Finite Group Databasebeta

← #27285190

Subgroups — #27285190

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810971C211yesmaximal, characteristic, complemented
2C2×#16810818C211yesmaximal, characteristic
3#20275642C211yesmaximal, characteristic, complemented
4C2×#16810369C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#2542C211yesmaximal, characteristic, complemented
8C22×#11719922—21no—
9C2×#13133214C2211yescharacteristic
10#16810971—22no—
11#16810971—22no—
12#16810818—22no—
13#16810818—22no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133137—22no—
17C2×#13133137—21no—
18C2×#13133137—22no—
19C2×C4×#88322C2211yescharacteristic
20#16810369—22no—
21D4⋊#88322—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C22⋊Q8×C53—22no—
24C2×C4×D4×C53C2211yescharacteristic
25C53×#399—22no—
26C53×#390—21no—
27C53×#384—21no—
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C2×#11719922—24no—
33C2×#11719922—42no—
34C2×#11719922—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×D4×C53D411yescharacteristic
53C23×C4×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C22⋊Q8×C53—44no—
59C22⋊Q8×C53—44no—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C2×C4⋊C4×C53—24no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C4⋊C4×C53D412yes—
68C2×C4⋊C4×C53D412yes—
69C2×C22⋊C4×C53—24no—
70C2×C22⋊C4×C53—21no—
71C2×C42×C53—21no—
72C2×C42×C53C2311yescharacteristic
73#95×C53—21no—
74C24×C53—21no—
75C23×D53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C2×Q8×C53—42no—
79C2×Q8×C53—42no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53SD1612yes—
86C22×C4×C53—22no—
87C22×C4×C53—22no—
88C22×C4×C53—21no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53—42no—
92C22×C4×C53—42no—
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C2×D212—22no—
97C2×D212—22no—
98C2×D212—22no—
99C2×D212—44no—
100C2×D212—42no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—42no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212—41no—
108C2×D212—41no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—48no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53D814yescomplemented
118C22⋊C4×C53—48no—
119C22⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C42×C53—41no—
122C42×C53—41no—
123C42×C53C2×D411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53#4111yescharacteristic
127#11719922—48no—
128C2×#88322—22no—
129C2×#88322—22no—
130C2×#88322—42no—
131C2×#88322—41no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—42no—
135C23×C53—22no—
136C23×C53—41no—
137C23×C53—42no—
138C23×C53—41no—
139C23×C53—42no—
140C23×C53#12011yescharacteristic
141C22×D53—42no—
142C22×D53—82no—
143C22×D53—81no—
144C22×D53—81no—
145C22×D53—82no—
146Q8×C53—82no—
147Q8×C53—81no—
148Q8×C53—81no—
149D4×C53—22no—
150D4×C53—22no—
151D4×C53—41no—
152D4×C53—41no—
153C2×C4×C53—22no—
154C2×C4×C53—24no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—44no—
161C2×C4×C53—82no—
162C2×C4×C53—81no—
163C2×C4×C53—81no—
164C2×C4×C53#12011yescharacteristic
165C2×C4×C53#12011yescharacteristic
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12711yescharacteristic
170C2×C4×C53C2×D812yes—
171C2×C4×C53C2×SD1611yescharacteristic
172C2×C4×C53#13511yescharacteristic
173C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C53AGL(1,ℤ/8)12yes—
175C2×C4×C53#13711yescharacteristic
176D212—44no—
177D212—42no—
178D212—42no—
179D212—44no—
180D212—82no—
181D212—81no—
182D212—81no—
183D212—81no—
184D212—81no—
185#88322—42no—
186#88322—42no—
187#88322—81no—
188#10052—531nomaximal, Sylow
189C22×C53—44no—
190C22×C53—42no—
191C22×C53—82no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53#39211yescharacteristic
195C22×C53#44712yes—
196C22×C53#44811yescharacteristic
197C22×C53#44911yescharacteristic
198C22×C53#45412yes—
199C2×D53—82no—
200C2×D53—82no—
201C2×D53—82no—
202C2×D53—162no—
203C2×D53—162no—
204C4×C53—21no—
205C4×C53—21no—
206C4×C53—44no—
207C4×C53—81no—
208C4×C53—82no—
209C4×C53#45011yescharacteristic
210C4×C53#45211yescharacteristic
211C4×C53#45912yes—
212C4×C53#46511yescharacteristic
213C4×C53#46811yescharacteristic
214C4×C53#49311yescharacteristic
215C4×C53#49611yescharacteristic
216C2×#486—531no—
217C2×#459—531no—
218#2542D5311yescharacteristic, complemented
219#1982—531no—
220C2×#332—531no—
221C2×D4⋊C8—531no—
222#1279—531no—
223C2×C53—82no—
224C2×C53#162512yes—
225C2×C53#162712yes—
226C2×C53#200011yescharacteristic
227C2×C53#200311yescharacteristic
228C2×C53#201111yescharacteristic
229D53—162no—
230C22×D8—1061no—
231C2×C4⋊Q8C2×D5311yescharacteristic
232C2×#127—531no—
233C2×C22⋊Q8—22no—
234C2×C4×D4C2×D5311yescharacteristic
235#486—1062no—
236#486—1062no—
237#459—1062no—
238#459—1062no—
239C2×C4⋊C8—531no—
240C2×C4⋊C8—531no—
241C2×#102—1062no—
242C2×#102—1061no—
243C2×#102—1062no—
244C2×C4×C8—531no—
245#399—22no—
246#390—21no—
247#384—21no—
248C4×C4⋊C4C2×D5311yescharacteristic
249#332—1062no—
250D4⋊C8—1062no—
251C53#1005211yescharacteristic, complemented, Sylow
252C22×Q8—21no—
253C22×D4—1061no—
254C22×D4—1061no—
255C22×D4—1061no—
256C22×D4C22⋊D5311yescharacteristic
257C23×C4—21no—
258C2×D8—1064no—
259C2×D8—2122no—
260C2×D8—2122no—
261C22×C8—1061no—
262C22×C8—1061no—
263C22×C8—1061no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268#127—532no—
269#127—1061no—
270#127—1061no—
271#127—1061no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4C22⋊D5311yescharacteristic
281C2×C4⋊C4C22⋊D5312yes—
282C2×C4⋊C4C22⋊D5312yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—21no—
285C2×C42—21no—
286C2×C42C22×D5311yescharacteristic
287C4⋊C8—532no—
288C4⋊C8—1061no—
289C4⋊C8—1061no—
290C4⋊C8—1061no—
291#102—1064no—
292#102—1068no—
293#102—2124no—
294C4×C8—532no—
295C4×C8—1061no—
296#95—21no—
297C24—21no—
298C24—2121no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—1062no—
309C2×D4—1062no—
310C2×D4—1062no—
311C2×D4—2124no—
312C2×D4—2121no—
313C2×D4—2121no—
314C2×D4—2121no—
315C2×D4—2122no—
316C2×D4—2121no—
317C2×D4—2121no—
318C2×D4—2121no—
319C2×D4—2122no—
320C2×D4—2121no—
321C2×D4#1171992412yes—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4C2×C22⋊D5311yescharacteristic
330C22×C4C2×C22⋊D5311yescharacteristic
331C22×C4C2×C22⋊D5311yescharacteristic
332D8—2128no—
333C2×C8—1062no—
334C2×C8—1062no—
335C2×C8—2121no—
336C2×C8—2122no—
337C2×C8—2122no—
338C2×C8—2121no—
339C2×C8—2121no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—48no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4#1171992214yescomplemented
349C22⋊C4—48no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C42—41no—
353C42—41no—
354C42D4×D5311yescharacteristic
355C42D4×D5311yescharacteristic
356C42D4×D5311yescharacteristic
357C42#1171994711yescharacteristic
358C23—22no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—4242no—
364C23—4241no—
365C23—4241no—
366C23—4242no—
367C23—2122no—
368C23#1313318611yescharacteristic
369Q8—81no—
370Q8—82no—
371Q8—81no—
372D4—22no—
373D4—22no—
374D4—41no—
375D4—41no—
376D4—4242no—
377D4—4241no—
378D4—4241no—
379D4—4241no—
380D4—4241no—
381D4—2124no—
382D4—2122no—
383D4—2122no—
384D4—2124no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—81no—
396C2×C4#1313318511yescharacteristic
397C2×C4#1313318511yescharacteristic
398C2×C4#1313318811yescharacteristic
399C2×C4#1313318911yescharacteristic
400C2×C4#1313318911yescharacteristic
401C2×C4#1313321311yescharacteristic
402C2×C4C2×#1171992212yes—
403C2×C4C2×#1171992411yescharacteristic
404C2×C4#1313323711yescharacteristic
405C2×C4#1313324211yescharacteristic
406C2×C4#1313324312yes—
407C2×C4#1313324811yescharacteristic
408C8—4241no—
409C8—2122no—
410C8—2122no—
411C22—42no—
412C22—44no—
413C22—81no—
414C22—81no—
415C22—82no—
416C22—8482no—
417C22—8482no—
418C22—4242no—
419C22—4242no—
420C22—4242no—
421C22#1681053111yescharacteristic
422C22#1681075712yes—
423C22#1681076211yescharacteristic
424C22#1681076811yescharacteristic
425C22#1681079612yes—
426C4—21no—
427C4—21no—
428C4—44no—
429C4—81no—
430C4—82no—
431C4#1681077311yescharacteristic
432C4#1681078311yescharacteristic
433C4#1681081512yes—
434C4#1681085411yescharacteristic
435C4#1681087111yescharacteristic
436C4#1681099911yescharacteristic
437C4#1681101011yescharacteristic
438C2—82no—
439C2—8482no—
440C2#2027380412yes—
441C2#2027381812yes—
442C2#2027571611yescharacteristic
443C2#2027573511yescharacteristic
444C2#2027578611yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2542C2
socleC23×C53#120
prime radicalC2×#16810818C2
hypercenter#2542D53
hypocenterC53#10052
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2542

Complements

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

NH ≅ G/Ntype
1. C2×#16810971C2semidirect
3. #20275642C2semidirect
4. C2×#16810369C2semidirect
5. C2×D4⋊#88322C2semidirect
6. #20272556C2semidirect
7. C53×#2542C2semidirect
117. C4⋊C4×C53D8semidirect
218. #2542D53semidirect
251. C53#10052semidirect
348. C4⋊C4#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168109712
C2×#168108182
#202756422
C2×#168103692
C2×D4⋊#883222
#202725562
C53×#25422
#1005253