Finite Group Databasebeta

← #27285197

Subgroups — #27285197

Order 13568.

634 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810971C211yesmaximal, characteristic
2C2×#16810823C211yesmaximal, characteristic
3#20275642C211yesmaximal, characteristic
4C2×#16810369C211yesmaximal, characteristic, complemented
5C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#2568C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×#13133214C2211yescharacteristic
10#16810971—22no—
11#16810971—22no—
12#16810823—22no—
13#16810823—22no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133137—22no—
17C2×#13133137—21no—
18C2×#13133137—22no—
19C2×C4×#88322C2211yescharacteristic
20#16810369—22no—
21Q8⋊#88322—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4×Q8×C53C2211yescharacteristic
24C53×#400—22no—
25C53×#391—21no—
26C53×#391—22no—
27C53×#382—21no—
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C2×#11719924—24no—
33C2×#11719924—42no—
34C2×#11719924—42no—
35C22×#88322—21no—
36C22×#88322—21no—
37C22×#88322—21no—
38#13133214—21no—
39#13133214—21no—
40#13133214—21no—
41#13133214D412yes—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D412yes—
46#13133137—24no—
47#13133137—28no—
48#13133137—44no—
49C4×#88322—21no—
50C4×#88322D412yes—
51C22×Q8×C53—21no—
52C22×Q8×C53D411yescharacteristic
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4×Q8×C53—21no—
58C4×Q8×C53—21no—
59C4×Q8×C53—21no—
60C4×Q8×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53D411yescharacteristic
64C2×C4⋊C4×C53D412yes—
65C2×C4⋊C4×C53D412yes—
66C2×C42×C53—21no—
67C2×C42×C53—21no—
68C2×C42×C53C2311yescharacteristic
69#95×C53—24no—
70#95×C53—24no—
71#95×C53—21no—
72C23×D53—41no—
73C2×Q8×C53—21no—
74C2×Q8×C53—21no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53—42no—
81C2×Q8×C53—42no—
82C2×Q8×C53SD1612yes—
83C22×C4×C53—21no—
84C22×C4×C53—22no—
85C22×C4×C53—21no—
86C22×C4×C53—22no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C2×D212—22no—
91C2×D212—22no—
92C2×D212—22no—
93C2×D212—44no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—42no—
97C2×D212—42no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—41no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C42×C53—41no—
114C42×C53—41no—
115C42×C53—41no—
116C42×C53—41no—
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53#4111yescharacteristic
121#11719924—48no—
122C2×#88322—22no—
123C2×#88322—22no—
124C2×#88322—42no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—42no—
129C23×C53#12011yescharacteristic
130C22×D53—42no—
131C22×D53—81no—
132C22×D53—81no—
133C22×D53—82no—
134C22×D53—82no—
135Q8×C53—22no—
136Q8×C53—22no—
137Q8×C53—41no—
138Q8×C53—41no—
139Q8×C53—82no—
140Q8×C53—81no—
141Q8×C53—81no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—41no—
148C2×C4×C53—44no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53#12011yescharacteristic
154C2×C4×C53#12011yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12711yescharacteristic
159C2×C4×C53C2×SD1611yescharacteristic
160C2×C4×C53#13511yescharacteristic
161C2×C4×C53#13512yes—
162C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)12yes—
164C2×C4×C53#13711yescharacteristic
165D212—44no—
166D212—42no—
167D212—42no—
168D212—44no—
169D212—82no—
170D212—81no—
171D212—81no—
172D212—81no—
173D212—81no—
174#88322—42no—
175#88322—42no—
176#88322—81no—
177#10053—531nomaximal, Sylow
178C22×C53#39211yescharacteristic
179C22×C53#44811yescharacteristic
180C22×C53#44912yes—
181C22×C53#44911yescharacteristic
182C22×C53#45412yes—
183C2×D53—82no—
184C2×D53—82no—
185C2×D53—82no—
186C2×D53—162no—
187C2×D53—162no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—42no—
191C4×C53—82no—
192C4×C53—81no—
193C4×C53—82no—
194C4×C53#45011yescharacteristic
195C4×C53#45211yescharacteristic
196C4×C53#46312yes—
197C4×C53#46511yescharacteristic
198C4×C53#46811yescharacteristic
199C4×C53#49511yescharacteristic
200C4×C53#49611yescharacteristic
201C2×#486—531no—
202C2×#460—531no—
203#2568D5311yescharacteristic, complemented
204#1982—531no—
205C2×#332—531no—
206C2×Q8⋊C8—531no—
207#1279—531no—
208C2×C53#162612yes—
209C2×C53#162812yes—
210C2×C53#200011yescharacteristic
211C2×C53#200311yescharacteristic
212C2×C53#201211yescharacteristic
213D53—162no—
214C22×SD16—1061no—
215C2×C4⋊Q8C2×D5311yescharacteristic
216C2×#127—531no—
217C2×C4×Q8C2×D5311yescharacteristic
218#486—1062no—
219#486—1062no—
220#460—1062no—
221#460—1062no—
222C2×C4⋊C8—531no—
223C2×C4⋊C8—531no—
224C2×#102—1062no—
225C2×#102—1061no—
226C2×#102—1062no—
227C2×C4×C8—531no—
228#400—22no—
229#391—21no—
230#391—22no—
231#382—21no—
232C4×C4⋊C4C2×D5311yescharacteristic
233#332—1062no—
234Q8⋊C8—1062no—
235C53#1005311yescharacteristic, complemented, Sylow
236C22×Q8—21no—
237C22×Q8C22⋊D5311yescharacteristic
238C22×D4—1061no—
239C22×D4—1061no—
240C22×D4—1061no—
241C2×SD16—1064no—
242C2×SD16—2122no—
243C2×SD16—2122no—
244C22×C8—1061no—
245C22×C8—1061no—
246C22×C8—1061no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251#127—532no—
252#127—1061no—
253#127—1061no—
254#127—1061no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C22⋊D5311yescharacteristic
262C2×C4⋊C4C22⋊D5312yes—
263C2×C4⋊C4C22⋊D5312yes—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42C22×D5311yescharacteristic
267C4⋊C8—532no—
268C4⋊C8—1061no—
269C4⋊C8—1061no—
270C4⋊C8—1061no—
271#102—1064no—
272#102—1068no—
273#102—2124no—
274C4×C8—532no—
275C4×C8—1061no—
276#95—24no—
277#95—24no—
278#95—21no—
279C24—2121no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8#1171992412yes—
290C2×D4—1062no—
291C2×D4—1062no—
292C2×D4—1062no—
293C2×D4—2124no—
294C2×D4—2121no—
295C2×D4—2121no—
296C2×D4—2121no—
297C2×D4—2121no—
298C2×D4—2121no—
299C2×D4—2122no—
300C2×D4—2122no—
301C2×D4—2121no—
302C2×D4—2121no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4C2×C22⋊D5311yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309C22×C4C2×C22⋊D5311yescharacteristic
310SD16—2128no—
311C2×C8—1062no—
312C2×C8—1062no—
313C2×C8—2121no—
314C2×C8—2122no—
315C2×C8—2122no—
316C2×C8—2121no—
317C2×C8—2121no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42D4×D5311yescharacteristic
333C42D4×D5311yescharacteristic
334C42D4×D5311yescharacteristic
335C42#1171994711yescharacteristic
336C23—4241no—
337C23—4242no—
338C23—4241no—
339C23—4242no—
340C23—2122no—
341C23#1313318611yescharacteristic
342Q8—22no—
343Q8—22no—
344Q8—41no—
345Q8—41no—
346Q8—81no—
347Q8—82no—
348Q8—81no—
349D4—4242no—
350D4—4241no—
351D4—4241no—
352D4—4241no—
353D4—4241no—
354D4—2124no—
355D4—2122no—
356D4—2122no—
357D4—2124no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4#1313318511yescharacteristic
370C2×C4#1313318511yescharacteristic
371C2×C4#1313318811yescharacteristic
372C2×C4#1313318911yescharacteristic
373C2×C4#1313318911yescharacteristic
374C2×C4#1313321311yescharacteristic
375C2×C4C2×#1171992411yescharacteristic
376C2×C4#1313323711yescharacteristic
377C2×C4#1313323712yes—
378C2×C4#1313324212yes—
379C2×C4#1313324211yescharacteristic
380C2×C4#1313324811yescharacteristic
381C8—4241no—
382C8—2122no—
383C8—2122no—
384C22—8482no—
385C22—8482no—
386C22—4242no—
387C22—4242no—
388C22—4242no—
389C22#1681053111yescharacteristic
390C22#1681076211yescharacteristic
391C22#1681076811yescharacteristic
392C22#1681076812yes—
393C22#1681079612yes—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—81no—
398C4—82no—
399C4—82no—
400C4#1681077311yescharacteristic
401C4#1681078311yescharacteristic
402C4#1681083512yes—
403C4#1681085411yescharacteristic
404C4#1681087111yescharacteristic
405C4#1681100511yescharacteristic
406C4#1681101311yescharacteristic
407C2—8482no—
408C2#2027381112yes—
409C2#2027382512yes—
410C2#2027571611yescharacteristic
411C2#2027573511yescharacteristic
412C2#2027579211yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2568C2
socleC23×C53#120
prime radicalC2×#13133214C22
hypercenter#2568D53
hypocenterC53#10053
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
4. C2×#16810369C2semidirect
5. C2×Q8⋊#88322C2semidirect
6. #20272556C2semidirect
7. C53×#2568C2semidirect
203. #2568D53semidirect
235. C53#10053semidirect

Maximal subgroups

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

subgroupindex
C2×#168109712
C2×#168108232
#202756422
C2×#168103692
C2×Q8⋊#883222
#202725562
C53×#25682
#1005353