Finite Group Databasebeta

← #28314109

Subgroups — #28314109

Order 13568.

1098 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291614C212yesmaximal, complemented
2#20289979C212yesmaximal, complemented
3#20289977C212yesmaximal, complemented
4#20289967C212yesmaximal
5#20289966C212yesmaximal
6#20289957C212yesmaximal, complemented
7#20289947C212yesmaximal
8C2×#16811100C211yesmaximal, characteristic
9#16811675—28no—
10#16811644C2214yes—
11#16811643C2214yes—
12#16811640C2214yes—
13#16811469C2214yes—
14#16811469C2214yes—
15#16811468C2214yes—
16#16811466C2214yes—
17C23×Dic106C411yescharacteristic
18#16811100—28no—
19C2×C4×Dic106C411yescharacteristic
20C2×C4×Dic106C411yescharacteristic
21C2×C4×Dic106C2211yescharacteristic
22C2×C4×Dic106C2211yescharacteristic
23C2×#13133182C411yescharacteristic
24C2×#13133182C411yescharacteristic
25C2×#13133182C2211yescharacteristic
26C2×#13133182C2211yescharacteristic
27C2×#13133176C2211yescharacteristic
28C2×#13133173C411yescharacteristic
29C2×#13133173C411yescharacteristic
30C2×#13133173C2211yescharacteristic
31C2×#13133173C2211yescharacteristic
32C2×#117×C53C411yescharacteristic
33#13133284—24no—
34#13133284—24no—
35C4⋊#88324—24no—
36C22⋊#88324C2318yes—
37#13133272—24no—
38#13133272—24no—
39C4×#88324—24no—
40C22×Dic106—24no—
41C22×Dic106—24no—
42C22×Dic106—24no—
43C22×Dic106C2×C411yescharacteristic
44C22×Dic106C2×C411yescharacteristic
45C22×Dic106C2×C411yescharacteristic
46C22×Dic106C2×C411yescharacteristic
47C23×Dic53C2×C411yescharacteristic
48C23×Dic53C2×C411yescharacteristic
49C4×Dic106—24no—
50C4×Dic106—24no—
51C4×Dic106—24no—
52C4×Dic106—24no—
53#13133182—24no—
54#13133182—24no—
55#13133182—24no—
56#13133182D418yes—
57#13133176—24no—
58#13133173—24no—
59#13133173—24no—
60#13133173—24no—
61#13133173—24no—
62C2×C4⋊Dic53C2×C411yescharacteristic
63C2×C4⋊Dic53C2311yescharacteristic
64C2×#11719919C2×C411yescharacteristic
65C2×#11719919C2×C411yescharacteristic
66C2×#11719919C2×C411yescharacteristic
67C2×#11719919C2×C411yescharacteristic
68C2×#11719919C2×C411yescharacteristic
69C2×#11719919C2×C411yescharacteristic
70C2×#11719919C2311yescharacteristic
71C2×#11719919C2311yescharacteristic
72C2×C22⋊Dic53C2×C411yescharacteristic
73C2×C22⋊Dic53C2311yescharacteristic
74C2×C4×Dic53C2×C411yescharacteristic
75C2×C4×Dic53C2×C411yescharacteristic
76C2×C4×Dic53C2×C411yescharacteristic
77C2×C4×Dic53C2×C411yescharacteristic
78C2×C4×Dic53C2×C411yescharacteristic
79C2×C4×Dic53C2×C411yescharacteristic
80C2×C4×Dic53C2×C411yescharacteristic
81C2×C4×Dic53C2311yescharacteristic
82C2×C4×Dic53C2311yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84C23×C4×C53C2×C411yescharacteristic
85#117×C53—24no—
86C2×C4⋊C4×C53C2×C411yescharacteristic
87C2×C4⋊C4×C53C2×C411yescharacteristic
88C2×C22⋊C4×C53C2×C411yescharacteristic
89C2×C22⋊C4×C53C2×C411yescharacteristic
90C2×C42×C53C2×C411yescharacteristic
91C2×C42×C53C2×C411yescharacteristic
92C24×C53C22×C411yescharacteristic
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53C22×C411yescharacteristic
97C22×C4×C53C22×C411yescharacteristic
98C22×C4×C53C22×C411yescharacteristic
99C22×C4×C53C22×C411yescharacteristic
100C22×C4×C53C22×C411yescharacteristic
101C22×C4×C53C22×C411yescharacteristic
102C22×Dic53—22no—
103C22×Dic53—22no—
104C22×Dic53—22no—
105C22×Dic53—22no—
106C22×Dic53—22no—
107C22×Dic53M1614yes—
108C22×Dic53C22×C411yescharacteristic
109C22×Dic53C22×C411yescharacteristic
110C22×Dic53C22×C411yescharacteristic
111C22×Dic53C22×C411yescharacteristic
112C22×Dic53C22×C411yescharacteristic
113C22×Dic53C22×C411yescharacteristic
114C22×Dic53C22×C411yescharacteristic
115C22×Dic53C2411yescharacteristic
116C2×Dic106—232no—
117C2×Dic106—22no—
118C2×Dic106—22no—
119C2×Dic106—22no—
120C2×Dic106—22no—
121C2×Dic106—22no—
122C2×Dic106—22no—
123C2×Dic106—22no—
124C2×Dic106—22no—
125C2×Dic106—22no—
126C2×Dic106—22no—
127C2×Dic106C22⋊C414yes—
128C2×Dic106M1614yes—
129C2×#88324—28no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53C22⋊C414yes—
132C22⋊C4×C53—22no—
133C22⋊C4×C53—22no—
134C42×C53—22no—
135C42×C53—22no—
136C22⋊Dic53—22no—
137C22⋊Dic53—22no—
138C4⋊Dic53—22no—
139C4⋊Dic53—22no—
140#11719919—22no—
141#11719919—22no—
142#11719919—22no—
143#11719919—22no—
144#11719919—22no—
145#11719919—22no—
146#11719919C22⋊C414yes—
147#11719919C22⋊C414yes—
148C4×Dic53—22no—
149C4×Dic53—22no—
150C4×Dic53—22no—
151C4×Dic53—22no—
152C4×Dic53—22no—
153C4×Dic53—22no—
154C4×Dic53—22no—
155C4×Dic53C2×D414yes—
156C4×Dic53C2×D414yes—
157C4×Dic53C2×D414yes—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—41no—
161C23×C53—41no—
162C23×C53C2×M1612yes—
163C23×C53C23×C411yescharacteristic
164C2×C4×C53—28no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53C2×C22⋊C412yes—
176C2×C4×C53C2×C22⋊C412yes—
177C2×C4×C53C2×C22⋊C412yes—
178C2×C4×C53C2×M1612yes—
179C2×Dic53—28no—
180C2×Dic53—21no—
181C2×Dic53—21no—
182C2×Dic53—21no—
183C2×Dic53—21no—
184C2×Dic53—21no—
185C2×Dic53—21no—
186C2×Dic53—28no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53C2×C22⋊C412yes—
194C2×Dic53C2×C22⋊C412yes—
195C2×Dic53C2×C22⋊C412yes—
196C2×Dic53C2×M1612yes—
197C2×Dic53C2×M1612yes—
198C2×Dic53C2×M1612yes—
199C2×Dic53C2×M1612yes—
200C2×Dic53C22×D412yes—
201Dic106—28no—
202Dic106—28no—
203Dic106—28no—
204Dic106—28no—
205#88324—48no—
206#17371—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—22no—
210C22×C53—42no—
211C22×C53—42no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53C22×C22⋊C411yescharacteristic
215C22×C53C22×M1611yescharacteristic
216C22×C53C22×M1611yescharacteristic
217C4×C53—22no—
218C4×C53—22no—
219C4×C53—22no—
220C4×C53—22no—
221C4×C53—41no—
222C4×C53—41no—
223Dic53—22no—
224Dic53—22no—
225Dic53—22no—
226Dic53—22no—
227Dic53—22no—
228Dic53—41no—
229Dic53—41no—
230Dic53#40714yes—
231C2×#519—531no—
232#2983—532no—
233#1467—532no—
234#1467—532no—
235#1464—532no—
236#1464—532no—
237#1461—532no—
238#1458—532no—
239C2×C53—21no—
240C2×C53—21no—
241C2×C53—42no—
242C2×C53C2×#40711yescharacteristic
243C2×C53#288411yescharacteristic
244C2×C53#288911yescharacteristic
245C23×Q8—531no—
246#519—1068no—
247C2×C4×Q8—531no—
248C2×C4×Q8—531no—
249C2×C4×Q8—531no—
250C2×C4×Q8—531no—
251C2×C4×Q8—531no—
252C2×C4×Q8—531no—
253C2×C4×Q8—531no—
254C2×C4×Q8—531no—
255C2×#117—531no—
256C2×#117—531no—
257C2×#117—531no—
258C2×#117—531no—
259C2×#117—531no—
260C2×#117#615711yescharacteristic
261#446—1068no—
262#432—534no—
263#432—534no—
264#431—534no—
265#324—534no—
266#324—534no—
267#324—534no—
268#323—534no—
269C53#1737111yescharacteristic, complemented, Sylow
270C22×Q8—531no—
271C22×Q8—531no—
272C22×Q8—531no—
273C22×Q8—531no—
274C22×Q8—1064no—
275C22×Q8—1064no—
276C22×Q8—1064no—
277C23×C4—531no—
278C23×C4—531no—
279C23×C4C2×#615711yescharacteristic
280C4×Q8—538no—
281C4×Q8—1064no—
282C4×Q8—1064no—
283C4×Q8—1064no—
284C4×Q8—1064no—
285C4×Q8—1064no—
286C4×Q8—1064no—
287C4×Q8—1064no—
288#117—24no—
289#117—1064no—
290#117—1064no—
291#117—1064no—
292#117—1064no—
293#117—1064no—
294C2×C4⋊C4—531no—
295C2×C4⋊C4—531no—
296C2×C4⋊C4—531no—
297C2×C4⋊C4—531no—
298C2×C4⋊C4—531no—
299C2×C4⋊C4—531no—
300C2×C4⋊C4—531no—
301C2×C4⋊C4—531no—
302C2×C4⋊C4—531no—
303C2×C4⋊C4—531no—
304C2×C4⋊C4C2×#615711yescharacteristic
305C2×C4⋊C4C2×#615711yescharacteristic
306C2×C22⋊C4—531no—
307C2×C22⋊C4—531no—
308C2×C22⋊C4C2×#615711yescharacteristic
309C2×C22⋊C4C2×#615711yescharacteristic
310C2×C42—531no—
311C2×C42—531no—
312C2×C42—531no—
313C2×C42—531no—
314C2×C42—531no—
315C2×C42—531no—
316C2×C42—531no—
317C2×C42—531no—
318C2×C42—531no—
319C2×C42—531no—
320C2×C42C2×#615711yescharacteristic
321C2×C42C2×#615711yescharacteristic
322C4⋊C8—1064no—
323C4⋊C8—1064no—
324C4⋊C8—1064no—
325C22⋊C8—538no—
326#97—1064no—
327#97—1064no—
328C4×C8—1064no—
329C24C22×#615711yescharacteristic
330C2×Q8—534no—
331C2×Q8—534no—
332C2×Q8—1062no—
333C2×Q8—1062no—
334C2×Q8—1062no—
335C2×Q8—1062no—
336C2×Q8—1062no—
337C2×Q8—1062no—
338C2×Q8—1062no—
339C2×Q8—1062no—
340C2×Q8—10632no—
341C2×Q8—1062no—
342C2×Q8—1062no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—531no—
347C22×C4—534no—
348C22×C4—531no—
349C22×C4—531no—
350C22×C4—531no—
351C22×C4—531no—
352C22×C4—531no—
353C22×C4—531no—
354C22×C4—531no—
355C22×C4—1062no—
356C22×C4—1062no—
357C22×C4—1062no—
358C22×C4—1062no—
359C22×C4—1062no—
360C22×C4C22×#615711yescharacteristic
361C22×C4C22×#615711yescharacteristic
362C22×C4C22×#615711yescharacteristic
363C22×C4C22×#615711yescharacteristic
364C22×C4C22×#615711yescharacteristic
365C22×C4C22×#615711yescharacteristic
366C2×C8—1068no—
367C4⋊C4—22no—
368C4⋊C4—534no—
369C4⋊C4—534no—
370C4⋊C4—1062no—
371C4⋊C4—1062no—
372C4⋊C4—1062no—
373C4⋊C4—1062no—
374C4⋊C4—1062no—
375C4⋊C4—1062no—
376C4⋊C4—1062no—
377C4⋊C4—1062no—
378C4⋊C4C22⋊#615714yes—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—1062no—
382C22⋊C4—1062no—
383C42—22no—
384C42—22no—
385C42—534no—
386C42—534no—
387C42—534no—
388C42—1062no—
389C42—1062no—
390C42—1062no—
391C42—1062no—
392C42—1062no—
393C42—1062no—
394C42—1062no—
395C23—21no—
396C23—21no—
397C23—41no—
398C23—41no—
399C23D53⋊M1612yes—
400C23C23×#615711yescharacteristic
401Q8—1068no—
402Q8—1068no—
403Q8—1068no—
404Q8—1068no—
405C2×C4—28no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—532no—
417C2×C4—532no—
418C2×C4—532no—
419C2×C4—532no—
420C2×C4—532no—
421C2×C4—532no—
422C2×C4—532no—
423C2×C4—532no—
424C2×C4—1068no—
425C2×C4—1068no—
426C2×C4—1061no—
427C2×C4—1061no—
428C2×C4—1061no—
429C2×C4—1061no—
430C2×C4—1061no—
431C2×C4—1061no—
432C2×C4—1061no—
433C2×C4—1061no—
434C2×C4—2121no—
435C2×C4—2121no—
436C2×C4—2121no—
437C2×C4—2121no—
438C2×C4C2×C22⋊#615712yes—
439C2×C4C2×C22⋊#615712yes—
440C2×C4C2×C22⋊#615712yes—
441C2×C4D53⋊M1612yes—
442C8—2128no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—42no—
447C22—42no—
448C22—41no—
449C22—41no—
450C22C22×C22⋊#615711yescharacteristic
451C22C2×D53⋊M1611yescharacteristic
452C22C2×D53⋊M1611yescharacteristic
453C4—22no—
454C4—22no—
455C4—22no—
456C4—22no—
457C4—41no—
458C4—41no—
459C4—534no—
460C4—1062no—
461C4—1062no—
462C4—1062no—
463C4—1062no—
464C4—1062no—
465C4—2121no—
466C4—2121no—
467C2—21no—
468C2—21no—
469C2—42no—
470C2D53⋊#40711yescharacteristic
471C2#2029131711yescharacteristic
472C2#2029132711yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC2×#117×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC24×C53C22×C4
cosocleC22×Dic53C24
hypercenterC2×#117#6157
hypocenterC53#17371
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #20291614C2semidirect
2. #20289979C2semidirect
3. #20289977C2semidirect
6. #20289957C2semidirect
269. C53#17371semidirect

Maximal subgroups

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

subgroupindex
#202916142
#202899792
#202899772
#202899672
#202899662
#202899572
#202899472
C2×#168111002
#1737153