Finite Group Databasebeta

← #27294970

Subgroups — #27294970

Order 13568.

620 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 91 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
2C2×#16810347C211yesmaximal, characteristic, complemented
3#20272739C212yesmaximal, complemented
4#20272712C212yesmaximal, complemented
5C53×#2539C211yesmaximal, characteristic
6C2×C22⋊#88322—21no—
7#16810479—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810347—21no—
11#16810347C2212yes—
12C2×#123×C53—21no—
13C2×#121×C53C411yescharacteristic, complemented
14C2×#120×C53—21no—
15C22×C22⋊C4×C53C2211yescharacteristic
16C53×#399C411yescharacteristic, complemented
17C53×#399C2211yescharacteristic
18C53×#392C411yescharacteristic, complemented
19C53×#392C2211yescharacteristic
20C53×#390—21no—
21C53×#390—21no—
22C4×C22⋊C4×C53C411yescharacteristic, complemented
23C22×#88322—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#123×C53—44no—
37#121×C53—41no—
38#121×C53—41no—
39#120×C53—44no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—41no—
50C2×C22⋊C4×C53—41no—
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53C2311yescharacteristic
55C2×C42×C53—21no—
56#95×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53—21no—
59C24×C53—21no—
60C24×C53—22no—
61C24×C53—21no—
62C24×C53—41no—
63C24×C53—42no—
64C24×C53—81no—
65C24×C53C22⋊C411yescharacteristic
66C24×C53C22⋊C411yescharacteristic
67C24×C53C2×D411yescharacteristic
68C2×D4×C53—42no—
69C2×D4×C53—42no—
70C2×D4×C53—41no—
71C2×D4×C53—42no—
72C2×D4×C53—42no—
73C2×D4×C53—42no—
74C2×D4×C53—41no—
75C2×D4×C53—41no—
76C2×D4×C53—42no—
77C2×D4×C53—81no—
78C2×D4×C53—81no—
79C2×D4×C53—81no—
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—21no—
87C22×C4×C53—42no—
88C22×C4×C53—81no—
89C22×C4×C53C22⋊C411yescharacteristic
90C22×C4×C53C22⋊C411yescharacteristic
91C22×C4×C53C22×C411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×Dic53—42no—
94C4⋊C4×C53—42no—
95C4⋊C4×C53—42no—
96C4⋊C4×C53—41no—
97C22⋊C4×C53—24no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—42no—
103C22⋊C4×C53—42no—
104C22⋊C4×C53—42no—
105C22⋊C4×C53—42no—
106C22⋊C4×C53—42no—
107C42×C53—81no—
108C22⋊Dic53—42no—
109C22⋊Dic53—42no—
110C22⋊Dic53—82no—
111C22⋊Dic53—82no—
112C2×#88322—42no—
113C2×#88322—82no—
114C23×C53—22no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—41no—
123C23×C53—42no—
124C23×C53—42no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—42no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—42no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53—81no—
139C23×C53—81no—
140C23×C53—84no—
141C23×C53—84no—
142C23×C53#9911yescharacteristic
143C23×C53#9911yescharacteristic
144C23×C53C2×C22⋊C411yescharacteristic
145C23×C53#13512yes—
146D4×C53—84no—
147D4×C53—84no—
148D4×C53—82no—
149D4×C53—82no—
150D4×C53—82no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
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—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—81no—
170C2×C4×C53—81no—
171C2×C4×C53—81no—
172C2×Dic53—82no—
173C2×Dic53—82no—
174C2×Dic53—82no—
175#88322—82no—
176#11843—531nomaximal, Sylow
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—22no—
184C22×C53—21no—
185C22×C53—42no—
186C22×C53—42no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—81no—
195C22×C53—84no—
196C22×C53—81no—
197C22×C53—81no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53—81no—
201C22×C53—84no—
202C22×C53—81no—
203C22×C53—81no—
204C22×C53—81no—
205C22×C53#35312yes—
206C22×C53C2×#9911yescharacteristic
207C22×C53#41611yescharacteristic
208C22×C53#42111yescharacteristic
209C4×C53—42no—
210C4×C53—42no—
211C4×C53—41no—
212C4×C53—41no—
213C4×C53—41no—
214C4×C53—41no—
215C4×C53—81no—
216Dic53—162no—
217#2539D5311yescharacteristic
218#1819—531no—
219C2×#323—531no—
220#1352—532no—
221#1343—532no—
222C2×C53—21no—
223C2×C53—21no—
224C2×C53—22no—
225C2×C53—42no—
226C2×C53—41no—
227C2×C53—81no—
228C2×C53—161no—
229C2×C53#151611yescharacteristic
230C2×C53#212111yescharacteristic
231C2×C53C2×#35311yescharacteristic
232C2×#123—21no—
233C2×#121Dic5311yescharacteristic, complemented
234C2×#120—21no—
235C22×C22⋊C4C2×D5311yescharacteristic
236C2×C22⋊C8—1061no—
237#399Dic5311yescharacteristic, complemented
238#399C2×D5311yescharacteristic
239#392Dic5311yescharacteristic, complemented
240#392C2×D5311yescharacteristic
241#390—21no—
242#390—21no—
243#379—1061no—
244C4×C22⋊C4Dic5311yescharacteristic, complemented
245#342—532no—
246#342—1061no—
247#323—532no—
248#323—1061no—
249C53#1184311yescharacteristic, complemented, Sylow
250C25C22⋊D5311yescharacteristic
251C22×D4—21no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4C2×Dic5311yescharacteristic
255C23×C4C2×Dic5311yescharacteristic
256C23×C4C22⋊D5311yescharacteristic
257C22×C8—2121no—
258#123—44no—
259#121—41no—
260#121—41no—
261#120—44no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C2×Dic5311yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—1062no—
274C2×C22⋊C4—2121no—
275C2×C22⋊C4—2121no—
276C2×C22⋊C4C2×Dic5311yescharacteristic
277C2×C22⋊C4C22⋊D5311yescharacteristic
278C2×C22⋊C4C22⋊D5311yescharacteristic
279C2×C22⋊C4C22×D5311yescharacteristic
280C2×C42—21no—
281C22⋊C8—1062no—
282C22⋊C8—2121no—
283#95C2×Dic5311yescharacteristic
284#95C2×Dic5311yescharacteristic
285C24—21no—
286C24—21no—
287C24—22no—
288C24—21no—
289C24—41no—
290C24—42no—
291C24—81no—
292C24C22⋊Dic5311yescharacteristic
293C24C22⋊Dic5311yescharacteristic
294C24C2×C22⋊D5311yescharacteristic
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—81no—
305C2×D4—81no—
306C2×D4—81no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—42no—
315C22×C4—81no—
316C22×C4—2122no—
317C22×C4C22⋊Dic5311yescharacteristic
318C22×C4C22⋊Dic5311yescharacteristic
319C22×C4C22×Dic5311yescharacteristic
320C22×C4C2×C22⋊D5311yescharacteristic
321C2×C8—4242no—
322C2×C8—2122no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C22⋊C4—24no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—4242no—
337C22⋊C4—4242no—
338C22⋊C4—2122no—
339C22⋊C4—2122no—
340C42—81no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—22no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—81no—
366C23—81no—
367C23—84no—
368C23—84no—
369C23#1313313211yescharacteristic
370C23#1313313211yescharacteristic
371C23C2×C22⋊Dic5311yescharacteristic
372C23#1313323712yes—
373D4—84no—
374D4—84no—
375D4—82no—
376D4—82no—
377D4—82no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—4242no—
400C2×C4—4242no—
401C2×C4—4242no—
402C8—4242no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—81no—
421C22—84no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—84no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22#1681041612yes—
432C22C2×#1313313211yescharacteristic
433C22#1681061911yescharacteristic
434C22#1681064011yescharacteristic
435C4—42no—
436C4—42no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—81no—
442C4—8482no—
443C2—21no—
444C2—21no—
445C2—22no—
446C2—42no—
447C2—41no—
448C2—81no—
449C2—161no—
450C2#2027327111yescharacteristic
451C2#2027636111yescharacteristic
452C2C2×#1681041611yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
Fitting, prime radicalC53×#2539C2
socleC22×C53C2×#99
cosocleC2×C22⋊C4×C53C23
hypercenter#2539D53
hypocenterC53#11843
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
1. #20274789C2semidirect
2. C2×#16810347C2semidirect
3. #20272739C2semidirect
4. #20272712C2semidirect
13. C2×#121×C53C4semidirect
16. C53×#399C4semidirect
18. C53×#392C4semidirect
22. C4×C22⋊C4×C53C4semidirect
233. C2×#121Dic53semidirect
237. #399Dic53semidirect
239. #392Dic53semidirect
244. C4×C22⋊C4Dic53semidirect
249. C53#11843semidirect

Maximal subgroups

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

subgroupindex
#202747892
C2×#168103472
#202727392
#202727122
C53×#25392
#1184353