Finite Group Databasebeta

← #27369313

Subgroups — #27369313

Order 13568.

674 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#20285856C211yesmaximal, characteristic, complemented
2#20284508C211yesmaximal, characteristic
3C2×#16810829C211yesmaximal, characteristic
4#20283738C211yesmaximal, characteristic
5#20274137C214yesmaximal, complemented
6#20274093C214yesmaximal, complemented
7#20273485C211yesmaximal, characteristic, complemented
8#20273386C211yesmaximal, characteristic, complemented
9C53×#1542C211yesmaximal, characteristic, complemented
10C22×#11719913—21no—
11C2×#13133217C2211yescharacteristic
12C2×C4×D212C2211yescharacteristic
13#16811019C2212yescomplemented
14#16810973C2212yescomplemented
15#16810951—22no—
16#16810920—22no—
17#16810910C2212yes—
18#16810908—22no—
19#16810900C2212yes—
20#16810829—22no—
21#16810829C2214yes—
22C8×D212—22no—
23#16810688C2214yes—
24#16810676C2211yescharacteristic
25#16810666—21no—
26#16810647C2211yescharacteristic
27#16810627—21no—
28C2×#13133142—21no—
29C22⋊C8×D53—22no—
30#16810373C2214yescomplemented
31#16810368C2214yescomplemented
32C53×#533C2211yescharacteristic
33C53×#431C2211yescharacteristic
34C2×C4⋊C8×C53C2211yescharacteristic
35C53×#333C2212yescomplemented
36C53×#332C2212yescomplemented
37C22×Dic106—21no—
38C22×D212D411yescharacteristic
39C22×C4×D53—21no—
40C2×#11719913—24no—
41C2×#11719913—44no—
42C2×#11719917—21no—
43C2×C8×D53—24no—
44#13133217—21no—
45#13133217—21no—
46#13133217—21no—
47#13133217C2312yes—
48C4×D212—21no—
49C4×D212—21no—
50C4×D212D412yes—
51C4×D212C2312yes—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53D411yescharacteristic
55C2×#11719921—21no—
56#13133150—22no—
57#13133148—22no—
58#13133148—24no—
59C4⋊#88322—21no—
60C4⋊#88322D412yes—
61C4⋊#88322C2312yes—
62#13133142—24no—
63#13133141—24no—
64#13133137—22no—
65#13133137—22no—
66C22⋊#88322D412yes—
67#13133129—24no—
68C4×#88322C2312yes—
69C22×C8×C53—21no—
70C4⋊Q8×C53C2311yescharacteristic
71C4⋊Q8×C53C2311yescharacteristic
72#125×C53—21no—
73C22⋊Q8×C53—21no—
74C22⋊Q8×C53—21no—
75C4×Q8×C53—21no—
76C4×Q8×C53—21no—
77#117×C53—21no—
78C2×C42×C53C2311yescharacteristic
79C4⋊C8×C53—21no—
80C4⋊C8×C53C2312yes—
81C4⋊C8×C53C2312yes—
82C22⋊C8×C53—21no—
83C4×C8×C53—21no—
84C23×D53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C22×Dic53—21no—
91C22×Dic53—41no—
92C2×D212—21no—
93C2×D212—22no—
94C2×D212SD1614yes—
95C2×D212C2×D412yes—
96C2×C4×D53—22no—
97C2×C4×D53—22no—
98C2×C4×D53—82no—
99C2×Dic106—22no—
100C2×Dic106—42no—
101C2×Dic106—42no—
102C2×Dic106—41no—
103C2×C8×C53—22no—
104C2×C8×C53—22no—
105C2×C8×C53—42no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C42×C53—41no—
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117C42×C53C2411yescharacteristic
118#11719921—22no—
119#11719921—22no—
120C4⋊Dic53—21no—
121C4⋊Dic53—22no—
122C4⋊Dic53—22no—
123C4⋊Dic53—41no—
124C4⋊Dic53—41no—
125C4⋊Dic53C2×D412yes—
126#11719917—42no—
127C2×#88322—22no—
128C2×#88322C2×D414yes—
129#11719913—48no—
130C8×D53—84no—
131C23×C53#12111yescharacteristic
132C22×D53—22no—
133C22×D53—24no—
134C22×D53—81no—
135C22×D53—81no—
136Q8×C53—42no—
137Q8×C53—81no—
138C2×C4×C53—22no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53C2×SD1612yes—
146C2×C4×C53#13512yes—
147C2×C4×C53C22×D411yescharacteristic
148C2×C4×C53C22×D411yescharacteristic
149C2×C4×C53C2×#4111yescharacteristic
150C2×Dic53—22no—
151C2×Dic53—42no—
152C2×Dic53—81no—
153C2×Dic53—82no—
154D212—22no—
155D212—24no—
156D212C2×SD1614yes—
157C4×D53—44no—
158C4×D53—82no—
159Dic106—42no—
160Dic106—42no—
161Dic106—82no—
162C8×C53—42no—
163C8×C53—42no—
164#88322—24no—
165#88322—42no—
166#19390—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53#46512yes—
170C22×C53C2×#12111yescharacteristic
171C2×D53—24no—
172C2×D53—42no—
173C2×D53—42no—
174C2×D53—84no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—81no—
179C4×C53—81no—
180C4×C53#46512yes—
181C4×C53C22×SD1611yescharacteristic
182C4×C53C2×#13511yescharacteristic
183C4×C53#57712yes—
184Dic53—42no—
185Dic53—82no—
186#3229—531no—
187#3077—531no—
188C2×#461—531no—
189#2972—531no—
190#1679—534no—
191#1671—534no—
192#1570—531no—
193#1546—531no—
194#1542D5311yescharacteristic, complemented
195C2×C53—22no—
196C2×C53#304211yescharacteristic
197C2×C53C2×#46511yescharacteristic
198C2×C53#306911yescharacteristic
199D53—44no—
200C22×SD16—1061no—
201#533C2×D5311yescharacteristic
202C2×C4⋊Q8—531no—
203C2×C4×D4—531no—
204#498—532no—
205#487—532no—
206#483—1062no—
207#478—1062no—
208#476—532no—
209#475—1062no—
210#474—532no—
211#461—534no—
212#461—1062no—
213C8×D4—534no—
214C8×D4—1062no—
215#431—531no—
216#431C2×D5311yescharacteristic
217#428—1061no—
218#423—531no—
219C2×C4⋊C8C2×D5311yescharacteristic
220#418—1061no—
221C2×#103—1061no—
222C2×C22⋊C8—1062no—
223#334—534no—
224#333C2×D5312yescomplemented
225#332—534no—
226#332C2×D5312yescomplemented
227C53#1939011yescharacteristic, complemented, Sylow
228C22×Q8—1061no—
229C22×D4—531no—
230C23×C4—1061no—
231C2×SD16—1064no—
232C2×SD16—2124no—
233C2×M16—1061no—
234C22×C8—21no—
235C22×C8—1064no—
236C4⋊Q8—532no—
237C4⋊Q8—1061no—
238C4⋊Q8—1061no—
239C4⋊Q8—1061no—
240C4⋊Q8C22×D5311yescharacteristic
241C4⋊Q8C22×D5311yescharacteristic
242#125—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×D4—532no—
248C4×D4—532no—
249C4×D4—1061no—
250C4×D4—1061no—
251#117—21no—
252C2×C4⋊C4—531no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4—1061no—
255C2×C22⋊C4—1061no—
256C2×C42C22×D5311yescharacteristic
257#107—1062no—
258#106—1062no—
259#106—1064no—
260C4⋊C8—21no—
261C4⋊C8—532no—
262C4⋊C8—532no—
263C4⋊C8—1061no—
264C4⋊C8C22×D5312yes—
265C4⋊C8C22×D5312yes—
266#103—1064no—
267#103—1064no—
268#102—1062no—
269#102—1062no—
270C22⋊C8—21no—
271C22⋊C8—532no—
272C22⋊C8—1064no—
273C4×C8—21no—
274C4×C8—532no—
275C24—1061no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—1062no—
279C2×Q8—2122no—
280C2×Q8—2122no—
281C2×Q8—2121no—
282C2×D4—534no—
283C2×D4—532no—
284C2×D4—1061no—
285C2×D4—1062no—
286C22×C4—1061no—
287C22×C4—1062no—
288C22×C4—1062no—
289C22×C4—4242no—
290C22×C4—2121no—
291C22×C4#1171994511yescharacteristic
292C22×C4D4×D5311yescharacteristic
293C22×C4D4×D5311yescharacteristic
294SD16—2128no—
295M16—2122no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—42no—
299C2×C8—534no—
300C2×C8—1062no—
301C2×C8—4244no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—532no—
309C4⋊C4—1061no—
310C4⋊C4—1062no—
311C4⋊C4—1062no—
312C4⋊C4—2121no—
313C4⋊C4—2121no—
314C22⋊C4—41no—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C42—41no—
318C42#1171994511yescharacteristic
319C42D4×D5311yescharacteristic
320C42D4×D5311yescharacteristic
321C42C23×D5311yescharacteristic
322C23—1064no—
323C23—1062no—
324C23—4241no—
325C23—4241no—
326C23#1313319011yescharacteristic
327Q8—42no—
328Q8—81no—
329Q8—4242no—
330Q8—2122no—
331Q8—2122no—
332D4—534no—
333D4—1064no—
334D4—1062no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—1062no—
340C2×C4—4242no—
341C2×C4—4242no—
342C2×C4—4241no—
343C2×C4—2122no—
344C2×C4—2124no—
345C2×C4#1313319011yescharacteristic
346C2×C4#1313319011yescharacteristic
347C2×C4#1313319011yescharacteristic
348C2×C4SD16×D5312yes—
349C2×C4#1313324012yes—
350C2×C4C2×D4×D5311yescharacteristic
351C2×C4C2×D4×D5311yescharacteristic
352C2×C4C2×#1171994511yescharacteristic
353C8—42no—
354C8—42no—
355C8—1064no—
356C8—2122no—
357C22—21no—
358C22—21no—
359C22—1064no—
360C22—4244no—
361C22—2122no—
362C22—2122no—
363C22#1681085512yes—
364C22C2×#1313319011yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—81no—
369C4—81no—
370C4—4242no—
371C4—2122no—
372C4#1681085512yes—
373C4C2×SD16×D5311yescharacteristic
374C4C2×#1313324011yescharacteristic
375C4#1681143012yes—
376C2—22no—
377C2—2124no—
378C2#2028423011yescharacteristic
379C2C2×#1681085511yescharacteristic
380C2#2028445511yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1542C2
socleC22×C53C2×#121
prime radicalC22×D212D4
hypercenter#1542D53
hypocenterC53#19390
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
1. #20285856C2semidirect
5. #20274137C2semidirect
6. #20274093C2semidirect
7. #20273485C2semidirect
8. #20273386C2semidirect
9. C53×#1542C2semidirect
13. #16811019C22semidirect
14. #16810973C22semidirect
30. #16810373C22semidirect
31. #16810368C22semidirect
35. C53×#333C22semidirect
36. C53×#332C22semidirect
194. #1542D53semidirect
224. #333C2×D53semidirect
226. #332C2×D53semidirect
227. C53#19390semidirect

Maximal subgroups

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

subgroupindex
#202858562
#202845082
C2×#168108292
#202837382
#202741372
#202740932
#202734852
#202733862
C53×#15422
#1939053