Finite Group Databasebeta

← #27369634

Subgroups — #27369634

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285653C211yesmaximal, characteristic, complemented
2C2×#16810818C211yesmaximal, characteristic
3#20283756C211yesmaximal, characteristic, complemented
4#20274250C214yesmaximal, complemented
5#20274182C214yesmaximal, complemented
6#20273491C211yesmaximal, characteristic, complemented
7#20273384C211yesmaximal, characteristic, complemented
8#20273372C211yesmaximal, characteristic, complemented
9C53×#3077C211yesmaximal, characteristic, complemented
10C22×#11719922—21no—
11C2×#13133214C2211yescharacteristic
12#16810997C2212yescomplemented
13#16810971C2212yescomplemented
14#16810823—22no—
15#16810818—22no—
16#16810818C2214yes—
17#16810773—22no—
18#16810701—22no—
19#16810701C2214yescomplemented
20#16810676C2211yescharacteristic
21#16810676C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133137—21no—
24#16810587—22no—
25#16810375C2214yescomplemented
26#16810372C2212yescomplemented
27#16810369C2212yescomplemented
28#16810366C2214yescomplemented
29C53×#533C2211yescharacteristic
30C2×C4×D4×C53C2211yescharacteristic
31C53×#478—22no—
32C53×#476C2212yescomplemented
33C53×#474C2212yescomplemented
34C53×#428—21no—
35C53×#423C2211yescharacteristic
36C53×#418—21no—
37C22×D212—21no—
38C22×D212—21no—
39C22×D212—21no—
40C2×#11719924—24no—
41C2×#11719922—24no—
42C2×#11719922—44no—
43C2×#11719917—24no—
44C22×#88322—21no—
45#13133214—21no—
46#13133214—21no—
47#13133214—21no—
48#13133214C2312yes—
49C4⋊#88322—21no—
50C4⋊#88322D412yes—
51C4⋊#88322C2312yes—
52C4⋊#88322C2312yes—
53#13133137—24no—
54#13133137—24no—
55C22⋊#88322—24no—
56C22⋊#88322—21no—
57C22⋊#88322D412yes—
58C4×#88322—21no—
59C4×#88322C2312yes—
60C22×D4×C53D411yescharacteristic
61C23×C4×C53—21no—
62C2×M16×C53—21no—
63C4⋊Q8×C53C2311yescharacteristic
64C4⋊Q8×C53C2311yescharacteristic
65#125×C53—21no—
66C22⋊Q8×C53—21no—
67C22⋊Q8×C53—21no—
68C4×Q8×C53—21no—
69C4×Q8×C53—21no—
70C4×D4×C53—21no—
71C4×D4×C53—21no—
72C4×D4×C53D412yes—
73C4×D4×C53C2312yes—
74#117×C53—21no—
75C2×C4⋊C4×C53D411yescharacteristic
76C2×C22⋊C4×C53—21no—
77C2×C42×C53C2311yescharacteristic
78#107×C53—22no—
79#106×C53—22no—
80C4⋊C8×C53—21no—
81C4⋊C8×C53C2312yes—
82#102×C53—22no—
83#102×C53—22no—
84C24×C53—21no—
85C23×D53—41no—
86C2×Q8×C53—21no—
87C2×Q8×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—22no—
90C2×D4×C53—22no—
91C2×D4×C53C2×D412yes—
92C22×C4×C53—21no—
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—82no—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53#4111yescharacteristic
99C2×D212—22no—
100C2×D212—22no—
101C2×D212—22no—
102C2×D212—42no—
103C2×D212—42no—
104C2×D212—41no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212—82no—
108C2×D212—82no—
109M16×C53—42no—
110C2×C8×C53—22no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53C2×D412yes—
119C22⋊C4×C53—22no—
120C22⋊C4×C53—22no—
121C22⋊C4×C53—41no—
122C42×C53—41no—
123C42×C53C2×D411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53#4111yescharacteristic
126C42×C53C2411yescharacteristic
127#11719924—84no—
128#11719922—48no—
129#11719917—84no—
130C2×#88322—22no—
131C2×#88322—22no—
132C2×#88322—42no—
133C2×#88322C2×D414yes—
134C23×C53—22no—
135C23×C53—42no—
136C23×C53—81no—
137C23×C53—81no—
138C23×C53#12111yescharacteristic
139C22×D53—42no—
140C22×D53—82no—
141C22×D53—82no—
142C22×D53—82no—
143Q8×C53—42no—
144Q8×C53—81no—
145D4×C53—22no—
146D4×C53—22no—
147D4×C53—42no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
151C2×C4×C53—44no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—82no—
155C2×C4×C53—81no—
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53AGL(1,ℤ/8)12yes—
160C2×C4×C53#13712yes—
161C2×C4×C53C22×D411yescharacteristic
162C2×C4×C53C22×D411yescharacteristic
163C2×C4×C53C2×#4111yescharacteristic
164D212—42no—
165D212—42no—
166D212—44no—
167D212—82no—
168D212—82no—
169D212—82no—
170D212—82no—
171C8×C53—42no—
172#88322—24no—
173#88322—42no—
174#88322—42no—
175#19420—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53—84no—
182C22×C53#46812yes—
183C22×C53C2×#12111yescharacteristic
184C2×D53—82no—
185C2×D53—82no—
186C2×D53—82no—
187C2×D53—164no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—42no—
192C4×C53—81no—
193C4×C53—81no—
194C4×C53#46812yes—
195C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
196C4×C53C2×#13711yescharacteristic
197C4×C53#57612yes—
198#3208—531no—
199#3077D5311yescharacteristic, complemented
200C2×#459—531no—
201#2975—531no—
202#1697—534no—
203#1685—534no—
204#1571—531no—
205#1545—531no—
206#1542—531no—
207C2×C53—22no—
208C2×C53—82no—
209C2×C53#304111yescharacteristic
210C2×C53C2×#46811yescharacteristic
211C2×C53#306811yescharacteristic
212D53—162no—
213C22×D8—1061no—
214#533C2×D5311yescharacteristic
215C2×#127—531no—
216C2×C4×D4C2×D5311yescharacteristic
217#492—532no—
218#486—532no—
219#478—22no—
220#476C2×D5312yescomplemented
221#474C2×D5312yescomplemented
222#460—1062no—
223#459—534no—
224#459—1062no—
225#450—1062no—
226#436—534no—
227#436—1062no—
228#431—531no—
229#431—531no—
230#428—21no—
231#423C2×D5311yescharacteristic
232C2×C4⋊C8—531no—
233#418—21no—
234C2×#102—1061no—
235#407—1062no—
236#335—534no—
237#333—532no—
238#332—532no—
239#331—534no—
240C53#1942011yescharacteristic, complemented, Sylow
241C22×D4—1061no—
242C22×D4—1061no—
243C22×D4—1061no—
244C22×D4C22⋊D5311yescharacteristic
245C23×C4—21no—
246C2×SD16—1064no—
247C2×D8—1064no—
248C2×D8—2124no—
249C2×M16—21no—
250C2×M16—1064no—
251C22×C8—1061no—
252C4⋊Q8C22×D5311yescharacteristic
253C4⋊Q8C22×D5311yescharacteristic
254#127—532no—
255#127—1061no—
256#127—1061no—
257#127—1061no—
258#125—21no—
259C22⋊Q8—21no—
260C22⋊Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4C22⋊D5312yes—
266C4×D4C22×D5312yes—
267#117—21no—
268C2×C4⋊C4C22⋊D5311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C42C22×D5311yescharacteristic
271#107—22no—
272#106—22no—
273C4⋊C8—21no—
274C4⋊C8—532no—
275C4⋊C8—532no—
276C4⋊C8—532no—
277C4⋊C8—1061no—
278C4⋊C8C22×D5312yes—
279#102—22no—
280#102—22no—
281#102—1064no—
282#102—1064no—
283C22⋊C8—532no—
284C22⋊C8—1064no—
285C22⋊C8—1061no—
286C4×C8—532no—
287C4×C8—1061no—
288C24—21no—
289C24—2121no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—1062no—
296C2×D4—1062no—
297C2×D4—1062no—
298C2×D4—4242no—
299C2×D4—4242no—
300C2×D4—2121no—
301C2×D4—2121no—
302C2×D4—2121no—
303C2×D4—2122no—
304C2×D4—2122no—
305C2×D4C2×C22⋊D5312yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—82no—
310C22×C4D4×D5311yescharacteristic
311C22×C4#1171994711yescharacteristic
312C22×C4C2×C22⋊D5311yescharacteristic
313SD16—4244no—
314D8—2128no—
315M16—42no—
316M16—4244no—
317C2×C8—22no—
318C2×C8—534no—
319C2×C8—1062no—
320C2×C8—1062no—
321C2×C8—2122no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4C2×C22⋊D5312yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—41no—
333C42—41no—
334C42D4×D5311yescharacteristic
335C42#1171994711yescharacteristic
336C42C2×C22⋊D5311yescharacteristic
337C42C23×D5311yescharacteristic
338C23—22no—
339C23—42no—
340C23—81no—
341C23—81no—
342C23—4242no—
343C23—4242no—
344C23—4242no—
345C23—2122no—
346C23#1313318911yescharacteristic
347Q8—42no—
348Q8—81no—
349D4—22no—
350D4—22no—
351D4—42no—
352D4—4242no—
353D4—4242no—
354D4—4242no—
355D4—4242no—
356D4—2122no—
357D4—2122no—
358D4—2124no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4#1313318911yescharacteristic
368C2×C4#1313318911yescharacteristic
369C2×C4#1313318911yescharacteristic
370C2×C4#1313324212yes—
371C2×C4#1313324812yes—
372C2×C4C2×D4×D5311yescharacteristic
373C2×C4C22×C22⋊D5311yescharacteristic
374C2×C4C2×#1171994711yescharacteristic
375C8—42no—
376C8—1064no—
377C8—2122no—
378C8—2122no—
379C22—21no—
380C22—21no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—84no—
385C22—8484no—
386C22—4242no—
387C22—4242no—
388C22—4242no—
389C22#1681087112yes—
390C22C2×#1313318911yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—81no—
397C4#1681087112yes—
398C4C2×#1313324211yescharacteristic
399C4C2×#1313324811yescharacteristic
400C4#1681142712yes—
401C2—22no—
402C2—82no—
403C2—8482no—
404C2#2028422011yescharacteristic
405C2C2×#1681087111yescharacteristic
406C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3077C2
socleC22×C53C2×#121
prime radicalC2×#16810818C2
hypercenter#3077D53
hypocenterC53#19420
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
1. #20285653C2semidirect
3. #20283756C2semidirect
4. #20274250C2semidirect
5. #20274182C2semidirect
6. #20273491C2semidirect
7. #20273384C2semidirect
8. #20273372C2semidirect
9. C53×#3077C2semidirect
12. #16810997C22semidirect
13. #16810971C22semidirect
19. #16810701C22semidirect
25. #16810375C22semidirect
26. #16810372C22semidirect
27. #16810369C22semidirect
28. #16810366C22semidirect
32. C53×#476C22semidirect
33. C53×#474C22semidirect
199. #3077D53semidirect
220. #476C2×D53semidirect
221. #474C2×D53semidirect
240. C53#19420semidirect

Maximal subgroups

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

subgroupindex
#202856532
C2×#168108182
#202837562
#202742502
#202741822
#202734912
#202733842
#202733722
C53×#30772
#1942053