Finite Group Databasebeta

← #27277269

Subgroups — #27277269

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#20283682C211yesmaximal, characteristic
2#20274735C211yesmaximal, characteristic
3C4×#13133148C211yesmaximal, characteristic, complemented
4#20273473C211yesmaximal, characteristic, complemented
5#20272618C211yesmaximal, characteristic, complemented
6#20272616C211yesmaximal, characteristic, complemented
7C53×#1806C211yesmaximal, characteristic
8#16810754C411yescharacteristic, complemented
9#16810754C411yescharacteristic, complemented
10#16810701C411yescharacteristic, complemented
11#16810701C411yescharacteristic, complemented
12#16810695—21no—
13#16810695—21no—
14#16810676—21no—
15#16810676C2211yescharacteristic
16C2×#13133148—21no—
17#16810647C2211yescharacteristic
18C2×#13133144—21no—
19C4×#11719917—21no—
20C2×C4×#88322C2211yescharacteristic
21C4×C4⋊Dic53C2211yescharacteristic
22#16810385—21no—
23#16810382—21no—
24#16810375—21no—
25#16810375—21no—
26C4×#41×C53C2211yescharacteristic
27C53×#423C2211yescharacteristic
28C2×#104×C53—21no—
29C4×C4⋊C4×C53C2211yescharacteristic
30C53×#339—21no—
31C2×#11719917—21no—
32C2×#11719917—21no—
33C2×#11719917—21no—
34C22×#88322D411yescharacteristic
35C22×#88322D411yescharacteristic
36C2×C4⋊Dic53—21no—
37C2×C4×Dic53—21no—
38#13133148—24no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322C2×C411yescharacteristic
44C4⋊#88322C2×C411yescharacteristic
45#13133144—42no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49#13133128—21no—
50#13133128—21no—
51C4×#88322—21no—
52C4×#88322C2×C411yescharacteristic
53C4×#88322C2×C411yescharacteristic
54C4×#88322D411yescharacteristic
55C4×#88322D411yescharacteristic
56#13133124—21no—
57C2×#41×C53D411yescharacteristic
58C2×M16×C53—21no—
59C4×Q8×C53C2×C411yescharacteristic
60C4×Q8×C53Q811yescharacteristic
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53C2×C411yescharacteristic
64C4×D4×C53Q811yescharacteristic
65#117×C53—21no—
66#117×C53D411yescharacteristic
67C2×C4⋊C4×C53—21no—
68C2×C42×C53—21no—
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71C4⋊C8×C53—21no—
72C4⋊C8×C53—21no—
73#104×C53—42no—
74#95×C53—21no—
75#41×C53—21no—
76#41×C53—21no—
77#41×C53—21no—
78#41×C53—21no—
79C2×Q8×C53C4⋊C411yescharacteristic
80C2×D4×C53—21no—
81C2×D4×C53C4⋊C411yescharacteristic
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—41no—
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C22×Dic53—41no—
90M16×C53—41no—
91M16×C53—41no—
92C2×C8×C53—41no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53C4⋊C411yescharacteristic
98C4⋊C4×C53C4⋊C411yescharacteristic
99C22⋊C4×C53—21no—
100C22⋊C4×C53—21no—
101C22⋊C4×C53—21no—
102C42×C53—21no—
103C42×C53—21no—
104C42×C53—21no—
105C42×C53—41no—
106C42×C53—41no—
107C42×C53C22×C411yescharacteristic
108C42×C53C2×D411yescharacteristic
109C42×C53C2×Q811yescharacteristic
110C42×C53#4111yescharacteristic
111C4⋊Dic53—22no—
112C4⋊Dic53—41no—
113C4×Dic53—41no—
114C4×Dic53—41no—
115#11719917—42no—
116#11719917—42no—
117#11719917—41no—
118#11719917—41no—
119C2×#88322—22no—
120C2×#88322—21no—
121C2×#88322—21no—
122C2×#88322—41no—
123C2×#88322—41no—
124C2×#88322C22⋊C411yescharacteristic
125C2×#88322C22⋊C411yescharacteristic
126C2×#88322C22⋊C411yescharacteristic
127C2×#88322C22⋊C411yescharacteristic
128C23×C53—21no—
129C23×C53#12111yescharacteristic
130Q8×C53—21no—
131Q8×C53—21no—
132D4×C53—21no—
133D4×C53—21no—
134D4×C53—41no—
135D4×C53—41no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—41no—
142C2×C4×C53—42no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
148C2×C4×C53—81no—
149C2×C4×C53—81no—
150C2×C4×C53C2×C22⋊C411yescharacteristic
151C2×C4×C53C2×C4⋊C411yescharacteristic
152C2×C4×C53#11711yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53C22⋊Q811yescharacteristic
155C2×C4×C53C22⋊Q811yescharacteristic
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C53#13711yescharacteristic
159C2×C4×C53#13711yescharacteristic
160C2×Dic53—42no—
161C2×Dic53—81no—
162C2×Dic53—81no—
163C8×C53—81no—
164#88322—21no—
165#88322—21no—
166#88322—21no—
167#88322—41no—
168#88322—41no—
169#88322—81no—
170#88322#10412yescomplemented
171#8669—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53#38011yescharacteristic
178C22×C53#46811yescharacteristic
179C22×C53#46811yescharacteristic
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—82no—
187C4×C53C2×#10411yescharacteristic
188C4×C53#42111yescharacteristic
189C4×C53#42811yescharacteristic
190C4×C53#42811yescharacteristic
191C4×C53#46911yescharacteristic
192C4×C53#47011yescharacteristic
193Dic53—82no—
194#2965—531no—
195#1806—531no—
196#1806D5311yescharacteristic
197C4×#106—531no—
198#1567—531no—
199#1303—531no—
200#1303—531no—
201C2×C53—21no—
202C2×C53—21no—
203C2×C53—81no—
204C2×C53#180611yescharacteristic
205C2×C53#180911yescharacteristic
206C2×C53#181411yescharacteristic
207C4×#41C2×D5311yescharacteristic
208#446—531no—
209#446—531no—
210#436—531no—
211#436—531no—
212#435—1061no—
213#435—1061no—
214#431—531no—
215#431—1061no—
216C2×#106—1061no—
217#423—531no—
218#423C2×D5311yescharacteristic
219C2×#104—21no—
220C2×#104—1061no—
221C4×M16—1061no—
222C2×C4×C8—531no—
223C4×C4⋊C4—531no—
224C4×C4⋊C4C2×D5311yescharacteristic
225#340—1061no—
226#339—21no—
227#339—1061no—
228#335—1061no—
229#335—1061no—
230C53#866911yescharacteristic, complemented, Sylow
231C2×#41C22⋊D5311yescharacteristic
232C2×M16—21no—
233C2×M16—1061no—
234C2×M16—1061no—
235C2×M16—1061no—
236C22×C8—531no—
237C22×C8—531no—
238C4×Q8Dic10611yescharacteristic
239C4×Q8C4×D5311yescharacteristic
240C4×D4—21no—
241C4×D4—21no—
242C4×D4Dic10611yescharacteristic
243C4×D4C4×D5311yescharacteristic
244#117—21no—
245#117C22⋊D5311yescharacteristic
246C2×C4⋊C4—21no—
247C2×C4⋊C4—1061no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—1061no—
251C2×C42C22×D5311yescharacteristic
252#106—1064no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—531no—
256C4⋊C8—531no—
257C4⋊C8—1061no—
258C4⋊C8—1061no—
259C4⋊C8—1061no—
260C4⋊C8—1061no—
261#104—42no—
262#104—2122no—
263C22⋊C8—1061no—
264C22⋊C8—1061no—
265C22⋊C8—1061no—
266#97—1061no—
267#97—1061no—
268C4×C8—531no—
269C4×C8—531no—
270C4×C8—531no—
271C4×C8—531no—
272C4×C8—1061no—
273#95—21no—
274#95—1061no—
275#41—21no—
276#41—21no—
277#41—21no—
278#41—21no—
279C2×Q8#1171991911yescharacteristic
280C2×D4—21no—
281C2×D4#1171991911yescharacteristic
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—2121no—
287C22×C4D4×D5311yescharacteristic
288C22×C4#1171994711yescharacteristic
289C22×C4C2×C22⋊D5311yescharacteristic
290M16—41no—
291M16—41no—
292M16—2122no—
293M16—2122no—
294M16—2121no—
295M16—2121no—
296C2×C8—41no—
297C2×C8—531no—
298C2×C8—531no—
299C2×C8—531no—
300C2×C8—531no—
301C2×C8—1062no—
302C2×C8—1061no—
303C2×C8—1061no—
304C2×C8—2121no—
305C2×C8—2121no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—1062no—
311C4⋊C4—2121no—
312C4⋊C4#1171991911yescharacteristic
313C4⋊C4#1171991911yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—41no—
321C42—41no—
322C42—2121no—
323C42—2121no—
324C42C2×Dic10611yescharacteristic
325C42C2×C4×D5311yescharacteristic
326C42D4×D5311yescharacteristic
327C42#1171994711yescharacteristic
328C23—21no—
329C23#1313318911yescharacteristic
330Q8—21no—
331Q8—21no—
332D4—21no—
333D4—21no—
334D4—41no—
335D4—41no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—4241no—
351C2×C4—4241no—
352C2×C4—2122no—
353C2×C4C22⋊C4×D5311yescharacteristic
354C2×C4C2×#1171991911yescharacteristic
355C2×C4#1313317311yescharacteristic
356C2×C4#1313318911yescharacteristic
357C2×C4#1313319411yescharacteristic
358C2×C4#1313319411yescharacteristic
359C2×C4#1313324211yescharacteristic
360C2×C4#1313324311yescharacteristic
361C2×C4#1313324711yescharacteristic
362C2×C4#1313324811yescharacteristic
363C8—81no—
364C8—532no—
365C8—1061no—
366C8—1061no—
367C8—1061no—
368C8—4241no—
369C8—2121no—
370C8—2121no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22#1681048111yescharacteristic
377C22#1681086711yescharacteristic
378C22#1681087111yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—82no—
386C4—4242no—
387C4#104×D5311yescharacteristic
388C4#1681063811yescharacteristic
389C4#1681066311yescharacteristic
390C4#1681066611yescharacteristic
391C4#1681087411yescharacteristic
392C4#1681087911yescharacteristic
393C2—21no—
394C2—21no—
395C2—81no—
396C2#2027473311yescharacteristic
397C2#2027474611yescharacteristic
398C2#2027476711yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1806C2
socleC22×C53#380
prime radicalC2×#41×C53D4
cosocleC2×C42×C53C23
hypercenter#1806D53
hypocenterC53#8669
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
3. C4×#13133148C2semidirect
4. #20273473C2semidirect
5. #20272618C2semidirect
6. #20272616C2semidirect
8. #16810754C4semidirect
9. #16810754C4semidirect
10. #16810701C4semidirect
11. #16810701C4semidirect
170. #88322#104semidirect
230. C53#8669semidirect

Maximal subgroups

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

subgroupindex
#202836822
#202747352
C4×#131331482
#202734732
#202726182
#202726162
C53×#18062
#866953