Finite Group Databasebeta

← #27287258

Subgroups — #27287258

Order 13568.

422 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 93 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#20275700C211yesmaximal, characteristic
2#20273286C211yesmaximal, characteristic
3C2×#16810359C211yesmaximal, characteristic
4#20272594C211yesmaximal, characteristic
5#20272582C211yesmaximal, characteristic
6#20272563C211yesmaximal, characteristic
7C53×#2569C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9C2×C22⋊#88322C2211yescharacteristic
10C2×C22⋊#88322C2211yescharacteristic
11#16810483C2211yescharacteristic
12#16810389—21no—
13#16810389—21no—
14#16810359—21no—
15#16810359—21no—
16#16810359—21no—
17#16810359—21no—
18C53×#400—21no—
19C53×#398—21no—
20C53×#387—21no—
21C53×#387—21no—
22C53×#385—21no—
23C53×#380—21no—
24C53×#380C2211yescharacteristic
25C4×C22⋊C4×C53C2211yescharacteristic
26C2×#95×C53C2211yescharacteristic
27C22×Dic106Q811yescharacteristic
28C22×#88322—21no—
29C22×#88322—21no—
30#13133196—21no—
31#13133196—21no—
32#13133196—21no—
33#13133196—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53Q811yescharacteristic
36C2×#11719919—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41C22⋊#88322—21no—
42C22⋊#88322—21no—
43#13133124—21no—
44C23×C4×C53—21no—
45C23×C4×C53C2311yescharacteristic
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C42×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53D411yescharacteristic
62#95×C53D411yescharacteristic
63#95×C53D411yescharacteristic
64#95×C53D411yescharacteristic
65C24×C53C2×D411yescharacteristic
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—42no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—81no—
76C22×C4×C53—81no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×Q811yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C22×C4×C53#4111yescharacteristic
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—41no—
86C2×Dic106—21no—
87C2×Dic106—21no—
88C2×Dic106—81no—
89C4⋊C4×C53—42no—
90C4⋊C4×C53—42no—
91C4⋊C4×C53—81no—
92C4⋊C4×C53—81no—
93C4⋊C4×C53—81no—
94C22⋊C4×C53—21no—
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊C4×C53—21no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C42×C53—81no—
101C22⋊Dic53—41no—
102C22⋊Dic53—41no—
103C22⋊Dic53—41no—
104C22⋊Dic53—41no—
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53—81no—
108#11719919—41no—
109#11719919—41no—
110C2×#88322—41no—
111C2×#88322—41no—
112C2×#88322—81no—
113C2×#88322—81no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53#12011yescharacteristic
117C23×C53C22⋊Q811yescharacteristic
118C23×C53#12411yescharacteristic
119C23×C53#13511yescharacteristic
120C23×C53#13511yescharacteristic
121C23×C53#13711yescharacteristic
122C23×C53#13711yescharacteristic
123C2×C4×C53—22no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—82no—
148C2×C4×C53—82no—
149C2×C4×C53C22⋊Q811yescharacteristic
150C2×C4×C53C22⋊Q811yescharacteristic
151C2×C4×C53#12411yescharacteristic
152C2×C4×C53#12411yescharacteristic
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—81no—
156C2×Dic53—81no—
157C2×Dic53—81no—
158C2×Dic53—81no—
159C2×Dic53—81no—
160Dic106—81no—
161Dic106—81no—
162#88322—81no—
163#88322—81no—
164#10377—531nomaximal, Sylow
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53#39311yescharacteristic
176C22×C53#45211yescharacteristic
177C22×C53#45211yescharacteristic
178C22×C53#47911yescharacteristic
179C22×C53#47911yescharacteristic
180C22×C53#48811yescharacteristic
181C22×C53#49111yescharacteristic
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—42no—
187C4×C53—81no—
188C4×C53—81no—
189C4×C53—81no—
190C4×C53—81no—
191C4×C53—81no—
192Dic53—81no—
193Dic53—81no—
194Dic53—161no—
195#2569D5311yescharacteristic
196#1996—531no—
197#1520—531no—
198C2×#328—531no—
199#1294—531no—
200#1289—531no—
201#1282—531no—
202C2×C53—21no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53—21no—
207C2×C53—21no—
208C2×C53#202611yescharacteristic
209C2×C53#203111yescharacteristic
210C2×C53#203211yescharacteristic
211C2×C22⋊Q8—531no—
212C2×C22⋊C8—531no—
213C2×C22⋊C8—531no—
214#400—21no—
215#398—21no—
216#387—21no—
217#387—21no—
218#385—21no—
219#380—21no—
220#380—531no—
221#380C2×D5311yescharacteristic
222C4×C22⋊C4C2×D5311yescharacteristic
223C2×#95C2×D5311yescharacteristic
224#342—1061no—
225#342—1061no—
226#328—1061no—
227#328—1061no—
228#328—1061no—
229#328—1061no—
230C53#1037711yescharacteristic, complemented, Sylow
231C22×Q8—531no—
232C23×C4—21no—
233C23×C4C22×D5311yescharacteristic
234C22×C8—1061no—
235C22×C8—1061no—
236C22⋊Q8—1061no—
237C22⋊Q8—1061no—
238C22⋊Q8—1061no—
239C22⋊Q8—1061no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4—1061no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—1061no—
250C2×C22⋊C4—1061no—
251C2×C22⋊C4C22⋊D5311yescharacteristic
252C2×C22⋊C4C22⋊D5311yescharacteristic
253C2×C42—21no—
254C22⋊C8—1061no—
255C22⋊C8—1061no—
256C22⋊C8—1061no—
257C22⋊C8—1061no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—1061no—
265#95C22⋊D5311yescharacteristic
266#95C22⋊D5311yescharacteristic
267#95C22⋊D5311yescharacteristic
268#95C22⋊D5311yescharacteristic
269C24C2×C22⋊D5311yescharacteristic
270C2×Q8—1061no—
271C2×Q8—1061no—
272C2×Q8—4241no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4—2121no—
287C22×C4Q8×D5311yescharacteristic
288C22×C4#1171994911yescharacteristic
289C22×C4#1171994911yescharacteristic
290C22×C4#1171994911yescharacteristic
291C22×C4C2×C22⋊D5311yescharacteristic
292C22×C4C2×C22⋊D5311yescharacteristic
293C2×C8—4241no—
294C2×C8—4241no—
295C2×C8—2121no—
296C2×C8—2121no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—81no—
300C4⋊C4—81no—
301C4⋊C4—81no—
302C4⋊C4—1061no—
303C4⋊C4—1061no—
304C4⋊C4—4241no—
305C4⋊C4—2121no—
306C4⋊C4—2121no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—2121no—
314C22⋊C4—2121no—
315C22⋊C4—2121no—
316C22⋊C4—2121no—
317C42—81no—
318C23—41no—
319C23—41no—
320C23#1313318611yescharacteristic
321C23#1313319311yescharacteristic
322C23#1313320611yescharacteristic
323C23#1313323711yescharacteristic
324C23#1313323711yescharacteristic
325C23#1313324711yescharacteristic
326C23#1313324711yescharacteristic
327Q8—4241no—
328Q8—4241no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—4241no—
356C2×C4—4241no—
357C2×C4—4241no—
358C2×C4—4241no—
359C2×C4—4241no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4#1313319311yescharacteristic
363C2×C4#1313319311yescharacteristic
364C2×C4#1313320611yescharacteristic
365C2×C4#1313320611yescharacteristic
366C8—4241no—
367C8—4241no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22#1681053411yescharacteristic
379C22#1681078511yescharacteristic
380C22#1681078511yescharacteristic
381C22#1681092711yescharacteristic
382C22#1681092711yescharacteristic
383C22#1681098011yescharacteristic
384C22#1681099311yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—42no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—8481no—
396C4—4241no—
397C4—4241no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2—21no—
403C2—21no—
404C2#2027586311yescharacteristic
405C2#2027588411yescharacteristic
406C2#2027589111yescharacteristic
subgroupG/H
centerC22#16810534
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2569C2
socleC22×C53#393
prime radicalC24×C53C2×D4
hypercenter#2569D53
hypocenterC53#10377
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2569

Complements

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

NH ≅ G/Ntype
230. C53#10377semidirect

Maximal subgroups

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

subgroupindex
#202757002
#202732862
C2×#168103592
#202725942
#202725822
#202725632
C53×#25692
#1037753