Finite Group Databasebeta

← #27284394

Subgroups — #27284394

Order 13568.

610 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16810958C211yesmaximal, characteristic
2#20280531C211yesmaximal, characteristic
3C2×#16810561C211yesmaximal, characteristic
4C2×#16810361C211yesmaximal, characteristic
5C2×#16810348C211yesmaximal, characteristic
6#20272593C211yesmaximal, characteristic
7C53×#1824C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9C22×#11719919C2211yescharacteristic
10#16810958—22no—
11#16810958—22no—
12C2×#13133150—21no—
13C2×#13133140—21no—
14C2×C22⋊#88322C2211yescharacteristic
15#16810563—21no—
16#16810561—22no—
17#16810561—22no—
18#16810553—22no—
19#16810544—21no—
20#16810498—21no—
21#16810484—21no—
22C2×#13133124—21no—
23C2×#13133124—21no—
24C2×#13133124C2211yescharacteristic
25#16810361—22no—
26#16810348—22no—
27C22×C4⋊C4×C53C2211yescharacteristic
28C2×C22⋊C8×C53C2211yescharacteristic
29C53×#380C2211yescharacteristic
30C53×#340—22no—
31C22×Dic106D411yescharacteristic
32C23×Dic53—21no—
33C23×Dic53D411yescharacteristic
34C23×Dic53D411yescharacteristic
35C22×#88322—21no—
36#13133196—21no—
37#13133196—21no—
38#13133196—21no—
39#13133196—21no—
40C2×C4⋊Dic53—22no—
41C2×C4⋊Dic53D411yescharacteristic
42C2×#11719919—21no—
43C2×#11719919—21no—
44C2×#11719919—21no—
45C2×#11719919—21no—
46C2×#11719919—21no—
47C2×#11719919—21no—
48C2×#11719919—21no—
49C2×C22⋊Dic53—21no—
50#13133150—24no—
51#13133140—24no—
52C22⋊#88322—21no—
53C22⋊#88322D412yes—
54#13133124—22no—
55#13133124—22no—
56#13133124—22no—
57#13133124—24no—
58#13133124—24no—
59#13133124—21no—
60#13133124D412yes—
61C23×C4×C53—21no—
62C23×C4×C53C2311yescharacteristic
63C22×C8×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53D412yes—
70C2×C4⋊C4×C53D412yes—
71C2×C22⋊C4×C53—21no—
72C22⋊C8×C53—21no—
73C22⋊C8×C53—21no—
74#95×C53—21no—
75C24×C53C2×D411yescharacteristic
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—42no—
79C22×C4×C53—44no—
80C22×C4×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C22×Dic53—21no—
89C22×Dic53—21no—
90C22×Dic53—21no—
91C22×Dic53—21no—
92C22×Dic53—21no—
93C22×Dic53—21no—
94C22×Dic53—22no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—21no—
99C22×Dic53—21no—
100C22×Dic53—42no—
101C22×Dic53—42no—
102C22×Dic53—42no—
103C22×Dic53—41no—
104C22×Dic53—41no—
105C22×Dic53Q1612yes—
106C2×Dic106—21no—
107C2×Dic106—21no—
108C2×Dic106—42no—
109C2×C8×C53—42no—
110C2×C8×C53—41no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—42no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C22⋊Dic53—41no—
119C22⋊Dic53—41no—
120C4⋊Dic53—21no—
121C4⋊Dic53—21no—
122C4⋊Dic53—82no—
123#11719919—41no—
124#11719919—41no—
125#11719919—41no—
126#11719919—41no—
127#11719919—41no—
128#11719919—41no—
129C2×#88322—22no—
130C2×#88322—42no—
131C23×C53—22no—
132C23×C53—22no—
133C23×C53#12011yescharacteristic
134C23×C53#12011yescharacteristic
135C23×C53#12111yescharacteristic
136C23×C53C2×Q1611yescharacteristic
137C23×C53#13511yescharacteristic
138C23×C53#13711yescharacteristic
139C23×C53#13711yescharacteristic
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—44no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—82no—
152C2×C4×C53—82no—
153C2×C4×C53#12011yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12711yescharacteristic
157C2×Dic53—22no—
158C2×Dic53—22no—
159C2×Dic53—22no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—42no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173C2×Dic53—42no—
174C2×Dic53—82no—
175C2×Dic53—82no—
176C2×Dic53—81no—
177Dic106—42no—
178Dic106—81no—
179C8×C53—81no—
180#88322—42no—
181#9929—531nomaximal, Sylow
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—22no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53#39211yescharacteristic
191C22×C53#45111yescharacteristic
192C22×C53#45111yescharacteristic
193C22×C53#45211yescharacteristic
194C22×C53#45211yescharacteristic
195C22×C53#45412yes—
196C22×C53#45812yes—
197C22×C53#46711yescharacteristic
198C22×C53#47011yescharacteristic
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—82no—
204C4×C53—82no—
205Dic53—42no—
206Dic53—82no—
207Dic53—81no—
208Dic53—81no—
209Dic53—81no—
210C2×#484—531no—
211#2627—531no—
212C2×#400—531no—
213#1824D5311yescharacteristic
214C2×#328—531no—
215C2×#324—531no—
216#1294—531no—
217C2×C53—21no—
218C2×C53—21no—
219C2×C53—21no—
220C2×C53—21no—
221C2×C53#161512yes—
222C2×C53#161712yes—
223C2×C53#200111yescharacteristic
224C2×C53#200311yescharacteristic
225C2×C53#202211yescharacteristic
226C2×C22⋊Q8—531no—
227C22×C4⋊C4—531no—
228C22×C4⋊C4C2×D5311yescharacteristic
229#484—1062no—
230#484—1062no—
231C2×#107—1061no—
232C2×#103—1061no—
233C2×C22⋊C8—531no—
234C2×C22⋊C8C2×D5311yescharacteristic
235#400—1061no—
236#400—1062no—
237#400—1062no—
238#398—1062no—
239#395—1061no—
240#385—1061no—
241#381—1061no—
242#380C2×D5311yescharacteristic
243C2×#95—531no—
244C2×#95—1061no—
245C2×#95—1061no—
246#340—22no—
247#328—1062no—
248#324—1062no—
249C53#992911yescharacteristic, complemented, Sylow
250C22×Q8—531no—
251C23×C4—21no—
252C23×C4—531no—
253C23×C4—531no—
254C23×C4—1061no—
255C23×C4C22×D5311yescharacteristic
256C22×C8—21no—
257C22×C8—1061no—
258C22⋊Q8—1061no—
259C22⋊Q8—1061no—
260C22⋊Q8—1061no—
261C22⋊Q8—1061no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—531no—
268C2×C4⋊C4—1062no—
269C2×C4⋊C4—1061no—
270C2×C4⋊C4—1061no—
271C2×C4⋊C4—1061no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4—1061no—
274C2×C4⋊C4—1061no—
275C2×C4⋊C4—1061no—
276C2×C4⋊C4D21212yes—
277C2×C4⋊C4C22⋊D5312yes—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—1061no—
280#107—1064no—
281#103—1064no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—532no—
285C22⋊C8—1061no—
286#95—21no—
287#95—532no—
288#95—1064no—
289#95—1064no—
290#95—1062no—
291#95—1062no—
292#95—1062no—
293#95—1061no—
294C24D4×D5311yescharacteristic
295C2×Q8—1061no—
296C2×Q8—1061no—
297C2×Q8—2122no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—532no—
305C22×C4—1061no—
306C22×C4—1061no—
307C22×C4—1061no—
308C22×C4—1061no—
309C22×C4—1061no—
310C22×C4—1061no—
311C22×C4—1061no—
312C22×C4—1061no—
313C22×C4—1061no—
314C22×C4—1061no—
315C22×C4—1062no—
316C22×C4—1061no—
317C22×C4—2122no—
318C22×C4—2122no—
319C22×C4—2121no—
320C22×C4—2121no—
321C22×C4—2122no—
322C22×C4C2×D21211yescharacteristic
323C22×C4#1171994511yescharacteristic
324C22×C4D4×D5311yescharacteristic
325C22×C4D4×D5311yescharacteristic
326C22×C4D4×D5311yescharacteristic
327C22×C4C2×C22⋊D5311yescharacteristic
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—1062no—
331C2×C8—2122no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—1061no—
338C4⋊C4—1061no—
339C4⋊C4—4242no—
340C4⋊C4—2121no—
341C4⋊C4—2121no—
342C4⋊C4—2121no—
343C4⋊C4—2121no—
344C4⋊C4—2121no—
345C4⋊C4—2121no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—2121no—
349C22⋊C4—2121no—
350C23—22no—
351C23—22no—
352C23#1313318511yescharacteristic
353C23#1313318711yescharacteristic
354C23#1313319011yescharacteristic
355C23Q16×D5311yescharacteristic
356C23#1313323811yescharacteristic
357C23#1313324611yescharacteristic
358C23#1313324911yescharacteristic
359Q8—4241no—
360Q8—2122no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—1062no—
375C2×C4—1062no—
376C2×C4—1062no—
377C2×C4—4242no—
378C2×C4—4242no—
379C2×C4—4241no—
380C2×C4—2122no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4—2121no—
384C2×C4—2121no—
385C2×C4—2121no—
386C2×C4—2122no—
387C2×C4—2122no—
388C2×C4—2122no—
389C2×C4—2122no—
390C2×C4—2121no—
391C2×C4—2121no—
392C2×C4—2121no—
393C2×C4—2121no—
394C2×C4#1313318711yescharacteristic
395C2×C4#1313319011yescharacteristic
396C2×C4#1313319111yescharacteristic
397C2×C4#1313321311yescharacteristic
398C8—81no—
399C8—2122no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—21no—
407C22—21no—
408C22#1681053211yescharacteristic
409C22#1681077711yescharacteristic
410C22#1681078011yescharacteristic
411C22#1681078211yescharacteristic
412C22#1681078811yescharacteristic
413C22#1681079412yes—
414C22#1681081212yes—
415C22#1681086311yescharacteristic
416C22#1681088011yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—82no—
422C4—82no—
423C4—4242no—
424C4—4241no—
425C4—4241no—
426C4—4241no—
427C4—2122no—
428C2—21no—
429C2—21no—
430C2—21no—
431C2—21no—
432C2#2027374312yes—
433C2#2027375512yes—
434C2#2027572411yescharacteristic
435C2#2027573811yescharacteristic
436C2#2027584611yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1824C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenter#1824D53
hypocenterC53#9929
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
249. C53#9929semidirect

Maximal subgroups

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

subgroupindex
C2×#168109582
#202805312
C2×#168105612
C2×#168103612
C2×#168103482
#202725932
C53×#18242
#992953