Finite Group Databasebeta

← #27368064

Subgroups — #27368064

Order 13568.

754 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810970C211yesmaximal, characteristic, complemented
2#20284202C211yesmaximal, characteristic, complemented
3#20284195C211yesmaximal, characteristic
4#20274159C214yesmaximal, complemented
5#20274152C214yesmaximal, complemented
6#20273474C211yesmaximal, characteristic, complemented
7#20273386C211yesmaximal, characteristic, complemented
8#20273382C211yesmaximal, characteristic, complemented
9C8×#41×C53C211yesmaximal, characteristic, complemented
10C2×#13133247—21no—
11C2×#13133242—21no—
12C2×#13133217C2211yescharacteristic
13C2×#13133214C2211yescharacteristic
14#16810970—21no—
15#16810970—21no—
16#16810970C2214yes—
17#16810833—21no—
18#16810833C2212yescomplemented
19#16810828—21no—
20#16810828C2212yescomplemented
21#16810823—21no—
22#16810823C2212yes—
23#16810818—21no—
24#16810818C2212yes—
25#16810647C2211yescharacteristic
26#16810647C2211yescharacteristic
27#16810631—21no—
28#16810627—21no—
29C2×#13133139—22no—
30#16810376C2214yescomplemented
31#16810368C2214yescomplemented
32#16810366C2214yes—
33C4×#41×C53C2211yescharacteristic
34C8×Q8×C53C2212yescomplemented
35C8×D4×C53—22no—
36C8×D4×C53C2212yescomplemented
37C53×#431—21no—
38C53×#431C2211yescharacteristic
39C2×C4×C8×C53—21no—
40C2×C4×C8×C53C2211yescharacteristic
41C22×Dic106—21no—
42C22×D212—21no—
43C22×D212—21no—
44C22×D212—21no—
45#13133247—44no—
46#13133242—44no—
47C2×#11719925—22no—
48C2×#11719923—22no—
49C2×#11719924—22no—
50C2×#11719922—22no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53#13133217—21no—
54#13133217—21no—
55#13133217—21no—
56#13133217C2312yes—
57#13133214—21no—
58#13133214—21no—
59#13133214—21no—
60#13133214C2312yes—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63C4⋊#88322—21no—
64C4⋊#88322—21no—
65C4⋊#88322C2312yes—
66C4⋊#88322C2312yes—
67#13133140—22no—
68#13133140—22no—
69#13133139—24no—
70#13133139—24no—
71#13133137—22no—
72#13133137—22no—
73C2×#41×C53D411yescharacteristic
74C22×C8×C53—22no—
75C22×C8×C53D412yes—
76C4×Q8×C53D411yescharacteristic
77C4×Q8×C53C2311yescharacteristic
78C4×D4×C53—21no—
79C4×D4×C53—21no—
80C4×D4×C53D411yescharacteristic
81C4×D4×C53C2311yescharacteristic
82#117×C53—21no—
83#117×C53D411yescharacteristic
84C2×C42×C53—21no—
85C2×C42×C53C2311yescharacteristic
86C4⋊C8×C53—22no—
87C4⋊C8×C53C2312yes—
88C22⋊C8×C53—22no—
89C22⋊C8×C53—21no—
90C4×C8×C53—22no—
91C4×C8×C53—21no—
92C4×C8×C53D411yescharacteristic
93C4×C8×C53D411yescharacteristic
94C4×C8×C53C2312yes—
95C23×D53—41no—
96#41×C53—22no—
97#41×C53D814yes—
98C2×Q8×C53C2×D411yescharacteristic
99C2×D4×C53—21no—
100C2×D4×C53C2×D411yescharacteristic
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×Dic53—41no—
108C2×D212—22no—
109C2×D212—22no—
110C2×D212—22no—
111C2×D212—44no—
112C2×D212—42no—
113C2×D212—42no—
114C2×D212—41no—
115C2×D212—41no—
116C2×D212—41no—
117C2×D212—82no—
118C2×Dic106—22no—
119C2×Dic106—42no—
120C2×Dic106—42no—
121C2×Dic106—41no—
122C2×C8×C53—22no—
123C2×C8×C53—42no—
124C2×C8×C53—44no—
125C2×C8×C53C2×D412yes—
126C2×C8×C53C2×D412yes—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53C2×D411yescharacteristic
130C4⋊C4×C53C2×D411yescharacteristic
131C22⋊C4×C53—21no—
132C22⋊C4×C53—21no—
133C22⋊C4×C53—21no—
134C42×C53—21no—
135C42×C53—21no—
136C42×C53—21no—
137C42×C53C2×D411yescharacteristic
138C42×C53C2×D411yescharacteristic
139C42×C53#4111yescharacteristic
140C42×C53C2411yescharacteristic
141#11719925—44no—
142#11719924—44no—
143#11719923—44no—
144#11719922—44no—
145C4⋊Dic53—22no—
146C4⋊Dic53—22no—
147C4⋊Dic53—41no—
148C4⋊Dic53—41no—
149#11719917—42no—
150#11719917—42no—
151C2×#88322—22no—
152C2×#88322—22no—
153C23×C53—21no—
154C23×C53#12111yescharacteristic
155C22×D53—42no—
156C22×D53—82no—
157C22×D53—82no—
158C22×D53—82no—
159Q8×C53—21no—
160Q8×C53C2×D812yes—
161D4×C53—24no—
162D4×C53—21no—
163D4×C53C2×D812yes—
164C2×C4×C53—21no—
165C2×C4×C53—22no—
166C2×C4×C53—24no—
167C2×C4×C53—22no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—42no—
171C2×C4×C53—42no—
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53C2×D812yes—
176C2×C4×C53#13512yes—
177C2×C4×C53C22×D411yescharacteristic
178C2×C4×C53C22×D411yescharacteristic
179C2×C4×C53C2×#4111yescharacteristic
180C2×Dic53—42no—
181C2×Dic53—82no—
182D212—42no—
183D212—42no—
184D212—44no—
185D212—82no—
186D212—82no—
187D212—82no—
188D212—82no—
189Dic106—42no—
190Dic106—42no—
191Dic106—82no—
192C8×C53—22no—
193C8×C53—22no—
194C8×C53—42no—
195#88322—42no—
196#88322—42no—
197#19276—531nomaximal, Sylow
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—22no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53#46612yes—
204C22×C53C2×#12111yescharacteristic
205C2×D53—82no—
206C2×D53—82no—
207C2×D53—82no—
208C2×D53—164no—
209C4×C53—21no—
210C4×C53—22no—
211C4×C53—21no—
212C4×C53—41no—
213C4×C53—41no—
214C4×C53#46612yes—
215C4×C53#47112yes—
216C4×C53C22×D811yescharacteristic
217C4×C53C2×#13511yescharacteristic
218C4×C53#57512yes—
219Dic53—82no—
220C2×#486—531no—
221#3039—531no—
222#3038—531no—
223C8×#41D5311yescharacteristic, complemented
224#1682—534no—
225#1681—534no—
226#1567—531no—
227#1546—531no—
228#1545—531no—
229C2×C53—22no—
230C2×C53—42no—
231C2×C53#304011yescharacteristic
232C2×C53C2×#46611yescharacteristic
233C2×C53C2×#47111yescharacteristic
234D53—162no—
235C2×#137—1061no—
236C2×AGL(1,ℤ/8)—1061no—
237C2×C4⋊Q8—531no—
238C2×#127—531no—
239C4×#41C2×D5311yescharacteristic
240#486—534no—
241#486—1061no—
242#486—1061no—
243#462—532no—
244#462—1061no—
245#461—532no—
246#461—1061no—
247#460—532no—
248#460—1061no—
249#459—532no—
250#459—1061no—
251C8×Q8C2×D5312yescomplemented
252C8×D4—22no—
253C8×D4C2×D5312yescomplemented
254#431—21no—
255#431C2×D5311yescharacteristic
256#423—531no—
257#423—531no—
258#419—1061no—
259#418—1061no—
260C2×#102—1062no—
261C2×C4×C8—21no—
262C2×C4×C8C2×D5311yescharacteristic
263#335—534no—
264#332—534no—
265#331—534no—
266C53#1927611yescharacteristic, complemented, Sylow
267C2×#41D21211yescharacteristic
268C22×Q8—1061no—
269C22×D4—1061no—
270C22×D4—1061no—
271C22×D4—1061no—
272#137—2124no—
273AGL(1,ℤ/8)—2124no—
274C2×Q16—1062no—
275C2×SD16—1062no—
276C2×SD16—1062no—
277C2×D8—1062no—
278C2×M16—1061no—
279C2×M16—1061no—
280C22×C8—22no—
281C22×C8C22⋊D5312yes—
282C4⋊Q8—532no—
283C4⋊Q8—1061no—
284C4⋊Q8—1061no—
285C4⋊Q8—1061no—
286#127—532no—
287#127—1061no—
288#127—1061no—
289#127—1061no—
290C4×Q8D21211yescharacteristic
291C4×Q8C22×D5311yescharacteristic
292C4×D4—21no—
293C4×D4—21no—
294C4×D4D21211yescharacteristic
295C4×D4C22×D5311yescharacteristic
296#117—21no—
297#117D21211yescharacteristic
298C2×C4⋊C4—1061no—
299C2×C4⋊C4—1061no—
300C2×C42—21no—
301C2×C42C22×D5311yescharacteristic
302C4⋊C8—22no—
303C4⋊C8—532no—
304C4⋊C8—532no—
305C4⋊C8—1061no—
306C4⋊C8—1061no—
307C4⋊C8C22×D5312yes—
308#103—1062no—
309#103—1062no—
310#102—1064no—
311#102—1064no—
312#102—1062no—
313#102—1062no—
314C22⋊C8—21no—
315C22⋊C8—22no—
316C4×C8—22no—
317C4×C8—21no—
318C4×C8C22⋊D5311yescharacteristic
319C4×C8C22⋊D5311yescharacteristic
320C4×C8C22×D5312yes—
321C24—2121no—
322#41—22no—
323#41D42414yes—
324C2×Q8—1062no—
325C2×Q8—2122no—
326C2×Q8—2122no—
327C2×Q8—2121no—
328C2×Q8C2×D21211yescharacteristic
329C2×D4—21no—
330C2×D4—1062no—
331C2×D4—1062no—
332C2×D4—1062no—
333C2×D4—4242no—
334C2×D4—2124no—
335C2×D4—2121no—
336C2×D4—2121no—
337C2×D4—2121no—
338C2×D4—2122no—
339C2×D4—2122no—
340C2×D4C2×D21211yescharacteristic
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—2121no—
345C22×C4C2×D21211yescharacteristic
346C22×C4#1171994511yescharacteristic
347C22×C4C2×C22⋊D5311yescharacteristic
348Q16—2124no—
349SD16—2124no—
350SD16—2124no—
351D8—2124no—
352M16—2122no—
353M16—2122no—
354C2×C8—22no—
355C2×C8—42no—
356C2×C8—44no—
357C2×C8—1062no—
358C2×C8—1062no—
359C2×C8C2×C22⋊D5312yes—
360C2×C8C2×C22⋊D5312yes—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—1062no—
364C4⋊C4—1062no—
365C4⋊C4—2121no—
366C4⋊C4—2121no—
367C4⋊C4C2×D21211yescharacteristic
368C4⋊C4C2×D21211yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42C2×D21211yescharacteristic
376C42#1171994511yescharacteristic
377C42C2×C22⋊D5311yescharacteristic
378C42C23×D5311yescharacteristic
379C23—21no—
380C23—4242no—
381C23—4242no—
382C23—4242no—
383C23—2122no—
384C23#1313319111yescharacteristic
385Q8—21no—
386Q8—4242no—
387Q8—2122no—
388Q8—2122no—
389Q8C2×D42412yes—
390D4—24no—
391D4—21no—
392D4—4242no—
393D4—4242no—
394D4—4242no—
395D4—4242no—
396D4—2122no—
397D4—2122no—
398D4—2124no—
399D4C2×D42412yes—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—4242no—
409C2×C4—2122no—
410C2×C4#1313319111yescharacteristic
411C2×C4#1313319111yescharacteristic
412C2×C4#1313319111yescharacteristic
413C2×C4C2×D42412yes—
414C2×C4#1313323912yes—
415C2×C4C22×C22⋊D5311yescharacteristic
416C2×C4C22×D21211yescharacteristic
417C2×C4C2×#1171994511yescharacteristic
418C8—22no—
419C8—22no—
420C8—42no—
421C8—2122no—
422C8—2122no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—8484no—
429C22—4242no—
430C22—4242no—
431C22—4242no—
432C22#1681085912yes—
433C22C2×#1313319111yescharacteristic
434C4—21no—
435C4—22no—
436C4—21no—
437C4—41no—
438C4—41no—
439C4—4242no—
440C4#1681085912yes—
441C4#1681088612yes—
442C4C22×D42411yescharacteristic
443C4C2×#1313323911yescharacteristic
444C4#1681142112yes—
445C2—22no—
446C2—42no—
447C2—8482no—
448C2#2028420911yescharacteristic
449C2C2×#1681085911yescharacteristic
450C2C2×#1681088611yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC8×#41×C53C2
socleC22×C53C2×#121
prime radical#20284195C2
hypercenterC8×#41D53
hypocenterC53#19276
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16810970C2semidirect
2. #20284202C2semidirect
4. #20274159C2semidirect
5. #20274152C2semidirect
6. #20273474C2semidirect
7. #20273386C2semidirect
8. #20273382C2semidirect
9. C8×#41×C53C2semidirect
18. #16810833C22semidirect
20. #16810828C22semidirect
30. #16810376C22semidirect
31. #16810368C22semidirect
34. C8×Q8×C53C22semidirect
36. C8×D4×C53C22semidirect
223. C8×#41D53semidirect
251. C8×Q8C2×D53semidirect
253. C8×D4C2×D53semidirect
266. C53#19276semidirect

Maximal subgroups

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

subgroupindex
C2×#168109702
#202842022
#202841952
#202741592
#202741522
#202734742
#202733862
#202733822
C8×#41×C532
#1927653