Finite Group Databasebeta

← #27288511

Subgroups — #27288511

Order 13568.

604 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 121 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#20280829C211yesmaximal, characteristic
2#20274910C212yesmaximal
3#20272979C211yesmaximal, characteristic
4#20272595C212yesmaximal
5C53×#1280C211yesmaximal, characteristic
6C2×#13133196—21no—
7C22×C4⋊Dic53C2211yescharacteristic
8C22×#11719919—21no—
9C2×#13133149—22no—
10#16810555—22no—
11#16810544—22no—
12#16810483—21no—
13#16810483C2212yes—
14C2×#13133124—21no—
15C2×#13133124C2211yescharacteristic
16#16810386—22no—
17C2×C22⋊C8×C53C2212yes—
18C2×#95×C53C2211yescharacteristic
19C22×Dic106—21no—
20C23×Dic53—21no—
21C23×Dic53—21no—
22#13133196—42no—
23#13133196—42no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53—42no—
29C2×C4⋊Dic53D412yes—
30C2×C4⋊Dic53D412yes—
31C2×#11719919—22no—
32C2×#11719919—42no—
33C2×#11719919—42no—
34C2×#11719919—42no—
35C2×C22⋊Dic53—22no—
36#13133149—44no—
37#13133124—22no—
38#13133124—22no—
39#13133124—21no—
40#13133124—42no—
41#13133124Q812yescomplemented
42C23×C4×C53D411yescharacteristic
43C23×C4×C53D411yescharacteristic
44C23×C4×C53C2311yescharacteristic
45C22×C8×C53—22no—
46C22⋊C8×C53—22no—
47C22⋊C8×C53Q814yescomplemented
48#95×C53—22no—
49C24×C53C2×D411yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—41no—
55C22×C4×C53—41no—
56C22×C4×C53—41no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53SD1612yes—
61C22×C4×C53SD1612yes—
62C22×C4×C53C2×D412yes—
63C22×C4×C53C2×Q812yes—
64C22×C4×C53C2×Q811yescharacteristic
65C22×C4×C53#4111yescharacteristic
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74C22×Dic53—41no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82C2×Dic106—42no—
83C2×Dic106—81no—
84C2×Dic106—81no—
85C2×C8×C53—24no—
86C2×C8×C53—42no—
87C2×C8×C53—42no—
88C22⋊Dic53—42no—
89C22⋊Dic53—42no—
90C4⋊Dic53—24no—
91C4⋊Dic53—42no—
92C4⋊Dic53—42no—
93C4⋊Dic53—42no—
94C4⋊Dic53—42no—
95C4⋊Dic53—81no—
96C4⋊Dic53—81no—
97#11719919—44no—
98#11719919—48no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53#12011yescharacteristic
104C23×C53C22⋊Q811yescharacteristic
105C23×C53C4⋊Q811yescharacteristic
106C23×C53C2×SD1611yescharacteristic
107C23×C53C2×SD1611yescharacteristic
108C23×C53AGL(1,ℤ/8)11yescharacteristic
109C23×C53#13711yescharacteristic
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—81no—
124C2×C4×C53—81no—
125C2×C4×C53C22⋊Q812yes—
126C2×C4×C53C4⋊Q812yes—
127C2×Dic53—24no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—81no—
144C2×Dic53—81no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147Dic106—84no—
148C8×C53—44no—
149#10570—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53#39511yescharacteristic
163C22×C53#44811yescharacteristic
164C22×C53#45011yescharacteristic
165C22×C53#45311yescharacteristic
166C22×C53#45611yescharacteristic
167C22×C53B(4,𝔽2)11yescharacteristic
168C22×C53#45811yescharacteristic
169C22×C53#47511yescharacteristic
170C22×C53#47611yescharacteristic
171C22×C53#49811yescharacteristic
172C22×C53#50111yescharacteristic
173C4×C53—22no—
174C4×C53—22no—
175C4×C53—42no—
176C4×C53—42no—
177Dic53—44no—
178Dic53—82no—
179Dic53—82no—
180Dic53—82no—
181#2659—531no—
182#1853—532no—
183#1444—531no—
184#1294—532no—
185#1280D5311yescharacteristic
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—21no—
190C2×C53#159611yescharacteristic
191C2×C53#160411yescharacteristic
192C2×C53#161111yescharacteristic
193C2×C53#161311yescharacteristic
194C2×C53#205811yescharacteristic
195C2×C53#206011yescharacteristic
196C2×C53#206211yescharacteristic
197C2×C22⋊Q8—1061no—
198C22×C4⋊C4—531no—
199C22×C4⋊C4—1061no—
200C2×#106—1062no—
201C2×C22⋊C8C2×D5312yes—
202#398—1062no—
203#395—1062no—
204#380—532no—
205#380—1061no—
206C2×#95—531no—
207C2×#95—1061no—
208C2×#95C2×D5311yescharacteristic
209#340—1062no—
210C53#1057011yescharacteristic, complemented, Sylow
211C22×Q8—1061no—
212C23×C4—1061no—
213C23×C4—1061no—
214C23×C4D21211yescharacteristic
215C23×C4D21211yescharacteristic
216C23×C4C22×D5311yescharacteristic
217C22×C8—22no—
218C22⋊Q8—2122no—
219C22⋊Q8—2122no—
220C2×C4⋊C4—532no—
221C2×C4⋊C4—532no—
222C2×C4⋊C4—1062no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1062no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—2122no—
228C2×C4⋊C4—2122no—
229C2×C4⋊C4—2122no—
230C2×C4⋊C4—2122no—
231C2×C22⋊C4—1062no—
232#106—2124no—
233C22⋊C8—22no—
234C22⋊C8Dic10614yescomplemented
235#95—22no—
236#95—532no—
237#95—1062no—
238#95—1062no—
239#95—1061no—
240#95—2122no—
241C24C2×D21211yescharacteristic
242C2×Q8—4241no—
243C2×Q8—4241no—
244C2×Q8—2122no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—1062no—
256C22×C4—1062no—
257C22×C4—1062no—
258C22×C4—1062no—
259C22×C4—1062no—
260C22×C4—1062no—
261C22×C4—2121no—
262C22×C4—2121no—
263C22×C4—2121no—
264C22×C4—2121no—
265C22×C4—2121no—
266C22×C4—2121no—
267C22×C4—2121no—
268C22×C4—2121no—
269C22×C4—2121no—
270C22×C4—2121no—
271C22×C4#1171991312yes—
272C22×C4#1171991312yes—
273C22×C4C2×Dic10612yes—
274C22×C4D4×D5312yes—
275C22×C4#1171994711yescharacteristic
276C22×C4Q8×D5311yescharacteristic
277C2×C8—24no—
278C2×C8—42no—
279C2×C8—42no—
280C4⋊C4—1064no—
281C4⋊C4—4241no—
282C4⋊C4—4241no—
283C4⋊C4—2122no—
284C4⋊C4—2122no—
285C4⋊C4—2122no—
286C4⋊C4—2124no—
287C4⋊C4—2122no—
288C4⋊C4—2128no—
289C22⋊C4—2122no—
290C22⋊C4—2122no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#1313318711yescharacteristic
296C23#1313319711yescharacteristic
297C23#1313321711yescharacteristic
298C23C2×#1171991311yescharacteristic
299C23C2×#1171991311yescharacteristic
300C23#1313324511yescharacteristic
301C23#1313325011yescharacteristic
302Q8—4244no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—1064no—
319C2×C4—4241no—
320C2×C4—4241no—
321C2×C4—4241no—
322C2×C4—4241no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4—2122no—
330C2×C4—2122no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4#1313319412yes—
339C2×C4#1313321612yes—
340C8—44no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22#1681054411yescharacteristic
354C22#1681076411yescharacteristic
355C22#1681077611yescharacteristic
356C22#1681079311yescharacteristic
357C22#1681080811yescharacteristic
358C22#1681081111yescharacteristic
359C22#1681081411yescharacteristic
360C22#1681090911yescharacteristic
361C22#1681091411yescharacteristic
362C22#1681102111yescharacteristic
363C22#1681103211yescharacteristic
364C4—22no—
365C4—22no—
366C4—42no—
367C4—42no—
368C4—4242no—
369C4—4242no—
370C4—4242no—
371C4—2124no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#2027364011yescharacteristic
377C2#2027368611yescharacteristic
378C2#2027372511yescharacteristic
379C2#2027373711yescharacteristic
380C2#2027604911yescharacteristic
381C2#2027606111yescharacteristic
382C2#2027606911yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1280C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenter#1280D53
hypocenterC53#10570
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
41. #13133124Q8semidirect
47. C22⋊C8×C53Q8semidirect
210. C53#10570semidirect
234. C22⋊C8Dic106semidirect

Maximal subgroups

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

subgroupindex
#202808292
#202749102
#202729792
#202725952
C53×#12802
#1057053