Finite Group Databasebeta

← #27275908

Subgroups — #27275908

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
1C2×#16811254C211yesmaximal, characteristic
2#20275020C212yesmaximal
3C2×#16810361C212yesmaximal
4#20272968C211yesmaximal, characteristic
5C2×C53×#324C211yesmaximal, characteristic
6#16811254—24no—
7#16811254—22no—
8#16811254C414yescomplemented
9C2×#13133216—22no—
10C2×#13133208—22no—
11C2×#13133196—21no—
12C2×#13133196C2212yes—
13C2×#13133176—21no—
14C22×C4⋊Dic53C2211yescharacteristic
15C2×#13133142—22no—
16C2×#13133124—21no—
17C2×#13133124—21no—
18C2×#13133124C2211yescharacteristic
19#16810386—22no—
20#16810361—24no—
21C22×C4⋊C4×C53C2211yescharacteristic
22C2×C22⋊C8×C53C2212yes—
23C53×#324—22no—
24C22×Dic106—21no—
25C22×Dic106D412yes—
26C23×Dic53—21no—
27C23×Dic53—21no—
28#13133216—28no—
29#13133216—28no—
30#13133208—28no—
31#13133208—28no—
32#13133196—24no—
33#13133196—24no—
34#13133196—22no—
35#13133196—22no—
36#13133196C2×C414yes—
37#13133196D414yes—
38#13133176—24no—
39#13133176—24no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—22no—
44C2×C4⋊Dic53—42no—
45C2×C4⋊Dic53D412yes—
46C2×#11719919—22no—
47C2×#11719919—22no—
48C2×C22⋊Dic53—22no—
49C2×C4×Dic53—22no—
50#13133142—44no—
51#13133124—22no—
52#13133124—42no—
53#13133124—42no—
54C23×C4×C53D411yescharacteristic
55C23×C4×C53D411yescharacteristic
56C23×C4×C53C2311yescharacteristic
57C22×C8×C53—22no—
58C2×C4⋊C4×C53—22no—
59C2×C4⋊C4×C53—22no—
60C2×C4⋊C4×C53C2×C412yes—
61C2×C4⋊C4×C53D412yes—
62C22⋊C8×C53—22no—
63C22⋊C8×C53—22no—
64C24×C53C2×D411yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53C22⋊C412yes—
76C22×C4×C53C22⋊C412yes—
77C22×C4×C53SD1612yes—
78C22×C4×C53Q1612yes—
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D412yes—
81C22×C4×C53C2×D412yes—
82C22×C4×C53C2×D411yescharacteristic
83C22×Dic53—22no—
84C22×Dic53—22no—
85C22×Dic53—22no—
86C22×Dic53—22no—
87C22×Dic53—41no—
88C22×Dic53—41no—
89C22×Dic53—41no—
90C22×Dic53—41no—
91C22×Dic53—41no—
92C22×Dic53—41no—
93C22×Dic53—41no—
94C22×Dic53—41no—
95C22×Dic53—41no—
96C22×Dic53—41no—
97C22×Dic53—41no—
98C22×Dic53—41no—
99C2×Dic106—24no—
100C2×Dic106—22no—
101C2×Dic106—42no—
102C2×Dic106—42no—
103C2×Dic106—81no—
104C2×Dic106—81no—
105C2×Dic106C22⋊C414yes—
106C2×C8×C53—42no—
107C2×C8×C53—42no—
108C2×C8×C53—42no—
109C4⋊C4×C53—24no—
110C4⋊C4×C53—24no—
111C22⋊Dic53—24no—
112C22⋊Dic53—24no—
113C4⋊Dic53—24no—
114C4⋊Dic53—24no—
115C4⋊Dic53—24no—
116C4⋊Dic53—22no—
117C4⋊Dic53—24no—
118C4⋊Dic53—42no—
119C4⋊Dic53C22⋊C414yes—
120#11719919—28no—
121#11719919—24no—
122#11719919—24no—
123C4×Dic53—28no—
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
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
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—41no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53—81no—
151C2×C4×C53#10314yescomplemented
152C2×C4×C53C2×C22⋊C412yes—
153C2×C4×C53#12012yes—
154C2×Dic53—24no—
155C2×Dic53—24no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—81no—
167C2×Dic53—81no—
168C2×Dic53—81no—
169C2×Dic53—81no—
170C2×Dic53—81no—
171C2×Dic53—81no—
172C2×Dic53—81no—
173C2×Dic53—81no—
174Dic106—44no—
175Dic106—82no—
176Dic106—84no—
177C8×C53—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—42no—
206Dic53—44no—
207Dic53—44no—
208Dic53—82no—
209Dic53—82no—
210C2×#544—531no—
211#1872—532no—
212C2×#328—532no—
213C2×#324D5311yescharacteristic
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—534no—
227#544—1064no—
228#544—1062no—
229C2×C4⋊Q8—1062no—
230C2×#125—1062no—
231C2×C22⋊Q8—532no—
232C2×C22⋊Q8—1061no—
233C2×#117—1061no—
234C22×C4⋊C4—531no—
235C22×C4⋊C4C2×D5311yescharacteristic
236C2×#103—1062no—
237C2×C22⋊C8C2×D5312yes—
238C2×#95—531no—
239C2×#95—1061no—
240C2×#95—1061no—
241#340—1062no—
242#328—1064no—
243#324—22no—
244C53#840611yescharacteristic, complemented, Sylow
245C22×Q8—532no—
246C22×Q8—1061no—
247C23×C4—1061no—
248C23×C4—1061no—
249C23×C4D21211yescharacteristic
250C23×C4D21211yescharacteristic
251C23×C4C22×D5311yescharacteristic
252C22×C8—22no—
253C4⋊Q8—1068no—
254C4⋊Q8—1068no—
255#125—1068no—
256#125—1068no—
257C22⋊Q8—534no—
258C22⋊Q8—534no—
259C22⋊Q8—1064no—
260C22⋊Q8—1062no—
261C22⋊Q8—1062no—
262C22⋊Q8—1064no—
263#117—1064no—
264#117—1064no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—532no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4—1061no—
270C2×C4⋊C4—1061no—
271C2×C4⋊C4—1062no—
272C2×C4⋊C4—1062no—
273C2×C4⋊C4—1062no—
274C2×C4⋊C4—2122no—
275C2×C4⋊C4C4×D5312yes—
276C2×C4⋊C4C22⋊D5312yes—
277C2×C22⋊C4—1062no—
278C2×C42—1062no—
279#103—2124no—
280C22⋊C8—22no—
281C22⋊C8—22no—
282#95—1062no—
283#95—2122no—
284#95—2122no—
285C24C2×D21211yescharacteristic
286C2×Q8—534no—
287C2×Q8—1064no—
288C2×Q8—1062no—
289C2×Q8—4241no—
290C2×Q8—4241no—
291C2×Q8—2122no—
292C2×Q8—2122no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—1062no—
304C22×C4—1062no—
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#1171991312yes—
320C22×C4Dic21212yes—
321C22×C4#1171992112yes—
322C22×C4#1171992112yes—
323C22×C4C2×C4×D5311yescharacteristic
324C22×C4D4×D5312yes—
325C22×C4D4×D5312yes—
326C22×C4C2×C22⋊D5311yescharacteristic
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—534no—
333C4⋊C4—1064no—
334C4⋊C4—1064no—
335C4⋊C4—1064no—
336C4⋊C4—1068no—
337C4⋊C4—1064no—
338C4⋊C4—1064no—
339C4⋊C4—1062no—
340C4⋊C4—1064no—
341C4⋊C4—2122no—
342C22⋊C4—1064no—
343C22⋊C4—1064no—
344C42—1068no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23C2×#1171992111yescharacteristic
350C23#1313318711yescharacteristic
351C23#1313318711yescharacteristic
352C23C2×#1171991311yescharacteristic
353C23C2×Dic21211yescharacteristic
354C23#1313325011yescharacteristic
355C23#1313325011yescharacteristic
356Q8—4242no—
357Q8—4244no—
358Q8—2124no—
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—22no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—1064no—
376C2×C4—1064no—
377C2×C4—4241no—
378C2×C4—4241no—
379C2×C4—4241no—
380C2×C4—4241no—
381C2×C4—4241no—
382C2×C4—4241no—
383C2×C4—4241no—
384C2×C4—4241no—
385C2×C4—2122no—
386C2×C4—2122no—
387C2×C4—2122no—
388C2×C4—2122no—
389C2×C4—2122no—
390C2×C4—2122no—
391C2×C4—2122no—
392C2×C4—2122no—
393C2×C4—2122no—
394C2×C4—2122no—
395C2×C4#1313314214yescomplemented
396C2×C4C22⋊C4×D5312yes—
397C2×C4#1313318512yes—
398C8—82no—
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#1681047811yescharacteristic
412C22C2×#1313314211yescharacteristic
413C22#1681063311yescharacteristic
414C22#1681076411yescharacteristic
415C22#1681076411yescharacteristic
416C22#1681078111yescharacteristic
417C22#1681078111yescharacteristic
418C22#1681079811yescharacteristic
419C22#1681080811yescharacteristic
420C22#1681081111yescharacteristic
421C22#1681081411yescharacteristic
422C4—24no—
423C4—22no—
424C4—22no—
425C4—42no—
426C4—4242no—
427C4—4242no—
428C4—2124no—
429C4—2124no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—21no—
434C2#2027365711yescharacteristic
435C2#2027369111yescharacteristic
436C2#2027372511yescharacteristic
437C2#2027375911yescharacteristic
438C2#2027465111yescharacteristic
439C2#2027466111yescharacteristic
440C2#2027467611yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#324C2
socleC23×C53#120
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenterC2×#324D53
hypocenterC53#8406
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
8. #16811254C4semidirect
151. C2×C4×C53#103semidirect
244. C53#8406semidirect
395. C2×C4#13133142semidirect

Maximal subgroups

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

subgroupindex
C2×#168112542
#202750202
C2×#168103612
#202729682
C2×C53×#3242
#840653