Finite Group Databasebeta

← #27293540

Subgroups — #27293540

Order 13568.

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

normal subgroups only

This group has 738 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#20273056C211yesmaximal, characteristic, complemented
2#20273053C211yesmaximal, characteristic, complemented
3#20272663C212yesmaximal, complemented
4#20272643C212yesmaximal, complemented
5C4×#120×C53C211yesmaximal, characteristic
6#16810680—21no—
7#16810587—21no—
8#16810349—21no—
9#16810347—21no—
10#16810347C2212yes—
11#16810347C2212yes—
12C24×C4×C53C411yescharacteristic
13C2×#120×C53C411yescharacteristic
14C2×C4×D4×C53—21no—
15C2×C4×D4×C53C2211yescharacteristic
16C53×#386—21no—
17C53×#386C411yescharacteristic
18C53×#385—21no—
19C53×#385C2211yescharacteristic
20C53×#379C2211yescharacteristic
21C4×C22⋊C4×C53—21no—
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×C53—21no—
33C23×C4×C53—21no—
34C23×C4×C53—44no—
35C23×C4×C53—41no—
36C23×C4×C53—41no—
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53C2×C411yescharacteristic
39C23×C4×C53D411yescharacteristic
40#120×C53—22no—
41#120×C53—22no—
42C4×D4×C53—41no—
43C4×D4×C53—41no—
44C4×D4×C53—44no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53C2311yescharacteristic
55C2×C42×C53—21no—
56C2×C42×C53D411yescharacteristic
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60C24×C53—21no—
61C24×C53—21no—
62C24×C53—42no—
63C24×C53—41no—
64C24×C53—42no—
65C24×C53—41no—
66C24×C53C22⋊C411yescharacteristic
67C24×C53M1612yescomplemented
68C24×C53C22×C411yescharacteristic
69C2×D4×C53—24no—
70C2×D4×C53—21no—
71C2×D4×C53—21no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C2×D4×C53—42no—
75C2×D4×C53—42no—
76C22×C4×C53—24no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—42no—
85C22×C4×C53—42no—
86C22×C4×C53—42no—
87C22×C4×C53—42no—
88C22×C4×C53—42no—
89C22×C4×C53—42no—
90C22×C4×C53—42no—
91C22×C4×C53—44no—
92C22×C4×C53—44no—
93C22×C4×C53—44no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—44no—
98C22×C4×C53C22⋊C411yescharacteristic
99C22×C4×C53C22⋊C411yescharacteristic
100C22×C4×C53C22⋊C411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C4⋊C4×C53—41no—
104C4⋊C4×C53—42no—
105C22⋊C4×C53—24no—
106C22⋊C4×C53—21no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109C22⋊C4×C53—42no—
110C22⋊C4×C53—42no—
111C22⋊C4×C53—42no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53M1612yescomplemented
114C42×C53—21no—
115C42×C53—21no—
116C42×C53—42no—
117#11719917—82no—
118C2×#88322—42no—
119C2×#88322—42no—
120C23×C53—22no—
121C23×C53—22no—
122C23×C53—22no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—41no—
129C23×C53—44no—
130C23×C53—42no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—44no—
135C23×C53—42no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53—42no—
139C23×C53—42no—
140C23×C53—42no—
141C23×C53—48no—
142C23×C53—42no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53C2×C22⋊C411yescharacteristic
146C23×C53C2×M1611yescharacteristic
147C23×C53C2×M1611yescharacteristic
148D4×C53—48no—
149D4×C53—42no—
150D4×C53—42no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—22no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—44no—
161C2×C4×C53—42no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—48no—
166C2×C4×C53—42no—
167C2×C4×C53—41no—
168C2×C4×C53—44no—
169C2×C4×C53—42no—
170C2×C4×C53—42no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—42no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—81no—
179C2×C4×C53#9911yescharacteristic
180C2×C4×C53#9911yescharacteristic
181#88322—82no—
182#88322—82no—
183#11525—531nomaximal, Sylow
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—24no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—42no—
192C22×C53—42no—
193C22×C53—42no—
194C22×C53—42no—
195C22×C53—42no—
196C22×C53—44no—
197C22×C53—44no—
198C22×C53—44no—
199C22×C53—44no—
200C22×C53—42no—
201C22×C53—41no—
202C22×C53—42no—
203C22×C53—41no—
204C22×C53—42no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—81no—
209C22×C53#40711yescharacteristic
210C22×C53C2×#9911yescharacteristic
211C22×C53#41311yescharacteristic
212C4×C53—21no—
213C4×C53—42no—
214C4×C53—41no—
215C4×C53—42no—
216C4×C53—41no—
217C4×C53—41no—
218C4×C53—41no—
219C4×C53—81no—
220C4×C53AΣL(1,𝔽16)12yes—
221C4×#120D5311yescharacteristic
222#1466—531no—
223#1465—531no—
224#1326—532no—
225#1316—532no—
226C2×C53—21no—
227C2×C53—21no—
228C2×C53—44no—
229C2×C53—41no—
230C2×C53—41no—
231C2×C53—81no—
232C2×C53#146511yescharacteristic
233C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
234C2×C53#212011yescharacteristic
235C24×C4Dic5311yescharacteristic
236C2×#120Dic5311yescharacteristic
237C2×C4×D4—21no—
238C2×C4×D4C2×D5311yescharacteristic
239#432—1061no—
240#407—1061no—
241#386—21no—
242#386Dic5311yescharacteristic
243#385—21no—
244#385C2×D5311yescharacteristic
245#379C2×D5311yescharacteristic
246C4×C22⋊C4—21no—
247C4×C22⋊C4Dic5311yescharacteristic
248#324—1061no—
249#323—532no—
250#323—532no—
251#323—1061no—
252C53#1152511yescharacteristic, complemented, Sylow
253C25C2×Dic5311yescharacteristic
254C22×D4—21no—
255C22×D4C2×Dic5311yescharacteristic
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—44no—
259C23×C4—41no—
260C23×C4—41no—
261C23×C4C2×Dic5311yescharacteristic
262C23×C4C2×Dic5311yescharacteristic
263C23×C4C22⋊D5311yescharacteristic
264C2×M16—2121no—
265#120—22no—
266#120—22no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—44no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C22⋊D5311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4C2×Dic5311yescharacteristic
277C2×C22⋊C4C2×Dic5311yescharacteristic
278C2×C22⋊C4C22⋊D5311yescharacteristic
279C2×C22⋊C4C22×D5311yescharacteristic
280C2×C42—21no—
281C2×C42C22⋊D5311yescharacteristic
282C4⋊C8—2121no—
283C22⋊C8—1062no—
284C22⋊C8—1062no—
285C22⋊C8—2121no—
286#97—2121no—
287#95—21no—
288#95—21no—
289#95—21no—
290C24—21no—
291C24—21no—
292C24—42no—
293C24—41no—
294C24—42no—
295C24—41no—
296C24#1171991712yescomplemented
297C24C22⋊Dic5311yescharacteristic
298C24C22×Dic5311yescharacteristic
299C2×D4—24no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—42no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—44no—
328C22×C4C22⋊Dic5311yescharacteristic
329C22×C4C22⋊Dic5311yescharacteristic
330C22×C4C22⋊Dic5311yescharacteristic
331C22×C4C2×C22⋊D5311yescharacteristic
332C22×C4C2×C22⋊D5311yescharacteristic
333M16—4242no—
334C2×C8—2122no—
335C2×C8—2122no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C22⋊C4—24no—
339C22⋊C4—21no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—41no—
346C22⋊C4#1171991712yescomplemented
347C42—21no—
348C42—21no—
349C42—42no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—41no—
359C23—41no—
360C23—44no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—44no—
365C23—42no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—48no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23C2×C22⋊Dic5311yescharacteristic
376C23C2×#1171991711yescharacteristic
377C23C2×#1171991711yescharacteristic
378D4—48no—
379D4—42no—
380D4—42no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—81no—
409C2×C4#1313313211yescharacteristic
410C2×C4#1313313211yescharacteristic
411C8—4242no—
412C8—4242no—
413C22—21no—
414C22—21no—
415C22—24no—
416C22—21no—
417C22—21no—
418C22—41no—
419C22—41no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—44no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22—42no—
430C22—41no—
431C22—42no—
432C22—41no—
433C22—42no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—81no—
438C22#1681058711yescharacteristic
439C22C2×#1313313211yescharacteristic
440C22#1681060711yescharacteristic
441C4—21no—
442C4—42no—
443C4—41no—
444C4—42no—
445C4—41no—
446C4—41no—
447C4—41no—
448C4—81no—
449C4#1681041012yes—
450C2—21no—
451C2—21no—
452C2—44no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2#2027305311yescharacteristic
457C2C2×#1681041011yescharacteristic
458C2#2027635611yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC4×#120×C53C2
socleC22×C53C2×#99
prime radicalC2×#120×C53C4
cosocleC2×C22⋊C4×C53C23
hypercenterC4×#120D53
hypocenterC53#11525
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#120

Complements

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

NH ≅ G/Ntype
1. #20273056C2semidirect
2. #20273053C2semidirect
3. #20272663C2semidirect
4. #20272643C2semidirect
67. C24×C53M16semidirect
113. C22⋊C4×C53M16semidirect
252. C53#11525semidirect
296. C24#11719917semidirect
346. 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
C4×#120×C532
#1152553