Finite Group Databasebeta

← #27281150

Subgroups — #27281150

Order 13568.

444 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283373C211yesmaximal, characteristic
2#20274810C211yesmaximal, characteristic
3#20273287C211yesmaximal, characteristic
4#20273079C211yesmaximal, characteristic
5#20272620C211yesmaximal, characteristic
6#20272602C211yesmaximal, characteristic
7C53×#3095C211yesmaximal, characteristic
8#16810684—22no—
9#16810680—21no—
10#16810676—21no—
11#16810650—21no—
12#16810587C2211yescharacteristic
13C2×C22⋊#88322C2211yescharacteristic
14C2×C22⋊#88322C2211yescharacteristic
15#16810579—21no—
16#16810483C2211yescharacteristic
17#16810385—21no—
18#16810385—21no—
19#16810360—21no—
20#16810360—21no—
21#16810349—22no—
22C2×C22⋊Q8×C53C2211yescharacteristic
23C2×#117×C53C2211yescharacteristic
24C53×#485—21no—
25C53×#485—21no—
26C53×#484—21no—
27C53×#483—21no—
28C53×#428C411yescharacteristic, complemented
29C53×#428C411yescharacteristic, complemented
30C53×#419C411yescharacteristic, complemented
31C53×#419C411yescharacteristic, complemented
32C53×#407C2211yescharacteristic
33C2×#11719917—21no—
34C22×#88322—21no—
35C22×#88322—21no—
36C2×C4⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C4⋊#88322—22no—
39C4⋊#88322—22no—
40C22⋊#88322—21no—
41C22⋊#88322—21no—
42C22⋊#88322—21no—
43C22⋊#88322—21no—
44C22⋊#88322D411yescharacteristic
45C22⋊#88322D411yescharacteristic
46C22⋊#88322D412yes—
47#13133128—22no—
48C4×#88322—22no—
49#13133124—21no—
50C22×Q8×C53C2×C411yescharacteristic
51C23×C4×C53C2311yescharacteristic
52C2×M16×C53C2×C411yescharacteristic
53C2×M16×C53C2×C411yescharacteristic
54C22⋊Q8×C53—21no—
55C22⋊Q8×C53—21no—
56C22⋊Q8×C53—21no—
57C22⋊Q8×C53—21no—
58#117×C53—21no—
59#117×C53—21no—
60#117×C53C2×C411yescharacteristic
61#117×C53C2×C411yescharacteristic
62#117×C53D411yescharacteristic
63#117×C53D411yescharacteristic
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53C2×C411yescharacteristic
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70C2×C42×C53—21no—
71#107×C53—21no—
72#107×C53—21no—
73#106×C53—21no—
74#106×C53—21no—
75#103×C53—21no—
76#103×C53—21no—
77#103×C53—21no—
78#103×C53—21no—
79C22⋊C8×C53—21no—
80C22⋊C8×C53—21no—
81C24×C53#4111yescharacteristic
82C2×Q8×C53—21no—
83C2×Q8×C53—21no—
84C2×Q8×C53—41no—
85C2×Q8×C53—41no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53C22×C411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53#4111yescharacteristic
98C22×Dic53—41no—
99M16×C53—41no—
100M16×C53—41no—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53C22⋊C411yescharacteristic
110C4⋊C4×C53C22⋊C411yescharacteristic
111C4⋊C4×C53C22⋊C411yescharacteristic
112C4⋊C4×C53C22⋊C411yescharacteristic
113C22⋊C4×C53—21no—
114C22⋊C4×C53—21no—
115C22⋊C4×C53—21no—
116C22⋊C4×C53—21no—
117C22⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C42×C53—21no—
120C42×C53—21no—
121C42×C53—21no—
122C42×C53—21no—
123C22⋊Dic53—81no—
124C4⋊Dic53—41no—
125C4⋊Dic53—41no—
126#11719917—41no—
127#11719917—41no—
128C2×#88322—21no—
129C2×#88322—22no—
130C2×#88322—21no—
131C2×#88322—41no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—41no—
136C23×C53—21no—
137C23×C53—21no—
138C23×C53—41no—
139C23×C53—41no—
140C23×C53C4×D411yescharacteristic
141C23×C53#12111yescharacteristic
142C23×C53#12311yescharacteristic
143Q8×C53—41no—
144Q8×C53—41no—
145Q8×C53—81no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—81no—
165C2×C4×C53—81no—
166C2×C4×C53C2×C22⋊C411yescharacteristic
167C2×C4×C53C4×D411yescharacteristic
168C2×C4×C53#12011yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×Dic53—81no—
171C2×Dic53—81no—
172C2×Dic53—81no—
173C8×C53—41no—
174C8×C53—41no—
175#88322—41no—
176#88322—41no—
177#88322—42no—
178#88322—81no—
179#9346—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—81no—
189C22×C53—81no—
190C22×C53#38611yescharacteristic
191C22×C53#47111yescharacteristic
192C22×C53#47311yescharacteristic
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—41no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—81no—
201C4×C53—81no—
202C4×C53#41011yescharacteristic
203C4×C53#41011yescharacteristic
204Dic53—161no—
205#3095D5311yescharacteristic
206#2924—531no—
207#1825—531no—
208#1520—531no—
209#1472—531no—
210#1305—531no—
211#1298—531no—
212C2×C53—21no—
213C2×C53—21no—
214C2×C53—41no—
215C2×C53—41no—
216C2×C53#188111yescharacteristic
217C2×C53#189911yescharacteristic
218C2×C53#190811yescharacteristic
219C2×C22⋊Q8C2×D5311yescharacteristic
220C2×#117C2×D5311yescharacteristic
221#485—21no—
222#485—21no—
223#484—21no—
224#483—21no—
225#433—1062no—
226#432—1061no—
227#431—1061no—
228#428Dic5311yescharacteristic, complemented
229#428Dic5311yescharacteristic, complemented
230#424—1061no—
231#419Dic5311yescharacteristic, complemented
232#419Dic5311yescharacteristic, complemented
233#407—531no—
234#407C2×D5311yescharacteristic
235C2×C22⋊C8—531no—
236C2×C22⋊C8—531no—
237#405—1061no—
238#380—531no—
239#340—1061no—
240#340—1061no—
241#328—1061no—
242#328—1061no—
243#324—1062no—
244C53#934611yescharacteristic, complemented, Sylow
245C22×Q8C2×Dic5311yescharacteristic
246C23×C4C22×D5311yescharacteristic
247C2×M16—1061no—
248C2×M16C2×Dic5311yescharacteristic
249C2×M16C2×Dic5311yescharacteristic
250C22×C8—1061no—
251C22×C8—1061no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256#117—21no—
257#117—21no—
258#117C2×Dic5311yescharacteristic
259#117C2×Dic5311yescharacteristic
260#117C22⋊D5311yescharacteristic
261#117C22⋊D5311yescharacteristic
262C2×C4⋊C4—21no—
263C2×C4⋊C4—1061no—
264C2×C4⋊C4C2×Dic5311yescharacteristic
265C2×C4⋊C4C22⋊D5311yescharacteristic
266C2×C4⋊C4C22⋊D5311yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—1061no—
270C2×C42—21no—
271#107—21no—
272#107—21no—
273#106—21no—
274#106—21no—
275C4⋊C8—1062no—
276C4⋊C8—1062no—
277#103—21no—
278#103—21no—
279#103—21no—
280#103—21no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—532no—
284C22⋊C8—531no—
285C22⋊C8—531no—
286C22⋊C8—1061no—
287C22⋊C8—1061no—
288C22⋊C8—1061no—
289C22⋊C8—1061no—
290#97—1062no—
291C4×C8—1062no—
292#95—1061no—
293C24#1171994711yescharacteristic
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—41no—
297C2×Q8—41no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—2121no—
305C22×C4D4×D5311yescharacteristic
306C22×C4D4×D5311yescharacteristic
307C22×C4#1171994711yescharacteristic
308C22×C4C22×Dic5311yescharacteristic
309C22×C4C2×C22⋊D5311yescharacteristic
310C22×C4C2×C22⋊D5311yescharacteristic
311M16—41no—
312M16—41no—
313M16—2121no—
314M16—2121no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—1061no—
318C2×C8—1062no—
319C2×C8—1061no—
320C2×C8—2121no—
321C2×C8—2121no—
322C2×C8—2121no—
323C2×C8—2121no—
324C2×C8—2121no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—2121no—
332C4⋊C4—2121no—
333C4⋊C4C22⋊Dic5311yescharacteristic
334C4⋊C4C22⋊Dic5311yescharacteristic
335C4⋊C4C22⋊Dic5311yescharacteristic
336C4⋊C4C22⋊Dic5311yescharacteristic
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—4241no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23D4×Dic5311yescharacteristic
353C23#1313318911yescharacteristic
354C23#1313319811yescharacteristic
355Q8—41no—
356Q8—41no—
357Q8—81no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—4241no—
379C2×C4—4241no—
380C2×C4—4241no—
381C2×C4C2×C22⋊Dic5311yescharacteristic
382C2×C4D4×Dic5311yescharacteristic
383C2×C4#1313318511yescharacteristic
384C2×C4#1313318911yescharacteristic
385C8—41no—
386C8—41no—
387C8—4241no—
388C8—2121no—
389C8—2121no—
390C8—2122no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—81no—
400C22—81no—
401C22#1681050311yescharacteristic
402C22#1681088811yescharacteristic
403C22#1681089611yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—81no—
412C4—81no—
413C4—8481no—
414C4#1681059711yescharacteristic
415C4#1681059711yescharacteristic
416C2—21no—
417C2—21no—
418C2—41no—
419C2—41no—
420C2#2027507911yescharacteristic
421C2#2027517411yescharacteristic
422C2#2027523211yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#3095C2
socleC22×C53#386
prime radicalC24×C53#41
cosocleC23×C4×C53C23
hypercenter#3095D53
hypocenterC53#9346
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3095

Complements

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

NH ≅ G/Ntype
28. C53×#428C4semidirect
29. C53×#428C4semidirect
30. C53×#419C4semidirect
31. C53×#419C4semidirect
228. #428Dic53semidirect
229. #428Dic53semidirect
231. #419Dic53semidirect
232. #419Dic53semidirect
244. C53#9346semidirect

Maximal subgroups

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

subgroupindex
#202833732
#202748102
#202732872
#202730792
#202726202
#202726022
C53×#30952
#934653