Finite Group Databasebeta

← #27272613

Subgroups — #27272613

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274783C211yesmaximal, characteristic
2#20274409C211yesmaximal, characteristic
3#20272541C212yesmaximal, complemented
4#20272538C212yesmaximal, complemented
5C53×#2855C211yesmaximal, characteristic
6C22×#11719917—21no—
7C2×C22⋊#88322—21no—
8#16810378—21no—
9#16810378—21no—
10#16810378C2212yes—
11#16810378C2212yes—
12C2×C22⋊Q8×C53C411yescharacteristic
13C22×C42×C53C2211yescharacteristic
14C53×#391C411yescharacteristic
15C53×#391C2211yescharacteristic
16C53×#389C411yescharacteristic
17C53×#389C2211yescharacteristic
18C53×#387—21no—
19C53×#387—21no—
20C53×#385—21no—
21C53×#381—21no—
22C53×#380C411yescharacteristic
23C2×#11719917—24no—
24C2×#11719917—42no—
25C2×#11719917—42no—
26C22×#88322—22no—
27C22×#88322—22no—
28C22⋊#88322—42no—
29C22×Q8×C53C2×C411yescharacteristic
30C23×C4×C53C2×C411yescharacteristic
31C23×C4×C53D411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C22⋊Q8×C53—42no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
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—
53#95×C53—21no—
54C24×C53C22⋊C411yescharacteristic
55C2×Q8×C53—22no—
56C2×Q8×C53—41no—
57C2×Q8×C53—41no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C4⋊C4×C53—22no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—42no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C42×C53—41no—
88C42×C53—41no—
89C42×C53—41no—
90C42×C53—41no—
91#11719917—48no—
92C2×#88322—24no—
93C2×#88322—42no—
94C2×#88322—42no—
95C2×#88322—42no—
96C2×#88322—42no—
97C2×#88322—42no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53#9912yes—
105C23×C53C2×C22⋊C411yescharacteristic
106Q8×C53—82no—
107C2×C4×C53—24no—
108C2×C4×C53—24no—
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—21no—
123C2×C4×C53—44no—
124C2×C4×C53—44no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53#10412yes—
138C2×C4×C53#13112yes—
139C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C53#13711yescharacteristic
141C2×C4×C53#13712yes—
142#88322—44no—
143#88322—82no—
144#7695—531nomaximal, Sylow
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53#40811yescharacteristic
153C22×C53C2×#9911yescharacteristic
154C22×C53#41311yescharacteristic
155C22×C53#41711yescharacteristic
156C22×C53#41911yescharacteristic
157C22×C53C2×#10411yescharacteristic
158C22×C53#42111yescharacteristic
159C4×C53—22no—
160C4×C53—21no—
161C4×C53—22no—
162C4×C53—21no—
163C4×C53—22no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53—81no—
167C4×C53—81no—
168C4×C53—81no—
169C4×C53—81no—
170#2855D5311yescharacteristic
171#1817—531no—
172#1728—531no—
173#1272—532no—
174#1271—532no—
175C2×C53—42no—
176C2×C53#146811yescharacteristic
177C2×C53#148511yescharacteristic
178C2×C53#149511yescharacteristic
179C2×C53#150411yescharacteristic
180C2×C53#151111yescharacteristic
181C2×C53#152911yescharacteristic
182C2×C53#155911yescharacteristic
183C22×M16—1061no—
184C2×C22⋊Q8Dic5311yescharacteristic
185C22×C42C2×D5311yescharacteristic
186C2×C22⋊C8—1061no—
187#391Dic5311yescharacteristic
188#391C2×D5311yescharacteristic
189#389Dic5311yescharacteristic
190#389C2×D5311yescharacteristic
191#387—21no—
192#387—21no—
193#385—21no—
194#381—21no—
195#380Dic5311yescharacteristic
196#336—532no—
197#336—532no—
198#336—1061no—
199#336—1061no—
200C53#769511yescharacteristic, complemented, Sylow
201C22×Q8C2×Dic5311yescharacteristic
202C23×C4C2×Dic5311yescharacteristic
203C23×C4C22⋊D5311yescharacteristic
204C23×C4C22⋊D5311yescharacteristic
205C2×M16—1064no—
206C2×M16—2122no—
207C2×M16—2122no—
208C22×C8—1062no—
209C22×C8—1062no—
210C22⋊Q8—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic5311yescharacteristic
215C2×C4⋊C4C2×Dic5311yescharacteristic
216C2×C4⋊C4C2×Dic5311yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—42no—
220C2×C42C2×Dic5311yescharacteristic
221C2×C42C22⋊D5311yescharacteristic
222C2×C42C22⋊D5311yescharacteristic
223C2×C42C22×D5311yescharacteristic
224C22⋊C8—2122no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232C24C22⋊Dic5311yescharacteristic
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4C22⋊Dic5311yescharacteristic
251C22×C4C22⋊Dic5311yescharacteristic
252C22×C4C22⋊Dic5311yescharacteristic
253C22×C4C22×Dic5311yescharacteristic
254C22×C4C2×C22⋊D5311yescharacteristic
255C22×C4C2×C22⋊D5311yescharacteristic
256M16—2128no—
257C2×C8—1064no—
258C2×C8—2122no—
259C2×C8—2122no—
260C2×C8—2122no—
261C2×C8—2122no—
262C2×C8—2122no—
263C4⋊C4—22no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42—41no—
273C42—41no—
274C42—41no—
275C42—41no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23#1313313212yes—
283C23C2×C22⋊Dic5311yescharacteristic
284Q8—82no—
285C2×C4—24no—
286C2×C4—24no—
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—21no—
300C2×C4—21no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4#1313314412yes—
316C2×C4#1313322412yes—
317C2×C4#1313324211yescharacteristic
318C2×C4#1313324711yescharacteristic
319C2×C4#1313324812yes—
320C8—4242no—
321C8—2124no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22#1681059011yescharacteristic
330C22C2×#1313313211yescharacteristic
331C22#1681060711yescharacteristic
332C22#1681062411yescharacteristic
333C22#1681063211yescharacteristic
334C22C2×#1313314411yescharacteristic
335C22#1681064011yescharacteristic
336C4—21no—
337C4—22no—
338C4—21no—
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C2—42no—
348C2#2027306311yescharacteristic
349C2#2027313611yescharacteristic
350C2#2027318011yescharacteristic
351C2#2027321911yescharacteristic
352C2#2027324811yescharacteristic
353C2#2027332211yescharacteristic
354C2#2027344511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2855C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2855D53
hypocenterC53#7695
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
3. #20272541C2semidirect
4. #20272538C2semidirect
200. C53#7695semidirect

Maximal subgroups

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

subgroupindex
#202747832
#202744092
#202725412
#202725382
C53×#28552
#769553