Finite Group Databasebeta

← #27294579

Subgroups — #27294579

Order 13568.

586 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279873C211yesmaximal, characteristic
2C2×#16810390C211yesmaximal, characteristic
3#20272737C212yesmaximal, complemented
4#20272716C212yesmaximal, complemented
5C53×#1462C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic, complemented
7C22×C22⋊Dic53—21no—
8#16810548C411yescharacteristic, complemented
9#16810548C2211yescharacteristic
10#16810535C411yescharacteristic, complemented
11#16810535C2211yescharacteristic
12#16810517—21no—
13#16810517—21no—
14#16810486—21no—
15#16810479—21no—
16C4×C22⋊Dic53C411yescharacteristic, complemented
17#16810390—21no—
18#16810390C2212yes—
19C22×C22⋊C4×C53C2211yescharacteristic
20C53×#407—21no—
21C53×#323—21no—
22C53×#323C2212yes—
23C22×Dic106C2×C411yescharacteristic, complemented
24C23×Dic53—41no—
25#13133196—41no—
26#13133196—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic, complemented
28C2×#11719919—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—24no—
36C2×C22⋊Dic53—22no—
37C2×C22⋊Dic53—41no—
38C2×C22⋊Dic53—42no—
39C2×C4×Dic53—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124C2×C411yescharacteristic, complemented
45#13133124C2×C411yescharacteristic, complemented
46C25×C53D411yescharacteristic
47C23×C4×C53C2×C411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C2×M16×C53—41no—
50C2×C22⋊C4×C53—41no—
51C2×C22⋊C4×C53—41no—
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53C2311yescharacteristic
56C22⋊C8×C53—22no—
57C22⋊C8×C53—41no—
58C24×C53—21no—
59C24×C53—21no—
60C24×C53—22no—
61C24×C53—21no—
62C24×C53—21no—
63C24×C53—42no—
64C24×C53—81no—
65C24×C53C22⋊C411yescharacteristic
66C24×C53C22⋊C411yescharacteristic
67C24×C53C2×D411yescharacteristic
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—42no—
73C22×C4×C53—81no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—44no—
83C22×Dic53—42no—
84C22×Dic53—82no—
85C22×Dic53—82no—
86C2×Dic106—41no—
87C2×Dic106—81no—
88M16×C53—82no—
89C2×C8×C53—42no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C22⋊Dic53—24no—
95C22⋊Dic53—42no—
96C22⋊Dic53—44no—
97C22⋊Dic53—41no—
98C22⋊Dic53—41no—
99C22⋊Dic53—42no—
100C22⋊Dic53—41no—
101C22⋊Dic53—42no—
102C22⋊Dic53—41no—
103C22⋊Dic53—81no—
104C22⋊Dic53—81no—
105C4⋊Dic53—41no—
106#11719919—42no—
107C4×Dic53—81no—
108C23×C53—22no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53—22no—
112C23×C53—22no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—41no—
120C23×C53—42no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—42no—
126C23×C53—81no—
127C23×C53—81no—
128C23×C53—84no—
129C23×C53—82no—
130C23×C53—82no—
131C23×C53#9911yescharacteristic
132C23×C53#9911yescharacteristic
133C23×C53#10412yes—
134C23×C53C2×C22⋊C411yescharacteristic
135C23×C53AGL(1,ℤ/8)11yescharacteristic
136C23×C53#13711yescharacteristic
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—81no—
147C2×C4×C53—81no—
148C2×C4×C53—81no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—44no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—88no—
163C2×Dic53—82no—
164C2×Dic53—82no—
165Dic106—82no—
166C8×C53—82no—
167#11759—531nomaximal, Sylow
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—22no—
177C22×C53—21no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—44no—
184C22×C53—44no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—81no—
190C22×C53—81no—
191C22×C53—82no—
192C22×C53—82no—
193C22×C53#35412yes—
194C22×C53C2×#9911yescharacteristic
195C22×C53#41711yescharacteristic
196C22×C53C2×#10411yescharacteristic
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—41no—
201Dic53—81no—
202Dic53—81no—
203Dic53—84no—
204Dic53—81no—
205Dic53—81no—
206#2561—531no—
207C2×#342—531no—
208#1462D5311yescharacteristic
209#1351—532no—
210#1344—532no—
211C2×C53—21no—
212C2×C53—21no—
213C2×C53—22no—
214C2×C53—41no—
215C2×C53—41no—
216C2×C53—42no—
217C2×C53—41no—
218C2×C53#150411yescharacteristic
219C2×C53#212111yescharacteristic
220C2×C53C2×#35411yescharacteristic
221C2×C22⋊Q8—531no—
222C22×C22⋊C4—1061no—
223C22×C22⋊C4C2×D5311yescharacteristic
224#407—21no—
225#397—531no—
226#397—531no—
227#393—531no—
228#393—531no—
229#388—1061no—
230#388—1061no—
231#381—1061no—
232#379—1061no—
233C4×C22⋊C4—531no—
234#342—532no—
235#342—1061no—
236#323—21no—
237#323C2×D5312yes—
238C53#1175911yescharacteristic, complemented, Sylow
239C25D21211yescharacteristic
240C22×Q8—531no—
241C23×C4—2121no—
242C23×C4C4×D5311yescharacteristic
243C23×C4C22⋊D5311yescharacteristic
244C2×M16—41no—
245C22⋊Q8—2121no—
246C22⋊Q8—2121no—
247C2×C4⋊C4—531no—
248C2×C4⋊C4—1061no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—1064no—
252C2×C22⋊C4—1062no—
253C2×C22⋊C4—1061no—
254C2×C22⋊C4—1061no—
255C2×C22⋊C4—1061no—
256C2×C22⋊C4—1061no—
257C2×C22⋊C4—1061no—
258C2×C22⋊C4—1061no—
259C2×C22⋊C4—2121no—
260C2×C22⋊C4—2122no—
261C2×C22⋊C4C4×D5311yescharacteristic
262C2×C22⋊C4D21211yescharacteristic
263C2×C22⋊C4C22⋊D5311yescharacteristic
264C2×C22⋊C4C22×D5311yescharacteristic
265C2×C42—1061no—
266C22⋊C8—22no—
267C22⋊C8—41no—
268#95—531no—
269#95—531no—
270#95—1061no—
271#95—1061no—
272#95—1061no—
273#95—1061no—
274C24—21no—
275C24—21no—
276C24—22no—
277C24—21no—
278C24—21no—
279C24—42no—
280C24—81no—
281C24#1171992111yescharacteristic
282C24#1171992111yescharacteristic
283C24C2×D21211yescharacteristic
284C2×Q8—4241no—
285C2×Q8—2121no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—81no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—4242no—
297C22×C4—4242no—
298C22×C4—2124no—
299C22×C4—2122no—
300C22×C4#1171992111yescharacteristic
301C22×C4#1171992111yescharacteristic
302C22×C4C2×C4×D5311yescharacteristic
303C22×C4C2×C22⋊D5311yescharacteristic
304M16—82no—
305C2×C8—42no—
306C4⋊C4—2121no—
307C4⋊C4—2122no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—1064no—
313C22⋊C4—4241no—
314C22⋊C4—4241no—
315C22⋊C4—2124no—
316C22⋊C4—2122no—
317C22⋊C4—2121no—
318C22⋊C4—2121no—
319C22⋊C4—2122no—
320C22⋊C4—2122no—
321C22⋊C4—2121no—
322C22⋊C4—2121no—
323C42—4241no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—22no—
333C23—21no—
334C23—21no—
335C23—41no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—81no—
343C23—81no—
344C23—84no—
345C23—82no—
346C23—82no—
347C23#1313313111yescharacteristic
348C23#1313313111yescharacteristic
349C23#1313314312yes—
350C23C2×#1171992111yescharacteristic
351C23#1313324511yescharacteristic
352C23#1313325011yescharacteristic
353Q8—4242no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—4248no—
367C2×C4—4242no—
368C2×C4—4242no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2124no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4—2121no—
380C2×C4—2121no—
381C2×C4—2121no—
382C8—82no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—44no—
399C22—44no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—81no—
405C22—81no—
406C22—82no—
407C22—82no—
408C22#1681041712yes—
409C22C2×#1313313111yescharacteristic
410C22#1681062511yescharacteristic
411C22C2×#1313314311yescharacteristic
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—4241no—
417C4—4241no—
418C4—4244no—
419C4—4241no—
420C4—4241no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—41no—
425C2—41no—
426C2—42no—
427C2—41no—
428C2#2027322011yescharacteristic
429C2#2027636211yescharacteristic
430C2C2×#1681041711yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1462C2
socleC22×C53C2×#99
prime radicalC25×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1462D53
hypocenterC53#11759
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #20272737C2semidirect
4. #20272716C2semidirect
6. C2×#13133196C4semidirect
8. #16810548C4semidirect
10. #16810535C4semidirect
16. C4×C22⋊Dic53C4semidirect
23. C22×Dic106C2×C4semidirect
27. C2×C4⋊Dic53C2×C4semidirect
44. #13133124C2×C4semidirect
45. #13133124C2×C4semidirect
238. C53#11759semidirect

Maximal subgroups

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

subgroupindex
#202798732
C2×#168103902
#202727372
#202727162
C53×#14622
#1175953