Finite Group Databasebeta

← #27278995

Subgroups — #27278995

Order 13568.

452 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283885C211yesmaximal, characteristic, complemented
2#20275548C211yesmaximal, characteristic, complemented
3#20273201C211yesmaximal, characteristic, complemented
4#20273184C211yesmaximal, characteristic, complemented
5#20272609C211yesmaximal, characteristic, complemented
6#20272605C211yesmaximal, characteristic, complemented
7C53×#3171C211yesmaximal, characteristic
8#16810754—21no—
9#16810701—21no—
10#16810695—21no—
11#16810695—21no—
12#16810684—21no—
13#16810680C2211yescharacteristic
14#16810676C2211yescharacteristic
15#16810650—21no—
16#16810647C2211yescharacteristic
17C2×#13133144—21no—
18C2×#13133144—21no—
19#16810469C2211yescharacteristic
20#16810360—21no—
21#16810357—21no—
22D4⋊#88322—21no—
23D4⋊#88322—21no—
24C53×#532C2211yescharacteristic
25C53×#529C2211yescharacteristic
26C53×#501C411yescharacteristic, complemented
27C53×#496C411yescharacteristic, complemented
28C53×#490C411yescharacteristic, complemented
29C53×#489C411yescharacteristic, complemented
30C53×#469—21no—
31C53×#468—21no—
32C4×SD16×C53—21no—
33C4×D8×C53—21no—
34C4×M16×C53C2211yescharacteristic
35C2×#11719917—21no—
36C2×#11719917D411yescharacteristic
37C2×#11719917D411yescharacteristic
38C22×#88322—21no—
39C2×C4×Dic53—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D411yescharacteristic
46C4⋊#88322D411yescharacteristic
47#13133144—42no—
48#13133144—42no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53#13133128—21no—
54#13133128—21no—
55C4×#88322—21no—
56C4×#88322—21no—
57#13133124—21no—
58#13133124—21no—
59C2×#41×C53—21no—
60C2×SD16×C53—21no—
61C2×D8×C53—21no—
62C2×M16×C53—21no—
63C4⋊Q8×C53C2×C411yescharacteristic
64#127×C53C2×C411yescharacteristic
65#126×C53—21no—
66#125×C53C2×C411yescharacteristic
67#124×C53—21no—
68#124×C53C2×C411yescharacteristic
69#123×C53—21no—
70C22⋊Q8×C53D411yescharacteristic
71#121×C53—21no—
72#121×C53—21no—
73#121×C53D411yescharacteristic
74C4×Q8×C53D411yescharacteristic
75C4×D4×C53—21no—
76C4×D4×C53—21no—
77C4×D4×C53—21no—
78C4×D4×C53D411yescharacteristic
79#117×C53—21no—
80C2×C42×C53C2311yescharacteristic
81#107×C53—21no—
82#106×C53—21no—
83#103×C53—21no—
84#102×C53—21no—
85#102×C53—21no—
86#102×C53—21no—
87#97×C53C2×C411yescharacteristic
88#97×C53C2×C411yescharacteristic
89C4×C8×C53—21no—
90#41×C53—41no—
91#41×C53—41no—
92#41×C53—41no—
93#41×C53—41no—
94C2×Q8×C53—21no—
95C2×Q8×C53C22⋊C411yescharacteristic
96C2×D4×C53—21no—
97C2×D4×C53—21no—
98C2×D4×C53—21no—
99C2×D4×C53—21no—
100C2×D4×C53—41no—
101C2×D4×C53C22⋊C411yescharacteristic
102C22×C4×C53—21no—
103C22×C4×C53—41no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×Dic53—41no—
108SD16×C53—42no—
109D8×C53—42no—
110M16×C53—42no—
111C2×C8×C53—21no—
112C2×C8×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53C22⋊C411yescharacteristic
120C4⋊C4×C53C22⋊C411yescharacteristic
121C22⋊C4×C53—21no—
122C22⋊C4×C53—21no—
123C22⋊C4×C53—21no—
124C22⋊C4×C53—21no—
125C22⋊C4×C53—21no—
126C22⋊C4×C53—41no—
127C22⋊C4×C53—41no—
128C42×C53—21no—
129C42×C53C22×C411yescharacteristic
130C42×C53C2×D411yescharacteristic
131C42×C53C2×D411yescharacteristic
132C42×C53#4111yescharacteristic
133C4×Dic53—22no—
134C4×Dic53—41no—
135#11719917—41no—
136#11719917—41no—
137#11719917—81no—
138C2×#88322—21no—
139C2×#88322—21no—
140C2×#88322—21no—
141C2×#88322—21no—
142C2×#88322—41no—
143C2×#88322—81no—
144C23×C53—21no—
145C23×C53—41no—
146C23×C53#12111yescharacteristic
147Q8×C53—41no—
148Q8×C53—41no—
149Q8×C53—41no—
150D4×C53—41no—
151D4×C53—42no—
152D4×C53—41no—
153D4×C53—41no—
154D4×C53—41no—
155D4×C53—41no—
156D4×C53—81no—
157D4×C53—81no—
158D4×C53—81no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—81no—
170C2×C4×C53—81no—
171C2×C4×C53C2×C22⋊C411yescharacteristic
172C2×C4×C53C4×D411yescharacteristic
173C2×C4×C53C4×D411yescharacteristic
174C2×C4×C53#12011yescharacteristic
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53#12311yescharacteristic
177C2×Dic53—42no—
178C2×Dic53—81no—
179C2×Dic53—81no—
180C8×C53—22no—
181C8×C53—41no—
182#88322—41no—
183#88322—41no—
184#88322—41no—
185#88322—41no—
186#88322—81no—
187#8997—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53#38611yescharacteristic
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—41no—
204C4×C53—41no—
205C4×C53—41no—
206C4×C53—81no—
207C4×C53#45311yescharacteristic
208C4×C53#45411yescharacteristic
209Dic53—82no—
210#3171D5311yescharacteristic
211#2995—531no—
212#1963—531no—
213#1500—531no—
214#1496—531no—
215#1300—531no—
216#1299—531no—
217C2×C53—41no—
218C2×C53—81no—
219C2×C53—161no—
220C2×C53#188111yescharacteristic
221C2×C53#188311yescharacteristic
222C2×C53#188411yescharacteristic
223#532C2×D5311yescharacteristic
224#529C2×D5311yescharacteristic
225#501Dic5311yescharacteristic, complemented
226#496Dic5311yescharacteristic, complemented
227#490Dic5311yescharacteristic, complemented
228#489Dic5311yescharacteristic, complemented
229#469—21no—
230#468—21no—
231#446—1061no—
232C4×SD16—21no—
233C4×D8—21no—
234#436—1061no—
235#435—1061no—
236#435—1061no—
237#433—1061no—
238#432—531no—
239#431—531no—
240#424—1061no—
241#423—531no—
242C2×#104—1061no—
243C2×#104—1061no—
244C4×M16C2×D5311yescharacteristic
245#376—531no—
246#328—1061no—
247#327—1061no—
248D4⋊C8—1061no—
249D4⋊C8—1061no—
250C53#899711yescharacteristic, complemented, Sylow
251C2×#41—21no—
252C2×SD16—21no—
253C2×D8—21no—
254C2×M16—21no—
255C2×M16—531no—
256C2×M16—531no—
257C2×M16—1061no—
258C22×C8—1061no—
259C4⋊Q8C2×Dic5311yescharacteristic
260#127C2×Dic5311yescharacteristic
261#126—21no—
262#125C2×Dic5311yescharacteristic
263#124—21no—
264#124C2×Dic5311yescharacteristic
265#123—21no—
266C22⋊Q8C22⋊D5311yescharacteristic
267#121—21no—
268#121—21no—
269#121C22⋊D5311yescharacteristic
270C4×Q8C22⋊D5311yescharacteristic
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4C22⋊D5311yescharacteristic
275#117—21no—
276C2×C42—1061no—
277C2×C42C22×D5311yescharacteristic
278#107—21no—
279#106—21no—
280C4⋊C8—531no—
281C4⋊C8—531no—
282C4⋊C8—1061no—
283C4⋊C8—1061no—
284C4⋊C8—1061no—
285C4⋊C8—1061no—
286C4⋊C8—1061no—
287#104—2122no—
288#104—2122no—
289#103—21no—
290#102—21no—
291#102—21no—
292#102—21no—
293C22⋊C8—1061no—
294C22⋊C8—1061no—
295C22⋊C8—1061no—
296C22⋊C8—1061no—
297#97—1061no—
298#97—1061no—
299#97C2×Dic5311yescharacteristic
300#97C2×Dic5311yescharacteristic
301C4×C8—21no—
302C4×C8—1061no—
303C4×C8—1061no—
304#95—1061no—
305#95—1061no—
306#41—41no—
307#41—41no—
308#41—41no—
309#41—41no—
310C2×Q8—21no—
311C2×Q8C22⋊Dic5311yescharacteristic
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4C22⋊Dic5311yescharacteristic
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—2121no—
321C22×C4D4×D5311yescharacteristic
322C22×C4#1171994711yescharacteristic
323C22×C4C2×C22⋊D5311yescharacteristic
324SD16—42no—
325D8—42no—
326M16—42no—
327M16—4241no—
328M16—2121no—
329M16—2121no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—1061no—
333C2×C8—1061no—
334C2×C8—1061no—
335C2×C8—1061no—
336C2×C8—4241no—
337C2×C8—2121no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4C22⋊Dic5311yescharacteristic
345C4⋊C4C22⋊Dic5311yescharacteristic
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C42—21no—
354C42—1062no—
355C42—2121no—
356C42D4×D5311yescharacteristic
357C42#1171994711yescharacteristic
358C42C22×Dic5311yescharacteristic
359C42C2×C22⋊D5311yescharacteristic
360C23—21no—
361C23—41no—
362C23#1313318911yescharacteristic
363Q8—41no—
364Q8—41no—
365Q8—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—42no—
370D4—41no—
371D4—41no—
372D4—81no—
373D4—81no—
374D4—81no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—4241no—
388C2×C4—4241no—
389C2×C4—2122no—
390C2×C4C2×C22⋊Dic5311yescharacteristic
391C2×C4D4×Dic5311yescharacteristic
392C2×C4D4×Dic5311yescharacteristic
393C2×C4#1313318511yescharacteristic
394C2×C4#1313318911yescharacteristic
395C2×C4#1313319811yescharacteristic
396C8—22no—
397C8—41no—
398C8—4241no—
399C8—2121no—
400C8—2121no—
401C8—2121no—
402C8—2121no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22#1681050311yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—81no—
422C4—4242no—
423C4#1681078911yescharacteristic
424C4#1681079411yescharacteristic
425C2—41no—
426C2—81no—
427C2—161no—
428C2#2027507911yescharacteristic
429C2#2027509311yescharacteristic
430C2#2027509911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3171C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3171D53
hypocenterC53#8997
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3171

Complements

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

NH ≅ G/Ntype
1. #20283885C2semidirect
2. #20275548C2semidirect
3. #20273201C2semidirect
4. #20273184C2semidirect
5. #20272609C2semidirect
6. #20272605C2semidirect
26. C53×#501C4semidirect
27. C53×#496C4semidirect
28. C53×#490C4semidirect
29. C53×#489C4semidirect
225. #501Dic53semidirect
226. #496Dic53semidirect
227. #490Dic53semidirect
228. #489Dic53semidirect
250. C53#8997semidirect

Maximal subgroups

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

subgroupindex
#202838852
#202755482
#202732012
#202731842
#202726092
#202726052
C53×#31712
#899753