Finite Group Databasebeta

← #27280655

Subgroups — #27280655

Order 13568.

522 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 107 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#20285487C211yesmaximal, characteristic
2#20277795C211yesmaximal, characteristic
3#20275701C211yesmaximal, characteristic
4#20273288C211yesmaximal, characteristic
5#20272594C211yesmaximal, characteristic
6#20272562C211yesmaximal, characteristic
7C2×C53×#328C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810958C412yescomplemented
10#16810956—21no—
11#16810862—22no—
12#16810779C412yescomplemented
13#16810778—21no—
14C2×C22⋊#88322C2211yescharacteristic
15#16810527—21no—
16#16810508—21no—
17#16810505—21no—
18#16810496—21no—
19#16810483C2211yescharacteristic
20#16810474—21no—
21C4×C22⋊Dic53—21no—
22C4×C22⋊Dic53C2211yescharacteristic
23#16810390—21no—
24#16810390—21no—
25#16810361—22no—
26C2×C22⋊Q8×C53C2211yescharacteristic
27C2×C22⋊C8×C53C2211yescharacteristic
28C2×#95×C53C2211yescharacteristic
29C53×#328—21no—
30C53×#328—21no—
31C22×Dic106D411yescharacteristic
32C2×#11719925—22no—
33C22×#88322—21no—
34#13133196—21no—
35#13133196—21no—
36#13133196C2×C412yes—
37#13133196D412yes—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53D411yescharacteristic
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53—21no—
45C2×C22⋊Dic53D411yescharacteristic
46C2×C22⋊Dic53D411yescharacteristic
47C2×C4×Dic53—21no—
48C2×C4×Dic53—21no—
49C2×C4×Dic53—21no—
50#13133150—22no—
51#13133141—22no—
52#13133140—22no—
53C22⋊#88322—21no—
54C22⋊#88322C2×C412yes—
55#13133124—21no—
56#13133124—21no—
57#13133124—21no—
58#13133124—21no—
59#13133124—21no—
60#13133124—21no—
61C22×Q8×C53C2×C411yescharacteristic
62C23×C4×C53—21no—
63C23×C4×C53C2311yescharacteristic
64C22×C8×C53—21no—
65C22⋊Q8×C53—21no—
66C22⋊Q8×C53—21no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53C2×C411yescharacteristic
71C2×C22⋊C4×C53—21no—
72C22⋊C8×C53—21no—
73C22⋊C8×C53—21no—
74#95×C53—21no—
75#95×C53D411yescharacteristic
76#95×C53D411yescharacteristic
77C24×C53C2×D411yescharacteristic
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—81no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—42no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—81no—
89C22×C4×C53—81no—
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53#4111yescharacteristic
95C22×C4×C53#4111yescharacteristic
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—21no—
99C22×Dic53—21no—
100C22×Dic53—41no—
101C2×Dic106—21no—
102C2×Dic106—81no—
103C2×Dic106C22⋊C412yes—
104C2×C8×C53—41no—
105C2×C8×C53—81no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—41no—
112C22⋊Dic53—22no—
113C22⋊Dic53—22no—
114C22⋊Dic53—21no—
115C22⋊Dic53—21no—
116C22⋊Dic53—21no—
117C22⋊Dic53—41no—
118C22⋊Dic53—41no—
119C22⋊Dic53Q1612yescomplemented
120#11719925—82no—
121C4⋊Dic53—21no—
122C4⋊Dic53—81no—
123C4⋊Dic53C22⋊C412yes—
124#11719919—22no—
125#11719919—22no—
126#11719919—42no—
127#11719919—81no—
128C4×Dic53—42no—
129C4×Dic53—81no—
130C4×Dic53—81no—
131C2×#88322—22no—
132C2×#88322—81no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53C4×D411yescharacteristic
136C23×C53#12011yescharacteristic
137C23×C53#12111yescharacteristic
138C23×C53C2×Q1611yescharacteristic
139C23×C53#13511yescharacteristic
140C23×C53#13711yescharacteristic
141C23×C53#13711yescharacteristic
142Q8×C53—42no—
143Q8×C53—81no—
144C2×C4×C53—22no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—82no—
159C2×C4×C53—82no—
160C2×C4×C53—81no—
161C2×C4×C53—81no—
162C2×C4×C53C2×C22⋊C411yescharacteristic
163C2×C4×C53C4×D411yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12311yescharacteristic
166C2×Dic53—22no—
167C2×Dic53—22no—
168C2×Dic53—22no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—81no—
179C2×Dic53—81no—
180C2×Dic53—81no—
181Dic106—42no—
182Dic106—81no—
183C8×C53—81no—
184#88322—42no—
185#9276—531nomaximal, Sylow
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53#38611yescharacteristic
197C22×C53C4×Q1611yescharacteristic
198C22×C53#44111yescharacteristic
199C22×C53#45111yescharacteristic
200C22×C53#45211yescharacteristic
201C22×C53#46211yescharacteristic
202C22×C53#46411yescharacteristic
203C4×C53—21no—
204C4×C53—21no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—42no—
208C4×C53—81no—
209C4×C53—81no—
210C4×C53—81no—
211Dic53—42no—
212Dic53—42no—
213Dic53—42no—
214Dic53—81no—
215Dic53—161no—
216#3189—531no—
217#2368—531no—
218#1996—531no—
219#1520—531no—
220C2×#328D5311yescharacteristic
221#1294—531no—
222#1282—531no—
223C2×C53—21no—
224C2×C53—21no—
225C2×C53—21no—
226C2×C53—21no—
227C2×C53—21no—
228C2×C53—21no—
229C2×C53#187811yescharacteristic
230C2×C53#190611yescharacteristic
231C2×C53#191311yescharacteristic
232C2×C22⋊Q8—531no—
233C2×C22⋊Q8C2×D5311yescharacteristic
234#484—532no—
235#484—1061no—
236#467—1062no—
237#451—532no—
238#451—1061no—
239C2×C22⋊C8—531no—
240C2×C22⋊C8C2×D5311yescharacteristic
241#391—1061no—
242#387—1061no—
243#387—1061no—
244#385—1061no—
245#380—531no—
246C4×C4⋊C4—1061no—
247C4×C22⋊C4—531no—
248C4×C22⋊C4—1061no—
249C2×#95C2×D5311yescharacteristic
250#342—1061no—
251#342—1061no—
252#328—21no—
253#328—21no—
254#328—1062no—
255C53#927611yescharacteristic, complemented, Sylow
256C22×Q8—531no—
257C22×Q8C4×D5311yescharacteristic
258C23×C4—21no—
259C23×C4C22×D5311yescharacteristic
260C2×Q16—1062no—
261C22×C8—21no—
262C22×C8—1061no—
263C22⋊Q8—21no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—532no—
268C22⋊Q8—532no—
269C22⋊Q8—1061no—
270C22⋊Q8—1061no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—531no—
273C2×C4⋊C4—1061no—
274C2×C4⋊C4—1061no—
275C2×C4⋊C4—1061no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4C4×D5311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—531no—
280C2×C22⋊C4—531no—
281C2×C22⋊C4—1061no—
282C2×C22⋊C4—1061no—
283C2×C42—1061no—
284C2×C42—1061no—
285C2×C42—1061no—
286#107—1062no—
287#103—1062no—
288#103—1062no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—532no—
292C22⋊C8—1061no—
293#95—21no—
294#95—1061no—
295#95—1061no—
296#95—1061no—
297#95—1061no—
298#95—1061no—
299#95—1061no—
300#95C22⋊D5311yescharacteristic
301#95C22⋊D5311yescharacteristic
302C24D4×D5311yescharacteristic
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—81no—
306C2×Q8—532no—
307C2×Q8—1061no—
308C2×Q8—4241no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4—1061no—
319C22×C4—1061no—
320C22×C4—1061no—
321C22×C4—1061no—
322C22×C4—2121no—
323C22×C4C2×C4×D5311yescharacteristic
324C22×C4#1171994511yescharacteristic
325C22×C4D4×D5311yescharacteristic
326C22×C4D4×D5311yescharacteristic
327C22×C4#1171994711yescharacteristic
328C22×C4C2×C22⋊D5311yescharacteristic
329Q16—4242no—
330C2×C8—41no—
331C2×C8—81no—
332C2×C8—1062no—
333C2×C8—4241no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—532no—
339C4⋊C4—1062no—
340C4⋊C4—1062no—
341C4⋊C4—1061no—
342C4⋊C4—4241no—
343C4⋊C4—4241no—
344C4⋊C4—2122no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—532no—
348C22⋊C4—1062no—
349C22⋊C4—1062no—
350C22⋊C4—1061no—
351C22⋊C4—1061no—
352C22⋊C4—1061no—
353C22⋊C4—2121no—
354C22⋊C4—2121no—
355C42—4241no—
356C42—4241no—
357C42—2122no—
358C23—41no—
359C23—41no—
360C23#1313317711yescharacteristic
361C23#1313318511yescharacteristic
362C23#1313319011yescharacteristic
363C23Q16×D5311yescharacteristic
364C23#1313323811yescharacteristic
365C23#1313324611yescharacteristic
366C23#1313324911yescharacteristic
367Q8—42no—
368Q8—81no—
369Q8—4241no—
370Q8—2122no—
371C2×C4—22no—
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—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—1062no—
390C2×C4—1062no—
391C2×C4—1062no—
392C2×C4—4241no—
393C2×C4—4241no—
394C2×C4—4241no—
395C2×C4—2121no—
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×C4C22⋊C4×D5311yescharacteristic
405C2×C4C4×C22⋊D5311yescharacteristic
406C2×C4#1313318811yescharacteristic
407C2×C4#1313319911yescharacteristic
408C8—81no—
409C8—2122no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22#1681049911yescharacteristic
421C22#1681071511yescharacteristic
422C22#1681072711yescharacteristic
423C22#1681077711yescharacteristic
424C22#1681078211yescharacteristic
425C22#1681083211yescharacteristic
426C22#1681084811yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—42no—
432C4—81no—
433C4—81no—
434C4—81no—
435C4—8481no—
436C4—4241no—
437C4—2122no—
438C4—2122no—
439C4—2122no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2#2027505311yescharacteristic
447C2#2027521911yescharacteristic
448C2#2027525411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#328C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenterC2×#328D53
hypocenterC53#9276
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
9. #16810958C4semidirect
12. #16810779C4semidirect
119. C22⋊Dic53Q16semidirect
255. C53#9276semidirect

Maximal subgroups

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

subgroupindex
#202854872
#202777952
#202757012
#202732882
#202725942
#202725622
C2×C53×#3282
#927653