Finite Group Databasebeta

← #27281420

Subgroups — #27281420

Order 13568.

618 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283744C211yesmaximal, characteristic, complemented
2#20279671C211yesmaximal, characteristic
3#20273282C212yesmaximal, complemented
4#20272593C212yesmaximal, complemented
5C53×#1799C211yesmaximal, characteristic, complemented
6C2×#13133191C2211yescharacteristic
7C2×#13133189—21no—
8#16810695—24no—
9#16810680—21no—
10#16810587—21no—
11C2×C22⋊#88322C2212yes—
12#16810553C411yescharacteristic, complemented
13#16810548C411yescharacteristic, complemented
14#16810540C411yescharacteristic, complemented
15#16810532C411yescharacteristic, complemented
16#16810514—21no—
17#16810502—21no—
18#16810500—21no—
19C2×#13133124C2211yescharacteristic
20#16810356—24no—
21C2×C4×D4×C53C2211yescharacteristic
22C2×#99×C53—21no—
23C53×#380C2212yes—
24C53×#342—21no—
25C22×D212C2×C411yescharacteristic
26C22×C22⋊D53—21no—
27C22×C22⋊D53—21no—
28C23×Dic53—21no—
29C2×#11719917—22no—
30C22×#88322—22no—
31#13133191—22no—
32#13133191—22no—
33#13133189—44no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×C22⋊Dic53—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C4⋊#88322—22no—
42C22⋊#88322—22no—
43C22⋊#88322—22no—
44C22⋊#88322D414yescomplemented
45#13133128—22no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124C2×C411yescharacteristic
50#13133124C2×C411yescharacteristic
51C22×D4×C53D411yescharacteristic
52C23×C4×C53D411yescharacteristic
53C23×C4×C53C2311yescharacteristic
54C4×D4×C53—22no—
55C4×D4×C53—22no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C22⋊C4×C53—22no—
59C2×C22⋊C4×C53C2×C411yescharacteristic
60C2×C22⋊C4×C53D411yescharacteristic
61C2×C42×C53D411yescharacteristic
62#99×C53—42no—
63#95×C53—22no—
64C24×C53C22⋊C411yescharacteristic
65C24×C53C2×D411yescharacteristic
66C23×D53—21no—
67C2×D4×C53—22no—
68C2×D4×C53—21no—
69C2×D4×C53—22no—
70C2×D4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—42no—
74C22×C4×C53—81no—
75C22×C4×C53—81no—
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D412yes—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53#4112yes—
83C2×C22⋊D53—42no—
84C2×C22⋊D53—42no—
85C2×C22⋊D53—42no—
86C2×C22⋊D53—42no—
87C2×C22⋊D53—41no—
88C2×C22⋊D53—41no—
89C2×C22⋊D53—81no—
90C22×Dic53—21no—
91C22×Dic53—21no—
92C22×Dic53—21no—
93C22×Dic53—42no—
94C22×Dic53—42no—
95C22×Dic53—81no—
96C2×D212—22no—
97C2×D212—81no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—82no—
100C22⋊C4×C53—22no—
101C22⋊C4×C53—22no—
102C22⋊C4×C53—42no—
103C22⋊C4×C53—42no—
104C42×C53—21no—
105C42×C53—21no—
106C22⋊Dic53—42no—
107#11719921—42no—
108#11719921—42no—
109#11719921—41no—
110#11719921—41no—
111#11719921—81no—
112C4⋊Dic53—22no—
113#11719919—42no—
114#11719917—82no—
115C2×#88322—24no—
116C2×#88322—82no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53C2×C22⋊C411yescharacteristic
125C23×C53#12112yes—
126C23×C53AGL(1,ℤ/8)12yes—
127C23×C53AGL(1,ℤ/8)11yescharacteristic
128C23×C53#13711yescharacteristic
129C22×D53—41no—
130C22×D53—41no—
131C22×D53—41no—
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—41no—
135C22×D53—41no—
136D4×C53—42no—
137D4×C53—42no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—81no—
151C2×C4×C53—82no—
152C2×C4×C53—82no—
153C2×C4×C53—82no—
154C2×C4×C53C4×D412yes—
155C2×C4×C53#12011yescharacteristic
156C2×C4×C53#12311yescharacteristic
157C22⋊D53—84no—
158C22⋊D53—82no—
159C22⋊D53—82no—
160C2×Dic53—22no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—84no—
170D212—82no—
171#88322—44no—
172#9386—531nomaximal, Sylow
173C22×C53—21no—
174C22×C53—22no—
175C22×C53—21no—
176C22×C53—22no—
177C22×C53—42no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—42no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—81no—
185C22×C53#38611yescharacteristic
186C22×C53#41711yescharacteristic
187C22×C53#41811yescharacteristic
188C22×C53#46812yes—
189C22×C53#46912yes—
190C2×D53—81no—
191C2×D53—81no—
192C2×D53—81no—
193C2×D53—81no—
194C2×D53—81no—
195C2×D53—81no—
196C2×D53—81no—
197C4×C53—21no—
198C4×C53—22no—
199C4×C53—21no—
200C4×C53—41no—
201C4×C53—41no—
202C4×C53—41no—
203C4×C53—162no—
204Dic53—42no—
205Dic53—81no—
206Dic53—81no—
207#2973—531no—
208#2543—531no—
209#1799D5311yescharacteristic, complemented
210#1519—532no—
211#1294—532no—
212C2×C53—21no—
213C2×C53—22no—
214C2×C53—21no—
215C2×C53—22no—
216C2×C53—81no—
217C2×C53#188411yescharacteristic
218C2×C53#189511yescharacteristic
219C2×C53#191311yescharacteristic
220D53—161no—
221C2×#121—531no—
222C2×#121—1061no—
223C2×C4×D4C2×D5311yescharacteristic
224#435—1064no—
225#432—1061no—
226C2×#99—21no—
227#407—1061no—
228C2×C22⋊C8—532no—
229#398—531no—
230#397—531no—
231#394—531no—
232#392—531no—
233#388—1061no—
234#386—1061no—
235#386—1061no—
236#380C2×D5312yes—
237C2×#95—531no—
238#342—21no—
239#327—1064no—
240C53#938611yescharacteristic, complemented, Sylow
241C22×D4—531no—
242C22×D4—1061no—
243C22×D4—1061no—
244C22×D4C22⋊D5311yescharacteristic
245C23×C4—1061no—
246C23×C4D21211yescharacteristic
247C23×C4C22×D5311yescharacteristic
248C2×M16—1062no—
249C22×C8—1062no—
250#121—1062no—
251#121—1062no—
252#121—2124no—
253C4×D4—22no—
254C4×D4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—531no—
257C2×C4⋊C4—1061no—
258C2×C4⋊C4C4×D5311yescharacteristic
259C2×C22⋊C4—22no—
260C2×C22⋊C4—1061no—
261C2×C22⋊C4—1061no—
262C2×C22⋊C4—1061no—
263C2×C22⋊C4—1061no—
264C2×C22⋊C4—1061no—
265C2×C22⋊C4C4×D5311yescharacteristic
266C2×C22⋊C4C22⋊D5311yescharacteristic
267C2×C42D21211yescharacteristic
268C4⋊C8—1062no—
269#99—42no—
270C22⋊C8—534no—
271C22⋊C8—1062no—
272C22⋊C8—1062no—
273#97—1062no—
274#95—22no—
275#95—531no—
276#95—531no—
277#95—1061no—
278#95—1061no—
279#95—1061no—
280C24—1061no—
281C24#1171992111yescharacteristic
282C24C2×C22⋊D5311yescharacteristic
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—1062no—
288C2×D4—4241no—
289C2×D4—4241no—
290C2×D4—2122no—
291C2×D4—2122no—
292C2×D4—2122no—
293C2×D4—2122no—
294C2×D4—2121no—
295C2×D4—2121no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—81no—
300C22×C4—81no—
301C22×C4—1061no—
302C22×C4—1061no—
303C22×C4—1061no—
304C22×C4—4241no—
305C22×C4—2122no—
306C22×C4—2122no—
307C22×C4#1171992111yescharacteristic
308C22×C4#1171992111yescharacteristic
309C22×C4#1171992111yescharacteristic
310C22×C4C2×C4×D5311yescharacteristic
311C22×C4C2×D21211yescharacteristic
312C22×C4D4×D5312yes—
313C22×C4#1171994712yes—
314M16—4242no—
315C2×C8—1064no—
316C2×C8—4242no—
317C4⋊C4—22no—
318C4⋊C4—82no—
319C4⋊C4—1062no—
320C4⋊C4—2122no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—4241no—
326C22⋊C4—2122no—
327C22⋊C4—2122no—
328C22⋊C4—2122no—
329C22⋊C4—2121no—
330C22⋊C4—2121no—
331C42—21no—
332C42—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—2121no—
341C23—2121no—
342C23—2121no—
343C23—2121no—
344C23—2121no—
345C23—2121no—
346C23—2121no—
347C23C2×#1171992111yescharacteristic
348C23#1313318912yes—
349C23#1313324212yes—
350C23#1313324311yescharacteristic
351C23#1313324811yescharacteristic
352D4—42no—
353D4—42no—
354D4—4242no—
355D4—4242no—
356D4—4242no—
357D4—4244no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—1062no—
375C2×C4—4244no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4—2121no—
380C2×C4—2121no—
381C2×C4—2121no—
382C2×C4—2121no—
383C2×C4—2121no—
384C2×C4#1313317712yes—
385C2×C4#1313318711yescharacteristic
386C2×C4#1313320111yescharacteristic
387C8—2124no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—22no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—81no—
400C22—4241no—
401C22—4241no—
402C22—4241no—
403C22—4241no—
404C22—4241no—
405C22—4241no—
406C22—4241no—
407C22#1681050011yescharacteristic
408C22#1681062211yescharacteristic
409C22#1681062711yescharacteristic
410C22#1681087112yes—
411C22#1681087412yes—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—162no—
419C4—4241no—
420C4—4241no—
421C4—2122no—
422C2—22no—
423C2—21no—
424C2—21no—
425C2—22no—
426C2—81no—
427C2—8481no—
428C2#2027509611yescharacteristic
429C2#2027515111yescharacteristic
430C2#2027525511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1799C2
socleC22×C53#386
prime radical#20279671C2
cosocleC23×C4×C53C23
hypercenter#1799D53
hypocenterC53#9386
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
1. #20283744C2semidirect
3. #20273282C2semidirect
4. #20272593C2semidirect
5. C53×#1799C2semidirect
12. #16810553C4semidirect
13. #16810548C4semidirect
14. #16810540C4semidirect
15. #16810532C4semidirect
44. C22⋊#88322D4semidirect
209. #1799D53semidirect
240. C53#9386semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202796712
#202732822
#202725932
C53×#17992
#938653