Finite Group Databasebeta

← #27279990

Subgroups — #27279990

Order 13568.

516 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284195C211yesmaximal, characteristic
2#20283381C211yesmaximal, characteristic, complemented
3#20273382C211yesmaximal, characteristic, complemented
4C2×#16810366C211yesmaximal, characteristic, complemented
5#20272611C211yesmaximal, characteristic, complemented
6#20272590C211yesmaximal, characteristic, complemented
7C53×#1477C211yesmaximal, characteristic, complemented
8C2×#13133242—21no—
9C2×#13133214C2211yescharacteristic
10#16810823—21no—
11#16810823C412yes—
12#16810818—21no—
13#16810818C412yes—
14#16810684—22no—
15#16810680—21no—
16#16810676—21no—
17#16810647C2211yescharacteristic
18#16810647C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20#16810627—21no—
21#16810590—21no—
22#16810579—21no—
23#16810366—21no—
24#16810366—21no—
25#16810366—22no—
26C4×#41×C53C2211yescharacteristic
27C53×#431C2211yescharacteristic
28C2×C4⋊C8×C53C2211yescharacteristic
29C53×Q8⋊C8—21no—
30C53×D4⋊C8—21no—
31C22×D212—21no—
32C22×D212—21no—
33C22×D212—21no—
34#13133242—42no—
35#13133242—42no—
36C2×#11719924—22no—
37C2×#11719922—22no—
38C2×#11719917—21no—
39C2×#11719917—21no—
40C22×#88322—21no—
41#13133214—21no—
42#13133214—21no—
43#13133214—21no—
44#13133214C2×C412yes—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322C2×C412yes—
49C4⋊#88322D411yescharacteristic
50C4⋊#88322D411yescharacteristic
51C4⋊#88322D412yes—
52#13133137—22no—
53#13133137—22no—
54C22⋊#88322—22no—
55C22⋊#88322—22no—
56#13133128—22no—
57C4×#88322—22no—
58C2×#41×C53D411yescharacteristic
59C22×C8×C53—21no—
60C4×Q8×C53C2×C411yescharacteristic
61C4×Q8×C53D411yescharacteristic
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53C2×C411yescharacteristic
65C4×D4×C53D411yescharacteristic
66#117×C53—21no—
67#117×C53D411yescharacteristic
68C2×C42×C53—21no—
69C2×C42×C53C2311yescharacteristic
70C4⋊C8×C53—21no—
71C4⋊C8×C53—21no—
72C4⋊C8×C53—21no—
73C22⋊C8×C53—21no—
74C4×C8×C53—21no—
75C23×D53—41no—
76#41×C53—21no—
77#41×C53—21no—
78#41×C53—21no—
79#41×C53—21no—
80C2×Q8×C53C22⋊C411yescharacteristic
81C2×D4×C53—21no—
82C2×D4×C53C22⋊C411yescharacteristic
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C2×D212—22no—
90C2×D212—22no—
91C2×D212—22no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—82no—
100C2×D212—81no—
101C2×D212—81no—
102C2×C8×C53—41no—
103C2×C8×C53—41no—
104C2×C8×C53—81no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53C22⋊C411yescharacteristic
108C4⋊C4×C53C22⋊C411yescharacteristic
109C22⋊C4×C53—21no—
110C22⋊C4×C53—21no—
111C22⋊C4×C53—21no—
112C42×C53—21no—
113C42×C53—21no—
114C42×C53—21no—
115C42×C53C22×C411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119#11719924—44no—
120#11719922—44no—
121#11719917—41no—
122#11719917—41no—
123#11719917—41no—
124#11719917—41no—
125C2×#88322—22no—
126C2×#88322—21no—
127C2×#88322—22no—
128C2×#88322—21no—
129C2×#88322—41no—
130C2×#88322—41no—
131C23×C53—21no—
132C23×C53#12311yescharacteristic
133C22×D53—42no—
134C22×D53—82no—
135C22×D53—82no—
136C22×D53—81no—
137C22×D53—81no—
138Q8×C53—21no—
139Q8×C53—21no—
140D4×C53—21no—
141D4×C53—21no—
142D4×C53—41no—
143D4×C53—41no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—81no—
155C2×C4×C53C2×C22⋊C411yescharacteristic
156C2×C4×C53C4×D411yescharacteristic
157C2×C4×C53C4×D411yescharacteristic
158C2×C4×C53#12011yescharacteristic
159C2×C4×C53#12111yescharacteristic
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#13511yescharacteristic
162C2×C4×C53#13511yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
165D212—42no—
166D212—42no—
167D212—82no—
168D212—82no—
169D212—82no—
170D212—81no—
171D212—81no—
172D212—81no—
173D212—81no—
174C8×C53—81no—
175C8×C53—81no—
176#88322—42no—
177#88322—41no—
178#88322—41no—
179#88322—42no—
180#9174—531nomaximal, Sylow
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53#38611yescharacteristic
187C22×C53#48211yescharacteristic
188C22×C53#48211yescharacteristic
189C2×D53—82no—
190C2×D53—82no—
191C2×D53—82no—
192C2×D53—162no—
193C2×D53—162no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53#41611yescharacteristic
201C4×C53#41811yescharacteristic
202C4×C53#44911yescharacteristic
203C4×C53#44911yescharacteristic
204C4×C53#47111yescharacteristic
205C4×C53#47211yescharacteristic
206#3038—531no—
207#2925—531no—
208#1545—531no—
209C2×#331—531no—
210#1477D5311yescharacteristic, complemented
211#1301—531no—
212#1292—531no—
213C2×C53—21no—
214C2×C53—21no—
215C2×C53—81no—
216C2×C53#187511yescharacteristic
217C2×C53#190311yescharacteristic
218C2×C53#190711yescharacteristic
219D53—162no—
220C2×AGL(1,ℤ/8)—1061no—
221C2×#127—531no—
222C4×#41C2×D5311yescharacteristic
223#460—532no—
224#460—1061no—
225#459—532no—
226#459—1061no—
227#433—1062no—
228#432—1061no—
229#431—1061no—
230#431C2×D5311yescharacteristic
231#423—531no—
232#423—531no—
233C2×C4⋊C8—531no—
234C2×C4⋊C8C2×D5311yescharacteristic
235#418—1061no—
236#408—1061no—
237#405—1061no—
238#331—1061no—
239#331—1061no—
240#331—1062no—
241Q8⋊C8—21no—
242D4⋊C8—21no—
243C53#917411yescharacteristic, complemented, Sylow
244C2×#41D21211yescharacteristic
245C22×D4—1061no—
246C22×D4—1061no—
247C22×D4—1061no—
248AGL(1,ℤ/8)—2122no—
249AGL(1,ℤ/8)—2122no—
250C2×SD16—1062no—
251C2×D8—1062no—
252C2×M16—1061no—
253C2×M16—1061no—
254C22×C8—21no—
255C22×C8—1061no—
256#127—532no—
257#127—1061no—
258#127—1061no—
259#127—1061no—
260C4×Q8C4×D5311yescharacteristic
261C4×Q8C22⋊D5311yescharacteristic
262C4×D4—21no—
263C4×D4—21no—
264C4×D4C4×D5311yescharacteristic
265C4×D4C22⋊D5311yescharacteristic
266#117—21no—
267#117D21211yescharacteristic
268C2×C42—21no—
269C2×C42C22×D5311yescharacteristic
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—532no—
274C4⋊C8—531no—
275C4⋊C8—531no—
276C4⋊C8—532no—
277C4⋊C8—1061no—
278C4⋊C8—1061no—
279C4⋊C8—1061no—
280#102—1062no—
281#102—1062no—
282C22⋊C8—21no—
283C22⋊C8—1062no—
284C22⋊C8—1062no—
285#97—1062no—
286C4×C8—21no—
287C4×C8—1062no—
288C24—2121no—
289#41—21no—
290#41—21no—
291#41—21no—
292#41—21no—
293C2×Q8#1171992111yescharacteristic
294C2×D4—21no—
295C2×D4—1062no—
296C2×D4—1062no—
297C2×D4—1062no—
298C2×D4—4241no—
299C2×D4—4241no—
300C2×D4—4242no—
301C2×D4—2121no—
302C2×D4—2121no—
303C2×D4—2121no—
304C2×D4—2121no—
305C2×D4—2121no—
306C2×D4—2121no—
307C2×D4—2121no—
308C2×D4#1171992111yescharacteristic
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4C2×D21211yescharacteristic
313C22×C4#1171994711yescharacteristic
314C22×C4#1171994711yescharacteristic
315SD16—2124no—
316D8—2124no—
317M16—2121no—
318M16—2121no—
319M16—2121no—
320M16—2121no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—81no—
324C2×C8—1062no—
325C2×C8—1061no—
326C2×C8—1062no—
327C2×C8—1061no—
328C2×C8—2121no—
329C2×C8—2121no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4#1171992111yescharacteristic
333C4⋊C4#1171992111yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42C2×C4×D5311yescharacteristic
341C42D4×D5311yescharacteristic
342C42D4×D5311yescharacteristic
343C42C2×C22⋊D5311yescharacteristic
344C23—21no—
345C23—4242no—
346C23—4242no—
347C23—4241no—
348C23—4241no—
349C23—2122no—
350C23#1313320111yescharacteristic
351Q8—21no—
352Q8—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—41no—
357D4—4242no—
358D4—4242no—
359D4—4242no—
360D4—4241no—
361D4—4241no—
362D4—4241no—
363D4—4241no—
364D4—2122no—
365D4—2122no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4C2×#1171992111yescharacteristic
378C2×C4#1313317711yescharacteristic
379C2×C4#1313317711yescharacteristic
380C2×C4#1313318711yescharacteristic
381C2×C4#1313318911yescharacteristic
382C2×C4#1313318911yescharacteristic
383C2×C4#1313323911yescharacteristic
384C2×C4#1313323911yescharacteristic
385C2×C4#1313324511yescharacteristic
386C2×C4#1313324511yescharacteristic
387C8—81no—
388C8—81no—
389C8—2121no—
390C8—2121no—
391C8—2122no—
392C8—2122no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—8482no—
399C22—8482no—
400C22—4242no—
401C22—4242no—
402C22—4242no—
403C22#1681050011yescharacteristic
404C22#1681094811yescharacteristic
405C22#1681094811yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4#1681062011yescharacteristic
413C4#1681062911yescharacteristic
414C4#1681077011yescharacteristic
415C4#1681077011yescharacteristic
416C4#1681088811yescharacteristic
417C4#1681089111yescharacteristic
418C2—21no—
419C2—21no—
420C2—81no—
421C2—8482no—
422C2#2027503511yescharacteristic
423C2#2027519311yescharacteristic
424C2#2027522111yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1477C2
socleC22×C53#386
prime radical#20284195C2
cosocleC2×C42×C53C23
hypercenter#1477D53
hypocenterC53#9174
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
2. #20283381C2semidirect
3. #20273382C2semidirect
4. C2×#16810366C2semidirect
5. #20272611C2semidirect
6. #20272590C2semidirect
7. C53×#1477C2semidirect
210. #1477D53semidirect
243. C53#9174semidirect

Maximal subgroups

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

subgroupindex
#202841952
#202833812
#202733822
C2×#168103662
#202726112
#202725902
C53×#14772
#917453