Finite Group Databasebeta

← #27590193

Subgroups — #27590193

Order 13568.

4086 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811170C216yesmaximal, complemented
2#20283112C211yesmaximal, characteristic, complemented
3#20283109C211yesmaximal, characteristic, complemented
4#20283069C216yesmaximal, complemented
5#20283061C211yesmaximal, characteristic, complemented
6#20283059C216yesmaximal, complemented
7#20283057C211yesmaximal, characteristic, complemented
8C4×#13133214C211yesmaximal, characteristic, complemented
9C4×#13133213C216yesmaximal, complemented
10C43×D53C211yesmaximal, characteristic, complemented
11C4×#127×C53C211yesmaximal, characteristic, complemented
12C22×#11719945C2214yes—
13#16811170—248no—
14C2×#13133216C2216yes—
15C2×#13133214C2211yescharacteristic
16C2×#13133213C2216yes—
17C2×#13133213C2216yes—
18C2×#13133203C22112yes—
19C2×#13133192C2216yes—
20C2×#13133190C22112yes—
21C2×#13133188C2216yes—
22C2×C4×D212C2216yes—
23C2×C4×C22⋊D53C22112yes—
24C2×C42×D53C2211yescharacteristic
25C2×C42×D53C2213yes—
26C2×C42×D53C2213yes—
27#16810526C2216yes—
28#16810525C2216yes—
29#16810524C2216yes—
30#16810523C22124yes—
31#16810493C2211yescharacteristic
32#16810492C2216yes—
33C42×Dic53C2213yes—
34C42×Dic53C2213yes—
35C42×Dic53C2211yescharacteristic
36C2×#127×C53C2211yescharacteristic
37C2×C4×D4×C53C2216yes—
38C53×#390C2216yes—
39C53×#384C2211yescharacteristic
40C43×C53C2211yescharacteristic
41C2×#11719945—248no—
42C22×Dic106C2314yes—
43C22×D212C2316yes—
44C22×D212C2314yes—
45C22×C22⋊D53C23112yes—
46C22×C22⋊D53C2314yes—
47C22×C22⋊D53C2314yes—
48C22×C4×D53C2314yes—
49C22×C4×D53C2313yes—
50C22×C4×D53C2313yes—
51C22×C4×D53C2311yescharacteristic
52#13133216—224no—
53#13133214—24no—
54#13133213—224no—
55#13133213—224no—
56#13133203—248no—
57#13133192—224no—
58#13133190—248no—
59#13133188—224no—
60C4×D212—224no—
61C4×C22⋊D53—248no—
62C2×C4⋊Dic53C2316yes—
63C2×#11719919C23112yes—
64C2×C22⋊Dic53C23112yes—
65C2×#11719921C23112yes—
66C2×#11719921C23112yes—
67C2×C4×Dic53C2316yes—
68C2×C4×Dic53C2313yes—
69C2×C4×Dic53C2313yes—
70C2×C4×Dic53C2313yes—
71C2×C4×Dic53C2313yes—
72C2×C4×Dic53C2313yes—
73C42×D53—24no—
74C42×D53—212no—
75C42×D53D4124yescomplemented
76C22×D4×C53C2316yes—
77C23×C4×C53C2314yes—
78#127×C53—24no—
79C4×D4×C53—224no—
80C2×C4⋊C4×C53C2316yes—
81C2×C22⋊C4×C53C23112yes—
82C2×C42×C53C2313yes—
83C2×C42×C53C2313yes—
84C2×C42×C53C2311yescharacteristic
85C24×C53C2414yes—
86C23×D53C2414yes—
87C23×D53C2411yescharacteristic
88C2×D4×C53—212no—
89C2×D4×C53—212no—
90C2×D4×C53—212no—
91C22×C4×C53—224no—
92C22×C4×C53C2414yes—
93C22×C4×C53C2413yes—
94C22×C4×C53C2411yescharacteristic
95C22×C4×C53C2413yes—
96C2×C22⋊D53—224no—
97C2×C22⋊D53—224no—
98C2×C22⋊D53—224no—
99C2×C22⋊D53—248no—
100C22×Dic53C2414yes—
101C22×Dic53C2413yes—
102C22×Dic53C2411yescharacteristic
103C22×Dic53C2413yes—
104#11719945—464no—
105C2×D212—224no—
106C2×D212—212no—
107C2×D212—212no—
108C2×D212—212no—
109C2×C4×D53—26no—
110C2×C4×D53—26no—
111C2×C4×D53—212no—
112C2×C4×D53—224no—
113C2×C4×D53C2×D4112yes—
114C2×C4×D53C2×D4112yes—
115C2×C4×D53C2×D4112yes—
116C2×Dic106—224no—
117C4⋊C4×C53—212no—
118C22⋊C4×C53—224no—
119C42×C53—26no—
120C42×C53C2×D4112yes—
121C42×C53#4114yescomplemented
122C22⋊Dic53—224no—
123#11719921—224no—
124#11719921—224no—
125C4⋊Dic53—212no—
126#11719919—224no—
127C4×Dic53—26no—
128C4×Dic53—212no—
129C4×Dic53C2×D4112yes—
130C4×Dic53C2×D4112yes—
131C4×Dic53C2×D4112yes—
132C4×Dic53#41112yescomplemented
133C23×C53—212no—
134C23×C53—212no—
135C23×C53—24no—
136C23×C53C2511yescharacteristic
137C22×D53—23no—
138C22×D53—21no—
139C22×D53—212no—
140C22×D53—24no—
141C22×D53—212no—
142C22×D53C22×D416yes—
143D4×C53—424no—
144C2×C4×C53—26no—
145C2×C4×C53—212no—
146C2×C4×C53—416no—
147C2×C4×C53C22×D416yes—
148C2×C4×C53C22×D416yes—
149C2×C4×C53C22×D416yes—
150C2×C4×C53C2×#4116yes—
151C2×C4×C53C2×#4116yes—
152C22⋊D53—416no—
153C22⋊D53—416no—
154C22⋊D53—448no—
155C2×Dic53—23no—
156C2×Dic53—23no—
157C2×Dic53—212no—
158C2×Dic53C22×D416yes—
159C2×Dic53C22×D416yes—
160C2×Dic53C22×D416yes—
161C2×Dic53C2×#41112yes—
162D212—416no—
163D212—424no—
164C4×D53—224no—
165C4×D53—224no—
166C4×D53—416no—
167C4×D53#127116yescomplemented
168Dic106—416no—
169#36644—531nomaximal, Sylow
170C22×C53—44no—
171C22×C53—412no—
172C22×C53—412no—
173C22×C53C23×D413yes—
174C22×C53C22×#4113yes—
175C22×C53C22×#4111yescharacteristic
176C2×D53—26no—
177C2×D53—26no—
178C2×D53—44no—
179C2×D53—412no—
180C2×D53—412no—
181C2×D53C2×#12714yes—
182C4×C53—26no—
183C4×C53—44no—
184C4×C53C2×#12714yes—
185C4×C53#532112yescomplemented
186Dic53—26no—
187Dic53—44no—
188Dic53C2×#12714yes—
189Dic53#532112yescomplemented
190C2×#532—536no—
191#2868—531no—
192#2867—531no—
193#2858—536no—
194#2856—536no—
195#2856—531no—
196#2856—531no—
197C4×#127—531no—
198C4×#127—536no—
199C4×#127D5311yescharacteristic, complemented
200C2×C43—531no—
201C2×C53—84no—
202C2×C53C22×#12711yescharacteristic
203C2×C53C2×#53213yes—
204C2×C53C2×#53213yes—
205D53—24no—
206D53—84no—
207C22×#41—534no—
208#532—10648no—
209C2×C4⋊Q8—536no—
210C2×#127—536no—
211C2×#127—536no—
212C2×#127—531no—
213C2×#127C2×D5311yescharacteristic
214C2×#124—5312no—
215C2×#121—536no—
216C2×#121—536no—
217C2×#121—5312no—
218C2×C4×D4—5312no—
219C2×C4×D4—536no—
220C2×C4×D4C2×D5316yes—
221C22×C42—533no—
222C22×C42—533no—
223C22×C42—531no—
224#390—536no—
225#390—536no—
226#390—5324no—
227#390—536no—
228#390C2×D5316yes—
229#384—536no—
230#384—531no—
231#384C2×D5311yescharacteristic
232C43—533no—
233C43—533no—
234C43—531no—
235C43C2×D5311yescharacteristic
236C53#3664411yescharacteristic, complemented, Sylow
237C2×#41—10648no—
238C22×Q8—534no—
239C22×D4—5312no—
240C22×D4—534no—
241C22×D4—534no—
242C22×D4—536no—
243C22×D4—534no—
244C22×D4C22×D5316yes—
245C23×C4—531no—
246C23×C4—533no—
247C23×C4—533no—
248C23×C4—534no—
249C23×C4C22×D5314yes—
250C4⋊Q8—10624no—
251#127—24no—
252#127—10624no—
253#127—10624no—
254#127—1064no—
255#124—10648no—
256#121—10648no—
257#121—10624no—
258#121—10624no—
259C4×D4—224no—
260C4×D4—10648no—
261C4×D4—10624no—
262C2×C4⋊C4—5312no—
263C2×C4⋊C4—536no—
264C2×C4⋊C4C22×D5316yes—
265C2×C22⋊C4—5312no—
266C2×C22⋊C4—5312no—
267C2×C22⋊C4—5312no—
268C2×C22⋊C4C22×D53112yes—
269C2×C42—533no—
270C2×C42—533no—
271C2×C42—533no—
272C2×C42—533no—
273C2×C42—5324no—
274C2×C42—536no—
275C2×C42—533no—
276C2×C42—10612no—
277C2×C42—1064no—
278C2×C42C22×D5313yes—
279C2×C42C22×D5311yescharacteristic
280C2×C42C22×D5313yes—
281C24—531no—
282C24—534no—
283C24C23×D5314yes—
284#41—21264no—
285C2×Q8—10624no—
286C2×D4—212no—
287C2×D4—212no—
288C2×D4—212no—
289C2×D4—10624no—
290C2×D4—10624no—
291C2×D4—10624no—
292C2×D4—10648no—
293C2×D4—10624no—
294C2×D4—10612no—
295C2×D4—10612no—
296C2×D4—10612no—
297C22×C4—224no—
298C22×C4—533no—
299C22×C4—531no—
300C22×C4—5312no—
301C22×C4—533no—
302C22×C4—5312no—
303C22×C4—5312no—
304C22×C4—534no—
305C22×C4—10612no—
306C22×C4—1066no—
307C22×C4—1066no—
308C22×C4—10624no—
309C22×C4C23×D5314yes—
310C22×C4C23×D5313yes—
311C22×C4C23×D5313yes—
312C22×C4C23×D5311yescharacteristic
313C4⋊C4—212no—
314C4⋊C4—10624no—
315C4⋊C4—10612no—
316C22⋊C4—224no—
317C22⋊C4—10624no—
318C22⋊C4—10624no—
319C22⋊C4—10624no—
320C42—26no—
321C42—5312no—
322C42—5312no—
323C42—5312no—
324C42—5312no—
325C42—10612no—
326C42—1066no—
327C42#1171994514yescomplemented
328C42D4×D53112yes—
329C23—212no—
330C23—212no—
331C23—24no—
332C23—536no—
333C23—1063no—
334C23—1061no—
335C23—10612no—
336C23—10612no—
337C23—1064no—
338C23C24×D5311yescharacteristic
339Q8—21216no—
340D4—424no—
341D4—21216no—
342D4—21248no—
343D4—21216no—
344D4—21216no—
345D4—21224no—
346C2×C4—26no—
347C2×C4—212no—
348C2×C4—416no—
349C2×C4—536no—
350C2×C4—536no—
351C2×C4—5312no—
352C2×C4—5316no—
353C2×C4—536no—
354C2×C4—10624no—
355C2×C4—1063no—
356C2×C4—1063no—
357C2×C4—10624no—
358C2×C4—10612no—
359C2×C4—21216no—
360C2×C4C2×D4×D5316yes—
361C2×C4C2×D4×D5316yes—
362C2×C4C2×D4×D5316yes—
363C2×C4#41×D5316yes—
364C2×C4C2×#1171994516yes—
365C22—44no—
366C22—412no—
367C22—412no—
368C22—534no—
369C22—1066no—
370C22—1066no—
371C22—2124no—
372C22—21212no—
373C22—21212no—
374C22C22×D4×D5313yes—
375C22C2×#41×D5313yes—
376C22C22×#1171994511yescharacteristic
377C4—26no—
378C4—44no—
379C4—534no—
380C4—5312no—
381C4—1066no—
382C4—2124no—
383C4#127×D5314yes—
384C4#16811170112yescomplemented
385C2—84no—
386C2—1064no—
387C2—4244no—
388C2C2×#127×D5311yescharacteristic
389C2#532×D5313yes—
390C2C2×#1681117013yes—
subgroupG/H
centerC22×C4C23×D53
commutator, socle, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC4×#127×C53C2
hypercenterC4×#127D53
hypocenterC53#36644
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
1. C2×#16811170C2semidirect
2. #20283112C2semidirect
3. #20283109C2semidirect
4. #20283069C2semidirect
5. #20283061C2semidirect
6. #20283059C2semidirect
7. #20283057C2semidirect
8. C4×#13133214C2semidirect
9. C4×#13133213C2semidirect
10. C43×D53C2semidirect
11. C4×#127×C53C2semidirect
75. C42×D53D4semidirect
121. C42×C53#41semidirect
132. C4×Dic53#41semidirect
167. C4×D53#127semidirect
185. C4×C53#532semidirect
189. Dic53#532semidirect
199. C4×#127D53semidirect
236. C53#36644semidirect
327. C42#11719945semidirect
384. C4#16811170semidirect

Maximal subgroups

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

subgroupindex
C2×#168111702
#202831122
#202831092
#202830692
#202830612
#202830592
#202830572
C4×#131332142
C4×#131332132
C43×D532
C4×#127×C532
#3664453