Finite Group Databasebeta

← #27294888

Subgroups — #27294888

Order 13568.

418 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 87 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#20274703C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3#20272739C212yesmaximal, complemented
4#20272715C212yesmaximal, complemented
5C53×#2633C211yesmaximal, characteristic
6C2×#13133132—21no—
7#16810587—21no—
8#16810483—21no—
9#16810390C2212yes—
10#16810349—21no—
11#16810347C2212yes—
12C2×#125×C53C411yescharacteristic, complemented
13C2×C4×D4×C53C2211yescharacteristic
14C53×#399C411yescharacteristic, complemented
15C53×#399C2211yescharacteristic
16C53×#398—21no—
17C53×#397C411yescharacteristic, complemented
18C53×#397C2211yescharacteristic
19C53×#394—21no—
20C53×#385—21no—
21C53×#385—21no—
22C53×#383C411yescharacteristic, complemented
23C2×#11719917—41no—
24C2×C4⋊Dic53—41no—
25C2×C22⋊Dic53—22no—
26#13133132—42no—
27C22⋊#88322—22no—
28C22⋊#88322—41no—
29#13133124—41no—
30C22×D4×C53D411yescharacteristic
31C23×C4×C53D411yescharacteristic
32C23×C4×C53D411yescharacteristic
33#125×C53—41no—
34#125×C53—41no—
35C4×D4×C53—41no—
36C4×D4×C53—41no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53C2311yescharacteristic
50C2×C42×C53C2×C411yescharacteristic
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—81no—
72C22×C4×C53—81no—
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×Dic53—42no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—81no—
86C4⋊C4×C53—81no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—42no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—81no—
93C22⋊C4×C53—81no—
94C42×C53—21no—
95C42×C53—21no—
96C22⋊Dic53—42no—
97C22⋊Dic53—42no—
98C4⋊Dic53—82no—
99#11719917—82no—
100C2×#88322—42no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53C2×C22⋊C411yescharacteristic
111C23×C53AGL(1,ℤ/8)11yescharacteristic
112C23×C53#13711yescharacteristic
113D4×C53—42no—
114D4×C53—42no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
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—81no—
140C2×C4×C53—81no—
141C2×C4×C53#9911yescharacteristic
142C2×C4×C53#9911yescharacteristic
143C2×Dic53—82no—
144C2×Dic53—82no—
145C2×Dic53—82no—
146#88322—82no—
147#11827—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—42no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53C2×#9911yescharacteristic
163C22×C53#41711yescharacteristic
164C22×C53#42111yescharacteristic
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—81no—
172C4×C53—81no—
173C4×C53—81no—
174C4×C53—81no—
175Dic53—162no—
176#2633D5311yescharacteristic
177#1799—531no—
178#1465—531no—
179#1352—532no—
180#1344—532no—
181C2×C53—21no—
182C2×C53—21no—
183C2×C53—41no—
184C2×C53—41no—
185C2×C53—81no—
186C2×C53#151311yescharacteristic
187C2×C53#212311yescharacteristic
188C2×C53#212711yescharacteristic
189C2×#125Dic5311yescharacteristic, complemented
190C2×C4×D4C2×D5311yescharacteristic
191C2×#99—1061no—
192#407—1061no—
193#399Dic5311yescharacteristic, complemented
194#399C2×D5311yescharacteristic
195#398—21no—
196#397Dic5311yescharacteristic, complemented
197#397C2×D5311yescharacteristic
198#394—21no—
199#385—21no—
200#385—21no—
201#383Dic5311yescharacteristic, complemented
202#380—1061no—
203#342—532no—
204#324—1061no—
205#323—532no—
206C53#1182711yescharacteristic, complemented, Sylow
207C22×D4C22⋊D5311yescharacteristic
208C23×C4C22⋊D5311yescharacteristic
209C23×C4C22⋊D5311yescharacteristic
210C2×M16—2121no—
211#125—41no—
212#125—41no—
213C4×D4—41no—
214C4×D4—41no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—2121no—
220C2×C4⋊C4C2×Dic5311yescharacteristic
221C2×C4⋊C4C2×Dic5311yescharacteristic
222C2×C4⋊C4C22⋊D5311yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—1062no—
228C2×C22⋊C4C2×Dic5311yescharacteristic
229C2×C22⋊C4C22×D5311yescharacteristic
230C2×C42C2×Dic5311yescharacteristic
231#99—2122no—
232C22⋊C8—1062no—
233C22⋊C8—2121no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—2121no—
239#95C2×Dic5311yescharacteristic
240#95C2×Dic5311yescharacteristic
241C24C22⋊Dic5311yescharacteristic
242C24C2×C22⋊D5311yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—41no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—2122no—
258C22×C4C22⋊Dic5311yescharacteristic
259C22×C4C22⋊Dic5311yescharacteristic
260C22×C4C22⋊Dic5311yescharacteristic
261C22×C4C22×Dic5311yescharacteristic
262C22×C4C2×C22⋊D5311yescharacteristic
263M16—4242no—
264C2×C8—2122no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—81no—
272C4⋊C4—81no—
273C4⋊C4—4242no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—81no—
280C22⋊C4—81no—
281C22⋊C4—2122no—
282C22⋊C4—2122no—
283C42—21no—
284C42—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23C2×C22⋊Dic5311yescharacteristic
295C23#1313324211yescharacteristic
296C23#1313324711yescharacteristic
297D4—42no—
298D4—42no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—4242no—
326C2×C4—4242no—
327C2×C4—4242no—
328C2×C4#1313313211yescharacteristic
329C2×C4#1313313211yescharacteristic
330C8—4242no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22C2×#1313313211yescharacteristic
346C22#1681062411yescharacteristic
347C22#1681064011yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—8482no—
359C2—21no—
360C2—21no—
361C2—41no—
362C2—41no—
363C2—81no—
364C2#2027325811yescharacteristic
365C2#2027636911yescharacteristic
366C2#2027638411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2633C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2633D53
hypocenterC53#11827
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2633

Complements

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

NH ≅ G/Ntype
3. #20272739C2semidirect
4. #20272715C2semidirect
12. C2×#125×C53C4semidirect
14. C53×#399C4semidirect
17. C53×#397C4semidirect
22. C53×#383C4semidirect
189. C2×#125Dic53semidirect
193. #399Dic53semidirect
196. #397Dic53semidirect
201. #383Dic53semidirect
206. C53#11827semidirect

Maximal subgroups

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

subgroupindex
#202747032
#202730532
#202727392
#202727152
C53×#26332
#1182753