Finite Group Databasebeta

← #27351962

Subgroups — #27351962

Order 13568.

874 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283800C211yesmaximal, characteristic
2#20277680C211yesmaximal, characteristic
3#20273279C212yesmaximal
4#20273266C214yesmaximal
5#20273227C214yesmaximal
6#20273214C212yesmaximal
7C53×#3502C211yesmaximal, characteristic
8#16810754—24no—
9#16810680C2214yes—
10#16810676C2212yes—
11#16810530—21no—
12#16810509—22no—
13#16810491C2212yes—
14#16810489C2211yescharacteristic
15C4⋊C4×Dic53—21no—
16C4×C4⋊Dic53C2211yescharacteristic
17#16810469C2212yes—
18#16810360C2214yes—
19#16810360C2214yes—
20#16810360C2218yes—
21Q82×C53C414yescomplemented
22C53×#557—24no—
23C53×#557C414yescomplemented
24C53×#544—22no—
25C53×#544—21no—
26C53×#533C2212yes—
27C53×#533C2214yes—
28C53×#519—21no—
29C2×C4×Q8×C53C2211yescharacteristic
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4×Dic53—22no—
33C2×C4×Dic53—21no—
34C4⋊#88322—22no—
35C4⋊#88322—24no—
36C22⋊#88322C2314yes—
37#13133128—24no—
38C4×#88322—22no—
39#13133124—21no—
40#13133124—22no—
41#13133124—22no—
42#13133124—22no—
43#13133124C2311yescharacteristic
44#13133124C2312yes—
45#13133124C2311yescharacteristic
46C22×Q8×C53D411yescharacteristic
47C4⋊Q8×C53—28no—
48C4⋊Q8×C53—24no—
49C4⋊Q8×C53C2×C412yes—
50C4⋊Q8×C53C2×C414yes—
51C4⋊Q8×C53C2×C412yes—
52C4⋊Q8×C53C2×C414yes—
53#125×C53—28no—
54#125×C53—24no—
55#125×C53—22no—
56#125×C53—24no—
57C22⋊Q8×C53—24no—
58C22⋊Q8×C53—22no—
59C22⋊Q8×C53C2314yes—
60C4×Q8×C53—22no—
61C4×Q8×C53—24no—
62C4×Q8×C53—24no—
63C4×Q8×C53—24no—
64C4×Q8×C53C2×C414yes—
65C4×Q8×C53C2×C414yes—
66C4×Q8×C53C2×C418yes—
67C4×Q8×C53D414yes—
68#117×C53—24no—
69#117×C53—22no—
70C2×C4⋊C4×C53D412yes—
71C2×C4⋊C4×C53D411yescharacteristic
72C2×C42×C53C2311yescharacteristic
73C2×C42×C53C2312yes—
74C2×Q8×C53—24no—
75C2×Q8×C53—22no—
76C2×Q8×C53—22no—
77C2×Q8×C53—22no—
78C2×Q8×C53C22⋊C414yes—
79C2×Q8×C53C22×C414yes—
80C22×C4×C53C2×D412yes—
81C22×C4×C53C2×D412yes—
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2411yescharacteristic
85C22×Dic53—21no—
86C22×Dic53—22no—
87C22×Dic53—21no—
88C4⋊C4×C53—24no—
89C4⋊C4×C53—22no—
90C4⋊C4×C53—24no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—24no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—28no—
95C4⋊C4×C53—24no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53C22⋊C418yes—
98C4⋊C4×C53C22⋊C414yes—
99C4⋊C4×C53C22×C414yes—
100C22⋊C4×C53—24no—
101C42×C53—28no—
102C42×C53—24no—
103C42×C53C22×C414yes—
104C42×C53C22×C412yes—
105C42×C53C2×D412yes—
106C42×C53C2×D414yes—
107C4⋊Dic53—42no—
108C4⋊Dic53—42no—
109C4×Dic53—44no—
110C4×Dic53—42no—
111C2×#88322—24no—
112C23×C53C22×D411yescharacteristic
113Q8×C53—28no—
114Q8×C53—44no—
115Q8×C53—84no—
116Q8×C53—82no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—24no—
120C2×C4×C53—24no—
121C2×C4×C53—42no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53C2×C22⋊C414yes—
125C2×C4×C53C2×C22⋊C412yes—
126C2×C4×C53C2×C22⋊C414yes—
127C2×C4×C53C2×C22⋊C412yes—
128C2×C4×C53#13712yes—
129C2×C4×C53#13712yes—
130C2×C4×C53C23×C411yescharacteristic
131C2×C4×C53C22×D411yescharacteristic
132C2×Dic53—22no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—42no—
141#88322—44no—
142#17632—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53C22×C22⋊C411yescharacteristic
146C22×C53C2×#13711yescharacteristic
147C22×C53C2×#13711yescharacteristic
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—24no—
151C4×C53—24no—
152C4×C53—22no—
153C4×C53—44no—
154C4×C53—44no—
155C4×C53#41914yes—
156Dic53—42no—
157Dic53—81no—
158Dic53—82no—
159#3502D5311yescharacteristic
160#2983—531no—
161#2356—531no—
162#1518—532no—
163#1515—534no—
164#1506—534no—
165#1503—532no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53#288411yescharacteristic
169C2×C53C2×#41911yescharacteristic
170C2×C53#290111yescharacteristic
171Q82Dic5314yescomplemented
172#557—24no—
173#557Dic5314yescomplemented
174#544—21no—
175#544—22no—
176#533C2×D5312yes—
177#533C2×D5314yes—
178#519—21no—
179C2×C4×Q8C2×D5311yescharacteristic
180#446—1064no—
181#432—534no—
182#431—532no—
183#391—1061no—
184#387—1062no—
185#383—532no—
186#382—531no—
187C4×C4⋊C4—531no—
188C4×C4⋊C4—1061no—
189#376—532no—
190#328—534no—
191#328—534no—
192#328—538no—
193C53#1763211yescharacteristic, complemented, Sylow
194C22×Q8C22⋊D5311yescharacteristic
195C4⋊Q8—28no—
196C4⋊Q8—24no—
197C4⋊Q8C2×Dic5312yes—
198C4⋊Q8C2×Dic5314yes—
199C4⋊Q8C2×Dic5312yes—
200C4⋊Q8C2×Dic5314yes—
201#125—28no—
202#125—24no—
203#125—22no—
204#125—24no—
205C22⋊Q8—24no—
206C22⋊Q8—22no—
207C22⋊Q8C22×D5314yes—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—22no—
211C4×Q8—24no—
212C4×Q8C2×Dic5314yes—
213C4×Q8C2×Dic5318yes—
214C4×Q8C2×Dic5314yes—
215C4×Q8C22⋊D5314yes—
216#117—24no—
217#117—22no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4C22⋊D5311yescharacteristic
221C2×C4⋊C4C22⋊D5312yes—
222C2×C42—1062no—
223C2×C42—1061no—
224C2×C42C22×D5312yes—
225C2×C42C22×D5311yescharacteristic
226C4⋊C8—1062no—
227C4⋊C8—1064no—
228C22⋊C8—534no—
229#97—1064no—
230C4×C8—1062no—
231#95—531no—
232#95—532no—
233#95—531no—
234#95—1062no—
235#95—1062no—
236#95—1061no—
237#95—1062no—
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8C22⋊Dic5314yes—
243C2×Q8C22×Dic5314yes—
244C22×C4—1061no—
245C22×C4—1062no—
246C22×C4—1061no—
247C22×C4C2×C22⋊D5312yes—
248C22×C4C2×C22⋊D5312yes—
249C22×C4C2×C22⋊D5311yescharacteristic
250C22×C4C2×C22⋊D5311yescharacteristic
251C22×C4C23×D5311yescharacteristic
252C2×C8—1064no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—28no—
261C4⋊C4—24no—
262C4⋊C4—2122no—
263C4⋊C4—2122no—
264C4⋊C4C22⋊Dic5318yes—
265C4⋊C4C22⋊Dic5314yes—
266C4⋊C4C22×Dic5314yes—
267C22⋊C4—24no—
268C42—28no—
269C42—24no—
270C42—2124no—
271C42—2122no—
272C42C22×Dic5314yes—
273C42C22×Dic5312yes—
274C42C2×C22⋊D5312yes—
275C42C2×C22⋊D5314yes—
276C23C22×C22⋊D5311yescharacteristic
277Q8—28no—
278Q8—44no—
279Q8—84no—
280Q8—82no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—1062no—
289C2×C4—2122no—
290C2×C4—2121no—
291C2×C4—2122no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2122no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4C2×C22⋊Dic5314yes—
298C2×C4C2×C22⋊Dic5312yes—
299C2×C4C2×C22⋊Dic5314yes—
300C2×C4C2×C22⋊Dic5312yes—
301C2×C4#1313324812yes—
302C2×C4#1313324812yes—
303C2×C4C23×Dic5311yescharacteristic
304C2×C4C22×C22⋊D5311yescharacteristic
305C8—2124no—
306C22—21no—
307C22—21no—
308C22C22×C22⋊Dic5311yescharacteristic
309C22C2×#1313324811yescharacteristic
310C22C2×#1313324811yescharacteristic
311C4—21no—
312C4—21no—
313C4—24no—
314C4—24no—
315C4—22no—
316C4—44no—
317C4—44no—
318C4—4242no—
319C4—4241no—
320C4—2122no—
321C4#1681063214yes—
322C2—21no—
323C2—21no—
324C2#2028317911yescharacteristic
325C2C2×#1681063211yescharacteristic
326C2#2028326211yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#3502C2
socleC22×C53C22×C22⋊C4
prime radicalC23×C53C22×D4
cosocleC22×C4×C53C24
hypercenter#3502D53
hypocenterC53#17632
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3502

Complements

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

NH ≅ G/Ntype
21. Q82×C53C4semidirect
23. C53×#557C4semidirect
171. Q82Dic53semidirect
173. #557Dic53semidirect
193. C53#17632semidirect

Maximal subgroups

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

subgroupindex
#202838002
#202776802
#202732792
#202732662
#202732272
#202732142
C53×#35022
#1763253