Finite Group Databasebeta

← #27440076

Subgroups — #27440076

Order 13568.

1038 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic106C211yesmaximal, characteristic
2C2×#16811021C211yesmaximal, characteristic
3C2×#16810908C211yesmaximal, characteristic
4#20279703C211yesmaximal, characteristic
5#20276047C214yesmaximal
6#20275663C211yesmaximal, characteristic
7#20275651C211yesmaximal, characteristic
8#20274985C214yesmaximal
9C4×#103×C53C211yesmaximal, characteristic
10Q8×Dic106—24no—
11Q8×Dic106—24no—
12C2×#13133217—21no—
13C2×#13133217—21no—
14C2×#13133217C2211yescharacteristic
15C2×#13133217C2211yescharacteristic
16C2×#13133216—22no—
17C2×#13133216C2212yes—
18C2×Q8×Dic53C2212yes—
19C2×C4×Dic106—21no—
20C2×C4×Dic106C2211yescharacteristic
21#16811021—24no—
22#16810908—24no—
23C2×#13133149C2214yes—
24C2×#13133148C2212yes—
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133141C2212yes—
27#16810555—22no—
28#16810529—21no—
29#16810522C2214yes—
30#16810385C2214yes—
31#16810377C2214yes—
32C2×C4×Q8×C53C2211yescharacteristic
33C2×#103×C53C2212yes—
34C2×C4×C8×C53C2211yescharacteristic
35C4×C4⋊C4×C53C2211yescharacteristic
36C53×#340C2212yes—
37C22×Dic106—21no—
38C22×Dic106—22no—
39C22×Dic106—21no—
40C22×Dic106D411yescharacteristic
41C22×#88322C2312yes—
42#13133217—28no—
43#13133217—22no—
44#13133217—22no—
45#13133217—22no—
46#13133217—44no—
47#13133217D414yes—
48#13133216—216no—
49#13133216—24no—
50#13133216—24no—
51Q8×Dic53—24no—
52Q8×Dic53—24no—
53C4×Dic106—28no—
54C4×Dic106—22no—
55C4×Dic106—22no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53—24no—
60C2×C4⋊Dic53D411yescharacteristic
61C2×C4⋊Dic53C2312yes—
62C2×C4⋊Dic53C2312yes—
63C2×#11719919—22no—
64C2×#11719919D412yes—
65C2×C4×Dic53—22no—
66C2×C4×Dic53C2312yes—
67#13133149—28no—
68#13133148—24no—
69C4⋊#88322—22no—
70#13133141—24no—
71#13133124—24no—
72C22×Q8×C53C2311yescharacteristic
73C22×C8×C53C2312yes—
74C4×Q8×C53—22no—
75C4×Q8×C53D414yes—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53C2311yescharacteristic
78C2×C4⋊C4×C53C2312yes—
79C2×C42×C53—21no—
80C2×C42×C53—21no—
81C2×C42×C53C2311yescharacteristic
82#103×C53Q818yescomplemented
83C4×C8×C53—22no—
84#95×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53—22no—
87C2×Q8×C53C2×D412yes—
88C2×Q8×C53C2×Q814yes—
89C22×C4×C53—21no—
90C22×C4×C53—22no—
91C22×C4×C53—21no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2411yescharacteristic
95C22×Dic53—22no—
96C22×Dic53—22no—
97C22×Dic53C2×D412yes—
98C22×Dic53C2×D412yes—
99C2×Dic106—22no—
100C2×Dic106—22no—
101C2×Dic106—28no—
102C2×Dic106—24no—
103C2×Dic106—42no—
104C2×Dic106—42no—
105C2×Dic106—42no—
106C2×Dic106—84no—
107C2×Dic106—82no—
108C2×Dic106SD1614yes—
109C2×C8×C53—22no—
110C2×C8×C53—22no—
111C2×C8×C53C2×Q814yes—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53C2×D412yes—
116C4⋊C4×C53C2×Q814yes—
117C42×C53—24no—
118C42×C53—42no—
119C42×C53C2×D412yes—
120C42×C53C2×D412yes—
121C4⋊Dic53—28no—
122C4⋊Dic53—24no—
123C4⋊Dic53—22no—
124C4⋊Dic53—22no—
125C4⋊Dic53—22no—
126C4⋊Dic53—22no—
127C4⋊Dic53—42no—
128C4⋊Dic53—48no—
129C4⋊Dic53C2×D414yes—
130#11719919—28no—
131#11719919—24no—
132C4×Dic53—28no—
133C4×Dic53—22no—
134C4×Dic53—22no—
135C2×#88322—22no—
136C2×#88322—24no—
137C23×C53C22×D411yescharacteristic
138Q8×C53—22no—
139Q8×C53—24no—
140Q8×C53C4⋊Q814yescomplemented
141C2×C4×C53—22no—
142C2×C4×C53—24no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—24no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53#12012yes—
150C2×C4×C53#12012yes—
151C2×C4×C53C2×SD1612yes—
152C2×C4×C53#13712yes—
153C2×C4×C53C22×D411yescharacteristic
154C2×C4×C53C22×D411yescharacteristic
155C2×C4×C53C22×Q812yes—
156C2×C4×C532−1+412yes—
157C2×Dic53—22no—
158C2×Dic53—22no—
159C2×Dic53—24no—
160C2×Dic53—22no—
161C2×Dic53—24no—
162C2×Dic53—44no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166C2×Dic53C2×SD1614yes—
167Dic106—28no—
168Dic106—84no—
169Dic106—88no—
170Dic106—84no—
171C8×C53—24no—
172#88322—42no—
173#25837—531nomaximal, Sylow
174C22×C53C2×#12011yescharacteristic
175C22×C53D4×Q812yes—
176C22×C53C22×SD1611yescharacteristic
177C22×C53C2×#13711yescharacteristic
178C22×C53#57712yes—
179C4×C53—24no—
180C4×C53—22no—
181C4×C53—44no—
182C4×C53—42no—
183C4×C53#44814yes—
184C4×C53C2×C4⋊Q811yescharacteristic
185C4×C53#53611yescharacteristic
186C4×C53D4×Q812yes—
187Dic53—24no—
188Dic53—42no—
189Dic53—44no—
190Dic53—82no—
191C2×Q82—531no—
192C2×#498—531no—
193C2×#475—531no—
194#2546—531no—
195#2058—534no—
196#1987—531no—
197#1984—531no—
198#1865—534no—
199C4×#103D5311yescharacteristic
200C2×C53#243411yescharacteristic
201C2×C53C2×#44811yescharacteristic
202C2×C53#300911yescharacteristic
203C2×C53Q8×SD1612yes—
204C2×C53#338712yes—
205Q82—1064no—
206Q82—1064no—
207C2×C4⋊Q8—532no—
208C2×C4⋊Q8—531no—
209C2×C4⋊Q8—531no—
210C2×C4⋊Q8—1061no—
211C2×C4⋊Q8—1061no—
212C2×C4⋊Q8—1062no—
213C2×C4×Q8—531no—
214C2×C4×Q8—532no—
215C2×C4×Q8—1061no—
216C2×C4×Q8C2×D5311yescharacteristic
217#498—1064no—
218#475—1064no—
219C2×#106—534no—
220C2×#106—532no—
221C2×C4⋊C8—531no—
222C2×#103—532no—
223C2×#103C2×D5312yes—
224C2×C4×C8C2×D5311yescharacteristic
225#398—1062no—
226#391—1061no—
227#389—534no—
228C4×C4⋊C4C2×D5311yescharacteristic
229#340—534no—
230#340C2×D5312yes—
231#336—534no—
232C53#2583711yescharacteristic, complemented, Sylow
233C22×Q8—531no—
234C22×Q8—1062no—
235C22×Q8—1061no—
236C22×Q8—1061no—
237C22×Q8C22×D5311yescharacteristic
238C22×C8—532no—
239C22×C8C22×D5312yes—
240C4⋊Q8—534no—
241C4⋊Q8—1068no—
242C4⋊Q8—1064no—
243C4⋊Q8—10616no—
244C4⋊Q8—1064no—
245C4⋊Q8—1062no—
246C4⋊Q8—1062no—
247C4⋊Q8—1062no—
248C4⋊Q8—2124no—
249C4×Q8—22no—
250C4×Q8—1068no—
251C4×Q8—1064no—
252C4×Q8—1064no—
253C4×Q8—1062no—
254C4×Q8—1062no—
255C4×Q8D21214yes—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—532no—
258C2×C4⋊C4—532no—
259C2×C4⋊C4—531no—
260C2×C4⋊C4—532no—
261C2×C4⋊C4—1062no—
262C2×C4⋊C4—1062no—
263C2×C4⋊C4—1061no—
264C2×C4⋊C4—1064no—
265C2×C4⋊C4—1061no—
266C2×C4⋊C4C22×D5312yes—
267C2×C4⋊C4C22×D5311yescharacteristic
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—532no—
271C2×C42—1062no—
272C2×C42C22×D5311yescharacteristic
273#106—1068no—
274#106—1064no—
275C4⋊C8—1062no—
276#103—1064no—
277#103Dic10618yescomplemented
278C4×C8—22no—
279#95—21no—
280#95—1064no—
281C2×Q8—21no—
282C2×Q8—22no—
283C2×Q8—534no—
284C2×Q8—1068no—
285C2×Q8—1062no—
286C2×Q8—1064no—
287C2×Q8—1062no—
288C2×Q8—4242no—
289C2×Q8—4244no—
290C2×Q8—2122no—
291C2×Q8—2122no—
292C2×Q8—2122no—
293C2×Q8C2×Dic10614yes—
294C2×Q8C2×D21212yes—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—532no—
299C22×C4—532no—
300C22×C4—1062no—
301C22×C4—1062no—
302C22×C4D4×D5311yescharacteristic
303C22×C4D4×D5311yescharacteristic
304C22×C4C23×D5311yescharacteristic
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—1062no—
308C2×C8—1064no—
309C2×C8C2×Dic10614yes—
310C4⋊C4—24no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—534no—
314C4⋊C4—1068no—
315C4⋊C4—1068no—
316C4⋊C4—1064no—
317C4⋊C4—1062no—
318C4⋊C4—1064no—
319C4⋊C4—1062no—
320C4⋊C4—1062no—
321C4⋊C4—1062no—
322C4⋊C4—2128no—
323C4⋊C4—2122no—
324C4⋊C4C2×Dic10614yes—
325C4⋊C4C2×D21212yes—
326C42—24no—
327C42—42no—
328C42—1068no—
329C42—1062no—
330C42—1062no—
331C42C2×D21212yes—
332C42D4×D5312yes—
333C23C2×D4×D5311yescharacteristic
334Q8—24no—
335Q8—22no—
336Q8—1068no—
337Q8—4244no—
338Q8—4248no—
339Q8—4244no—
340Q8#1313321714yescomplemented
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—534no—
350C2×C4—1062no—
351C2×C4—1062no—
352C2×C4—1064no—
353C2×C4—1062no—
354C2×C4—1064no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4—2122no—
358C2×C4—2124no—
359C2×C4#1313318712yes—
360C2×C4#1313318712yes—
361C2×C4SD16×D5312yes—
362C2×C4#1313324912yes—
363C2×C4C2×D4×D5311yescharacteristic
364C2×C4C22×D21211yescharacteristic
365C2×C4C22×Dic10612yes—
366C2×C4#1313326412yes—
367C8—24no—
368C8—2122no—
369C22C2×#1313318711yescharacteristic
370C22D4×Dic10612yes—
371C22C2×SD16×D5311yescharacteristic
372C22C2×#1313324911yescharacteristic
373C22#1681143012yes—
374C4—24no—
375C4—22no—
376C4—44no—
377C4—42no—
378C4—1064no—
379C4—4242no—
380C4—2122no—
381C4—2124no—
382C4#1681076314yes—
383C4C2×#1313321711yescharacteristic
384C4#1681119911yescharacteristic
385C4D4×Dic10612yes—
386C2#2027835911yescharacteristic
387C2C2×#1681076311yescharacteristic
388C2#2028399511yescharacteristic
389C2SD16×Dic10612yes—
390C2#2028764512yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×#103×C53C2
socle, prime radicalC23×C53C22×D4
hypercenterC4×#103D53
hypocenterC53#25837
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
82. #103×C53Q8semidirect
140. Q8×C53C4⋊Q8semidirect
232. C53#25837semidirect
277. #103Dic106semidirect
340. Q8#13133217semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1062
C2×#168110212
C2×#168109082
#202797032
#202760472
#202756632
#202756512
#202749852
C4×#103×C532
#2583753