Finite Group Databasebeta

← #27412640

Subgroups — #27412640

Order 13568.

994 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133152C211yesmaximal, characteristic, complemented
2C2×#16810650C211yesmaximal, characteristic, complemented
3#20283232C211yesmaximal, characteristic
4C2×#16810617C211yesmaximal, characteristic
5C2×#16810605C211yesmaximal, characteristic
6#20276079C214yesmaximal, complemented
7#20276072C214yesmaximal, complemented
8#20274454C211yesmaximal, characteristic, complemented
9C53×#2911C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13C23×#88322C2211yescharacteristic
14C2×#13133152—24no—
15C2×#13133152C2214yes—
16#16810650—22no—
17#16810650C2214yes—
18C2×C4⋊#88322—22no—
19#16810631C2212yes—
20#16810627C2212yes—
21#16810622—22no—
22#16810617—22no—
23#16810617—22no—
24#16810617—22no—
25C2×#13133140C2212yes—
26C2×#13133137C2212yes—
27#16810605—24no—
28C2×#13133135C2212yes—
29C2×#13133133C2212yes—
30#16810392C2214yes—
31C22×M16×C53C2211yescharacteristic
32C2×#117×C53C2211yescharacteristic
33C2×#117×C53C2211yescharacteristic
34C53×#428—22no—
35C53×#428C2214yes—
36C53×#427—22no—
37C2×#11719945—22no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—22no—
43C2×#11719945—22no—
44C2×#11719945Q814yes—
45C22×Dic106C2311yescharacteristic
46C22×D212C2311yescharacteristic
47C22×C22⋊D53—21no—
48C22×C4×D53—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917—22no—
52C2×#11719917—22no—
53C2×#11719917—22no—
54C2×#11719917—22no—
55C2×#11719917C2312yes—
56C2×#11719917C2312yes—
57C22×#88322—22no—
58C22×#88322—21no—
59C22×#88322—21no—
60C22×#88322—22no—
61C22×#88322D412yes—
62C22×#88322D412yes—
63C22×#88322C2312yes—
64#13133152—28no—
65C4⋊#88322—28no—
66#13133140—24no—
67#13133140—24no—
68#13133137—24no—
69#13133137—24no—
70#13133135—24no—
71#13133133—24no—
72C23×C4×C53C2311yescharacteristic
73C2×M16×C53—21no—
74C2×M16×C53—22no—
75C2×M16×C53D414yes—
76C2×M16×C53C2312yes—
77C22×C8×C53—21no—
78#117×C53—21no—
79#117×C53—21no—
80#117×C53—21no—
81#117×C53—21no—
82#117×C53—21no—
83#117×C53—21no—
84#117×C53D412yes—
85#117×C53C2312yes—
86C2×C4⋊C4×C53D412yes—
87C2×C4⋊C4×C53C2312yes—
88C2×C22⋊C4×C53—21no—
89C2×C22⋊C4×C53—21no—
90C2×C42×C53—21no—
91C2×C42×C53—21no—
92#107×C53—24no—
93#106×C53—24no—
94C24×C53#4111yescharacteristic
95C23×D53—21no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53C2×D412yes—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×Q812yes—
104C22×C4×C53#4111yescharacteristic
105C22×C4×C53#4112yes—
106C22×C4×C53#4112yes—
107C22×C4×C53C2411yescharacteristic
108C2×C22⋊D53—24no—
109C2×C22⋊D53—41no—
110C2×C22⋊D53—41no—
111C2×C22⋊D53—42no—
112C22×Dic53—21no—
113#11719945—24no—
114#11719945—24no—
115#11719945—24no—
116#11719945—24no—
117#11719945—42no—
118#11719945—42no—
119#11719945—42no—
120#11719945—42no—
121C2×D212—21no—
122C2×D212—21no—
123C2×D212—21no—
124C2×D212—21no—
125C2×D212—21no—
126C2×D212C2×Q812yes—
127C2×C4×D53—24no—
128C2×C4×D53—41no—
129C2×C4×D53—41no—
130C2×C4×D53—42no—
131C2×Dic106—21no—
132C2×Dic106—21no—
133C2×Dic106—21no—
134C2×Dic106—21no—
135C2×Dic106—21no—
136C2×Dic106C2×Q812yes—
137M16×C53—24no—
138M16×C53—22no—
139M16×C53C2×D414yes—
140C2×C8×C53—24no—
141C2×C8×C53—22no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53C2×D414yes—
146C22⋊C4×C53—22no—
147C22⋊C4×C53—22no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C42×C53—22no—
151C42×C53—22no—
152C42×C53—41no—
153C42×C53—41no—
154#11719917—22no—
155#11719917—22no—
156#11719917—22no—
157#11719917—22no—
158#11719917—42no—
159#11719917—42no—
160C2×#88322—24no—
161C2×#88322—24no—
162C2×#88322—22no—
163C2×#88322—22no—
164C2×#88322—22no—
165C2×#88322—22no—
166C2×#88322—42no—
167C2×#88322—42no—
168C2×#88322C2×D414yes—
169C23×C53—42no—
170C23×C53#12111yescharacteristic
171C23×C53#12111yescharacteristic
172C23×C53C22⋊Q812yes—
173C23×C53#12612yes—
174C23×C53C2×#4111yescharacteristic
175C22×D53—22no—
176C22×D53—41no—
177C22×D53—41no—
178C22×D53—82no—
179C2×C4×C53—24no—
180C2×C4×C53—44no—
181C2×C4×C53—42no—
182C2×C4×C53—82no—
183C2×C4×C53#12111yescharacteristic
184C2×C4×C53#12111yescharacteristic
185C2×C4×C53#12112yes—
186C2×C4×C53#12112yes—
187C2×C4×C53#12112yes—
188C2×C4×C53#12112yes—
189C2×C4×C53C22⋊Q812yes—
190C2×C4×C53C22⋊Q812yes—
191C2×C4×C53C22⋊Q812yes—
192C2×C4×C53C22⋊Q812yes—
193C2×C4×C53C22⋊Q812yes—
194C2×C4×C53#12612yes—
195C2×C4×C53C22×D411yescharacteristic
196C2×C4×C53C22×D411yescharacteristic
197C2×C4×C53C22×D411yescharacteristic
198C2×C4×C53C22×Q811yescharacteristic
199C2×C4×C53C2×#4111yescharacteristic
200C2×C4×C53C2×#4111yescharacteristic
201C22⋊D53—44no—
202C22⋊D53—44no—
203C2×Dic53—22no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206D212—22no—
207D212—22no—
208D212—41no—
209D212—41no—
210C4×D53—44no—
211C4×D53—44no—
212Dic106—22no—
213Dic106—22no—
214Dic106—41no—
215Dic106—41no—
216C8×C53—24no—
217#88322—24no—
218#88322—42no—
219#88322—42no—
220#23601—531nomaximal, Sylow
221C22×C53—22no—
222C22×C53—22no—
223C22×C53—22no—
224C22×C53—42no—
225C22×C53—42no—
226C22×C53#39612yes—
227C22×C53#39612yes—
228C22×C53C2×#12111yescharacteristic
229C22×C53C2×C22⋊Q811yescharacteristic
230C22×C53C2×#12611yescharacteristic
231C2×D53—42no—
232C2×D53—82no—
233C2×D53—82no—
234C2×D53—82no—
235C4×C53—44no—
236C4×C53—82no—
237C4×C53#39612yes—
238C4×C53#39612yes—
239C4×C53C2×#12111yescharacteristic
240C4×C53C2×#12111yescharacteristic
241C4×C53C2×C22⋊Q811yescharacteristic
242C4×C53C2×C22⋊Q811yescharacteristic
243Dic53—42no—
244C22×#108—531no—
245#2911D5311yescharacteristic, complemented
246C2×#424—531no—
247#2896—531no—
248C2×#416—531no—
249C2×#413—531no—
250#2064—534no—
251#2063—534no—
252#1742—531no—
253C2×C53—21no—
254C2×C53—21no—
255C2×C53—22no—
256C2×C53—22no—
257C2×C53C2×#39611yescharacteristic
258C2×C53#306712yes—
259D53—82no—
260C22×#41—531no—
261C22×M16—531no—
262C22×M16—531no—
263C22×M16C2×D5311yescharacteristic
264C23×C8—531no—
265C2×#117C2×D5311yescharacteristic
266C2×#117C2×D5311yescharacteristic
267C2×#108—534no—
268C2×#108—1064no—
269#428—22no—
270#428C2×D5314yes—
271#427—22no—
272#424—534no—
273#424—1062no—
274C2×C4⋊C8—1062no—
275#419—532no—
276#418—532no—
277#417—1062no—
278#416—1062no—
279#416—1062no—
280#416—1062no—
281C2×#103—532no—
282C2×#102—532no—
283#413—1064no—
284C2×#101—532no—
285C2×#100—532no—
286#343—534no—
287C53#2360111yescharacteristic, complemented, Sylow
288C2×#41—534no—
289C2×#41—1062no—
290C2×#41—1062no—
291C2×#41—1061no—
292C2×#41—1061no—
293C2×#41—1062no—
294C2×#41—1061no—
295C2×#41—1061no—
296C22×Q8—531no—
297C22×D4—531no—
298C22×D4—1061no—
299C23×C4—1061no—
300C23×C4C22×D5311yescharacteristic
301C2×M16—22no—
302C2×M16—21no—
303C2×M16—532no—
304C2×M16—532no—
305C2×M16—1061no—
306C2×M16—1061no—
307C2×M16—1062no—
308C2×M16—1062no—
309C2×M16—1062no—
310C2×M16—1062no—
311C2×M16D21214yes—
312C2×M16C22×D5312yes—
313C22×C8—21no—
314C22×C8—532no—
315C22×C8—532no—
316C22×C8—532no—
317C22×C8—1062no—
318C22×C8—1062no—
319C22×C8—1061no—
320C22×C8—1061no—
321#117—21no—
322#117—21no—
323#117—21no—
324#117—21no—
325#117—21no—
326#117—21no—
327#117C22⋊D5312yes—
328#117C22×D5312yes—
329C2×C4⋊C4C22⋊D5312yes—
330C2×C4⋊C4C22×D5312yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C42—21no—
334C2×C42—21no—
335#108—1068no—
336#107—24no—
337#106—24no—
338C4⋊C8—1068no—
339#103—1064no—
340#103—1064no—
341#102—1064no—
342#102—1064no—
343#101—1064no—
344#100—1064no—
345C24—1061no—
346C24#1171994711yescharacteristic
347#41—1064no—
348#41—1064no—
349#41—1064no—
350#41—1064no—
351#41—2122no—
352#41—2122no—
353#41—2122no—
354#41—2122no—
355C2×Q8—532no—
356C2×Q8—1061no—
357C2×Q8—1061no—
358C2×Q8—1061no—
359C2×Q8—1061no—
360C2×Q8—1061no—
361C2×D4—532no—
362C2×D4—1064no—
363C2×D4—1061no—
364C2×D4—1061no—
365C2×D4—1061no—
366C2×D4—1061no—
367C2×D4—1061no—
368C2×D4—2121no—
369C2×D4—2121no—
370C2×D4—2122no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—1061no—
374C22×C4—1064no—
375C22×C4—2121no—
376C22×C4—2121no—
377C22×C4—2122no—
378C22×C4C2×D21212yes—
379C22×C4#1171994512yes—
380C22×C4D4×D5311yescharacteristic
381C22×C4D4×D5311yescharacteristic
382C22×C4#1171994711yescharacteristic
383C22×C4Q8×D5312yes—
384C22×C4#1171994912yes—
385C22×C4C2×C22⋊D5311yescharacteristic
386C22×C4C2×C22⋊D5311yescharacteristic
387C22×C4C23×D5311yescharacteristic
388M16—24no—
389M16—22no—
390M16—1062no—
391M16—1062no—
392M16—1062no—
393M16—1062no—
394M16—2122no—
395M16—2122no—
396M16C2×D21214yes—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—534no—
400C2×C8—1064no—
401C2×C8—1064no—
402C2×C8—1062no—
403C2×C8—1062no—
404C2×C8—1062no—
405C2×C8—1062no—
406C2×C8—2122no—
407C2×C8—2122no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4C2×C22⋊D5314yes—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C42—22no—
417C42—22no—
418C42—41no—
419C42—41no—
420C23—42no—
421C23—1062no—
422C23—4242no—
423C23—2121no—
424C23—2121no—
425C23#1313318911yescharacteristic
426C23#1313318911yescharacteristic
427C23#1313319712yes—
428C23#1313321212yes—
429C23C2×#1171994711yescharacteristic
430Q8—1062no—
431Q8—1062no—
432Q8—2121no—
433Q8—2121no—
434D4—1062no—
435D4—1062no—
436D4—2124no—
437D4—2124no—
438D4—2121no—
439D4—2121no—
440C2×C4—24no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—82no—
444C2×C4—1062no—
445C2×C4—2121no—
446C2×C4—2121no—
447C2×C4—2124no—
448C2×C4—2124no—
449C2×C4#1313318911yescharacteristic
450C2×C4#1313318911yescharacteristic
451C2×C4#1313319112yes—
452C2×C4#1313319112yes—
453C2×C4#1313319212yes—
454C2×C4#1313319212yes—
455C2×C4#1313319312yes—
456C2×C4#1313319312yes—
457C2×C4#1313319512yes—
458C2×C4#1313319512yes—
459C2×C4#1313319712yes—
460C2×C4#1313321212yes—
461C2×C4C2×D4×D5311yescharacteristic
462C2×C4C22×C22⋊D5311yescharacteristic
463C2×C4C22×D21211yescharacteristic
464C2×C4C2×Q8×D5311yescharacteristic
465C2×C4C2×#1171994511yescharacteristic
466C2×C4C2×#1171994911yescharacteristic
467C8—24no—
468C8—1064no—
469C8—2122no—
470C8—2122no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—42no—
475C22—42no—
476C22—4242no—
477C22—4242no—
478C22—4242no—
479C22—2122no—
480C22#1681054512yes—
481C22#1681054512yes—
482C22C2×#1313318911yescharacteristic
483C22C2×#1313319711yescharacteristic
484C22C2×#1313321211yescharacteristic
485C4—44no—
486C4—82no—
487C4—2122no—
488C4#1681054512yes—
489C4#1681054512yes—
490C4C2×#1313319111yescharacteristic
491C4C2×#1313319211yescharacteristic
492C4C2×#1313319311yescharacteristic
493C4C2×#1313319511yescharacteristic
494C2—21no—
495C2—21no—
496C2—22no—
497C2—22no—
498C2—4242no—
499C2C2×#1681054511yescharacteristic
500C2#2028443212yes—
subgroupG/H
centerC2×C4#13133189
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2911C2
socleC22×C53C2×#121
prime radicalC22×#11719945C22
hypercenter#2911D53
hypocenterC53#23601
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2911

Complements

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

NH ≅ G/Ntype
1. C22×#13133152C2semidirect
2. C2×#16810650C2semidirect
6. #20276079C2semidirect
7. #20276072C2semidirect
8. #20274454C2semidirect
9. C53×#2911C2semidirect
245. #2911D53semidirect
287. C53#23601semidirect

Maximal subgroups

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

subgroupindex
C22×#131331522
C2×#168106502
#202832322
C2×#168106172
C2×#168106052
#202760792
#202760722
#202744542
C53×#29112
#2360153