Finite Group Databasebeta

← #27277270

Subgroups — #27277270

Order 13568.

432 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283676C211yesmaximal, characteristic
2#20274736C211yesmaximal, characteristic
3#20274732C211yesmaximal, characteristic
4C4×#13133148C211yesmaximal, characteristic, complemented
5#20273473C211yesmaximal, characteristic, complemented
6#20272617C211yesmaximal, characteristic, complemented
7C53×#1303C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810750C411yescharacteristic, complemented
10#16810750C411yescharacteristic, complemented
11#16810700C411yescharacteristic, complemented
12#16810700C411yescharacteristic, complemented
13#16810691—21no—
14#16810691—21no—
15#16810674—21no—
16C2×#13133148—21no—
17#16810647C2211yescharacteristic
18C2×#13133145—21no—
19C2×#13133143—21no—
20C4×#11719912—21no—
21C2×C4×#88322C2211yescharacteristic
22C4×C4⋊Dic53C2211yescharacteristic
23#16810385—21no—
24#16810384—21no—
25#16810383—21no—
26#16810375—21no—
27#16810375—21no—
28C53×#431C2211yescharacteristic
29C53×#423C2211yescharacteristic
30C4×C4⋊C4×C53C2211yescharacteristic
31C2×#11719945D411yescharacteristic
32C2×#11719917—21no—
33C2×#11719912—21no—
34C2×#11719912—21no—
35C22×#88322D411yescharacteristic
36C22×#88322D411yescharacteristic
37C4×Dic106C2×C411yescharacteristic
38C4×Dic106Q811yescharacteristic
39C4×D212C2×C411yescharacteristic
40C4×D212Q811yescharacteristic
41C4×C22⋊D53—21no—
42C4×C22⋊D53—21no—
43#13133176D411yescharacteristic
44#13133174—21no—
45C2×C4⋊Dic53—21no—
46C2×C4×Dic53—21no—
47C42×D53—21no—
48#13133148—24no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51#13133146—21no—
52#13133146—21no—
53#13133145—42no—
54#13133143—42no—
55#13133129—21no—
56#13133129—21no—
57#13133127—21no—
58#13133127—21no—
59C4×#88322C2×C411yescharacteristic
60C4×#88322C2×C411yescharacteristic
61C4×#88322D411yescharacteristic
62C4×#88322D411yescharacteristic
63#13133124—21no—
64C2×M16×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C42×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53C2×C411yescharacteristic
71C4⋊C8×C53C2×C411yescharacteristic
72C22⋊C8×C53—21no—
73C4×C8×C53—21no—
74#95×C53—21no—
75C22×C4×C53—41no—
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C2×C22⋊D53—21no—
80C22×Dic53—41no—
81#11719945—21no—
82#11719945—21no—
83#11719945—21no—
84#11719945—21no—
85C2×D212C4⋊C411yescharacteristic
86C2×C4×D53—21no—
87C2×C4×D53—21no—
88C2×C4×D53—21no—
89C2×Dic106C4⋊C411yescharacteristic
90M16×C53—41no—
91M16×C53—41no—
92C2×C8×C53—21no—
93C2×C8×C53—21no—
94C2×C8×C53—41no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—41no—
97C42×C53—41no—
98C42×C53—41no—
99C42×C53C22×C411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53C2×Q811yescharacteristic
102C42×C53#4111yescharacteristic
103C22⋊Dic53—21no—
104#11719921—21no—
105#11719921—21no—
106C4⋊Dic53—22no—
107C4⋊Dic53—41no—
108C4⋊Dic53C4⋊C411yescharacteristic
109C4⋊Dic53C4⋊C411yescharacteristic
110#11719919—21no—
111#11719919—21no—
112C4×Dic53—21no—
113C4×Dic53—21no—
114C4×Dic53—21no—
115C4×Dic53—41no—
116C4×Dic53—41no—
117#11719917—41no—
118#11719917—41no—
119C2×#88322—22no—
120C2×#88322—41no—
121C2×#88322—41no—
122C2×#88322C22⋊C411yescharacteristic
123C2×#88322C22⋊C411yescharacteristic
124C2×#88322C22⋊C411yescharacteristic
125C2×#88322C22⋊C411yescharacteristic
126#11719912—42no—
127#11719912—42no—
128C23×C53#12111yescharacteristic
129C22×D53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53C2×C22⋊C411yescharacteristic
137C2×C4×C53C2×C4⋊C411yescharacteristic
138C2×C4×C53#11711yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53C22⋊Q811yescharacteristic
141C2×C4×C53C22⋊Q811yescharacteristic
142C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C53#13711yescharacteristic
145C2×C4×C53#13711yescharacteristic
146C22⋊D53—41no—
147C22⋊D53—41no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—42no—
152C2×Dic53—81no—
153C2×Dic53—81no—
154D212—21no—
155D212—21no—
156C4×D53—42no—
157C4×D53—41no—
158C4×D53—41no—
159C4×D53—41no—
160C4×D53—41no—
161Dic106—21no—
162Dic106—21no—
163C8×C53—41no—
164C8×C53—41no—
165C8×C53—81no—
166#88322—21no—
167#88322—21no—
168#88322—21no—
169#88322—81no—
170#88322#10412yescomplemented
171#8669—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53#38011yescharacteristic
175C22×C53#46811yescharacteristic
176C22×C53#46811yescharacteristic
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—41no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—82no—
184C4×C53C2×#10411yescharacteristic
185C4×C53#42111yescharacteristic
186C4×C53#42811yescharacteristic
187C4×C53#42811yescharacteristic
188C4×C53#46911yescharacteristic
189C4×C53#47011yescharacteristic
190Dic53—41no—
191Dic53—41no—
192Dic53—41no—
193Dic53—82no—
194#2965—531no—
195#1806—531no—
196#1806—531no—
197C4×#106—531no—
198#1567—531no—
199#1303—531no—
200#1303D5311yescharacteristic, complemented
201C2×C53—21no—
202C2×C53—21no—
203C2×C53#180611yescharacteristic
204C2×C53#180911yescharacteristic
205C2×C53#181411yescharacteristic
206D53—81no—
207C4×#41—531no—
208#446—531no—
209#446—531no—
210#436—531no—
211#436—531no—
212#435—1061no—
213#435—1061no—
214#431—1061no—
215#431C2×D5311yescharacteristic
216C2×#106—1061no—
217#423—531no—
218#423C2×D5311yescharacteristic
219C2×#104—1061no—
220C2×#104—1061no—
221C4×M16—1061no—
222C2×C4×C8—531no—
223C4×C4⋊C4—531no—
224C4×C4⋊C4C2×D5311yescharacteristic
225#340—1061no—
226#339—1061no—
227#339—1061no—
228#335—1061no—
229#335—1061no—
230C53#866911yescharacteristic, complemented, Sylow
231C2×#41—531no—
232C2×M16—21no—
233C2×M16—1061no—
234C2×M16—1061no—
235C2×M16—1061no—
236C22×C8—531no—
237C22×C8—531no—
238C4×Q8—531no—
239C4×Q8—531no—
240C4×D4—531no—
241C4×D4—531no—
242C4×D4—1061no—
243C4×D4—1061no—
244#117—531no—
245#117—1061no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—1061no—
248C2×C42—21no—
249C2×C42—1061no—
250C2×C42—1061no—
251C2×C42C22×D5311yescharacteristic
252#106—1064no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—1061no—
256C4⋊C8—1061no—
257C4⋊C8—1061no—
258C4⋊C8—1061no—
259C4⋊C8C4×D5311yescharacteristic
260C4⋊C8C4×D5311yescharacteristic
261#104—2122no—
262#104—2122no—
263C22⋊C8—21no—
264C22⋊C8—1061no—
265C22⋊C8—1061no—
266#97—1061no—
267#97—1061no—
268C4×C8—21no—
269C4×C8—531no—
270C4×C8—531no—
271C4×C8—531no—
272C4×C8—531no—
273#95—21no—
274#95—1061no—
275#41—1061no—
276#41—1061no—
277#41—1061no—
278#41—1061no—
279C2×Q8—531no—
280C2×D4—531no—
281C2×D4—1061no—
282C22×C4—41no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4—2121no—
287C22×C4#1171994511yescharacteristic
288C22×C4D4×D5311yescharacteristic
289C22×C4D4×D5311yescharacteristic
290M16—41no—
291M16—41no—
292M16—2122no—
293M16—2122no—
294M16—2121no—
295M16—2121no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—41no—
299C2×C8—531no—
300C2×C8—531no—
301C2×C8—531no—
302C2×C8—531no—
303C2×C8—1062no—
304C2×C8—2121no—
305C2×C8—2121no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—531no—
309C4⋊C4—531no—
310C4⋊C4—1062no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C4⋊C4—2121no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C42—41no—
318C42—41no—
319C42—1061no—
320C42—1061no—
321C42—1061no—
322C42—2121no—
323C42—2121no—
324C42C2×C4×D5311yescharacteristic
325C42#1171994511yescharacteristic
326C42D4×D5311yescharacteristic
327C42Q8×D5311yescharacteristic
328C23—1061no—
329C23#1313319011yescharacteristic
330Q8—1061no—
331Q8—1061no—
332D4—1061no—
333D4—1061no—
334D4—2121no—
335D4—2121no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—4241no—
346C2×C4—4241no—
347C2×C4—2122no—
348C2×C4—2122no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4C22⋊C4×D5311yescharacteristic
354C2×C4C4⋊C4×D5311yescharacteristic
355C2×C4#1313317411yescharacteristic
356C2×C4#1313319011yescharacteristic
357C2×C4#1313319511yescharacteristic
358C2×C4#1313319511yescharacteristic
359C2×C4#1313324111yescharacteristic
360C2×C4#1313324411yescharacteristic
361C2×C4#1313324611yescharacteristic
362C2×C4#1313324911yescharacteristic
363C8—41no—
364C8—41no—
365C8—81no—
366C8—532no—
367C8—1061no—
368C8—1061no—
369C8—1061no—
370C8—4241no—
371C22—21no—
372C22—21no—
373C22—2121no—
374C22—2121no—
375C22—2121no—
376C22#1681048211yescharacteristic
377C22#1681086811yescharacteristic
378C22#1681087211yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—82no—
383C4—4242no—
384C4—2121no—
385C4—2121no—
386C4—2121no—
387C4#104×D5311yescharacteristic
388C4#1681063811yescharacteristic
389C4#1681066211yescharacteristic
390C4#1681066411yescharacteristic
391C4#1681087511yescharacteristic
392C4#1681088011yescharacteristic
393C2—21no—
394C2—21no—
395C2—4241no—
396C2#2027473411yescharacteristic
397C2#2027474711yescharacteristic
398C2#2027476811yescharacteristic
subgroupG/H
centerC2×C4#13133190
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1303C2
socleC22×C53#380
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenter#1303D53
hypocenterC53#8669
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
4. C4×#13133148C2semidirect
5. #20273473C2semidirect
6. #20272617C2semidirect
7. C53×#1303C2semidirect
9. #16810750C4semidirect
10. #16810750C4semidirect
11. #16810700C4semidirect
12. #16810700C4semidirect
170. #88322#104semidirect
200. #1303D53semidirect
230. C53#8669semidirect

Maximal subgroups

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

subgroupindex
#202836762
#202747362
#202747322
C4×#131331482
#202734732
#202726172
C53×#13032
#866953