Finite Group Databasebeta

← #27280147

Subgroups — #27280147

Order 13568.

516 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 105 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#20280530C211yesmaximal, characteristic
2#20277317C211yesmaximal, characteristic
3#20273287C211yesmaximal, characteristic
4C2×#16810348C211yesmaximal, characteristic
5#20272583C211yesmaximal, characteristic
6#20272582C211yesmaximal, characteristic
7C53×#1520C211yesmaximal, characteristic
8C22×#11719919C2211yescharacteristic
9C2×C22⋊#88322C2211yescharacteristic
10#16810561C412yescomplemented
11#16810556—22no—
12#16810542C412yescomplemented
13#16810510—22no—
14#16810504—22no—
15#16810495—21no—
16#16810484—21no—
17#16810484—21no—
18#16810483C2211yescharacteristic
19#16810483C2211yescharacteristic
20C22⋊C4×Dic53—21no—
21#16810360—22no—
22#16810348—22no—
23C2×C22⋊Q8×C53C2211yescharacteristic
24C2×C22⋊C8×C53C2211yescharacteristic
25C53×#380C2211yescharacteristic
26C53×#328—22no—
27C23×Dic53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C22×#88322—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—22no—
37C2×#11719919C2×C412yes—
38C2×#11719919D412yes—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C4×Dic53—22no—
42C22⋊#88322—21no—
43C22⋊#88322—21no—
44#13133124—22no—
45#13133124—24no—
46#13133124—22no—
47#13133124—22no—
48#13133124—22no—
49#13133124C2×C412yes—
50#13133124D412yes—
51C22×Q8×C53C2×C411yescharacteristic
52C23×C4×C53C2311yescharacteristic
53C22×C8×C53—21no—
54C22⋊Q8×C53—21no—
55C22⋊Q8×C53—21no—
56C22⋊Q8×C53—21no—
57C22⋊Q8×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53C2×C411yescharacteristic
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C22⋊C8×C53—21no—
64C22⋊C8×C53—21no—
65#95×C53D412yes—
66C24×C53C2×D411yescharacteristic
67C2×Q8×C53—21no—
68C2×Q8×C53—21no—
69C2×Q8×C53—81no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—22no—
73C22×C4×C53—81no—
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—22no—
89C22×Dic53—22no—
90C22×Dic53—81no—
91C22×Dic53C22⋊C412yes—
92C22×Dic53C22⋊C412yes—
93C2×C8×C53—41no—
94C2×C8×C53—81no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—81no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C22⋊Dic53—41no—
105C22⋊Dic53—41no—
106C22⋊Dic53—41no—
107C22⋊Dic53—41no—
108C4⋊Dic53—81no—
109C4⋊Dic53—81no—
110#11719919—22no—
111#11719919—22no—
112#11719919—22no—
113#11719919—22no—
114#11719919—82no—
115C4×Dic53—82no—
116C2×#88322—41no—
117C2×#88322—81no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53C2×C22⋊C411yescharacteristic
121C23×C53#12011yescharacteristic
122C23×C53#12311yescharacteristic
123C23×C53#13511yescharacteristic
124C23×C53#13511yescharacteristic
125C23×C53#13711yescharacteristic
126C23×C53#13711yescharacteristic
127Q8×C53—81no—
128Q8×C53—81no—
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—42no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53C4×D411yescharacteristic
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×Dic53—22no—
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53—22no—
150C2×Dic53—42no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—81no—
163C8×C53—81no—
164#88322—81no—
165#9202—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53#38611yescharacteristic
177C22×C53#41611yescharacteristic
178C22×C53#41911yescharacteristic
179C22×C53#45211yescharacteristic
180C22×C53#45211yescharacteristic
181C22×C53#48511yescharacteristic
182C22×C53#48511yescharacteristic
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—82no—
188C4×C53—81no—
189C4×C53—81no—
190Dic53—42no—
191Dic53—42no—
192Dic53—82no—
193Dic53—82no—
194#2627—531no—
195#2313—531no—
196#1520—531no—
197#1520D5311yescharacteristic
198C2×#324—531no—
199#1289—531no—
200#1289—531no—
201C2×C53—21no—
202C2×C53—21no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53—21no—
207C2×C53#187311yescharacteristic
208C2×C53#191211yescharacteristic
209C2×C53#191311yescharacteristic
210C2×C22⋊Q8C2×D5311yescharacteristic
211C22×C4⋊C4—531no—
212C2×C22⋊C8—531no—
213C2×C22⋊C8C2×D5311yescharacteristic
214#400—532no—
215#398—1062no—
216#395—532no—
217#387—1062no—
218#387—1062no—
219#385—1061no—
220#381—1061no—
221#381—1061no—
222#380—531no—
223#380—531no—
224#380C2×D5311yescharacteristic
225C4×C22⋊C4—1061no—
226#328—22no—
227#328—1062no—
228#324—1062no—
229C53#920211yescharacteristic, complemented, Sylow
230C22×Q8C4×D5311yescharacteristic
231C23×C4—531no—
232C23×C4—531no—
233C23×C4C22×D5311yescharacteristic
234C22×C8—21no—
235C22×C8—1061no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—532no—
243C2×C4⋊C4—532no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1062no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4C4×D5311yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—1061no—
255C2×C22⋊C4—1061no—
256C2×C42—1062no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259C22⋊C8—1061no—
260C22⋊C8—1061no—
261#95—532no—
262#95—532no—
263#95—1062no—
264#95—1062no—
265#95—1062no—
266#95—1062no—
267#95—1064no—
268#95C22⋊D5312yes—
269C24D4×D5311yescharacteristic
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—81no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—81no—
277C22×C4—532no—
278C22×C4—532no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—1062no—
288C22×C4—1062no—
289C22×C4—4241no—
290C22×C4C2×C4×D5311yescharacteristic
291C22×C4#1171994511yescharacteristic
292C22×C4D4×D5311yescharacteristic
293C22×C4D4×D5311yescharacteristic
294C22×C4#1171994711yescharacteristic
295C22×C4C2×C22⋊D5311yescharacteristic
296C2×C8—41no—
297C2×C8—81no—
298C2×C8—4241no—
299C2×C8—2121no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—81no—
305C4⋊C4—1062no—
306C4⋊C4—1062no—
307C4⋊C4—1062no—
308C4⋊C4—1062no—
309C4⋊C4—4242no—
310C4⋊C4—4241no—
311C4⋊C4—4241no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—2121no—
317C22⋊C4—2121no—
318C22⋊C4—2121no—
319C22⋊C4—2121no—
320C42—4242no—
321C23—41no—
322C23—41no—
323C23C22⋊C4×D5311yescharacteristic
324C23#1313318511yescharacteristic
325C23#1313319911yescharacteristic
326C23#1313323811yescharacteristic
327C23#1313324011yescharacteristic
328C23#1313324611yescharacteristic
329C23#1313324911yescharacteristic
330Q8—81no—
331Q8—81no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—1062no—
346C2×C4—1062no—
347C2×C4—1062no—
348C2×C4—1062no—
349C2×C4—4241no—
350C2×C4—2122no—
351C2×C4—2122no—
352C2×C4—2122no—
353C2×C4—2122no—
354C2×C4—2122no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2122no—
362C2×C4#1313317711yescharacteristic
363C2×C4C4×C22⋊D5311yescharacteristic
364C2×C4#1313318811yescharacteristic
365C2×C4#1313319011yescharacteristic
366C8—81no—
367C8—4241no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22#1681049911yescharacteristic
379C22#1681061811yescharacteristic
380C22#1681063011yescharacteristic
381C22#1681078211yescharacteristic
382C22#1681078411yescharacteristic
383C22#1681096211yescharacteristic
384C22#1681096311yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—82no—
390C4—81no—
391C4—81no—
392C4—4242no—
393C4—4242no—
394C4—2122no—
395C4—2122no—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2#2027502411yescharacteristic
403C2#2027524911yescharacteristic
404C2#2027525411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1520C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1520D53
hypocenterC53#9202
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
10. #16810561C4semidirect
12. #16810542C4semidirect
229. C53#9202semidirect

Maximal subgroups

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

subgroupindex
#202805302
#202773172
#202732872
C2×#168103482
#202725832
#202725822
C53×#15202
#920253