Finite Group Databasebeta

← #28306784

Subgroups — #28306784

Order 13568.

678 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×#16811504C212yesmaximal
2C2×#16811504C212yesmaximal
3C2×#16811503C211yesmaximal, characteristic
4C2×#16811503C211yesmaximal, characteristic
5#20277070C211yesmaximal, characteristic
6C22×#11719940C2212yes—
7C22×#11719940C2212yes—
8#16811504—24no—
9#16811504C418yescomplemented
10#16811503—21no—
11#16811503—21no—
12#16811503—21no—
13#16811503—21no—
14C2×#13133173C2211yescharacteristic
15C2×#13133173C2211yescharacteristic
16C22×#11719919C2211yescharacteristic
17C4×#11719919—21no—
18C4×#11719919—21no—
19#16810474—21no—
20#16810474—21no—
21C22⋊C4×Dic53C411yescharacteristic
22C22⋊C4×Dic53C411yescharacteristic
23C4×C22⋊Dic53C411yescharacteristic
24C4×C22⋊C4×C53C411yescharacteristic
25C2×#11719940—22no—
26C2×#11719940—22no—
27C2×#11719940—22no—
28C2×#11719940—22no—
29C2×#11719940—22no—
30C2×#11719940—22no—
31C2×#11719940—22no—
32C2×#11719940—22no—
33C2×#11719940—22no—
34C2×#11719940—22no—
35C2×#11719940C2×C414yes—
36C2×#11719940C2×C414yes—
37C22×#88324—22no—
38C22×#88324—22no—
39C23×Dic53C2×C411yescharacteristic
40C23×Dic53C2311yescharacteristic
41#13133173—21no—
42#13133173—21no—
43#13133173—22no—
44#13133173C2×C414yes—
45#13133173D412yes—
46#13133173Q812yes—
47C2×#11719919—21no—
48C2×#11719919—21no—
49C2×#11719919—22no—
50C2×#11719919—22no—
51C2×#11719919D411yescharacteristic
52C2×#11719919D411yescharacteristic
53C2×#11719919D411yescharacteristic
54C2×#11719919D411yescharacteristic
55C2×C22⋊Dic53C2×C411yescharacteristic
56C2×C22⋊Dic53C2×C411yescharacteristic
57C2×C4×Dic53—21no—
58C2×C4×Dic53—21no—
59C2×C4×Dic53—21no—
60C2×C4×Dic53—21no—
61C2×C4×Dic53—21no—
62#13133124—21no—
63#13133124—21no—
64#13133124—21no—
65C23×C4×C53C2×C411yescharacteristic
66C2×C22⋊C4×C53C2×C411yescharacteristic
67C2×C22⋊C4×C53C2×C411yescharacteristic
68C2×C42×C53—21no—
69#95×C53—21no—
70C24×C53C22×C411yescharacteristic
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—81no—
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—81no—
83C22×Dic53C22⋊C411yescharacteristic
84C22×Dic53C22⋊C411yescharacteristic
85C22×Dic53C22×C412yes—
86C22×Dic53C2×D411yescharacteristic
87C22×Dic53C2×D411yescharacteristic
88C22×Dic53C2×D411yescharacteristic
89C22×Dic53C2×Q811yescharacteristic
90C22×Dic53#4111yescharacteristic
91C22×Dic53#4111yescharacteristic
92C22×Dic53#4112yes—
93C22×Dic53#4112yes—
94C22×Dic53#4112yes—
95#11719940—24no—
96#11719940—24no—
97#11719940—24no—
98#11719940—24no—
99#11719940—42no—
100#11719940—42no—
101#11719940—42no—
102#11719940—42no—
103C2×#88324—24no—
104C2×#88324—24no—
105C2×#88324—42no—
106C2×#88324—42no—
107C2×#88324—42no—
108C2×#88324—42no—
109C22⋊C4×C53—21no—
110C22⋊C4×C53—21no—
111C22⋊C4×C53C4212yes—
112C22⋊C4×C53C4⋊C412yes—
113C42×C53—81no—
114C22⋊Dic53—21no—
115C22⋊Dic53—21no—
116C22⋊Dic53C4212yes—
117C22⋊Dic53C4⋊C412yes—
118#11719919—24no—
119#11719919—22no—
120#11719919—22no—
121#11719919—22no—
122#11719919—22no—
123#11719919—22no—
124#11719919—22no—
125C4×Dic53—24no—
126C4×Dic53—22no—
127C4×Dic53—22no—
128C4×Dic53—81no—
129C4×Dic53—81no—
130C4×Dic53—81no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53C2×C4211yescharacteristic
134C23×C53C2×C22⋊C411yescharacteristic
135C23×C53C2×C4⋊C411yescharacteristic
136C23×C53#11711yescharacteristic
137C23×C53#11711yescharacteristic
138C23×C53#11711yescharacteristic
139C23×C53#11711yescharacteristic
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—81no—
151C2×C4×C53—81no—
152C2×Dic53—22no—
153C2×Dic53—22no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—81no—
163C2×Dic53—81no—
164C2×Dic53#11712yes—
165C2×Dic53#11712yes—
166C2×Dic53C4×D412yes—
167C2×Dic53C4×D412yes—
168C2×Dic53C4×D412yes—
169C2×Dic53C4×Q812yes—
170C2×Dic53#12111yescharacteristic
171C2×Dic53#12111yescharacteristic
172C2×Dic53C22⋊Q811yescharacteristic
173C2×Dic53C22⋊Q811yescharacteristic
174C2×Dic53#12312yes—
175C2×Dic53#12312yes—
176C2×Dic53#12412yes—
177C2×Dic53#12512yes—
178C2×Dic53#12612yes—
179C2×Dic53#12612yes—
180#88324—44no—
181#88324—44no—
182#10830—531nomaximal, Sylow
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53#37611yescharacteristic
194C22×C53C4×C22⋊C411yescharacteristic
195C22×C53C4×C4⋊C411yescharacteristic
196C22×C53#38011yescharacteristic
197C22×C53#38111yescharacteristic
198C22×C53#38211yescharacteristic
199C22×C53#38311yescharacteristic
200C4×C53—42no—
201C4×C53—42no—
202C4×C53—41no—
203C4×C53—41no—
204Dic53—42no—
205Dic53—42no—
206Dic53—41no—
207Dic53—41no—
208Dic53#38712yes—
209Dic53#38712yes—
210Dic53#38812yes—
211Dic53#38812yes—
212#2274—531no—
213C2×#344—532no—
214C2×#344—532no—
215C2×#343—531no—
216C2×#343—531no—
217C2×C53—21no—
218C2×C53—21no—
219C2×C53—21no—
220C2×C53—21no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53#143411yescharacteristic
224C2×C53#182911yescharacteristic
225C2×C53#183011yescharacteristic
226C22×M16—532no—
227C22×M16—532no—
228C2×#117—531no—
229C2×#117—531no—
230C22×C4⋊C4—531no—
231C4×C4⋊C4—1061no—
232C4×C4⋊C4—1061no—
233C4×C4⋊C4—1061no—
234C4×C4⋊C4—1061no—
235C4×C22⋊C4—531no—
236C4×C22⋊C4—531no—
237C4×C22⋊C4—531no—
238C4×C22⋊C4#615711yescharacteristic
239#344—538no—
240#344—1064no—
241#343—1061no—
242#343—1061no—
243#343—1061no—
244#343—1061no—
245C53#1083011yescharacteristic, complemented, Sylow
246C23×C4—531no—
247C23×C4—531no—
248C23×C4C2×#615711yescharacteristic
249C2×M16—534no—
250C2×M16—534no—
251C2×M16—1062no—
252C2×M16—1062no—
253C2×M16—1062no—
254C2×M16—1062no—
255C2×M16—1062no—
256C2×M16—1062no—
257C2×M16—1062no—
258C2×M16—1062no—
259C2×M16—1062no—
260C2×M16—1062no—
261C22×C8—1062no—
262C22×C8—1062no—
263#117—534no—
264#117—532no—
265#117—532no—
266#117—1062no—
267#117—1061no—
268#117—1061no—
269C2×C4⋊C4—531no—
270C2×C4⋊C4—531no—
271C2×C4⋊C4—531no—
272C2×C4⋊C4—531no—
273C2×C4⋊C4—1062no—
274C2×C4⋊C4—1062no—
275C2×C4⋊C4—1061no—
276C2×C4⋊C4—1061no—
277C2×C22⋊C4—531no—
278C2×C22⋊C4—531no—
279C2×C22⋊C4C2×#615711yescharacteristic
280C2×C22⋊C4C2×#615711yescharacteristic
281C2×C42—21no—
282C2×C42—1061no—
283C2×C42—1061no—
284C2×C42—1061no—
285C2×C42—1061no—
286C2×C42—1061no—
287#95—21no—
288#95—1061no—
289#95—1061no—
290#95—1061no—
291C24C22×#615711yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—81no—
297C22×C4—532no—
298C22×C4—531no—
299C22×C4—531no—
300C22×C4—531no—
301C22×C4—531no—
302C22×C4—532no—
303C22×C4—532no—
304C22×C4—531no—
305C22×C4—531no—
306C22×C4—532no—
307C22×C4—531no—
308C22×C4—531no—
309C22×C4—1061no—
310C22×C4—1061no—
311C22×C4—1061no—
312C22×C4—1061no—
313C22×C4—4241no—
314C22×C4C22⋊#615711yescharacteristic
315C22×C4C22⋊#615711yescharacteristic
316M16—1064no—
317M16—1064no—
318M16—1064no—
319M16—1064no—
320M16—2122no—
321M16—2122no—
322M16—2122no—
323M16—2122no—
324C2×C8—1064no—
325C2×C8—1064no—
326C2×C8—2122no—
327C2×C8—2122no—
328C2×C8—2122no—
329C2×C8—2122no—
330C4⋊C4—1062no—
331C4⋊C4—1062no—
332C4⋊C4—1062no—
333C4⋊C4—1062no—
334C4⋊C4—1064no—
335C4⋊C4—1062no—
336C4⋊C4—1062no—
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—532no—
340C22⋊C4—532no—
341C22⋊C4—1061no—
342C22⋊C4—1061no—
343C22⋊C4C4×#615712yes—
344C22⋊C4C4⋊#615712yes—
345C42—81no—
346C42—1064no—
347C42—1062no—
348C42—1062no—
349C42—4241no—
350C42—4241no—
351C42—4241no—
352C23—41no—
353C23—41no—
354C23C2×C4×#615711yescharacteristic
355C23C2×C22⋊#615711yescharacteristic
356C23C2×C4⋊#615711yescharacteristic
357C23#1313329611yescharacteristic
358C23#1313329611yescharacteristic
359C23#1313329611yescharacteristic
360C23#1313329611yescharacteristic
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—532no—
374C2×C4—532no—
375C2×C4—532no—
376C2×C4—532no—
377C2×C4—532no—
378C2×C4—532no—
379C2×C4—531no—
380C2×C4—531no—
381C2×C4—531no—
382C2×C4—531no—
383C2×C4—532no—
384C2×C4—532no—
385C2×C4—532no—
386C2×C4—532no—
387C2×C4—532no—
388C2×C4—532no—
389C2×C4—1062no—
390C2×C4—1062no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—4241no—
394C2×C4—4241no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4—2121no—
399C2×C4—2121no—
400C2×C4—2121no—
401C8—2124no—
402C8—2124no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22#1681155911yescharacteristic
414C22C4×C22⋊#615711yescharacteristic
415C22C4×C4⋊#615711yescharacteristic
416C22#1681156511yescharacteristic
417C22#1681156611yescharacteristic
418C22#1681156711yescharacteristic
419C22#1681156811yescharacteristic
420C4—42no—
421C4—42no—
422C4—41no—
423C4—41no—
424C4—532no—
425C4—532no—
426C4—532no—
427C4—532no—
428C4—2122no—
429C4—2122no—
430C4—2121no—
431C4—2121no—
432C2—21no—
433C2—21no—
434C2—21no—
435C2—21no—
436C2—21no—
437C2—21no—
438C2#2028989111yescharacteristic
439C2#2029062811yescharacteristic
440C2#2029062911yescharacteristic
subgroupG/H
centerC22#16811566
commutatorC23×C53C2×C42
FrattiniC24C22×#6157
FittingC4×C22⋊C4×C53C4
socleC22×C53#381
prime radicalC24×C53C22×C4
cosocleC23×Dic53C23
hypercenterC4×C22⋊C4#6157
hypocenterC53#10830
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
9. #16811504C4semidirect
245. C53#10830semidirect

Maximal subgroups

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

subgroupindex
C2×#168115042
C2×#168115032
#202770702
#1083053