Finite Group Databasebeta

← #27306755

Subgroups — #27306755

Order 13568.

1342 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810491C211yesmaximal, characteristic
2C2×C4×C4⋊Dic53C211yesmaximal, characteristic
3#20272980C212yesmaximal
4#20272972C212yesmaximal
5#20272949C214yesmaximal
6#20272945C212yesmaximal
7#20272942C212yesmaximal
8C2×C53×#376C211yesmaximal, characteristic
9C22×C4⋊Dic53C2212yes—
10C22×C4⋊Dic53C2214yes—
11C22×C4×Dic53C2212yes—
12#16810491—24no—
13C4×C4⋊Dic53—24no—
14C2×#13133124C2211yescharacteristic
15C2×#13133124C2212yes—
16C2×#13133124C2214yes—
17C2×#13133124C2214yes—
18C2×#13133124C2211yescharacteristic
19C2×#13133124C2212yes—
20C2×#13133124C2212yes—
21C2×#13133124C2214yes—
22C22×C42×C53C2212yes—
23C22×C42×C53C2211yescharacteristic
24C53×#376C418yescomplemented
25C2×#95×C53C2212yes—
26C2×#95×C53C2211yescharacteristic
27C2×#95×C53C2211yescharacteristic
28C23×Dic53C2312yes—
29C23×Dic53C2312yes—
30C23×Dic53C2314yes—
31C2×C4⋊Dic53—24no—
32C2×C4⋊Dic53—28no—
33C2×C4⋊Dic53—28no—
34C2×C4⋊Dic53—28no—
35C2×C4⋊Dic53—44no—
36C2×C4×Dic53—24no—
37C2×C4×Dic53—42no—
38C2×C4×Dic53—42no—
39#13133124—24no—
40#13133124—22no—
41#13133124—22no—
42#13133124—24no—
43#13133124—24no—
44#13133124—28no—
45#13133124—28no—
46#13133124—44no—
47C23×C4×C53C2312yes—
48C23×C4×C53C2311yescharacteristic
49C23×C4×C53C2311yescharacteristic
50C23×C4×C53C2311yescharacteristic
51C23×C4×C53C2312yes—
52C2×C42×C53—28no—
53C2×C42×C53C2×C418yes—
54C2×C42×C53C2×C414yes—
55C2×C42×C53D414yes—
56C2×C42×C53Q814yes—
57#95×C53C2×C414yes—
58#95×C53C2×C418yes—
59#95×C53C2×C414yes—
60C24×C53C2411yescharacteristic
61C22×C4×C53—24no—
62C22×C4×C53—24no—
63C22×C4×C53—24no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—22no—
70C22×C4×C53—24no—
71C22×C4×C53—24no—
72C22×C4×C53—22no—
73C22×C4×C53—22no—
74C22×C4×C53C22×C414yes—
75C22×C4×C53C22×C412yes—
76C22×C4×C53C22×C412yes—
77C22×C4×C53C22×C412yes—
78C22×C4×C53C22×C414yes—
79C22×C4×C53C2×D412yes—
80C22×C4×C53C2×D412yes—
81C22×C4×C53C2×D412yes—
82C22×C4×C53C2×Q812yes—
83C22×C4×C53C2×Q812yes—
84C22×C4×C53C2×Q812yes—
85C22×C4×C53#4118yes—
86C22×Dic53—24no—
87C22×Dic53—24no—
88C22×Dic53—24no—
89C22×Dic53—24no—
90C22×Dic53—24no—
91C22×Dic53—24no—
92C22×Dic53—22no—
93C22×Dic53—22no—
94C22×Dic53—24no—
95C22×Dic53—24no—
96C22×Dic53—24no—
97C22×Dic53—24no—
98C22×Dic53—22no—
99C22×Dic53—22no—
100C22×Dic53—22no—
101C22×Dic53—22no—
102C22×Dic53#4118yes—
103C42×C53—216no—
104C42×C53C4⋊C4116yescomplemented
105C4⋊Dic53—216no—
106C4⋊Dic53—416no—
107C4×Dic53—48no—
108C23×C53C23×C411yescharacteristic
109C23×C53C22×D411yescharacteristic
110C23×C53C22×Q811yescharacteristic
111C23×C53C2×#4114yes—
112C23×C53C2×#4114yes—
113C23×C53H(5,𝔽2)11yescharacteristic
114C23×C53H(5,𝔽2)11yescharacteristic
115C23×C532−1+411yescharacteristic
116C23×C532−1+411yescharacteristic
117C2×C4×C53—24no—
118C2×C4×C53—24no—
119C2×C4×C53—24no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—24no—
123C2×C4×C53—24no—
124C2×C4×C53—24no—
125C2×C4×C53—24no—
126C2×C4×C53—24no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—28no—
132C2×C4×C53—28no—
133C2×C4×C53—24no—
134C2×C4×C53—24no—
135C2×C4×C53—24no—
136C2×C4×C53—44no—
137C2×C4×C53—44no—
138C2×C4×C53C2×C4⋊C414yes—
139C2×C4×C53C2×C4⋊C414yes—
140C2×C4×C53C2×C4⋊C414yes—
141C2×C4×C53#11718yes—
142C2×Dic53—28no—
143C2×Dic53—28no—
144C2×Dic53—28no—
145C2×Dic53—44no—
146C2×Dic53—44no—
147C2×Dic53—44no—
148C2×Dic53—44no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—44no—
156C2×Dic53—44no—
157C2×Dic53—44no—
158C2×Dic53—44no—
159#13952—531nomaximal, Sylow
160C22×C53C22×C4⋊C411yescharacteristic
161C22×C53C2×#11712yes—
162C22×C53C4×#4112yes—
163C22×C53#52011yescharacteristic
164C22×C53#52011yescharacteristic
165C22×C53#52512yes—
166C22×C53#53212yes—
167C22×C53#53312yes—
168C22×C53#53312yes—
169C22×C53#53511yescharacteristic
170C22×C53#53611yescharacteristic
171C22×C53#54411yescharacteristic
172C22×C53#54411yescharacteristic
173C22×C53#55114yes—
174C22×C53#55214yes—
175C22×C53#55314yes—
176C22×C53#55614yes—
177C4×C53—24no—
178C4×C53—28no—
179C4×C53—24no—
180C4×C53—24no—
181C4×C53—44no—
182Dic53—48no—
183Dic53—82no—
184Dic53—82no—
185C2×#383—531no—
186C2×C4×C4⋊C4—531no—
187C2×#376D5311yescharacteristic
188#1444—532no—
189#1442—532no—
190#1436—534no—
191C4⋊#95—532no—
192#1434—532no—
193C2×C53#228511yescharacteristic
194C2×C53#229611yescharacteristic
195C2×C53#231711yescharacteristic
196C2×C53#234312yes—
197C2×C53#234712yes—
198C2×C53#260312yes—
199C2×C53#260412yes—
200C2×C53#260512yes—
201C2×C53#260612yes—
202C22×C4⋊C4—532no—
203C22×C4⋊C4—534no—
204C22×C42—532no—
205C22×C42C2×D5312yes—
206C22×C42C2×D5311yescharacteristic
207#383—1064no—
208C4×C4⋊C4—1064no—
209#376Dic5318yescomplemented
210C2×#95—531no—
211C2×#95—532no—
212C2×#95—534no—
213C2×#95—534no—
214C2×#95—531no—
215C2×#95—532no—
216C2×#95—532no—
217C2×#95—534no—
218C2×#95C2×D5312yes—
219C2×#95C2×D5311yescharacteristic
220C2×#95C2×D5311yescharacteristic
221C53#1395211yescharacteristic, complemented, Sylow
222C23×C4—534no—
223C23×C4—532no—
224C23×C4—532no—
225C23×C4C22×D5312yes—
226C23×C4C22×D5311yescharacteristic
227C23×C4C22×D5311yescharacteristic
228C23×C4C22×D5311yescharacteristic
229C23×C4C22×D5312yes—
230C2×C4⋊C4—1068no—
231C2×C4⋊C4—1064no—
232C2×C4⋊C4—1068no—
233C2×C4⋊C4—1068no—
234C2×C4⋊C4—2124no—
235C2×C42—28no—
236C2×C42—1064no—
237C2×C42—2122no—
238C2×C42—2122no—
239C2×C42Dic10614yes—
240C2×C42D21214yes—
241C2×C42C2×Dic5318yes—
242C2×C42C2×Dic5314yes—
243#95—1068no—
244#95—1064no—
245#95—1062no—
246#95—1062no—
247#95—1064no—
248#95—1064no—
249#95—1068no—
250#95—2124no—
251#95C2×Dic5314yes—
252#95C2×Dic5318yes—
253#95C2×Dic5314yes—
254C24C23×D5311yescharacteristic
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—538no—
269C22×C4—1064no—
270C22×C4—1064no—
271C22×C4—1064no—
272C22×C4—1064no—
273C22×C4—1064no—
274C22×C4—1064no—
275C22×C4—1062no—
276C22×C4—1062no—
277C22×C4—1064no—
278C22×C4—1064no—
279C22×C4—1064no—
280C22×C4—1064no—
281C22×C4—1062no—
282C22×C4—1062no—
283C22×C4—1062no—
284C22×C4—1062no—
285C22×C4C2×Dic10612yes—
286C22×C4C2×Dic10612yes—
287C22×C4C2×Dic10612yes—
288C22×C4C2×D21212yes—
289C22×C4C2×D21212yes—
290C22×C4C2×D21212yes—
291C22×C4#1171994518yes—
292C22×C4C22×Dic5314yes—
293C22×C4C22×Dic5312yes—
294C22×C4C22×Dic5312yes—
295C22×C4C22×Dic5312yes—
296C22×C4C22×Dic5314yes—
297C4⋊C4—10616no—
298C4⋊C4—21216no—
299C42—216no—
300C42—2128no—
301C42C4⋊Dic53116yescomplemented
302C23C23×Dic5311yescharacteristic
303C23C22×D21211yescharacteristic
304C23C22×Dic10611yescharacteristic
305C23#41×D5314yes—
306C23C2×#1171994514yes—
307C23#1313326311yescharacteristic
308C23#1313326311yescharacteristic
309C23#1313326411yescharacteristic
310C23#1313326411yescharacteristic
311C2×C4—28no—
312C2×C4—28no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—1068no—
333C2×C4—1068no—
334C2×C4—1068no—
335C2×C4—2124no—
336C2×C4—2124no—
337C2×C4—2124no—
338C2×C4—2124no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4—2124no—
346C2×C4—2124no—
347C2×C4—2124no—
348C2×C4—2124no—
349C2×C4C2×C4⋊Dic5314yes—
350C2×C4C2×C4⋊Dic5314yes—
351C2×C4C2×C4⋊Dic5314yes—
352C2×C4#1313317618yes—
353C22C22×C4⋊Dic5311yescharacteristic
354C22C2×#1313317612yes—
355C22#41×Dic5312yes—
356C22#1681110611yescharacteristic
357C22#1681110611yescharacteristic
358C22#1681113512yes—
359C22#1681117312yes—
360C22#1681117612yes—
361C22#1681118012yes—
362C22#1681119211yescharacteristic
363C22#1681119911yescharacteristic
364C22#1681125111yescharacteristic
365C22#1681125111yescharacteristic
366C22#1681129714yes—
367C22#1681130614yes—
368C22#1681131714yes—
369C22#1681134114yes—
370C4—28no—
371C4—24no—
372C4—24no—
373C4—24no—
374C4—44no—
375C4—4242no—
376C4—4242no—
377C4—2128no—
378C2#2027712211yescharacteristic
379C2#2027719911yescharacteristic
380C2#2027733811yescharacteristic
381C2#2027757512yes—
382C2#2027760812yes—
383C2#2028034212yes—
384C2#2028035312yes—
385C2#2028036312yes—
386C2#2028037212yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#376C2
socle, prime radical, cosocleC24×C53C24
hypercenterC2×#376D53
hypocenterC53#13952
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
24. C53×#376C4semidirect
104. C42×C53C4⋊C4semidirect
209. #376Dic53semidirect
221. C53#13952semidirect
301. C42C4⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104912
C2×C4×C4⋊Dic532
#202729802
#202729722
#202729492
#202729452
#202729422
C2×C53×#3762
#1395253