Finite Group Databasebeta

← #27272458

Subgroups — #27272458

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283054C211yesmaximal, characteristic
2C4×C22⋊#88322C211yesmaximal, characteristic
3#20272540C212yesmaximal, complemented
4#20272537C212yesmaximal, complemented
5C53×#1797C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic
7C2×C22⋊#88322—21no—
8C2×C4×#88322—21no—
9#16810529C411yescharacteristic
10#16810529C2211yescharacteristic
11#16810521C411yescharacteristic
12#16810521C2211yescharacteristic
13#16810507—21no—
14#16810507—21no—
15#16810496—21no—
16#16810486—21no—
17#16810483C411yescharacteristic
18#16810378C2212yes—
19C22×C42×C53C2211yescharacteristic
20C2×C4⋊C8×C53—21no—
21C2×C22⋊C8×C53—21no—
22C53×#336C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C22×#88322—22no—
25#13133196—42no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C22⋊#88322—24no—
35C4×#88322—42no—
36#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C22×C8×C53—22no—
47C2×C42×C53—42no—
48C2×C42×C53C2×C411yescharacteristic
49C2×C42×C53D411yescharacteristic
50C2×C42×C53D411yescharacteristic
51C2×C42×C53C2311yescharacteristic
52C4⋊C8×C53—42no—
53C22⋊C8×C53—42no—
54C24×C53C22⋊C411yescharacteristic
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53SD1612yes—
67C22×C4×C53Q1612yes—
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C2×Dic106—22no—
76C2×Dic106—41no—
77C2×Dic106—41no—
78C2×C8×C53—42no—
79C2×C8×C53—42no—
80C2×C8×C53—42no—
81C42×C53—22no—
82C42×C53—22no—
83C42×C53—41no—
84C42×C53—41no—
85C22⋊Dic53—42no—
86C22⋊Dic53—42no—
87C4⋊Dic53—22no—
88C4⋊Dic53—41no—
89C4⋊Dic53—41no—
90#11719919—42no—
91#11719919—42no—
92#11719919—42no—
93C2×#88322—24no—
94C2×#88322—42no—
95C2×#88322—42no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53#10312yes—
103C23×C53C2×C22⋊C411yescharacteristic
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—21no—
115C2×C4×C53—21no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—44no—
119C2×C4×C53—44no—
120C2×C4×C53#9912yes—
121C2×C4×C53#10312yes—
122C2×C4×C53#10412yes—
123C2×C4×C53#13112yes—
124C2×C4×C53C2×SD1611yescharacteristic
125C2×C4×C53C2×Q1611yescharacteristic
126C2×C4×C53#13512yes—
127C2×Dic53—41no—
128C2×Dic53—41no—
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—
139Dic106—82no—
140C8×C53—82no—
141#88322—44no—
142#7661—531nomaximal, Sylow
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53#40811yescharacteristic
151C22×C53C2×#9911yescharacteristic
152C22×C53#41311yescharacteristic
153C22×C53C2×#10311yescharacteristic
154C22×C53#41611yescharacteristic
155C22×C53C2×#10411yescharacteristic
156C22×C53#42111yescharacteristic
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—22no—
160C4×C53—22no—
161C4×C53—21no—
162C4×C53—22no—
163C4×C53—21no—
164Dic53—81no—
165Dic53—81no—
166Dic53—81no—
167Dic53—81no—
168#2855—531no—
169#1797D5311yescharacteristic
170C4×C22⋊C8—531no—
171#1272—532no—
172#1271—532no—
173C2×C53—42no—
174C2×C53#146811yescharacteristic
175C2×C53#148211yescharacteristic
176C2×C53#149311yescharacteristic
177C2×C53#150111yescharacteristic
178C2×C53#152011yescharacteristic
179C2×C53#152911yescharacteristic
180C2×C53#154211yescharacteristic
181C2×C22⋊Q8—531no—
182C22×C42C2×D5311yescharacteristic
183C2×C4⋊C8—21no—
184C2×C22⋊C8—21no—
185C2×C22⋊C8—1061no—
186C2×C4×C8—1061no—
187#391—531no—
188#391—531no—
189#389—531no—
190#389—531no—
191#387—1061no—
192#387—1061no—
193#385—1061no—
194#381—1061no—
195#380—531no—
196#336—532no—
197#336C2×D5312yes—
198C53#766111yescharacteristic, complemented, Sylow
199C22×Q8—531no—
200C23×C4C4×D5311yescharacteristic
201C23×C4D21211yescharacteristic
202C23×C4C22⋊D5311yescharacteristic
203C22×C8—22no—
204C22×C8—1062no—
205C22⋊Q8—2122no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4—1061no—
212C2×C22⋊C4—1061no—
213C2×C22⋊C4—1061no—
214C2×C42—42no—
215C2×C42C4×D5311yescharacteristic
216C2×C42D21211yescharacteristic
217C2×C42C22⋊D5311yescharacteristic
218C2×C42C22×D5311yescharacteristic
219C4⋊C8—42no—
220C22⋊C8—42no—
221C22⋊C8—1064no—
222C4×C8—2122no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230C24#1171992111yescharacteristic
231C2×Q8—1062no—
232C2×Q8—2121no—
233C2×Q8—2121no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4#1171992111yescharacteristic
247C22×C4#1171992111yescharacteristic
248C22×C4#1171992111yescharacteristic
249C22×C4#1171992312yes—
250C22×C4#1171992512yes—
251C22×C4C2×C4×D5311yescharacteristic
252C22×C4C2×D21211yescharacteristic
253C22×C4C2×C22⋊D5311yescharacteristic
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—1064no—
258C2×C8—2122no—
259C2×C8—2122no—
260C4⋊C4—1062no—
261C4⋊C4—2121no—
262C4⋊C4—2121no—
263C4⋊C4—2122no—
264C4⋊C4—2122no—
265C4⋊C4—2122no—
266C22⋊C4—2122no—
267C22⋊C4—2122no—
268C42—22no—
269C42—22no—
270C42—41no—
271C42—41no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23#1313314012yes—
279C23C2×#1171992111yescharacteristic
280Q8—4242no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
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—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
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#1313313112yes—
310C2×C4#1313314012yes—
311C2×C4#1313314312yes—
312C2×C4#1313322512yes—
313C2×C4C2×#1171992311yescharacteristic
314C2×C4C2×#1171992511yescharacteristic
315C2×C4#1313323912yes—
316C8—82no—
317C8—2124no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22#1681059111yescharacteristic
326C22C2×#1313313111yescharacteristic
327C22#1681060511yescharacteristic
328C22C2×#1313314011yescharacteristic
329C22#1681062011yescharacteristic
330C22C2×#1313314311yescharacteristic
331C22#1681064111yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—22no—
336C4—21no—
337C4—22no—
338C4—22no—
339C4—4241no—
340C4—4241no—
341C4—4241no—
342C4—4241no—
343C2—42no—
344C2#2027306411yescharacteristic
345C2#2027312411yescharacteristic
346C2#2027316911yescharacteristic
347C2#2027320711yescharacteristic
348C2#2027328611yescharacteristic
349C2#2027332311yescharacteristic
350C2#2027337311yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1797C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1797D53
hypocenterC53#7661
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #20272540C2semidirect
4. #20272537C2semidirect
198. C53#7661semidirect

Maximal subgroups

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

subgroupindex
#202830542
C4×C22⋊#883222
#202725402
#202725372
C53×#17972
#766153