Finite Group Databasebeta

← #27288608

Subgroups — #27288608

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275642C211yesmaximal, characteristic
2#20275521C211yesmaximal, characteristic
3#20275473C211yesmaximal, characteristic, complemented
4C2×#16810374C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic
6#20272599C211yesmaximal, characteristic, complemented
7C53×#1939C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9C2×#13133150—21no—
10C2×C4⋊#88322C2211yescharacteristic
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133139—21no—
13C2×#13133139—21no—
14C2×#13133137—21no—
15C2×#13133137—21no—
16#16810493C2211yescharacteristic
17#16810385—21no—
18#16810374—22no—
19#16810366—22no—
20C2×C4⋊C8×C53—21no—
21C2×C4⋊C8×C53C2211yescharacteristic
22C2×C4×C8×C53C2211yescharacteristic
23C4×C4⋊C4×C53C2211yescharacteristic
24C53×#336—21no—
25C22×D212—21no—
26C22×D212—21no—
27C22×D212—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30#13133214—21no—
31#13133214—21no—
32#13133214—21no—
33#13133214Q812yes—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37#13133150—42no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322D412yes—
42#13133139—42no—
43#13133139—42no—
44#13133137—42no—
45#13133137—42no—
46C22×C8×C53—21no—
47C22×C8×C53D411yescharacteristic
48C22×C8×C53D411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C42×C53—21no—
52C2×C42×C53C2311yescharacteristic
53C4⋊C8×C53—21no—
54C4⋊C8×C53—21no—
55C4⋊C8×C53—42no—
56C4×C8×C53—21no—
57C4×C8×C53—21no—
58#95×C53—21no—
59C23×D53—41no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53#4111yescharacteristic
65C22×Dic53—41no—
66C2×D212—22no—
67C2×D212—22no—
68C2×D212—22no—
69C2×D212—41no—
70C2×D212—41no—
71C2×D212—41no—
72C2×D212—41no—
73C2×D212—41no—
74C2×D212—41no—
75C2×D212—41no—
76C2×D212—41no—
77C2×D212—41no—
78C2×D212—41no—
79C2×D212—41no—
80C2×D212—81no—
81C2×D212—81no—
82C2×C8×C53—21no—
83C2×C8×C53—21no—
84C2×C8×C53—21no—
85C2×C8×C53—21no—
86C2×C8×C53—21no—
87C2×C8×C53—41no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53D812yes—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53SD1612yes—
95C42×C53—41no—
96C42×C53—41no—
97C42×C53C2×D411yescharacteristic
98C42×C53C2×Q811yescharacteristic
99C42×C53#4111yescharacteristic
100C42×C53#4111yescharacteristic
101C4⋊Dic53—41no—
102C4⋊Dic53—41no—
103C4⋊Dic53—41no—
104C4⋊Dic53—41no—
105C4⋊Dic53—81no—
106C2×#88322—22no—
107C2×#88322—41no—
108C2×#88322—41no—
109C2×#88322—41no—
110C2×#88322—41no—
111C2×#88322—41no—
112C23×C53#12111yescharacteristic
113C22×D53—42no—
114C22×D53—81no—
115C22×D53—81no—
116C22×D53—81no—
117C22×D53—81no—
118C22×D53—81no—
119C22×D53—81no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—22no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53#12111yescharacteristic
129C2×C4×C53#12111yescharacteristic
130C2×C4×C53C22⋊Q811yescharacteristic
131C2×C4×C53C22⋊Q811yescharacteristic
132C2×C4×C53C22⋊Q811yescharacteristic
133C2×C4×C53#12611yescharacteristic
134C2×C4×C53C2×D811yescharacteristic
135C2×C4×C53C2×SD1611yescharacteristic
136C2×C4×C53#13511yescharacteristic
137C2×C4×C53#13511yescharacteristic
138C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C53#13711yescharacteristic
142C2×Dic53—81no—
143C2×Dic53—81no—
144C2×Dic53—81no—
145D212—42no—
146D212—42no—
147D212—42no—
148D212—42no—
149D212—82no—
150D212—82no—
151D212—81no—
152D212—81no—
153D212—81no—
154D212—81no—
155C8×C53—22no—
156C8×C53—41no—
157C8×C53—81no—
158#88322—42no—
159#88322—81no—
160#10587—531nomaximal, Sylow
161C22×C53#39611yescharacteristic
162C22×C53#45911yescharacteristic
163C22×C53#46411yescharacteristic
164C22×C53#46511yescharacteristic
165C22×C53#46911yescharacteristic
166C22×C53#47111yescharacteristic
167C22×C53#47211yescharacteristic
168C2×D53—82no—
169C2×D53—82no—
170C2×D53—82no—
171C2×D53—161no—
172C2×D53—161no—
173C2×D53—161no—
174C2×D53—161no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—42no—
178C4×C53—81no—
179C4×C53#46011yescharacteristic
180C4×C53#46311yescharacteristic
181C4×C53#46611yescharacteristic
182C4×C53#46811yescharacteristic
183C4×C53#47411yescharacteristic
184C4×C53#47611yescharacteristic
185C4×C53#47811yescharacteristic
186C4×C53#47811yescharacteristic
187Dic53—161no—
188#1982—531no—
189#1957—531no—
190#1947—531no—
191#1939D5311yescharacteristic, complemented
192C2×#334—531no—
193C2×#331—531no—
194#1296—531no—
195C2×C53#166511yescharacteristic
196C2×C53#167811yescharacteristic
197C2×C53#168711yescharacteristic
198C2×C53#170211yescharacteristic
199C2×C53#206311yescharacteristic
200C2×C53#206511yescharacteristic
201C2×C53#206711yescharacteristic
202D53—162no—
203C2×#127—531no—
204C2×#107—1061no—
205C2×C4⋊C8—21no—
206C2×C4⋊C8—531no—
207C2×C4⋊C8—531no—
208C2×C4⋊C8C2×D5311yescharacteristic
209C2×#102—1061no—
210C2×#102—1061no—
211C2×#102—1061no—
212C2×#102—1061no—
213C2×C4×C8C2×D5311yescharacteristic
214#384—531no—
215C4×C4⋊C4C2×D5311yescharacteristic
216#340—1061no—
217#336—21no—
218#334—1062no—
219#331—1062no—
220C53#1058711yescharacteristic, complemented, Sylow
221C22×D4—1061no—
222C22×D4—1061no—
223C22×D4—1061no—
224C22×C8—21no—
225C22×C8—1061no—
226C22×C8—1061no—
227C22×C8C22⋊D5311yescharacteristic
228C22×C8C22⋊D5311yescharacteristic
229#127—532no—
230#127—1061no—
231#127—1061no—
232#127—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4D21211yescharacteristic
237C2×C4⋊C4D21211yescharacteristic
238C2×C42—21no—
239C2×C42C22×D5311yescharacteristic
240#107—2122no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—42no—
244C4⋊C8—532no—
245C4⋊C8—1061no—
246C4⋊C8—1061no—
247C4⋊C8—1061no—
248#102—2122no—
249#102—2122no—
250#102—2122no—
251#102—2122no—
252C4×C8—21no—
253C4×C8—21no—
254#95—21no—
255C24—2121no—
256C2×D4—1062no—
257C2×D4—1062no—
258C2×D4—1062no—
259C2×D4—4241no—
260C2×D4—4241no—
261C2×D4—2121no—
262C2×D4—2121no—
263C2×D4—2121no—
264C2×D4—2121no—
265C2×D4—2121no—
266C2×D4—2121no—
267C2×D4—2121no—
268C2×D4—2121no—
269C2×D4—2121no—
270C2×D4—2121no—
271C2×D4—2121no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—2121no—
275C22×C4C2×D21211yescharacteristic
276C22×C4#1171994511yescharacteristic
277C22×C4C2×C22⋊D5311yescharacteristic
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—1062no—
287C2×C8—2121no—
288C2×C8—2121no—
289C2×C8—2121no—
290C2×C8—2121no—
291C2×C8—2121no—
292C2×C8#1171992212yes—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—4241no—
297C4⋊C4—2121no—
298C4⋊C4—2121no—
299C4⋊C4—2121no—
300C4⋊C4—2121no—
301C4⋊C4#1171991312yes—
302C42—41no—
303C42—41no—
304C42D4×D5311yescharacteristic
305C42#1171994711yescharacteristic
306C42Q8×D5311yescharacteristic
307C42#1171994911yescharacteristic
308C23—4241no—
309C23—4241no—
310C23—4241no—
311C23—4241no—
312C23—4241no—
313C23—4241no—
314C23—2122no—
315C23#1313319111yescharacteristic
316D4—4242no—
317D4—4242no—
318D4—4241no—
319D4—4241no—
320D4—4241no—
321D4—4241no—
322D4—2122no—
323D4—2122no—
324D4—2122no—
325D4—2122no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—4241no—
335C2×C4—4241no—
336C2×C4—4241no—
337C2×C4#1313318911yescharacteristic
338C2×C4#1313319211yescharacteristic
339C2×C4#1313319311yescharacteristic
340C2×C4#1313319511yescharacteristic
341C2×C4#1313319711yescharacteristic
342C2×C4#1313321211yescharacteristic
343C2×C4C2×#1171992211yescharacteristic
344C2×C4C2×#1171991311yescharacteristic
345C2×C4#1313323711yescharacteristic
346C2×C4#1313323911yescharacteristic
347C2×C4#1313324211yescharacteristic
348C2×C4#1313324511yescharacteristic
349C2×C4#1313324511yescharacteristic
350C2×C4#1313324811yescharacteristic
351C8—22no—
352C8—41no—
353C8—81no—
354C8—4241no—
355C8—2122no—
356C22—8481no—
357C22—8481no—
358C22—8481no—
359C22—8481no—
360C22—4242no—
361C22—4242no—
362C22—4242no—
363C22#1681054511yescharacteristic
364C22#1681081811yescharacteristic
365C22#1681084511yescharacteristic
366C22#1681085211yescharacteristic
367C22#1681087611yescharacteristic
368C22#1681088611yescharacteristic
369C22#1681089311yescharacteristic
370C4—21no—
371C4—21no—
372C4—42no—
373C4—81no—
374C4—8481no—
375C4#1681082411yescharacteristic
376C4#1681083911yescharacteristic
377C4#1681085711yescharacteristic
378C4#1681087111yescharacteristic
379C4#1681090011yescharacteristic
380C4#1681091411yescharacteristic
381C4#1681092011yescharacteristic
382C4#1681092411yescharacteristic
383C2—8482no—
384C2#2027405611yescharacteristic
385C2#2027413111yescharacteristic
386C2#2027419411yescharacteristic
387C2#2027428311yescharacteristic
388C2#2027607411yescharacteristic
389C2#2027608811yescharacteristic
390C2#2027610011yescharacteristic
subgroupG/H
centerC23#13133191
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1939C2
socleC23×C53#121
prime radicalC2×#13133214C22
hypercenter#1939D53
hypocenterC53#10587
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1939

Complements

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

NH ≅ G/Ntype
3. #20275473C2semidirect
4. C2×#16810374C2semidirect
6. #20272599C2semidirect
7. C53×#1939C2semidirect
191. #1939D53semidirect
220. C53#10587semidirect

Maximal subgroups

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

subgroupindex
#202756422
#202755212
#202754732
C2×#168103742
C2×#168103662
#202725992
C53×#19392
#1058753