Finite Group Databasebeta

← #27469708

Subgroups — #27469708

Order 13568.

810 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283023C211yesmaximal, characteristic
2C2×#16810492C211yesmaximal, characteristic
3#20276276C212yesmaximal, complemented
4#20276275C212yesmaximal, complemented
5#20275682C212yesmaximal, complemented
6#20274847C212yesmaximal, complemented
7#20274806C211yesmaximal, characteristic
8#20274805C211yesmaximal, characteristic
9#20274803C211yesmaximal, characteristic
10#20274479C211yesmaximal, characteristic
11C53×#1824C211yesmaximal, characteristic
12C2×#13133208C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919—21no—
15C22×#11719919—21no—
16C22×C4×Dic53C2211yescharacteristic
17C2×C22⋊#88322C2211yescharacteristic
18C2×#13133127C2212yes—
19#16810522C2211yescharacteristic
20#16810519C2211yescharacteristic
21#16810510—21no—
22#16810506—21no—
23#16810498—21no—
24#16810495—21no—
25#16810492—21no—
26#16810492—21no—
27#16810492C2212yes—
28#16810492C2212yes—
29#16810488C2211yescharacteristic
30#16810483C2211yescharacteristic
31#16810386C2212yes—
32#16810386C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810377C2212yes—
38C22×C4⋊C4×C53C2211yescharacteristic
39C2×C22⋊C8×C53C2211yescharacteristic
40C53×#380C2211yescharacteristic
41C53×#340C2212yes—
42C53×#340C2212yes—
43C23×Dic53—21no—
44C23×Dic53D411yescharacteristic
45C23×Dic53D411yescharacteristic
46C22×#88322C2312yes—
47#13133208—24no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53—22no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2311yescharacteristic
55C2×C4⋊Dic53C2312yes—
56C2×#11719919—21no—
57C2×#11719919—21no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919—22no—
61C2×#11719919—22no—
62C2×#11719919—44no—
63C2×#11719919—44no—
64C2×C22⋊Dic53—21no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53D411yescharacteristic
67C2×C4×Dic53D411yescharacteristic
68C2×C4×Dic53C2311yescharacteristic
69C2×C4×Dic53C2311yescharacteristic
70C22⋊#88322—22no—
71#13133127—24no—
72#13133124—21no—
73#13133124—21no—
74#13133124—21no—
75#13133124—21no—
76#13133124—21no—
77C23×C4×C53—21no—
78C23×C4×C53C2311yescharacteristic
79C22×C8×C53C2312yes—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53—22no—
82C2×C4⋊C4×C53—22no—
83C2×C4⋊C4×C53C2311yescharacteristic
84C2×C4⋊C4×C53C2311yescharacteristic
85C2×C4⋊C4×C53C2312yes—
86C2×C22⋊C4×C53—21no—
87C22⋊C8×C53—22no—
88#95×C53—21no—
89C24×C53C2×D411yescharacteristic
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—22no—
95C22×C4×C53—42no—
96C22×C4×C53—42no—
97C22×C4×C53—42no—
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53#4112yes—
100C22×C4×C53#4112yes—
101C22×C4×C53C2411yescharacteristic
102C22×Dic53—22no—
103C22×Dic53—22no—
104C22×Dic53—22no—
105C22×Dic53—22no—
106C22×Dic53—22no—
107C22×Dic53—22no—
108C22×Dic53—21no—
109C22×Dic53—21no—
110C22×Dic53—22no—
111C22×Dic53—42no—
112C22×Dic53—42no—
113C22×Dic53—42no—
114C22×Dic53C2×D411yescharacteristic
115C22×Dic53C2×D411yescharacteristic
116C22×Dic53C2×D411yescharacteristic
117C22×Dic53C2×D411yescharacteristic
118C2×C8×C53—22no—
119C2×C8×C53—22no—
120C2×C8×C53—22no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—42no—
126C22⋊C4×C53—42no—
127C22⋊Dic53—42no—
128C4⋊Dic53—22no—
129C4⋊Dic53—22no—
130C4⋊Dic53—22no—
131C4⋊Dic53—22no—
132C4⋊Dic53—42no—
133C4⋊Dic53—42no—
134C4⋊Dic53—42no—
135#11719919—44no—
136#11719919—44no—
137#11719919—42no—
138#11719919—42no—
139#11719919—44no—
140#11719919—44no—
141C4×Dic53—22no—
142C4×Dic53—22no—
143C4×Dic53—22no—
144C4×Dic53#4114yes—
145C2×#88322—22no—
146C2×#88322—22no—
147C2×#88322—22no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53#12412yes—
153C23×C53#13712yes—
154C23×C53#13712yes—
155C23×C53C22×D411yescharacteristic
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—24no—
159C2×C4×C53—24no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—84no—
170C2×C4×C53#12412yes—
171C2×C4×C53C2×#4112yes—
172C2×C4×C53C2×#4111yescharacteristic
173C2×C4×C53C2×#4111yescharacteristic
174C2×C4×C532−1+412yes—
175C2×Dic53—28no—
176C2×Dic53—28no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—21no—
181C2×Dic53—21no—
182C2×Dic53—21no—
183C2×Dic53—21no—
184C2×Dic53—21no—
185C2×Dic53—41no—
186C2×Dic53—42no—
187C2×Dic53—41no—
188C2×Dic53—42no—
189C2×Dic53—42no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—84no—
195C2×Dic53#12312yes—
196C2×Dic53#12312yes—
197C2×Dic53AGL(1,ℤ/8)12yes—
198C2×Dic53AGL(1,ℤ/8)12yes—
199C8×C53—42no—
200#88322—42no—
201#27965—531nomaximal, Sylow
202C22×C53—22no—
203C22×C53—22no—
204C22×C53—22no—
205C22×C53—22no—
206C22×C53—22no—
207C22×C53—22no—
208C22×C53#49114yes—
209C22×C53C2×#12311yescharacteristic
210C22×C53C2×#12411yescharacteristic
211C22×C53#53611yescharacteristic
212C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
213C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
214C22×C53C2×#13711yescharacteristic
215C22×C53C2×#13711yescharacteristic
216C4×C53—22no—
217C4×C53—81no—
218C4×C53—82no—
219C4×C53—81no—
220C4×C53#55212yes—
221C4×C53#55212yes—
222Dic53—22no—
223Dic53—22no—
224Dic53—22no—
225Dic53—22no—
226Dic53—81no—
227Dic53—82no—
228Dic53—81no—
229#2851—531no—
230C2×#384—531no—
231#2099—532no—
232#2099—532no—
233#1992—532no—
234#1836—532no—
235#1824—531no—
236#1824—531no—
237#1824—531no—
238#1824D5311yescharacteristic
239#1749—531no—
240C2×C53—24no—
241C2×C53#250311yescharacteristic
242C2×C53#309211yescharacteristic
243C2×C53#309211yescharacteristic
244C2×C53C2×#49111yescharacteristic
245C2×C53#313811yescharacteristic
246C2×C53#323911yescharacteristic
247C2×C53#323911yescharacteristic
248C2×#125—531no—
249C22×C4⋊C4—531no—
250C22×C4⋊C4—1061no—
251C22×C4⋊C4—1061no—
252C22×C4⋊C4C2×D5311yescharacteristic
253C22×C42—531no—
254C2×C22⋊C8—531no—
255C2×C22⋊C8C2×D5311yescharacteristic
256C2×#97—532no—
257#389—531no—
258#389—531no—
259#387—1061no—
260#387—1061no—
261#385—1061no—
262#385—1061no—
263#384—532no—
264#384—532no—
265#384—1061no—
266#384—1061no—
267#382—531no—
268#380—531no—
269#380C2×D5311yescharacteristic
270#340—532no—
271#340—532no—
272#340—532no—
273#340—532no—
274#340—532no—
275#340—532no—
276#340C2×D5312yes—
277#340C2×D5312yes—
278#336—532no—
279C53#2796511yescharacteristic, complemented, Sylow
280C23×C4—21no—
281C23×C4—531no—
282C23×C4—531no—
283C23×C4—1061no—
284C23×C4C22×D5311yescharacteristic
285C22×C8—532no—
286C22×C8C22×D5312yes—
287#125—1064no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—532no—
292C2×C4⋊C4—531no—
293C2×C4⋊C4—531no—
294C2×C4⋊C4—1062no—
295C2×C4⋊C4—1062no—
296C2×C4⋊C4—1062no—
297C2×C4⋊C4—1062no—
298C2×C4⋊C4—1062no—
299C2×C4⋊C4—1062no—
300C2×C4⋊C4—1061no—
301C2×C4⋊C4—1061no—
302C2×C4⋊C4—1061no—
303C2×C4⋊C4—1061no—
304C2×C4⋊C4—1061no—
305C2×C4⋊C4—2124no—
306C2×C4⋊C4—2124no—
307C2×C4⋊C4C22×D5312yes—
308C2×C4⋊C4C22×D5311yescharacteristic
309C2×C4⋊C4C22×D5311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—1061no—
312C2×C42—531no—
313C2×C42—531no—
314C2×C42—531no—
315C2×C42—531no—
316C2×C42—1064no—
317C22⋊C8—22no—
318C22⋊C8—1062no—
319#97—1064no—
320#95—21no—
321#95—1061no—
322#95—1061no—
323#95—1061no—
324#95—1061no—
325#95—1061no—
326C24D4×D5311yescharacteristic
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—531no—
336C22×C4—531no—
337C22×C4—531no—
338C22×C4—531no—
339C22×C4—1062no—
340C22×C4—1062no—
341C22×C4—1062no—
342C22×C4—1062no—
343C22×C4—1062no—
344C22×C4—1062no—
345C22×C4—1061no—
346C22×C4—1061no—
347C22×C4—1062no—
348C22×C4—2122no—
349C22×C4—2122no—
350C22×C4—2122no—
351C22×C4#1171994512yes—
352C22×C4D4×D5311yescharacteristic
353C22×C4#1171994712yes—
354C22×C4C23×D5311yescharacteristic
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—1062no—
359C2×C8—1062no—
360C2×C8—1062no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—42no—
366C4⋊C4—1062no—
367C4⋊C4—1062no—
368C4⋊C4—1062no—
369C4⋊C4—1062no—
370C4⋊C4—2124no—
371C4⋊C4—2124no—
372C4⋊C4—2122no—
373C4⋊C4—2122no—
374C4⋊C4—2122no—
375C4⋊C4—2122no—
376C4⋊C4—2124no—
377C4⋊C4—2124no—
378C4⋊C4—2122no—
379C22⋊C4—42no—
380C22⋊C4—2122no—
381C42—534no—
382C42—1062no—
383C42—1062no—
384C42—1062no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23#1313320312yes—
390C23#1313324612yes—
391C23#1313324912yes—
392C23C2×D4×D5311yescharacteristic
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—84no—
407C2×C4—532no—
408C2×C4—532no—
409C2×C4—532no—
410C2×C4—532no—
411C2×C4—1068no—
412C2×C4—1068no—
413C2×C4—1061no—
414C2×C4—1061no—
415C2×C4—1061no—
416C2×C4—1061no—
417C2×C4—1061no—
418C2×C4—1061no—
419C2×C4—1061no—
420C2×C4—1061no—
421C2×C4—4244no—
422C2×C4—2121no—
423C2×C4—2122no—
424C2×C4—2122no—
425C2×C4—2122no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4—2121no—
429C2×C4—2121no—
430C2×C4—2121no—
431C2×C4#1313320312yes—
432C2×C4#41×D5312yes—
433C2×C4C2×#1171994711yescharacteristic
434C2×C4C2×#1171994511yescharacteristic
435C2×C4#1313326412yes—
436C8—42no—
437C8—2122no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22#1681099014yes—
445C22#123×D5311yescharacteristic
446C22C2×#1313320311yescharacteristic
447C22#1681119511yescharacteristic
448C22AGL(1,ℤ/8)×D5311yescharacteristic
449C22AGL(1,ℤ/8)×D5311yescharacteristic
450C22C2×#1313324611yescharacteristic
451C22C2×#1313324911yescharacteristic
452C4—22no—
453C4—82no—
454C4—81no—
455C4—81no—
456C4—1062no—
457C4—1062no—
458C4—1062no—
459C4—1062no—
460C4—4242no—
461C4—4241no—
462C4—4241no—
463C4#1681130612yes—
464C4#1681131012yes—
465C2—24no—
466C2#2027919811yescharacteristic
467C2#2028463711yescharacteristic
468C2#2028464211yescharacteristic
469C2C2×#1681099011yescharacteristic
470C2#2028508711yescharacteristic
471C2#2028594311yescharacteristic
472C2#2028594811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1824C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1824D53
hypocenterC53#27965
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
3. #20276276C2semidirect
4. #20276275C2semidirect
5. #20275682C2semidirect
6. #20274847C2semidirect
279. C53#27965semidirect

Maximal subgroups

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

subgroupindex
#202830232
C2×#168104922
#202762762
#202762752
#202756822
#202748472
#202748062
#202748052
#202748032
#202744792
C53×#18242
#2796553