Finite Group Databasebeta

← #28301334

Subgroups — #28301334

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291392C211yesmaximal, characteristic
2#20291164C211yesmaximal, characteristic
3#20289615C214yesmaximal
4#20283347C211yesmaximal, characteristic
5C2×#13133296C412yes—
6#16811645—21no—
7#16811644C412yescomplemented
8#16811642—21no—
9#16811639C412yescomplemented
10#16811627—21no—
11#16811594C2212yes—
12C4×#11719936—21no—
13#16811569—21no—
14#16811567—21no—
15#16811565C412yes—
16#16811565C2212yes—
17C4×C4⋊#6157—21no—
18C4×C4⋊#6157—21no—
19C2×C42×D53C2211yescharacteristic
20#16810679—21no—
21#16810679—21no—
22#16810675—21no—
23#16810675—21no—
24#16810647C411yescharacteristic, complemented
25#16810586C2212yes—
26M16×Dic53C412yescomplemented
27C53×#423C411yescharacteristic, complemented
28C2×#11719936—22no—
29#13133296—44no—
30C2×C4⋊#6157—22no—
31C2×C4⋊#6157—22no—
32C2×C4⋊#6157—22no—
33C2×C22⋊#6157C2×C414yes—
34C2×C4×#6157—22no—
35C4⋊#88324—22no—
36C4⋊#88324—22no—
37#13133274—22no—
38C22⋊#88324C2×C414yes—
39#13133272—22no—
40C4×#88324—22no—
41(C2×#6157)⋊C4—22no—
42(C2×#6157)⋊C4—22no—
43C22×C4×D53C2×C411yescharacteristic
44C22×C4×D53C2×C411yescharacteristic
45C22×C4×D53C2311yescharacteristic
46C2×#11719917C2×C412yes—
47C2×C4×Dic53C2×C411yescharacteristic
48C2×C4×Dic53C2×C411yescharacteristic
49C2×C4×Dic53C2×C411yescharacteristic
50C42×D53D411yescharacteristic
51C42×D53D411yescharacteristic
52C42×D53D411yescharacteristic
53C42×D53D411yescharacteristic
54C42×D53D411yescharacteristic
55C42×D53D411yescharacteristic
56C42×D53Q811yescharacteristic
57C42×D53Q811yescharacteristic
58C4⋊#88322—21no—
59C4⋊#88322—21no—
60#13133129—22no—
61#13133129—22no—
62#13133127—22no—
63C8×Dic53—22no—
64C2×M16×C53C2×C412yes—
65C2×C42×C53C2×C411yescharacteristic
66C4⋊C8×C53—21no—
67C4⋊C8×C53—21no—
68C23×D53C22×C411yescharacteristic
69C22×#6157—24no—
70C22×C4×C53C4211yescharacteristic
71C22×C4×C53C4211yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×Dic53C4211yescharacteristic
74C22×Dic53C4211yescharacteristic
75C22×Dic53C22×C411yescharacteristic
76C2×C4×D53—22no—
77C2×C4×D53—22no—
78C2×C4×D53—42no—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C4×D53C22⋊C411yescharacteristic
82C2×C4×D53C22⋊C411yescharacteristic
83C2×C4×D53C22⋊C411yescharacteristic
84C2×C4×D53C22⋊C411yescharacteristic
85C2×C4×D53C4⋊C411yescharacteristic
86C2×C4×D53C4⋊C411yescharacteristic
87C2×C4×D53C4⋊C411yescharacteristic
88C2×C4×D53C4⋊C411yescharacteristic
89C2×C4×D53C2×D411yescharacteristic
90C2×C4×D53C2×D411yescharacteristic
91C2×C4×D53C2×D411yescharacteristic
92C2×C4×D53C2×Q811yescharacteristic
93C22⋊#6157—44no—
94C2×#88324—24no—
95C4⋊#6157—44no—
96C4⋊#6157—42no—
97C4⋊#6157—42no—
98C4⋊#6157—44no—
99C4×#6157—44no—
100#11719936—42no—
101#11719936—42no—
102M16×C53—42no—
103C2×C8×C53—22no—
104C42×C53C22⋊C411yescharacteristic
105C42×C53C22⋊C411yescharacteristic
106C42×C53C4⋊C411yescharacteristic
107C42×C53C4⋊C411yescharacteristic
108C4×Dic53C22⋊C411yescharacteristic
109C4×Dic53C22⋊C411yescharacteristic
110C4×Dic53C22⋊C411yescharacteristic
111C4×Dic53C22⋊C411yescharacteristic
112C4×Dic53C22⋊C411yescharacteristic
113C4×Dic53C22⋊C411yescharacteristic
114C4×Dic53C4⋊C411yescharacteristic
115C4×Dic53C4⋊C411yescharacteristic
116C4×Dic53C4⋊C411yescharacteristic
117C4×Dic53C4⋊C411yescharacteristic
118C4×Dic53C4⋊C411yescharacteristic
119C4×Dic53C4⋊C411yescharacteristic
120#11719917—42no—
121C2×#88322—22no—
122C23×C53C2×C4211yescharacteristic
123C22×D53—21no—
124C22×D53—21no—
125C22×D53—41no—
126C22×D53—41no—
127C22×D53C2×C22⋊C411yescharacteristic
128C22×D53C2×C4⋊C411yescharacteristic
129C2×#6157—44no—
130C2×#6157—44no—
131C2×#6157—44no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53#9511yescharacteristic
137C2×C4×C53#9511yescharacteristic
138C2×C4×C53#9511yescharacteristic
139C2×C4×C53#9511yescharacteristic
140C2×C4×C53C2×C22⋊C411yescharacteristic
141C2×C4×C53C2×C4⋊C411yescharacteristic
142C2×Dic53—22no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53#9511yescharacteristic
146C2×Dic53#9511yescharacteristic
147C2×Dic53#9511yescharacteristic
148C2×Dic53#9511yescharacteristic
149C2×Dic53C2×C22⋊C411yescharacteristic
150C2×Dic53C2×C4⋊C411yescharacteristic
151C4×D53—21no—
152C4×D53—21no—
153C4×D53—22no—
154C4×D53—22no—
155C4×D53—22no—
156C4×D53—21no—
157C4×D53—21no—
158C4×D53—21no—
159C4×D53—21no—
160C4×D53—22no—
161C4×D53—41no—
162C4×D53—41no—
163C4×D53—41no—
164C4×D53—41no—
165C4×D53#10012yes—
166C4×D53#10112yes—
167#88324—44no—
168C8×C53—42no—
169#88322—42no—
170#8105—531nomaximal, Sylow
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53C2×#9511yescharacteristic
175C2×D53—21no—
176C2×D53—21no—
177C2×D53—21no—
178C2×D53—21no—
179C2×D53—21no—
180C2×D53—42no—
181C2×D53—42no—
182C2×D53C2×#10011yescharacteristic
183C2×D53C2×#10111yescharacteristic
184#6157—84no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53#34312yes—
189C4×C53#34412yes—
190C4×C53#41011yescharacteristic
191C4×C53#41011yescharacteristic
192Dic53—21no—
193Dic53—21no—
194Dic53—21no—
195Dic53—21no—
196Dic53—21no—
197Dic53#34312yes—
198#2920—531no—
199#2920—531no—
200#2285—531no—
201#1305—534no—
202C2×C53—41no—
203C2×C53#174011yescharacteristic
204C2×C53C2×#34311yescharacteristic
205C2×C53C2×#34411yescharacteristic
206D53—21no—
207D53—21no—
208D53—41no—
209C2×#117—532no—
210C22×C42—531no—
211#432—532no—
212#432—1061no—
213#432—1061no—
214#432—1061no—
215#431—532no—
216#431—1061no—
217#431—1061no—
218#431—1061no—
219#423—531no—
220#423—1061no—
221#423#615711yescharacteristic, complemented
222#407—532no—
223#407—532no—
224C4×M16—532no—
225C4×M16—1061no—
226#384—1061no—
227#382—1061no—
228#380—532no—
229#380—532no—
230C4×C4⋊C4—1061no—
231C4×C4⋊C4—1061no—
232C53#810511yescharacteristic, complemented, Sylow
233C23×C4—531no—
234C23×C4—531no—
235C23×C4—531no—
236C2×M16—532no—
237C2×M16—1062no—
238C2×M16C2×#615712yes—
239#117—2124no—
240C2×C4⋊C4—1062no—
241C2×C4⋊C4—1062no—
242C2×C4⋊C4—1062no—
243C2×C22⋊C4—534no—
244C2×C42—531no—
245C2×C42—531no—
246C2×C42—531no—
247C2×C42—531no—
248C2×C42—531no—
249C2×C42—531no—
250C2×C42—531no—
251C2×C42—531no—
252C2×C42—531no—
253C2×C42—531no—
254C2×C42—531no—
255C2×C42—1062no—
256C2×C42C2×#615711yescharacteristic
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—1062no—
260C4⋊C8—1062no—
261C4⋊C8—1061no—
262C4⋊C8—1061no—
263C22⋊C8—534no—
264C22⋊C8—1062no—
265C22⋊C8—1062no—
266C22⋊C8—1062no—
267#97—1062no—
268#97—1062no—
269C4×C8—1062no—
270C4×C8—1062no—
271#95—1062no—
272#95—1062no—
273C24—531no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—1062no—
290C22×C4—1062no—
291C22×C4—1064no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4—2122no—
295C22×C4C4×#615711yescharacteristic
296C22×C4C4×#615711yescharacteristic
297C22×C4C22×#615711yescharacteristic
298M16—42no—
299M16—2122no—
300M16—2122no—
301M16—2122no—
302C2×C8—22no—
303C2×C8—1064no—
304C2×C8—1062no—
305C4⋊C4—2122no—
306C4⋊C4—2122no—
307C4⋊C4—2124no—
308C4⋊C4—2124no—
309C22⋊C4—2124no—
310C42—531no—
311C42—531no—
312C42—531no—
313C42—531no—
314C42—531no—
315C42—531no—
316C42—531no—
317C42—531no—
318C42—531no—
319C42—531no—
320C42—531no—
321C42—531no—
322C42—2124no—
323C42C4⋊#615711yescharacteristic
324C42C4⋊#615711yescharacteristic
325C42C22⋊#615711yescharacteristic
326C42C22⋊#615711yescharacteristic
327C23—531no—
328C23—531no—
329C23—1061no—
330C23—1061no—
331C23—2121no—
332C23—2121no—
333C23C2×C4×#615711yescharacteristic
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—531no—
339C2×C4—532no—
340C2×C4—532no—
341C2×C4—531no—
342C2×C4—531no—
343C2×C4—531no—
344C2×C4—531no—
345C2×C4—531no—
346C2×C4—1062no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1062no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1062no—
355C2×C4—1062no—
356C2×C4—1062no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4—2124no—
364C2×C4—2124no—
365C2×C4—2124no—
366C2×C4(C2×#6157)⋊C411yescharacteristic
367C2×C4(C2×#6157)⋊C411yescharacteristic
368C2×C4(C2×#6157)⋊C411yescharacteristic
369C2×C4(C2×#6157)⋊C411yescharacteristic
370C2×C4C2×C22⋊#615711yescharacteristic
371C2×C4C2×C4⋊#615711yescharacteristic
372C8—42no—
373C8—2124no—
374C8—2122no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—531no—
379C22—531no—
380C22—1061no—
381C22—1061no—
382C22—1061no—
383C22—1061no—
384C22—1061no—
385C22—2122no—
386C22—2122no—
387C22C2×(C2×#6157)⋊C411yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—532no—
392C4—1061no—
393C4—1061no—
394C4—1061no—
395C4—1061no—
396C4—1061no—
397C4—4244no—
398C4#1681150212yes—
399C4#1681150612yes—
400C4#1681160311yescharacteristic
401C4#1681160411yescharacteristic
402C2—41no—
403C2—1061no—
404C2—1061no—
405C2—2121no—
406C2#2029041511yescharacteristic
407C2C2×#1681150211yescharacteristic
408C2C2×#1681150611yescharacteristic
subgroupG/H
centerC22C2×(C2×#6157)⋊C4
commutatorC23×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#423C4
socleC22×C53C2×#95
prime radicalC23×D53C22×C4
cosocleC22×C4×D53C23
hypercenter#423#6157
hypocenterC53#8105
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
7. #16811644C4semidirect
9. #16811639C4semidirect
24. #16810647C4semidirect
26. M16×Dic53C4semidirect
27. C53×#423C4semidirect
221. #423#6157semidirect
232. C53#8105semidirect

Maximal subgroups

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

subgroupindex
#202913922
#202911642
#202896152
#202833472
#810553