Finite Group Databasebeta

← #28318377

Subgroups — #28318377

Order 13568.

1126 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811633C211yesmaximal, characteristic
2C2×#16811586C211yesmaximal, characteristic
3#20291153C212yesmaximal
4#20290531C214yesmaximal, complemented
5#106×#6157C214yesmaximal, complemented
6#20290402C212yesmaximal
7C2×#106×D53C211yesmaximal, characteristic
8C2×D53⋊M16C2211yescharacteristic
9C2×#13133296C2212yes—
10C2×#13133296C2211yescharacteristic
11#16811633C418yescomplemented
12C2×#13133286C2212yes—
13C2×#13133285C2212yes—
14#16811586C418yescomplemented
15C2×#13133271C2212yes—
16C2×C8×#6157C2212yes—
17C4⋊C4×#6157—24no—
18C4⋊C4×#6157C2218yes—
19C4×C22⋊#6157—24no—
20#16811496C2218yes—
21C22×C8×D53C2211yescharacteristic
22C2×C4⋊C4×D53C2212yes—
23C2×#13133149C411yescharacteristic, complemented
24C2×#13133148C412yescomplemented
25#106×D53C412yescomplemented
26#106×D53C412yescomplemented
27#106×D53C2212yes—
28#106×D53C2212yes—
29C2×#106×C53C411yescharacteristic, complemented
30C2×#11719936—21no—
31D53⋊M16—22no—
32D53⋊M16C2×C414yes—
33C2×#11719940—21no—
34C2×D53⋊C8C2312yes—
35#13133296—24no—
36#13133296C2×C418yes—
37#13133296C2×C414yes—
38#13133296Q814yes—
39C2×C4⋊#6157D412yes—
40C2×C4⋊#6157C2314yes—
41C2×C22⋊#6157—22no—
42C2×C22⋊#6157D412yes—
43C2×C4×#6157—22no—
44C2×C4×#6157—28no—
45C2×C4×#6157C2312yes—
46#13133286C2×C418yescomplemented
47#13133285C2×C418yescomplemented
48#13133271C2×C418yescomplemented
49C8×#6157C2×C418yescomplemented
50(C2×#6157)⋊C4—28no—
51C22×C4×D53—22no—
52C22×C4×D53C2311yescharacteristic
53C22×#88322C2×C411yescharacteristic
54C2×C8×D53C2×C411yescharacteristic
55C2×C8×D53C2×C411yescharacteristic
56C2×C8×D53C2×C414yes—
57C2×C8×D53C2×C414yes—
58C2×C8×D53C2311yescharacteristic
59C2×C8×D53C2311yescharacteristic
60C2×C4⋊Dic53C2×C412yes—
61C2×#11719919—22no—
62C4⋊C4×D53—22no—
63C4⋊C4×D53—22no—
64C4⋊C4×D53C2×C414yes—
65C4⋊C4×D53C2314yes—
66#13133149C2×C412yes—
67#13133149C2×C412yes—
68#13133148C2×C414yes—
69#13133148C2×C414yes—
70C22×C8×C53C2×C411yescharacteristic
71C2×C4⋊C4×C53C2×C412yes—
72#106×C53C2×C412yes—
73#106×C53C2×C412yes—
74C23×D53C2×D411yescharacteristic
75C22×#6157—24no—
76C22×#6157C2×D414yes—
77C22×C4×C53—22no—
78C22×C4×C53C22×C411yescharacteristic
79C22×Dic53—22no—
80C22×Dic53#4111yescharacteristic
81C2×C4×D53—24no—
82C2×C4×D53—24no—
83C2×C4×D53—84no—
84C2×C4×D53C22×C411yescharacteristic
85C2×C4×D53C22×C412yes—
86C2×C4×D53C22×C412yes—
87C2×C4×D53C2×D411yescharacteristic
88C2×C4×D53C2×Q812yes—
89C2×C4×D53#4111yescharacteristic
90C2×C4×D53#4112yes—
91C2×C4×D53C2411yescharacteristic
92C22⋊#6157—28no—
93C22⋊#6157C4⋊C418yes—
94#11719940—24no—
95C2×#88324—22no—
96C4⋊#6157—24no—
97C4⋊#6157C4⋊C418yes—
98C4⋊#6157C22×C418yes—
99C4×#6157—28no—
100C4×#6157—88no—
101C4×#6157C22×C414yes—
102C4×#6157C2×Q814yes—
103#11719936—24no—
104D53⋊C8—22no—
105D53⋊C8C22×C414yes—
106C2×C8×C53C4214yescomplemented
107C2×C8×C53C22×C411yescharacteristic
108C2×C8×C53C22×C411yescharacteristic
109C4⋊C4×C53—22no—
110C4⋊C4×C53C22×C414yes—
111C4⋊Dic53—22no—
112C4⋊Dic53C22×C414yes—
113#11719919—24no—
114#11719919—24no—
115C2×#88322C4214yescomplemented
116C2×#88322C22×C411yescharacteristic
117C2×#88322C22×C411yescharacteristic
118C8×D53C4214yescomplemented
119C8×D53C4214yescomplemented
120C8×D53C22×C412yes—
121C8×D53C22×C412yes—
122C8×D53C22×C412yes—
123C8×D53C22×C412yes—
124C23×C53C4×D411yescharacteristic
125C22×D53—24no—
126C22×D53C2×C4⋊C412yes—
127C22×D53C4×D412yes—
128C22×D53C4×D411yescharacteristic
129C22×D53C22×D411yescharacteristic
130C2×#6157—28no—
131C2×#6157—28no—
132C2×#6157—44no—
133C2×#6157—44no—
134C2×#6157C2×C4⋊C418yes—
135C2×C4×C53—24no—
136C2×C4×C53—82no—
137C2×C4×C53C2×C4212yes—
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53C4×Q812yes—
140C2×C4×C53C23×C411yescharacteristic
141C2×Dic53—24no—
142C2×Dic53—82no—
143C2×Dic53#11712yes—
144C2×Dic53C4×D411yescharacteristic
145C2×Dic53C4×Q812yes—
146C2×Dic53C2×#4111yescharacteristic
147C4×D53—48no—
148C4×D53—84no—
149C4×D53C2×C4212yes—
150C4×D53C2×C4⋊C412yes—
151C4×D53#11712yes—
152C4×D53C4×D412yes—
153C4×D53C4×Q812yes—
154C4×D53C4×Q812yes—
155C4×D53C23×C411yescharacteristic
156C4×D53C23×C411yescharacteristic
157C4×D53C22×Q811yescharacteristic
158C4×D53C2×#4111yescharacteristic
159#88324—24no—
160C8×C53C2×C4212yes—
161C8×C53C2×C4212yes—
162#88322C2×C4212yes—
163#88322C2×C4212yes—
164#20454—531nomaximal, Sylow
165C22×C53—22no—
166C22×C53C4×C4⋊C412yes—
167C22×C53C2×C4×D411yescharacteristic
168C2×D53—22no—
169C2×D53—24no—
170C2×D53—24no—
171C2×D53C4×C4⋊C412yes—
172C2×D53C22×C4⋊C411yescharacteristic
173C2×D53C2×C4×D411yescharacteristic
174C2×D53#57714yes—
175#6157—28no—
176#6157—48no—
177C4×C53—44no—
178C4×C53C4×C4⋊C412yes—
179C4×C53C22×C4211yescharacteristic
180C4×C53C2×C4×Q811yescharacteristic
181Dic53—44no—
182Dic53C4×C4⋊C412yes—
183Dic53C2×#11711yescharacteristic
184Dic53C2×C4×Q811yescharacteristic
185C2×#427—531no—
186C22×#106—531no—
187C2×#405—531no—
188#2274—532no—
189#1776—534no—
190C4×#106—534no—
191#1736—532no—
192C2×C53—22no—
193C2×C53C2×C4×C4⋊C411yescharacteristic
194C2×C53#294912yes—
195D53—22no—
196D53#291212yes—
197D53#295212yes—
198C22×M16—531no—
199C23×C8—531no—
200C2×#117—531no—
201C2×#117—532no—
202C22×C4⋊C4—532no—
203#427—538no—
204C2×#107—532no—
205C2×#106—532no—
206C2×#106—531no—
207C2×#106—532no—
208C2×#106—532no—
209C2×#106—532no—
210C2×#106—532no—
211C2×#106—532no—
212C2×#106#615711yescharacteristic, complemented
213#405—538no—
214C2×#97—532no—
215C2×C4×C8—532no—
216C4×C4⋊C4—538no—
217C4×C4⋊C4—1064no—
218C4×C22⋊C4—1064no—
219#340—538no—
220C53#2045411yescharacteristic, complemented, Sylow
221C23×C4—531no—
222C23×C4—1062no—
223C2×M16—534no—
224C2×M16—1061no—
225C2×M16—1062no—
226C2×M16—1061no—
227C22×C8—531no—
228C22×C8—534no—
229C22×C8—534no—
230C22×C8—531no—
231C22×C8—531no—
232C22×C8—531no—
233C22×C8—531no—
234C22×C8—532no—
235C22×C8C2×#615711yescharacteristic
236#117—534no—
237#117—534no—
238#117—538no—
239#117—1064no—
240C2×C4⋊C4—532no—
241C2×C4⋊C4—534no—
242C2×C4⋊C4—532no—
243C2×C4⋊C4—534no—
244C2×C4⋊C4—534no—
245C2×C4⋊C4—1062no—
246C2×C4⋊C4—1062no—
247C2×C4⋊C4—1062no—
248C2×C4⋊C4C2×#615712yes—
249C2×C22⋊C4—532no—
250C2×C22⋊C4—1062no—
251C2×C42—532no—
252C2×C42—1062no—
253C2×C42—1068no—
254#107—538no—
255#106—538no—
256#106—532no—
257#106—532no—
258#106—534no—
259#106—534no—
260#106C2×#615712yes—
261#106C2×#615712yes—
262#97—538no—
263C4×C8—538no—
264#95—1068no—
265C24—531no—
266C22×C4—22no—
267C22×C4—531no—
268C22×C4—532no—
269C22×C4—532no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—532no—
273C22×C4—532no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—534no—
277C22×C4—1064no—
278C22×C4—1064no—
279C22×C4—1062no—
280C22×C4—1064no—
281C22×C4—4244no—
282C22×C4C22×#615711yescharacteristic
283M16—1064no—
284M16—1064no—
285C2×C8—534no—
286C2×C8—531no—
287C2×C8—531no—
288C2×C8—532no—
289C2×C8—532no—
290C2×C8—532no—
291C2×C8—532no—
292C2×C8—534no—
293C2×C8—534no—
294C2×C8—534no—
295C2×C8—1062no—
296C2×C8—1062no—
297C2×C8C4×#615714yescomplemented
298C2×C8C22×#615711yescharacteristic
299C2×C8C22×#615711yescharacteristic
300C4⋊C4—22no—
301C4⋊C4—538no—
302C4⋊C4—538no—
303C4⋊C4—534no—
304C4⋊C4—1064no—
305C4⋊C4—1064no—
306C4⋊C4—1064no—
307C4⋊C4—1062no—
308C4⋊C4C22×#615714yes—
309C22⋊C4—538no—
310C22⋊C4—1068no—
311C42—534no—
312C42—534no—
313C42—1068no—
314C42—4248no—
315C23—531no—
316C23—532no—
317C23—532no—
318C23—531no—
319C23—1064no—
320C23D4×#615711yescharacteristic
321C2×C4—24no—
322C2×C4—82no—
323C2×C4—532no—
324C2×C4—532no—
325C2×C4—531no—
326C2×C4—531no—
327C2×C4—532no—
328C2×C4—532no—
329C2×C4—532no—
330C2×C4—532no—
331C2×C4—532no—
332C2×C4—531no—
333C2×C4—531no—
334C2×C4—532no—
335C2×C4—531no—
336C2×C4—531no—
337C2×C4—538no—
338C2×C4—1068no—
339C2×C4—1068no—
340C2×C4—1064no—
341C2×C4—4244no—
342C2×C4—4242no—
343C2×C4—2128no—
344C2×C4—2124no—
345C2×C4—2124no—
346C2×C4C2×C4×#615712yes—
347C2×C4D4×#615711yescharacteristic
348C2×C4Q8×#615712yes—
349C2×C4C23×#615711yescharacteristic
350C8—532no—
351C8—532no—
352C8—1064no—
353C8C2×C4×#615712yes—
354C8C2×C4×#615712yes—
355C22—22no—
356C22—531no—
357C22—531no—
358C22—534no—
359C22—532no—
360C22—1064no—
361C22—1064no—
362C22—1062no—
363C22C4⋊C4×#615712yes—
364C22C2×D4×#615711yescharacteristic
365C4—44no—
366C4—531no—
367C4—531no—
368C4—532no—
369C4—1068no—
370C4—2128no—
371C4—2124no—
372C4C4⋊C4×#615712yes—
373C4C22×C4×#615711yescharacteristic
374C4C2×Q8×#615711yescharacteristic
375C2—22no—
376C2—532no—
377C2—532no—
378C2—1062no—
379C2C2×C4⋊C4×#615711yescharacteristic
380C2#2029147512yes—
subgroupG/H
centerC22C2×D4×#6157
commutatorC4×C53C22×C42
FrattiniC2×C4C23×#6157
FittingC2×#106×C53C4
socleC22×C53C2×C4×D4
prime radicalC23×D53C2×D4
cosocleC2×C4×D53C24
hypercenterC2×#106#6157
hypocenterC53#20454
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
4. #20290531C2semidirect
5. #106×#6157C2semidirect
11. #16811633C4semidirect
14. #16811586C4semidirect
23. C2×#13133149C4semidirect
24. C2×#13133148C4semidirect
25. #106×D53C4semidirect
26. #106×D53C4semidirect
29. C2×#106×C53C4semidirect
46. #13133286C2×C4semidirect
47. #13133285C2×C4semidirect
48. #13133271C2×C4semidirect
49. C8×#6157C2×C4semidirect
106. C2×C8×C53C42semidirect
115. C2×#88322C42semidirect
118. C8×D53C42semidirect
119. C8×D53C42semidirect
212. C2×#106#6157semidirect
220. C53#20454semidirect
297. C2×C8C4×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168116332
C2×#168115862
#202911532
#202905312
#106×#61572
#202904022
C2×#106×D532
#2045453