Finite Group Databasebeta

← #27275910

Subgroups — #27275910

Order 13568.

868 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275023C212yesmaximal
2C2×#16810360C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic
4#20272968C211yesmaximal, characteristic
5C2×C53×#544C211yesmaximal, characteristic
6C22×C4⋊Dic53C2211yescharacteristic
7C2×#13133141—22no—
8C2×C22⋊#88322C2212yes—
9C2×#13133124—21no—
10C2×#13133124—21no—
11C2×#13133124C2211yescharacteristic
12#16810385—22no—
13#16810360—24no—
14#16810349—22no—
15C53×#544—24no—
16C53×#544—22no—
17C53×#544C414yescomplemented
18C2×C4⋊Q8×C53—22no—
19C2×#125×C53—22no—
20C2×C22⋊Q8×C53—21no—
21C2×C22⋊Q8×C53C2212yes—
22C2×#117×C53—21no—
23C22×C4⋊C4×C53C2211yescharacteristic
24C23×Dic53—21no—
25C23×Dic53—21no—
26C22×#88322—22no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53—42no—
29#13133141—44no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32#13133124—22no—
33#13133124—42no—
34#13133124—42no—
35C22×Q8×C53—21no—
36C22×Q8×C53D412yes—
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40C4⋊Q8×C53—28no—
41C4⋊Q8×C53—28no—
42#125×C53—28no—
43#125×C53—28no—
44C22⋊Q8×C53—24no—
45C22⋊Q8×C53—24no—
46C22⋊Q8×C53—22no—
47C22⋊Q8×C53—22no—
48C22⋊Q8×C53C2×C414yes—
49C22⋊Q8×C53D414yes—
50#117×C53—24no—
51#117×C53—24no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—22no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53—22no—
58C2×C4⋊C4×C53—22no—
59C2×C4⋊C4×C53C2×C412yes—
60C2×C4⋊C4×C53D412yes—
61C2×C4⋊C4×C53D412yes—
62C2×C22⋊C4×C53—22no—
63C2×C42×C53—22no—
64C24×C53C2×D411yescharacteristic
65C2×Q8×C53—24no—
66C2×Q8×C53—22no—
67C2×Q8×C53—42no—
68C2×Q8×C53—42no—
69C2×Q8×C53—81no—
70C2×Q8×C53—81no—
71C2×Q8×C53C22⋊C414yes—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53C22⋊C412yes—
85C22×C4×C53C22⋊C412yes—
86C22×C4×C53SD1612yes—
87C22×C4×C53Q1612yes—
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C2×D412yes—
90C22×C4×C53C2×D412yes—
91C22×C4×C53C2×D411yescharacteristic
92C22×Dic53—22no—
93C22×Dic53—22no—
94C22×Dic53—41no—
95C22×Dic53—41no—
96C22×Dic53—41no—
97C22×Dic53—41no—
98C22×Dic53—41no—
99C22×Dic53—41no—
100C22×Dic53—41no—
101C22×Dic53—41no—
102C22×Dic53—41no—
103C22×Dic53—41no—
104C22×Dic53—41no—
105C22×Dic53—41no—
106C4⋊C4×C53—24no—
107C4⋊C4×C53—24no—
108C4⋊C4×C53—24no—
109C4⋊C4×C53—28no—
110C4⋊C4×C53—24no—
111C4⋊C4×C53—24no—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53C22⋊C414yes—
116C22⋊C4×C53—24no—
117C22⋊C4×C53—24no—
118C42×C53—28no—
119C4⋊Dic53—24no—
120C4⋊Dic53—42no—
121C2×#88322—42no—
122C2×#88322—42no—
123C2×#88322—42no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53C2×C22⋊C411yescharacteristic
129C23×C53#12011yescharacteristic
130C23×C53#12011yescharacteristic
131C23×C53C2×SD1611yescharacteristic
132C23×C53C2×Q1611yescharacteristic
133C23×C53#13711yescharacteristic
134C23×C53#13711yescharacteristic
135Q8×C53—44no—
136Q8×C53—82no—
137Q8×C53—84no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—24no—
148C2×C4×C53—24no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—41no—
155C2×C4×C53—42no—
156C2×C4×C53—41no—
157C2×C4×C53—42no—
158C2×C4×C53—42no—
159C2×C4×C53—81no—
160C2×C4×C53#10314yescomplemented
161C2×C4×C53C2×C22⋊C412yes—
162C2×C4×C53#12012yes—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53—81no—
170C2×Dic53—81no—
171C2×Dic53—81no—
172C2×Dic53—81no—
173C2×Dic53—81no—
174C2×Dic53—81no—
175C2×Dic53—81no—
176C2×Dic53—81no—
177#88322—82no—
178#8406—531nomaximal, Sylow
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53#37911yescharacteristic
192C22×C53C2×#10311yescharacteristic
193C22×C53#41911yescharacteristic
194C22×C53#44811yescharacteristic
195C22×C53#44811yescharacteristic
196C22×C53#45111yescharacteristic
197C22×C53#45111yescharacteristic
198C22×C53#45411yescharacteristic
199C22×C53#45611yescharacteristic
200C22×C53B(4,𝔽2)11yescharacteristic
201C22×C53#45811yescharacteristic
202C4×C53—24no—
203C4×C53—22no—
204C4×C53—22no—
205C4×C53—44no—
206C4×C53—42no—
207C4×C53—44no—
208Dic53—82no—
209Dic53—82no—
210C2×#544D5311yescharacteristic
211#1872—532no—
212C2×#328—532no—
213C2×#324—531no—
214#1441—531no—
215C2×C53—21no—
216C2×C53—21no—
217C2×C53—21no—
218C2×C53—21no—
219C2×C53#159911yescharacteristic
220C2×C53#160511yescharacteristic
221C2×C53#161111yescharacteristic
222C2×C53#161711yescharacteristic
223C2×C53#178711yescharacteristic
224C2×C53#179011yescharacteristic
225C2×C53#179311yescharacteristic
226#544—24no—
227#544—22no—
228#544Dic5314yescomplemented
229C2×C4⋊Q8—22no—
230C2×#125—22no—
231C2×C22⋊Q8—21no—
232C2×C22⋊Q8C2×D5312yes—
233C2×#117—21no—
234C22×C4⋊C4—531no—
235C22×C4⋊C4C2×D5311yescharacteristic
236C2×#103—1062no—
237C2×C22⋊C8—532no—
238C2×#95—531no—
239C2×#95—1061no—
240C2×#95—1061no—
241#340—1062no—
242#328—1064no—
243#324—1062no—
244C53#840611yescharacteristic, complemented, Sylow
245C22×Q8—21no—
246C22×Q8C22⋊D5312yes—
247C23×C4—1061no—
248C23×C4—1061no—
249C23×C4C22⋊D5311yescharacteristic
250C23×C4C22⋊D5311yescharacteristic
251C23×C4C22×D5311yescharacteristic
252C22×C8—1062no—
253C4⋊Q8—28no—
254C4⋊Q8—28no—
255#125—28no—
256#125—28no—
257C22⋊Q8—24no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—24no—
261C22⋊Q8C2×Dic5314yes—
262C22⋊Q8C22⋊D5314yes—
263#117—24no—
264#117—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—1062no—
273C2×C4⋊C4—2122no—
274C2×C4⋊C4C2×Dic5312yes—
275C2×C4⋊C4C22⋊D5312yes—
276C2×C4⋊C4C22⋊D5312yes—
277C2×C22⋊C4—22no—
278C2×C42—22no—
279#103—2124no—
280C22⋊C8—1062no—
281C22⋊C8—1062no—
282#95—1062no—
283#95—2122no—
284#95—2122no—
285C24C2×C22⋊D5311yescharacteristic
286C2×Q8—24no—
287C2×Q8—22no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—81no—
291C2×Q8—81no—
292C2×Q8C22⋊Dic5314yes—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—1062no—
306C22×C4—1062no—
307C22×C4—2121no—
308C22×C4—2121no—
309C22×C4—2121no—
310C22×C4—2121no—
311C22×C4—2121no—
312C22×C4—2121no—
313C22×C4—2121no—
314C22×C4—2121no—
315C22×C4—2121no—
316C22×C4—2121no—
317C22×C4—2121no—
318C22×C4—2121no—
319C22×C4#1171992412yes—
320C22×C4#1171992512yes—
321C22×C4C22⋊Dic5312yes—
322C22×C4C22⋊Dic5312yes—
323C22×C4C22×Dic5311yescharacteristic
324C22×C4C2×C22⋊D5312yes—
325C22×C4C2×C22⋊D5312yes—
326C22×C4C2×C22⋊D5311yescharacteristic
327C2×C8—2122no—
328C2×C8—2122no—
329C2×C8—2122no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—28no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—1064no—
340C4⋊C4—2122no—
341C4⋊C4C22⋊Dic5314yes—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C42—28no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23C2×C22⋊Dic5311yescharacteristic
350C23#1313318611yescharacteristic
351C23#1313318611yescharacteristic
352C23C2×#1171992411yescharacteristic
353C23C2×#1171992511yescharacteristic
354C23#1313324811yescharacteristic
355C23#1313324811yescharacteristic
356Q8—44no—
357Q8—82no—
358Q8—84no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—81no—
381C2×C4—4241no—
382C2×C4—4241no—
383C2×C4—4241no—
384C2×C4—4241no—
385C2×C4—4241no—
386C2×C4—4241no—
387C2×C4—4241no—
388C2×C4—4241no—
389C2×C4—2122no—
390C2×C4—2122no—
391C2×C4—2122no—
392C2×C4—2122no—
393C2×C4—2122no—
394C2×C4—2122no—
395C2×C4#1313314114yescomplemented
396C2×C4C2×C22⋊Dic5312yes—
397C2×C4#1313318612yes—
398C8—4242no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22#1681047911yescharacteristic
412C22C2×#1313314111yescharacteristic
413C22#1681063211yescharacteristic
414C22#1681076211yescharacteristic
415C22#1681076211yescharacteristic
416C22#1681077911yescharacteristic
417C22#1681077911yescharacteristic
418C22#1681079611yescharacteristic
419C22#1681080611yescharacteristic
420C22#1681081011yescharacteristic
421C22#1681081311yescharacteristic
422C4—24no—
423C4—22no—
424C4—22no—
425C4—42no—
426C4—44no—
427C4—44no—
428C4—4242no—
429C4—4242no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—21no—
434C2#2027365511yescharacteristic
435C2#2027368911yescharacteristic
436C2#2027372311yescharacteristic
437C2#2027375711yescharacteristic
438C2#2027465011yescharacteristic
439C2#2027466311yescharacteristic
440C2#2027467411yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#544C2
socleC23×C53#120
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenterC2×#544D53
hypocenterC53#8406
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
17. C53×#544C4semidirect
160. C2×C4×C53#103semidirect
228. #544Dic53semidirect
244. C53#8406semidirect
395. C2×C4#13133141semidirect

Maximal subgroups

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

subgroupindex
#202750232
C2×#168103602
C2×#168103492
#202729682
C2×C53×#5442
#840653