Finite Group Databasebeta

← #27278813

Subgroups — #27278813

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283840C211yesmaximal, characteristic
2#20275693C211yesmaximal, characteristic, complemented
3C4×#13133144C211yesmaximal, characteristic
4#20273136C211yesmaximal, characteristic, complemented
5#20272606C211yesmaximal, characteristic, complemented
6#20272547C211yesmaximal, characteristic, complemented
7C53×#3107C211yesmaximal, characteristic
8#16810754—21no—
9#16810695—22no—
10D4×#88322—21no—
11#16810680—21no—
12#16810676C2211yescharacteristic
13#16810647C2211yescharacteristic
14C2×C4⋊#88322—21no—
15C2×#13133144—21no—
16C4×#11719917C2211yescharacteristic
17C42×Dic53C2211yescharacteristic
18#16810382—22no—
19#16810382—21no—
20Q8⋊#88322—21no—
21Q8⋊#88322—21no—
22C53×#533C2211yescharacteristic
23C4×#41×C53C2211yescharacteristic
24C53×#485—21no—
25C53×#475C411yescharacteristic, complemented
26C53×#474C411yescharacteristic, complemented
27C53×#462C411yescharacteristic, complemented
28C53×#461C411yescharacteristic, complemented
29C53×#452—21no—
30C4×Q16×C53—21no—
31C4×SD16×C53—21no—
32C53×#431C2211yescharacteristic
33C2×#11719917—21no—
34C2×#11719917D412yes—
35C22×#88322—22no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C4⋊#88322—22no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322D411yescharacteristic
43C4⋊#88322D411yescharacteristic
44#13133144—24no—
45C22⋊#88322—22no—
46C22⋊#88322—21no—
47#13133128—22no—
48#13133128—21no—
49C4×#88322—21no—
50C4×#88322—21no—
51C2×#41×C53D411yescharacteristic
52C2×Q16×C53—21no—
53C2×SD16×C53—21no—
54C4⋊Q8×C53C2×C411yescharacteristic
55C4⋊Q8×C53C2×C411yescharacteristic
56#125×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C4×Q8×C53—21no—
60C4×Q8×C53—21no—
61C4×Q8×C53C2×C411yescharacteristic
62C4×Q8×C53D411yescharacteristic
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53C2×C411yescharacteristic
66C4×D4×C53D411yescharacteristic
67#117×C53—21no—
68#117×C53—21no—
69#117×C53D411yescharacteristic
70C2×C42×C53—21no—
71C2×C42×C53C2311yescharacteristic
72#107×C53—21no—
73#106×C53—21no—
74C4⋊C8×C53C2×C411yescharacteristic
75C4⋊C8×C53C2×C411yescharacteristic
76#103×C53—21no—
77#103×C53—21no—
78#103×C53—21no—
79#102×C53—21no—
80C22⋊C8×C53—21no—
81C4×C8×C53—21no—
82#41×C53—22no—
83#41×C53—41no—
84C2×Q8×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53C22⋊C411yescharacteristic
87C2×D4×C53—21no—
88C2×D4×C53C22⋊C411yescharacteristic
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53#4111yescharacteristic
95C22×Dic53—41no—
96Q16×C53—42no—
97SD16×C53—42no—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53C22⋊C411yescharacteristic
109C4⋊C4×C53C22⋊C411yescharacteristic
110C22⋊C4×C53—21no—
111C22⋊C4×C53—21no—
112C22⋊C4×C53—21no—
113C22⋊C4×C53—41no—
114C42×C53—21no—
115C42×C53—21no—
116C42×C53—21no—
117C42×C53—41no—
118C42×C53C22×C411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53#4111yescharacteristic
122C4×Dic53—22no—
123C4×Dic53—22no—
124C4×Dic53—22no—
125C4×Dic53—41no—
126C4×Dic53—41no—
127C4×Dic53—41no—
128#11719917—22no—
129#11719917—41no—
130#11719917—42no—
131C2×#88322—21no—
132C2×#88322—21no—
133C2×#88322—22no—
134C2×#88322—41no—
135C2×#88322—82no—
136C23×C53—21no—
137C23×C53#12111yescharacteristic
138Q8×C53—21no—
139Q8×C53—21no—
140Q8×C53—41no—
141Q8×C53—41no—
142Q8×C53—41no—
143Q8×C53—41no—
144D4×C53—21no—
145D4×C53—42no—
146D4×C53#10412yes—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—42no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53C2×C22⋊C411yescharacteristic
164C2×C4×C53C4×D411yescharacteristic
165C2×C4×C53C4×D411yescharacteristic
166C2×C4×C53#12011yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12311yescharacteristic
169C2×Dic53—42no—
170C2×Dic53—42no—
171C2×Dic53—42no—
172C8×C53—41no—
173C8×C53—41no—
174#88322—42no—
175#88322—41no—
176#88322—42no—
177#88322—41no—
178#8971—531nomaximal, Sylow
179C22×C53—21no—
180C22×C53—22no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53#38611yescharacteristic
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—41no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53—81no—
193C4×C53—81no—
194C4×C53C2×#10411yescharacteristic
195C4×C53#42111yescharacteristic
196C4×C53#44311yescharacteristic
197C4×C53#44411yescharacteristic
198C4×C53#45511yescharacteristic
199C4×C53#45611yescharacteristic
200Dic53—44no—
201#3107D5311yescharacteristic
202#2990—531no—
203#1994—531no—
204C4×#104—531no—
205#1485—531no—
206#1299—531no—
207#1276—531no—
208C2×C53—41no—
209C2×C53—42no—
210C2×C53#187111yescharacteristic
211C2×C53#187611yescharacteristic
212C2×C53#188211yescharacteristic
213#533C2×D5311yescharacteristic
214C4×#41C2×D5311yescharacteristic
215#485—21no—
216#475Dic5311yescharacteristic, complemented
217#474Dic5311yescharacteristic, complemented
218#462Dic5311yescharacteristic, complemented
219#461Dic5311yescharacteristic, complemented
220#452—21no—
221#446—1061no—
222C4×Q16—21no—
223C4×SD16—21no—
224#435—1062no—
225C8×D4—1061no—
226#432—1061no—
227#431—531no—
228#431C2×D5311yescharacteristic
229#423—531no—
230C2×C4⋊C8—1061no—
231C2×#104—1061no—
232C4×M16—531no—
233C43—531no—
234#339—1062no—
235#339—1061no—
236Q8⋊C8—1061no—
237Q8⋊C8—1061no—
238C53#897111yescharacteristic, complemented, Sylow
239C2×#41C22⋊D5311yescharacteristic
240C2×Q16—21no—
241C2×SD16—21no—
242C2×M16—532no—
243C2×M16—1061no—
244C22×C8—1062no—
245C4⋊Q8C2×Dic5311yescharacteristic
246C4⋊Q8C2×Dic5311yescharacteristic
247#125—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8C2×Dic5311yescharacteristic
253C4×Q8C22⋊D5311yescharacteristic
254C4×D4—21no—
255C4×D4—21no—
256C4×D4C2×Dic5311yescharacteristic
257C4×D4C22⋊D5311yescharacteristic
258#117—21no—
259#117—21no—
260#117C22⋊D5311yescharacteristic
261C2×C42—21no—
262C2×C42—1061no—
263C2×C42—1061no—
264C2×C42—1061no—
265C2×C42C22×D5311yescharacteristic
266#107—21no—
267#106—21no—
268C4⋊C8—531no—
269C4⋊C8—531no—
270C4⋊C8—1062no—
271C4⋊C8—1061no—
272C4⋊C8—1061no—
273C4⋊C8C2×Dic5311yescharacteristic
274C4⋊C8C2×Dic5311yescharacteristic
275#104—1064no—
276#103—21no—
277#103—21no—
278#103—21no—
279#102—21no—
280C22⋊C8—21no—
281C22⋊C8—1062no—
282C22⋊C8—1061no—
283#97—1062no—
284#97—1061no—
285C4×C8—21no—
286C4×C8—1061no—
287C4×C8—1061no—
288#41—22no—
289#41—41no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8C22⋊Dic5311yescharacteristic
293C2×D4—21no—
294C2×D4C22⋊Dic5311yescharacteristic
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—2121no—
299C22×C4D4×D5311yescharacteristic
300C22×C4#1171994711yescharacteristic
301C22×C4C2×C22⋊D5311yescharacteristic
302Q16—42no—
303SD16—42no—
304M16—1062no—
305M16—2121no—
306M16—2122no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—1061no—
310C2×C8—1061no—
311C2×C8—1062no—
312C2×C8—4242no—
313C2×C8—2121no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4C22⋊Dic5311yescharacteristic
323C4⋊C4C22⋊Dic5311yescharacteristic
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—41no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—41no—
332C42—1062no—
333C42—1062no—
334C42—1062no—
335C42—2121no—
336C42—2121no—
337C42—2121no—
338C42D4×D5311yescharacteristic
339C42#1171994711yescharacteristic
340C42C22×Dic5311yescharacteristic
341C42C2×C22⋊D5311yescharacteristic
342C23—21no—
343C23#1313318911yescharacteristic
344Q8—21no—
345Q8—21no—
346Q8—41no—
347Q8—41no—
348Q8—41no—
349Q8—41no—
350D4—21no—
351D4—42no—
352D4#1313314412yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4—2122no—
372C2×C4C2×C22⋊Dic5311yescharacteristic
373C2×C4D4×Dic5311yescharacteristic
374C2×C4D4×Dic5311yescharacteristic
375C2×C4#1313318511yescharacteristic
376C2×C4#1313318911yescharacteristic
377C2×C4#1313319811yescharacteristic
378C8—41no—
379C8—41no—
380C8—2122no—
381C8—2121no—
382C8—2122no—
383C8—2121no—
384C22—21no—
385C22—22no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22#1681050311yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—81no—
398C4—81no—
399C4—2124no—
400C4C2×#1313314411yescharacteristic
401C4#1681064011yescharacteristic
402C4#1681074011yescharacteristic
403C4#1681074511yescharacteristic
404C4#1681079911yescharacteristic
405C4#1681080411yescharacteristic
406C2—41no—
407C2—42no—
408C2#2027501811yescharacteristic
409C2#2027504411yescharacteristic
410C2#2027508611yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3107C2
socleC22×C53#386
prime radicalC2×#41×C53D4
cosocleC2×C42×C53C23
hypercenter#3107D53
hypocenterC53#8971
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
2. #20275693C2semidirect
4. #20273136C2semidirect
5. #20272606C2semidirect
6. #20272547C2semidirect
25. C53×#475C4semidirect
26. C53×#474C4semidirect
27. C53×#462C4semidirect
28. C53×#461C4semidirect
216. #475Dic53semidirect
217. #474Dic53semidirect
218. #462Dic53semidirect
219. #461Dic53semidirect
238. C53#8971semidirect

Maximal subgroups

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

subgroupindex
#202838402
#202756932
C4×#131331442
#202731362
#202726062
#202725472
C53×#31072
#897153