Finite Group Databasebeta

← #27276224

Subgroups — #27276224

Order 13568.

838 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 139 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#20274795C211yesmaximal, characteristic, complemented
2#20273219C212yesmaximal, complemented
3#20273219C212yesmaximal, complemented
4#20272983C211yesmaximal, characteristic, complemented
5C53×#3492C211yesmaximal, characteristic
6#16810587C2212yes—
7C2×#13133124—22no—
8C2×#13133124C2211yescharacteristic
9C2×#13133124C2211yescharacteristic
10#16810392—22no—
11#16810360—22no—
12#16810360—22no—
13#16810357—22no—
14#16810357—22no—
15C53×#554—22no—
16C53×#553—22no—
17C53×#552—22no—
18C53×#547—22no—
19C53×#544C411yescharacteristic, complemented
20C53×#539—22no—
21C53×#537C411yescharacteristic, complemented
22C53×#525—22no—
23C53×#525C2212yes—
24C53×#520C412yescomplemented
25C22×C4⋊C4×C53C2211yescharacteristic
26C23×Dic53—21no—
27C23×Dic53—21no—
28C2×#11719917—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133124—22no—
32#13133124—22no—
33#13133124—44no—
34C2×#41×C53—22no—
35C2×#41×C53C2×C412yes—
36C23×C4×C53D412yes—
37C23×C4×C53C2311yescharacteristic
38C4⋊Q8×C53—22no—
39#126×C53—24no—
40#125×C53—24no—
41#125×C53—22no—
42#123×C53—24no—
43#123×C53—24no—
44#123×C53—22no—
45C22⋊Q8×C53—22no—
46C22⋊Q8×C53—24no—
47C22⋊Q8×C53—22no—
48C22⋊Q8×C53—22no—
49C22⋊Q8×C53D412yes—
50C22⋊Q8×C53D412yes—
51#121×C53—22no—
52#121×C53—22no—
53#121×C53—22no—
54#121×C53D412yes—
55#121×C53D412yes—
56#120×C53—22no—
57C4×Q8×C53—22no—
58C4×D4×C53—24no—
59C4×D4×C53—22no—
60C4×D4×C53—24no—
61C4×D4×C53—22no—
62C4×D4×C53—22no—
63#117×C53—22no—
64#117×C53—22no—
65#117×C53C2×C412yes—
66C2×C4⋊C4×C53—22no—
67C2×C4⋊C4×C53—22no—
68C2×C4⋊C4×C53—22no—
69C2×C4⋊C4×C53C2×C411yescharacteristic
70C2×C4⋊C4×C53C2×C411yescharacteristic
71C2×C4⋊C4×C53D412yes—
72C24×C53C2×D411yescharacteristic
73#41×C53—44no—
74#41×C53—82no—
75#41×C53—82no—
76C2×Q8×C53—22no—
77C2×Q8×C53C22⋊C412yes—
78C2×D4×C53—22no—
79C2×D4×C53—22no—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53C22⋊C412yes—
83C22×C4×C53—22no—
84C22×C4×C53—24no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53—42no—
88C22×C4×C53—42no—
89C22×C4×C53—42no—
90C22×C4×C53C22⋊C412yes—
91C22×C4×C53C22⋊C412yes—
92C22×C4×C53C22×C411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D412yes—
95C22×C4×C53C2×D412yes—
96C22×Dic53—22no—
97C22×Dic53—22no—
98C22×Dic53—42no—
99C22×Dic53—42no—
100C22×Dic53—42no—
101C22×Dic53—41no—
102C22×Dic53—41no—
103C22×Dic53—42no—
104C22×Dic53—41no—
105C22×Dic53—41no—
106C4⋊C4×C53—24no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—24no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53C22⋊C412yes—
116C4⋊C4×C53C22⋊C412yes—
117C22⋊C4×C53—24no—
118C22⋊C4×C53—24no—
119C22⋊C4×C53—22no—
120C22⋊C4×C53—22no—
121C22⋊C4×C53—22no—
122C22⋊C4×C53—22no—
123C22⋊C4×C53—22no—
124C42×C53—24no—
125C42×C53—22no—
126#11719917—44no—
127C2×#88322—42no—
128C23×C53—22no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—22no—
134C23×C53C2×C22⋊C411yescharacteristic
135C23×C53#12011yescharacteristic
136C23×C53#12011yescharacteristic
137Q8×C53—42no—
138Q8×C53—82no—
139D4×C53—42no—
140D4×C53—44no—
141D4×C53—42no—
142D4×C53—42no—
143D4×C53—42no—
144D4×C53—42no—
145D4×C53—82no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—44no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—82no—
161C2×C4×C53—84no—
162C2×C4×C53C2×C22⋊C412yes—
163C2×C4×C53#12012yes—
164C2×Dic53—44no—
165C2×Dic53—44no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—82no—
169C2×Dic53—82no—
170C2×Dic53—81no—
171C2×Dic53—81no—
172C2×Dic53—81no—
173C2×Dic53—81no—
174#88322—82no—
175#8454—531nomaximal, Sylow
176C22×C53—22no—
177C22×C53—22no—
178C22×C53—22no—
179C22×C53—22no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—42no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53#37911yescharacteristic
190C22×C53#41012yes—
191C22×C53#45311yescharacteristic
192C22×C53#45411yescharacteristic
193C22×C53#45511yescharacteristic
194C22×C53#45611yescharacteristic
195C4×C53—22no—
196C4×C53—22no—
197C4×C53—42no—
198C4×C53—42no—
199C4×C53—42no—
200C4×C53—42no—
201C4×C53—42no—
202Dic53—82no—
203Dic53—82no—
204#3492D5311yescharacteristic
205#1821—531no—
206#1504—532no—
207#1504—532no—
208#1445—531no—
209C2×C53—21no—
210C2×C53—21no—
211C2×C53—22no—
212C2×C53—21no—
213C2×C53—21no—
214C2×C53—82no—
215C2×C53#178211yescharacteristic
216C2×C53#179211yescharacteristic
217C2×C53#179311yescharacteristic
218#554—22no—
219#553—22no—
220#552—22no—
221#547—22no—
222#544Dic5311yescharacteristic, complemented
223#539—22no—
224#537Dic5311yescharacteristic, complemented
225#525—22no—
226#525C2×D5312yes—
227#520Dic5312yescomplemented
228C22×C4⋊C4C2×D5311yescharacteristic
229#407—532no—
230C2×#95—531no—
231C2×#95—531no—
232C2×#95—1062no—
233#343—1062no—
234#328—1062no—
235#328—1062no—
236#327—1062no—
237#327—1062no—
238C53#845411yescharacteristic, complemented, Sylow
239C2×#41—22no—
240C2×#41C2×Dic5312yes—
241C23×C4—1061no—
242C23×C4—1061no—
243C23×C4C22⋊D5312yes—
244C23×C4C22×D5311yescharacteristic
245C2×M16—1062no—
246C4⋊Q8—22no—
247#126—24no—
248#125—24no—
249#125—22no—
250#123—24no—
251#123—24no—
252#123—22no—
253C22⋊Q8—22no—
254C22⋊Q8—24no—
255C22⋊Q8—22no—
256C22⋊Q8—22no—
257C22⋊Q8C22⋊D5312yes—
258C22⋊Q8C22⋊D5312yes—
259#121—22no—
260#121—22no—
261#121—22no—
262#121C22⋊D5312yes—
263#121C22⋊D5312yes—
264#120—22no—
265C4×Q8—22no—
266C4×D4—24no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—24no—
271#117—22no—
272#117—22no—
273#117C2×Dic5312yes—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4C2×Dic5311yescharacteristic
278C2×C4⋊C4C2×Dic5311yescharacteristic
279C2×C4⋊C4C22⋊D5312yes—
280C22⋊C8—1062no—
281C22⋊C8—1062no—
282#95—1062no—
283#95—1062no—
284#95—2124no—
285C24C2×C22⋊D5311yescharacteristic
286#41—44no—
287#41—82no—
288#41—82no—
289C2×Q8—22no—
290C2×Q8C22⋊Dic5312yes—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4C22⋊Dic5312yes—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—1062no—
304C22×C4—1062no—
305C22×C4—2122no—
306C22×C4—2121no—
307C22×C4—2121no—
308C22×C4—2122no—
309C22×C4—2122no—
310C22×C4—2122no—
311C22×C4—2121no—
312C22×C4—2121no—
313C22×C4C22⋊Dic5312yes—
314C22×C4C22⋊Dic5312yes—
315C22×C4C22×Dic5311yescharacteristic
316C22×C4C2×C22⋊D5311yescharacteristic
317C22×C4C2×C22⋊D5312yes—
318C22×C4C2×C22⋊D5312yes—
319M16—2124no—
320C2×C8—2122no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4C22⋊Dic5312yes—
331C4⋊C4C22⋊Dic5312yes—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C42—24no—
340C42—22no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—22no—
346C23—22no—
347C23C2×C22⋊Dic5311yescharacteristic
348C23#1313318611yescharacteristic
349C23#1313318611yescharacteristic
350Q8—42no—
351Q8—82no—
352D4—42no—
353D4—42no—
354D4—42no—
355D4—42no—
356D4—42no—
357D4—44no—
358D4—82no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—84no—
375C2×C4—4242no—
376C2×C4—4242no—
377C2×C4—4241no—
378C2×C4—4241no—
379C2×C4—4241no—
380C2×C4—4241no—
381C2×C4—2124no—
382C2×C4—2124no—
383C2×C4—2122no—
384C2×C4—2122no—
385C2×C4C2×C22⋊Dic5312yes—
386C2×C4#1313318612yes—
387C8—4242no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22#1681047911yescharacteristic
402C22#1681059712yes—
403C22#1681079111yescharacteristic
404C22#1681079611yescharacteristic
405C22#1681080111yescharacteristic
406C22#1681080611yescharacteristic
407C4—22no—
408C4—22no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—4242no—
415C4—4242no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—22no—
421C2—82no—
422C2#2027463311yescharacteristic
423C2#2027466911yescharacteristic
424C2#2027467411yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#3492C2
socleC22×C53#379
cosocleC23×C4×C53C23
hypercenter#3492D53
hypocenterC53#8454
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3492

Complements

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

NH ≅ G/Ntype
1. #20274795C2semidirect
2. #20273219C2semidirect
3. #20273219C2semidirect
4. #20272983C2semidirect
19. C53×#544C4semidirect
21. C53×#537C4semidirect
24. C53×#520C4semidirect
222. #544Dic53semidirect
224. #537Dic53semidirect
227. #520Dic53semidirect
238. C53#8454semidirect

Maximal subgroups

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

subgroupindex
#202747952
#202732192
#202729832
C53×#34922
#845453