Finite Group Databasebeta

← #27286422

Subgroups — #27286422

Order 13568.

644 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810904C211yesmaximal, characteristic
2C2×#16810833C211yesmaximal, characteristic, complemented
3#20279633C211yesmaximal, characteristic
4#20275676C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic
6#20272599C211yesmaximal, characteristic, complemented
7C2×C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C22×#11719925—21no—
9C2×#13133217C2211yescharacteristic
10C2×#13133193—21no—
11C2×C4×D212C2211yescharacteristic
12#16810904—22no—
13#16810904—22no—
14#16810833—22no—
15#16810833—22no—
16C2×#13133151—21no—
17C2×C4⋊#88322C2211yescharacteristic
18C2×#13133140—21no—
19C2×#13133139—21no—
20#16810545—22no—
21#16810525—21no—
22#16810521—21no—
23#16810493C2211yescharacteristic
24#16810480—21no—
25#16810386—22no—
26#16810352—22no—
27C2×C4×Q8×C53C2211yescharacteristic
28C2×C4⋊C8×C53C2211yescharacteristic
29C2×C4×C8×C53C2211yescharacteristic
30C53×Q8⋊C8—22no—
31C22×Dic106—21no—
32C22×D212D411yescharacteristic
33C22×C4×D53—21no—
34C2×#11719925—24no—
35C2×#11719925—42no—
36C2×#11719925—42no—
37C22×#88322—21no—
38#13133217—21no—
39#13133217—21no—
40#13133217—21no—
41#13133217D412yes—
42#13133193—42no—
43#13133193—42no—
44C4×D212—21no—
45C4×D212—21no—
46C4×D212—21no—
47C4×D212—21no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53D411yescharacteristic
53C2×C4⋊Dic53D412yes—
54C2×#11719919—22no—
55C2×#11719921—22no—
56C2×#11719921—22no—
57C2×#11719921—21no—
58#13133151—24no—
59C4⋊#88322—21no—
60C4⋊#88322D412yes—
61#13133140—24no—
62#13133139—24no—
63#13133124—22no—
64C22×Q8×C53D411yescharacteristic
65C22×C8×C53—21no—
66C22×C8×C53—21no—
67C4×Q8×C53—21no—
68C4×Q8×C53—21no—
69C4×Q8×C53—21no—
70C4×Q8×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53D411yescharacteristic
73C2×C42×C53—21no—
74C2×C42×C53C2311yescharacteristic
75C4⋊C8×C53—21no—
76C4⋊C8×C53D412yes—
77C4×C8×C53—21no—
78C4×C8×C53—21no—
79C23×D53—21no—
80C2×Q8×C53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C2×Q8×C53—21no—
84C2×Q8×C53—21no—
85C2×Q8×C53D812yes—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×Dic53—21no—
92C22×Dic53—22no—
93C22×Dic53—41no—
94C2×D212—21no—
95C2×D212—21no—
96C2×D212—21no—
97C2×D212—21no—
98C2×D212—21no—
99C2×D212Q1612yescomplemented
100C2×C4×D53—41no—
101C2×C4×D53—41no—
102C2×C4×D53—41no—
103C2×C4×D53—41no—
104C2×C4×D53—41no—
105C2×C4×D53—41no—
106C2×Dic106—22no—
107C2×Dic106—41no—
108C2×Dic106—41no—
109C2×Dic106—41no—
110C2×Dic106—41no—
111C2×Dic106—41no—
112C2×C8×C53—22no—
113C2×C8×C53—22no—
114C2×C8×C53—41no—
115C2×C8×C53—41no—
116C2×C8×C53—41no—
117C2×C8×C53—41no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53—41no—
122C42×C53—41no—
123C42×C53—41no—
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53#4111yescharacteristic
128#11719925—48no—
129#11719921—44no—
130#11719921—41no—
131#11719921—41no—
132#11719921—82no—
133C4⋊Dic53—22no—
134C4⋊Dic53—21no—
135C4⋊Dic53—22no—
136C4⋊Dic53—21no—
137C4⋊Dic53—41no—
138C4⋊Dic53—41no—
139C4⋊Dic53—41no—
140C4⋊Dic53—41no—
141C4⋊Dic53—41no—
142C4⋊Dic53—41no—
143C4⋊Dic53—42no—
144#11719919—44no—
145C2×#88322—22no—
146C2×#88322—41no—
147C2×#88322—41no—
148C23×C53#12011yescharacteristic
149C22×D53—22no—
150C22×D53—41no—
151C22×D53—41no—
152C22×D53—41no—
153C22×D53—41no—
154C22×D53—41no—
155C22×D53—41no—
156Q8×C53—22no—
157Q8×C53—22no—
158Q8×C53—41no—
159Q8×C53—41no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—22no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53#12011yescharacteristic
169C2×C4×C53#12011yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12711yescharacteristic
174C2×C4×C53C2×D811yescharacteristic
175C2×C4×C53C2×Q1611yescharacteristic
176C2×C4×C53#13511yescharacteristic
177C2×C4×C53#13511yescharacteristic
178C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C53#13711yescharacteristic
181C2×C4×C53#13711yescharacteristic
182C2×Dic53—42no—
183C2×Dic53—41no—
184C2×Dic53—41no—
185C2×Dic53—42no—
186C2×Dic53—41no—
187C2×Dic53—42no—
188C2×Dic53—42no—
189C2×Dic53—81no—
190C2×Dic53—81no—
191D212—22no—
192D212—22no—
193D212—41no—
194D212—41no—
195C4×D53—81no—
196C4×D53—81no—
197C4×D53—81no—
198C4×D53—81no—
199Dic106—42no—
200Dic106—42no—
201Dic106—81no—
202Dic106—81no—
203C8×C53—42no—
204C8×C53—42no—
205#88322—42no—
206#10233—531nomaximal, Sylow
207C22×C53#39211yescharacteristic
208C22×C53#44711yescharacteristic
209C22×C53#44911yescharacteristic
210C22×C53#45111yescharacteristic
211C22×C53#45211yescharacteristic
212C22×C53#45511yescharacteristic
213C22×C53#45611yescharacteristic
214C2×D53—42no—
215C2×D53—42no—
216C2×D53—42no—
217C2×D53—81no—
218C2×D53—81no—
219C2×D53—81no—
220C2×D53—81no—
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—42no—
224C4×C53—81no—
225C4×C53#44711yescharacteristic
226C4×C53#44911yescharacteristic
227C4×C53#45111yescharacteristic
228C4×C53#45211yescharacteristic
229C4×C53#46611yescharacteristic
230C4×C53#46711yescharacteristic
231C4×C53#46911yescharacteristic
232C4×C53#47011yescharacteristic
233Dic53—82no—
234Dic53—82no—
235Dic53—81no—
236C2×#474—531no—
237C2×#462—531no—
238#2538—531no—
239#1990—531no—
240C2×Q8⋊C8D5311yescharacteristic, complemented
241C2×D4⋊C8—531no—
242#1296—531no—
243C2×C53#164111yescharacteristic
244C2×C53#164411yescharacteristic
245C2×C53#164911yescharacteristic
246C2×C53#165211yescharacteristic
247C2×C53#199911yescharacteristic
248C2×C53#200111yescharacteristic
249C2×C53#200711yescharacteristic
250D53—82no—
251C22×Q16—1061no—
252C2×C4⋊Q8—531no—
253C2×C22⋊Q8—1061no—
254C2×C4×Q8C2×D5311yescharacteristic
255C2×C4×D4—531no—
256#474—1062no—
257#474—1062no—
258#462—1062no—
259#462—1062no—
260C2×#107—1061no—
261C2×C4⋊C8—531no—
262C2×C4⋊C8C2×D5311yescharacteristic
263C2×#103—1061no—
264C2×#102—1061no—
265C2×C4×C8C2×D5311yescharacteristic
266#396—1062no—
267#390—1061no—
268#389—1061no—
269#384—531no—
270#380—1061no—
271#340—1062no—
272Q8⋊C8—22no—
273D4⋊C8—1062no—
274C53#1023311yescharacteristic, complemented, Sylow
275C22×Q8—1061no—
276C22×Q8D21211yescharacteristic
277C22×D4—531no—
278C23×C4—1061no—
279C2×Q16—1064no—
280C2×Q16—2122no—
281C2×Q16—2122no—
282C22×C8—21no—
283C22×C8—21no—
284C22×C8—1061no—
285C4⋊Q8—532no—
286C4⋊Q8—1061no—
287C4⋊Q8—1061no—
288C4⋊Q8—1061no—
289C22⋊Q8—2122no—
290C22⋊Q8—2122no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×D4—1061no—
296C4×D4—1061no—
297C4×D4—1061no—
298C4×D4—1061no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—531no—
301C2×C4⋊C4—532no—
302C2×C4⋊C4—1061no—
303C2×C4⋊C4—1061no—
304C2×C4⋊C4—1062no—
305C2×C4⋊C4—1061no—
306C2×C4⋊C4—1061no—
307C2×C4⋊C4D21211yescharacteristic
308C2×C22⋊C4—1062no—
309C2×C22⋊C4—1062no—
310C2×C22⋊C4—1061no—
311C2×C42—21no—
312C2×C42C22×D5311yescharacteristic
313#107—1064no—
314C4⋊C8—21no—
315C4⋊C8—532no—
316C4⋊C8—1061no—
317C4⋊C8C22⋊D5312yes—
318#103—1064no—
319#102—1064no—
320C4×C8—21no—
321C4×C8—21no—
322#95—1062no—
323C24—1061no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—1062no—
330C2×Q8—2121no—
331C2×Q8—2121no—
332C2×Q8—2121no—
333C2×Q8—2121no—
334C2×Q8—2121no—
335C2×Q8D42412yes—
336C2×D4—532no—
337C2×D4—1061no—
338C2×D4—1061no—
339C2×D4—1061no—
340C2×D4—1061no—
341C2×D4—1061no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—1061no—
345C22×C4—1062no—
346C22×C4—2121no—
347C22×C4—2121no—
348C22×C4—2121no—
349C22×C4—2121no—
350C22×C4—2121no—
351C22×C4—2121no—
352C22×C4—2121no—
353C22×C4C2×D21211yescharacteristic
354C22×C4D4×D5311yescharacteristic
355C22×C4D4×D5311yescharacteristic
356Q16—2128no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—1062no—
364C2×C8—2121no—
365C2×C8—2121no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—1062no—
371C4⋊C4—1061no—
372C4⋊C4—1061no—
373C4⋊C4—1062no—
374C4⋊C4—2122no—
375C4⋊C4—2124no—
376C4⋊C4—2121no—
377C4⋊C4—2121no—
378C4⋊C4—2121no—
379C4⋊C4—2121no—
380C4⋊C4—2121no—
381C4⋊C4—2121no—
382C22⋊C4—4242no—
383C22⋊C4—2124no—
384C22⋊C4—2121no—
385C22⋊C4—2121no—
386C42—41no—
387C42—41no—
388C42#1171994511yescharacteristic
389C42D4×D5311yescharacteristic
390C42D4×D5311yescharacteristic
391C42C2×C22⋊D5311yescharacteristic
392C23—1062no—
393C23—2121no—
394C23—2121no—
395C23—2121no—
396C23—2121no—
397C23—2121no—
398C23—2121no—
399C23#1313318711yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—41no—
403Q8—41no—
404Q8—4241no—
405Q8—4241no—
406Q8—2122no—
407Q8—2122no—
408D4—1062no—
409D4—1062no—
410D4—2121no—
411D4—2121no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—4241no—
421C2×C4—4241no—
422C2×C4—4241no—
423C2×C4—4241no—
424C2×C4—4241no—
425C2×C4—4241no—
426C2×C4—2122no—
427C2×C4—2121no—
428C2×C4—2121no—
429C2×C4—2121no—
430C2×C4—2122no—
431C2×C4—2122no—
432C2×C4—2122no—
433C2×C4#1313318511yescharacteristic
434C2×C4#1313318711yescharacteristic
435C2×C4#1313319011yescharacteristic
436C2×C4#1313319011yescharacteristic
437C2×C4#1313319111yescharacteristic
438C2×C4#1313321311yescharacteristic
439C2×C4C2×D42411yescharacteristic
440C2×C4Q16×D5311yescharacteristic
441C2×C4#1313323811yescharacteristic
442C2×C4#1313323911yescharacteristic
443C2×C4#1313324511yescharacteristic
444C2×C4#1313324511yescharacteristic
445C2×C4#1313324611yescharacteristic
446C2×C4#1313324911yescharacteristic
447C8—42no—
448C8—42no—
449C8—2122no—
450C22—4241no—
451C22—4241no—
452C22—4241no—
453C22—4241no—
454C22—2122no—
455C22—2122no—
456C22—2122no—
457C22#1681053211yescharacteristic
458C22#1681075911yescharacteristic
459C22#1681077011yescharacteristic
460C22#1681078011yescharacteristic
461C22#1681078811yescharacteristic
462C22#1681080211yescharacteristic
463C22#1681080711yescharacteristic
464C4—21no—
465C4—21no—
466C4—42no—
467C4—81no—
468C4—4241no—
469C4—4242no—
470C4—4242no—
471C4#1681075911yescharacteristic
472C4#1681077011yescharacteristic
473C4#1681077711yescharacteristic
474C4#1681078211yescharacteristic
475C4#1681085911yescharacteristic
476C4#1681086311yescharacteristic
477C4#1681087611yescharacteristic
478C4#1681088011yescharacteristic
479C2—4242no—
480C2#2027391011yescharacteristic
481C2#2027393111yescharacteristic
482C2#2027396611yescharacteristic
483C2#2027398711yescharacteristic
484C2#2027571311yescharacteristic
485C2#2027572411yescharacteristic
486C2#2027576311yescharacteristic
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#10233
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. C2×#16810833C2semidirect
4. #20275676C2semidirect
6. #20272599C2semidirect
7. C2×C53×Q8⋊C8C2semidirect
99. C2×D212Q16semidirect
240. C2×Q8⋊C8D53semidirect
274. C53#10233semidirect

Maximal subgroups

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

subgroupindex
C2×#168109042
C2×#168108332
#202796332
#202756762
C2×#168103522
#202725992
C2×C53×Q8⋊C82
#1023353