Finite Group Databasebeta

← #27358152

Subgroups — #27358152

Order 13568.

888 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285863C212yesmaximal, complemented
2#20280536C211yesmaximal, characteristic
3#20273757C214yesmaximal, complemented
4#20273740C214yesmaximal
5#20273279C212yesmaximal, complemented
6#20273029C211yesmaximal, characteristic
7C53×#3458C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9#16811026—22no—
10#16810973—22no—
11#16810956C2218yes—
12#16810918—24no—
13#16810676C2212yes—
14#16810562C2212yes—
15#16810559C2212yes—
16#16810543—21no—
17#16810535—21no—
18#16810521—21no—
19#16810489C2211yescharacteristic
20#16810483—21no—
21#16810360C2218yes—
22#16810349C2212yes—
23#16810347C2212yes—
24C53×#544—22no—
25C53×#544—22no—
26C53×#544—21no—
27C53×#544C2212yes—
28C53×#543—21no—
29C53×#543C2212yes—
30C53×#533—24no—
31C53×#533—21no—
32C53×#533C2212yes—
33C2×#117×C53—21no—
34C2×#117×C53C2211yescharacteristic
35C22×Dic106—21no—
36#13133217—22no—
37#13133217—22no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53C2312yes—
41C2×#11719919—22no—
42C2×C22⋊Dic53—22no—
43#13133150—28no—
44C4⋊#88322—22no—
45#13133141—28no—
46C22⋊#88322C2314yes—
47C4×#88322—22no—
48#13133124—22no—
49#13133124—21no—
50#13133124—22no—
51#13133124C2312yes—
52C23×C4×C53D411yescharacteristic
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—24no—
56C4⋊Q8×C53—24no—
57C4⋊Q8×C53D412yes—
58C4⋊Q8×C53D412yes—
59#125×C53—24no—
60#125×C53—22no—
61#125×C53—24no—
62#125×C53—22no—
63C22⋊Q8×C53—24no—
64C22⋊Q8×C53—24no—
65C22⋊Q8×C53—24no—
66C22⋊Q8×C53—24no—
67C22⋊Q8×C53—22no—
68C22⋊Q8×C53—22no—
69C22⋊Q8×C53—22no—
70C22⋊Q8×C53C2314yes—
71C4×Q8×C53—22no—
72C4×Q8×C53—22no—
73C4×Q8×C53—22no—
74C4×Q8×C53D414yes—
75#117×C53—22no—
76#117×C53—22no—
77#117×C53—22no—
78#117×C53—22no—
79#117×C53—22no—
80#117×C53—22no—
81C2×C4⋊C4×C53—21no—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53D411yescharacteristic
84C2×C4⋊C4×C53C2311yescharacteristic
85C2×C22⋊C4×C53—22no—
86C2×C22⋊C4×C53D411yescharacteristic
87C2×C22⋊C4×C53C2311yescharacteristic
88C2×C42×C53—22no—
89C2×C42×C53D411yescharacteristic
90C2×C42×C53C2311yescharacteristic
91C24×C53C2×D411yescharacteristic
92C2×Q8×C53—22no—
93C2×Q8×C53—22no—
94C2×Q8×C53—22no—
95C2×Q8×C53C2×D414yes—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53Q1614yes—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2411yescharacteristic
108C22×Dic53—22no—
109C22×Dic53—22no—
110C2×Dic106—42no—
111C2×Dic106—82no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—24no—
116C4⋊C4×C53—24no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—24no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—24no—
125C4⋊C4×C53C2×D414yes—
126C22⋊C4×C53—24no—
127C22⋊C4×C53—24no—
128C22⋊C4×C53—22no—
129C22⋊C4×C53—22no—
130C42×C53—24no—
131C42×C53—24no—
132C42×C53—21no—
133C42×C53—21no—
134C42×C53C2×D412yes—
135C42×C53C2×D412yes—
136C22⋊Dic53—82no—
137C4⋊Dic53—24no—
138C4⋊Dic53—41no—
139C4⋊Dic53—41no—
140C4⋊Dic53—42no—
141#11719919—82no—
142C2×#88322—24no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53C2×Q1612yes—
148C23×C53C22×D411yescharacteristic
149Q8×C53—44no—
150Q8×C53—82no—
151Q8×C53—82no—
152Q8×C53—82no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—24no—
156C2×C4×C53—24no—
157C2×C4×C53—24no—
158C2×C4×C53—24no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—82no—
166C2×C4×C53—82no—
167C2×C4×C53#12012yes—
168C2×C4×C53#12011yescharacteristic
169C2×C4×C53#12011yescharacteristic
170C2×C4×C53C2×Q1612yes—
171C2×C4×C53C2×Q1612yes—
172C2×C4×C53#13712yes—
173C2×C4×C53C22×D412yes—
174C2×Dic53—42no—
175C2×Dic53—42no—
176C2×Dic53—42no—
177C2×Dic53—42no—
178C2×Dic53—42no—
179C2×Dic53—42no—
180Dic106—84no—
181#88322—44no—
182#18311—531nomaximal, Sylow
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—22no—
186C22×C53—42no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—81no—
190C22×C53C2×#12011yescharacteristic
191C22×C53C22×Q1611yescharacteristic
192C22×C53C2×#13711yescharacteristic
193C4×C53—22no—
194C4×C53—22no—
195C4×C53—22no—
196C4×C53—41no—
197C4×C53—44no—
198C4×C53—41no—
199C4×C53—44no—
200C4×C53#45112yes—
201C4×C53#45112yes—
202Dic53—82no—
203Dic53—82no—
204#3458D5311yescharacteristic
205#3230—532no—
206#2628—531no—
207#1617—534no—
208#1614—534no—
209#1518—532no—
210#1458—531no—
211C2×C53—21no—
212C2×C53—21no—
213C2×C53—42no—
214C2×C53C2×#45111yescharacteristic
215C2×C53#302111yescharacteristic
216C2×C53#302811yescharacteristic
217#544—22no—
218#544—22no—
219#544—21no—
220#544C2×D5312yes—
221#543—21no—
222#543C2×D5312yes—
223#533—24no—
224#533—21no—
225#533C2×D5312yes—
226C2×C4⋊Q8—531no—
227C2×#117—21no—
228C2×#117C2×D5311yescharacteristic
229#500—1062no—
230#487—1062no—
231#484—538no—
232#477—1064no—
233#431—532no—
234#400—532no—
235#399—532no—
236#395—1061no—
237#393—1061no—
238#389—1061no—
239#382—531no—
240#380—1061no—
241#328—538no—
242#324—532no—
243#323—532no—
244C53#1831111yescharacteristic, complemented, Sylow
245C22×Q8—1061no—
246C23×C4C22⋊D5311yescharacteristic
247C4⋊Q8—24no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—24no—
251C4⋊Q8—1062no—
252C4⋊Q8—1062no—
253C4⋊Q8C22⋊D5312yes—
254C4⋊Q8C22⋊D5312yes—
255#125—24no—
256#125—22no—
257#125—24no—
258#125—22no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—24no—
266C22⋊Q8C22×D5314yes—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8C22⋊D5314yes—
271#117—22no—
272#117—22no—
273#117—22no—
274#117—22no—
275#117—22no—
276#117—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—532no—
280C2×C4⋊C4—1062no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4—1061no—
283C2×C4⋊C4C22⋊D5311yescharacteristic
284C2×C4⋊C4C22×D5311yescharacteristic
285C2×C22⋊C4—22no—
286C2×C22⋊C4—1062no—
287C2×C22⋊C4C22⋊D5311yescharacteristic
288C2×C22⋊C4C22×D5311yescharacteristic
289C2×C42—22no—
290C2×C42C22⋊D5311yescharacteristic
291C2×C42C22×D5311yescharacteristic
292#107—1068no—
293C4⋊C8—1062no—
294#103—1068no—
295C22⋊C8—534no—
296C4×C8—1062no—
297#95—532no—
298#95—1062no—
299#95—1061no—
300#95—1062no—
301C24C2×C22⋊D5311yescharacteristic
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—4242no—
306C2×Q8—2122no—
307C2×Q8C2×C22⋊D5314yes—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—1062no—
314C22×C4—1062no—
315C22×C4#1171992514yes—
316C22×C4C2×C22⋊D5311yescharacteristic
317C22×C4C2×C22⋊D5311yescharacteristic
318C22×C4C2×C22⋊D5311yescharacteristic
319C22×C4C2×C22⋊D5311yescharacteristic
320C22×C4C2×C22⋊D5311yescharacteristic
321C22×C4C23×D5311yescharacteristic
322C2×C8—1064no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—1064no—
337C4⋊C4—4242no—
338C4⋊C4—2121no—
339C4⋊C4—2121no—
340C4⋊C4—2122no—
341C4⋊C4C2×C22⋊D5314yes—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—4242no—
347C42—24no—
348C42—24no—
349C42—21no—
350C42—21no—
351C42C2×C22⋊D5312yes—
352C42C2×C22⋊D5312yes—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23C2×#1171992512yes—
358C23C22×C22⋊D5311yescharacteristic
359Q8—44no—
360Q8—82no—
361Q8—82no—
362Q8—82no—
363Q8—4244no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—2122no—
379C2×C4—2122no—
380C2×C4—2122no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4—2122no—
384C2×C4#1313318612yes—
385C2×C4#1313318611yescharacteristic
386C2×C4#1313318611yescharacteristic
387C2×C4C2×#1171992512yes—
388C2×C4C2×#1171992512yes—
389C2×C4#1313324812yes—
390C2×C4C22×C22⋊D5312yes—
391C8—2124no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—81no—
399C22C2×#1313318611yescharacteristic
400C22C22×#1171992511yescharacteristic
401C22C2×#1313324811yescharacteristic
402C4—22no—
403C4—22no—
404C4—22no—
405C4—41no—
406C4—44no—
407C4—41no—
408C4—44no—
409C4—4242no—
410C4—4242no—
411C4#1681077912yes—
412C4#1681077912yes—
413C2—21no—
414C2—21no—
415C2—42no—
416C2C2×#1681077911yescharacteristic
417C2#2028408711yescharacteristic
418C2#2028412111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3458C2
socleC22×C53C2×#120
prime radicalC24×C53C2×D4
hypercenter#3458D53
hypocenterC53#18311
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3458

Complements

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

NH ≅ G/Ntype
1. #20285863C2semidirect
3. #20273757C2semidirect
5. #20273279C2semidirect
244. C53#18311semidirect

Maximal subgroups

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

subgroupindex
#202858632
#202805362
#202737572
#202737402
#202732792
#202730292
C53×#34582
#1831153