Finite Group Databasebeta

← #27367404

Subgroups — #27367404

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286057C211yesmaximal, characteristic, complemented
2#20286027C211yesmaximal, characteristic
3#20285038C211yesmaximal, characteristic
4#20274297C212yesmaximal, complemented
5#20274283C212yesmaximal, complemented
6#20274145C212yesmaximal, complemented
7#20274131C212yesmaximal, complemented
8C2×#16810374C211yesmaximal, characteristic, complemented
9#20273406C211yesmaximal, characteristic, complemented
10#20273393C211yesmaximal, characteristic
11C53×#2960C211yesmaximal, characteristic, complemented
12#16811180C2211yescharacteristic
13#16811173C2211yescharacteristic
14#16810978—21no—
15#16810978—21no—
16#16810974C2211yescharacteristic
17#16810974C2211yescharacteristic
18#16810970C2211yescharacteristic
19#16810970C2211yescharacteristic
20#16810918C2212yes—
21#16810915—21no—
22#16810910—21no—
23#16810908C2212yes—
24#16810862—22no—
25#16810854—22no—
26#16810833C2212yes—
27#16810830—21no—
28#16810825—21no—
29#16810823C2212yes—
30C2×C4⋊#88322C2211yescharacteristic
31C2×C4⋊#88322C2211yescharacteristic
32#16810620—21no—
33#16810620—21no—
34#16810374C2212yes—
35#16810374C2212yes—
36#16810371C2212yes—
37#16810370C2212yes—
38#16810368C2212yes—
39#16810366C2212yes—
40C53×#533C2211yescharacteristic
41C53×#446—21no—
42C53×#446C2212yes—
43C8×Q8×C53—21no—
44C8×Q8×C53C2212yes—
45C53×#424—21no—
46C53×#423C2211yescharacteristic
47C53×#405—21no—
48C2×C4×C8×C53C2211yescharacteristic
49C2×#11719945—21no—
50C2×#11719925—22no—
51C2×#11719924—22no—
52C22×#88322—21no—
53C22×#88322—21no—
54#13133217C2311yescharacteristic
55#13133217C2311yescharacteristic
56#13133217C2311yescharacteristic
57#13133214C2311yescharacteristic
58#13133209—21no—
59#13133205—21no—
60#13133196—21no—
61#13133196—21no—
62#13133191—21no—
63#13133191—21no—
64C4×Dic106—21no—
65C4×Dic106—21no—
66C4×D212—21no—
67C4×D212—21no—
68#13133176—21no—
69#13133150—22no—
70#13133148—22no—
71C4⋊#88322—21no—
72C4⋊#88322—21no—
73C4⋊#88322C2312yes—
74C4⋊#88322C2312yes—
75#13133142—21no—
76#13133142—21no—
77#13133142—21no—
78#13133142—21no—
79#13133141—22no—
80#13133141—22no—
81#13133140—22no—
82#13133139—21no—
83#13133139—21no—
84#13133139—21no—
85#13133139—21no—
86#13133137—22no—
87C2×M16×C53—21no—
88C22×C8×C53D411yescharacteristic
89C22×C8×C53D411yescharacteristic
90C4⋊Q8×C53D411yescharacteristic
91C4⋊Q8×C53D411yescharacteristic
92#125×C53—21no—
93C22⋊Q8×C53—21no—
94C22⋊Q8×C53D412yes—
95C4×Q8×C53—21no—
96C4×Q8×C53C2312yes—
97#117×C53—21no—
98C2×C42×C53C2311yescharacteristic
99C4⋊C8×C53—22no—
100C4⋊C8×C53—22no—
101C4⋊C8×C53—21no—
102C4⋊C8×C53C2312yes—
103#97×C53—22no—
104C4×C8×C53—22no—
105C4×C8×C53D411yescharacteristic
106C4×C8×C53D411yescharacteristic
107C4×C8×C53C2311yescharacteristic
108C4×C8×C53C2311yescharacteristic
109C2×Q8×C53—21no—
110C2×Q8×C53C2×D412yes—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C2×C22⋊D53—41no—
115#11719945—42no—
116#11719945—42no—
117C2×D212—21no—
118C2×D212—21no—
119C2×D212—21no—
120C2×C4×D53—41no—
121C2×Dic106—21no—
122C2×Dic106—21no—
123C2×Dic106—21no—
124M16×C53—42no—
125C2×C8×C53—22no—
126C2×C8×C53—22no—
127C2×C8×C53—22no—
128C2×C8×C53C2×D411yescharacteristic
129C2×C8×C53C2×D411yescharacteristic
130C2×C8×C53C2×D411yescharacteristic
131C2×C8×C53C2×D411yescharacteristic
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53C2×D412yes—
138C22⋊C4×C53—22no—
139C42×C53—22no—
140C42×C53C2×D411yescharacteristic
141C42×C53C2×D411yescharacteristic
142C42×C53#4111yescharacteristic
143C42×C53C2411yescharacteristic
144C22⋊Dic53—41no—
145#11719925—82no—
146#11719924—82no—
147#11719921—41no—
148#11719921—41no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153C4⋊Dic53—21no—
154C4⋊Dic53—21no—
155#11719919—41no—
156#11719919—41no—
157C4×Dic53—41no—
158C2×#88322—22no—
159C2×#88322—22no—
160C2×#88322—81no—
161C2×#88322—81no—
162C23×C53#12111yescharacteristic
163C22×D53—41no—
164Q8×C53—22no—
165Q8×C53—22no—
166Q8×C53—42no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—22no—
170C2×C4×C53—22no—
171C2×C4×C53—41no—
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53#13512yes—
176C2×C4×C53#13712yes—
177C2×C4×C53C22×D411yescharacteristic
178C2×C4×C53C22×D411yescharacteristic
179C2×C4×C53C2×#4111yescharacteristic
180C22⋊D53—82no—
181C2×Dic53—41no—
182C2×Dic53—41no—
183C2×Dic53—41no—
184D212—41no—
185D212—41no—
186D212—41no—
187D212—41no—
188D212—81no—
189C4×D53—82no—
190Dic106—41no—
191Dic106—41no—
192Dic106—41no—
193Dic106—41no—
194Dic106—81no—
195C8×C53—21no—
196C8×C53—21no—
197C8×C53—21no—
198C8×C53—21no—
199C8×C53—42no—
200#88322—42no—
201#88322—42no—
202#19216—531nomaximal, Sylow
203C22×C53—21no—
204C22×C53—21no—
205C22×C53#46412yes—
206C22×C53C2×#12111yescharacteristic
207C2×D53—81no—
208C2×D53—81no—
209C2×D53—81no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—42no—
214C4×C53—42no—
215C4×C53#46412yes—
216C4×C53C2×#13511yescharacteristic
217C4×C53C2×#13711yescharacteristic
218C4×C53#57711yescharacteristic
219C4×C53#57711yescharacteristic
220Dic53—81no—
221Dic53—81no—
222Dic53—81no—
223#3249—531no—
224#3246—531no—
225#3131—531no—
226#2960D5311yescharacteristic, complemented
227#1704—532no—
228#1702—532no—
229#1680—532no—
230#1678—532no—
231C2×#334—531no—
232#1550—531no—
233#1547—531no—
234C2×C53—22no—
235C2×C53C2×#46411yescharacteristic
236C2×C53#306011yescharacteristic
237C2×C53#306711yescharacteristic
238D53—161no—
239#533—531no—
240#533C2×D5311yescharacteristic
241#532—531no—
242#488—1061no—
243#488—1061no—
244#487—531no—
245#487—531no—
246#486—531no—
247#486—531no—
248#477—532no—
249#477—1061no—
250#476—1061no—
251#475—532no—
252#467—1062no—
253#465—1062no—
254#462—532no—
255#462—1061no—
256#461—1061no—
257#460—532no—
258#446—21no—
259#446C2×D5312yes—
260C8×Q8—21no—
261C8×Q8C2×D5312yes—
262#424—21no—
263#423C2×D5311yescharacteristic
264C2×C4⋊C8—531no—
265C2×C4⋊C8—531no—
266#416—1061no—
267#416—1061no—
268#405—21no—
269C2×C4×C8C2×D5311yescharacteristic
270#334—532no—
271#334—532no—
272#333—532no—
273#332—532no—
274#332—532no—
275#331—532no—
276C53#1921611yescharacteristic, complemented, Sylow
277C2×#41—1061no—
278C2×Q16—1062no—
279C2×SD16—1062no—
280C2×M16—21no—
281C22×C8—1061no—
282C22×C8—1061no—
283C22×C8C22⋊D5311yescharacteristic
284C22×C8C22⋊D5311yescharacteristic
285C4⋊Q8—531no—
286C4⋊Q8—531no—
287C4⋊Q8—531no—
288C4⋊Q8D21211yescharacteristic
289C4⋊Q8D21211yescharacteristic
290#127—531no—
291#125—21no—
292#125—1061no—
293#124—1061no—
294C22⋊Q8—21no—
295C22⋊Q8—1061no—
296C22⋊Q8—1061no—
297C22⋊Q8D21212yes—
298#121—1061no—
299#121—1061no—
300C4×Q8—21no—
301C4×Q8—1061no—
302C4×Q8—1061no—
303C4×Q8C22×D5312yes—
304C4×D4—1061no—
305C4×D4—1061no—
306#117—21no—
307#117—1061no—
308C2×C42C22×D5311yescharacteristic
309#107—1062no—
310#106—1062no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—21no—
314C4⋊C8—532no—
315C4⋊C8—532no—
316C4⋊C8—1061no—
317C4⋊C8—1061no—
318C4⋊C8C22×D5312yes—
319#103—1062no—
320#103—1062no—
321#103—1061no—
322#103—1061no—
323#103—1061no—
324#103—1061no—
325#103—1062no—
326#102—1061no—
327#102—1061no—
328#102—1061no—
329#102—1061no—
330#102—1062no—
331#97—22no—
332C4×C8—22no—
333C4×C8C22⋊D5311yescharacteristic
334C4×C8C22⋊D5311yescharacteristic
335C4×C8C22×D5311yescharacteristic
336C4×C8C22×D5311yescharacteristic
337#41—2122no—
338#41—2122no—
339C2×Q8—21no—
340C2×Q8—1061no—
341C2×Q8—1061no—
342C2×Q8—1061no—
343C2×Q8C2×D21212yes—
344C2×D4—1061no—
345C2×D4—1061no—
346C2×D4—1061no—
347C2×D4—2121no—
348C22×C4—2121no—
349C22×C4C2×D21211yescharacteristic
350C22×C4#1171994511yescharacteristic
351C22×C4C2×C22⋊D5311yescharacteristic
352Q16—4242no—
353SD16—4242no—
354M16—42no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—1062no—
359C2×C8—1062no—
360C2×C8—4241no—
361C2×C8—4241no—
362C2×C8C2×C22⋊D5311yescharacteristic
363C2×C8C2×C22⋊D5311yescharacteristic
364C2×C8C2×C22⋊D5311yescharacteristic
365C2×C8C2×C22⋊D5311yescharacteristic
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—1061no—
372C4⋊C4—1061no—
373C4⋊C4—1061no—
374C4⋊C4—1061no—
375C4⋊C4—1061no—
376C4⋊C4—1061no—
377C4⋊C4—2121no—
378C4⋊C4—2121no—
379C4⋊C4C2×D21212yes—
380C22⋊C4—22no—
381C22⋊C4—2121no—
382C22⋊C4—2121no—
383C22⋊C4—2121no—
384C42—22no—
385C42—2121no—
386C42C2×D21211yescharacteristic
387C42#1171994511yescharacteristic
388C42C2×C22⋊D5311yescharacteristic
389C42C23×D5311yescharacteristic
390C23—2121no—
391C23#1313319111yescharacteristic
392Q8—22no—
393Q8—22no—
394Q8—42no—
395Q8—4241no—
396Q8—2121no—
397Q8—2121no—
398Q8—2121no—
399Q8—2121no—
400D4—4241no—
401D4—4242no—
402D4—2121no—
403D4—2121no—
404D4—2121no—
405D4—2121no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—4242no—
412C2×C4—2121no—
413C2×C4—2121no—
414C2×C4—2121no—
415C2×C4#1313319111yescharacteristic
416C2×C4#1313319111yescharacteristic
417C2×C4#1313319111yescharacteristic
418C2×C4#1313323712yes—
419C2×C4#1313324812yes—
420C2×C4C22×C22⋊D5311yescharacteristic
421C2×C4C22×D21211yescharacteristic
422C2×C4C2×#1171994511yescharacteristic
423C8—21no—
424C8—21no—
425C8—21no—
426C8—21no—
427C8—42no—
428C8—2122no—
429C8—2122no—
430C22—21no—
431C22—21no—
432C22—4241no—
433C22—4241no—
434C22—4241no—
435C22#1681084512yes—
436C22C2×#1313319111yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—42no—
441C4—42no—
442C4—4241no—
443C4—4241no—
444C4—4241no—
445C4#1681084512yes—
446C4C2×#1313323711yescharacteristic
447C4C2×#1313324811yescharacteristic
448C4#1681143511yescharacteristic
449C4#1681143511yescharacteristic
450C2—22no—
451C2—8481no—
452C2C2×#1681084511yescharacteristic
453C2#2028437711yescharacteristic
454C2#2028443511yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2960C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#2960D53
hypocenterC53#19216
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960

Complements

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

NH ≅ G/Ntype
1. #20286057C2semidirect
4. #20274297C2semidirect
5. #20274283C2semidirect
6. #20274145C2semidirect
7. #20274131C2semidirect
8. C2×#16810374C2semidirect
9. #20273406C2semidirect
11. C53×#2960C2semidirect
226. #2960D53semidirect
276. C53#19216semidirect

Maximal subgroups

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

subgroupindex
#202860572
#202860272
#202850382
#202742972
#202742832
#202741452
#202741312
C2×#168103742
#202734062
#202733932
C53×#29602
#1921653