Finite Group Databasebeta

← #27272501

Subgroups — #27272501

Order 13568.

468 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274783C211yesmaximal, characteristic
2C4×C22⋊#88322C211yesmaximal, characteristic
3#20272538C212yesmaximal, complemented
4#20272538C212yesmaximal, complemented
5C53×#2851C211yesmaximal, characteristic
6C2×C22⋊#88322—21no—
7C2×C22⋊#88322—21no—
8C2×C4×#88322—21no—
9#16810378—21no—
10#16810378C2212yes—
11#16810378C2212yes—
12C2×#125×C53C411yescharacteristic
13C22×C42×C53C2211yescharacteristic
14C53×#389C2211yescharacteristic
15C53×#389C2211yescharacteristic
16C53×#387—21no—
17C53×#387—21no—
18C53×#385—21no—
19C53×#385—21no—
20C53×#382C411yescharacteristic
21C53×#380C411yescharacteristic
22C53×#380C411yescharacteristic
23C22×#88322—22no—
24C22×#88322—22no—
25C22⋊#88322—24no—
26C22⋊#88322—42no—
27C4×#88322—42no—
28C23×C4×C53C2×C411yescharacteristic
29C23×C4×C53D411yescharacteristic
30C23×C4×C53D411yescharacteristic
31#125×C53—42no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—21no—
42C2×C42×C53—42no—
43C2×C42×C53C2×C411yescharacteristic
44C2×C42×C53D411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C2×C42×C53C2311yescharacteristic
47#95×C53—21no—
48#95×C53—21no—
49#95×C53—21no—
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53C24×C53C22⋊C411yescharacteristic
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C4⋊C4×C53—22no—
74C4⋊C4×C53—22no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—42no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—42no—
83C42×C53—22no—
84C42×C53—22no—
85C42×C53—41no—
86C42×C53—41no—
87C2×#88322—24no—
88C2×#88322—42no—
89C2×#88322—42no—
90C2×#88322—42no—
91C2×#88322—42no—
92C2×#88322—42no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53#10412yes—
100C23×C53C2×C22⋊C411yescharacteristic
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—44no—
115C2×C4×C53—44no—
116C2×C4×C53—44no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53#9912yes—
130C2×C4×C53#10112yes—
131C2×C4×C53#13112yes—
132C2×C4×C53#13512yes—
133C2×C4×C53#13512yes—
134#88322—44no—
135#88322—82no—
136#7671—531nomaximal, Sylow
137C22×C53—22no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—41no—
144C22×C53#40811yescharacteristic
145C22×C53C2×#9911yescharacteristic
146C22×C53C2×#10111yescharacteristic
147C22×C53#41611yescharacteristic
148C22×C53#41611yescharacteristic
149C22×C53C2×#10411yescharacteristic
150C22×C53#42111yescharacteristic
151C4×C53—22no—
152C4×C53—22no—
153C4×C53—22no—
154C4×C53—22no—
155C4×C53—22no—
156C4×C53—41no—
157C4×C53—81no—
158C4×C53—81no—
159C4×C53—81no—
160C4×C53—81no—
161#2851D5311yescharacteristic
162#1817—531no—
163C4×C22⋊C8—531no—
164#1271—532no—
165#1271—532no—
166C2×C53—42no—
167C2×C53#147211yescharacteristic
168C2×C53#148211yescharacteristic
169C2×C53#149411yescharacteristic
170C2×C53#150111yescharacteristic
171C2×C53#151611yescharacteristic
172C2×C53#152811yescharacteristic
173C2×C53#154411yescharacteristic
174C2×#125Dic5311yescharacteristic
175C22×C42C2×D5311yescharacteristic
176C2×C22⋊C8—1061no—
177C2×C22⋊C8—1061no—
178C2×C4×C8—1061no—
179#389C2×D5311yescharacteristic
180#389C2×D5311yescharacteristic
181#387—21no—
182#387—21no—
183#385—21no—
184#385—21no—
185#382Dic5311yescharacteristic
186#380Dic5311yescharacteristic
187#380Dic5311yescharacteristic
188#336—532no—
189#336—532no—
190#336—1061no—
191C53#767111yescharacteristic, complemented, Sylow
192C23×C4C2×Dic5311yescharacteristic
193C23×C4C22⋊D5311yescharacteristic
194C23×C4C22⋊D5311yescharacteristic
195C22×C8—1062no—
196C22×C8—1062no—
197#125—42no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C2×Dic5311yescharacteristic
203C2×C4⋊C4C2×Dic5311yescharacteristic
204C2×C4⋊C4C2×Dic5311yescharacteristic
205C2×C4⋊C4C2×Dic5311yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C42—42no—
209C2×C42C2×Dic5311yescharacteristic
210C2×C42C22⋊D5311yescharacteristic
211C2×C42C22⋊D5311yescharacteristic
212C2×C42C22×D5311yescharacteristic
213C22⋊C8—1064no—
214C22⋊C8—2122no—
215C4×C8—2122no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222C24C22⋊Dic5311yescharacteristic
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4C22⋊Dic5311yescharacteristic
237C22×C4C22⋊Dic5311yescharacteristic
238C22×C4C22⋊Dic5311yescharacteristic
239C22×C4C22×Dic5311yescharacteristic
240C22×C4C2×C22⋊D5311yescharacteristic
241C22×C4C2×C22⋊D5311yescharacteristic
242C2×C8—1064no—
243C2×C8—2122no—
244C2×C8—2122no—
245C2×C8—2122no—
246C2×C8—2122no—
247C2×C8—2122no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C42—22no—
259C42—22no—
260C42—41no—
261C42—41no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#1313314412yes—
269C23C2×C22⋊Dic5311yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4#1313313212yes—
299C2×C4#1313313612yes—
300C2×C4#1313322412yes—
301C2×C4#1313323712yes—
302C2×C4#1313323712yes—
303C8—4242no—
304C8—2124no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22#1681059011yescharacteristic
313C22C2×#1313313211yescharacteristic
314C22C2×#1313313611yescharacteristic
315C22#1681061911yescharacteristic
316C22#1681061911yescharacteristic
317C22C2×#1313314411yescharacteristic
318C22#1681064011yescharacteristic
319C4—22no—
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—41no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4—81no—
329C2—42no—
330C2#2027307911yescharacteristic
331C2#2027312311yescharacteristic
332C2#2027317511yescharacteristic
333C2#2027320611yescharacteristic
334C2#2027327111yescharacteristic
335C2#2027331711yescharacteristic
336C2#2027338011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2851C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2851D53
hypocenterC53#7671
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
3. #20272538C2semidirect
4. #20272538C2semidirect
191. C53#7671semidirect

Maximal subgroups

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

subgroupindex
#202747832
C4×C22⋊#883222
#202725382
C53×#28512
#767153