Finite Group Databasebeta

← #27281997

Subgroups — #27281997

Order 13568.

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

normal subgroups only

This group has 900 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#20277313C211yesmaximal, characteristic, complemented
2#20274789C211yesmaximal, characteristic, complemented
3#20274630C212yesmaximal
4#20272619C212yesmaximal, complemented
5C53×#2405C211yesmaximal, characteristic
6C2×#13133132—24no—
7#16810516—24no—
8#16810483—21no—
9#16810479C2212yes—
10#16810479C2212yes—
11C4×C22⋊Dic53—21no—
12#16810390—22no—
13C23×D4×C53C2211yescharacteristic
14C2×#123×C53—21no—
15C22×C22⋊C4×C53C2211yescharacteristic
16C53×#394—22no—
17C53×#386—21no—
18C53×#385C412yescomplemented
19C53×#379C412yescomplemented
20C53×#379C2211yescharacteristic
21C2×C4⋊Dic53—22no—
22C2×C22⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53D414yes—
28C2×C4×Dic53—22no—
29#13133132—48no—
30#13133124—22no—
31#13133124—22no—
32C25×C53C2×C411yescharacteristic
33C25×C53C2311yescharacteristic
34C22×D4×C53—21no—
35C22×D4×C53—22no—
36C22×D4×C53—21no—
37C22×D4×C53—22no—
38C22×D4×C53—44no—
39C23×C4×C53C2×C411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C23×C4×C53D411yescharacteristic
42#123×C53—44no—
43C2×C4⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53—22no—
50C2×C22⋊C4×C53C2×C412yes—
51C2×C22⋊C4×C53C2×C412yes—
52C2×C22⋊C4×C53D412yes—
53#95×C53—22no—
54C24×C53—21no—
55C24×C53—22no—
56C24×C53—42no—
57C24×C53—42no—
58C24×C53—42no—
59C24×C53—42no—
60C24×C53—81no—
61C24×C53—81no—
62C24×C53C22×C411yescharacteristic
63C24×C53C2×D411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C24×C53C2×D412yes—
66C24×C53#4112yes—
67C2×D4×C53—24no—
68C2×D4×C53—28no—
69C2×D4×C53—44no—
70C2×D4×C53—42no—
71C2×D4×C53—42no—
72C2×D4×C53—44no—
73C2×D4×C53—42no—
74C2×D4×C53—44no—
75C2×D4×C53—41no—
76C2×D4×C53—42no—
77C2×D4×C53—41no—
78C22×C4×C53—21no—
79C22×C4×C53—22no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—22no—
85C22×C4×C53—42no—
86C22×C4×C53—42no—
87C22×C4×C53—81no—
88C22×C4×C53C22⋊C412yes—
89C22×C4×C53C22⋊C412yes—
90C22×Dic53—24no—
91C22×Dic53—42no—
92C4⋊C4×C53—44no—
93C22⋊C4×C53—24no—
94C22⋊C4×C53—24no—
95C22⋊C4×C53—44no—
96C22⋊C4×C53—42no—
97C22⋊C4×C53—42no—
98C22⋊C4×C53—82no—
99C22⋊C4×C53—81no—
100C22⋊Dic53—44no—
101C22⋊Dic53—44no—
102C22⋊Dic53—44no—
103C22⋊Dic53—44no—
104C22⋊Dic53—82no—
105C22⋊Dic53—82no—
106C4⋊Dic53—82no—
107C4×Dic53—82no—
108C23×C53—22no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53—22no—
112C23×C53—24no—
113C23×C53—22no—
114C23×C53—24no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—42no—
122C23×C53—42no—
123C23×C53—41no—
124C23×C53—42no—
125C23×C53—41no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—41no—
129C23×C53—42no—
130C23×C53—82no—
131C23×C53—82no—
132C23×C53—88no—
133C23×C53—88no—
134C23×C53C2×C22⋊C411yescharacteristic
135C23×C53C4×D412yes—
136C23×C53#12011yescharacteristic
137C23×C53#12112yes—
138C23×C53#12311yescharacteristic
139D4×C53—48no—
140D4×C53—44no—
141D4×C53—44no—
142C2×C4×C53—24no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53—81no—
157C2×C4×C53—81no—
158C2×Dic53—44no—
159C2×Dic53—44no—
160C2×Dic53—44no—
161C2×Dic53—82no—
162C2×Dic53—82no—
163C2×Dic53—82no—
164#9496—531nomaximal, Sylow
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—21no—
172C22×C53—22no—
173C22×C53—22no—
174C22×C53—21no—
175C22×C53—22no—
176C22×C53—21no—
177C22×C53—22no—
178C22×C53—44no—
179C22×C53—44no—
180C22×C53—44no—
181C22×C53—42no—
182C22×C53—84no—
183C22×C53—82no—
184C22×C53—82no—
185C22×C53—84no—
186C22×C53—84no—
187C22×C53—84no—
188C22×C53—81no—
189C22×C53—81no—
190C22×C53#38611yescharacteristic
191C22×C53#41012yes—
192C22×C53B(4,𝔽2)12yes—
193C4×C53—42no—
194C4×C53—42no—
195C4×C53—42no—
196C4×C53—82no—
197Dic53—84no—
198Dic53—162no—
199#2405D5311yescharacteristic
200#2312—531no—
201#1819—531no—
202#1781—532no—
203#1304—532no—
204C2×C53—22no—
205C2×C53—22no—
206C2×C53—42no—
207C2×C53—41no—
208C2×C53—42no—
209C2×C53—82no—
210C2×C53#188911yescharacteristic
211C2×C53#189911yescharacteristic
212C2×C53#191311yescharacteristic
213C23×D4C2×D5311yescharacteristic
214C2×#123—21no—
215C22×C22⋊C4C2×D5311yescharacteristic
216C2×#99—1064no—
217#394—22no—
218#388—1064no—
219#386—21no—
220#385Dic5312yescomplemented
221#380—1061no—
222#379—532no—
223#379—532no—
224#379Dic5312yescomplemented
225#379C2×D5311yescharacteristic
226C4×C22⋊C4—1061no—
227#342—1062no—
228C53#949611yescharacteristic, complemented, Sylow
229C25C2×Dic5311yescharacteristic
230C25C22×D5311yescharacteristic
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—21no—
235C22×D4—44no—
236C23×C4C2×Dic5311yescharacteristic
237C23×C4C22⋊D5311yescharacteristic
238C23×C4C22⋊D5311yescharacteristic
239#123—44no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—1062no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—534no—
249C2×C22⋊C4—1062no—
250C2×C22⋊C4—1062no—
251C2×C22⋊C4—1062no—
252C2×C22⋊C4—1062no—
253C2×C22⋊C4—1062no—
254C2×C22⋊C4C2×Dic5312yes—
255C2×C22⋊C4C2×Dic5312yes—
256C2×C22⋊C4C22⋊D5312yes—
257C2×C42—1062no—
258#99—2128no—
259#95—22no—
260#95—1062no—
261#95—1062no—
262C24—21no—
263C24—22no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—81no—
269C24—81no—
270C24D4×D5312yes—
271C24#1171994712yes—
272C24C22×Dic5311yescharacteristic
273C24C2×C22⋊D5311yescharacteristic
274C24C2×C22⋊D5311yescharacteristic
275C2×D4—28no—
276C2×D4—24no—
277C2×D4—42no—
278C2×D4—44no—
279C2×D4—42no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—44no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—41no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—81no—
296C22×C4—1064no—
297C22×C4—2122no—
298C22×C4C22⋊Dic5312yes—
299C22×C4C22⋊Dic5312yes—
300C4⋊C4—44no—
301C4⋊C4—4242no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—82no—
308C22⋊C4—81no—
309C22⋊C4—4242no—
310C22⋊C4—4242no—
311C22⋊C4—2124no—
312C22⋊C4—2124no—
313C22⋊C4—2124no—
314C22⋊C4—2124no—
315C42—4242no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—24no—
320C23—22no—
321C23—22no—
322C23—24no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—41no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—42no—
338C23—82no—
339C23—82no—
340C23—88no—
341C23—88no—
342C23C2×C22⋊Dic5311yescharacteristic
343C23D4×Dic5312yes—
344C23#1313318511yescharacteristic
345C23#1313318912yes—
346C23#1313319811yescharacteristic
347D4—44no—
348D4—48no—
349D4—44no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—4242no—
367C2×C4—4242no—
368C2×C4—4242no—
369C2×C4—2124no—
370C2×C4—2124no—
371C2×C4—2124no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—21no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—21no—
385C22—44no—
386C22—44no—
387C22—44no—
388C22—42no—
389C22—82no—
390C22—84no—
391C22—82no—
392C22—84no—
393C22—84no—
394C22—84no—
395C22—81no—
396C22—81no—
397C22#1681050311yescharacteristic
398C22#1681059712yes—
399C22#1681080912yes—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—82no—
404C4—8482no—
405C4—4244no—
406C2—22no—
407C2—22no—
408C2—42no—
409C2—42no—
410C2—41no—
411C2—82no—
412C2#2027512411yescharacteristic
413C2#2027517411yescharacteristic
414C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC24×C53C22×C4
FrattiniC25C22×D53
FittingC53×#2405C2
socleC22×C53#386
prime radicalC23×D4×C53C22
cosocleC25×C53C23
hypercenter#2405D53
hypocenterC53#9496
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #2405

Complements

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

NH ≅ G/Ntype
1. #20277313C2semidirect
2. #20274789C2semidirect
4. #20272619C2semidirect
18. C53×#385C4semidirect
19. C53×#379C4semidirect
220. #385Dic53semidirect
224. #379Dic53semidirect
228. C53#9496semidirect

Maximal subgroups

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

subgroupindex
#202773132
#202747892
#202746302
#202726192
C53×#24052
#949653