Finite Group Databasebeta

← #27302374

Subgroups — #27302374

Order 13568.

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

normal subgroups only

This group has 536 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#20279023C211yesmaximal, characteristic
2C2×#16810412C214yesmaximal
3C4×#13133134C211yesmaximal, characteristic
4C53×#1799C211yesmaximal, characteristic
5C2×#13133198C2212yes—
6C22×C4⋊Dic53C411yescharacteristic, complemented
7C2×#13133134C2212yes—
8C4×#11719917—21no—
9#16810553—21no—
10#16810548—21no—
11#16810521—21no—
12#16810484C412yescomplemented
13#16810483—21no—
14#16810483C411yescharacteristic, complemented
15#16810412—28no—
16#16810392—21no—
17C2×C4×D4×C53C2211yescharacteristic
18C2×#99×C53C2212yes—
19C53×#380—21no—
20C53×#342—21no—
21C23×Dic53C2×C412yes—
22C2×#11719917—22no—
23#13133198—24no—
24#13133198—24no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—41no—
29C2×C4⋊Dic53—41no—
30C2×#11719919—22no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53C2×C412yes—
33#13133134—22no—
34#13133134D414yes—
35#13133128—41no—
36C4×#88322—41no—
37#13133124—22no—
38#13133124—21no—
39#13133124—22no—
40C22×D4×C53C2311yescharacteristic
41C23×C4×C53C2×C411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C4×D4×C53—41no—
44C4×D4×C53—41no—
45C2×C4⋊C4×C53—41no—
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C42×C53D411yescharacteristic
51#99×C53—24no—
52#95×C53—41no—
53C24×C53C22×C411yescharacteristic
54C24×C53C2×D411yescharacteristic
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53C2×D412yes—
58C2×D4×C53#4112yes—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—24no—
74C22×Dic53—42no—
75C22×Dic53—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—82no—
78C22⋊C4×C53—41no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—42no—
81C22⋊C4×C53—42no—
82C42×C53—21no—
83C42×C53—21no—
84C22⋊Dic53—24no—
85C22⋊Dic53—42no—
86C22⋊Dic53—42no—
87C4⋊Dic53—42no—
88C4⋊Dic53—42no—
89C4⋊Dic53—41no—
90C4⋊Dic53—41no—
91#11719919—44no—
92#11719917—24no—
93#11719917—42no—
94C2×#88322—42no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53C4×D412yes—
105C23×C53#12411yescharacteristic
106C23×C53#12711yescharacteristic
107D4×C53—42no—
108D4×C53—42no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—81no—
122C2×C4×C53—82no—
123C2×C4×C53—82no—
124C2×C4×C53—82no—
125C2×C4×C53#9911yescharacteristic
126C2×C4×C53#9911yescharacteristic
127C2×C4×C53#12112yes—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—84no—
135#88322—44no—
136#13217—531nomaximal, Sylow
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—22no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—42no—
143C22×C53—41no—
144C22×C53—41no—
145C22×C53—42no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—81no—
151C22×C53#39011yescharacteristic
152C22×C53C2×#9911yescharacteristic
153C22×C53#41011yescharacteristic
154C4×C53—21no—
155C4×C53—22no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—162no—
160C4×C53#35212yescomplemented
161Dic53—82no—
162Dic53—82no—
163#2486—531no—
164C2×#352—534no—
165#1799D5311yescharacteristic
166C4×#100—531no—
167C2×C53—21no—
168C2×C53—21no—
169C2×C53—41no—
170C2×C53—42no—
171C2×C53—41no—
172C2×C53#196411yescharacteristic
173C2×C53C2×#35211yescharacteristic
174C2×C53#212311yescharacteristic
175C2×#123—532no—
176C2×C4×D4C2×D5311yescharacteristic
177C22×C4⋊C4—531no—
178C2×#100—532no—
179C2×#99C2×D5312yes—
180C4×M16—1061no—
181#398—1061no—
182#397—1061no—
183#389—1061no—
184#381—532no—
185#380—21no—
186#380—531no—
187#380—1061no—
188#352—1068no—
189#343—1061no—
190#342—21no—
191C53#1321711yescharacteristic, complemented, Sylow
192C22×D4C22×D5311yescharacteristic
193C23×C4—532no—
194C23×C4C4×D5311yescharacteristic
195C23×C4C22⋊D5311yescharacteristic
196C2×M16—1062no—
197#123—1064no—
198#123—1064no—
199C4×D4—41no—
200C4×D4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—1062no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—2121no—
207C2×C4⋊C4—2121no—
208C2×C4⋊C4D21211yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—532no—
211C2×C22⋊C4—1062no—
212C2×C22⋊C4C4×D5311yescharacteristic
213C2×C22⋊C4C22⋊D5311yescharacteristic
214C2×C42D21211yescharacteristic
215#100—534no—
216#100—1062no—
217#99—24no—
218#97—2121no—
219C4×C8—2121no—
220#95—41no—
221#95—1061no—
222#95—1062no—
223#95—1062no—
224C24C2×C4×D5311yescharacteristic
225C24C2×C22⋊D5311yescharacteristic
226C2×D4—41no—
227C2×D4—41no—
228C2×D4D4×D5312yes—
229C2×D4#1171994912yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—1064no—
238C22×C4—1062no—
239C22×C4—1062no—
240C22×C4—2122no—
241C22×C4—2122no—
242C22×C4#1171992111yescharacteristic
243C22×C4#1171992111yescharacteristic
244C22×C4#1171992111yescharacteristic
245C22×C4#1171992111yescharacteristic
246C22×C4C2×D21211yescharacteristic
247M16—1064no—
248M16—2122no—
249C2×C8—2122no—
250C4⋊C4—41no—
251C4⋊C4—82no—
252C4⋊C4—2122no—
253C4⋊C4—2122no—
254C4⋊C4—2121no—
255C4⋊C4—2121no—
256C4⋊C4—2124no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—1064no—
262C22⋊C4—2122no—
263C22⋊C4—2122no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23C2×#1171992111yescharacteristic
275C23#1313317912yes—
276C23#1313320611yescharacteristic
277C23#1313321311yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—4244no—
297C2×C4—2122no—
298C2×C4—2122no—
299C2×C4—2122no—
300C2×C4—2122no—
301C2×C4—2122no—
302C2×C4—2122no—
303C2×C4#1313313111yescharacteristic
304C2×C4#1313313111yescharacteristic
305C2×C4#1313319212yes—
306C8—2124no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1681052411yescharacteristic
322C22C2×#1313313111yescharacteristic
323C22#1681059611yescharacteristic
324C4—22no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—162no—
330C4—4242no—
331C4—4242no—
332C4#1681041212yescomplemented
333C2—21no—
334C2—21no—
335C2—41no—
336C2—42no—
337C2—41no—
338C2#2027555011yescharacteristic
339C2C2×#1681041211yescharacteristic
340C2#2027636811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1799C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#1799D53
hypocenterC53#13217
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
6. C22×C4⋊Dic53C4semidirect
12. #16810484C4semidirect
14. #16810483C4semidirect
160. C4×C53#352semidirect
191. C53#13217semidirect
332. C4#16810412semidirect

Maximal subgroups

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

subgroupindex
#202790232
C2×#168104122
C4×#131331342
C53×#17992
#1321753