
294C2×C42—2121no—
295#100—21no—
296#100—532no—
297#100—1061no—
298#100C4×D5312yes—
299#99—21no—
300#99—532no—
301#99—1061no—
302#99C4×D5312yes—
303#97—2121no—
304C4×C8—2121no—
305#95—21no—
306#95—1061no—
307#95—1061no—
308#95—1061no—
309#95—1061no—
310#95—2121no—
311C24C2×C4×D5311yescharacteristic
312C24D4×D5311yescharacteristic
313C2×D4—41no—
314C2×D4—41no—
315C2×D4C2×C4×D5311yescharacteristic
316C2×D4C2×C4×D5311yescharacteristic
317C2×D4#1171994711yescharacteristic
318C2×D4#1171994911yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—81no—
324C22×C4—531no—
325C22×C4—532no—
326C22×C4—531no—
327C22×C4—531no—
328C22×C4—531no—
329C22×C4—1061no—
330C22×C4—1061no—
331C22×C4—1061no—
332C22×C4—1061no—
333C22×C4—1061no—
334C22×C4—1061no—
335C22×C4—1061no—
336C22×C4—4241no—
337C22×C4—4241no—
338C22×C4—2121no—
339C22×C4—2121no—
340C22×C4—2121no—
341C22×C4D4×D5311yescharacteristic
342M16—22no—
343M16—41no—
344M16—1062no—
345M16—2121no—
346C2×C8—41no—
347C2×C8—2121no—
348C4⋊C4—1062no—
349C4⋊C4—1062no—
350C4⋊C4—4241no—
351C4⋊C4—2121no—
352C4⋊C4—2121no—
353C4⋊C4—2121no—
354C4⋊C4—2122no—
355C22⋊C4—22no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—81no—
359C22⋊C4—532no—
360C22⋊C4—1061no—
361C22⋊C4—1062no—
362C22⋊C4—1062no—
363C22⋊C4—1062no—
364C22⋊C4—2121no—
365C22⋊C4—2121no—
366C22⋊C4—2121no—
367C42—1061no—
368C42—1061no—
369C42—4241no—
370C42—4242no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23C42×D5311yescharacteristic
380C23C22⋊C4×D5311yescharacteristic
381C23#1313317511yescharacteristic
382C23#1313317711yescharacteristic
383C23#1313317911yescharacteristic
384D4—41no—
385D4—41no—
386D4—41no—
387D4—41no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—82no—
400C2×C4—531no—
401C2×C4—531no—
402C2×C4—1061no—
403C2×C4—1061no—
404C2×C4—1061no—
405C2×C4—1062no—
406C2×C4—1062no—
407C2×C4—4241no—
408C2×C4—4241no—
409C2×C4—4241no—
410C2×C4—4241no—
411C2×C4—4241no—
412C2×C4—2121no—
413C2×C4—2122no—
414C2×C4—2122no—
415C2×C4—2121no—
416C2×C4—2121no—
417C2×C4—2122no—
418C2×C4—2121no—
419C2×C4—2121no—
420C2×C4—2121no—
421C2×C4—2121no—
422C2×C4—2121no—
423C2×C4—2122no—
424C2×C4#1313317711yescharacteristic
425C2×C4#1313317911yescharacteristic
426C8—42no—
427C8—2122no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—42no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—81no—
444C22#1681047011yescharacteristic
445C22#99×D5311yescharacteristic
446C22#1681059511yescharacteristic
447C4—21no—
448C4—21no—
449C4—81no—
450C4—82no—
451C4—81no—
452C4—1061no—
453C4—1061no—
454C4—4242no—
455C4—2122no—
456C4—2122no—
457C4—2121no—
458C4—2121no—
459C4—2121no—
460C2—21no—
461C2—21no—
462C2—41no—
463C2—41no—
464C2—41no—
465C2—41no—
466C2#2027450211yescharacteristic
467C2#2027635811yescharacteristic
468C2#2027636711yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC23×C53C2×C42
FrattiniC22×D4C22×D53
FittingC53×#1390C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#1390D53
hypocenterC53#12621
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1390

Complements

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

NH ≅ G/Ntype
10. C22×#11719919C4semidirect
17. #16810496C4semidirect
19. #16810484C4semidirect
21. C22⋊C4×Dic53C4semidirect
22. #16810414C4semidirect
23. #16810412C4semidirect
102. C22×Dic53C42semidirect
121. C22⋊Dic53C42semidirect
263. C53#12621semidirect

Maximal subgroups

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

subgroupindex
#202773892
C2×#168104142
C2×#168104122
#100×Dic532
#99×Dic532
#202728332
C53×#13902
#1262153