Finite Group Databasebeta

← #27293550

Subgroups — #27293550

Order 13568.

530 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 95 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#20273056C211yesmaximal, characteristic, complemented
2#20273053C211yesmaximal, characteristic, complemented
3#20272663C212yesmaximal, complemented
4#20272643C212yesmaximal, complemented
5C53×#2321C211yesmaximal, characteristic
6#16810680—21no—
7#16810587—21no—
8#16810349—21no—
9#16810347—21no—
10#16810347C2212yes—
11#16810347C2212yes—
12C2×#120×C53C411yescharacteristic
13C2×C4×D4×C53C2211yescharacteristic
14C22×C22⋊C4×C53C411yescharacteristic
15C53×#390—21no—
16C53×#388—21no—
17C53×#388—21no—
18C53×#386—21no—
19C53×#386C411yescharacteristic
20C53×#385C2211yescharacteristic
21C53×#379C2211yescharacteristic
22C4×C22⋊C4×C53C411yescharacteristic
23C2×#11719917—41no—
24C4⋊#88322—41no—
25C22⋊#88322—22no—
26C22⋊#88322—22no—
27C22⋊#88322—41no—
28#13133128—41no—
29C25×C53C2×C411yescharacteristic
30C22×D4×C53—21no—
31C22×D4×C53C2×C411yescharacteristic
32C23×C4×C53C2×C411yescharacteristic
33C23×C4×C53C2×C411yescharacteristic
34C23×C4×C53D411yescharacteristic
35#120×C53—22no—
36#120×C53—22no—
37C4×D4×C53—41no—
38C4×D4×C53—41no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—41no—
48C2×C22⋊C4×C53—41no—
49C2×C22⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53—21no—
54C2×C42×C53D411yescharacteristic
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58C24×C53—21no—
59C24×C53—21no—
60C24×C53—41no—
61C24×C53—41no—
62C24×C53—81no—
63C24×C53—81no—
64C24×C53C22⋊C411yescharacteristic
65C24×C53M1612yescomplemented
66C24×C53C22×C411yescharacteristic
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—24no—
70C2×D4×C53—41no—
71C2×D4×C53—41no—
72C2×D4×C53—81no—
73C2×D4×C53—81no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—41no—
82C22×C4×C53—81no—
83C22×C4×C53—81no—
84C22×C4×C53—81no—
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22⋊C411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C4⋊C4×C53—41no—
91C4⋊C4×C53—81no—
92C22⋊C4×C53—24no—
93C22⋊C4×C53—21no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—42no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—81no—
101C22⋊C4×C53—81no—
102C22⋊C4×C53—81no—
103C22⋊C4×C53M1612yescomplemented
104C42×C53—21no—
105C42×C53—21no—
106C42×C53—81no—
107#11719917—82no—
108C2×#88322—42no—
109C2×#88322—42no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—22no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—42no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—42no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—82no—
129C23×C53—82no—
130C23×C53—82no—
131C23×C53—82no—
132C23×C53—84no—
133C23×C53C2×C22⋊C411yescharacteristic
134C23×C53C2×M1611yescharacteristic
135C23×C53C2×M1611yescharacteristic
136D4×C53—42no—
137D4×C53—42no—
138D4×C53—84no—
139C2×C4×C53—22no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
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—41no—
159C2×C4×C53—81no—
160C2×C4×C53—81no—
161C2×C4×C53—81no—
162C2×C4×C53—82no—
163C2×C4×C53#9911yescharacteristic
164C2×C4×C53#9911yescharacteristic
165#88322—82no—
166#88322—82no—
167#11527—531nomaximal, Sylow
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53—42no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—42no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53—84no—
188C22×C53—84no—
189C22×C53—81no—
190C22×C53—81no—
191C22×C53#40711yescharacteristic
192C22×C53C2×#9911yescharacteristic
193C22×C53#41311yescharacteristic
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—42no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—41no—
201C4×C53—81no—
202C4×C53—81no—
203#2321D5311yescharacteristic
204#1466—531no—
205#1465—531no—
206#1326—532no—
207#1316—532no—
208C2×C53—21no—
209C2×C53—21no—
210C2×C53—41no—
211C2×C53—41no—
212C2×C53—81no—
213C2×C53—82no—
214C2×C53#146511yescharacteristic
215C2×C53#212111yescharacteristic
216C2×C53#212211yescharacteristic
217C2×#120Dic5311yescharacteristic
218C2×C4×D4C2×D5311yescharacteristic
219C22×C22⋊C4Dic5311yescharacteristic
220#432—1061no—
221#407—1061no—
222#390—21no—
223#388—21no—
224#388—21no—
225#386—21no—
226#386Dic5311yescharacteristic
227#385C2×D5311yescharacteristic
228#379C2×D5311yescharacteristic
229C4×C22⋊C4Dic5311yescharacteristic
230#324—1061no—
231#323—532no—
232#323—532no—
233#323—1061no—
234C53#1152711yescharacteristic, complemented, Sylow
235C25C2×Dic5311yescharacteristic
236C22×D4—21no—
237C22×D4C2×Dic5311yescharacteristic
238C23×C4C2×Dic5311yescharacteristic
239C23×C4C2×Dic5311yescharacteristic
240C23×C4C22⋊D5311yescharacteristic
241C2×M16—2121no—
242#120—22no—
243#120—22no—
244C4×D4—41no—
245C4×D4—41no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22⋊D5311yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4C2×Dic5311yescharacteristic
257C2×C22⋊C4C2×Dic5311yescharacteristic
258C2×C22⋊C4C22⋊D5311yescharacteristic
259C2×C22⋊C4C22×D5311yescharacteristic
260C2×C42—21no—
261C2×C42C22⋊D5311yescharacteristic
262C4⋊C8—2121no—
263C22⋊C8—1062no—
264C22⋊C8—1062no—
265C22⋊C8—2121no—
266#97—2121no—
267#95—21no—
268#95—21no—
269#95—21no—
270C24—21no—
271C24—21no—
272C24—41no—
273C24—41no—
274C24—81no—
275C24—81no—
276C24#1171991712yescomplemented
277C24C22⋊Dic5311yescharacteristic
278C24C22×Dic5311yescharacteristic
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—24no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—81no—
285C2×D4—81no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4C22⋊Dic5311yescharacteristic
298C22×C4C22⋊Dic5311yescharacteristic
299C22×C4C22⋊Dic5311yescharacteristic
300C22×C4C2×C22⋊D5311yescharacteristic
301C22×C4C2×C22⋊D5311yescharacteristic
302M16—4242no—
303C2×C8—2122no—
304C2×C8—2122no—
305C4⋊C4—41no—
306C4⋊C4—81no—
307C22⋊C4—24no—
308C22⋊C4—21no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—81no—
316C22⋊C4—81no—
317C22⋊C4—81no—
318C22⋊C4#1171991712yescomplemented
319C42—21no—
320C42—21no—
321C42—81no—
322C23—21no—
323C23—21no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—82no—
341C23—82no—
342C23—82no—
343C23—82no—
344C23—84no—
345C23C2×C22⋊Dic5311yescharacteristic
346C23C2×#1171991711yescharacteristic
347C23C2×#1171991711yescharacteristic
348D4—42no—
349D4—42no—
350D4—84no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4#1313313211yescharacteristic
376C2×C4#1313313211yescharacteristic
377C8—4242no—
378C8—4242no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—84no—
399C22—84no—
400C22—81no—
401C22—81no—
402C22#1681058711yescharacteristic
403C22C2×#1313313211yescharacteristic
404C22#1681060711yescharacteristic
405C4—21no—
406C4—21no—
407C4—42no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—81no—
413C4—81no—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—41no—
418C2—81no—
419C2—82no—
420C2#2027305311yescharacteristic
421C2#2027636111yescharacteristic
422C2#2027636511yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2321C2
socleC22×C53C2×#99
prime radicalC2×#120×C53C4
cosocleC2×C22⋊C4×C53C23
hypercenter#2321D53
hypocenterC53#11527
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2321

Complements

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

NH ≅ G/Ntype
1. #20273056C2semidirect
2. #20273053C2semidirect
3. #20272663C2semidirect
4. #20272643C2semidirect
65. C24×C53M16semidirect
103. C22⋊C4×C53M16semidirect
234. C53#11527semidirect
276. C24#11719917semidirect
318. C22⋊C4#11719917semidirect

Maximal subgroups

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

subgroupindex
#202730562
#202730532
#202726632
#202726432
C53×#23212
#1152753