Finite Group Databasebeta

← #27282419

Subgroups — #27282419

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 390 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×#16810366C211yesmaximal, characteristic
5#20272599C211yesmaximal, characteristic, complemented
6#20272550C211yesmaximal, characteristic, complemented
7C8×C4⋊C4×C53C211yesmaximal, 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#16810366C414yescomplemented
19C2×C4⋊C8×C53C2211yescharacteristic
20C2×C4×C8×C53—21no—
21C2×C4×C8×C53C2211yescharacteristic
22C4×C4⋊C4×C53C2211yescharacteristic
23C53×#336—21no—
24C22×D212—21no—
25C22×D212—21no—
26C22×D212—21no—
27C22×#88322—21no—
28C22×#88322—21no—
29#13133214—21no—
30#13133214—21no—
31#13133214—21no—
32#13133214C2×C412yes—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36#13133150—42no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322C2×C412yes—
41#13133139—42no—
42#13133139—42no—
43#13133137—42no—
44#13133137—42no—
45C22×C8×C53—21no—
46C22×C8×C53D411yescharacteristic
47C22×C8×C53D411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C42×C53—21no—
51C2×C42×C53C2311yescharacteristic
52C4⋊C8×C53—21no—
53C4⋊C8×C53C2×C412yes—
54C4×C8×C53—21no—
55C4×C8×C53—21no—
56C4×C8×C53—42no—
57#95×C53—21no—
58C23×D53—41no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53#4111yescharacteristic
64C22×Dic53—41no—
65C2×D212—22no—
66C2×D212—22no—
67C2×D212—22no—
68C2×D212—41no—
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—81no—
80C2×D212—81no—
81C2×C8×C53—21no—
82C2×C8×C53—21no—
83C2×C8×C53—22no—
84C2×C8×C53—21no—
85C2×C8×C53—21no—
86C2×C8×C53—21no—
87C2×C8×C53—41no—
88C2×C8×C53—41no—
89C2×C8×C53D812yes—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53D812yes—
93C4⋊C4×C53SD1612yes—
94C42×C53—41no—
95C42×C53—41no—
96C42×C53C22×C411yescharacteristic
97C42×C53#4111yescharacteristic
98C42×C53#4111yescharacteristic
99C42×C53#4111yescharacteristic
100C4⋊Dic53—41no—
101C4⋊Dic53—41no—
102C4⋊Dic53—41no—
103C4⋊Dic53—41no—
104C4⋊Dic53—81no—
105C2×#88322—22no—
106C2×#88322—41no—
107C2×#88322—41no—
108C2×#88322—41no—
109C2×#88322—41no—
110C2×#88322—41no—
111C23×C53#12111yescharacteristic
112C22×D53—42no—
113C22×D53—81no—
114C22×D53—81no—
115C22×D53—81no—
116C22×D53—81no—
117C22×D53—81no—
118C22×D53—81no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53#11711yescharacteristic
128C2×C4×C53C4×D411yescharacteristic
129C2×C4×C53C4×D411yescharacteristic
130C2×C4×C53#12311yescharacteristic
131C2×C4×C53#12411yescharacteristic
132C2×C4×C53#12611yescharacteristic
133C2×C4×C53C2×D811yescharacteristic
134C2×C4×C53C2×D811yescharacteristic
135C2×C4×C53C2×SD1611yescharacteristic
136C2×C4×C53#13511yescharacteristic
137C2×C4×C53#13511yescharacteristic
138C2×C4×C53#13511yescharacteristic
139C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C53#13711yescharacteristic
141C2×Dic53—81no—
142C2×Dic53—81no—
143C2×Dic53—81no—
144D212—42no—
145D212—42no—
146D212—42no—
147D212—42no—
148D212—82no—
149D212—82no—
150D212—81no—
151D212—81no—
152D212—81no—
153D212—81no—
154C8×C53—22no—
155C8×C53—42no—
156C8×C53—41no—
157#88322—42no—
158#88322—81no—
159#9563—531nomaximal, Sylow
160C22×C53#38811yescharacteristic
161C22×C53#45911yescharacteristic
162C22×C53#46411yescharacteristic
163C22×C53#46511yescharacteristic
164C22×C53#46611yescharacteristic
165C22×C53#47111yescharacteristic
166C22×C53#47211yescharacteristic
167C2×D53—82no—
168C2×D53—82no—
169C2×D53—82no—
170C2×D53—161no—
171C2×D53—161no—
172C2×D53—161no—
173C2×D53—161no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—42no—
177C4×C53—42no—
178C4×C53C4×D811yescharacteristic
179C4×C53C4×D811yescharacteristic
180C4×C53C4×SD1611yescharacteristic
181C4×C53#44011yescharacteristic
182C4×C53#48011yescharacteristic
183C4×C53#48211yescharacteristic
184C4×C53#48611yescharacteristic
185C4×C53#48811yescharacteristic
186Dic53—161no—
187#1982—531no—
188#1957—531no—
189#1947—531no—
190C8×C4⋊C4D5311yescharacteristic, complemented
191C2×#331—531no—
192#1296—531no—
193#1277—531no—
194C2×C53#166511yescharacteristic
195C2×C53#166711yescharacteristic
196C2×C53#167811yescharacteristic
197C2×C53#168211yescharacteristic
198C2×C53#192511yescharacteristic
199C2×C53#193111yescharacteristic
200C2×C53#193411yescharacteristic
201D53—162no—
202C2×#127—531no—
203C2×#107—1061no—
204C2×C4⋊C8—531no—
205C2×C4⋊C8—531no—
206C2×C4⋊C8C2×D5311yescharacteristic
207C2×#102—1061no—
208C2×#102—1061no—
209C2×#102—1061no—
210C2×#102—1061no—
211C2×C4×C8—21no—
212C2×C4×C8C2×D5311yescharacteristic
213#384—531no—
214C4×C4⋊C4C2×D5311yescharacteristic
215#340—1061no—
216#336—21no—
217#331—534no—
218C53#956311yescharacteristic, complemented, Sylow
219C22×D4—1061no—
220C22×D4—1061no—
221C22×D4—1061no—
222C22×C8—21no—
223C22×C8—1061no—
224C22×C8—1061no—
225C22×C8C22⋊D5311yescharacteristic
226C22×C8C22⋊D5311yescharacteristic
227#127—532no—
228#127—1061no—
229#127—1061no—
230#127—1061no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4D21211yescharacteristic
235C2×C4⋊C4D21211yescharacteristic
236C2×C42—21no—
237C2×C42C22×D5311yescharacteristic
238#107—2122no—
239C4⋊C8—21no—
240C4⋊C8—532no—
241C4⋊C8—1061no—
242C4⋊C8—1061no—
243C4⋊C8—1061no—
244C4⋊C8C4×D5312yes—
245#102—2122no—
246#102—2122no—
247#102—2122no—
248#102—2122no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—42no—
252#95—21no—
253C24—2121no—
254C2×D4—1062no—
255C2×D4—1062no—
256C2×D4—1062no—
257C2×D4—4241no—
258C2×D4—4241no—
259C2×D4—2121no—
260C2×D4—2121no—
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—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—2121no—
273C22×C4C2×D21211yescharacteristic
274C22×C4#1171994511yescharacteristic
275C22×C4C2×C22⋊D5311yescharacteristic
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—22no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—1062no—
285C2×C8—2121no—
286C2×C8—2121no—
287C2×C8—2121no—
288C2×C8—2121no—
289C2×C8—2121no—
290C2×C8#1171992212yes—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—4241no—
294C4⋊C4—2121no—
295C4⋊C4—2121no—
296C4⋊C4—2121no—
297C4⋊C4—2121no—
298C4⋊C4#1171991312yes—
299C4⋊C4D42412yes—
300C42—41no—
301C42—41no—
302C42C2×C4×D5311yescharacteristic
303C42#1171994511yescharacteristic
304C42#1171994511yescharacteristic
305C42#1171994511yescharacteristic
306C23—4241no—
307C23—4241no—
308C23—4241no—
309C23—4241no—
310C23—4241no—
311C23—4241no—
312C23—2122no—
313C23#1313319111yescharacteristic
314D4—4242no—
315D4—4242no—
316D4—4241no—
317D4—4241no—
318D4—4241no—
319D4—4241no—
320D4—2122no—
321D4—2122no—
322D4—2122no—
323D4—2122no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—4241no—
333C2×C4—4241no—
334C2×C4—4241no—
335C2×C4#1313317411yescharacteristic
336C2×C4C4×C22⋊D5311yescharacteristic
337C2×C4C4×D21211yescharacteristic
338C2×C4#1313320011yescharacteristic
339C2×C4#1313320511yescharacteristic
340C2×C4#1313321111yescharacteristic
341C2×C4C2×#1171992211yescharacteristic
342C2×C4C2×D42411yescharacteristic
343C2×C4C2×#1171991311yescharacteristic
344C2×C4#1313323711yescharacteristic
345C2×C4#1313323911yescharacteristic
346C2×C4#1313323911yescharacteristic
347C2×C4#1313324211yescharacteristic
348C2×C4#1313324811yescharacteristic
349C8—22no—
350C8—41no—
351C8—42no—
352C8—4241no—
353C8—2122no—
354C22—8481no—
355C22—8481no—
356C22—8481no—
357C22—8481no—
358C22—4242no—
359C22—4242no—
360C22—4242no—
361C22#1681051411yescharacteristic
362C22#1681081811yescharacteristic
363C22#1681084511yescharacteristic
364C22#1681085211yescharacteristic
365C22#1681085911yescharacteristic
366C22#1681088611yescharacteristic
367C22#1681089311yescharacteristic
368C4—21no—
369C4—21no—
370C4—42no—
371C4—42no—
372C4—8481no—
373C4C4×#1171992211yescharacteristic
374C4C4×D42411yescharacteristic
375C4C4×#1171991311yescharacteristic
376C4#1681072111yescharacteristic
377C4#1681093611yescharacteristic
378C4#1681094711yescharacteristic
379C4#1681097011yescharacteristic
380C4#1681097811yescharacteristic
381C2—8482no—
382C2#2027405611yescharacteristic
383C2#2027407011yescharacteristic
384C2#2027413111yescharacteristic
385C2#2027415911yescharacteristic
386C2#2027533511yescharacteristic
387C2#2027537711yescharacteristic
388C2#2027539811yescharacteristic
subgroupG/H
centerC23#13133191
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC8×C4⋊C4×C53C2
socleC23×C53#121
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenterC8×C4⋊C4D53
hypocenterC53#9563
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20275473C2semidirect
5. #20272599C2semidirect
6. #20272550C2semidirect
7. C8×C4⋊C4×C53C2semidirect
18. #16810366C4semidirect
190. C8×C4⋊C4D53semidirect
218. C53#9563semidirect

Maximal subgroups

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

subgroupindex
#202756422
#202755212
#202754732
C2×#168103662
#202725992
#202725502
C8×C4⋊C4×C532
#956353