Finite Group Databasebeta

← #27283243

Subgroups — #27283243

Order 13568.

468 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275660C211yesmaximal, characteristic, complemented
2#20275635C211yesmaximal, characteristic
3C4×#13133149C211yesmaximal, characteristic, complemented
4#20273214C211yesmaximal, characteristic, complemented
5#20273210C211yesmaximal, characteristic
6C8×#11719917C211yesmaximal, characteristic, complemented
7C53×#3167C211yesmaximal, characteristic
8#16810676C2211yescharacteristic
9C2×#13133149—21no—
10C2×#13133144—22no—
11C4×#11719917C2211yescharacteristic
12C4×C4⋊Dic53C2211yescharacteristic
13C4×C4⋊Dic53C2211yescharacteristic
14#16810386—21no—
15#16810382—22no—
16#16810360—22no—
17#16810357—22no—
18#16810345—21no—
19C8×#88322—21no—
20C53×#533C2211yescharacteristic
21C53×#532C2211yescharacteristic
22C53×#498C411yescharacteristic, complemented
23C53×#492C411yescharacteristic, complemented
24C53×#487C411yescharacteristic, complemented
25C53×#486C411yescharacteristic, complemented
26C53×#465—21no—
27C53×#465—21no—
28C4×SD16×C53—21no—
29C4×SD16×C53—21no—
30C2×C4×C8×C53C2211yescharacteristic
31C2×#11719917D412yes—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53—21no—
36#13133149—24no—
37C4⋊#88322—21no—
38#13133144—44no—
39C22⋊#88322D412yes—
40#13133128—21no—
41C4×#88322—21no—
42C4×#88322—21no—
43#13133124—21no—
44#13133124—21no—
45C2×#41×C53—21no—
46C2×SD16×C53—21no—
47C2×SD16×C53—21no—
48C22×C8×C53D411yescharacteristic
49C22×C8×C53D411yescharacteristic
50C4⋊Q8×C53C2×C411yescharacteristic
51C4⋊Q8×C53C2×C411yescharacteristic
52C4⋊Q8×C53C2×C411yescharacteristic
53#127×C53C2×C411yescharacteristic
54#125×C53—21no—
55#124×C53—21no—
56C22⋊Q8×C53—21no—
57C22⋊Q8×C53—21no—
58#121×C53—21no—
59#121×C53—21no—
60C4×Q8×C53—21no—
61C4×Q8×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64#117×C53—21no—
65C2×C42×C53C2311yescharacteristic
66#106×C53—21no—
67#106×C53—21no—
68#103×C53—21no—
69#103×C53—21no—
70#102×C53—21no—
71#102×C53—21no—
72C4×C8×C53C2×C411yescharacteristic
73C4×C8×C53C2×C411yescharacteristic
74C4×C8×C53D411yescharacteristic
75C4×C8×C53D411yescharacteristic
76#41×C53—42no—
77#41×C53—42no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×Dic53—41no—
90C22×Dic53—41no—
91SD16×C53—42no—
92SD16×C53—42no—
93C2×C8×C53—22no—
94C2×C8×C53—41no—
95C2×C8×C53C22⋊C411yescharacteristic
96C2×C8×C53C22⋊C411yescharacteristic
97C2×C8×C53C22⋊C411yescharacteristic
98C2×C8×C53C22⋊C411yescharacteristic
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109C22⋊C4×C53—41no—
110C42×C53—41no—
111C42×C53C22×C411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53#4111yescharacteristic
114C42×C53#4111yescharacteristic
115C4⋊Dic53—22no—
116C4⋊Dic53—22no—
117C4⋊Dic53—41no—
118C4⋊Dic53—41no—
119C4×Dic53—42no—
120C4×Dic53—42no—
121#11719917—22no—
122#11719917SD1614yescomplemented
123C2×#88322—22no—
124C2×#88322—22no—
125C23×C53—41no—
126C23×C53#12711yescharacteristic
127Q8×C53—41no—
128Q8×C53—41no—
129Q8×C53—42no—
130Q8×C53—81no—
131D4×C53—41no—
132D4×C53—41no—
133D4×C53—42no—
134D4×C53—81no—
135D4×C53—81no—
136D4×C53—81no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—82no—
144C2×C4×C53C2×C22⋊C411yescharacteristic
145C2×C4×C53C4×D411yescharacteristic
146C2×C4×C53C4×D411yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12411yescharacteristic
150C2×C4×C53C2×SD1612yes—
151C2×C4×C53#13512yes—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—81no—
155C2×Dic53—81no—
156C2×Dic53—81no—
157C2×Dic53—81no—
158C8×C53—21no—
159C8×C53—21no—
160C8×C53—21no—
161C8×C53#10412yescomplemented
162#88322—24no—
163#88322—42no—
164#9717—531nomaximal, Sylow
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53—81no—
170C22×C53#39011yescharacteristic
171C22×C53#49211yescharacteristic
172C22×C53#49511yescharacteristic
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—81no—
177C4×C53—81no—
178C4×C53—81no—
179C4×C53C2×#10411yescharacteristic
180C4×C53#42111yescharacteristic
181C4×C53C4×SD1612yes—
182C4×C53#46512yes—
183Dic53—82no—
184Dic53—82no—
185#3167D5311yescharacteristic
186#1986—531no—
187#1980—531no—
188C4×#106—531no—
189#1503—531no—
190#1502—531no—
191C8×M16—531no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—161no—
195C2×C53#196811yescharacteristic
196C2×C53#197611yescharacteristic
197C2×C53#198011yescharacteristic
198#533C2×D5311yescharacteristic
199#532C2×D5311yescharacteristic
200#498Dic5311yescharacteristic, complemented
201#492Dic5311yescharacteristic, complemented
202#487Dic5311yescharacteristic, complemented
203#486Dic5311yescharacteristic, complemented
204#465—21no—
205#465—21no—
206C4×SD16—21no—
207C4×SD16—21no—
208#431—531no—
209C2×#106—1061no—
210C2×#104—1062no—
211C4×M16—531no—
212C2×C4×C8C2×D5311yescharacteristic
213C4×C4⋊C4—531no—
214C4×C4⋊C4—531no—
215#340—1061no—
216#339—1062no—
217#328—1062no—
218#327—1062no—
219#322—1061no—
220C82—1061no—
221C53#971711yescharacteristic, complemented, Sylow
222C2×#41—21no—
223C2×SD16—21no—
224C2×SD16—21no—
225C2×M16—532no—
226C22×C8C22⋊D5311yescharacteristic
227C22×C8C22⋊D5311yescharacteristic
228C4⋊Q8C2×Dic5311yescharacteristic
229C4⋊Q8C2×Dic5311yescharacteristic
230C4⋊Q8C2×Dic5311yescharacteristic
231#127C2×Dic5311yescharacteristic
232#125—21no—
233#124—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236#121—21no—
237#121—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×D4—21no—
241C4×D4—21no—
242#117—21no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—1061no—
245C2×C42—1061no—
246C2×C42—1061no—
247C2×C42C22×D5311yescharacteristic
248#106—21no—
249#106—21no—
250#106—1064no—
251C4⋊C8—1061no—
252#104—2124no—
253#103—21no—
254#103—21no—
255#102—21no—
256#102—21no—
257C22⋊C8—532no—
258#97—1061no—
259C4×C8—1061no—
260C4×C8—1061no—
261C4×C8C2×Dic5311yescharacteristic
262C4×C8C2×Dic5311yescharacteristic
263C4×C8C22⋊D5311yescharacteristic
264C4×C8C22⋊D5311yescharacteristic
265#95—1061no—
266#95—1061no—
267#41—42no—
268#41—42no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—41no—
276C22×C4—41no—
277C22×C4—2121no—
278C22×C4—2121no—
279C22×C4D4×D5311yescharacteristic
280C22×C4D4×D5311yescharacteristic
281C22×C4C2×C22⋊D5311yescharacteristic
282SD16—42no—
283SD16—42no—
284M16—534no—
285M16—1062no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—1062no—
289C2×C8—1062no—
290C2×C8C22⋊Dic5311yescharacteristic
291C2×C8C22⋊Dic5311yescharacteristic
292C2×C8C22⋊Dic5311yescharacteristic
293C2×C8C22⋊Dic5311yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—1062no—
303C4⋊C4—1062no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C42—41no—
310C42—2122no—
311C42—2122no—
312C42#1171994711yescharacteristic
313C42#1171994711yescharacteristic
314C42C22×Dic5311yescharacteristic
315C42C2×C22⋊D5311yescharacteristic
316C23—41no—
317C23#1313321311yescharacteristic
318Q8—41no—
319Q8—41no—
320Q8—42no—
321Q8—81no—
322D4—41no—
323D4—41no—
324D4—42no—
325D4—81no—
326D4—81no—
327D4—81no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—82no—
335C2×C4—4241no—
336C2×C4—4241no—
337C2×C4—4241no—
338C2×C4—4241no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4C2×C22⋊Dic5311yescharacteristic
342C2×C4D4×Dic5311yescharacteristic
343C2×C4D4×Dic5311yescharacteristic
344C2×C4#1313318811yescharacteristic
345C2×C4#1313318811yescharacteristic
346C2×C4#1313320411yescharacteristic
347C2×C4SD16×D5312yes—
348C2×C4#1313323812yes—
349C8—21no—
350C8—21no—
351C8—21no—
352C8—1064no—
353C8—2122no—
354C8#1313314412yescomplemented
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#1681052611yescharacteristic
361C22#1681099411yescharacteristic
362C22#1681100411yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—4242no—
370C4—4242no—
371C4C2×#1313314411yescharacteristic
372C4#1681064011yescharacteristic
373C4SD16×Dic5312yes—
374C4#1681084912yes—
375C2—21no—
376C2—21no—
377C2—161no—
378C2#2027557111yescharacteristic
379C2#2027561411yescharacteristic
380C2#2027563511yescharacteristic
subgroupG/H
centerC2×C4#13133188
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3167C2
socleC22×C53#390
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#3167D53
hypocenterC53#9717
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #20275660C2semidirect
3. C4×#13133149C2semidirect
4. #20273214C2semidirect
6. C8×#11719917C2semidirect
22. C53×#498C4semidirect
23. C53×#492C4semidirect
24. C53×#487C4semidirect
25. C53×#486C4semidirect
122. #11719917SD16semidirect
161. C8×C53#104semidirect
200. #498Dic53semidirect
201. #492Dic53semidirect
202. #487Dic53semidirect
203. #486Dic53semidirect
221. C53#9717semidirect
354. C8#13133144semidirect

Maximal subgroups

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

subgroupindex
#202756602
#202756352
C4×#131331492
#202732142
#202732102
C8×#117199172
C53×#31672
#971753