Finite Group Databasebeta

← #27465297

Subgroups — #27465297

Order 13568.

816 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810521C211yesmaximal, characteristic
2#20276272C212yesmaximal, complemented
3#20276187C212yesmaximal, complemented
4#20275286C214yesmaximal, complemented
5#20274902C212yesmaximal
6#20274799C212yesmaximal
7#20274783C211yesmaximal, characteristic
8C53×#2285C211yesmaximal, characteristic
9C22×C4⋊Dic53C2211yescharacteristic
10C22×C4⋊Dic53C2211yescharacteristic
11C22×#11719919—21no—
12C2×#13133150C2214yes—
13C2×C22⋊#88322C2212yes—
14#16810521—21no—
15#16810521—21no—
16#16810521C2212yes—
17#16810521C2212yes—
18C2×#13133124—21no—
19#16810385C2214yes—
20#16810385C2214yes—
21#16810385C2214yes—
22#16810378C2212yes—
23#16810378C2212yes—
24C2×#117×C53—21no—
25C22×C42×C53C2211yescharacteristic
26C53×#384C2211yescharacteristic
27C53×#384C2211yescharacteristic
28C53×#382—21no—
29C53×#380—21no—
30C53×#380C2212yes—
31C4×C4⋊C4×C53—21no—
32C4×C4⋊C4×C53C2212yes—
33C23×Dic53—21no—
34C23×Dic53—21no—
35C22×#88322C2314yes—
36C2×C4⋊Dic53—24no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—24no—
40C2×C4⋊Dic53C2312yes—
41C2×C4⋊Dic53C2312yes—
42C2×#11719919—22no—
43C2×#11719919—22no—
44C2×#11719919—44no—
45#13133150—28no—
46C22⋊#88322—24no—
47#13133124—22no—
48#13133124—22no—
49C23×C4×C53D411yescharacteristic
50C23×C4×C53D411yescharacteristic
51C23×C4×C53C2311yescharacteristic
52#117×C53—44no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53C2312yes—
58C2×C4⋊C4×C53C2312yes—
59C2×C22⋊C4×C53—22no—
60C2×C42×C53—22no—
61C2×C42×C53—24no—
62C2×C42×C53D411yescharacteristic
63C2×C42×C53D411yescharacteristic
64C2×C42×C53C2311yescharacteristic
65C2×C42×C53C2311yescharacteristic
66#95×C53—22no—
67#95×C53—22no—
68C24×C53C2×D411yescharacteristic
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—22no—
73C22×C4×C53—22no—
74C22×C4×C53—22no—
75C22×C4×C53—22no—
76C22×C4×C53—22no—
77C22×C4×C53—41no—
78C22×C4×C53D814yes—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53#4114yes—
85C22×C4×C53C2411yescharacteristic
86C22×Dic53—24no—
87C22×Dic53—22no—
88C22×Dic53—22no—
89C22×Dic53—42no—
90C22×Dic53—42no—
91C22×Dic53—42no—
92C22×Dic53—42no—
93C22×Dic53—42no—
94C4⋊C4×C53—24no—
95C4⋊C4×C53—24no—
96C4⋊C4×C53—42no—
97C4⋊C4×C53—42no—
98C4⋊C4×C53—42no—
99C4⋊C4×C53—42no—
100C22⋊C4×C53—44no—
101C42×C53—22no—
102C42×C53—22no—
103C42×C53—22no—
104C42×C53—22no—
105C42×C53—44no—
106C4⋊Dic53—24no—
107C4⋊Dic53—22no—
108C4⋊Dic53—24no—
109C4⋊Dic53—42no—
110C4⋊Dic53#4114yes—
111#11719919—44no—
112#11719919—44no—
113C2×#88322—24no—
114C2×#88322—24no—
115C2×#88322—24no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53#12312yes—
121C23×C53C2×D812yes—
122C23×C53#13712yes—
123C23×C53C22×D411yescharacteristic
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—28no—
133C2×C4×C53—28no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—44no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53#12312yes—
145C2×C4×C53C2×D812yes—
146C2×C4×C53C2×D812yes—
147C2×C4×C53#13712yes—
148C2×C4×C53C2×#4112yes—
149C2×C4×C53C2×#4112yes—
150C2×C4×C532−1+412yes—
151C2×Dic53—24no—
152C2×Dic53—48no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—84no—
159#88322—44no—
160#27610—531nomaximal, Sylow
161C22×C53—22no—
162C22×C53—22no—
163C22×C53—22no—
164C22×C53—22no—
165C22×C53—22no—
166C22×C53—22no—
167C22×C53#48014yes—
168C22×C53C2×#12311yescharacteristic
169C22×C53#52511yescharacteristic
170C22×C53#53611yescharacteristic
171C22×C53C22×D811yescharacteristic
172C22×C53C2×#13711yescharacteristic
173C22×C53C2×#13711yescharacteristic
174C22×C53#57511yescharacteristic
175C4×C53—22no—
176C4×C53—22no—
177C4×C53—22no—
178C4×C53—22no—
179C4×C53—41no—
180C4×C53—82no—
181C4×C53—82no—
182C4×C53#55214yes—
183Dic53—44no—
184Dic53—82no—
185C2×#389—531no—
186#2285D5311yescharacteristic
187#2099—532no—
188#2083—532no—
189#1918—534no—
190#1852—532no—
191#1823—532no—
192#1817—531no—
193C2×C53—24no—
194C2×C53#248411yescharacteristic
195C2×C53C2×#48011yescharacteristic
196C2×C53#309511yescharacteristic
197C2×C53#310011yescharacteristic
198C2×C53#311311yescharacteristic
199C2×C53#322611yescharacteristic
200C2×C53#323611yescharacteristic
201C2×#117—21no—
202C22×C4⋊C4—531no—
203C22×C4⋊C4—531no—
204C22×C4⋊C4—1061no—
205C22×C42C2×D5311yescharacteristic
206C2×#107—534no—
207C2×C22⋊C8—532no—
208#389—532no—
209#389—532no—
210#389—1061no—
211#389—1061no—
212#384C2×D5311yescharacteristic
213#384C2×D5311yescharacteristic
214#382—21no—
215#380—21no—
216#380C2×D5312yes—
217C4×C4⋊C4—21no—
218C4×C4⋊C4C2×D5312yes—
219C2×#95—1061no—
220#340—534no—
221#340—534no—
222#340—534no—
223#336—532no—
224#336—532no—
225C53#2761011yescharacteristic, complemented, Sylow
226C23×C4—1061no—
227C23×C4—1061no—
228C23×C4C22⋊D5311yescharacteristic
229C23×C4C22⋊D5311yescharacteristic
230C23×C4C22×D5311yescharacteristic
231C22×C8—534no—
232#117—44no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—532no—
238C2×C4⋊C4—532no—
239C2×C4⋊C4—1064no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—1064no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1062no—
244C2×C4⋊C4—1062no—
245C2×C4⋊C4—2124no—
246C2×C4⋊C4C22×D5312yes—
247C2×C4⋊C4C22×D5312yes—
248C2×C22⋊C4—22no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42C22⋊D5311yescharacteristic
252C2×C42C22⋊D5311yescharacteristic
253C2×C42C22×D5311yescharacteristic
254C2×C42C22×D5311yescharacteristic
255#107—1068no—
256C22⋊C8—1064no—
257#95—22no—
258#95—22no—
259#95—1062no—
260#95—1062no—
261C24C2×C22⋊D5311yescharacteristic
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—1064no—
272C22×C4—1062no—
273C22×C4—1062no—
274C22×C4—2122no—
275C22×C4—2122no—
276C22×C4—2122no—
277C22×C4—2122no—
278C22×C4—2122no—
279C22×C4#1171992214yes—
280C22×C4#1171994514yes—
281C22×C4C2×C22⋊D5311yescharacteristic
282C22×C4C2×C22⋊D5311yescharacteristic
283C22×C4C2×C22⋊D5311yescharacteristic
284C22×C4C2×C22⋊D5311yescharacteristic
285C22×C4C2×C22⋊D5311yescharacteristic
286C22×C4C23×D5311yescharacteristic
287C2×C8—1064no—
288C2×C8—1064no—
289C2×C8—1064no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—534no—
297C4⋊C4—1064no—
298C4⋊C4—1062no—
299C4⋊C4—1064no—
300C4⋊C4—2122no—
301C4⋊C4—2124no—
302C4⋊C4—2124no—
303C22⋊C4—44no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—44no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23#1313320012yes—
314C23C2×#1171992212yes—
315C23#1313324812yes—
316C23C22×C22⋊D5311yescharacteristic
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—28no—
325C2×C4—21no—
326C2×C4—28no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—1064no—
338C2×C4—4244no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2128no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4#1313320012yes—
346C2×C4C2×#1171992212yes—
347C2×C4C2×#1171992212yes—
348C2×C4#1313324712yes—
349C2×C4#41×D5312yes—
350C2×C4C2×#1171994512yes—
351C2×C4#1313326412yes—
352C8—2124no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22#1681093614yes—
360C22C2×#1313320011yescharacteristic
361C22#1681113011yescharacteristic
362C22#1681119411yescharacteristic
363C22C22×#1171992211yescharacteristic
364C22C2×#1313324711yescharacteristic
365C22C2×#1313324811yescharacteristic
366C22#1681142111yescharacteristic
367C4—22no—
368C4—22no—
369C4—22no—
370C4—22no—
371C4—41no—
372C4—82no—
373C4—82no—
374C4—4242no—
375C4—2124no—
376C4#1681130614yes—
377C2—24no—
378C2#2027899811yescharacteristic
379C2C2×#1681093611yescharacteristic
380C2#2028466311yescharacteristic
381C2#2028471111yescharacteristic
382C2#2028484111yescharacteristic
383C2#2028583311yescharacteristic
384C2#2028591711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2285C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2285D53
hypocenterC53#27610
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
2. #20276272C2semidirect
3. #20276187C2semidirect
4. #20275286C2semidirect
225. C53#27610semidirect

Maximal subgroups

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

subgroupindex
C2×#168105212
#202762722
#202761872
#202752862
#202749022
#202747992
#202747832
C53×#22852
#2761053