Finite Group Databasebeta

← #27275705

Subgroups — #27275705

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274814C212yesmaximal, complemented
2C4×#13133132C211yesmaximal, characteristic
3#20273235C212yesmaximal, complemented
4#20273053C211yesmaximal, characteristic
5C2×C53×#556C211yesmaximal, characteristic
6C2×#13133132—21no—
7#16810587C2212yes—
8C4×C22⋊Dic53C2212yes—
9#16810390—21no—
10#16810382—24no—
11#16810360—24no—
12#16810349—21no—
13#16810347—21no—
14C53×#556—24no—
15C53×#556—24no—
16C2×C4⋊Q8×C53C411yescharacteristic, complemented
17C2×#126×C53—21no—
18C2×#126×C53—21no—
19C2×#124×C53—21no—
20C2×C22⋊Q8×C53C411yescharacteristic, complemented
21C2×C22⋊Q8×C53C2211yescharacteristic
22C2×C4×Q8×C53—21no—
23C2×C4×Q8×C53C411yescharacteristic, complemented
24C2×C4×D4×C53C2211yescharacteristic
25C2×#117×C53C411yescharacteristic, complemented
26C2×#117×C53C2211yescharacteristic
27C2×#11719917—22no—
28C2×C22⋊Dic53—22no—
29C2×C4×Dic53—22no—
30#13133132—24no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33#13133124—22no—
34C22×Q8×C53—21no—
35C22×Q8×C53C2×C411yescharacteristic
36C22×D4×C53D411yescharacteristic
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53C2311yescharacteristic
39C4⋊Q8×C53—22no—
40C4⋊Q8×C53—22no—
41#126×C53—28no—
42#126×C53—28no—
43#124×C53—28no—
44C22⋊Q8×C53—22no—
45C22⋊Q8×C53—22no—
46C22⋊Q8×C53—22no—
47C22⋊Q8×C53D414yes—
48C4×Q8×C53—28no—
49C4×Q8×C53—22no—
50C4×Q8×C53—22no—
51C4×D4×C53—22no—
52C4×D4×C53—22no—
53#117×C53—22no—
54#117×C53—22no—
55#117×C53—22no—
56#117×C53D414yes—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53C2×C411yescharacteristic
62C2×C4⋊C4×C53C2×C411yescharacteristic
63C2×C4⋊C4×C53C2×C411yescharacteristic
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53D411yescharacteristic
70C2×C22⋊C4×C53D411yescharacteristic
71C2×C42×C53—21no—
72C2×C42×C53—21no—
73C2×C42×C53—21no—
74C2×C42×C53C2×C411yescharacteristic
75C24×C53C22⋊C411yescharacteristic
76C24×C53C2×D411yescharacteristic
77C2×Q8×C53—24no—
78C2×Q8×C53—21no—
79C2×Q8×C53—42no—
80C2×Q8×C53—42no—
81C2×Q8×C53—42no—
82C2×Q8×C53C22⋊C412yes—
83C2×D4×C53—22no—
84C2×D4×C53—22no—
85C2×D4×C53—41no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—42no—
93C22×C4×C53—42no—
94C22×C4×C53C22⋊C411yescharacteristic
95C22×C4×C53C22⋊C411yescharacteristic
96C22×C4×C53C22⋊C411yescharacteristic
97C22×C4×C53C22×C411yescharacteristic
98C22×C4×C53C2×D412yes—
99C22×C4×C53C2×D412yes—
100C22×C4×C53C2×D411yescharacteristic
101C22×Dic53—42no—
102C4⋊C4×C53—24no—
103C4⋊C4×C53—24no—
104C4⋊C4×C53—24no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—24no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53C22⋊C412yes—
111C4⋊C4×C53C22⋊C412yes—
112C4⋊C4×C53C22⋊C412yes—
113C22⋊C4×C53—24no—
114C22⋊C4×C53—24no—
115C22⋊C4×C53—24no—
116C22⋊C4×C53—24no—
117C22⋊C4×C53—22no—
118C22⋊C4×C53—22no—
119C42×C53—24no—
120C42×C53—24no—
121C42×C53—21no—
122C42×C53—21no—
123C42×C53—24no—
124C22⋊Dic53—24no—
125C22⋊Dic53—42no—
126C4×Dic53—44no—
127#11719917—44no—
128C2×#88322—42no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53C2×C22⋊C411yescharacteristic
139C23×C53#12012yes—
140Q8×C53—48no—
141Q8×C53—42no—
142Q8×C53—42no—
143D4×C53—44no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—22no—
147C2×C4×C53—24no—
148C2×C4×C53—21no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—21no—
154C2×C4×C53—41no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—42no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53#10412yes—
167C2×C4×C53C2×C22⋊C411yescharacteristic
168C2×C4×C53C2×C22⋊C411yescharacteristic
169C2×C4×C53#12012yes—
170C2×Dic53—44no—
171C2×Dic53—82no—
172C2×Dic53—82no—
173#88322—82no—
174#8365—531nomaximal, Sylow
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—22no—
178C22×C53—22no—
179C22×C53—22no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—81no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53#37911yescharacteristic
188C22×C53C2×#10411yescharacteristic
189C22×C53#42111yescharacteristic
190C4×C53—22no—
191C4×C53—21no—
192C4×C53—22no—
193C4×C53—41no—
194C4×C53—42no—
195C4×C53—41no—
196C4×C53—42no—
197C4×C53—42no—
198C4×C53—42no—
199C4×C53#41012yes—
200Dic53—84no—
201C2×#556D5311yescharacteristic
202#1826—532no—
203C4×#99—531no—
204#1508—532no—
205#1465—531no—
206C2×C53—21no—
207C2×C53—21no—
208C2×C53—42no—
209C2×C53—81no—
210C2×C53#178211yescharacteristic
211C2×C53#178911yescharacteristic
212C2×C53#179611yescharacteristic
213#556—24no—
214#556—24no—
215C2×C4⋊Q8Dic5311yescharacteristic, complemented
216C2×#126—21no—
217C2×#126—21no—
218C2×#124—21no—
219C2×C22⋊Q8Dic5311yescharacteristic, complemented
220C2×C22⋊Q8C2×D5311yescharacteristic
221C2×C4×Q8—21no—
222C2×C4×Q8Dic5311yescharacteristic, complemented
223C2×C4×D4C2×D5311yescharacteristic
224C2×#117Dic5311yescharacteristic, complemented
225C2×#117C2×D5311yescharacteristic
226C2×#99—1061no—
227#407—532no—
228C4×C22⋊C4—532no—
229#342—1061no—
230#339—1064no—
231#328—1064no—
232#324—1061no—
233#323—1061no—
234C53#836511yescharacteristic, complemented, Sylow
235C22×Q8—21no—
236C22×Q8C2×Dic5311yescharacteristic
237C22×D4C22⋊D5311yescharacteristic
238C23×C4C2×Dic5311yescharacteristic
239C23×C4C22×D5311yescharacteristic
240C2×M16—1062no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243#126—28no—
244#126—28no—
245#124—28no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8C22⋊D5314yes—
250C4×Q8—28no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×D4—22no—
254C4×D4—22no—
255#117—22no—
256#117—22no—
257#117—22no—
258#117C22⋊D5314yes—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C2×Dic5311yescharacteristic
264C2×C4⋊C4C2×Dic5311yescharacteristic
265C2×C4⋊C4C2×Dic5311yescharacteristic
266C2×C4⋊C4C22⋊D5311yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—1062no—
272C2×C22⋊C4C22⋊D5311yescharacteristic
273C2×C22⋊C4C22⋊D5311yescharacteristic
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—1062no—
278C2×C42C2×Dic5311yescharacteristic
279#99—1064no—
280C22⋊C8—1062no—
281C22⋊C8—1062no—
282#95—1062no—
283C24C22⋊Dic5311yescharacteristic
284C24C2×C22⋊D5311yescharacteristic
285C2×Q8—24no—
286C2×Q8—21no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8C22⋊Dic5312yes—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—41no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—2122no—
303C22×C4C22⋊Dic5311yescharacteristic
304C22×C4C22⋊Dic5311yescharacteristic
305C22×C4C22⋊Dic5311yescharacteristic
306C22×C4C22×Dic5311yescharacteristic
307C22×C4C2×C22⋊D5312yes—
308C22×C4C2×C22⋊D5312yes—
309C22×C4C2×C22⋊D5311yescharacteristic
310M16—2124no—
311C2×C8—2122no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4C22⋊Dic5312yes—
321C4⋊C4C22⋊Dic5312yes—
322C4⋊C4C22⋊Dic5312yes—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—1064no—
330C22⋊C4—2122no—
331C42—24no—
332C42—24no—
333C42—21no—
334C42—21no—
335C42—24no—
336C42—2124no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23C2×C22⋊Dic5311yescharacteristic
347C23#1313318612yes—
348Q8—42no—
349Q8—48no—
350Q8—42no—
351D4—44no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—4242no—
375C2×C4—4242no—
376C2×C4—2124no—
377C2×C4#1313314412yes—
378C2×C4C2×C22⋊Dic5311yescharacteristic
379C2×C4C2×C22⋊Dic5311yescharacteristic
380C2×C4#1313318612yes—
381C8—4242no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22#1681047911yescharacteristic
395C22C2×#1313314411yescharacteristic
396C22#1681064011yescharacteristic
397C4—22no—
398C4—21no—
399C4—22no—
400C4—42no—
401C4—41no—
402C4—41no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—4244no—
407C4#1681059712yes—
408C2—21no—
409C2—21no—
410C2—42no—
411C2—81no—
412C2#2027463311yescharacteristic
413C2#2027465911yescharacteristic
414C2#2027469011yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#556C2
socleC22×C53#379
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenterC2×#556D53
hypocenterC53#8365
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. #20274814C2semidirect
3. #20273235C2semidirect
16. C2×C4⋊Q8×C53C4semidirect
20. C2×C22⋊Q8×C53C4semidirect
23. C2×C4×Q8×C53C4semidirect
25. C2×#117×C53C4semidirect
215. C2×C4⋊Q8Dic53semidirect
219. C2×C22⋊Q8Dic53semidirect
222. C2×C4×Q8Dic53semidirect
224. C2×#117Dic53semidirect
234. C53#8365semidirect

Maximal subgroups

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

subgroupindex
#202748142
C4×#131331322
#202732352
#202730532
C2×C53×#5562
#836553