Finite Group Databasebeta

← #27300060

Subgroups — #27300060

Order 13568.

540 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810413C212yesmaximal
2#20274707C211yesmaximal, characteristic
3#20274703C211yesmaximal, characteristic
4#20272833C212yesmaximal
5C4×#123×C53C211yesmaximal, characteristic
6#16810647—21no—
7C2×#13133134C2212yes—
8C2×#13133132C2212yes—
9#16810483—21no—
10#16810413—24no—
11#16810392—21no—
12#16810390—21no—
13C2×#123×C53C2211yescharacteristic
14C2×C4×D4×C53C2211yescharacteristic
15C22×C42×C53C411yescharacteristic, complemented
16C53×#388—21no—
17C53×#387—21no—
18C53×#386C2211yescharacteristic
19C53×#385C411yescharacteristic, complemented
20C53×#381C411yescharacteristic, complemented
21C4×C4⋊C4×C53—21no—
22C4×C22⋊C4×C53—21no—
23C4×C22⋊C4×C53C411yescharacteristic, complemented
24C2×#11719917—22no—
25C2×C4⋊Dic53—41no—
26C2×C22⋊Dic53—22no—
27C4⋊#88322—41no—
28C4⋊#88322—41no—
29#13133134—22no—
30#13133134—22no—
31#13133132—22no—
32#13133132—22no—
33#13133124—41no—
34C22×D4×C53C2311yescharacteristic
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53C2×C411yescharacteristic
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53D411yescharacteristic
39#123×C53—21no—
40#123×C53—21no—
41#123×C53D412yes—
42#123×C53Q812yes—
43C4×D4×C53—41no—
44C4×D4×C53—41no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C42×C53—21no—
55C2×C42×C53—21no—
56C2×C42×C53—21no—
57C2×C42×C53—41no—
58C2×C42×C53—41no—
59C2×C42×C53D411yescharacteristic
60#95×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63#95×C53—21no—
64#95×C53—21no—
65C24×C53C22×C411yescharacteristic
66C24×C53C2×D411yescharacteristic
67C2×D4×C53—41no—
68C2×D4×C53—41no—
69C2×D4×C53C2×D411yescharacteristic
70C2×D4×C53C2×Q811yescharacteristic
71C2×D4×C53#4111yescharacteristic
72C2×D4×C53#4111yescharacteristic
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—42no—
83C22×C4×C53—42no—
84C22×C4×C53—42no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53C22⋊C411yescharacteristic
89C22×C4×C53C22⋊C411yescharacteristic
90C22×C4×C53C22⋊C411yescharacteristic
91C22×C4×C53C22⋊C411yescharacteristic
92C22×C4×C53C4⋊C412yescomplemented
93C22×C4×C53C2×D411yescharacteristic
94C22×Dic53—42no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—42no—
99C22⋊C4×C53—21no—
100C22⋊C4×C53—22no—
101C22⋊C4×C53—22no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—42no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53C4⋊C412yescomplemented
107C42×C53—21no—
108C42×C53—21no—
109C42×C53—42no—
110C42×C53—42no—
111C42×C53—41no—
112C42×C53—41no—
113C22⋊Dic53—42no—
114C22⋊Dic53—42no—
115C4⋊Dic53—82no—
116#11719917—42no—
117#11719917—42no—
118C2×#88322—42no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53C2×C22⋊C411yescharacteristic
128C23×C53C2×C4⋊C411yescharacteristic
129C23×C53#11711yescharacteristic
130C23×C53#12111yescharacteristic
131C23×C53C22⋊Q811yescharacteristic
132D4×C53—41no—
133D4×C53—41no—
134D4×C53—41no—
135D4×C53—41no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53—41no—
160C2×C4×C53—42no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—81no—
165C2×C4×C53#9911yescharacteristic
166C2×C4×C53#9911yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53C22⋊Q811yescharacteristic
169C2×Dic53—82no—
170C2×Dic53—82no—
171C2×Dic53—82no—
172#88322—82no—
173#12794—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—42no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—81no—
190C22×C53#38011yescharacteristic
191C22×C53C2×#9911yescharacteristic
192C22×C53#41011yescharacteristic
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—42no—
197C4×C53—42no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—41no—
201C4×C53—42no—
202C4×C53—81no—
203C4×C53#35212yescomplemented
204Dic53—162no—
205C4×#123D5311yescharacteristic
206C2×#352—532no—
207#1800—531no—
208#1799—531no—
209#1390—532no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—41no—
213C2×C53—41no—
214C2×C53—41no—
215C2×C53—41no—
216C2×C53#179911yescharacteristic
217C2×C53C2×#35211yescharacteristic
218C2×C53#212011yescharacteristic
219C2×#123C2×D5311yescharacteristic
220C2×C4×D4C2×D5311yescharacteristic
221C22×C42Dic5311yescharacteristic, complemented
222#423—1061no—
223C2×#100—532no—
224C2×#99—532no—
225#388—21no—
226#387—21no—
227#386C2×D5311yescharacteristic
228#385Dic5311yescharacteristic, complemented
229#381Dic5311yescharacteristic, complemented
230#380—1061no—
231C4×C4⋊C4—21no—
232C4×C22⋊C4—21no—
233C4×C22⋊C4Dic5311yescharacteristic, complemented
234#352—1064no—
235#343—1061no—
236#342—1061no—
237C53#1279411yescharacteristic, complemented, Sylow
238C22×D4C22×D5311yescharacteristic
239C23×C4C2×Dic5311yescharacteristic
240C23×C4C2×Dic5311yescharacteristic
241C23×C4C2×Dic5311yescharacteristic
242C23×C4C22⋊D5311yescharacteristic
243C2×M16—1062no—
244#123—21no—
245#123—21no—
246#123Dic10612yes—
247#123D21212yes—
248C4×D4—41no—
249C4×D4—41no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—2121no—
253C2×C4⋊C4C22⋊D5311yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—1062no—
257C2×C22⋊C4C2×Dic5311yescharacteristic
258C2×C22⋊C4C2×Dic5311yescharacteristic
259C2×C22⋊C4C2×Dic5311yescharacteristic
260C2×C22⋊C4C22⋊D5311yescharacteristic
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—41no—
265C2×C42—41no—
266C2×C42C22⋊D5311yescharacteristic
267C4⋊C8—2121no—
268C4⋊C8—2121no—
269#100—1062no—
270#100—1062no—
271#99—1062no—
272#99—1062no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—2121no—
279C24C22×Dic5311yescharacteristic
280C24C2×C22⋊D5311yescharacteristic
281C2×D4—41no—
282C2×D4—41no—
283C2×D4C2×Dic10611yescharacteristic
284C2×D4C2×D21211yescharacteristic
285C2×D4#1171994511yescharacteristic
286C2×D4#1171994511yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—2122no—
303C22×C4C4⋊Dic5312yescomplemented
304C22×C4C22⋊Dic5311yescharacteristic
305C22×C4C22⋊Dic5311yescharacteristic
306C22×C4C22⋊Dic5311yescharacteristic
307C22×C4C22⋊Dic5311yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309M16—2122no—
310M16—2122no—
311C2×C8—2122no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—4242no—
317C22⋊C4—21no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—42no—
323C22⋊C4—41no—
324C22⋊C4—2122no—
325C22⋊C4—2122no—
326C22⋊C4C4⋊Dic5312yescomplemented
327C42—21no—
328C42—21no—
329C42—42no—
330C42—42no—
331C42—41no—
332C42—41no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23C2×C22⋊Dic5311yescharacteristic
342C23C2×C4⋊Dic5311yescharacteristic
343C23#1313317611yescharacteristic
344C23#1313319111yescharacteristic
345C23#1313319611yescharacteristic
346D4—41no—
347D4—41no—
348D4—41no—
349D4—41no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—81no—
379C2×C4—4242no—
380C2×C4—4242no—
381C2×C4—4242no—
382C2×C4#1313313211yescharacteristic
383C2×C4#1313313211yescharacteristic
384C2×C4#1313319111yescharacteristic
385C2×C4#1313319611yescharacteristic
386C8—4242no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22#1681048311yescharacteristic
404C22C2×#1313313211yescharacteristic
405C22#1681059711yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—42no—
410C4—42no—
411C4—41no—
412C4—41no—
413C4—42no—
414C4—41no—
415C4—81no—
416C4—8482no—
417C4#1681041312yescomplemented
418C2—21no—
419C2—21no—
420C2—41no—
421C2—41no—
422C2—41no—
423C2—41no—
424C2#2027470311yescharacteristic
425C2C2×#1681041311yescharacteristic
426C2#2027635611yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC4×#123×C53C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenterC4×#123D53
hypocenterC53#12794
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#123

Complements

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

NH ≅ G/Ntype
15. C22×C42×C53C4semidirect
19. C53×#385C4semidirect
20. C53×#381C4semidirect
23. C4×C22⋊C4×C53C4semidirect
92. C22×C4×C53C4⋊C4semidirect
106. C22⋊C4×C53C4⋊C4semidirect
203. C4×C53#352semidirect
221. C22×C42Dic53semidirect
228. #385Dic53semidirect
229. #381Dic53semidirect
233. C4×C22⋊C4Dic53semidirect
237. C53#12794semidirect
303. C22×C4C4⋊Dic53semidirect
326. C22⋊C4C4⋊Dic53semidirect
417. C4#16810413semidirect

Maximal subgroups

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

subgroupindex
C2×#168104132
#202747072
#202747032
#202728332
C4×#123×C532
#1279453