Finite Group Databasebeta

← #27384694

Subgroups — #27384694

Order 13568.

842 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810575C211yesmaximal, characteristic
2#20277111C212yesmaximal
3#20274878C211yesmaximal, characteristic, complemented
4#20274876C211yesmaximal, characteristic, complemented
5#20274844C212yesmaximal, complemented
6#20274749C211yesmaximal, characteristic
7#20274747C212yesmaximal
8#20274745C211yesmaximal, characteristic
9#20274613C212yesmaximal, complemented
10#20274612C211yesmaximal, characteristic, complemented
11C53×#1776C211yesmaximal, characteristic, complemented
12C22×#11719912C2211yescharacteristic
13C22×#11719912C2211yescharacteristic
14C2×C42×D53C2211yescharacteristic
15C2×#13133151C2211yescharacteristic
16C2×#13133150C2212yes—
17C2×#13133149C2211yescharacteristic
18C2×#13133148C2212yes—
19#16810575C418yescomplemented
20C2×#13133128C2211yescharacteristic
21C2×C8×Dic53C2211yescharacteristic
22C2×C8×Dic53C2211yescharacteristic
23#16810490—22no—
24#16810488C2212yes—
25#16810488C2212yes—
26#16810485—22no—
27#16810480C2212yes—
28#16810480C2212yes—
29C4×C4⋊Dic53C2212yes—
30C4×#11719921—22no—
31#16810469—22no—
32#16810386C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35C2×#107×C53C2211yescharacteristic
36C2×#106×C53C2211yescharacteristic
37C2×#97×C53C2211yescharacteristic
38C4×C4⋊C4×C53C2212yes—
39C53×#340C2212yes—
40C22×C4×D53D412yes—
41C22×C4×D53C2311yescharacteristic
42C2×#11719912—22no—
43C2×#11719912—22no—
44C2×#11719912—22no—
45C2×#11719912—22no—
46C2×#11719912C2×C414yes—
47C2×#11719912C2×C414yes—
48C22×#88322C2311yescharacteristic
49C22×#88322C2311yescharacteristic
50C2×C4⋊Dic53C2312yes—
51C2×C4⋊Dic53C2312yes—
52C2×#11719921—22no—
53C2×#11719921—22no—
54C2×C4×Dic53—22no—
55C2×C4×Dic53D412yes—
56C2×C4×Dic53C2311yescharacteristic
57C42×D53C2×C414yes—
58C42×D53Q814yes—
59#13133151—22no—
60#13133150—24no—
61#13133149—22no—
62#13133148—24no—
63#13133128C2×C414yes—
64C8×Dic53C2×C414yes—
65C8×Dic53C2×C414yes—
66#13133124—22no—
67#13133124—22no—
68#13133124—22no—
69#13133124—22no—
70#13133124—22no—
71#13133124—22no—
72#13133124—22no—
73C22×C8×C53C2311yescharacteristic
74C22×C8×C53C2311yescharacteristic
75C2×C4⋊C4×C53C2312yes—
76C2×C4⋊C4×C53C2312yes—
77C2×C42×C53—22no—
78C2×C42×C53C2311yescharacteristic
79#107×C53—22no—
80#106×C53—22no—
81#97×C53C2×C414yes—
82#95×C53—22no—
83C23×D53C2×D411yescharacteristic
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53C2×D412yes—
87C22×C4×C53C2411yescharacteristic
88C22×Dic53—22no—
89C22×Dic53—22no—
90C22×Dic53C2×D411yescharacteristic
91C22×Dic53C2×D412yes—
92C2×C4×D53—21no—
93C2×C4×D53—21no—
94C2×C4×D53—21no—
95C2×C4×D53—21no—
96C2×C4×D53—42no—
97C2×C4×D53—42no—
98C2×C4×D53C4⋊C418yes—
99C2×C4×D53C22×C412yes—
100C2×C4×D53C2×Q812yes—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C2×C8×C53C22×C412yes—
104C2×C8×C53C22×C412yes—
105C2×C8×C53#4112yes—
106C2×C8×C53#4112yes—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—22no—
111C42×C53—44no—
112C42×C53C22×C412yes—
113C42×C53C2×Q812yes—
114#11719921—44no—
115#11719921—44no—
116C4⋊Dic53—22no—
117C4⋊Dic53—22no—
118C4⋊Dic53—22no—
119C4⋊Dic53—22no—
120C4×Dic53—44no—
121C4×Dic53C4⋊C418yes—
122C4×Dic53C22×C412yes—
123C4×Dic53C2×Q812yes—
124C2×#88322—21no—
125C2×#88322—21no—
126C2×#88322C22×C412yes—
127C2×#88322C22×C412yes—
128C2×#88322#4112yes—
129C2×#88322#4112yes—
130#11719912—28no—
131#11719912—28no—
132C23×C53C22×D411yescharacteristic
133C22×D53—21no—
134C22×D53—21no—
135C22×D53—22no—
136C22×D53—22no—
137C22×D53C2×C4⋊C412yes—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53C2×C4⋊C414yes—
147C2×C4×C53C23×C411yescharacteristic
148C2×C4×C53C22×Q811yescharacteristic
149C2×C4×C53C2×#4111yescharacteristic
150C2×C4×C53C2×#4111yescharacteristic
151C2×C4×C53C2×#4111yescharacteristic
152C2×C4×C53C2×#4111yescharacteristic
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—22no—
156C2×Dic53—22no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53C2×C4⋊C414yes—
164C2×Dic53C2×C4⋊C412yes—
165C4×D53—22no—
166C4×D53—22no—
167C4×D53—22no—
168C4×D53—22no—
169C4×D53—48no—
170C8×C53—22no—
171C8×C53#11714yes—
172#88322—22no—
173#88322#11714yes—
174#20825—531nomaximal, Sylow
175C22×C53C22×C4⋊C411yescharacteristic
176C22×C53#53211yescharacteristic
177C22×C53#53211yescharacteristic
178C22×C53#57512yes—
179C22×C53#57512yes—
180C2×D53—24no—
181C2×D53—22no—
182C2×D53—42no—
183C2×D53—41no—
184C2×D53—41no—
185C4×C53—24no—
186C4×C53—82no—
187C4×C53—82no—
188C4×C53C2×#11711yescharacteristic
189C4×C53C2×#11711yescharacteristic
190C4×C53#53311yescharacteristic
191C4×C53#53311yescharacteristic
192Dic53—24no—
193Dic53—22no—
194Dic53—82no—
195Dic53—82no—
196C2×C4×M16—531no—
197#2283—532no—
198#1846—531no—
199#1846—531no—
200#1835—532no—
201#1809—531no—
202#1809—531no—
203#1809—532no—
204#1776—531no—
205#1776—532no—
206#1776D5311yescharacteristic, complemented
207C2×C53#228511yescharacteristic
208C2×C53#291112yes—
209C2×C53#318312yes—
210C2×C53#318311yescharacteristic
211C2×C53#318311yescharacteristic
212D53—42no—
213C22×M16—531no—
214C22×M16—531no—
215C22×C42—531no—
216C2×#107—531no—
217C2×#107—532no—
218C2×#107C2×D5311yescharacteristic
219C2×#106—531no—
220C2×#106—532no—
221C2×#106C2×D5311yescharacteristic
222C4×M16—538no—
223C2×#97—531no—
224C2×#97C2×D5311yescharacteristic
225C2×C4×C8—531no—
226C2×C4×C8—531no—
227#383—1062no—
228#382—532no—
229#382—532no—
230#381—1062no—
231#380—532no—
232#380—532no—
233C4×C4⋊C4—532no—
234C4×C4⋊C4C2×D5312yes—
235C4×C22⋊C4—1062no—
236#376—1062no—
237#340—532no—
238#340—532no—
239#340—532no—
240#340C2×D5312yes—
241C53#2082511yescharacteristic, complemented, Sylow
242C23×C4—531no—
243C23×C4—532no—
244C2×M16—534no—
245C2×M16—534no—
246C2×M16—1062no—
247C2×M16—1062no—
248C2×M16—1062no—
249C2×M16—1062no—
250C22×C8—531no—
251C22×C8—531no—
252C22×C8C22×D5311yescharacteristic
253C22×C8C22×D5311yescharacteristic
254C2×C4⋊C4—532no—
255C2×C4⋊C4—532no—
256C2×C4⋊C4C22×D5312yes—
257C2×C4⋊C4C22×D5312yes—
258C2×C22⋊C4—1062no—
259C2×C22⋊C4—1062no—
260C2×C42—22no—
261C2×C42—532no—
262C2×C42—531no—
263C2×C42—534no—
264C2×C42—534no—
265C2×C42—1062no—
266C2×C42C22×D5311yescharacteristic
267#107—22no—
268#107—1064no—
269#107—1062no—
270#106—22no—
271#106—1064no—
272#106—1062no—
273#97—534no—
274#97C4×D5314yes—
275C4×C8—534no—
276C4×C8—534no—
277#95—22no—
278#95—1062no—
279#95—1062no—
280#95—1062no—
281#95—1062no—
282#95—1062no—
283#95—1062no—
284#95—1062no—
285C24—531no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—532no—
289C22×C4—531no—
290C22×C4—532no—
291C22×C4—532no—
292C22×C4—538no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4—1062no—
298C22×C4—1062no—
299C22×C4—2122no—
300C22×C4—2122no—
301C22×C4D4×D5312yes—
302C22×C4C23×D5311yescharacteristic
303M16—1068no—
304M16—1068no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—532no—
308C2×C8—532no—
309C2×C8—532no—
310C2×C8—532no—
311C2×C8—1061no—
312C2×C8—1061no—
313C2×C8C2×C4×D5312yes—
314C2×C8C2×C4×D5312yes—
315C2×C8#1171994512yes—
316C2×C8#1171994512yes—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—1062no—
322C4⋊C4—1062no—
323C4⋊C4—1062no—
324C4⋊C4—1062no—
325C22⋊C4—2124no—
326C22⋊C4—2124no—
327C42—44no—
328C42—538no—
329C42—532no—
330C42—532no—
331C42—2124no—
332C42C2×C4×D5312yes—
333C42Q8×D5312yes—
334C23—532no—
335C23—1062no—
336C23—1062no—
337C23—1061no—
338C23—1061no—
339C23C2×D4×D5311yescharacteristic
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—532no—
349C2×C4—534no—
350C2×C4—1062no—
351C2×C4—1062no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1062no—
355C2×C4—1062no—
356C2×C4—1062no—
357C2×C4—1062no—
358C2×C4—2128no—
359C2×C4—2122no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4—2122no—
365C2×C4C4⋊C4×D5314yes—
366C2×C4C22×C4×D5311yescharacteristic
367C2×C4C2×Q8×D5311yescharacteristic
368C2×C4#41×D5311yescharacteristic
369C2×C4#41×D5311yescharacteristic
370C2×C4C2×#1171994511yescharacteristic
371C2×C4C2×#1171994511yescharacteristic
372C8—22no—
373C8—534no—
374C8—1062no—
375C8#1313317414yes—
376C22—1062no—
377C22—1064no—
378C22—2122no—
379C22—2121no—
380C22—2121no—
381C22C2×C4⋊C4×D5311yescharacteristic
382C22#1681117011yescharacteristic
383C22#1681117011yescharacteristic
384C22#1681141912yes—
385C22#1681142212yes—
386C4—24no—
387C4—82no—
388C4—82no—
389C4—1062no—
390C4—1064no—
391C4—4242no—
392C4—4242no—
393C4#117×D5311yescharacteristic
394C4C2×#1313317411yescharacteristic
395C4#1681117711yescharacteristic
396C4#1681117711yescharacteristic
397C2—2122no—
398C2#2027711911yescharacteristic
399C2#2028330212yes—
400C2#2028544012yes—
401C2#2028544211yescharacteristic
402C2#2028544711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1776C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1776D53
hypocenterC53#20825
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
3. #20274878C2semidirect
4. #20274876C2semidirect
5. #20274844C2semidirect
9. #20274613C2semidirect
10. #20274612C2semidirect
11. C53×#1776C2semidirect
19. #16810575C4semidirect
206. #1776D53semidirect
241. C53#20825semidirect

Maximal subgroups

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

subgroupindex
C2×#168105752
#202771112
#202748782
#202748762
#202748442
#202747492
#202747472
#202747452
#202746132
#202746122
C53×#17762
#2082553