Finite Group Databasebeta

← #27277633

Subgroups — #27277633

Order 13568.

474 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 83 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#20277312C211yesmaximal, characteristic
2#20273041C211yesmaximal, characteristic
3#20273041C211yesmaximal, characteristic
4#20272622C211yesmaximal, characteristic, complemented
5#20272621C211yesmaximal, characteristic, complemented
6#20272620C211yesmaximal, characteristic, complemented
7C53×#1305C211yesmaximal, characteristic, complemented
8C22×#11719921C2211yescharacteristic
9#16810587C2211yescharacteristic
10#16810587C2211yescharacteristic
11#16810515—22no—
12#16810513—22no—
13#16810483C2211yescharacteristic
14#16810478C412yes—
15C22⋊C4×Dic53C412yes—
16#16810346—22no—
17#16810346—22no—
18C53×#407C2211yescharacteristic
19C53×#407C2211yescharacteristic
20C53×#380C2211yescharacteristic
21C24×D53C2×C411yescharacteristic
22C23×Dic53C2×C411yescharacteristic
23C2×#11719917—21no—
24C2×#11719917—21no—
25C2×C4⋊Dic53—21no—
26C2×C22⋊Dic53C2×C412yes—
27C2×#11719921—22no—
28C2×#11719921—22no—
29C2×#11719921—21no—
30C2×#11719921—21no—
31C2×#11719921—21no—
32C2×#11719921—21no—
33C2×#11719921D412yes—
34C2×#11719921D412yes—
35C2×C4×Dic53—22no—
36C22⋊#88322—21no—
37C22⋊#88322—21no—
38C22⋊#88322—21no—
39C22⋊#88322—21no—
40#13133124—21no—
41#13133124—22no—
42C23×C4×C53C2311yescharacteristic
43C2×M16×C53—21no—
44C2×M16×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C22⋊C4×C53C2×C412yes—
47C22⋊C8×C53—21no—
48C22⋊C8×C53—21no—
49C22⋊C8×C53—21no—
50C22⋊C8×C53—21no—
51#95×C53—21no—
52C24×C53C22×C411yescharacteristic
53C23×D53—21no—
54C23×D53—21no—
55C23×D53—41no—
56C23×D53—41no—
57C23×D53—81no—
58C23×D53—81no—
59C23×D53C22⋊C412yes—
60C22×C4×C53—22no—
61C22×C4×C53—81no—
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53#4111yescharacteristic
65C22×C4×C53#4111yescharacteristic
66C22×C4×C53#4111yescharacteristic
67C22×C4×C53#4111yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—22no—
71C22×Dic53—81no—
72C22×Dic53C22⋊C412yes—
73M16×C53—81no—
74M16×C53—81no—
75C2×C8×C53—41no—
76C2×C8×C53—41no—
77C4⋊C4×C53—81no—
78C22⋊C4×C53—42no—
79C22⋊Dic53—42no—
80#11719921—22no—
81#11719921—22no—
82#11719921—22no—
83#11719921—22no—
84#11719921—82no—
85#11719921—82no—
86C4⋊Dic53—81no—
87C4×Dic53—82no—
88#11719917—81no—
89#11719917—81no—
90C2×#88322—41no—
91C2×#88322—41no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53C2×C22⋊C411yescharacteristic
97C23×C53#11711yescharacteristic
98C23×C53#11711yescharacteristic
99C22×D53—22no—
100C22×D53—22no—
101C22×D53—22no—
102C22×D53—42no—
103C22×D53—42no—
104C22×D53—42no—
105C22×D53—42no—
106C22×D53—81no—
107C22×D53—81no—
108C22×D53—81no—
109C22×D53—81no—
110C22×D53—84no—
111C22×D53—84no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53#12311yescharacteristic
122C2×C4×C53#12311yescharacteristic
123C2×C4×C53#12311yescharacteristic
124C2×C4×C53#12311yescharacteristic
125C2×Dic53—22no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—81no—
131C2×Dic53—81no—
132C8×C53—81no—
133C8×C53—81no—
134#88322—81no—
135#88322—81no—
136#8759—531nomaximal, Sylow
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—42no—
146C22×C53—42no—
147C22×C53#38111yescharacteristic
148C22×C53#41311yescharacteristic
149C22×C53#41311yescharacteristic
150C2×D53—42no—
151C2×D53—42no—
152C2×D53—42no—
153C2×D53—84no—
154C2×D53—84no—
155C2×D53—82no—
156C2×D53—82no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—82no—
162Dic53—42no—
163Dic53—82no—
164#2312—531no—
165#1462—531no—
166#1462—531no—
167#1305—531no—
168#1305—531no—
169#1305—531no—
170#1305D5311yescharacteristic, complemented
171C2×C53—21no—
172C2×C53—21no—
173C2×C53—41no—
174C2×C53—41no—
175C2×C53#182211yescharacteristic
176C2×C53#182811yescharacteristic
177C2×C53#182911yescharacteristic
178D53—82no—
179C22×C22⋊C4—531no—
180#407—531no—
181#407—531no—
182#407C2×D5311yescharacteristic
183#407C2×D5311yescharacteristic
184#388—1062no—
185#388—1062no—
186#380—531no—
187#380C2×D5311yescharacteristic
188#379—532no—
189C4×C22⋊C4—532no—
190#323—1062no—
191#323—1062no—
192C53#875911yescharacteristic, complemented, Sylow
193C25—531no—
194C23×C4—531no—
195C23×C4C22×D5311yescharacteristic
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—1061no—
199C2×M16—1061no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—1061no—
202C2×C22⋊C4—532no—
203C2×C22⋊C4—532no—
204C2×C22⋊C4—532no—
205C2×C22⋊C4—1061no—
206C2×C22⋊C4—1061no—
207C2×C22⋊C4—1061no—
208C2×C22⋊C4—1061no—
209C2×C22⋊C4—1062no—
210C2×C22⋊C4—1062no—
211C2×C22⋊C4C4×D5312yes—
212C2×C42—1062no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217C22⋊C8—1061no—
218C22⋊C8—1061no—
219C22⋊C8—1061no—
220C22⋊C8—1061no—
221#95—21no—
222#95—1062no—
223#95—1061no—
224C24—532no—
225C24—1061no—
226C24—1061no—
227C24—4241no—
228C24—4241no—
229C24—2121no—
230C24—2121no—
231C24C2×C4×D5311yescharacteristic
232C22×C4—22no—
233C22×C4—81no—
234C22×C4—532no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4—1062no—
238C22×C4—4241no—
239C22×C4#1171994511yescharacteristic
240C22×C4#1171994511yescharacteristic
241C22×C4D4×D5311yescharacteristic
242C22×C4D4×D5311yescharacteristic
243C22×C4#1171994711yescharacteristic
244C22×C4#1171994911yescharacteristic
245M16—81no—
246M16—81no—
247M16—4241no—
248M16—4241no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—2121no—
252C2×C8—2121no—
253C4⋊C4—81no—
254C4⋊C4—4241no—
255C22⋊C4—42no—
256C22⋊C4—1062no—
257C22⋊C4—1062no—
258C22⋊C4—1062no—
259C22⋊C4—1062no—
260C22⋊C4—4242no—
261C22⋊C4—4242no—
262C22⋊C4—2122no—
263C42—4242no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—1062no—
269C23—1062no—
270C23—1062no—
271C23—4241no—
272C23—4241no—
273C23—4241no—
274C23—4241no—
275C23—4244no—
276C23—4244no—
277C23—2122no—
278C23—2122no—
279C23—2122no—
280C23—2122no—
281C23C22⋊C4×D5311yescharacteristic
282C23#1313317411yescharacteristic
283C23#1313317511yescharacteristic
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—1062no—
294C2×C4—4241no—
295C2×C4—4241no—
296C2×C4—2122no—
297C2×C4—2122no—
298C2×C4—2122no—
299C2×C4—2122no—
300C2×C4#1313319911yescharacteristic
301C2×C4#1313319911yescharacteristic
302C2×C4#1313320211yescharacteristic
303C2×C4#1313320211yescharacteristic
304C8—81no—
305C8—81no—
306C8—4241no—
307C8—4241no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—42no—
317C22—42no—
318C22—4242no—
319C22—4242no—
320C22—4244no—
321C22—4244no—
322C22—2122no—
323C22—2122no—
324C22—2122no—
325C22#1681048511yescharacteristic
326C22#1681060411yescharacteristic
327C22#1681060611yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—82no—
333C4—4242no—
334C4—2122no—
335C2—21no—
336C2—21no—
337C2—41no—
338C2—41no—
339C2—4242no—
340C2#2027479711yescharacteristic
341C2#2027482211yescharacteristic
342C2#2027482511yescharacteristic
subgroupG/H
centerC22#16810485
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1305C2
socleC22×C53#381
prime radicalC24×D53C2×C4
cosocleC23×C4×C53C23
hypercenter#1305D53
hypocenterC53#8759
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
4. #20272622C2semidirect
5. #20272621C2semidirect
6. #20272620C2semidirect
7. C53×#1305C2semidirect
170. #1305D53semidirect
192. C53#8759semidirect

Maximal subgroups

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

subgroupindex
#202773122
#202730412
#202726222
#202726212
#202726202
C53×#13052
#875953