Finite Group Databasebeta

← #27287626

Subgroups — #27287626

Order 13568.

546 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
1#20279633C211yesmaximal, characteristic
2#20274872C211yesmaximal, characteristic, complemented
3#20274720C211yesmaximal, characteristic, complemented
4#20274716C211yesmaximal, characteristic
5C2×#16810352C211yesmaximal, characteristic
6#20272599C211yesmaximal, characteristic, complemented
7C2×C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#13133193—21no—
9C2×C4×D212C2211yescharacteristic
10C2×#13133149—22no—
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133141—21no—
13C2×#13133139—21no—
14#16810545—22no—
15#16810525—21no—
16#16810521—21no—
17#16810493C2211yescharacteristic
18#16810493C2211yescharacteristic
19#16810480—21no—
20#16810386—21no—
21#16810385—21no—
22#16810352—22no—
23C2×C4×Q8×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C53×Q8⋊C8—22no—
27C22×D212D411yescharacteristic
28C22×C4×D53—21no—
29C22×#88322—21no—
30#13133193—42no—
31#13133193—42no—
32C4×D212—21no—
33C4×D212—21no—
34C4×D212—21no—
35C4×D212—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53D411yescharacteristic
42C2×C4⋊Dic53D412yes—
43C2×#11719919—22no—
44C2×#11719921—22no—
45C2×#11719921—22no—
46C2×#11719921—21no—
47#13133149—44no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50#13133141—42no—
51#13133139—42no—
52#13133124—22no—
53C22×Q8×C53D411yescharacteristic
54C22×C8×C53—21no—
55C22×C8×C53—21no—
56C4×Q8×C53—21no—
57C4×Q8×C53—21no—
58C4×Q8×C53—21no—
59C4×Q8×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C42×C53—21no—
63C2×C42×C53C2311yescharacteristic
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53D412yes—
68C23×D53—21no—
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53SD1612yes—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—22no—
82C22×Dic53—41no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212SD1612yes—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×C4×D53—41no—
93C2×C4×D53—41no—
94C2×C4×D53—41no—
95C2×C8×C53—22no—
96C2×C8×C53—22no—
97C2×C8×C53—22no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C42×C53—41no—
106C42×C53—41no—
107C42×C53C2×D411yescharacteristic
108C42×C53#4111yescharacteristic
109C42×C53#4111yescharacteristic
110C42×C53#4111yescharacteristic
111#11719921—44no—
112#11719921—41no—
113#11719921—41no—
114#11719921—82no—
115C4⋊Dic53—21no—
116C4⋊Dic53—21no—
117C4⋊Dic53—41no—
118C4⋊Dic53—41no—
119C4⋊Dic53—41no—
120C4⋊Dic53—41no—
121C4⋊Dic53—42no—
122C4⋊Dic53—42no—
123C4⋊Dic53—41no—
124C4⋊Dic53—41no—
125C4⋊Dic53—41no—
126C4⋊Dic53—41no—
127#11719919—44no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322—41no—
131C23×C53#12011yescharacteristic
132C22×D53—22no—
133C22×D53—41no—
134C22×D53—41no—
135C22×D53—41no—
136C22×D53—41no—
137C22×D53—41no—
138C22×D53—41no—
139Q8×C53—22no—
140Q8×C53—22no—
141Q8×C53—41no—
142Q8×C53—41no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12311yescharacteristic
155C2×C4×C53#12311yescharacteristic
156C2×C4×C53#12411yescharacteristic
157C2×C4×C53C2×SD1611yescharacteristic
158C2×C4×C53C2×SD1611yescharacteristic
159C2×C4×C53#13511yescharacteristic
160C2×C4×C53#13511yescharacteristic
161C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C53#13711yescharacteristic
164C2×C4×C53#13711yescharacteristic
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—42no—
169C2×Dic53—42no—
170C2×Dic53—42no—
171C2×Dic53—81no—
172C2×Dic53—81no—
173C2×Dic53—81no—
174D212—22no—
175D212—22no—
176D212—41no—
177D212—41no—
178C4×D53—81no—
179C4×D53—81no—
180C4×D53—81no—
181C4×D53—81no—
182C8×C53—24no—
183C8×C53—81no—
184#88322—81no—
185#10435—531nomaximal, Sylow
186C22×C53#39411yescharacteristic
187C22×C53#44811yescharacteristic
188C22×C53#44911yescharacteristic
189C22×C53#45011yescharacteristic
190C22×C53#45211yescharacteristic
191C22×C53#45311yescharacteristic
192C22×C53#45611yescharacteristic
193C2×D53—42no—
194C2×D53—42no—
195C2×D53—42no—
196C2×D53—81no—
197C2×D53—81no—
198C2×D53—81no—
199C2×D53—81no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—42no—
203C4×C53—81no—
204C4×C53#46511yescharacteristic
205C4×C53#46511yescharacteristic
206C4×C53#46911yescharacteristic
207C4×C53#47011yescharacteristic
208C4×C53#48111yescharacteristic
209C4×C53#48211yescharacteristic
210C4×C53#48311yescharacteristic
211C4×C53#48511yescharacteristic
212Dic53—82no—
213Dic53—81no—
214Dic53—161no—
215#2538—531no—
216#1844—531no—
217#1803—531no—
218#1802—531no—
219C2×Q8⋊C8D5311yescharacteristic, complemented
220C2×D4⋊C8—531no—
221#1296—531no—
222C2×C53#162211yescharacteristic
223C2×C53#163111yescharacteristic
224C2×C53#165111yescharacteristic
225C2×C53#165211yescharacteristic
226C2×C53#203411yescharacteristic
227C2×C53#203511yescharacteristic
228C2×C53#203911yescharacteristic
229D53—82no—
230C2×C22⋊Q8—1061no—
231C2×C4×Q8C2×D5311yescharacteristic
232C2×C4×D4—531no—
233C2×#106—1062no—
234C2×C4⋊C8—531no—
235C2×C4⋊C8C2×D5311yescharacteristic
236C2×#103—1061no—
237C2×#102—1061no—
238C2×C4×C8C2×D5311yescharacteristic
239#396—1062no—
240#390—1061no—
241#389—1061no—
242#384—531no—
243#384—531no—
244#380—1061no—
245#340—1061no—
246#340—1061no—
247Q8⋊C8—22no—
248D4⋊C8—1062no—
249C53#1043511yescharacteristic, complemented, Sylow
250C22×Q8D21211yescharacteristic
251C22×D4—531no—
252C23×C4—1061no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—1061no—
256C22⋊Q8—2122no—
257C22⋊Q8—2122no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×D4—1061no—
263C4×D4—1061no—
264C4×D4—1061no—
265C4×D4—1061no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—531no—
268C2×C4⋊C4—532no—
269C2×C4⋊C4—1062no—
270C2×C4⋊C4—1061no—
271C2×C4⋊C4—1061no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4—1061no—
274C2×C4⋊C4—1061no—
275C2×C4⋊C4D21211yescharacteristic
276C2×C22⋊C4—1062no—
277C2×C22⋊C4—1062no—
278C2×C22⋊C4—1061no—
279C2×C42—21no—
280C2×C42C22×D5311yescharacteristic
281#106—2124no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—1061no—
285C4⋊C8—1061no—
286#103—2122no—
287#102—2122no—
288C4×C8—21no—
289C4×C8C22⋊D5312yes—
290#95—1062no—
291C24—1061no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8#1171991312yes—
298C2×D4—532no—
299C2×D4—1061no—
300C2×D4—1061no—
301C2×D4—1061no—
302C2×D4—1061no—
303C2×D4—1061no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—1061no—
307C22×C4—1062no—
308C22×C4—2121no—
309C22×C4—2121no—
310C22×C4—2121no—
311C22×C4—2121no—
312C22×C4—2121no—
313C22×C4—2121no—
314C22×C4—2121no—
315C22×C4C2×D21211yescharacteristic
316C22×C4D4×D5311yescharacteristic
317C22×C4D4×D5311yescharacteristic
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—2121no—
325C2×C8—2121no—
326C2×C8—2121no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—1061no—
332C4⋊C4—1061no—
333C4⋊C4—2122no—
334C4⋊C4—2122no—
335C4⋊C4—2124no—
336C4⋊C4—2121no—
337C4⋊C4—2121no—
338C4⋊C4—2121no—
339C4⋊C4—2121no—
340C4⋊C4—2121no—
341C4⋊C4—2121no—
342C4⋊C4—2121no—
343C4⋊C4—2121no—
344C22⋊C4—4242no—
345C22⋊C4—2124no—
346C22⋊C4—2121no—
347C22⋊C4—2121no—
348C42—41no—
349C42—41no—
350C42#1171994511yescharacteristic
351C42#1171994711yescharacteristic
352C42#1171994711yescharacteristic
353C42C2×C22⋊D5311yescharacteristic
354C23—1062no—
355C23—2121no—
356C23—2121no—
357C23—2121no—
358C23—2121no—
359C23—2121no—
360C23—2121no—
361C23#1313318711yescharacteristic
362Q8—22no—
363Q8—22no—
364Q8—41no—
365Q8—41no—
366D4—1062no—
367D4—1062no—
368D4—2121no—
369D4—2121no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—4241no—
379C2×C4—4241no—
380C2×C4—4241no—
381C2×C4—4241no—
382C2×C4—4241no—
383C2×C4—4241no—
384C2×C4—4241no—
385C2×C4—2121no—
386C2×C4—2122no—
387C2×C4—2121no—
388C2×C4—2122no—
389C2×C4—2122no—
390C2×C4—2121no—
391C2×C4#1313318811yescharacteristic
392C2×C4#1313318911yescharacteristic
393C2×C4#1313319111yescharacteristic
394C2×C4#1313319911yescharacteristic
395C2×C4#1313320111yescharacteristic
396C2×C4#1313320311yescharacteristic
397C2×C4SD16×D5311yescharacteristic
398C2×C4C2×#1171991311yescharacteristic
399C2×C4#1313323811yescharacteristic
400C2×C4#1313323911yescharacteristic
401C2×C4#1313324511yescharacteristic
402C2×C4#1313324511yescharacteristic
403C2×C4#1313324911yescharacteristic
404C2×C4#1313324911yescharacteristic
405C8—24no—
406C8—81no—
407C8—4241no—
408C22—4241no—
409C22—4241no—
410C22—4241no—
411C22—4241no—
412C22—2122no—
413C22—2122no—
414C22—2122no—
415C22#1681053911yescharacteristic
416C22#1681076311yescharacteristic
417C22#1681077011yescharacteristic
418C22#1681077611yescharacteristic
419C22#1681078811yescharacteristic
420C22#1681079211yescharacteristic
421C22#1681080711yescharacteristic
422C4—21no—
423C4—21no—
424C4—42no—
425C4—81no—
426C4—8481no—
427C4—4242no—
428C4—4241no—
429C4#1681084911yescharacteristic
430C4#1681085211yescharacteristic
431C4#1681087611yescharacteristic
432C4#1681087811yescharacteristic
433C4#1681094211yescharacteristic
434C4#1681094811yescharacteristic
435C4#1681095211yescharacteristic
436C4#1681096211yescharacteristic
437C2—4242no—
438C2#2027379011yescharacteristic
439C2#2027384411yescharacteristic
440C2#2027398111yescharacteristic
441C2#2027398711yescharacteristic
442C2#2027590511yescharacteristic
443C2#2027591011yescharacteristic
444C2#2027593811yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×Q8⋊C8C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenterC2×Q8⋊C8D53
hypocenterC53#10435
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #20274872C2semidirect
3. #20274720C2semidirect
6. #20272599C2semidirect
7. C2×C53×Q8⋊C8C2semidirect
219. C2×Q8⋊C8D53semidirect
249. C53#10435semidirect

Maximal subgroups

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

subgroupindex
#202796332
#202748722
#202747202
#202747162
C2×#168103522
#202725992
C2×C53×Q8⋊C82
#1043553