Finite Group Databasebeta

← #27393359

Subgroups — #27393359

Order 13568.

1334 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811201C211yesmaximal, characteristic
2#20277316C212yesmaximal, complemented
3#20275549C218yesmaximal, complemented
4#20274633C211yesmaximal, characteristic, complemented
5#20274632C212yesmaximal, complemented
6C53×#1782C211yesmaximal, characteristic, complemented
7C22×#11719945C2211yescharacteristic
8#16811201—24no—
9#16811201—24no—
10C2×#13133196C412yescomplemented
11C2×#13133193C412yescomplemented
12C2×#13133191C412yescomplemented
13C2×#13133189C2214yes—
14C2×#13133188C412yescomplemented
15C22×C4⋊Dic53C2211yescharacteristic
16C2×#13133132C2214yes—
17C2×#13133131C2218yes—
18#16810508—24no—
19#16810506—24no—
20#16810486C2212yes—
21#16810481C2214yes—
22C22⋊C4×Dic53C2214yes—
23C22×#41×C53C2211yescharacteristic
24C2×#99×C53C2214yes—
25C53×#381C2212yes—
26C2×#11719945—22no—
27C2×#11719945—22no—
28C2×#11719945—44no—
29C22×Dic106C2×C411yescharacteristic
30C22×D212C2×C411yescharacteristic
31C22×C22⋊D53C2×C414yes—
32C22×C22⋊D53C2312yes—
33C23×Dic53C2312yes—
34C22×C4×D53C2×C412yes—
35#13133196—24no—
36#13133196—24no—
37#13133193—28no—
38#13133191—24no—
39#13133191—24no—
40#13133189—28no—
41#13133189—28no—
42#13133188—28no—
43C2×C4⋊Dic53C2×C412yes—
44C2×C4⋊Dic53C2×C412yes—
45C2×C4⋊Dic53D414yes—
46C2×C4⋊Dic53D414yes—
47C2×#11719919—24no—
48C2×#11719919C2×C414yes—
49C2×C22⋊Dic53C2×C414yes—
50C2×C22⋊Dic53C2314yes—
51C2×#11719921C2×C414yes—
52C2×C4×Dic53—24no—
53#13133132—44no—
54#13133131—48no—
55#13133124—22no—
56#13133124—22no—
57#13133124—24no—
58#13133124—24no—
59C2×#41×C53—22no—
60C2×#41×C53—22no—
61C2×#41×C53—44no—
62C22×Q8×C53C2×C411yescharacteristic
63C22×D4×C53C2×C411yescharacteristic
64C22×D4×C53C2312yes—
65C23×C4×C53C2×C412yes—
66C23×C4×C53C2311yescharacteristic
67C2×C22⋊C4×C53C2314yes—
68#99×C53—44no—
69#95×C53—22no—
70#95×C53—22no—
71C24×C53C22×C412yes—
72C24×C53C2411yescharacteristic
73C23×D53C22×C412yes—
74#41×C53—48no—
75#41×C53—48no—
76C2×Q8×C53—22no—
77C2×Q8×C53—42no—
78C2×D4×C53—24no—
79C2×D4×C53—24no—
80C2×D4×C53—24no—
81C2×D4×C53—22no—
82C2×D4×C53—42no—
83C22×C4×C53—24no—
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53—24no—
87C22×C4×C53—42no—
88C22×C4×C53—42no—
89C22×C4×C53C22×C412yes—
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C22×C411yescharacteristic
92C22×C4×C53C2×D412yes—
93C22×C4×C53C2×D412yes—
94C2×C22⋊D53—24no—
95C2×C22⋊D53—28no—
96C2×C22⋊D53—28no—
97C2×C22⋊D53—84no—
98C22×Dic53—24no—
99C22×Dic53—82no—
100C22×Dic53C22×C412yes—
101C22×Dic53C22×C414yes—
102C22×Dic53C2×D414yes—
103C22×Dic53C2×D414yes—
104#11719945—48no—
105#11719945—48no—
106C2×D212—21no—
107C2×D212—21no—
108C2×D212—42no—
109C2×C4×D53—24no—
110C2×C4×D53—44no—
111C2×Dic106—21no—
112C2×Dic106—21no—
113C2×Dic106—42no—
114C22⋊C4×C53—28no—
115C22⋊Dic53—28no—
116C22⋊Dic53—28no—
117#11719921—28no—
118C4⋊Dic53C22⋊C418yes—
119C4⋊Dic53C22⋊C414yes—
120C4⋊Dic53C22⋊C414yes—
121#11719919—28no—
122#11719919—84no—
123C4×Dic53—84no—
124C23×C53—22no—
125C23×C53—22no—
126C23×C53—22no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53C23×C411yescharacteristic
132C23×C53C22×D412yes—
133C23×C53H(5,𝔽2)14yes—
134C22×D53—22no—
135C22×D53—24no—
136C22×D53—42no—
137C22×D53—42no—
138Q8×C53—44no—
139D4×C53—44no—
140D4×C53—48no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—24no—
144C2×C4×C53—24no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—44no—
149C2×C4×C53—44no—
150C2×C4×C53—44no—
151C2×C4×C53—44no—
152C2×C4×C53—44no—
153C2×C4×C53C2×C22⋊C411yescharacteristic
154C2×C4×C53C2×C22⋊C411yescharacteristic
155C2×C4×C53C2×C22⋊C412yes—
156C22⋊D53—48no—
157C22⋊D53—88no—
158C2×Dic53—22no—
159C2×Dic53—24no—
160C2×Dic53—44no—
161C2×Dic53—44no—
162C2×Dic53—44no—
163C2×Dic53—44no—
164C2×Dic53—44no—
165C2×Dic53—44no—
166C2×Dic53C2×C22⋊C418yes—
167D212—42no—
168D212—42no—
169C4×D53—48no—
170Dic106—42no—
171Dic106—42no—
172#21796—531nomaximal, Sylow
173C22×C53—22no—
174C22×C53—21no—
175C22×C53—22no—
176C22×C53—21no—
177C22×C53—44no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53—82no—
182C22×C53—82no—
183C22×C53C22×C22⋊C411yescharacteristic
184C22×C53#51812yes—
185C22×C53#53514yes—
186C2×D53—42no—
187C2×D53—42no—
188C2×D53—42no—
189C2×D53—84no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—22no—
193C4×C53—42no—
194C4×C53—84no—
195Dic53—28no—
196Dic53—42no—
197Dic53—84no—
198C2×#537—531no—
199#2313—532no—
200#1964—538no—
201#1782—532no—
202#1782—531no—
203#1782D5311yescharacteristic, complemented
204C2×C53—24no—
205C2×C53—22no—
206C2×C53—82no—
207C2×C53#230911yescharacteristic
208C2×C53#288311yescharacteristic
209C2×C53#288411yescharacteristic
210D53—82no—
211C22×#41—531no—
212C22×#41C2×D5311yescharacteristic
213#537—1064no—
214#537—1064no—
215C2×C22⋊Q8—532no—
216C2×C22⋊Q8—532no—
217C2×#121—532no—
218C2×#121—534no—
219C2×#121—532no—
220C22×C4⋊C4—531no—
221C2×#99—534no—
222C2×#99—538no—
223C2×#99C2×D5314yes—
224#387—1064no—
225#387—1064no—
226#381—532no—
227#381C2×D5312yes—
228#380—534no—
229C4×C22⋊C4—534no—
230C53#2179611yescharacteristic, complemented, Sylow
231C2×#41—22no—
232C2×#41—22no—
233C2×#41—44no—
234C2×#41—1062no—
235C2×#41—1062no—
236C2×#41—2124no—
237C22×Q8—531no—
238C22×Q8C4×D5311yescharacteristic
239C22×D4—532no—
240C22×D4—534no—
241C22×D4—531no—
242C22×D4C4×D5311yescharacteristic
243C22×D4C22×D5312yes—
244C23×C4—532no—
245C23×C4—532no—
246C23×C4C4×D5312yes—
247C23×C4C22×D5311yescharacteristic
248C22⋊Q8—1064no—
249C22⋊Q8—1064no—
250C22⋊Q8—1068no—
251#121—1064no—
252#121—1064no—
253#121—1068no—
254#121—1068no—
255#121—1068no—
256C2×C4⋊C4—532no—
257C2×C4⋊C4—532no—
258C2×C4⋊C4—534no—
259C2×C4⋊C4—534no—
260C2×C4⋊C4—534no—
261C2×C4⋊C4—1064no—
262C2×C22⋊C4—534no—
263C2×C22⋊C4—534no—
264C2×C22⋊C4—534no—
265C2×C22⋊C4C22×D5314yes—
266C2×C42—1064no—
267#99—44no—
268#99—2128no—
269#99—2124no—
270#95—22no—
271#95—22no—
272#95—1064no—
273#95—1064no—
274#95—1062no—
275#95—1062no—
276C24—532no—
277C24C2×C4×D5312yes—
278C24C23×D5311yescharacteristic
279#41—48no—
280#41—48no—
281#41—2128no—
282#41—2128no—
283C2×Q8—22no—
284C2×Q8—42no—
285C2×Q8—1061no—
286C2×Q8—1061no—
287C2×Q8—2122no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—42no—
293C2×D4—1068no—
294C2×D4—1064no—
295C2×D4—1068no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4—4244no—
299C2×D4—2122no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—534no—
307C22×C4—534no—
308C22×C4—534no—
309C22×C4—532no—
310C22×C4—1064no—
311C22×C4—1064no—
312C22×C4—4242no—
313C22×C4—2124no—
314C22×C4C2×C4×D5312yes—
315C22×C4C2×C4×D5311yescharacteristic
316C22×C4C2×C4×D5311yescharacteristic
317C22×C4D4×D5312yes—
318C22×C4D4×D5312yes—
319C4⋊C4—534no—
320C4⋊C4—534no—
321C4⋊C4—538no—
322C4⋊C4—1068no—
323C4⋊C4—4244no—
324C22⋊C4—28no—
325C22⋊C4—1068no—
326C22⋊C4—1068no—
327C22⋊C4—1068no—
328C42—4244no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—41no—
333C23—41no—
334C23—42no—
335C23—42no—
336C23—1064no—
337C23—1062no—
338C23—2122no—
339C23—2122no—
340C23C22×C4×D5311yescharacteristic
341C23C2×D4×D5312yes—
342C23#1313326214yes—
343Q8—44no—
344Q8—2122no—
345Q8—2122no—
346D4—44no—
347D4—48no—
348D4—4248no—
349D4—2122no—
350D4—2122no—
351D4—2128no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—538no—
365C2×C4—1062no—
366C2×C4—1064no—
367C2×C4—2128no—
368C2×C4—2124no—
369C2×C4—2124no—
370C2×C4—2124no—
371C2×C4—2124no—
372C2×C4—2124no—
373C2×C4—2124no—
374C2×C4C22⋊C4×D5311yescharacteristic
375C2×C4C22⋊C4×D5311yescharacteristic
376C2×C4C22⋊C4×D5312yes—
377C22—22no—
378C22—21no—
379C22—22no—
380C22—21no—
381C22—44no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—82no—
386C22—82no—
387C22—4244no—
388C22—2122no—
389C22—2122no—
390C22—2122no—
391C22C2×C22⋊C4×D5311yescharacteristic
392C22#1681109612yes—
393C22#1681118814yes—
394C4—21no—
395C4—21no—
396C4—22no—
397C4—42no—
398C4—84no—
399C4—1068no—
400C4—4244no—
401C4—2122no—
402C2—24no—
403C2—22no—
404C2—82no—
405C2—4242no—
406C2#2027729611yescharacteristic
407C2#2028317411yescharacteristic
408C2#2028317711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1782C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#16811201C2
cosocleC24×C53C24
hypercenter#1782D53
hypocenterC53#21796
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. #20277316C2semidirect
3. #20275549C2semidirect
4. #20274633C2semidirect
5. #20274632C2semidirect
6. C53×#1782C2semidirect
10. C2×#13133196C4semidirect
11. C2×#13133193C4semidirect
12. C2×#13133191C4semidirect
14. C2×#13133188C4semidirect
203. #1782D53semidirect
230. C53#21796semidirect

Maximal subgroups

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

subgroupindex
C2×#168112012
#202773162
#202755492
#202746332
#202746322
C53×#17822
#2179653