Finite Group Databasebeta

← #27287139

Subgroups — #27287139

Order 13568.

536 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279750C211yesmaximal, characteristic, complemented
2#20275642C211yesmaximal, characteristic
3#20275641C211yesmaximal, characteristic
4#20275521C211yesmaximal, characteristic
5#20272599C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9C2×#13133209—21no—
10C2×C4⋊#88322C2211yescharacteristic
11C2×#13133139—22no—
12C2×#13133139—21no—
13C2×#13133139—21no—
14C2×#13133137—21no—
15C2×#13133137—21no—
16#16810557—22no—
17#16810521—21no—
18#16810519—21no—
19#16810493C2211yescharacteristic
20#16810492—21no—
21C4×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊C8×C53C2211yescharacteristic
23C2×C4×C8×C53C2211yescharacteristic
24C4×C4⋊C4×C53C2211yescharacteristic
25C22×D212—21no—
26C22×D212—21no—
27C22×D212—21no—
28C22×#88322—21no—
29#13133214—21no—
30#13133214—21no—
31#13133214—21no—
32#13133214Q812yes—
33#13133209—42no—
34#13133209—42no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53D411yescharacteristic
38C2×C4⋊Dic53D411yescharacteristic
39C2×C4⋊Dic53D412yes—
40C2×#11719919—22no—
41C2×#11719919—22no—
42C2×#11719919—22no—
43C2×C4×Dic53—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46#13133139—44no—
47#13133139—42no—
48#13133139—42no—
49#13133137—42no—
50#13133137—42no—
51#13133124—22no—
52#13133124—21no—
53C22×C8×C53—21no—
54C22×C8×C53—21no—
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C42×C53—21no—
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61C4×C8×C53—21no—
62C4×C8×C53—21no—
63#95×C53—21no—
64C23×D53—41no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—22no—
73C2×D212—22no—
74C2×D212—22no—
75C2×D212—22no—
76C2×D212—42no—
77C2×D212—42no—
78C2×D212—41no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×C8×C53—22no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C42×C53—41no—
100C42×C53—41no—
101C42×C53C2×Q811yescharacteristic
102C42×C53#4111yescharacteristic
103C42×C53#4111yescharacteristic
104C42×C53#4111yescharacteristic
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53—41no—
108C4⋊Dic53—41no—
109C4⋊Dic53—41no—
110C4⋊Dic53—41no—
111C4⋊Dic53—42no—
112C4⋊Dic53D812yes—
113C4⋊Dic53SD1612yes—
114#11719919—44no—
115#11719919—44no—
116#11719919—82no—
117C4×Dic53—41no—
118C4×Dic53—41no—
119C2×#88322—41no—
120C2×#88322—41no—
121C2×#88322—41no—
122C23×C53#12011yescharacteristic
123C22×D53—42no—
124C22×D53—81no—
125C22×D53—81no—
126C22×D53—81no—
127C22×D53—81no—
128C22×D53—81no—
129C22×D53—81no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53C22⋊Q811yescharacteristic
139C2×C4×C53C22⋊Q811yescharacteristic
140C2×C4×C53C22⋊Q811yescharacteristic
141C2×C4×C53#12411yescharacteristic
142C2×C4×C53#12411yescharacteristic
143C2×C4×C53#12411yescharacteristic
144C2×C4×C53C2×D811yescharacteristic
145C2×C4×C53C2×SD1611yescharacteristic
146C2×C4×C53#13511yescharacteristic
147C2×C4×C53#13511yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C53#13711yescharacteristic
152C2×Dic53—22no—
153C2×Dic53—22no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161D212—44no—
162D212—42no—
163D212—42no—
164D212—42no—
165D212—42no—
166D212—82no—
167D212—81no—
168D212—81no—
169D212—81no—
170D212—81no—
171C8×C53—42no—
172C8×C53—81no—
173#88322—81no—
174#10354—531nomaximal, Sylow
175C22×C53#39311yescharacteristic
176C22×C53#44711yescharacteristic
177C22×C53#44811yescharacteristic
178C22×C53#44911yescharacteristic
179C22×C53#44911yescharacteristic
180C22×C53#45311yescharacteristic
181C22×C53#45411yescharacteristic
182C2×D53—82no—
183C2×D53—82no—
184C2×D53—82no—
185C2×D53—161no—
186C2×D53—161no—
187C2×D53—161no—
188C2×D53—161no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—81no—
192C4×C53—81no—
193C4×C53#47411yescharacteristic
194C4×C53#47611yescharacteristic
195C4×C53#47811yescharacteristic
196C4×C53#47811yescharacteristic
197C4×C53#48611yescharacteristic
198C4×C53#48811yescharacteristic
199C4×C53#48911yescharacteristic
200C4×C53#49011yescharacteristic
201Dic53—42no—
202Dic53—42no—
203Dic53—82no—
204#2552—531no—
205#1982—531no—
206#1982—531no—
207#1957—531no—
208#1296—531no—
209#1279—531no—
210#1279D5311yescharacteristic, complemented
211C2×C53#162111yescharacteristic
212C2×C53#162211yescharacteristic
213C2×C53#162511yescharacteristic
214C2×C53#162611yescharacteristic
215C2×C53#202311yescharacteristic
216C2×C53#202511yescharacteristic
217C2×C53#202711yescharacteristic
218D53—162no—
219C2×#127—531no—
220C2×#125—1061no—
221C2×C4⋊C8—531no—
222C2×C4⋊C8C2×D5311yescharacteristic
223C2×#102—1062no—
224C2×#102—1061no—
225C2×#102—1061no—
226C2×#102—1061no—
227C2×#102—1061no—
228C2×C4×C8C2×D5311yescharacteristic
229#398—1062no—
230#389—1061no—
231#389—1061no—
232#384—531no—
233#384—1061no—
234C4×C4⋊C4—531no—
235C4×C4⋊C4C2×D5311yescharacteristic
236C53#1035411yescharacteristic, complemented, Sylow
237C22×D4—1061no—
238C22×D4—1061no—
239C22×D4—1061no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—1061no—
243#127—532no—
244#127—1061no—
245#127—1061no—
246#127—1061no—
247#125—2122no—
248#125—2122no—
249C2×C4⋊C4—531no—
250C2×C4⋊C4—531no—
251C2×C4⋊C4—532no—
252C2×C4⋊C4—1062no—
253C2×C4⋊C4—1062no—
254C2×C4⋊C4—1062no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4—1061no—
257C2×C4⋊C4D21211yescharacteristic
258C2×C4⋊C4D21211yescharacteristic
259C2×C42—21no—
260C2×C42—1061no—
261C2×C42C22×D5311yescharacteristic
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—1061no—
265C4⋊C8—1061no—
266#102—2122no—
267#102—2122no—
268#102—2122no—
269#102—2122no—
270#102—2124no—
271C4×C8—21no—
272C4×C8—21no—
273#95—21no—
274#95—1062no—
275#95—1061no—
276C24—2121no—
277C2×D4—1062no—
278C2×D4—1062no—
279C2×D4—1062no—
280C2×D4—2122no—
281C2×D4—2122no—
282C2×D4—2121no—
283C2×D4—2121no—
284C2×D4—2121no—
285C2×D4—2121no—
286C2×D4—2121no—
287C2×D4—2121no—
288C2×D4—2121no—
289C2×D4—2121no—
290C2×D4—2121no—
291C2×D4—2121no—
292C2×D4—2121no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4—1062no—
298C22×C4C2×D21211yescharacteristic
299C22×C4D4×D5311yescharacteristic
300C22×C4D4×D5311yescharacteristic
301C2×C8—22no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—2121no—
308C2×C8—2121no—
309C2×C8—2121no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—532no—
315C4⋊C4—532no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C4⋊C4—4242no—
319C4⋊C4—2122no—
320C4⋊C4—2124no—
321C4⋊C4—2124no—
322C4⋊C4—2121no—
323C4⋊C4—2121no—
324C4⋊C4—2121no—
325C4⋊C4—2121no—
326C42—41no—
327C42—41no—
328C42—2121no—
329C42—2121no—
330C42#1171994511yescharacteristic
331C42#1171994511yescharacteristic
332C42#1171994711yescharacteristic
333C42Q8×D5311yescharacteristic
334C23—4241no—
335C23—4241no—
336C23—4241no—
337C23—4241no—
338C23—4241no—
339C23—4241no—
340C23—2122no—
341C23#1313318711yescharacteristic
342D4—4242no—
343D4—4241no—
344D4—4241no—
345D4—4241no—
346D4—4241no—
347D4—2124no—
348D4—2122no—
349D4—2122no—
350D4—2122no—
351D4—2122no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—1062no—
361C2×C4—1062no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4—2122no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4#1313319511yescharacteristic
370C2×C4#1313319511yescharacteristic
371C2×C4#1313319711yescharacteristic
372C2×C4#1313320311yescharacteristic
373C2×C4#1313320311yescharacteristic
374C2×C4#1313320511yescharacteristic
375C2×C4D8×D5311yescharacteristic
376C2×C4SD16×D5311yescharacteristic
377C2×C4#1313323911yescharacteristic
378C2×C4#1313323911yescharacteristic
379C2×C4#1313324411yescharacteristic
380C2×C4#1313324511yescharacteristic
381C2×C4#1313324511yescharacteristic
382C2×C4#1313324911yescharacteristic
383C8—42no—
384C8—81no—
385C8—4241no—
386C22—8481no—
387C22—8481no—
388C22—8481no—
389C22—8481no—
390C22—4242no—
391C22—4242no—
392C22—4242no—
393C22#1681053611yescharacteristic
394C22#1681075811yescharacteristic
395C22#1681076311yescharacteristic
396C22#1681077011yescharacteristic
397C22#1681077011yescharacteristic
398C22#1681079211yescharacteristic
399C22#1681079711yescharacteristic
400C4—21no—
401C4—21no—
402C4—81no—
403C4—81no—
404C4—4242no—
405C4—2122no—
406C4—2122no—
407C4#1681090211yescharacteristic
408C4#1681091211yescharacteristic
409C4#1681092411yescharacteristic
410C4#1681092411yescharacteristic
411C4#1681096811yescharacteristic
412C4#1681097811yescharacteristic
413C4#1681098411yescharacteristic
414C4#1681098811yescharacteristic
415C2—8482no—
416C2#2027378311yescharacteristic
417C2#2027379011yescharacteristic
418C2#2027380711yescharacteristic
419C2#2027381311yescharacteristic
420C2#2027585011yescharacteristic
421C2#2027586211yescharacteristic
422C2#2027587011yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53#120
prime radicalC2×#13133214C22
hypercenter#1279D53
hypocenterC53#10354
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
1. #20279750C2semidirect
5. #20272599C2semidirect
6. #20272555C2semidirect
7. C53×#1279C2semidirect
210. #1279D53semidirect
236. C53#10354semidirect

Maximal subgroups

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

subgroupindex
#202797502
#202756422
#202756412
#202755212
#202725992
#202725552
C53×#12792
#1035453