Finite Group Databasebeta

← #27294373

Subgroups — #27294373

Order 13568.

468 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277983C211yesmaximal, characteristic
2#20274448C211yesmaximal, characteristic
3#20272740C212yesmaximal
4#20272737C212yesmaximal
5C53×#1458C211yesmaximal, characteristic
6C2×C4×Dic106C411yescharacteristic, complemented
7C2×#13133176—21no—
8#16810562C411yescharacteristic, complemented
9#16810559C2211yescharacteristic
10#16810556—21no—
11#16810546—21no—
12#16810543C411yescharacteristic, complemented
13#16810535C2211yescharacteristic
14C4×#11719919—21no—
15C4×C4⋊Dic53C411yescharacteristic, complemented
16C4×C22⋊Dic53—21no—
17#16810390—21no—
18#16810390C2212yes—
19C2×#117×C53C2211yescharacteristic
20C53×#431—21no—
21C53×#324—21no—
22C53×#323C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C4×Dic106—41no—
25C4×Dic106—41no—
26#13133176—24no—
27#13133176—42no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—41no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—22no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53—41no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124C2×C411yescharacteristic
45#13133124C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47#117×C53—41no—
48#117×C53—41no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53C2×C411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C4⋊C8×C53—41no—
56C22⋊C8×C53—22no—
57C4×C8×C53—41no—
58C24×C53C2×D411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53SD1612yes—
67C22×C4×C53Q1612yes—
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—42no—
75C2×Dic106—41no—
76C2×Dic106—42no—
77C2×C8×C53—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C22⋊Dic53—24no—
87C22⋊Dic53—42no—
88C22⋊Dic53—42no—
89C22⋊Dic53—42no—
90C4⋊Dic53—42no—
91C4⋊Dic53—44no—
92C4⋊Dic53—41no—
93#11719919—42no—
94#11719919—42no—
95#11719919—42no—
96C4×Dic53—44no—
97C4×Dic53—41no—
98C4×Dic53—41no—
99C4×Dic53—42no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105C23×C53C2×SD1611yescharacteristic
106C23×C53C2×Q1611yescharacteristic
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53#9911yescharacteristic
122C2×C4×C53#9911yescharacteristic
123C2×C4×C53#10312yes—
124C2×C4×C53#10312yes—
125C2×C4×C53#10412yes—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—44no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—82no—
140C2×Dic53—82no—
141Dic106—42no—
142Dic106—42no—
143C8×C53—82no—
144#11715—531nomaximal, Sylow
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—81no—
154C22×C53C2×#9911yescharacteristic
155C22×C53C2×#10311yescharacteristic
156C22×C53C2×#10411yescharacteristic
157C4×C53—22no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53#32812yes—
163C4×C53#32812yes—
164Dic53—42no—
165Dic53—42no—
166Dic53—81no—
167Dic53—84no—
168Dic53—81no—
169#2396—531no—
170#1740—531no—
171#1458D5311yescharacteristic
172#1352—532no—
173#1351—532no—
174C2×C53—21no—
175C2×C53—21no—
176C2×C53—41no—
177C2×C53—41no—
178C2×C53C2×#32811yescharacteristic
179C2×C53#212011yescharacteristic
180C2×C53#212611yescharacteristic
181C2×C4×Q8—531no—
182C2×#117—1061no—
183C2×#117C2×D5311yescharacteristic
184#431—21no—
185#400—531no—
186#399—531no—
187#398—1061no—
188#396—1061no—
189#395—531no—
190#393—531no—
191C4×C4⋊C4—531no—
192C4×C4⋊C4—1061no—
193C4×C22⋊C4—1061no—
194#342—532no—
195#342—1061no—
196#324—21no—
197#323C2×D5312yes—
198C53#1171511yescharacteristic, complemented, Sylow
199C22×Q8—531no—
200C23×C4D21211yescharacteristic
201C4×Q8—2121no—
202C4×Q8—2121no—
203#117—41no—
204#117—41no—
205#117—1064no—
206#117—2122no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—2121no—
213C2×C4⋊C4C4×D5311yescharacteristic
214C2×C4⋊C4C22⋊D5311yescharacteristic
215C2×C22⋊C4—1062no—
216C2×C22⋊C4—1061no—
217C2×C22⋊C4—1061no—
218C2×C22⋊C4D21211yescharacteristic
219C2×C22⋊C4C22×D5311yescharacteristic
220C2×C42—1061no—
221C2×C42—1061no—
222C2×C42—2121no—
223C2×C42C4×D5311yescharacteristic
224C2×C42C22⋊D5311yescharacteristic
225C4⋊C8—41no—
226C22⋊C8—22no—
227C4×C8—41no—
228#95—531no—
229#95—531no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233#95—1061no—
234C24C2×D21211yescharacteristic
235C2×Q8—2121no—
236C2×Q8—2122no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—2122no—
245C22×C4#1171991312yes—
246C22×C4Dic21212yes—
247C22×C4#1171992111yescharacteristic
248C22×C4#1171992111yescharacteristic
249C22×C4#1171992111yescharacteristic
250C22×C4#1171992111yescharacteristic
251C22×C4C2×C4×D5311yescharacteristic
252C22×C4C2×C22⋊D5311yescharacteristic
253C2×C8—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—2121no—
257C4⋊C4—2122no—
258C4⋊C4—2124no—
259C4⋊C4—2122no—
260C4⋊C4—2122no—
261C4⋊C4—2122no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—1064no—
265C22⋊C4—2122no—
266C22⋊C4—2122no—
267C22⋊C4—2122no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—2124no—
273C42—2121no—
274C42—2121no—
275C42—2122no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23C2×#1171992111yescharacteristic
281C23C2×#1171991311yescharacteristic
282C23C2×Dic21211yescharacteristic
283Q8—2122no—
284Q8—2122no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—1062no—
300C2×C4—1062no—
301C2×C4—4242no—
302C2×C4—4242no—
303C2×C4—2124no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4#1313313111yescharacteristic
315C2×C4#1313313111yescharacteristic
316C2×C4#1313314212yes—
317C2×C4#1313314212yes—
318C2×C4#1313314312yes—
319C8—82no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22C2×#1313313111yescharacteristic
330C22C2×#1313314211yescharacteristic
331C22C2×#1313314311yescharacteristic
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—4241no—
338C4—4244no—
339C4—4241no—
340C4—2122no—
341C4—2122no—
342C4#1681036112yes—
343C4#1681036112yes—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C2×#1681036111yescharacteristic
349C2#2027635711yescharacteristic
350C2#2027637811yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1458C2
socleC22×C53C2×#99
prime radicalC24×C53C2×D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1458D53
hypocenterC53#11715
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic106C4semidirect
8. #16810562C4semidirect
12. #16810543C4semidirect
15. C4×C4⋊Dic53C4semidirect
198. C53#11715semidirect

Maximal subgroups

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

subgroupindex
#202779832
#202744482
#202727402
#202727372
C53×#14582
#1171553