Finite Group Databasebeta

← #27277476

Subgroups — #27277476

Order 13568.

868 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277098C211yesmaximal, characteristic
2#20274625C211yesmaximal, characteristic
3#20272594C212yesmaximal, complemented
4#20272594C212yesmaximal, complemented
5C53×C22⋊#95C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic, complemented
7C2×#13133176C411yescharacteristic, complemented
8C2×C22⋊#88322—21no—
9C2×C22⋊#88322—21no—
10C2×C22⋊#88322C2212yes—
11#16810486—21no—
12#16810483C411yescharacteristic, complemented
13#16810483C411yescharacteristic, complemented
14#16810483C2211yescharacteristic
15#16810483C2211yescharacteristic
16#16810479—21no—
17C4×C22⋊Dic53—21no—
18C4×C22⋊Dic53—21no—
19C24×C4×C53C2211yescharacteristic
20C22×C22⋊C4×C53—21no—
21C2×#95×C53—21no—
22C2×#95×C53C2212yes—
23C22×#88322—22no—
24#13133176—42no—
25#13133176—42no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C22⋊#88322—24no—
39C22⋊#88322—42no—
40C22⋊#88322—42no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45C25×C53D411yescharacteristic
46C23×C4×C53—24no—
47C23×C4×C53—22no—
48C23×C4×C53—42no—
49C23×C4×C53—42no—
50C23×C4×C53C2×C411yescharacteristic
51C23×C4×C53C2×C411yescharacteristic
52C23×C4×C53D411yescharacteristic
53C23×C4×C53D411yescharacteristic
54C23×C4×C53D411yescharacteristic
55C23×C4×C53C2311yescharacteristic
56C2×C22⋊C4×C53—24no—
57C2×C22⋊C4×C53—42no—
58C2×C22⋊C4×C53—42no—
59#95×C53—24no—
60#95×C53—42no—
61C24×C53—22no—
62C24×C53—22no—
63C24×C53—21no—
64C24×C53—21no—
65C24×C53—22no—
66C24×C53—22no—
67C24×C53—41no—
68C24×C53—41no—
69C24×C53C22⋊C411yescharacteristic
70C24×C53C22⋊C411yescharacteristic
71C24×C53C2×D411yescharacteristic
72C22×C4×C53—22no—
73C22×C4×C53—22no—
74C22×C4×C53—22no—
75C22×C4×C53—22no—
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—22no—
79C22×C4×C53—22no—
80C22×C4×C53—22no—
81C22×C4×C53—22no—
82C22×C4×C53—24no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—416no—
91C22×C4×C53—42no—
92C22×C4×C53—42no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53—42no—
96C22×C4×C53—42no—
97C22×C4×C53C22⋊C411yescharacteristic
98C22×C4×C53C22⋊C411yescharacteristic
99C22×C4×C53C22×C411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53#4114yes—
102C22×Dic53—21no—
103C22×Dic53—21no—
104C22×Dic53—21no—
105C22×Dic53—21no—
106C22⋊C4×C53—48no—
107C22⋊Dic53—42no—
108C22⋊Dic53—42no—
109C22⋊Dic53—42no—
110C22⋊Dic53—42no—
111C22⋊Dic53—42no—
112C22⋊Dic53—41no—
113C22⋊Dic53—41no—
114C4⋊Dic53—41no—
115C4⋊Dic53—41no—
116C4⋊Dic53—41no—
117C4⋊Dic53—41no—
118C4×Dic53—42no—
119C4×Dic53—42no—
120C2×#88322—42no—
121C2×#88322—42no—
122C2×#88322—42no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—24no—
128C23×C53—24no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—21no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—44no—
142C23×C53—44no—
143C23×C53—44no—
144C23×C53—44no—
145C23×C53—42no—
146C23×C53—42no—
147C23×C53—42no—
148C23×C53—42no—
149C23×C53#9912yes—
150C23×C53#10412yes—
151C23×C53C2×C22⋊C411yescharacteristic
152C23×C53#12312yes—
153C23×C53#13512yes—
154C23×C53AGL(1,ℤ/8)11yescharacteristic
155C23×C53#13711yescharacteristic
156C2×C4×C53—28no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—44no—
167C2×C4×C53—44no—
168C2×C4×C53—44no—
169C2×C4×C53—44no—
170C2×C4×C53—44no—
171C2×C4×C53—44no—
172C2×C4×C53—44no—
173C2×C4×C53—44no—
174C2×C4×C53—82no—
175C2×C4×C53—82no—
176C2×C4×C53—82no—
177C2×C4×C53—82no—
178C2×C4×C53#9912yes—
179C2×C4×C53#11712yes—
180C2×C4×C53#12312yes—
181C2×Dic53—41no—
182C2×Dic53—41no—
183C2×Dic53—41no—
184C2×Dic53—41no—
185C2×Dic53—41no—
186C2×Dic53—41no—
187C2×Dic53—41no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193#88322—82no—
194#8715—531nomaximal, Sylow
195C22×C53—22no—
196C22×C53—22no—
197C22×C53—22no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—22no—
203C22×C53—22no—
204C22×C53—22no—
205C22×C53—22no—
206C22×C53—22no—
207C22×C53—22no—
208C22×C53—22no—
209C22×C53—48no—
210C22×C53—48no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—41no—
219C22×C53#38111yescharacteristic
220C22×C53C2×#9911yescharacteristic
221C22×C53C2×#9911yescharacteristic
222C22×C53#41611yescharacteristic
223C22×C53#41711yescharacteristic
224C22×C53C2×#10411yescharacteristic
225C22×C53#42111yescharacteristic
226C22×C53#48212yes—
227C22×C53#48512yes—
228C4×C53—22no—
229C4×C53—22no—
230C4×C53—22no—
231C4×C53—41no—
232C4×C53—84no—
233Dic53—81no—
234Dic53—81no—
235Dic53—81no—
236Dic53—81no—
237#2280—531no—
238#1779—531no—
239C22⋊#95D5311yescharacteristic
240#1294—532no—
241#1294—532no—
242C2×C53—22no—
243C2×C53—22no—
244C2×C53—42no—
245C2×C53—42no—
246C2×C53#150111yescharacteristic
247C2×C53#150411yescharacteristic
248C2×C53#151311yescharacteristic
249C2×C53#151611yescharacteristic
250C2×C53#181911yescharacteristic
251C2×C53#182511yescharacteristic
252C2×C53#182611yescharacteristic
253C24×C4C2×D5311yescharacteristic
254C2×#117—531no—
255C2×#117—531no—
256C22×C22⋊C4—21no—
257C2×C22⋊C8—532no—
258C2×C22⋊C8—1061no—
259C2×C22⋊C8—1061no—
260#381—1061no—
261#380—531no—
262#380—531no—
263#380—531no—
264#380—531no—
265#379—1061no—
266C4×C22⋊C4—1061no—
267C4×C22⋊C4—1061no—
268C2×#95—21no—
269C2×#95C2×D5312yes—
270C53#871511yescharacteristic, complemented, Sylow
271C25C22⋊D5311yescharacteristic
272C23×C4—24no—
273C23×C4—22no—
274C23×C4—42no—
275C23×C4—42no—
276C23×C4C4×D5311yescharacteristic
277C23×C4C4×D5311yescharacteristic
278C23×C4D21211yescharacteristic
279C23×C4D21211yescharacteristic
280C23×C4C22⋊D5311yescharacteristic
281C23×C4C22×D5311yescharacteristic
282C22×C8—1062no—
283#117—2122no—
284#117—2122no—
285C2×C4⋊C4—531no—
286C2×C4⋊C4—531no—
287C2×C4⋊C4—531no—
288C2×C4⋊C4—531no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—1061no—
293C2×C22⋊C4—1061no—
294C2×C22⋊C4—1061no—
295C2×C22⋊C4—1061no—
296C2×C22⋊C4—1061no—
297C2×C22⋊C4—1061no—
298C2×C42—1061no—
299C2×C42—1061no—
300C22⋊C8—1064no—
301C22⋊C8—2122no—
302C22⋊C8—2122no—
303#95—24no—
304#95—42no—
305#95—1061no—
306#95—1061no—
307#95—1061no—
308#95—1061no—
309C24—22no—
310C24—22no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—41no—
316C24—41no—
317C24#1171992111yescharacteristic
318C24#1171992111yescharacteristic
319C24C2×C22⋊D5311yescharacteristic
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—416no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—1061no—
346C22×C4—1061no—
347C22×C4—1061no—
348C22×C4—1061no—
349C22×C4#1171992111yescharacteristic
350C22×C4#1171992111yescharacteristic
351C22×C4C2×C4×D5311yescharacteristic
352C22×C4C2×D21211yescharacteristic
353C22×C4#1171994714yes—
354C2×C8—2122no—
355C2×C8—2122no—
356C2×C8—2122no—
357C4⋊C4—2121no—
358C4⋊C4—2121no—
359C4⋊C4—2121no—
360C4⋊C4—2121no—
361C22⋊C4—48no—
362C22⋊C4—2122no—
363C22⋊C4—2122no—
364C22⋊C4—2122no—
365C22⋊C4—2122no—
366C22⋊C4—2122no—
367C22⋊C4—2121no—
368C22⋊C4—2121no—
369C42—2122no—
370C42—2122no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—24no—
376C23—24no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—44no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23#1313313112yes—
398C23#1313314512yes—
399C23C2×#1171992111yescharacteristic
400C23#1313319812yes—
401C23#1313323712yes—
402C23#1313324311yescharacteristic
403C23#1313324811yescharacteristic
404C2×C4—28no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4—2121no—
429C2×C4—2121no—
430C2×C4—2121no—
431C2×C4—2121no—
432C2×C4—2121no—
433C2×C4—2121no—
434C2×C4—2121no—
435C2×C4—2121no—
436C2×C4—2121no—
437C2×C4—2121no—
438C2×C4#1313313112yes—
439C2×C4#1313317312yes—
440C2×C4#1313320112yes—
441C8—4242no—
442C22—22no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—48no—
457C22—48no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22#1681048411yescharacteristic
467C22C2×#1313313111yescharacteristic
468C22C2×#1313313111yescharacteristic
469C22#1681061711yescharacteristic
470C22#1681062211yescharacteristic
471C22C2×#1313314511yescharacteristic
472C22#1681064111yescharacteristic
473C22#1681094412yes—
474C22#1681096112yes—
475C4—22no—
476C4—22no—
477C4—22no—
478C4—41no—
479C4—84no—
480C4—4241no—
481C4—4241no—
482C4—4241no—
483C4—4241no—
484C2—22no—
485C2—22no—
486C2—42no—
487C2—42no—
488C2#2027320411yescharacteristic
489C2#2027321711yescharacteristic
490C2#2027325611yescharacteristic
491C2#2027326911yescharacteristic
492C2#2027478711yescharacteristic
493C2#2027480711yescharacteristic
494C2#2027481511yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×C22⋊#95C2
socleC23×C53C2×C22⋊C4
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenterC22⋊#95D53
hypocenterC53#8715
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
3. #20272594C2semidirect
4. #20272594C2semidirect
6. C2×#13133176C4semidirect
7. C2×#13133176C4semidirect
12. #16810483C4semidirect
13. #16810483C4semidirect
270. C53#8715semidirect

Maximal subgroups

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

subgroupindex
#202770982
#202746252
#202725942
C53×C22⋊#952
#871553