Finite Group Databasebeta

← #28302902

Subgroups — #28302902

Order 13568.

898 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290571C212yesmaximal, complemented
2C2×#16811466C212yesmaximal
3#20289553C212yesmaximal, complemented
4C2×C4×C22⋊Dic53C211yesmaximal, characteristic
5C2×#13133284—22no—
6C2×C22⋊#88324C2214yes—
7#16811489—22no—
8#16811466—24no—
9C22×C22⋊Dic53C411yescharacteristic
10C22×C22⋊Dic53C2211yescharacteristic
11C22×C4×Dic53C411yescharacteristic
12C22×C4×Dic53C2211yescharacteristic
13C4×C22⋊Dic53—22no—
14C4×C22⋊Dic53—22no—
15C2×#13133124C411yescharacteristic
16C2×#13133124C2211yescharacteristic
17C24×C4×C53C411yescharacteristic
18C22×#88324—24no—
19#13133284—44no—
20C22⋊#88324—24no—
21C22⋊#88324—24no—
22C23×Dic53C2×C411yescharacteristic
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2311yescharacteristic
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53C814yescomplemented
31C2×C22⋊Dic53D412yes—
32C2×C22⋊Dic53Q812yes—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53—42no—
36C2×C4×Dic53—42no—
37C2×C4×Dic53D412yes—
38C2×C4×Dic53D412yes—
39#13133124—22no—
40#13133124—22no—
41C25×C53C2×C411yescharacteristic
42C23×C4×C53—22no—
43C23×C4×C53—21no—
44C23×C4×C53—21no—
45C23×C4×C53—22no—
46C23×C4×C53—42no—
47C23×C4×C53—42no—
48C23×C4×C53C2×C411yescharacteristic
49C23×C4×C53C2×C411yescharacteristic
50C24×C53—21no—
51C24×C53—21no—
52C24×C53—21no—
53C24×C53—21no—
54C24×C53—41no—
55C24×C53—41no—
56C24×C53—41no—
57C24×C53—41no—
58C24×C53C4⋊C412yes—
59C24×C53C2×C812yes—
60C24×C53M1612yes—
61C24×C53C22×C411yescharacteristic
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—24no—
67C22×C4×C53—24no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—48no—
71C22×C4×C53—48no—
72C22×C4×C53—42no—
73C22×C4×C53—42no—
74C22×C4×C53—42no—
75C22×C4×C53—42no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53M1612yes—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—41no—
86C22×Dic53—41no—
87C22×Dic53—41no—
88C22×Dic53—41no—
89C22×Dic53—41no—
90C22×Dic53—41no—
91C22×Dic53—41no—
92C22×Dic53—41no—
93C22×Dic53C22⋊C412yes—
94C22×Dic53C4⋊C412yes—
95C22×Dic53C2×C812yes—
96C22×Dic53C2×C812yes—
97C22×Dic53M1612yes—
98C22×Dic53C2×D412yes—
99C22×Dic53C2×D411yescharacteristic
100C22×Dic53C2×Q811yescharacteristic
101C22×Dic53#4112yes—
102C2×#88324—44no—
103C2×#88324—44no—
104C2×#88324—44no—
105C22⋊Dic53—28no—
106C22⋊Dic53—24no—
107C22⋊Dic53—24no—
108C4×Dic53—24no—
109C4×Dic53—42no—
110C4×Dic53—42no—
111C4×Dic53—42no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—22no—
117C23×C53—22no—
118C23×C53—22no—
119C23×C53—22no—
120C23×C53—22no—
121C23×C53—22no—
122C23×C53—42no—
123C23×C53—42no—
124C23×C53—42no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53—42no—
132C23×C53—42no—
133C23×C53—42no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53C4⋊C814yescomplemented
139C23×C53C2×C22⋊C411yescharacteristic
140C23×C53C2×C4⋊C411yescharacteristic
141C23×C53#11711yescharacteristic
142C23×C53C22×C811yescharacteristic
143C23×C53C2×M1611yescharacteristic
144C23×C53C2×M1611yescharacteristic
145C23×C53C2×M1611yescharacteristic
146C2×C4×C53—24no—
147C2×C4×C53—24no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—22no—
153C2×C4×C53—48no—
154C2×C4×C53—48no—
155C2×C4×C53—42no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53#9912yes—
160C2×C4×C53#10012yes—
161C2×Dic53—22no—
162C2×Dic53—22no—
163C2×Dic53—22no—
164C2×Dic53—22no—
165C2×Dic53—22no—
166C2×Dic53—22no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173C2×Dic53—42no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—42no—
179C2×Dic53—42no—
180C2×Dic53C22⋊C814yes—
181C2×Dic53#12112yes—
182C2×Dic53C22⋊Q812yes—
183#88324—84no—
184#8559—531nomaximal, Sylow
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—24no—
198C22×C53—44no—
199C22×C53—44no—
200C22×C53—44no—
201C22×C53—44no—
202C22×C53—42no—
203C22×C53—42no—
204C22×C53—42no—
205C22×C53—42no—
206C22×C53—42no—
207C22×C53—42no—
208C22×C53#38011yescharacteristic
209C22×C53C2×C22⋊C811yescharacteristic
210C22×C53#40711yescharacteristic
211C22×C53C2×#9911yescharacteristic
212C22×C53#41011yescharacteristic
213C22×C53C2×#10011yescharacteristic
214C22×C53#41311yescharacteristic
215C22×C53C2×C4⋊C811yescharacteristic
216C22×C53#42311yescharacteristic
217C22×C53#43111yescharacteristic
218C22×C53#43211yescharacteristic
219C4×C53—22no—
220C4×C53—22no—
221C4×C53—42no—
222Dic53—22no—
223Dic53—22no—
224Dic53—24no—
225Dic53—42no—
226Dic53—42no—
227C2×C4×C22⋊C4—531no—
228#1797—532no—
229C2×#323—532no—
230#1280—532no—
231C2×C53—21no—
232C2×C53—21no—
233C2×C53—21no—
234C2×C53—21no—
235C2×C53—44no—
236C2×C53#145911yescharacteristic
237C2×C53#146011yescharacteristic
238C2×C53#146211yescharacteristic
239C2×C53#146411yescharacteristic
240C2×C53#179711yescharacteristic
241C2×C53#179911yescharacteristic
242C2×C53#180011yescharacteristic
243C24×C4#615711yescharacteristic
244C22×C22⋊C4—531no—
245C22×C22⋊C4—531no—
246C22×C42—531no—
247C22×C42—531no—
248C2×C4⋊C8—1062no—
249C2×C22⋊C8—534no—
250C4×C22⋊C4—1062no—
251C4×C22⋊C4—1062no—
252C2×#95—531no—
253C2×#95—531no—
254#336—1062no—
255#323—1064no—
256C53#855911yescharacteristic, complemented, Sylow
257C25C2×#615711yescharacteristic
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—42no—
263C23×C4—42no—
264C23×C4—531no—
265C23×C4—531no—
266C23×C4—531no—
267C23×C4—531no—
268C23×C4C2×#615711yescharacteristic
269C23×C4C2×#615711yescharacteristic
270C22×C8—1064no—
271C2×C22⋊C4—532no—
272C2×C22⋊C4—532no—
273C2×C22⋊C4—534no—
274C2×C22⋊C4—1062no—
275C2×C22⋊C4—1062no—
276C2×C22⋊C4—1062no—
277C2×C22⋊C4—1062no—
278C2×C42—532no—
279C2×C42—532no—
280C2×C42—1061no—
281C2×C42—1061no—
282C2×C42—2122no—
283C2×C42—2122no—
284C4⋊C8—2124no—
285C22⋊C8—1064no—
286C22⋊C8—1064no—
287#95—1062no—
288#95—1062no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24D53⋊C812yes—
298C24C4⋊#615712yes—
299C24#1171994012yes—
300C24C22×#615711yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—48no—
310C22×C4—48no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—532no—
318C22×C4—532no—
319C22×C4—531no—
320C22×C4—531no—
321C22×C4—532no—
322C22×C4—532no—
323C22×C4—532no—
324C22×C4—532no—
325C22×C4—532no—
326C22×C4—1061no—
327C22×C4—1061no—
328C22×C4—1061no—
329C22×C4—1061no—
330C22×C4—2121no—
331C22×C4—2121no—
332C22×C4—2121no—
333C22×C4—2121no—
334C22×C4—2121no—
335C22×C4—2121no—
336C22×C4—2121no—
337C22×C4—2121no—
338C22×C4#1171994012yes—
339C22×C4C22⋊#615711yescharacteristic
340C22×C4C22⋊#615711yescharacteristic
341C2×C8—2124no—
342C2×C8—2124no—
343C2×C8—2124no—
344C22⋊C4—1064no—
345C22⋊C4—1064no—
346C22⋊C4—1068no—
347C42—1064no—
348C42—2122no—
349C42—2122no—
350C42—2122no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23#1313328414yescomplemented
378C23C2×C22⋊#615711yescharacteristic
379C23C2×C4⋊#615711yescharacteristic
380C23#1313329611yescharacteristic
381C23C2×D53⋊C811yescharacteristic
382C23C2×#1171994011yescharacteristic
383C23C2×#1171994011yescharacteristic
384C23D53⋊M1611yescharacteristic
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—532no—
399C2×C4—532no—
400C2×C4—534no—
401C2×C4—1062no—
402C2×C4—1062no—
403C2×C4—1062no—
404C2×C4—1062no—
405C2×C4—1062no—
406C2×C4—1062no—
407C2×C4—2121no—
408C2×C4—2122no—
409C2×C4—2121no—
410C2×C4—2122no—
411C2×C4—2122no—
412C2×C4—2122no—
413C2×C4—2122no—
414C2×C4—2121no—
415C2×C4—2121no—
416C2×C4—2121no—
417C2×C4—2121no—
418C2×C4—2122no—
419C2×C4—2122no—
420C2×C4#1313327512yes—
421C2×C4#1313327712yes—
422C8—4244no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—24no—
436C22—44no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22#1681156511yescharacteristic
447C22D53⋊(C22⋊C8)11yescharacteristic
448C22#1681159311yescharacteristic
449C22C2×#1313327511yescharacteristic
450C22#1681160211yescharacteristic
451C22C2×#1313327711yescharacteristic
452C22#1681160911yescharacteristic
453C22C2×#1313328411yescharacteristic
454C22#1681162611yescharacteristic
455C22#1681164011yescharacteristic
456C22#1681164311yescharacteristic
457C4—22no—
458C4—22no—
459C4—42no—
460C4—1062no—
461C4—1062no—
462C4—1064no—
463C4—2122no—
464C4—2122no—
465C2—21no—
466C2—21no—
467C2—21no—
468C2—21no—
469C2—44no—
470C2#2028995011yescharacteristic
471C2#2028995411yescharacteristic
472C2#2028996011yescharacteristic
473C2#2028996611yescharacteristic
474C2#2029057111yescharacteristic
475C2#2029057511yescharacteristic
476C2#2029057911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC24×C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC23×Dic53C23
hypercenterC24×C4#6157
hypocenterC53#8559
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. #20290571C2semidirect
3. #20289553C2semidirect
30. C2×C22⋊Dic53C8semidirect
138. C23×C53C4⋊C8semidirect
256. C53#8559semidirect
377. C23#13133284semidirect

Maximal subgroups

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

subgroupindex
#202905712
C2×#168114662
#202895532
C2×C4×C22⋊Dic532
#855953