Finite Group Databasebeta

← #27272594

Subgroups — #27272594

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283025C211yesmaximal, characteristic
2C2×#16810378C211yesmaximal, characteristic
3#20272539C212yesmaximal, complemented
4#20272537C212yesmaximal, complemented
5C53×#1749C211yesmaximal, characteristic
6C23×#88322—21no—
7C2×#13133209C411yescharacteristic
8#16810521C2211yescharacteristic
9#16810521C2211yescharacteristic
10#16810507—21no—
11#16810507—21no—
12#16810496—21no—
13#16810496—21no—
14#16810489C411yescharacteristic
15#16810483C411yescharacteristic
16#16810483C411yescharacteristic
17#16810378—21no—
18#16810378C2212yes—
19C22×C42×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
21C2×#97×C53—21no—
22C53×#336C2212yes—
23C22×#88322—24no—
24C22×#88322—22no—
25C22×#88322—42no—
26C22×#88322—42no—
27#13133209—42no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C22×C8×C53—22no—
48C2×C42×C53—42no—
49C2×C42×C53C2×C411yescharacteristic
50C2×C42×C53D411yescharacteristic
51C2×C42×C53D411yescharacteristic
52C2×C42×C53C2311yescharacteristic
53C22⋊C8×C53—24no—
54#97×C53—42no—
55C24×C53C22⋊C411yescharacteristic
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C2×C8×C53—24no—
77C2×C8×C53—42no—
78C2×C8×C53—42no—
79C42×C53—22no—
80C42×C53—22no—
81C42×C53—41no—
82C42×C53—41no—
83C22⋊Dic53—42no—
84C22⋊Dic53—42no—
85C4⋊Dic53—22no—
86C4⋊Dic53—22no—
87C4⋊Dic53—41no—
88C4⋊Dic53—41no—
89#11719919—42no—
90#11719919—42no—
91#11719919—42no—
92#11719919—41no—
93#11719919—41no—
94C2×#88322—24no—
95C2×#88322—48no—
96C2×#88322—42no—
97C2×#88322—42no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53#10112yes—
103C23×C53#10412yes—
104C23×C53#10412yes—
105C23×C53C2×C22⋊C411yescharacteristic
106C2×C4×C53—24no—
107C2×C4×C53—24no—
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—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—44no—
123C2×C4×C53—44no—
124C2×C4×C53#9912yes—
125C2×C4×C53#13112yes—
126C2×C4×C53#13512yes—
127C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C53#13711yescharacteristic
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C8×C53—44no—
142#88322—44no—
143#7691—531nomaximal, Sylow
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—22no—
149C22×C53—22no—
150C22×C53#33014yes—
151C22×C53#40811yescharacteristic
152C22×C53C2×#9911yescharacteristic
153C22×C53C2×#10111yescharacteristic
154C22×C53#41611yescharacteristic
155C22×C53#41711yescharacteristic
156C22×C53C2×#10411yescharacteristic
157C22×C53C2×#10411yescharacteristic
158C4×C53—22no—
159C4×C53—22no—
160C4×C53—21no—
161C4×C53—22no—
162C4×C53—21no—
163C4×C53—41no—
164C4×C53—41no—
165Dic53—81no—
166Dic53—81no—
167Dic53—81no—
168Dic53—81no—
169#2851—531no—
170#1749D5311yescharacteristic
171C2×#336—531no—
172#1271—532no—
173#1271—532no—
174C2×C53—24no—
175C2×C53#147211yescharacteristic
176C2×C53#148311yescharacteristic
177C2×C53#149411yescharacteristic
178C2×C53#150111yescharacteristic
179C2×C53#150411yescharacteristic
180C2×C53C2×#33011yescharacteristic
181C2×C53#155211yescharacteristic
182C23×C8—1061no—
183C2×#125—531no—
184C22×C42C2×D5311yescharacteristic
185C2×C22⋊C8—21no—
186C2×#97—21no—
187#389—531no—
188#389—531no—
189#387—1061no—
190#387—1061no—
191#385—1061no—
192#385—1061no—
193#382—531no—
194#380—531no—
195#380—531no—
196#336—532no—
197#336—1061no—
198#336C2×D5312yes—
199C53#769111yescharacteristic, complemented, Sylow
200C23×C4C4×D5311yescharacteristic
201C23×C4D21211yescharacteristic
202C23×C4C22⋊D5311yescharacteristic
203C22×C8—22no—
204C22×C8—1064no—
205C22×C8—1062no—
206C22×C8—2122no—
207C22×C8—2122no—
208#125—2122no—
209C2×C4⋊C4—531no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4—1061no—
217C2×C22⋊C4—1061no—
218C2×C22⋊C4—1061no—
219C2×C42—42no—
220C2×C42C4×D5311yescharacteristic
221C2×C42D21211yescharacteristic
222C2×C42C22⋊D5311yescharacteristic
223C2×C42C22×D5311yescharacteristic
224C22⋊C8—24no—
225#97—42no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231#95—1061no—
232C24#1171992111yescharacteristic
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4#1171992111yescharacteristic
248C22×C4#1171992111yescharacteristic
249C22×C4#1171992111yescharacteristic
250C22×C4C2×C4×D5311yescharacteristic
251C22×C4C2×D21211yescharacteristic
252C22×C4C2×C22⋊D5311yescharacteristic
253C2×C8—24no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—1064no—
257C2×C8—2128no—
258C2×C8—2122no—
259C2×C8—2122no—
260C4⋊C4—1062no—
261C4⋊C4—1062no—
262C4⋊C4—2121no—
263C4⋊C4—2121no—
264C4⋊C4—2122no—
265C4⋊C4—2122no—
266C4⋊C4—2122no—
267C4⋊C4—2121no—
268C4⋊C4—2121no—
269C22⋊C4—2122no—
270C22⋊C4—2122no—
271C42—22no—
272C42—22no—
273C42—41no—
274C42—41no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23#1313313512yes—
280C23#1313314312yes—
281C23#1313314512yes—
282C23C2×#1171992111yescharacteristic
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4#1313313112yes—
314C2×C4#1313322512yes—
315C2×C4#1313323712yes—
316C2×C4#1313324511yescharacteristic
317C2×C4#1313325011yescharacteristic
318C8—44no—
319C8—2124no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22#1681036414yes—
327C22#1681059111yescharacteristic
328C22C2×#1313313111yescharacteristic
329C22C2×#1313313511yescharacteristic
330C22#1681061711yescharacteristic
331C22#1681062511yescharacteristic
332C22C2×#1313314311yescharacteristic
333C22C2×#1313314511yescharacteristic
334C4—22no—
335C4—21no—
336C4—22no—
337C4—21no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—4241no—
342C4—4241no—
343C4—4241no—
344C4—4241no—
345C2—24no—
346C2#2027308011yescharacteristic
347C2#2027312911yescharacteristic
348C2#2027317311yescharacteristic
349C2#2027320411yescharacteristic
350C2#2027322011yescharacteristic
351C2C2×#1681036411yescharacteristic
352C2#2027341611yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1749C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1749D53
hypocenterC53#7691
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. #20272539C2semidirect
4. #20272537C2semidirect
199. C53#7691semidirect

Maximal subgroups

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

subgroupindex
#202830252
C2×#168103782
#202725392
#202725372
C53×#17492
#769153