Finite Group Databasebeta

← #27358015

Subgroups — #27358015

Order 13568.

940 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285487C212yesmaximal
2#20280528C211yesmaximal, characteristic
3#20273757C214yesmaximal, complemented
4#20273689C214yesmaximal, complemented
5#20273287C212yesmaximal
6C2×#16810349C211yesmaximal, characteristic
7C2×C53×#544C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810958—22no—
10#16810956C2214yes—
11#16810862—24no—
12#16810779—22no—
13#16810778C2214yes—
14C2×C22⋊#88322C2212yes—
15#16810562C2212yes—
16#16810556—22no—
17#16810543C2212yes—
18#16810496—21no—
19#16810486—21no—
20#16810483C2211yescharacteristic
21#16810360C2218yes—
22#16810349C2214yes—
23C53×#544—24no—
24C53×#544—22no—
25C53×#544C2214yes—
26C2×C4⋊Q8×C53—22no—
27C2×#125×C53—22no—
28C2×C22⋊Q8×C53—21no—
29C2×C22⋊Q8×C53C2212yes—
30C2×#117×C53—21no—
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×Dic106C2311yescharacteristic
33C2×#11719925—24no—
34C22×#88322—22no—
35#13133196—22no—
36#13133196—22no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53C2311yescharacteristic
39C2×#11719919—21no—
40C2×#11719919—22no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43#13133150—24no—
44#13133141—24no—
45#13133140—24no—
46C22⋊#88322—22no—
47C22⋊#88322C2314yes—
48#13133124—22no—
49#13133124—22no—
50#13133124—22no—
51#13133124C2312yes—
52C22×Q8×C53—21no—
53C22×Q8×C53D412yes—
54C23×C4×C53D411yescharacteristic
55C23×C4×C53D411yescharacteristic
56C23×C4×C53C2311yescharacteristic
57C4⋊Q8×C53—28no—
58C4⋊Q8×C53—28no—
59#125×C53—28no—
60#125×C53—28no—
61C22⋊Q8×C53—24no—
62C22⋊Q8×C53—24no—
63C22⋊Q8×C53—22no—
64C22⋊Q8×C53—22no—
65C22⋊Q8×C53D414yes—
66C22⋊Q8×C53C2314yes—
67#117×C53—24no—
68#117×C53—24no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—22no—
73C2×C4⋊C4×C53—22no—
74C2×C4⋊C4×C53—22no—
75C2×C4⋊C4×C53—22no—
76C2×C4⋊C4×C53D412yes—
77C2×C4⋊C4×C53D412yes—
78C2×C4⋊C4×C53C2312yes—
79C2×C22⋊C4×C53—22no—
80C2×C42×C53—22no—
81C24×C53C2×D411yescharacteristic
82C2×Q8×C53—24no—
83C2×Q8×C53—22no—
84C2×Q8×C53—82no—
85C2×Q8×C53—82no—
86C2×Q8×C53C2×D414yes—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—22no—
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—81no—
95C22×C4×C53—81no—
96C22×C4×C53Q1614yescomplemented
97C22×C4×C53C2×D412yes—
98C22×C4×C53C2×D412yes—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D412yes—
101C22×C4×C53C2×D412yes—
102C22×C4×C53C2411yescharacteristic
103C22×Dic53—21no—
104C22×Dic53—22no—
105C22×Dic53—21no—
106C2×Dic106—22no—
107C2×Dic106—81no—
108C4⋊C4×C53—24no—
109C4⋊C4×C53—24no—
110C4⋊C4×C53—24no—
111C4⋊C4×C53—28no—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—24no—
116C4⋊C4×C53—24no—
117C4⋊C4×C53C2×D414yes—
118C22⋊C4×C53—24no—
119C22⋊C4×C53—24no—
120C42×C53—28no—
121C22⋊Dic53—41no—
122C22⋊Dic53—41no—
123C22⋊Dic53—42no—
124#11719925—84no—
125C4⋊Dic53—22no—
126C4⋊Dic53—81no—
127#11719919—42no—
128#11719919—82no—
129C2×#88322—24no—
130C2×#88322—82no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53#12011yescharacteristic
134C23×C53#12011yescharacteristic
135C23×C53C2×Q1612yes—
136C23×C53#13712yes—
137C23×C53C22×D411yescharacteristic
138Q8×C53—44no—
139Q8×C53—82no—
140Q8×C53—84no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—24no—
144C2×C4×C53—24no—
145C2×C4×C53—22no—
146C2×C4×C53—24no—
147C2×C4×C53—24no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—41no—
152C2×C4×C53—44no—
153C2×C4×C53—42no—
154C2×C4×C53—44no—
155C2×C4×C53—42no—
156C2×C4×C53—81no—
157C2×C4×C53#12012yes—
158C2×C4×C53C2×Q1614yes—
159C2×C4×C53C22×D412yes—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—42no—
167Dic106—44no—
168#88322—44no—
169#18295—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—42no—
177C22×C53—42no—
178C22×C53#45112yes—
179C22×C53#45112yes—
180C22×C53C2×#12011yescharacteristic
181C22×C53C22×Q1611yescharacteristic
182C22×C53C2×#13711yescharacteristic
183C4×C53—24no—
184C4×C53—22no—
185C4×C53—22no—
186C4×C53—44no—
187C4×C53—42no—
188C4×C53—44no—
189Dic53—82no—
190Dic53—81no—
191Dic53—81no—
192C2×#544D5311yescharacteristic
193#3189—532no—
194#2627—531no—
195#1617—534no—
196#1605—534no—
197#1520—532no—
198C2×#324—531no—
199C2×C53—21no—
200C2×C53—21no—
201C2×C53—22no—
202C2×C53—22no—
203C2×C53C2×#45111yescharacteristic
204C2×C53#302011yescharacteristic
205C2×C53#302611yescharacteristic
206#544—24no—
207#544—22no—
208#544C2×D5314yes—
209C2×C4⋊Q8—22no—
210C2×#125—22no—
211C2×C22⋊Q8—21no—
212C2×C22⋊Q8—531no—
213C2×C22⋊Q8C2×D5312yes—
214C2×#117—21no—
215C22×C4⋊C4C2×D5311yescharacteristic
216#484—534no—
217#484—1062no—
218#467—1064no—
219#451—534no—
220#451—1062no—
221C2×C22⋊C8—532no—
222#400—532no—
223#398—1062no—
224#395—532no—
225#385—1061no—
226#381—1061no—
227#380—531no—
228#328—538no—
229#324—534no—
230C53#1829511yescharacteristic, complemented, Sylow
231C22×Q8—21no—
232C22×Q8—531no—
233C22×Q8C22⋊D5312yes—
234C23×C4C22⋊D5311yescharacteristic
235C23×C4C22⋊D5311yescharacteristic
236C23×C4C22×D5311yescharacteristic
237C2×Q16—1064no—
238C22×C8—1062no—
239C4⋊Q8—28no—
240C4⋊Q8—28no—
241#125—28no—
242#125—28no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—1062no—
248C22⋊Q8—1062no—
249C22⋊Q8C22⋊D5314yes—
250C22⋊Q8C22×D5314yes—
251#117—24no—
252#117—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—531no—
261C2×C4⋊C4—1062no—
262C2×C4⋊C4—1061no—
263C2×C4⋊C4—1061no—
264C2×C4⋊C4C22⋊D5312yes—
265C2×C4⋊C4C22⋊D5312yes—
266C2×C4⋊C4C22×D5312yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—1061no—
269C2×C22⋊C4—1061no—
270C2×C42—22no—
271#107—1064no—
272#103—1064no—
273#103—1064no—
274C22⋊C8—534no—
275C22⋊C8—1062no—
276#95—532no—
277#95—1062no—
278#95—1062no—
279#95—1062no—
280C24C2×C22⋊D5311yescharacteristic
281C2×Q8—24no—
282C2×Q8—22no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×Q8—1062no—
286C2×Q8—4241no—
287C2×Q8C2×C22⋊D5314yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—1062no—
298C22×C4—1061no—
299C22×C4—1061no—
300C22×C4#1171992514yescomplemented
301C22×C4C2×C22⋊D5312yes—
302C22×C4C2×C22⋊D5312yes—
303C22×C4C2×C22⋊D5311yescharacteristic
304C22×C4C2×C22⋊D5312yes—
305C22×C4C2×C22⋊D5312yes—
306C22×C4C23×D5311yescharacteristic
307Q16—4244no—
308C2×C8—1064no—
309C2×C8—4242no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—28no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—1062no—
320C4⋊C4—4242no—
321C4⋊C4—4241no—
322C4⋊C4—2122no—
323C4⋊C4C2×C22⋊D5314yes—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—2121no—
327C22⋊C4—2121no—
328C22⋊C4—2122no—
329C42—28no—
330C23—41no—
331C23—41no—
332C23#1313318611yescharacteristic
333C23#1313318611yescharacteristic
334C23C2×#1171992512yes—
335C23#1313324812yes—
336C23C22×C22⋊D5311yescharacteristic
337Q8—44no—
338Q8—82no—
339Q8—84no—
340Q8—2124no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4—2122no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4—2122no—
364C2×C4#1313318612yes—
365C2×C4C2×#1171992514yes—
366C2×C4C22×C22⋊D5312yes—
367C8—2124no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#1681077912yes—
377C22#1681077912yes—
378C22C2×#1313318611yescharacteristic
379C22C22×#1171992511yescharacteristic
380C22C2×#1313324811yescharacteristic
381C4—24no—
382C4—22no—
383C4—22no—
384C4—44no—
385C4—42no—
386C4—44no—
387C4—4242no—
388C4—4241no—
389C4—4241no—
390C2—21no—
391C2—22no—
392C2—22no—
393C2—21no—
394C2C2×#1681077911yescharacteristic
395C2#2028407911yescharacteristic
396C2#2028411211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#544C2
socleC22×C53C2×#120
prime radicalC24×C53C2×D4
hypercenterC2×#544D53
hypocenterC53#18295
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #20273757C2semidirect
4. #20273689C2semidirect
96. C22×C4×C53Q16semidirect
230. C53#18295semidirect
300. C22×C4#11719925semidirect

Maximal subgroups

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

subgroupindex
#202854872
#202805282
#202737572
#202736892
#202732872
C2×#168103492
C2×C53×#5442
#1829553