Finite Group Databasebeta

← #27277652

Subgroups — #27277652

Order 13568.

472 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 99 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#20277651C211yesmaximal, characteristic
2#20274700C211yesmaximal, characteristic
3#20273031C211yesmaximal, characteristic
4#20272594C211yesmaximal, characteristic, complemented
5#20272583C211yesmaximal, characteristic, complemented
6#20272582C211yesmaximal, characteristic, complemented
7C53×#1294C211yesmaximal, characteristic, complemented
8C2×C4×C22⋊D53C2211yescharacteristic
9C2×#13133131—21no—
10C2×C22⋊#88322C2211yescharacteristic
11#16810515—21no—
12#16810510—21no—
13#16810502—21no—
14#16810490C411yescharacteristic, complemented
15#16810485C411yescharacteristic, complemented
16#16810483C2211yescharacteristic
17#16810483C2211yescharacteristic
18C4×C22⋊Dic53—21no—
19#16810470C411yescharacteristic, complemented
20#16810468C411yescharacteristic, complemented
21#16810389—21no—
22#16810348—21no—
23#16810346—21no—
24C2×C22⋊C8×C53C2211yescharacteristic
25C53×#380C2211yescharacteristic
26C2×#95×C53C2211yescharacteristic
27C22×C22⋊D53D411yescharacteristic
28C22×C4×D53C2×C411yescharacteristic
29C22×#88322—21no—
30C4×C22⋊D53—21no—
31C4×C22⋊D53—21no—
32C4×C22⋊D53—21no—
33C4×C22⋊D53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×#11719919D411yescharacteristic
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53D411yescharacteristic
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921D411yescharacteristic
43C2×C4×Dic53—21no—
44C2×C4×Dic53—21no—
45C2×C4×Dic53C2×C411yescharacteristic
46#13133131—42no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124C2×C411yescharacteristic
55#13133124C2×C411yescharacteristic
56C23×C4×C53—21no—
57C23×C4×C53C2311yescharacteristic
58C22×C8×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C22⋊C8×C53—21no—
62C22⋊C8×C53—21no—
63#95×C53—21no—
64#95×C53—21no—
65#95×C53C2×C411yescharacteristic
66#95×C53C2×C411yescharacteristic
67C24×C53C2×D411yescharacteristic
68C23×D53C22⋊C411yescharacteristic
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—42no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—81no—
76C22×C4×C53—81no—
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C22×C4×C53#4111yescharacteristic
83C2×C22⋊D53—21no—
84C2×C22⋊D53—21no—
85C2×C22⋊D53—21no—
86C2×C22⋊D53—21no—
87C2×C22⋊D53—21no—
88C2×C22⋊D53—21no—
89C22×Dic53—21no—
90C22×Dic53—21no—
91C22×Dic53—41no—
92C22×Dic53C22⋊C411yescharacteristic
93C22×Dic53C22⋊C411yescharacteristic
94C22×Dic53C22⋊C411yescharacteristic
95C2×C4×D53—21no—
96C2×C4×D53—21no—
97C2×C4×D53—81no—
98C2×C8×C53—41no—
99C2×C8×C53—81no—
100C4⋊C4×C53—81no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C22⋊Dic53—21no—
104C22⋊Dic53—21no—
105C22⋊Dic53—41no—
106C22⋊Dic53—41no—
107C22⋊Dic53—41no—
108C22⋊Dic53—41no—
109#11719921—21no—
110#11719921—21no—
111#11719921—42no—
112#11719921—81no—
113C4⋊Dic53—81no—
114C4⋊Dic53—81no—
115#11719919—21no—
116#11719919—21no—
117C4×Dic53—21no—
118C4×Dic53—21no—
119C4×Dic53—42no—
120C4×Dic53—81no—
121C2×#88322—41no—
122C2×#88322—81no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53C2×C22⋊C411yescharacteristic
126C23×C53#12311yescharacteristic
127C23×C53#12311yescharacteristic
128C23×C53#13511yescharacteristic
129C23×C53#13511yescharacteristic
130C23×C53AGL(1,ℤ/8)11yescharacteristic
131C23×C53#13711yescharacteristic
132C22×D53—21no—
133C22×D53—21no—
134C22×D53—21no—
135C22×D53—41no—
136C22×D53—41no—
137C2×C4×C53—22no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—82no—
150C2×C4×C53—82no—
151C2×C4×C53—81no—
152C2×C4×C53—81no—
153C2×C4×C53—81no—
154C2×C4×C53#11711yescharacteristic
155C2×C4×C53#11711yescharacteristic
156C2×C4×C53#12311yescharacteristic
157C2×C4×C53#12311yescharacteristic
158C22⋊D53—41no—
159C22⋊D53—41no—
160C22⋊D53—41no—
161C22⋊D53—41no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—81no—
176C2×Dic53—81no—
177C2×Dic53—81no—
178C4×D53—81no—
179C4×D53—81no—
180C8×C53—81no—
181#88322—81no—
182#8763—531nomaximal, Sylow
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53#38111yescharacteristic
194C22×C53#41611yescharacteristic
195C22×C53#41711yescharacteristic
196C22×C53#48211yescharacteristic
197C22×C53#48211yescharacteristic
198C22×C53#48511yescharacteristic
199C22×C53#48511yescharacteristic
200C2×D53—41no—
201C2×D53—41no—
202C2×D53—41no—
203C2×D53—81no—
204C2×D53—81no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—42no—
210C4×C53—81no—
211C4×C53—161no—
212Dic53—41no—
213Dic53—41no—
214Dic53—41no—
215Dic53—81no—
216Dic53—81no—
217Dic53—161no—
218#2353—531no—
219#1799—531no—
220#1459—531no—
221#1294—531no—
222#1294D5311yescharacteristic, complemented
223#1289—531no—
224#1289—531no—
225C2×C53—21no—
226C2×C53—21no—
227C2×C53—21no—
228C2×C53—21no—
229C2×C53—21no—
230C2×C53—21no—
231C2×C53#182511yescharacteristic
232C2×C53#182711yescharacteristic
233C2×C53#182811yescharacteristic
234D53—81no—
235C2×C4×D4—531no—
236C2×#99—1061no—
237C2×C22⋊C8—531no—
238C2×C22⋊C8C2×D5311yescharacteristic
239#388—1061no—
240#387—1061no—
241#386—1061no—
242#383—531no—
243#381—531no—
244#380—531no—
245#380—531no—
246#380C2×D5311yescharacteristic
247C4×C22⋊C4—531no—
248C4×C22⋊C4—1061no—
249#376—531no—
250C2×#95C2×D5311yescharacteristic
251#342—1061no—
252#324—1061no—
253#323—1061no—
254C53#876311yescharacteristic, complemented, Sylow
255C22×D4—531no—
256C23×C4—21no—
257C23×C4—531no—
258C23×C4C22×D5311yescharacteristic
259C22×C8—21no—
260C22×C8—1061no—
261C4×D4—1061no—
262C4×D4—1061no—
263C4×D4—1061no—
264C4×D4—1061no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—1061no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—531no—
271C2×C22⋊C4—531no—
272C2×C22⋊C4—1061no—
273C2×C22⋊C4—1061no—
274C2×C22⋊C4—1061no—
275C2×C22⋊C4—1061no—
276C2×C42—531no—
277C2×C42—1061no—
278C2×C42—1061no—
279#99—2122no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—1061no—
283C22⋊C8—1061no—
284#95—21no—
285#95—21no—
286#95—531no—
287#95—531no—
288#95—1061no—
289#95—1061no—
290#95—1061no—
291#95—1061no—
292#95—1061no—
293#95C4×D5311yescharacteristic
294#95C4×D5311yescharacteristic
295C24—531no—
296C24D4×D5311yescharacteristic
297C2×D4—1061no—
298C2×D4—1061no—
299C2×D4—1061no—
300C2×D4—1061no—
301C2×D4—1061no—
302C2×D4—1061no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4—531no—
312C22×C4—531no—
313C22×C4—531no—
314C22×C4—1061no—
315C22×C4—1061no—
316C22×C4—1061no—
317C22×C4—1061no—
318C22×C4—4241no—
319C22×C4—2121no—
320C22×C4C2×C4×D5311yescharacteristic
321C22×C4#1171994511yescharacteristic
322C22×C4#1171994511yescharacteristic
323C22×C4D4×D5311yescharacteristic
324C22×C4#1171994711yescharacteristic
325C22×C4#1171994911yescharacteristic
326C2×C8—41no—
327C2×C8—81no—
328C2×C8—4241no—
329C2×C8—2121no—
330C4⋊C4—81no—
331C4⋊C4—1061no—
332C4⋊C4—1061no—
333C4⋊C4—4241no—
334C4⋊C4—4241no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—1061no—
338C22⋊C4—1061no—
339C22⋊C4—1061no—
340C22⋊C4—1061no—
341C22⋊C4—4241no—
342C22⋊C4—2122no—
343C22⋊C4—2121no—
344C22⋊C4—2121no—
345C22⋊C4—2121no—
346C22⋊C4—2121no—
347C42—1061no—
348C42—1061no—
349C42—4241no—
350C42—2122no—
351C23—41no—
352C23—41no—
353C23—1061no—
354C23—1061no—
355C23—1061no—
356C23—2121no—
357C23—2121no—
358C23C22⋊C4×D5311yescharacteristic
359C23#1313319911yescharacteristic
360C23#1313320211yescharacteristic
361C23#1313323611yescharacteristic
362C23#1313323811yescharacteristic
363C23#1313324411yescharacteristic
364C23#1313324911yescharacteristic
365D4—2121no—
366D4—2121no—
367D4—2121no—
368D4—2121no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—1061no—
387C2×C4—1061no—
388C2×C4—1061no—
389C2×C4—1061no—
390C2×C4—1061no—
391C2×C4—4241no—
392C2×C4—4241no—
393C2×C4—4241no—
394C2×C4—4241no—
395C2×C4—4241no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4—2121no—
399C2×C4—2121no—
400C2×C4—2121no—
401C2×C4—2121no—
402C2×C4—2121no—
403C2×C4—2121no—
404C2×C4#1313317411yescharacteristic
405C2×C4#1313317511yescharacteristic
406C2×C4#1313319911yescharacteristic
407C2×C4#1313320211yescharacteristic
408C8—81no—
409C8—4241no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—4241no—
421C22—4241no—
422C22—2121no—
423C22—2121no—
424C22—2121no—
425C22#1681048511yescharacteristic
426C22#1681061611yescharacteristic
427C22#1681062311yescharacteristic
428C22#1681094511yescharacteristic
429C22#1681094911yescharacteristic
430C22#1681096211yescharacteristic
431C22#1681096611yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—81no—
438C4—161no—
439C4—8481no—
440C4—4241no—
441C4—4241no—
442C4—2121no—
443C4—2121no—
444C4—2121no—
445C2—21no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—4241no—
452C2#2027480811yescharacteristic
453C2#2027481811yescharacteristic
454C2#2027482211yescharacteristic
subgroupG/H
centerC22#16810485
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1294C2
socleC22×C53#381
prime radicalC22×C22⋊D53D4
cosocleC23×C4×C53C23
hypercenter#1294D53
hypocenterC53#8763
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
4. #20272594C2semidirect
5. #20272583C2semidirect
6. #20272582C2semidirect
7. C53×#1294C2semidirect
14. #16810490C4semidirect
15. #16810485C4semidirect
19. #16810470C4semidirect
20. #16810468C4semidirect
222. #1294D53semidirect
254. C53#8763semidirect

Maximal subgroups

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

subgroupindex
#202776512
#202747002
#202730312
#202725942
#202725832
#202725822
C53×#12942
#876353