Finite Group Databasebeta

← #27276989

Subgroups — #27276989

Order 13568.

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

normal subgroups only

This group has 550 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
1C2×#16810686C211yesmaximal, characteristic
2#20274742C212yesmaximal, complemented
3#20274700C212yesmaximal
4#20272583C211yesmaximal, characteristic, complemented
5C53×#1289C211yesmaximal, characteristic, complemented
6C23×#88322C2211yescharacteristic
7C22×C8×D53C411yescharacteristic, complemented
8C2×C4×C22⋊D53C2211yescharacteristic
9#16810686—22no—
10#16810686—22no—
11C2×#13133150—22no—
12C2×#13133146C411yescharacteristic, complemented
13C2×#13133131—22no—
14C2×#13133129C411yescharacteristic, complemented
15C2×C8×Dic53C411yescharacteristic, complemented
16#16810483C2212yes—
17#16810389—22no—
18#16810385—22no—
19C2×C22⋊C8×C53C2211yescharacteristic
20C53×#380C2212yes—
21C22×C22⋊D53D411yescharacteristic
22C22×C4×D53C2×C411yescharacteristic
23C22×#88322—42no—
24C22×#88322C2×C411yescharacteristic
25C22×#88322C2×C411yescharacteristic
26C22×#88322D412yes—
27C22×#88322D412yes—
28C2×C8×D53—21no—
29C2×C8×D53—21no—
30C2×C8×D53—42no—
31C4×C22⋊D53—22no—
32C4×C22⋊D53—22no—
33C2×C4⋊Dic53—22no—
34C2×#11719919Q811yescharacteristic
35C2×C22⋊Dic53—22no—
36C2×C22⋊Dic53D411yescharacteristic
37C2×#11719921Q811yescharacteristic
38C2×C4×Dic53C2×C411yescharacteristic
39#13133150—44no—
40#13133146—22no—
41#13133131—44no—
42#13133129—22no—
43C8×Dic53—21no—
44C8×Dic53—21no—
45#13133124—22no—
46C23×C4×C53C2311yescharacteristic
47C22×C8×C53C2×C411yescharacteristic
48C22×C8×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53—22no—
50C2×C22⋊C4×C53—22no—
51C22⋊C8×C53—22no—
52#95×C53—22no—
53C24×C53C2×D411yescharacteristic
54C23×D53C4⋊C411yescharacteristic
55C22×C4×C53—41no—
56C22×C4×C53—41no—
57C22×C4×C53—42no—
58C22×C4×C53C22×C411yescharacteristic
59C22×C4×C53C2×D412yes—
60C22×C4×C53C2×Q811yescharacteristic
61C22×C4×C53#4112yes—
62C2×C22⋊D53—22no—
63C2×C22⋊D53—22no—
64C2×C22⋊D53D812yes—
65C2×C22⋊D53Q1612yes—
66C22×Dic53—42no—
67C22×Dic53C4⋊C411yescharacteristic
68C22×Dic53C4⋊C411yescharacteristic
69C22×Dic53C4⋊C411yescharacteristic
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—41no—
73C2×C4×D53—41no—
74C2×C8×C53—21no—
75C2×C8×C53—21no—
76C2×C8×C53—41no—
77C2×C8×C53—41no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C22⋊C4×C53—42no—
81C22⋊C4×C53—42no—
82C22⋊Dic53—22no—
83C22⋊Dic53—42no—
84C22⋊Dic53—42no—
85#11719921—22no—
86C4⋊Dic53—42no—
87C4⋊Dic53—42no—
88#11719919—22no—
89C4×Dic53—21no—
90C4×Dic53—21no—
91C2×#88322—24no—
92C2×#88322—22no—
93C2×#88322—41no—
94C2×#88322—42no—
95C2×#88322C22⋊C411yescharacteristic
96C2×#88322C22⋊C412yes—
97C2×#88322C22⋊C411yescharacteristic
98C8×D53—42no—
99C8×D53—42no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53C2×C4⋊C411yescharacteristic
103C23×C53#12112yes—
104C23×C53C2×D811yescharacteristic
105C23×C53C2×Q1611yescharacteristic
106C23×C53#13512yes—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—41no—
110C22×D53—41no—
111C22×D53#10712yescomplemented
112C2×C4×C53—21no—
113C2×C4×C53—22no—
114C2×C4×C53—21no—
115C2×C4×C53—22no—
116C2×C4×C53—41no—
117C2×C4×C53—42no—
118C2×C4×C53—82no—
119C2×C4×C53—82no—
120C2×C4×C53—82no—
121C2×C4×C53C2×C22⋊C411yescharacteristic
122C2×C4×C53#11711yescharacteristic
123C2×C4×C53C22⋊Q812yes—
124C22⋊D53—24no—
125C22⋊D53—24no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—82no—
133C2×Dic53—82no—
134C2×Dic53—82no—
135C2×Dic53#10712yescomplemented
136C4×D53—42no—
137C4×D53—41no—
138C4×D53—41no—
139C8×C53—41no—
140C8×C53—41no—
141#88322—21no—
142#88322—22no—
143#88322#9912yescomplemented
144#8609—531nomaximal, Sylow
145C22×C53—21no—
146C22×C53—22no—
147C22×C53—21no—
148C22×C53—22no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—42no—
152C22×C53#38011yescharacteristic
153C22×C53C2×#10711yescharacteristic
154C22×C53#42711yescharacteristic
155C22×C53#46612yes—
156C22×C53#46712yes—
157C2×D53—22no—
158C2×D53—42no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—42no—
162C4×C53—21no—
163C4×C53—22no—
164C4×C53—162no—
165C4×C53C2×#9911yescharacteristic
166C4×C53#41011yescharacteristic
167Dic53—22no—
168Dic53—41no—
169Dic53—41no—
170Dic53—162no—
171C2×C8×D4—531no—
172#1808—532no—
173#1799—532no—
174#1289—531no—
175#1289D5311yescharacteristic, complemented
176C2×C53—22no—
177C2×C53—21no—
178C2×C53—21no—
179C2×C53—22no—
180C2×C53#179911yescharacteristic
181C2×C53#180811yescharacteristic
182C2×C53#181211yescharacteristic
183D53—42no—
184C23×C8—531no—
185C23×C8—531no—
186C2×C4×D4—531no—
187C8×D4—1062no—
188C8×D4—1062no—
189C2×#107—1062no—
190C2×C4⋊C8—531no—
191C2×#99—1062no—
192C2×C22⋊C8—531no—
193C2×C22⋊C8C2×D5311yescharacteristic
194C2×C4×C8—531no—
195#380—532no—
196#380C2×D5312yes—
197#342—1062no—
198#340—1062no—
199C53#860911yescharacteristic, complemented, Sylow
200C22×D4—531no—
201C23×C4—531no—
202C23×C4C22×D5311yescharacteristic
203C22×C8—531no—
204C22×C8—532no—
205C22×C8—531no—
206C22×C8—532no—
207C22×C8—1061no—
208C22×C8—1061no—
209C22×C8—2122no—
210C22×C8—2122no—
211C22×C8C4×D5311yescharacteristic
212C22×C8C4×D5311yescharacteristic
213C4×D4—1062no—
214C4×D4—1062no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—1062no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—531no—
220C2×C22⋊C4—531no—
221C2×C22⋊C4—1062no—
222C2×C42—531no—
223#107—2124no—
224C4⋊C8—1062no—
225#99—2124no—
226C22⋊C8—22no—
227C22⋊C8—1062no—
228C4×C8—1061no—
229C4×C8—1061no—
230#95—22no—
231#95—1062no—
232C24—531no—
233C24D4×D5311yescharacteristic
234C2×D4—532no—
235C2×D4—532no—
236C2×D4—1062no—
237C2×D4—1062no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—2121no—
247C22×C4—2121no—
248C22×C4—2122no—
249C22×C4C2×C4×D5311yescharacteristic
250C22×C4#1171994512yes—
251C22×C4D4×D5312yes—
252C22×C4Q8×D5311yescharacteristic
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—531no—
258C2×C8—531no—
259C2×C8—532no—
260C2×C8—1064no—
261C2×C8—1062no—
262C2×C8—2122no—
263C2×C8—2122no—
264C2×C8—2121no—
265C2×C8—2122no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—1062no—
269C4⋊C4—2122no—
270C4⋊C4—2122no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—1062no—
274C22⋊C4—1062no—
275C22⋊C4—2122no—
276C22⋊C4—2122no—
277C42—1061no—
278C42—1061no—
279C23—41no—
280C23—41no—
281C23—532no—
282C23—1061no—
283C23—1061no—
284C23—2121no—
285C23—2121no—
286C23C4⋊C4×D5311yescharacteristic
287C23#1313319012yes—
288C23D8×D5311yescharacteristic
289C23Q16×D5311yescharacteristic
290C23#1313323812yes—
291D4—1064no—
292D4—1064no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—532no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—4242no—
308C2×C4—4242no—
309C2×C4—4242no—
310C2×C4—2122no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4C22⋊C4×D5311yescharacteristic
316C2×C4#1313317411yescharacteristic
317C2×C4#1313319512yes—
318C8—41no—
319C8—41no—
320C8—532no—
321C8—1062no—
322C8—1061no—
323C22—22no—
324C22—21no—
325C22—22no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—42no—
330C22—1062no—
331C22—2122no—
332C22—2122no—
333C22—2121no—
334C22—2121no—
335C22#1681048211yescharacteristic
336C22#107×D5311yescharacteristic
337C22#1681065811yescharacteristic
338C22#1681085812yes—
339C22#1681086312yes—
340C4—21no—
341C4—22no—
342C4—162no—
343C4—8482no—
344C4—1062no—
345C4—2121no—
346C4—2121no—
347C4#99×D5311yescharacteristic
348C4#1681059511yescharacteristic
349C2—22no—
350C2—21no—
351C2—21no—
352C2—22no—
353C2—2122no—
354C2#2027470211yescharacteristic
355C2#2027474311yescharacteristic
356C2#2027476011yescharacteristic
subgroupG/H
centerC22#16810482
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1289C2
socleC22×C53#380
prime radicalC22×C22⋊D53D4
cosocleC23×C4×C53C23
hypercenter#1289D53
hypocenterC53#8609
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
2. #20274742C2semidirect
4. #20272583C2semidirect
5. C53×#1289C2semidirect
7. C22×C8×D53C4semidirect
12. C2×#13133146C4semidirect
14. C2×#13133129C4semidirect
15. C2×C8×Dic53C4semidirect
111. C22×D53#107semidirect
135. C2×Dic53#107semidirect
143. #88322#99semidirect
175. #1289D53semidirect
199. C53#8609semidirect

Maximal subgroups

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

subgroupindex
C2×#168106862
#202747422
#202747002
#202725832
C53×#12892
#860953