Finite Group Databasebeta

← #27299264

Subgroups — #27299264

Order 13568.

564 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 125 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#20277428C211yesmaximal, characteristic
2C2×#16810418C211yesmaximal, characteristic
3#99×Dic53C212yesmaximal
4#20272839C212yesmaximal
5C2×C53×#354C211yesmaximal, characteristic
6C2×Q8×Dic53C411yescharacteristic, complemented
7C2×D4×Dic53C2211yescharacteristic
8C2×#13133132C2212yes—
9#16810530C411yescharacteristic, complemented
10#16810526C2211yescharacteristic
11#16810516—22no—
12#16810498—22no—
13#16810493C411yescharacteristic, complemented
14C22⋊C4×Dic53—22no—
15#16810469C411yescharacteristic, complemented
16#16810418C414yescomplemented
17#16810389—22no—
18C2×#124×C53C2211yescharacteristic
19C2×#99×C53C2212yes—
20C53×#354C414yescomplemented
21C23×Dic53D412yes—
22Q8×Dic53—42no—
23D4×Dic53—42no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53D412yes—
31C2×C4×Dic53—21no—
32C2×C4×Dic53—42no—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35#13133132—22no—
36#13133132C2×C414yes—
37#13133124—21no—
38#13133124—22no—
39#13133124—21no—
40#13133124—42no—
41C22×Q8×C53C2×C411yescharacteristic
42C22×D4×C53C2311yescharacteristic
43#124×C53—22no—
44#124×C53C2×C412yes—
45#124×C53C2×C412yes—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53—22no—
48C2×C42×C53C2×C411yescharacteristic
49#99×C53—22no—
50#99×C53C2×C414yes—
51C24×C53C2×D412yes—
52C2×Q8×C53—21no—
53C2×Q8×C53—41no—
54C2×Q8×C53—41no—
55C2×Q8×C53C4212yescomplemented
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C2×D4×C53C22×C411yescharacteristic
59C2×D4×C53C22×C411yescharacteristic
60C2×D4×C53#4112yes—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—42no—
64C22×C4×C53C22×C411yescharacteristic
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—42no—
70C22×Dic53—82no—
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C22⋊C412yes—
73C22×Dic53C22⋊C411yescharacteristic
74C22⋊C4×C53—24no—
75C22⋊C4×C53—24no—
76C22⋊C4×C53—24no—
77C22⋊C4×C53—42no—
78C42×C53—21no—
79C42×C53C4212yescomplemented
80C22⋊Dic53—24no—
81C22⋊Dic53—42no—
82C22⋊Dic53—42no—
83C22⋊Dic53—82no—
84C4⋊Dic53—41no—
85C4⋊Dic53—42no—
86C4⋊Dic53—41no—
87C4⋊Dic53—82no—
88C4×Dic53—22no—
89C4×Dic53—42no—
90C4×Dic53—41no—
91C4×Dic53—84no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53C2×C22⋊C411yescharacteristic
97C23×C53C4×D412yes—
98C23×C53C4×D412yes—
99Q8×C53—42no—
100Q8×C53—42no—
101D4×C53—41no—
102D4×C53—41no—
103D4×C53—42no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—44no—
113C2×C4×C53—82no—
114C2×C4×C53—82no—
115C2×C4×C53C2×C4211yescharacteristic
116C2×C4×C53#11711yescharacteristic
117C2×Dic53—22no—
118C2×Dic53—21no—
119C2×Dic53—41no—
120C2×Dic53—44no—
121C2×Dic53—42no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—42no—
125C2×Dic53—41no—
126C2×Dic53—44no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—82no—
130C2×Dic53—82no—
131C2×Dic53—82no—
132C2×Dic53#9912yes—
133#12627—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—22no—
137C22×C53—22no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—42no—
142C22×C53—82no—
143C22×C53C4×C22⋊C411yescharacteristic
144C22×C53C2×#9911yescharacteristic
145C22×C53#41011yescharacteristic
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—42no—
149C4×C53—42no—
150C4×C53—84no—
151Dic53—22no—
152Dic53—42no—
153Dic53—41no—
154Dic53—81no—
155Dic53—84no—
156Dic53—81no—
157#2326—531no—
158C2×#354—531no—
159C2×#354D5311yescharacteristic
160C4×#99—532no—
161#1393—532no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53—42no—
165C2×C53—42no—
166C2×C53C4×#9911yescharacteristic
167C2×C53#212711yescharacteristic
168C2×C53#212911yescharacteristic
169C2×#124C2×D5311yescharacteristic
170C2×C4×Q8—531no—
171C2×C4×D4—531no—
172C2×#99—532no—
173C2×#99C2×D5312yes—
174#391—531no—
175#390—531no—
176#388—1062no—
177#385—1062no—
178#384—531no—
179C4×C22⋊C4—1062no—
180#376—531no—
181#354—534no—
182#354Dic5314yescomplemented
183#342—1062no—
184C53#1262711yescharacteristic, complemented, Sylow
185C22×Q8C4×D5311yescharacteristic
186C22×D4C22×D5311yescharacteristic
187C23×C4—532no—
188#124—22no—
189#124C4×D5312yes—
190#124C2×Dic5312yes—
191C4×Q8—2122no—
192C4×D4—2122no—
193C2×C4⋊C4—531no—
194C2×C4⋊C4—531no—
195C2×C4⋊C4—1062no—
196C2×C4⋊C4—1061no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—532no—
200C2×C22⋊C4—1062no—
201C2×C22⋊C4—1062no—
202C2×C42—531no—
203C2×C42—531no—
204C2×C42—1061no—
205C2×C42—2122no—
206C2×C42C4×D5311yescharacteristic
207#99—22no—
208#99—534no—
209#99—1062no—
210#99C2×Dic5314yes—
211#95—1061no—
212#95—1061no—
213#95—1062no—
214#95—2122no—
215C24D4×D5312yes—
216C2×Q8—21no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8C4×Dic5312yescomplemented
220C2×D4—41no—
221C2×D4—41no—
222C2×D4C2×C4×D5311yescharacteristic
223C2×D4#1171994712yes—
224C2×D4C22×Dic5311yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—531no—
229C22×C4—531no—
230C22×C4—532no—
231C22×C4—1062no—
232C22×C4—1062no—
233C22×C4—1061no—
234C22×C4—1061no—
235C22×C4—4242no—
236C22×C4—2122no—
237C22×C4C2×C4×D5311yescharacteristic
238C4⋊C4—4242no—
239C4⋊C4—2121no—
240C4⋊C4—2121no—
241C4⋊C4—2122no—
242C22⋊C4—24no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—42no—
246C22⋊C4—1064no—
247C22⋊C4—4242no—
248C22⋊C4—2122no—
249C22⋊C4—2122no—
250C42—21no—
251C42—1062no—
252C42—4244no—
253C42—2121no—
254C42—2122no—
255C42C4×Dic5312yescomplemented
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23C22⋊C4×D5311yescharacteristic
261C23#1313317712yes—
262C23D4×Dic5312yes—
263Q8—42no—
264Q8—42no—
265D4—41no—
266D4—42no—
267D4—41no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—44no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—532no—
280C2×C4—1061no—
281C2×C4—1062no—
282C2×C4—4242no—
283C2×C4—4242no—
284C2×C4—4242no—
285C2×C4—2121no—
286C2×C4—2124no—
287C2×C4—2124no—
288C2×C4—2121no—
289C2×C4—2122no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2122no—
294C2×C4—2121no—
295C2×C4C2×C4×Dic5311yescharacteristic
296C2×C4#1313317311yescharacteristic
297C22—21no—
298C22—21no—
299C22—22no—
300C22—22no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22C22⋊C4×Dic5311yescharacteristic
307C22#99×D5311yescharacteristic
308C22#1681059511yescharacteristic
309C4—21no—
310C4—21no—
311C4—42no—
312C4—42no—
313C4—84no—
314C4—1062no—
315C4—4241no—
316C4—4244no—
317C4—4241no—
318C4—2122no—
319C4—2121no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2#99×Dic5311yescharacteristic
325C2#2027638311yescharacteristic
326C2#2027638811yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC2×C4×C53C2×C42
FrattiniC22×D4C22×D53
FittingC2×C53×#354C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenterC2×#354D53
hypocenterC53#12627
LCSG ▷ C2×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
13. #16810493C4semidirect
15. #16810469C4semidirect
16. #16810418C4semidirect
20. C53×#354C4semidirect
55. C2×Q8×C53C42semidirect
79. C42×C53C42semidirect
182. #354Dic53semidirect
184. C53#12627semidirect
219. C2×Q8C4×Dic53semidirect
255. C42C4×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202774282
C2×#168104182
#99×Dic532
#202728392
C2×C53×#3542
#1262753