Finite Group Databasebeta

← #27279999

Subgroups — #27279999

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284512C211yesmaximal, characteristic
2#20273278C211yesmaximal, characteristic, complemented
3#20273213C211yesmaximal, characteristic, complemented
4#20273106C211yesmaximal, characteristic
5#20272618C211yesmaximal, characteristic, complemented
6#20272573C211yesmaximal, characteristic, complemented
7C53×#2349C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810923—22no—
10#16810913C412yescomplemented
11#16810903C412yescomplemented
12#16810676C2211yescharacteristic
13#16810676C2211yescharacteristic
14#16810663—21no—
15#16810647C2211yescharacteristic
16#16810628—21no—
17#16810359—21no—
18#16810359—21no—
19#16810359—22no—
20D4⋊#88322—22no—
21C2×C4×D4×C53—21no—
22C2×C4×D4×C53C2211yescharacteristic
23C53×#389—21no—
24C53×#385—22no—
25C53×#382C2211yescharacteristic
26C53×#380—21no—
27C4×C4⋊C4×C53C2211yescharacteristic
28C2×#95×C53—21no—
29C2×#11719917—21no—
30#13133217C2×C411yescharacteristic, complemented
31#13133217C2×C411yescharacteristic, complemented
32#13133209—21no—
33#13133196—21no—
34#13133196—21no—
35C4×Dic106—21no—
36C4×Dic106—21no—
37#13133176—21no—
38#13133150—22no—
39#13133148—22no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322C2×C412yescomplemented
44#13133138—22no—
45#13133138—22no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C4×#88322—21no—
49C4×#88322—21no—
50C22×D4×C53D411yescharacteristic
51C23×C4×C53—21no—
52C23×C4×C53—22no—
53C4×D4×C53—21no—
54C4×D4×C53—21no—
55C4×D4×C53—42no—
56C4×D4×C53—42no—
57C4×D4×C53C2×C412yes—
58C4×D4×C53D412yes—
59C2×C4⋊C4×C53—22no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53—22no—
64C2×C22⋊C4×C53—21no—
65C2×C42×C53—21no—
66C2×C42×C53C2311yescharacteristic
67#95×C53—22no—
68#95×C53—21no—
69#95×C53D411yescharacteristic
70#95×C53D411yescharacteristic
71#95×C53D412yes—
72C24×C53—21no—
73C2×D4×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53C22⋊C412yes—
77C2×D4×C53SD1612yes—
78C2×D4×C53Q1612yes—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—21no—
85C22×C4×C53—44no—
86C22×C4×C53—44no—
87C22×C4×C53—82no—
88C22×C4×C53—82no—
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C2×Dic106—21no—
93C2×Dic106—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—42no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—82no—
99C4⋊C4×C53C22⋊C412yes—
100C22⋊C4×C53—22no—
101C22⋊C4×C53—22no—
102C22⋊C4×C53—44no—
103C42×C53—41no—
104C42×C53—41no—
105C42×C53C22×C411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53#4111yescharacteristic
108C42×C53#4111yescharacteristic
109C22⋊Dic53—41no—
110C4⋊Dic53—21no—
111C4⋊Dic53—21no—
112C4⋊Dic53—21no—
113C4⋊Dic53—21no—
114#11719919—41no—
115#11719919—41no—
116C4×Dic53—41no—
117#11719917—41no—
118#11719917—41no—
119C2×#88322—22no—
120C2×#88322—41no—
121C2×#88322—41no—
122C23×C53—22no—
123C23×C53—22no—
124C23×C53—22no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53#12011yescharacteristic
128D4×C53—22no—
129D4×C53—22no—
130D4×C53—22no—
131D4×C53#10314yes—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53—41no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—41no—
145C2×C4×C53—82no—
146C2×C4×C53—82no—
147C2×C4×C53—88no—
148C2×C4×C53C2×C22⋊C411yescharacteristic
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53C4×D411yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12311yescharacteristic
154C2×C4×C53C2×SD1611yescharacteristic
155C2×C4×C53C2×Q1611yescharacteristic
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C53#13711yescharacteristic
158C2×Dic53—41no—
159C2×Dic53—41no—
160Dic106—81no—
161Dic106—81no—
162#88322—42no—
163#88322—81no—
164#88322—81no—
165#9175—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—24no—
169C22×C53—44no—
170C22×C53—44no—
171C22×C53—42no—
172C22×C53—42no—
173C22×C53#38611yescharacteristic
174C22×C53#45011yescharacteristic
175C22×C53#45111yescharacteristic
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—42no—
181C4×C53—42no—
182C4×C53—81no—
183C4×C53—82no—
184C4×C53C2×#10311yescharacteristic
185C4×C53#41711yescharacteristic
186C4×C53#48311yescharacteristic
187C4×C53#48411yescharacteristic
188Dic53—81no—
189Dic53—81no—
190#3077—531no—
191#2349D5311yescharacteristic
192#1518—531no—
193#1503—531no—
194#1478—531no—
195#1303—531no—
196#1286—531no—
197C2×C53—21no—
198C2×C53—21no—
199C2×C53—44no—
200C2×C53#187211yescharacteristic
201C2×C53#190411yescharacteristic
202C2×C53#190611yescharacteristic
203#533—531no—
204C2×C4×D4—21no—
205C2×C4×D4C2×D5311yescharacteristic
206#478—1062no—
207#476—532no—
208#474—532no—
209#431—531no—
210#431—531no—
211#428—1061no—
212#423—531no—
213#418—1061no—
214#389—21no—
215#385—22no—
216#382C2×D5311yescharacteristic
217#380—21no—
218C4×C4⋊C4C2×D5311yescharacteristic
219C2×#95—21no—
220#328—1061no—
221#328—1061no—
222#328—1062no—
223D4⋊C8—1062no—
224C53#917511yescharacteristic, complemented, Sylow
225C22×D4C22⋊D5311yescharacteristic
226C23×C4—21no—
227C23×C4—22no—
228C2×M16—1061no—
229C4⋊Q8—531no—
230C4⋊Q8—531no—
231#125—1061no—
232C22⋊Q8—1061no—
233C22⋊Q8—1061no—
234C4×Q8—1061no—
235C4×Q8—1061no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4C4×D5312yes—
241C4×D4D21212yes—
242#117—1061no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4C22⋊D5311yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C42—21no—
250C2×C42C22×D5311yescharacteristic
251#107—1062no—
252#106—1062no—
253C4⋊C8—532no—
254C4⋊C8—1061no—
255C4⋊C8—1061no—
256C4⋊C8—1061no—
257#102—1062no—
258#102—1062no—
259C22⋊C8—1061no—
260C22⋊C8—1061no—
261C4×C8—1061no—
262C4×C8—1061no—
263#95—22no—
264#95—21no—
265#95C22⋊D5311yescharacteristic
266#95C22⋊D5311yescharacteristic
267#95C22⋊D5312yes—
268C24—21no—
269C2×Q8—1061no—
270C2×Q8—1061no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4#1171992112yes—
275C2×D4#1171992312yes—
276C2×D4#1171992512yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4C2×C22⋊D5311yescharacteristic
288C22×C4C2×C22⋊D5311yescharacteristic
289C22×C4C2×C22⋊D5311yescharacteristic
290M16—2121no—
291M16—2121no—
292C2×C8—1062no—
293C2×C8—2121no—
294C2×C8—2121no—
295C4⋊C4—21no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—82no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4—1061no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C4⋊C4#1171992112yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—44no—
310C22⋊C4—2121no—
311C42—41no—
312C42—41no—
313C42—2121no—
314C42C2×C4×D5311yescharacteristic
315C42C2×D21211yescharacteristic
316C42#1171994511yescharacteristic
317C42#1171994511yescharacteristic
318C23—22no—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23#1313318611yescharacteristic
324Q8—4241no—
325Q8—4241no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4#1313314014yes—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—88no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4C2×#1171992111yescharacteristic
349C2×C4C4×C22⋊D5311yescharacteristic
350C2×C4C4×C22⋊D5311yescharacteristic
351C2×C4#1313319111yescharacteristic
352C2×C4#1313319111yescharacteristic
353C2×C4#1313320011yescharacteristic
354C2×C4C2×#1171992311yescharacteristic
355C2×C4C2×#1171992511yescharacteristic
356C2×C4#1313324311yescharacteristic
357C2×C4#1313324811yescharacteristic
358C8—4241no—
359C8—4241no—
360C8—2122no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22#1681050211yescharacteristic
369C22#1681077411yescharacteristic
370C22#1681077911yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—42no—
377C4—82no—
378C4—81no—
379C4—4241no—
380C4—4241no—
381C4C2×#1313314011yescharacteristic
382C4#1681062211yescharacteristic
383C4#1681095311yescharacteristic
384C4#1681095811yescharacteristic
385C2—21no—
386C2—21no—
387C2—44no—
388C2#2027502211yescharacteristic
389C2#2027520211yescharacteristic
390C2#2027521611yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2349C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#2349D53
hypocenterC53#9175
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #20273278C2semidirect
3. #20273213C2semidirect
5. #20272618C2semidirect
6. #20272573C2semidirect
10. #16810913C4semidirect
11. #16810903C4semidirect
30. #13133217C2×C4semidirect
31. #13133217C2×C4semidirect
43. C4⋊#88322C2×C4semidirect
224. C53#9175semidirect

Maximal subgroups

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

subgroupindex
#202845122
#202732782
#202732132
#202731062
#202726182
#202725732
C53×#23492
#917553