Finite Group Databasebeta

← #27294762

Subgroups — #27294762

Order 13568.

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

normal subgroups only

This group has 574 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#20274789C211yesmaximal, characteristic, complemented
2#20273042C211yesmaximal, characteristic, complemented
3#20272715C212yesmaximal, complemented
4#20272712C212yesmaximal, complemented
5C53×#2617C211yesmaximal, characteristic
6#16810587—21no—
7#16810479—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810347—21no—
11#16810347C2212yes—
12C2×#121×C53C411yescharacteristic, complemented
13C2×#120×C53—21no—
14C22×C22⋊C4×C53C2211yescharacteristic
15C53×#397C411yescharacteristic, complemented
16C53×#397C2211yescharacteristic
17C53×#394—21no—
18C53×#394—21no—
19C53×#392C411yescharacteristic, complemented
20C53×#392C2211yescharacteristic
21C53×#386—21no—
22C53×#380C411yescharacteristic, complemented
23C2×#11719917—41no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—41no—
26C2×C22⋊Dic53—41no—
27C22⋊#88322—22no—
28C22⋊#88322—41no—
29C25×C53D411yescharacteristic
30C22×D4×C53—21no—
31C22×D4×C53—21no—
32C22×D4×C53—21no—
33C22×D4×C53C2×C411yescharacteristic
34C23×C4×C53C2×C411yescharacteristic
35C23×C4×C53D411yescharacteristic
36#121×C53—41no—
37#121×C53—41no—
38#120×C53—44no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—41no—
48C2×C22⋊C4×C53—41no—
49C2×C22⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53#95×C53—21no—
54#95×C53—21no—
55#95×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C24×C53—21no—
58C24×C53—21no—
59C24×C53—22no—
60C24×C53—21no—
61C24×C53—41no—
62C24×C53—81no—
63C24×C53—81no—
64C24×C53C22⋊C411yescharacteristic
65C24×C53C22⋊C411yescharacteristic
66C24×C53C2×D411yescharacteristic
67C2×D4×C53—41no—
68C2×D4×C53—42no—
69C2×D4×C53—42no—
70C2×D4×C53—42no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—42no—
74C2×D4×C53—81no—
75C2×D4×C53—81no—
76C2×D4×C53—81no—
77C2×D4×C53—81no—
78C2×D4×C53—81no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—81no—
87C22×C4×C53—81no—
88C22×C4×C53C22⋊C411yescharacteristic
89C22×C4×C53C22⋊C411yescharacteristic
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×Dic53—42no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—81no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—42no—
100C22⋊C4×C53—42no—
101C22⋊C4×C53—42no—
102C22⋊C4×C53—42no—
103C22⋊C4×C53—81no—
104C22⋊C4×C53—81no—
105C22⋊Dic53—42no—
106C22⋊Dic53—42no—
107C22⋊Dic53—82no—
108C22⋊Dic53—82no—
109#11719917—82no—
110C2×#88322—42no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—42no—
121C23×C53—42no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—81no—
136C23×C53—81no—
137C23×C53—84no—
138C23×C53—84no—
139C23×C53#9911yescharacteristic
140C23×C53#9911yescharacteristic
141C23×C53C2×C22⋊C411yescharacteristic
142C23×C53AGL(1,ℤ/8)12yes—
143D4×C53—82no—
144D4×C53—82no—
145D4×C53—82no—
146D4×C53—84no—
147D4×C53—84no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—42no—
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—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—81no—
167C2×C4×C53—81no—
168C2×C4×C53—81no—
169C2×Dic53—82no—
170C2×Dic53—82no—
171C2×Dic53—82no—
172#88322—82no—
173#11799—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—22no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—42no—
184C22×C53—42no—
185C22×C53—42no—
186C22×C53—42no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53—81no—
198C22×C53—84no—
199C22×C53—84no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53—81no—
203C22×C53C2×#9911yescharacteristic
204C22×C53#41811yescharacteristic
205C22×C53#42111yescharacteristic
206C4×C53—41no—
207C4×C53—41no—
208C4×C53—41no—
209C4×C53—41no—
210C4×C53—81no—
211C4×C53—81no—
212C4×C53—81no—
213Dic53—162no—
214#2617D5311yescharacteristic
215#1819—531no—
216#1462—531no—
217#1344—532no—
218#1343—532no—
219C2×C53—21no—
220C2×C53—21no—
221C2×C53—42no—
222C2×C53—41no—
223C2×C53—41no—
224C2×C53—81no—
225C2×C53—161no—
226C2×C53#151011yescharacteristic
227C2×C53#212111yescharacteristic
228C2×C53#212711yescharacteristic
229C2×#121Dic5311yescharacteristic, complemented
230C2×#120—21no—
231C22×C22⋊C4C2×D5311yescharacteristic
232#407—1061no—
233#397Dic5311yescharacteristic, complemented
234#397C2×D5311yescharacteristic
235#394—21no—
236#394—21no—
237#392Dic5311yescharacteristic, complemented
238#392C2×D5311yescharacteristic
239#386—21no—
240#380Dic5311yescharacteristic, complemented
241#379—1061no—
242#342—532no—
243#342—1061no—
244#323—532no—
245#323—1061no—
246C53#1179911yescharacteristic, complemented, Sylow
247C25C22⋊D5311yescharacteristic
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2×Dic5311yescharacteristic
252C23×C4C2×Dic5311yescharacteristic
253C23×C4C22⋊D5311yescharacteristic
254C2×M16—2121no—
255#121—41no—
256#121—41no—
257#120—44no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4C2×Dic5311yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—1062no—
269C2×C22⋊C4—2121no—
270C2×C22⋊C4—2121no—
271C2×C22⋊C4C2×Dic5311yescharacteristic
272C2×C22⋊C4C22⋊D5311yescharacteristic
273C2×C22⋊C4C22⋊D5311yescharacteristic
274C2×C22⋊C4C22×D5311yescharacteristic
275C22⋊C8—1062no—
276C22⋊C8—2121no—
277#95—21no—
278#95—21no—
279#95C2×Dic5311yescharacteristic
280#95C2×Dic5311yescharacteristic
281C24—21no—
282C24—21no—
283C24—22no—
284C24—21no—
285C24—41no—
286C24—81no—
287C24—81no—
288C24C22⋊Dic5311yescharacteristic
289C24C22⋊Dic5311yescharacteristic
290C24C2×C22⋊D5311yescharacteristic
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—81no—
299C2×D4—81no—
300C2×D4—81no—
301C2×D4—81no—
302C2×D4—81no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—2122no—
313C22×C4C22⋊Dic5311yescharacteristic
314C22×C4C22⋊Dic5311yescharacteristic
315C22×C4C22×Dic5311yescharacteristic
316C22×C4C2×C22⋊D5311yescharacteristic
317M16—4242no—
318C2×C8—2122no—
319C4⋊C4—41no—
320C4⋊C4—81no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—81no—
330C22⋊C4—81no—
331C22⋊C4—4242no—
332C22⋊C4—4242no—
333C22⋊C4—2122no—
334C22⋊C4—2122no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23—42no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—81no—
360C23—81no—
361C23—84no—
362C23—84no—
363C23#1313313211yescharacteristic
364C23#1313313211yescharacteristic
365C23C2×C22⋊Dic5311yescharacteristic
366C23#1313324312yes—
367D4—82no—
368D4—82no—
369D4—82no—
370D4—84no—
371D4—84no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—4242no—
394C2×C4—4242no—
395C2×C4—4242no—
396C8—4242no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—84no—
422C22—84no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22C2×#1313313211yescharacteristic
427C22#1681062811yescharacteristic
428C22#1681064011yescharacteristic
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—81no—
434C4—81no—
435C4—81no—
436C4—8482no—
437C2—21no—
438C2—21no—
439C2—42no—
440C2—41no—
441C2—41no—
442C2—81no—
443C2—161no—
444C2#2027324411yescharacteristic
445C2#2027636111yescharacteristic
446C2#2027638411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
Fitting, prime radicalC53×#2617C2
socleC22×C53C2×#99
cosocleC2×C22⋊C4×C53C23
hypercenter#2617D53
hypocenterC53#11799
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2617

Complements

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

NH ≅ G/Ntype
1. #20274789C2semidirect
2. #20273042C2semidirect
3. #20272715C2semidirect
4. #20272712C2semidirect
12. C2×#121×C53C4semidirect
15. C53×#397C4semidirect
19. C53×#392C4semidirect
22. C53×#380C4semidirect
229. C2×#121Dic53semidirect
233. #397Dic53semidirect
237. #392Dic53semidirect
240. #380Dic53semidirect
246. C53#11799semidirect

Maximal subgroups

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

subgroupindex
#202747892
#202730422
#202727152
#202727122
C53×#26172
#1179953