Finite Group Databasebeta

← #27413080

Subgroups — #27413080

Order 13568.

946 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810662C212yesmaximal
2#389×D53C211yesmaximal, characteristic
3#20277120C211yesmaximal, characteristic
4#20276202C212yesmaximal, complemented
5#20276200C212yesmaximal, complemented
6#20276192C211yesmaximal, characteristic, complemented
7#20276188C211yesmaximal, characteristic, complemented
8#20276186C211yesmaximal, characteristic, complemented
9#20274433C212yesmaximal
10#20274407C211yesmaximal, characteristic
11C53×#2084C211yesmaximal, characteristic, complemented
12C22×#11719912C2212yes—
13C2×#13133175C2212yes—
14C2×C4⋊C4×D53—21no—
15C2×C4⋊C4×D53C2211yescharacteristic
16C2×C4⋊C4×D53C2211yescharacteristic
17C2×C42×D53C2211yescharacteristic
18#16810662—28no—
19C2×#13133151C2212yes—
20C2×#13133150C2212yes—
21C2×#13133148C2212yes—
22#16810522C2211yescharacteristic
23#16810521C2211yescharacteristic
24#16810520—21no—
25#16810519C2211yescharacteristic
26#16810518—21no—
27#16810493C2211yescharacteristic
28#16810488C2212yes—
29#16810480—21no—
30#16810480—21no—
31C4⋊C4×Dic53—21no—
32C4⋊C4×Dic53—21no—
33#95×D53—21no—
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810378C2211yescharacteristic
38#16810377C2212yes—
39C2×#106×C53C2212yes—
40C53×#389C2211yescharacteristic
41C53×#384C2211yescharacteristic
42C53×#340C2212yes—
43C53×#336C2211yescharacteristic
44C22×C4×D53—21no—
45C22×C4×D53—21no—
46C22×C4×D53D411yescharacteristic
47C22×C4×D53D411yescharacteristic
48C22×C4×D53C2311yescharacteristic
49C2×#11719912—24no—
50C2×#11719912—24no—
51C2×#11719912Q818yes—
52C22×#88322C2312yes—
53#13133175—24no—
54#13133175—24no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53C2311yescharacteristic
59C2×C4⋊Dic53C2312yes—
60C2×C4⋊Dic53C2311yescharacteristic
61C2×#11719919—21no—
62C2×#11719919—21no—
63C2×#11719919—21no—
64C2×#11719919—21no—
65C4⋊C4×D53—22no—
66C4⋊C4×D53—24no—
67C4⋊C4×D53—44no—
68C4⋊C4×D53D414yes—
69C2×#11719921—22no—
70C2×C4×Dic53—22no—
71C2×C4×Dic53D411yescharacteristic
72C2×C4×Dic53D411yescharacteristic
73C2×C4×Dic53C2311yescharacteristic
74C42×D53—24no—
75#13133151—24no—
76#13133150—24no—
77#13133148—24no—
78#13133124—21no—
79#13133124—21no—
80#13133124—21no—
81#13133124—22no—
82#13133124—22no—
83C22×C8×C53C2312yes—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53C2311yescharacteristic
88C2×C4⋊C4×C53C2312yes—
89C2×C4⋊C4×C53C2311yescharacteristic
90C2×C42×C53C2311yescharacteristic
91#106×C53—24no—
92#95×C53—21no—
93C23×D53C2×D411yescharacteristic
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—22no—
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2411yescharacteristic
100C22×Dic53—21no—
101C22×Dic53—22no—
102C22×Dic53—21no—
103C22×Dic53C2×D411yescharacteristic
104C22×Dic53C2×D411yescharacteristic
105C22×Dic53C2×D411yescharacteristic
106C2×C4×D53—24no—
107C2×C4×D53—22no—
108C2×C4×D53—22no—
109C2×C4×D53—22no—
110C2×C4×D53—22no—
111C2×C4×D53—41no—
112C2×C4×D53—41no—
113C2×C4×D53—42no—
114C2×C4×D53—42no—
115C2×C4×D53—42no—
116C2×C4×D53—42no—
117C2×C4×D53—42no—
118C2×C4×D53C2×D412yes—
119C2×C4×D53C2×Q814yes—
120C2×C4×D53#4114yes—
121C2×C4×D53#4112yes—
122C2×C8×C53—22no—
123C2×C8×C53—22no—
124C2×C8×C53C2×Q814yes—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—42no—
130C4⋊C4×C53—42no—
131C4⋊C4×C53—42no—
132C4⋊C4×C53C2×D412yes—
133C42×C53—22no—
134#11719921—44no—
135C4⋊Dic53—21no—
136C4⋊Dic53—22no—
137C4⋊Dic53—22no—
138C4⋊Dic53—22no—
139C4⋊Dic53—42no—
140C4⋊Dic53—42no—
141C4⋊Dic53—42no—
142C4⋊Dic53C2×D412yes—
143#11719919—24no—
144#11719919—42no—
145#11719919—42no—
146#11719919—42no—
147C4×Dic53—22no—
148C4×Dic53—22no—
149C4×Dic53—22no—
150C4×Dic53—44no—
151C2×#88322—22no—
152C2×#88322—22no—
153C2×#88322C2×Q814yes—
154#11719912—216no—
155C23×C53C22×D411yescharacteristic
156C22×D53—21no—
157C22×D53—21no—
158C22×D53—21no—
159C22×D53—21no—
160C22×D53#12112yes—
161C22×D53#12312yes—
162C22×D53C4⋊Q812yes—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—22no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—42no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—42no—
174C2×C4×C53—41no—
175C2×C4×C53—42no—
176C2×C4×C53AGL(1,ℤ/8)12yes—
177C2×C4×C53#13712yes—
178C2×C4×C53C22×D411yescharacteristic
179C2×C4×C53C22×Q812yes—
180C2×C4×C53C2×#4112yes—
181C2×C4×C53C2×#4111yescharacteristic
182C2×Dic53—22no—
183C2×Dic53—21no—
184C2×Dic53—21no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—42no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—42no—
195C2×Dic53—41no—
196C2×Dic53—42no—
197C2×Dic53#12112yes—
198C2×Dic53#12312yes—
199C2×Dic53C4⋊Q812yes—
200C4×D53—44no—
201C4×D53—44no—
202C4×D53—48no—
203C4×D53—84no—
204C4×D53C22⋊Q818yes—
205C4×D53#12518yes—
206C8×C53—24no—
207#88322—24no—
208#23657—531nomaximal, Sylow
209C22×C53C2×#12111yescharacteristic
210C22×C53C2×#12311yescharacteristic
211C22×C53C2×C4⋊Q811yescharacteristic
212C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
213C22×C53C2×#13711yescharacteristic
214C22×C53#57511yescharacteristic
215C22×C53#57511yescharacteristic
216C2×D53—22no—
217C2×D53—22no—
218C2×D53—22no—
219C2×D53—22no—
220C2×D53—22no—
221C2×D53—22no—
222C2×D53#39814yes—
223C4×C53—22no—
224C4×C53—22no—
225C4×C53—42no—
226C4×C53—81no—
227C4×C53—82no—
228C4×C53C2×C22⋊Q812yes—
229C4×C53C2×#12512yes—
230Dic53—41no—
231Dic53—41no—
232Dic53—42no—
233Dic53—81no—
234Dic53—82no—
235Dic53#39814yes—
236C2×#428—532no—
237C2×#389—531no—
238#2285—531no—
239#2085—532no—
240#2085—532no—
241#2084—531no—
242#2084D5311yescharacteristic, complemented
243#2083—531no—
244#2083—531no—
245#1736—532no—
246#1728—531no—
247C2×C53C2×#39811yescharacteristic
248C2×C53#303611yescharacteristic
249C2×C53#303711yescharacteristic
250C2×C53#309411yescharacteristic
251C2×C53#309511yescharacteristic
252C2×C53#316511yescharacteristic
253C2×C53#316611yescharacteristic
254D53—24no—
255C22×M16—532no—
256C2×#117—532no—
257C22×C4⋊C4—531no—
258C22×C4⋊C4—531no—
259C22×C4⋊C4—1061no—
260C22×C42—531no—
261#428—1068no—
262C2×#107—532no—
263C2×#107—532no—
264C2×#106—532no—
265C2×#106C2×D5312yes—
266#389—531no—
267#389—531no—
268#389—531no—
269#389—1061no—
270#389—1061no—
271#389C2×D5311yescharacteristic
272#384—531no—
273#384C2×D5311yescharacteristic
274#382—532no—
275#380—1061no—
276#380—1061no—
277C4×C4⋊C4—1061no—
278C4×C4⋊C4—1061no—
279C2×#95—1061no—
280#340—532no—
281#340—532no—
282#340—532no—
283#340C2×D5312yes—
284#336—532no—
285#336—531no—
286#336C2×D5311yescharacteristic
287C53#2365711yescharacteristic, complemented, Sylow
288C23×C4—531no—
289C23×C4—531no—
290C23×C4—531no—
291C23×C4—1061no—
292C23×C4—1061no—
293C2×M16—538no—
294C2×M16—1064no—
295C2×M16—1064no—
296C22×C8—532no—
297C22×C8C22×D5312yes—
298#117—1064no—
299#117—1064no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—531no—
304C2×C4⋊C4—534no—
305C2×C4⋊C4—531no—
306C2×C4⋊C4—532no—
307C2×C4⋊C4—1062no—
308C2×C4⋊C4—1064no—
309C2×C4⋊C4—1061no—
310C2×C4⋊C4—1061no—
311C2×C4⋊C4—1061no—
312C2×C4⋊C4—1061no—
313C2×C4⋊C4—1061no—
314C2×C4⋊C4—1061no—
315C2×C4⋊C4—1061no—
316C2×C4⋊C4—2124no—
317C2×C4⋊C4C22×D5311yescharacteristic
318C2×C4⋊C4C22×D5312yes—
319C2×C4⋊C4C22×D5311yescharacteristic
320C2×C22⋊C4—1062no—
321C2×C42—531no—
322C2×C42—531no—
323C2×C42—531no—
324C2×C42—1064no—
325C2×C42—1062no—
326C2×C42C22×D5311yescharacteristic
327#107—1064no—
328#107—1064no—
329#106—24no—
330#106—1064no—
331#95—21no—
332#95—1061no—
333#95—1061no—
334#95—1062no—
335#95—1062no—
336#95—1061no—
337C24—531no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—531no—
342C22×C4—532no—
343C22×C4—534no—
344C22×C4—534no—
345C22×C4—532no—
346C22×C4—531no—
347C22×C4—531no—
348C22×C4—1062no—
349C22×C4—1062no—
350C22×C4—1064no—
351C22×C4—1061no—
352C22×C4—1062no—
353C22×C4—1062no—
354C22×C4—1062no—
355C22×C4—1061no—
356C22×C4—2121no—
357C22×C4—2121no—
358C22×C4—2122no—
359C22×C4—2122no—
360C22×C4—2122no—
361C22×C4—2122no—
362C22×C4—2122no—
363C22×C4D4×D5311yescharacteristic
364C22×C4D4×D5311yescharacteristic
365C22×C4C23×D5311yescharacteristic
366M16—10616no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—534no—
370C2×C8—1062no—
371C2×C8—1062no—
372C2×C8Q8×D5314yes—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—532no—
381C4⋊C4—1064no—
382C4⋊C4—1061no—
383C4⋊C4—1062no—
384C4⋊C4—1062no—
385C4⋊C4—1062no—
386C4⋊C4—2122no—
387C4⋊C4—2122no—
388C4⋊C4—2122no—
389C4⋊C4—2122no—
390C4⋊C4—2122no—
391C4⋊C4—2122no—
392C4⋊C4D4×D5312yes—
393C22⋊C4—2124no—
394C42—22no—
395C42—1062no—
396C42—1062no—
397C42—1062no—
398C42—2124no—
399C23—532no—
400C23—532no—
401C23—532no—
402C23—1061no—
403C23—1061no—
404C23—1061no—
405C23—1061no—
406C23C2×D4×D5311yescharacteristic
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—532no—
421C2×C4—532no—
422C2×C4—532no—
423C2×C4—538no—
424C2×C4—538no—
425C2×C4—1062no—
426C2×C4—1061no—
427C2×C4—1061no—
428C2×C4—1061no—
429C2×C4—1061no—
430C2×C4—1061no—
431C2×C4—1061no—
432C2×C4—4244no—
433C2×C4—2124no—
434C2×C4—2124no—
435C2×C4—2128no—
436C2×C4—2122no—
437C2×C4—2121no—
438C2×C4—2121no—
439C2×C4—2122no—
440C2×C4—2121no—
441C2×C4—2122no—
442C2×C4—2121no—
443C2×C4—2121no—
444C2×C4#1313324412yes—
445C2×C4#1313324612yes—
446C2×C4C2×D4×D5311yescharacteristic
447C2×C4C2×Q8×D5312yes—
448C2×C4#41×D5312yes—
449C2×C4#41×D5311yescharacteristic
450C8—24no—
451C8—1064no—
452C22—534no—
453C22—1062no—
454C22—1062no—
455C22—1062no—
456C22—1062no—
457C22—1062no—
458C22—1062no—
459C22#121×D5311yescharacteristic
460C22#123×D5311yescharacteristic
461C22C4⋊Q8×D5311yescharacteristic
462C22C2×#1313324411yescharacteristic
463C22C2×#1313324611yescharacteristic
464C22#1681141911yescharacteristic
465C22#1681142211yescharacteristic
466C4—22no—
467C4—22no—
468C4—42no—
469C4—82no—
470C4—81no—
471C4—534no—
472C4—4242no—
473C4—4241no—
474C4—2122no—
475C4—2121no—
476C4—2121no—
477C4C22⋊Q8×D5312yes—
478C4#125×D5312yes—
479C2—1064no—
480C2#398×D5311yescharacteristic
481C2#2028418011yescharacteristic
482C2#2028418211yescharacteristic
483C2#2028465211yescharacteristic
484C2#2028465811yescharacteristic
485C2#2028525611yescharacteristic
486C2#2028526611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2084C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2084D53
hypocenterC53#23657
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2084

Complements

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

NH ≅ G/Ntype
4. #20276202C2semidirect
5. #20276200C2semidirect
6. #20276192C2semidirect
7. #20276188C2semidirect
8. #20276186C2semidirect
11. C53×#2084C2semidirect
242. #2084D53semidirect
287. C53#23657semidirect

Maximal subgroups

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

subgroupindex
C2×#168106622
#389×D532
#202771202
#202762022
#202762002
#202761922
#202761882
#202761862
#202744332
#202744072
C53×#20842
#2365753