Finite Group Databasebeta

← #27294578

Subgroups — #27294578

Order 13568.

586 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#16810390C211yesmaximal, characteristic
2#20273042C211yesmaximal, characteristic
3#20272736C212yesmaximal, complemented
4#20272715C212yesmaximal, complemented
5C53×#2561C211yesmaximal, characteristic
6C22×C22⋊Dic53—21no—
7#16810587—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810347—21no—
11#16810347C2212yes—
12C2×C22⋊Q8×C53C411yescharacteristic, complemented
13C22×C22⋊C4×C53C2211yescharacteristic
14C53×#397C411yescharacteristic, complemented
15C53×#397C2211yescharacteristic
16C53×#393C411yescharacteristic, complemented
17C53×#393C2211yescharacteristic
18C53×#388—21no—
19C53×#388—21no—
20C53×#381—21no—
21C53×#379—21no—
22C4×C22⋊C4×C53C411yescharacteristic, complemented
23C23×Dic53—41no—
24C2×#11719917—41no—
25C2×C22⋊Dic53—24no—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53—41no—
28C2×C22⋊Dic53—42no—
29C22⋊#88322—22no—
30C22⋊#88322—41no—
31C25×C53D411yescharacteristic
32C22×Q8×C53C2×C411yescharacteristic, complemented
33C23×C4×C53C2×C411yescharacteristic
34C23×C4×C53D411yescharacteristic
35C22⋊Q8×C53—41no—
36C22⋊Q8×C53—41no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—41no—
46C2×C22⋊C4×C53—41no—
47C2×C22⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53C2311yescharacteristic
51C2×C42×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic, complemented
57#95×C53C2×C411yescharacteristic, complemented
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
68C2×Q8×C53—41no—
69C2×Q8×C53—81no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—42no—
79C22×C4×C53—81no—
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—44no—
85C22×Dic53—42no—
86C22×Dic53—82no—
87C22×Dic53—82no—
88C4⋊C4×C53—42no—
89C4⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—42no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—42no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—81no—
101C22⋊C4×C53—81no—
102C42×C53—81no—
103C22⋊Dic53—24no—
104C22⋊Dic53—44no—
105C22⋊Dic53—42no—
106#11719917—82no—
107C2×#88322—42no—
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
137Q8×C53—82no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—42no—
143C2×C4×C53—41no—
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—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—81no—
160C2×C4×C53—81no—
161C2×C4×C53—81no—
162C2×Dic53—44no—
163C2×Dic53—88no—
164C2×Dic53—82no—
165C2×Dic53—82no—
166#88322—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—
201C4×C53—81no—
202C4×C53—81no—
203C4×C53—81no—
204C4×C53—81no—
205Dic53—84no—
206#2561D5311yescharacteristic
207C2×#342—531no—
208#1462—531no—
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⋊Q8Dic5311yescharacteristic, complemented
222C22×C22⋊C4—1061no—
223C22×C22⋊C4C2×D5311yescharacteristic
224#407—1061no—
225#397Dic5311yescharacteristic, complemented
226#397C2×D5311yescharacteristic
227#393Dic5311yescharacteristic, complemented
228#393C2×D5311yescharacteristic
229#388—21no—
230#388—21no—
231#381—21no—
232#379—21no—
233C4×C22⋊C4Dic5311yescharacteristic, complemented
234#342—532no—
235#342—1061no—
236#323—532no—
237#323—1061no—
238C53#1175911yescharacteristic, complemented, Sylow
239C25C22⋊D5311yescharacteristic
240C22×Q8C2×Dic5311yescharacteristic, complemented
241C23×C4—2121no—
242C23×C4C2×Dic5311yescharacteristic
243C23×C4C22⋊D5311yescharacteristic
244C2×M16—2121no—
245C22⋊Q8—41no—
246C22⋊Q8—41no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C2×Dic5311yescharacteristic, complemented
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—1064no—
258C2×C22⋊C4—1062no—
259C2×C22⋊C4—2121no—
260C2×C22⋊C4—2122no—
261C2×C22⋊C4C2×Dic5311yescharacteristic
262C2×C22⋊C4C22⋊D5311yescharacteristic
263C2×C22⋊C4C22⋊D5311yescharacteristic
264C2×C22⋊C4C22×D5311yescharacteristic
265C2×C42—21no—
266C22⋊C8—1062no—
267C22⋊C8—2121no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95C2×Dic5311yescharacteristic, complemented
273#95C2×Dic5311yescharacteristic, complemented
274C24—21no—
275C24—21no—
276C24—22no—
277C24—21no—
278C24—21no—
279C24—42no—
280C24—81no—
281C24C22⋊Dic5311yescharacteristic
282C24C22⋊Dic5311yescharacteristic
283C24C2×C22⋊D5311yescharacteristic
284C2×Q8—41no—
285C2×Q8—81no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—81no—
296C22×C4—4242no—
297C22×C4—4242no—
298C22×C4—2124no—
299C22×C4—2122no—
300C22×C4C22⋊Dic5311yescharacteristic
301C22×C4C22⋊Dic5311yescharacteristic
302C22×C4C22×Dic5311yescharacteristic
303C22×C4C2×C22⋊D5311yescharacteristic
304M16—4242no—
305C2×C8—2122no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—42no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—81no—
319C22⋊C4—81no—
320C22⋊C4—1064no—
321C22⋊C4—2124no—
322C22⋊C4—2122no—
323C42—81no—
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#1313313211yescharacteristic
348C23#1313313211yescharacteristic
349C23#1313314412yes—
350C23C2×C22⋊Dic5311yescharacteristic
351C23#1313324211yescharacteristic
352C23#1313324711yescharacteristic
353Q8—82no—
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—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—4248no—
379C2×C4—4242no—
380C2×C4—4242no—
381C2×C4—2124no—
382C8—4242no—
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#1681041812yes—
409C22C2×#1313313211yescharacteristic
410C22#1681062411yescharacteristic
411C22C2×#1313314411yescharacteristic
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4—4244no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—41no—
425C2—41no—
426C2—42no—
427C2—41no—
428C2#2027321911yescharacteristic
429C2#2027636111yescharacteristic
430C2C2×#1681041811yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2561C2
socleC22×C53C2×#99
prime radicalC25×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2561D53
hypocenterC53#11759
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
3. #20272736C2semidirect
4. #20272715C2semidirect
12. C2×C22⋊Q8×C53C4semidirect
14. C53×#397C4semidirect
16. C53×#393C4semidirect
22. C4×C22⋊C4×C53C4semidirect
32. C22×Q8×C53C2×C4semidirect
38. C2×C4⋊C4×C53C2×C4semidirect
56. #95×C53C2×C4semidirect
57. #95×C53C2×C4semidirect
221. C2×C22⋊Q8Dic53semidirect
225. #397Dic53semidirect
227. #393Dic53semidirect
233. C4×C22⋊C4Dic53semidirect
238. C53#11759semidirect
240. C22×Q8C2×Dic53semidirect
248. C2×C4⋊C4C2×Dic53semidirect
272. #95C2×Dic53semidirect
273. #95C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168103902
#202730422
#202727362
#202727152
C53×#25612
#1175953