Finite Group Databasebeta

← #27280139

Subgroups — #27280139

Order 13568.

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

normal subgroups only

This group has 560 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
1#20280531C211yesmaximal, characteristic
2#20277332C211yesmaximal, characteristic
3#20273288C211yesmaximal, characteristic
4#20273286C211yesmaximal, characteristic
5C2×#16810348C211yesmaximal, characteristic
6#20272582C211yesmaximal, characteristic
7C53×#1289C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9C2×#13133173—21no—
10C22×#11719919C2211yescharacteristic
11C2×C22⋊#88322C2211yescharacteristic
12#16810563C412yescomplemented
13#16810553—22no—
14#16810544C412yescomplemented
15#16810520—22no—
16#16810518—22no—
17#16810498—21no—
18#16810484—21no—
19#16810483C2211yescharacteristic
20#16810481—21no—
21#16810361—22no—
22#16810359—22no—
23#16810348—22no—
24C2×C22⋊C8×C53C2211yescharacteristic
25C53×#380C2211yescharacteristic
26C53×#380C2211yescharacteristic
27C22×Dic106C2×C411yescharacteristic
28C23×Dic53D411yescharacteristic
29C23×Dic53D411yescharacteristic
30C22×#88322—21no—
31#13133196—21no—
32#13133196—21no—
33#13133196—21no—
34#13133196—21no—
35#13133173—42no—
36#13133173—42no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—22no—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—22no—
41C2×#11719919—22no—
42C2×#11719919—21no—
43C2×#11719919—21no—
44C2×#11719919—21no—
45C2×#11719919—21no—
46C2×#11719919—21no—
47C2×#11719919C2×C412yes—
48C2×#11719919D412yes—
49C2×C22⋊Dic53—21no—
50C2×C22⋊Dic53—21no—
51C2×C4×Dic53—22no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54#13133124—22no—
55#13133124—22no—
56#13133124—22no—
57#13133124—22no—
58#13133124D412yes—
59C23×C4×C53C2311yescharacteristic
60C22×C8×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C22⋊C8×C53—21no—
66C22⋊C8×C53—21no—
67#95×C53C2×C412yes—
68#95×C53D412yes—
69C24×C53C2×D411yescharacteristic
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—81no—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53#4111yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—22no—
88C22×Dic53—81no—
89C22×Dic53C22⋊C412yes—
90C22×Dic53C22⋊C412yes—
91C22×Dic53SD1612yes—
92C22×Dic53Q1612yes—
93C2×Dic106—21no—
94C2×Dic106—21no—
95C2×Dic106—81no—
96C2×C8×C53—41no—
97C2×C8×C53—81no—
98C4⋊C4×C53—81no—
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—21no—
109C4⋊Dic53—21no—
110C4⋊Dic53—82no—
111C4⋊Dic53—81no—
112#11719919—22no—
113#11719919—22no—
114#11719919—22no—
115#11719919—22no—
116#11719919—44no—
117#11719919—41no—
118#11719919—41no—
119#11719919—82no—
120C4×Dic53—44no—
121C2×#88322—41no—
122C2×#88322—81no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53C2×C22⋊C411yescharacteristic
126C23×C53#12011yescharacteristic
127C23×C53#12311yescharacteristic
128C23×C53C2×SD1611yescharacteristic
129C23×C53C2×Q1611yescharacteristic
130C23×C53AGL(1,ℤ/8)11yescharacteristic
131C23×C53#13711yescharacteristic
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—42no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53C4×D411yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×Dic53—22no—
149C2×Dic53—22no—
150C2×Dic53—22no—
151C2×Dic53—22no—
152C2×Dic53—22no—
153C2×Dic53—22no—
154C2×Dic53—22no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—42no—
164C2×Dic53—81no—
165C2×Dic53#10314yes—
166Dic106—81no—
167Dic106—81no—
168C8×C53—81no—
169#88322—81no—
170#9200—531nomaximal, Sylow
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53#38611yescharacteristic
182C22×C53C2×#10311yescharacteristic
183C22×C53#41711yescharacteristic
184C22×C53#45011yescharacteristic
185C22×C53#45111yescharacteristic
186C22×C53#48311yescharacteristic
187C22×C53#48411yescharacteristic
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—82no—
193C4×C53—82no—
194Dic53—24no—
195Dic53—42no—
196Dic53—82no—
197Dic53—81no—
198Dic53—81no—
199#2627—531no—
200#2316—531no—
201#1520—531no—
202#1520—531no—
203C2×#324—531no—
204#1289—531no—
205#1289D5311yescharacteristic
206C2×C53—21no—
207C2×C53—21no—
208C2×C53—21no—
209C2×C53—21no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53#187211yescharacteristic
213C2×C53#191211yescharacteristic
214C2×C53#191311yescharacteristic
215C2×C22⋊Q8—531no—
216C2×#117—1061no—
217C22×C4⋊C4—531no—
218C2×C22⋊C8—531no—
219C2×C22⋊C8C2×D5311yescharacteristic
220#400—532no—
221#398—1062no—
222#395—532no—
223#389—1062no—
224#389—1062no—
225#385—1061no—
226#381—1061no—
227#380—531no—
228#380—1061no—
229#380C2×D5311yescharacteristic
230#380C2×D5311yescharacteristic
231#328—1062no—
232#328—1062no—
233#324—1062no—
234C53#920011yescharacteristic, complemented, Sylow
235C22×Q8—531no—
236C23×C4—531no—
237C23×C4—531no—
238C23×C4C22×D5311yescharacteristic
239C22×C8—21no—
240C22×C8—1061no—
241C22⋊Q8—1061no—
242C22⋊Q8—1061no—
243C22⋊Q8—1061no—
244C22⋊Q8—1061no—
245#117—2122no—
246#117—2122no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—532no—
250C2×C4⋊C4—532no—
251C2×C4⋊C4—531no—
252C2×C4⋊C4—1061no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4—1062no—
257C2×C4⋊C4—1062no—
258C2×C4⋊C4—1061no—
259C2×C4⋊C4—1062no—
260C2×C4⋊C4—1061no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—1061no—
264C2×C22⋊C4—1061no—
265C2×C42—1062no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—1061no—
269C22⋊C8—1061no—
270#95—532no—
271#95—1062no—
272#95—1062no—
273#95—1062no—
274#95—1062no—
275#95C4×D5312yes—
276#95D21212yes—
277C24D4×D5311yescharacteristic
278C2×Q8—1061no—
279C2×Q8—1061no—
280C2×Q8—4241no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—81no—
284C22×C4—532no—
285C22×C4—532no—
286C22×C4—532no—
287C22×C4—532no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—1062no—
297C22×C4—4241no—
298C22×C4C2×C4×D5311yescharacteristic
299C22×C4C2×D21211yescharacteristic
300C22×C4#1171994511yescharacteristic
301C22×C4D4×D5311yescharacteristic
302C22×C4D4×D5311yescharacteristic
303C22×C4#1171994911yescharacteristic
304C2×C8—41no—
305C2×C8—81no—
306C2×C8—4241no—
307C2×C8—2121no—
308C4⋊C4—81no—
309C4⋊C4—81no—
310C4⋊C4—1062no—
311C4⋊C4—1062no—
312C4⋊C4—1062no—
313C4⋊C4—1062no—
314C4⋊C4—1061no—
315C4⋊C4—1061no—
316C4⋊C4—4242no—
317C4⋊C4—4241no—
318C4⋊C4—4242no—
319C4⋊C4—2124no—
320C4⋊C4—2121no—
321C4⋊C4—2121no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—2121no—
327C22⋊C4—2121no—
328C22⋊C4—2121no—
329C22⋊C4—2121no—
330C42—2124no—
331C23—41no—
332C23—41no—
333C23C22⋊C4×D5311yescharacteristic
334C23#1313318711yescharacteristic
335C23#1313320211yescharacteristic
336C23SD16×D5311yescharacteristic
337C23Q16×D5311yescharacteristic
338C23#1313324111yescharacteristic
339C23#1313324611yescharacteristic
340Q8—4241no—
341Q8—4241no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—534no—
355C2×C4—1062no—
356C2×C4—1062no—
357C2×C4—1062no—
358C2×C4—1062no—
359C2×C4—1062no—
360C2×C4—1062no—
361C2×C4—1062no—
362C2×C4—4241no—
363C2×C4—2122no—
364C2×C4—2122no—
365C2×C4—2122no—
366C2×C4—2122no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4—2122no—
372C2×C4#1313317911yescharacteristic
373C2×C4C4×D21211yescharacteristic
374C2×C4#1313319011yescharacteristic
375C2×C4#1313319211yescharacteristic
376C8—81no—
377C8—4241no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22#1681050111yescharacteristic
389C22#103×D5311yescharacteristic
390C22#1681062111yescharacteristic
391C22#1681077511yescharacteristic
392C22#1681078011yescharacteristic
393C22#1681095511yescharacteristic
394C22#1681096011yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—82no—
400C4—82no—
401C4—1064no—
402C4—4242no—
403C4—4241no—
404C4—4241no—
405C4—2122no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—21no—
412C2#2027502111yescharacteristic
413C2#2027525111yescharacteristic
414C2#2027525611yescharacteristic
subgroupG/H
centerC22#16810501
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1289C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1289D53
hypocenterC53#9200
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
12. #16810563C4semidirect
14. #16810544C4semidirect
234. C53#9200semidirect

Maximal subgroups

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

subgroupindex
#202805312
#202773322
#202732882
#202732862
C2×#168103482
#202725822
C53×#12892
#920053