Finite Group Databasebeta

← #27300077

Subgroups — #27300077

Order 13568.

544 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 103 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#124×Dic53C211yesmaximal, characteristic
2C2×#16810418C211yesmaximal, characteristic
3#20274701C212yesmaximal
4#20272839C212yesmaximal
5C2×C53×#354C211yesmaximal, characteristic
6C2×Q8×Dic53C411yescharacteristic, complemented
7C2×D4×Dic53C2211yescharacteristic
8C2×#13133132C2212yes—
9#16810530C411yescharacteristic, complemented
10#16810526C2211yescharacteristic
11#16810516—22no—
12#16810489C411yescharacteristic, complemented
13#16810481—22no—
14C22⋊C4×Dic53—22no—
15C42×Dic53C411yescharacteristic, complemented
16#16810418—22no—
17#16810389—22no—
18C2×#124×C53C2211yescharacteristic
19C2×#99×C53C2212yes—
20C53×#354—22no—
21C23×Dic53D412yes—
22Q8×Dic53—41no—
23Q8×Dic53—41no—
24D4×Dic53—41no—
25D4×Dic53—41no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—42no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53D412yes—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53C2×C411yescharacteristic
38#13133132—22no—
39#13133132—22no—
40#13133124—22no—
41#13133124—22no—
42#13133124—42no—
43C22×Q8×C53C2×C411yescharacteristic
44C22×D4×C53C2311yescharacteristic
45#124×C53—22no—
46#124×C53D412yes—
47#124×C53Q812yes—
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53—22no—
50C2×C42×C53C2×C411yescharacteristic
51#99×C53—22no—
52#99×C53—22no—
53C24×C53C2×D412yes—
54C2×Q8×C53—21no—
55C2×Q8×C53—41no—
56C2×Q8×C53—41no—
57C2×Q8×C53C4⋊C412yescomplemented
58C2×D4×C53—41no—
59C2×D4×C53—41no—
60C2×D4×C53C2×D411yescharacteristic
61C2×D4×C53C2×Q811yescharacteristic
62C2×D4×C53#4112yes—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—42no—
66C22×C4×C53C22×C411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—21no—
71C22×Dic53—44no—
72C22×Dic53—42no—
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C22⋊C412yes—
75C22×Dic53C22⋊C411yescharacteristic
76C22⋊C4×C53—24no—
77C22⋊C4×C53—24no—
78C22⋊C4×C53—42no—
79C22⋊C4×C53—42no—
80C42×C53—21no—
81C42×C53C4⋊C412yescomplemented
82C22⋊Dic53—42no—
83C22⋊Dic53—44no—
84C22⋊Dic53—42no—
85C22⋊Dic53—42no—
86C4⋊Dic53—41no—
87C4⋊Dic53—42no—
88C4⋊Dic53—41no—
89#11719919—84no—
90C4×Dic53—21no—
91C4×Dic53—21no—
92C4×Dic53—42no—
93C4×Dic53—42no—
94C4×Dic53—41no—
95C4×Dic53—41no—
96C4×Dic53—41no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—42no—
100C23×C53—42no—
101C23×C53C2×C22⋊C411yescharacteristic
102C23×C53#12112yes—
103C23×C53C22⋊Q812yes—
104Q8×C53—42no—
105Q8×C53—42no—
106D4×C53—41no—
107D4×C53—41no—
108D4×C53—42no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—82no—
118C2×C4×C53—82no—
119C2×C4×C53—82no—
120C2×C4×C53C2×C4⋊C411yescharacteristic
121C2×C4×C53#11711yescharacteristic
122C2×Dic53—22no—
123C2×Dic53—22no—
124C2×Dic53—22no—
125C2×Dic53—21no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—44no—
129C2×Dic53—42no—
130C2×Dic53—41no—
131C2×Dic53—42no—
132C2×Dic53—41no—
133C2×Dic53—82no—
134C2×Dic53—82no—
135C2×Dic53—82no—
136C2×Dic53—82no—
137C2×Dic53#9911yescharacteristic
138C2×Dic53#9911yescharacteristic
139#12798—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—22no—
143C22×C53—22no—
144C22×C53—42no—
145C22×C53—42no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—82no—
149C22×C53#38011yescharacteristic
150C22×C53C2×#9911yescharacteristic
151C22×C53#41011yescharacteristic
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—42no—
155C4×C53—42no—
156C4×C53—162no—
157Dic53—21no—
158Dic53—42no—
159Dic53—42no—
160Dic53—41no—
161Dic53—81no—
162Dic53—162no—
163Dic53#35412yescomplemented
164C4×#124—531no—
165C2×#354—531no—
166C2×#354D5311yescharacteristic
167#1799—532no—
168#1393—532no—
169C2×C53—21no—
170C2×C53—21no—
171C2×C53—42no—
172C2×C53—42no—
173C2×C53#179911yescharacteristic
174C2×C53C2×#35411yescharacteristic
175C2×C53#212611yescharacteristic
176C2×#124C2×D5311yescharacteristic
177C2×C4×Q8—531no—
178C2×C4×D4—531no—
179C2×#99—532no—
180C2×#99C2×D5312yes—
181#391—531no—
182#390—531no—
183#388—1062no—
184#382—531no—
185#380—1062no—
186C4×C22⋊C4—1062no—
187C43—531no—
188#354—22no—
189#354—1062no—
190#342—1062no—
191C53#1279811yescharacteristic, complemented, Sylow
192C22×Q8C4×D5311yescharacteristic
193C22×D4C22×D5311yescharacteristic
194C23×C4—532no—
195#124—22no—
196#124Dic10612yes—
197#124C22⋊D5312yes—
198C4×Q8—2121no—
199C4×Q8—2121no—
200C4×D4—2121no—
201C4×D4—2121no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—2122no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—532no—
209C2×C22⋊C4—1062no—
210C2×C22⋊C4—1062no—
211C2×C42—531no—
212C2×C42—531no—
213C2×C42—1061no—
214C2×C42—1061no—
215C2×C42—1061no—
216C2×C42C4×D5311yescharacteristic
217#99—22no—
218#99—22no—
219#99—1062no—
220#99—1062no—
221#95—1062no—
222#95—1062no—
223#95—2122no—
224C24D4×D5312yes—
225C2×Q8—21no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8#1171991912yescomplemented
229C2×D4—41no—
230C2×D4—41no—
231C2×D4C2×Dic10611yescharacteristic
232C2×D4#1171994712yes—
233C2×D4C2×C22⋊D5311yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—531no—
238C22×C4—532no—
239C22×C4—531no—
240C22×C4—1062no—
241C22×C4—1062no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—2124no—
245C22×C4—2122no—
246C22×C4C2×C4×D5311yescharacteristic
247C4⋊C4—4244no—
248C4⋊C4—2121no—
249C4⋊C4—2121no—
250C4⋊C4—2122no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—2122no—
256C22⋊C4—2124no—
257C22⋊C4—2122no—
258C22⋊C4—2122no—
259C42—21no—
260C42—1061no—
261C42—1061no—
262C42—2121no—
263C42—2121no—
264C42—2121no—
265C42—2122no—
266C42—2122no—
267C42#1171991912yescomplemented
268C23—22no—
269C23—22no—
270C23—42no—
271C23—42no—
272C23C22⋊C4×D5311yescharacteristic
273C23#1313318912yes—
274C23#1313319412yes—
275Q8—42no—
276Q8—42no—
277D4—41no—
278D4—42no—
279D4—41no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—531no—
292C2×C4—531no—
293C2×C4—1062no—
294C2×C4—1062no—
295C2×C4—1061no—
296C2×C4—1062no—
297C2×C4—4242no—
298C2×C4—4242no—
299C2×C4—4242no—
300C2×C4—4242no—
301C2×C4—2124no—
302C2×C4—2122no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2122no—
308C2×C4C2×#1171991911yescharacteristic
309C2×C4#1313317311yescharacteristic
310C22—21no—
311C22—21no—
312C22—22no—
313C22—22no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22#1681048111yescharacteristic
320C22#99×D5311yescharacteristic
321C22#1681059511yescharacteristic
322C4—21no—
323C4—21no—
324C4—42no—
325C4—42no—
326C4—162no—
327C4—532no—
328C4—8482no—
329C4—1061no—
330C4—4241no—
331C4—2122no—
332C4—2122no—
333C4—2121no—
334C2—21no—
335C2—21no—
336C2—42no—
337C2—42no—
338C2#2027470111yescharacteristic
339C2#354×D5311yescharacteristic
340C2#2027637911yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC2×C53×#354C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenterC2×#354D53
hypocenterC53#12798
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
6. C2×Q8×Dic53C4semidirect
9. #16810530C4semidirect
12. #16810489C4semidirect
15. C42×Dic53C4semidirect
57. C2×Q8×C53C4⋊C4semidirect
81. C42×C53C4⋊C4semidirect
163. Dic53#354semidirect
191. C53#12798semidirect
228. C2×Q8#11719919semidirect
267. C42#11719919semidirect

Maximal subgroups

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

subgroupindex
#124×Dic532
C2×#168104182
#202747012
#202728392
C2×C53×#3542
#1279853