Finite Group Databasebeta

← #27276991

Subgroups — #27276991

Order 13568.

524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810694C211yesmaximal, characteristic
2#20274744C211yesmaximal, characteristic, complemented
3#20274700C212yesmaximal
4#20272582C212yesmaximal, complemented
5C53×#1807C211yesmaximal, characteristic, complemented
6C22×#11719912C411yescharacteristic, complemented
7C2×C4×C22⋊D53C2211yescharacteristic
8#16810694—21no—
9#16810694—21no—
10#16810694—21no—
11#16810694—21no—
12C2×#13133151—21no—
13C2×#13133146C411yescharacteristic, complemented
14C2×#13133131—22no—
15C2×C22⋊#88322C2211yescharacteristic
16C2×#13133129C411yescharacteristic, complemented
17C2×#13133127C411yescharacteristic, complemented
18#16810483C2212yes—
19#16810389—22no—
20#16810386—21no—
21C23×C8×C53C2211yescharacteristic
22C2×#106×C53—21no—
23C53×#380C2212yes—
24C53×#340—21no—
25C22×C22⋊D53D411yescharacteristic
26C22×C4×D53C2×C411yescharacteristic
27C2×#11719912—21no—
28C2×#11719912—21no—
29C2×#11719912—42no—
30C22×#88322C2×C411yescharacteristic
31C22×#88322C2×C411yescharacteristic
32C4×C22⋊D53—21no—
33C4×C22⋊D53—21no—
34C4×C22⋊D53—21no—
35C4×C22⋊D53—21no—
36C2×C4⋊Dic53—22no—
37C2×#11719919Q811yescharacteristic
38C2×C22⋊Dic53—22no—
39C2×C22⋊Dic53D411yescharacteristic
40C2×#11719921Q811yescharacteristic
41C2×C4×Dic53C2×C411yescharacteristic
42#13133151—42no—
43#13133146—21no—
44#13133146—21no—
45#13133131—44no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48#13133129—21no—
49#13133129—21no—
50#13133127—21no—
51#13133127—21no—
52#13133124—22no—
53C23×C4×C53C2311yescharacteristic
54C22×C8×C53—42no—
55C22×C8×C53C2×C411yescharacteristic
56C22×C8×C53C2×C411yescharacteristic
57C22×C8×C53D411yescharacteristic
58C22×C8×C53D411yescharacteristic
59C22×C8×C53D411yescharacteristic
60C22×C8×C53D411yescharacteristic
61C2×C4⋊C4×C53—22no—
62C2×C22⋊C4×C53—22no—
63#106×C53—42no—
64#95×C53—22no—
65C24×C53C2×D411yescharacteristic
66C23×D53C4⋊C411yescharacteristic
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53—42no—
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×Q811yescharacteristic
74C22×C4×C53#4111yescharacteristic
75C22×C4×C53#4111yescharacteristic
76C2×C22⋊D53—21no—
77C2×C22⋊D53—21no—
78C2×C22⋊D53—22no—
79C2×C22⋊D53—22no—
80C22×Dic53—42no—
81C22×Dic53C4⋊C411yescharacteristic
82C22×Dic53C4⋊C411yescharacteristic
83C22×Dic53C4⋊C411yescharacteristic
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C8×C53—22no—
89C2×C8×C53—22no—
90C2×C8×C53—22no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53C22⋊C411yescharacteristic
95C2×C8×C53C22⋊C411yescharacteristic
96C2×C8×C53C22⋊C411yescharacteristic
97C2×C8×C53C22⋊C411yescharacteristic
98C4⋊C4×C53—42no—
99C4⋊C4×C53—42no—
100C22⋊C4×C53—42no—
101C22⋊C4×C53—42no—
102C22⋊Dic53—21no—
103C22⋊Dic53—21no—
104C22⋊Dic53—42no—
105C22⋊Dic53—42no—
106#11719921—21no—
107#11719921—21no—
108C4⋊Dic53—42no—
109C4⋊Dic53—42no—
110#11719919—21no—
111#11719919—21no—
112C4×Dic53—21no—
113C4×Dic53—21no—
114C2×#88322—21no—
115C2×#88322—21no—
116C2×#88322—41no—
117C2×#88322—41no—
118#11719912—42no—
119#11719912—42no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53C2×C4⋊C411yescharacteristic
123C23×C53#12111yescharacteristic
124C23×C53#12111yescharacteristic
125C23×C53AGL(1,ℤ/8)12yes—
126C23×C53#13712yes—
127C22×D53—21no—
128C22×D53—22no—
129C22×D53—41no—
130C22×D53—41no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—82no—
141C2×C4×C53—82no—
142C2×C4×C53—82no—
143C2×C4×C53C2×C22⋊C411yescharacteristic
144C2×C4×C53#11711yescharacteristic
145C2×C4×C53C22⋊Q811yescharacteristic
146C2×C4×C53C22⋊Q811yescharacteristic
147C22⋊D53—42no—
148C22⋊D53—41no—
149C22⋊D53—41no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—22no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—82no—
157C2×Dic53—82no—
158C2×Dic53—82no—
159C4×D53—42no—
160C4×D53—41no—
161C4×D53—41no—
162C8×C53—21no—
163C8×C53—21no—
164C8×C53—21no—
165C8×C53#9912yescomplemented
166#88322—41no—
167#88322—41no—
168#8610—531nomaximal, Sylow
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53#38011yescharacteristic
178C22×C53#42812yes—
179C22×C53#46812yes—
180C22×C53#46911yescharacteristic
181C22×C53#47011yescharacteristic
182C2×D53—41no—
183C2×D53—41no—
184C2×D53—41no—
185C2×D53—82no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—162no—
190C4×C53C2×#9911yescharacteristic
191C4×C53#41011yescharacteristic
192Dic53—41no—
193Dic53—41no—
194Dic53—41no—
195Dic53—162no—
196C2×#435—531no—
197#1808—531no—
198#1807D5311yescharacteristic, complemented
199#1799—532no—
200#1289—532no—
201C2×C53—21no—
202C2×C53—22no—
203C2×C53—22no—
204C2×C53—21no—
205C2×C53#179911yescharacteristic
206C2×C53#180911yescharacteristic
207C2×C53#181211yescharacteristic
208D53—81no—
209C22×M16—531no—
210C23×C8C2×D5311yescharacteristic
211C2×C4×D4—531no—
212#435—1061no—
213#435—1061no—
214#435—1061no—
215#435—1061no—
216C2×#107—1061no—
217C2×#106—21no—
218C2×C4⋊C8—531no—
219C2×#99—1062no—
220C2×C22⋊C8—531no—
221C2×C22⋊C8—531no—
222C2×#97—531no—
223#380—532no—
224#380C2×D5312yes—
225#342—1062no—
226#340—21no—
227#340—1061no—
228C53#861011yescharacteristic, complemented, Sylow
229C22×D4—531no—
230C23×C4—531no—
231C23×C4C22×D5311yescharacteristic
232C2×M16—1061no—
233C2×M16—1061no—
234C2×M16—2122no—
235C22×C8—42no—
236C22×C8—531no—
237C22×C8—531no—
238C22×C8C4×D5311yescharacteristic
239C22×C8C4×D5311yescharacteristic
240C22×C8D21211yescharacteristic
241C22×C8D21211yescharacteristic
242C22×C8C22⋊D5311yescharacteristic
243C22×C8C22⋊D5311yescharacteristic
244C4×D4—1061no—
245C4×D4—1061no—
246C4×D4—1061no—
247C4×D4—1061no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—531no—
250C2×C4⋊C4—1062no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—531no—
253C2×C22⋊C4—531no—
254C2×C22⋊C4—1062no—
255C2×C42—531no—
256#107—2122no—
257#106—42no—
258C4⋊C8—1061no—
259C4⋊C8—1061no—
260#99—2124no—
261C22⋊C8—1061no—
262C22⋊C8—1061no—
263C22⋊C8—1061no—
264C22⋊C8—1061no—
265#97—1061no—
266#97—1061no—
267#95—22no—
268#95—1062no—
269C24—531no—
270C24D4×D5311yescharacteristic
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1062no—
274C2×D4—1062no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—2121no—
284C22×C4—2121no—
285C22×C4—2122no—
286C22×C4C2×C4×D5311yescharacteristic
287C22×C4C2×D21211yescharacteristic
288C22×C4#1171994711yescharacteristic
289C22×C4Q8×D5311yescharacteristic
290C22×C4#1171994911yescharacteristic
291C22×C4C2×C22⋊D5311yescharacteristic
292M16—2122no—
293M16—2122no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—1061no—
301C2×C8—1061no—
302C2×C8—2121no—
303C2×C8—2121no—
304C2×C8#1171992111yescharacteristic
305C2×C8#1171992111yescharacteristic
306C2×C8#1171992111yescharacteristic
307C2×C8#1171992111yescharacteristic
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4—2122no—
313C4⋊C4—2122no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—1061no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C22⋊C4—1061no—
320C22⋊C4—2122no—
321C22⋊C4—2122no—
322C42—1061no—
323C42—1061no—
324C23—41no—
325C23—41no—
326C23—1062no—
327C23—1061no—
328C23—2121no—
329C23—2121no—
330C23C4⋊C4×D5311yescharacteristic
331C23#1313318811yescharacteristic
332C23#1313319211yescharacteristic
333C23#1313324412yes—
334C23#1313324612yes—
335D4—2122no—
336D4—2121no—
337D4—2121no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—1062no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—4242no—
355C2×C4—4242no—
356C2×C4—4242no—
357C2×C4—2122no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4C2×#1171992111yescharacteristic
363C2×C4#1313317511yescharacteristic
364C2×C4#1313319311yescharacteristic
365C2×C4#1313319711yescharacteristic
366C8—21no—
367C8—21no—
368C8—21no—
369C8—2121no—
370C8—2121no—
371C8#1313313112yescomplemented
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—42no—
379C22—42no—
380C22—4242no—
381C22—2121no—
382C22—2121no—
383C22—2121no—
384C22#1681048011yescharacteristic
385C22#1681066212yes—
386C22#1681086612yes—
387C22#1681087711yescharacteristic
388C22#1681088211yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—162no—
393C4—8482no—
394C4—2121no—
395C4—2121no—
396C4—2121no—
397C4C2×#1313313111yescharacteristic
398C4#1681059611yescharacteristic
399C2—22no—
400C2—21no—
401C2—22no—
402C2—21no—
403C2—4241no—
404C2#2027470011yescharacteristic
405C2#2027474511yescharacteristic
406C2#2027475811yescharacteristic
subgroupG/H
centerC22#16810480
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1807C2
socleC22×C53#380
prime radicalC22×C22⋊D53D4
cosocleC23×C4×C53C23
hypercenter#1807D53
hypocenterC53#8610
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
2. #20274744C2semidirect
4. #20272582C2semidirect
5. C53×#1807C2semidirect
6. C22×#11719912C4semidirect
13. C2×#13133146C4semidirect
16. C2×#13133129C4semidirect
17. C2×#13133127C4semidirect
165. C8×C53#99semidirect
198. #1807D53semidirect
228. C53#8610semidirect
371. C8#13133131semidirect

Maximal subgroups

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

subgroupindex
C2×#168106942
#202747442
#202747002
#202725822
C53×#18072
#861053