Finite Group Databasebeta

← #27277074

Subgroups — #27277074

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274753C211yesmaximal, characteristic, complemented
2#20274744C211yesmaximal, characteristic, complemented
3#20273258C212yesmaximal
4#20272620C212yesmaximal, complemented
5C53×#2929C211yesmaximal, characteristic
6C2×#13133153—21no—
7C2×#13133151—21no—
8#16810587C2212yes—
9#16810483C2212yes—
10#16810388—21no—
11#16810386—21no—
12#16810360—22no—
13#16810357—22no—
14C2×#131×C53C411yescharacteristic, complemented
15C23×C8×C53C2211yescharacteristic
16C53×#525C2211yescharacteristic
17C53×#435—21no—
18C53×#435—21no—
19C8×D4×C53—21no—
20C8×D4×C53—21no—
21C53×#424C411yescharacteristic, complemented
22C53×#408C411yescharacteristic, complemented
23C53×#407C2211yescharacteristic
24C53×#405C411yescharacteristic, complemented
25C2×#11719917—22no—
26C2×C4⋊Dic53—22no—
27C2×C22⋊Dic53—22no—
28#13133153—42no—
29#13133151—42no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32#13133124—22no—
33C2×#41×C53C2×C411yescharacteristic
34C23×C4×C53C2311yescharacteristic
35#131×C53—42no—
36C2×M16×C53—21no—
37C2×M16×C53C2×C411yescharacteristic
38C2×M16×C53C2×C411yescharacteristic
39C22×C8×C53—21no—
40C22×C8×C53—42no—
41C22×C8×C53C2×C411yescharacteristic
42C22×C8×C53C2×C411yescharacteristic
43C22×C8×C53D411yescharacteristic
44C22×C8×C53D411yescharacteristic
45C22×C8×C53D411yescharacteristic
46C22×C8×C53D411yescharacteristic
47#123×C53—21no—
48C22⋊Q8×C53D411yescharacteristic
49C22⋊Q8×C53Q811yescharacteristic
50#121×C53D411yescharacteristic
51#121×C53Q811yescharacteristic
52#120×C53—21no—
53C4×D4×C53—21no—
54C4×D4×C53—21no—
55#117×C53C2×C411yescharacteristic
56C4⋊C8×C53—21no—
57C4⋊C8×C53—21no—
58C22⋊C8×C53—21no—
59C22⋊C8×C53—21no—
60C22⋊C8×C53—21no—
61C22⋊C8×C53—21no—
62#97×C53—21no—
63C4×C8×C53—21no—
64C24×C53#4111yescharacteristic
65#41×C53—41no—
66#41×C53—41no—
67C2×Q8×C53C4⋊C411yescharacteristic
68C2×D4×C53—21no—
69C2×D4×C53—21no—
70C2×D4×C53—21no—
71C2×D4×C53C4⋊C411yescharacteristic
72C22×C4×C53—21no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×Q811yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×Dic53—42no—
82M16×C53—42no—
83M16×C53—41no—
84M16×C53—41no—
85C2×C8×C53—22no—
86C2×C8×C53—22no—
87C2×C8×C53—22no—
88C2×C8×C53—21no—
89C2×C8×C53—21no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C2×C8×C53—42no—
94C2×C8×C53C22⋊C411yescharacteristic
95C2×C8×C53C22⋊C411yescharacteristic
96C2×C8×C53C22⋊C411yescharacteristic
97C2×C8×C53C22⋊C411yescharacteristic
98C4⋊C4×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53C4⋊C411yescharacteristic
101C4⋊C4×C53C4⋊C411yescharacteristic
102C22⋊C4×C53—21no—
103C22⋊C4×C53—21no—
104C22⋊C4×C53—21no—
105C22⋊C4×C53—21no—
106C22⋊C4×C53—21no—
107C42×C53—21no—
108C22⋊Dic53—82no—
109C4⋊Dic53—42no—
110C4⋊Dic53—42no—
111#11719917—42no—
112#11719917—42no—
113C2×#88322—42no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53#11711yescharacteristic
120C23×C53#12111yescharacteristic
121C23×C53C22⋊Q811yescharacteristic
122Q8×C53—41no—
123D4×C53—41no—
124D4×C53—42no—
125D4×C53—42no—
126D4×C53—41no—
127D4×C53—41no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53C2×C22⋊C411yescharacteristic
143C2×C4×C53C2×C4⋊C411yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53C22⋊Q811yescharacteristic
146C2×Dic53—82no—
147C2×Dic53—82no—
148C2×Dic53—82no—
149C8×C53—21no—
150C8×C53—21no—
151C8×C53—21no—
152C8×C53—41no—
153C8×C53—41no—
154C8×C53#9912yes—
155#88322—82no—
156#8628—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—42no—
166C22×C53—42no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53#38011yescharacteristic
171C22×C53#47211yescharacteristic
172C22×C53#47311yescharacteristic
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53C2×#9911yescharacteristic
180C4×C53#41011yescharacteristic
181Dic53—162no—
182#2929D5311yescharacteristic
183#1810—531no—
184#1808—531no—
185#1513—532no—
186#1305—532no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—41no—
190C2×C53—41no—
191C2×C53—81no—
192C2×C53#179911yescharacteristic
193C2×C53#181411yescharacteristic
194C2×C53#181611yescharacteristic
195C2×#131Dic5311yescharacteristic, complemented
196C23×C8C2×D5311yescharacteristic
197#525C2×D5311yescharacteristic
198#435—21no—
199#435—21no—
200C8×D4—21no—
201C8×D4—21no—
202C2×#108—1061no—
203C2×#107—1061no—
204#424Dic5311yescharacteristic, complemented
205#408Dic5311yescharacteristic, complemented
206#407—532no—
207#407C2×D5311yescharacteristic
208#405Dic5311yescharacteristic, complemented
209#380—532no—
210#341—1061no—
211#340—1061no—
212#328—1062no—
213#327—1062no—
214C53#862811yescharacteristic, complemented, Sylow
215C2×#41C2×Dic5311yescharacteristic
216C23×C4C22×D5311yescharacteristic
217#131—42no—
218C2×M16—21no—
219C2×M16—1062no—
220C2×M16C2×Dic5311yescharacteristic
221C2×M16C2×Dic5311yescharacteristic
222C22×C8—21no—
223C22×C8—42no—
224C22×C8C2×Dic5311yescharacteristic
225C22×C8C2×Dic5311yescharacteristic
226C22×C8C22⋊D5311yescharacteristic
227C22×C8C22⋊D5311yescharacteristic
228C22×C8C22⋊D5311yescharacteristic
229C22×C8C22⋊D5311yescharacteristic
230#123—21no—
231C22⋊Q8Dic10611yescharacteristic
232C22⋊Q8D21211yescharacteristic
233#121Dic10611yescharacteristic
234#121D21211yescharacteristic
235#120—21no—
236C4×D4—21no—
237C4×D4—21no—
238#117C2×Dic5311yescharacteristic
239C2×C4⋊C4—1062no—
240C2×C22⋊C4—1062no—
241#108—2122no—
242#107—2122no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—1062no—
250C22⋊C8—1062no—
251#97—21no—
252C4×C8—21no—
253#95—1062no—
254C24#1171994511yescharacteristic
255#41—41no—
256#41—41no—
257C2×Q8C4⋊Dic5311yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4C4⋊Dic5311yescharacteristic
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—2122no—
266C22×C4C2×Dic10611yescharacteristic
267C22×C4C2×D21211yescharacteristic
268C22×C4#1171994511yescharacteristic
269C22×C4C22×Dic5311yescharacteristic
270C22×C4C2×C22⋊D5311yescharacteristic
271C22×C4C2×C22⋊D5311yescharacteristic
272M16—42no—
273M16—41no—
274M16—41no—
275M16—2122no—
276M16—2122no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—2122no—
287C2×C8C22⋊Dic5311yescharacteristic
288C2×C8C22⋊Dic5311yescharacteristic
289C2×C8C22⋊Dic5311yescharacteristic
290C2×C8C22⋊Dic5311yescharacteristic
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—2122no—
294C4⋊C4—2122no—
295C4⋊C4C4⋊Dic5311yescharacteristic
296C4⋊C4C4⋊Dic5311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—4242no—
303C42—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23#1313317611yescharacteristic
310C23#1313319111yescharacteristic
311C23#1313319611yescharacteristic
312Q8—41no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—41no—
317D4—41no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—4242no—
333C2×C4—4242no—
334C2×C4—4242no—
335C2×C4C2×C22⋊Dic5311yescharacteristic
336C2×C4C2×C4⋊Dic5311yescharacteristic
337C2×C4#1313319111yescharacteristic
338C2×C4#1313319611yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—41no—
343C8—41no—
344C8—4242no—
345C8#1313313212yes—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#1681048311yescharacteristic
360C22#1681089311yescharacteristic
361C22#1681089811yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—8482no—
369C4C2×#1313313211yescharacteristic
370C4#1681059711yescharacteristic
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—81no—
376C2#2027470311yescharacteristic
377C2#2027476911yescharacteristic
378C2#2027477911yescharacteristic
subgroupG/H
centerC22#16810483
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2929C2
socleC22×C53#380
prime radicalC53×#525C22
cosocleC23×C4×C53C23
hypercenter#2929D53
hypocenterC53#8628
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. #20274753C2semidirect
2. #20274744C2semidirect
4. #20272620C2semidirect
14. C2×#131×C53C4semidirect
21. C53×#424C4semidirect
22. C53×#408C4semidirect
24. C53×#405C4semidirect
195. C2×#131Dic53semidirect
204. #424Dic53semidirect
205. #408Dic53semidirect
208. #405Dic53semidirect
214. C53#8628semidirect

Maximal subgroups

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

subgroupindex
#202747532
#202747442
#202732582
#202726202
C53×#29292
#862853