Finite Group Databasebeta

← #27276994

Subgroups — #27276994

Order 13568.

524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274744C211yesmaximal, characteristic, complemented
2#20274740C211yesmaximal, characteristic, complemented
3#20274703C212yesmaximal
4#20272583C212yesmaximal, complemented
5C2×C53×#435C211yesmaximal, characteristic
6C2×#13133151—21no—
7C2×#13133149—21no—
8C2×#13133132—22no—
9#16810483C2212yes—
10#16810483C2212yes—
11#16810390—22no—
12#16810386—21no—
13#16810386—21no—
14C22×M16×C53C411yescharacteristic, complemented
15C23×C8×C53C2211yescharacteristic
16C2×C4×D4×C53C2211yescharacteristic
17C53×#435—21no—
18C53×#435—21no—
19C53×#435—21no—
20C53×#435—21no—
21C2×C4⋊C8×C53C411yescharacteristic, complemented
22C2×C22⋊C8×C53C411yescharacteristic, complemented
23C2×C22⋊C8×C53C2211yescharacteristic
24C2×#97×C53C411yescharacteristic, complemented
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53—22no—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—22no—
29#13133151—42no—
30#13133149—42no—
31#13133132—44no—
32#13133124—22no—
33#13133124—22no—
34C22×D4×C53D411yescharacteristic
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53C2311yescharacteristic
37C2×M16×C53—21no—
38C2×M16×C53—21no—
39C2×M16×C53—42no—
40C22×C8×C53—42no—
41C22×C8×C53C2×C411yescharacteristic
42C22×C8×C53C2×C411yescharacteristic
43C22×C8×C53C2×C411yescharacteristic
44C22×C8×C53C2×C411yescharacteristic
45C22×C8×C53D411yescharacteristic
46C22×C8×C53D411yescharacteristic
47C22×C8×C53D411yescharacteristic
48C22×C8×C53D411yescharacteristic
49C4×D4×C53—21no—
50C4×D4×C53—21no—
51C4×D4×C53—21no—
52C4×D4×C53—21no—
53C2×C4⋊C4×C53Q811yescharacteristic
54C2×C22⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53Q811yescharacteristic
56C2×C42×C53C2×C411yescharacteristic
57C4⋊C8×C53—21no—
58C4⋊C8×C53—21no—
59C22⋊C8×C53—21no—
60C22⋊C8×C53—21no—
61C22⋊C8×C53—21no—
62C22⋊C8×C53—21no—
63#97×C53—21no—
64#97×C53—21no—
65C24×C53C4⋊C411yescharacteristic
66C24×C53C2×D411yescharacteristic
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53C4⋊C411yescharacteristic
78C22×C4×C53C4⋊C411yescharacteristic
79C22×C4×C53C4⋊C411yescharacteristic
80C22×C4×C53C22×C411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×Q811yescharacteristic
84C22×C4×C53#4111yescharacteristic
85C22×C4×C53#4111yescharacteristic
86C22×Dic53—42no—
87C22×Dic53—42no—
88M16×C53—42no—
89M16×C53—42no—
90C2×C8×C53—22no—
91C2×C8×C53—22no—
92C2×C8×C53—22no—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C2×C8×C53C22⋊C411yescharacteristic
101C2×C8×C53C22⋊C411yescharacteristic
102C2×C8×C53C22⋊C411yescharacteristic
103C2×C8×C53C22⋊C411yescharacteristic
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C22⋊C4×C53—21no—
107C22⋊C4×C53—21no—
108C22⋊C4×C53—21no—
109C22⋊C4×C53—21no—
110C42×C53—21no—
111C42×C53—21no—
112C22⋊Dic53—42no—
113C22⋊Dic53—42no—
114C22⋊Dic53—42no—
115C22⋊Dic53—42no—
116C4⋊Dic53—42no—
117C4⋊Dic53—42no—
118C4⋊Dic53—42no—
119C4⋊Dic53—42no—
120C23×C53—21no—
121C23×C53—22no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53C2×C4⋊C411yescharacteristic
127C23×C53#12111yescharacteristic
128C23×C53#12111yescharacteristic
129C23×C53AGL(1,ℤ/8)12yes—
130C23×C53#13712yes—
131D4×C53—42no—
132D4×C53—41no—
133D4×C53—41no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—22no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53C2×C22⋊C411yescharacteristic
153C2×C4×C53#11711yescharacteristic
154C2×C4×C53C22⋊Q811yescharacteristic
155C2×C4×C53C22⋊Q811yescharacteristic
156C2×Dic53—82no—
157C2×Dic53—82no—
158C2×Dic53—82no—
159C2×Dic53—82no—
160C2×Dic53—82no—
161C2×Dic53—82no—
162C8×C53—21no—
163C8×C53—21no—
164C8×C53—21no—
165C8×C53—41no—
166C8×C53—41no—
167C8×C53#9912yescomplemented
168#8610—531nomaximal, Sylow
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—82no—
181C22×C53#38011yescharacteristic
182C22×C53#42812yes—
183C22×C53#46812yes—
184C22×C53#46911yescharacteristic
185C22×C53#47011yescharacteristic
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—41no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53C2×#9911yescharacteristic
193C4×C53#41011yescharacteristic
194Dic53—162no—
195Dic53—162no—
196C2×#435D5311yescharacteristic
197#1808—531no—
198#1807—531no—
199#1799—532no—
200#1289—532no—
201C2×C53—21no—
202C2×C53—22no—
203C2×C53—22no—
204C2×C53—21no—
205C2×C53—81no—
206C2×C53#179911yescharacteristic
207C2×C53#180911yescharacteristic
208C2×C53#181211yescharacteristic
209C22×M16Dic5311yescharacteristic, complemented
210C23×C8C2×D5311yescharacteristic
211C2×C4×D4C2×D5311yescharacteristic
212#435—21no—
213#435—21no—
214#435—21no—
215#435—21no—
216C2×#107—1061no—
217C2×#106—1061no—
218C2×C4⋊C8Dic5311yescharacteristic, complemented
219C2×#99—1062no—
220C2×C22⋊C8Dic5311yescharacteristic, complemented
221C2×C22⋊C8C2×D5311yescharacteristic
222C2×#97Dic5311yescharacteristic, complemented
223#380—532no—
224#380—532no—
225#342—1062no—
226#340—1061no—
227#340—1061no—
228C53#861011yescharacteristic, complemented, Sylow
229C22×D4D21211yescharacteristic
230C23×C4C2×Dic5311yescharacteristic
231C23×C4C22×D5311yescharacteristic
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—42no—
235C22×C8—42no—
236C22×C8C2×Dic5311yescharacteristic
237C22×C8C2×Dic5311yescharacteristic
238C22×C8C2×Dic5311yescharacteristic
239C22×C8C2×Dic5311yescharacteristic
240C22×C8C22⋊D5311yescharacteristic
241C22×C8C22⋊D5311yescharacteristic
242C22×C8C22⋊D5311yescharacteristic
243C22×C8C22⋊D5311yescharacteristic
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C2×C4⋊C4—1062no—
249C2×C4⋊C4—1062no—
250C2×C4⋊C4Dic10611yescharacteristic
251C2×C22⋊C4—1062no—
252C2×C22⋊C4—1062no—
253C2×C22⋊C4Dic10611yescharacteristic
254C2×C22⋊C4D21211yescharacteristic
255C2×C42C2×Dic5311yescharacteristic
256#107—2122no—
257#106—2122no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260#99—2124no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265#97—21no—
266#97—21no—
267#95—1062no—
268#95—1062no—
269C24C4⋊Dic5311yescharacteristic
270C24C2×D21211yescharacteristic
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—2122no—
282C22×C4—2122no—
283C22×C4C4⋊Dic5311yescharacteristic
284C22×C4C4⋊Dic5311yescharacteristic
285C22×C4C4⋊Dic5311yescharacteristic
286C22×C4C2×Dic10611yescharacteristic
287C22×C4#1171994511yescharacteristic
288C22×C4#1171994511yescharacteristic
289C22×C4C22×Dic5311yescharacteristic
290C22×C4C2×C22⋊D5311yescharacteristic
291C22×C4C2×C22⋊D5311yescharacteristic
292M16—42no—
293M16—42no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8C22⋊Dic5311yescharacteristic
305C2×C8C22⋊Dic5311yescharacteristic
306C2×C8C22⋊Dic5311yescharacteristic
307C2×C8C22⋊Dic5311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—2122no—
311C4⋊C4—2122no—
312C4⋊C4—2122no—
313C4⋊C4—2122no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—2122no—
319C22⋊C4—2122no—
320C22⋊C4—2122no—
321C22⋊C4—2122no—
322C42—21no—
323C42—21no—
324C23—21no—
325C23—22no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23C2×C4⋊Dic5311yescharacteristic
331C23#1313319111yescharacteristic
332C23#1313319111yescharacteristic
333C23#1313324512yes—
334C23#1313325012yes—
335D4—42no—
336D4—41no—
337D4—41no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—4242no—
357C2×C4—4242no—
358C2×C4—4242no—
359C2×C4—4242no—
360C2×C4—4242no—
361C2×C4—4242no—
362C2×C4C2×C22⋊Dic5311yescharacteristic
363C2×C4#1313317611yescharacteristic
364C2×C4#1313319611yescharacteristic
365C2×C4#1313319611yescharacteristic
366C8—21no—
367C8—21no—
368C8—21no—
369C8—41no—
370C8—41no—
371C8#1313313212yescomplemented
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—42no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—82no—
384C22#1681048311yescharacteristic
385C22#1681066512yes—
386C22#1681086912yes—
387C22#1681087611yescharacteristic
388C22#1681088111yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—8482no—
396C4—8482no—
397C4C2×#1313313211yescharacteristic
398C4#1681059711yescharacteristic
399C2—21no—
400C2—22no—
401C2—22no—
402C2—21no—
403C2—81no—
404C2#2027470311yescharacteristic
405C2#2027474811yescharacteristic
406C2#2027476111yescharacteristic
subgroupG/H
centerC22#16810483
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#435C2
socleC22×C53#380
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenterC2×#435D53
hypocenterC53#8610
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#435

Complements

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

NH ≅ G/Ntype
1. #20274744C2semidirect
2. #20274740C2semidirect
4. #20272583C2semidirect
14. C22×M16×C53C4semidirect
21. C2×C4⋊C8×C53C4semidirect
22. C2×C22⋊C8×C53C4semidirect
24. C2×#97×C53C4semidirect
167. C8×C53#99semidirect
209. C22×M16Dic53semidirect
218. C2×C4⋊C8Dic53semidirect
220. C2×C22⋊C8Dic53semidirect
222. C2×#97Dic53semidirect
228. C53#8610semidirect
371. C8#13133132semidirect

Maximal subgroups

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

subgroupindex
#202747442
#202747402
#202747032
#202725832
C2×C53×#4352
#861053