Finite Group Databasebeta

← #27412665

Subgroups — #27412665

Order 13568.

1002 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810671C211yesmaximal, characteristic, complemented
2#20283279C211yesmaximal, characteristic, complemented
3#20283232C211yesmaximal, characteristic
4C2×#16810622C211yesmaximal, characteristic
5C2×#16810605C211yesmaximal, characteristic
6#20276079C214yesmaximal, complemented
7#20276072C214yesmaximal, complemented
8C2×#16810392C211yesmaximal, characteristic, complemented
9C2×C53×#428C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14#16810671—24no—
15C2×#13133152C2214yes—
16#16810650—22no—
17#16810650C2214yes—
18#16810647—22no—
19#16810631C2212yes—
20#16810627C2212yes—
21#16810622—22no—
22#16810622—22no—
23#16810622—21no—
24#16810622—21no—
25#16810617—22no—
26C2×#13133140C2212yes—
27C2×#13133137C2212yes—
28#16810605—24no—
29C2×#13133135C2212yes—
30C2×#13133133C2212yes—
31#16810392C2214yes—
32C22×M16×C53C2211yescharacteristic
33C2×#117×C53C2211yescharacteristic
34C22×C4⋊C4×C53C2211yescharacteristic
35C53×#428—22no—
36C53×#428C2214yes—
37C2×#107×C53—21no—
38C2×#106×C53—21no—
39C2×#11719945—22no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945—21no—
44C2×#11719945—22no—
45C2×#11719945—22no—
46C2×#11719945Q814yes—
47C22×Dic106C2311yescharacteristic
48C22×D212C2311yescharacteristic
49C22×C22⋊D53—21no—
50C22×C4×D53—21no—
51C2×#11719917—22no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917—22no—
55C2×#11719917—22no—
56C2×#11719917—22no—
57C2×#11719917—22no—
58C2×#11719917—22no—
59C2×#11719917D412yes—
60C2×#11719917D412yes—
61C2×#11719917C2312yes—
62C2×#11719917C2312yes—
63C22×#88322—21no—
64C22×#88322—21no—
65C22×#88322C2312yes—
66#13133152—28no—
67C4⋊#88322—28no—
68#13133140—22no—
69#13133140—22no—
70#13133140—24no—
71#13133137—22no—
72#13133137—22no—
73#13133137—24no—
74#13133135—24no—
75#13133133—24no—
76C23×C4×C53—21no—
77C23×C4×C53C2311yescharacteristic
78C2×M16×C53—21no—
79C2×M16×C53—22no—
80C2×M16×C53D414yes—
81C2×M16×C53C2312yes—
82C22×C8×C53—21no—
83#117×C53—21no—
84#117×C53—21no—
85#117×C53D412yes—
86#117×C53C2312yes—
87C2×C4⋊C4×C53—21no—
88C2×C4⋊C4×C53—21no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—21no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53D411yescharacteristic
93C2×C4⋊C4×C53D411yescharacteristic
94C2×C4⋊C4×C53C2312yes—
95C2×C22⋊C4×C53—21no—
96C2×C42×C53—21no—
97#107×C53—24no—
98#106×C53—24no—
99C24×C53#4111yescharacteristic
100C23×D53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—22no—
104C22×C4×C53—41no—
105C22×C4×C53—41no—
106C22×C4×C53—82no—
107C22×C4×C53C2×D412yes—
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×Q812yes—
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53#4112yes—
115C22×C4×C53#4112yes—
116C22×C4×C53C2411yescharacteristic
117C2×C22⋊D53—24no—
118C2×C22⋊D53—41no—
119C2×C22⋊D53—41no—
120C2×C22⋊D53—42no—
121C22×Dic53—21no—
122#11719945—24no—
123#11719945—24no—
124#11719945—24no—
125#11719945—24no—
126#11719945—42no—
127#11719945—42no—
128#11719945—42no—
129#11719945—42no—
130C2×D212—21no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212—21no—
135C2×D212C2×Q812yes—
136C2×C4×D53—24no—
137C2×C4×D53—41no—
138C2×C4×D53—41no—
139C2×C4×D53—42no—
140C2×Dic106—21no—
141C2×Dic106—21no—
142C2×Dic106—21no—
143C2×Dic106—21no—
144C2×Dic106—21no—
145C2×Dic106C2×Q812yes—
146M16×C53—24no—
147M16×C53—22no—
148M16×C53C2×D414yes—
149C2×C8×C53—24no—
150C2×C8×C53—22no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—41no—
156C4⋊C4×C53—41no—
157C4⋊C4×C53C2×D412yes—
158C4⋊C4×C53C2×D412yes—
159C22⋊C4×C53—22no—
160C22⋊C4×C53—22no—
161C42×C53—22no—
162C42×C53—22no—
163#11719917—24no—
164#11719917—24no—
165#11719917—22no—
166#11719917—22no—
167#11719917—22no—
168#11719917—22no—
169#11719917—42no—
170#11719917—42no—
171C2×#88322—22no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322—22no—
175C2×#88322—42no—
176C2×#88322—42no—
177C2×#88322C2×D414yes—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53#12111yescharacteristic
181C23×C53#12111yescharacteristic
182C23×C53C22⋊Q812yes—
183C23×C53#12612yes—
184C23×C53C2×#4111yescharacteristic
185C22×D53—22no—
186C22×D53—41no—
187C22×D53—41no—
188C22×D53—81no—
189C22×D53—81no—
190C2×C4×C53—22no—
191C2×C4×C53—22no—
192C2×C4×C53—42no—
193C2×C4×C53—81no—
194C2×C4×C53—81no—
195C2×C4×C53—82no—
196C2×C4×C53—82no—
197C2×C4×C53—82no—
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53#12112yes—
201C2×C4×C53#12112yes—
202C2×C4×C53#12112yes—
203C2×C4×C53#12112yes—
204C2×C4×C53C22⋊Q812yes—
205C2×C4×C53C22⋊Q812yes—
206C2×C4×C53C22⋊Q812yes—
207C2×C4×C53C22⋊Q812yes—
208C2×C4×C53C22⋊Q812yes—
209C2×C4×C53#12612yes—
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C2×C4×C53C22×D411yescharacteristic
213C2×C4×C53C22×Q811yescharacteristic
214C2×C4×C53C2×#4111yescharacteristic
215C2×C4×C53C2×#4111yescharacteristic
216C22⋊D53—44no—
217C22⋊D53—44no—
218C2×Dic53—22no—
219C2×Dic53—41no—
220C2×Dic53—41no—
221D212—22no—
222D212—22no—
223D212—41no—
224D212—41no—
225C4×D53—44no—
226C4×D53—44no—
227Dic106—22no—
228Dic106—22no—
229Dic106—41no—
230Dic106—41no—
231C8×C53—24no—
232#88322—24no—
233#88322—42no—
234#88322—42no—
235#23604—531nomaximal, Sylow
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—42no—
243C22×C53—42no—
244C22×C53#39612yes—
245C22×C53#39612yes—
246C22×C53C2×#12111yescharacteristic
247C22×C53C2×C22⋊Q811yescharacteristic
248C22×C53C2×#12611yescharacteristic
249C2×D53—42no—
250C2×D53—82no—
251C2×D53—81no—
252C2×D53—81no—
253C2×D53—82no—
254C4×C53—42no—
255C4×C53—42no—
256C4×C53—82no—
257C4×C53#39612yes—
258C4×C53#39612yes—
259C4×C53C2×#12111yescharacteristic
260C4×C53C2×#12111yescharacteristic
261C4×C53C2×C22⋊Q811yescharacteristic
262C4×C53C2×C22⋊Q811yescharacteristic
263Dic53—42no—
264C2×#430—531no—
265C2×#428D5311yescharacteristic, complemented
266#2905—531no—
267#2896—531no—
268C2×#417—531no—
269C2×#413—531no—
270#2064—534no—
271#2063—534no—
272C2×#343—531no—
273C2×C53—21no—
274C2×C53—21no—
275C2×C53—22no—
276C2×C53—22no—
277C2×C53C2×#39611yescharacteristic
278C2×C53#306811yescharacteristic
279C2×C53#306911yescharacteristic
280D53—82no—
281C22×#41—531no—
282C22×M16—531no—
283C22×M16—531no—
284C22×M16—531no—
285C22×M16C2×D5311yescharacteristic
286C2×#117C2×D5311yescharacteristic
287C22×C4⋊C4C2×D5311yescharacteristic
288#430—1064no—
289C2×#108—534no—
290#428—22no—
291#428C2×D5314yes—
292C2×#107—21no—
293C2×#106—21no—
294#424—534no—
295#424—1062no—
296#423—1062no—
297#419—532no—
298#418—532no—
299#417—1062no—
300#417—1062no—
301#417—1061no—
302#417—1061no—
303#416—1062no—
304C2×#103—532no—
305C2×#102—532no—
306#413—1064no—
307C2×#101—532no—
308C2×#100—532no—
309#343—534no—
310C53#2360411yescharacteristic, complemented, Sylow
311C2×#41—534no—
312C2×#41—1062no—
313C2×#41—1062no—
314C2×#41—1061no—
315C2×#41—1061no—
316C2×#41—1062no—
317C2×#41—1061no—
318C2×#41—1061no—
319C22×Q8—531no—
320C22×D4—531no—
321C22×D4—1061no—
322C23×C4—21no—
323C23×C4—1061no—
324C23×C4C22×D5311yescharacteristic
325C2×M16—22no—
326C2×M16—21no—
327C2×M16—532no—
328C2×M16—532no—
329C2×M16—532no—
330C2×M16—532no—
331C2×M16—1062no—
332C2×M16—1062no—
333C2×M16—1061no—
334C2×M16—1061no—
335C2×M16—1062no—
336C2×M16—1062no—
337C2×M16—1062no—
338C2×M16—1062no—
339C2×M16D21214yes—
340C2×M16C22×D5312yes—
341C22×C8—21no—
342C22×C8—532no—
343C22×C8—1061no—
344C22×C8—1061no—
345#117—21no—
346#117—21no—
347#117C22⋊D5312yes—
348#117C22×D5312yes—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4C22⋊D5311yescharacteristic
355C2×C4⋊C4C22⋊D5311yescharacteristic
356C2×C4⋊C4C22×D5312yes—
357C2×C22⋊C4—21no—
358C2×C42—21no—
359#108—1068no—
360#107—24no—
361#106—24no—
362C4⋊C8—1068no—
363#103—1064no—
364#103—1062no—
365#103—1062no—
366#102—1064no—
367#102—1062no—
368#102—1062no—
369#101—1064no—
370#100—1064no—
371C24—1061no—
372C24#1171994711yescharacteristic
373#41—1064no—
374#41—1064no—
375#41—1064no—
376#41—1064no—
377#41—2122no—
378#41—2122no—
379#41—2122no—
380#41—2122no—
381C2×Q8—532no—
382C2×Q8—1061no—
383C2×Q8—1061no—
384C2×Q8—1061no—
385C2×Q8—1061no—
386C2×Q8—1061no—
387C2×D4—532no—
388C2×D4—1064no—
389C2×D4—1061no—
390C2×D4—1061no—
391C2×D4—1061no—
392C2×D4—1061no—
393C2×D4—1061no—
394C2×D4—2121no—
395C2×D4—2121no—
396C2×D4—2122no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—82no—
403C22×C4—1061no—
404C22×C4—1064no—
405C22×C4—2121no—
406C22×C4—2121no—
407C22×C4—2122no—
408C22×C4C2×D21212yes—
409C22×C4#1171994512yes—
410C22×C4D4×D5311yescharacteristic
411C22×C4D4×D5311yescharacteristic
412C22×C4#1171994711yescharacteristic
413C22×C4Q8×D5312yes—
414C22×C4#1171994912yes—
415C22×C4C2×C22⋊D5311yescharacteristic
416C22×C4C2×C22⋊D5311yescharacteristic
417C22×C4C23×D5311yescharacteristic
418M16—24no—
419M16—22no—
420M16—1064no—
421M16—1064no—
422M16—1062no—
423M16—1062no—
424M16—1062no—
425M16—1062no—
426M16—2122no—
427M16—2122no—
428M16C2×D21214yes—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—534no—
432C2×C8—1062no—
433C2×C8—1062no—
434C2×C8—1062no—
435C2×C8—1062no—
436C2×C8—2122no—
437C2×C8—2122no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4C2×C22⋊D5312yes—
445C4⋊C4C2×C22⋊D5312yes—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C42—22no—
449C42—22no—
450C23—41no—
451C23—41no—
452C23—1062no—
453C23—4241no—
454C23—4241no—
455C23—2121no—
456C23—2121no—
457C23#1313318911yescharacteristic
458C23#1313318911yescharacteristic
459C23#1313319712yes—
460C23#1313321212yes—
461C23C2×#1171994711yescharacteristic
462Q8—1062no—
463Q8—1062no—
464Q8—2121no—
465Q8—2121no—
466D4—1062no—
467D4—1062no—
468D4—2124no—
469D4—2124no—
470D4—2121no—
471D4—2121no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—1062no—
481C2×C4—2121no—
482C2×C4—2121no—
483C2×C4—2124no—
484C2×C4—2124no—
485C2×C4#1313318911yescharacteristic
486C2×C4#1313318911yescharacteristic
487C2×C4#1313319112yes—
488C2×C4#1313319112yes—
489C2×C4#1313319212yes—
490C2×C4#1313319212yes—
491C2×C4#1313319312yes—
492C2×C4#1313319312yes—
493C2×C4#1313319512yes—
494C2×C4#1313319512yes—
495C2×C4#1313319712yes—
496C2×C4#1313321212yes—
497C2×C4C2×D4×D5311yescharacteristic
498C2×C4C22×C22⋊D5311yescharacteristic
499C2×C4C22×D21211yescharacteristic
500C2×C4C2×Q8×D5311yescharacteristic
501C2×C4C2×#1171994511yescharacteristic
502C2×C4C2×#1171994911yescharacteristic
503C8—24no—
504C8—1064no—
505C8—2122no—
506C8—2122no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—42no—
514C22—42no—
515C22—4242no—
516C22—4242no—
517C22—4241no—
518C22—4241no—
519C22—2122no—
520C22#1681054512yes—
521C22#1681054512yes—
522C22C2×#1313318911yescharacteristic
523C22C2×#1313319711yescharacteristic
524C22C2×#1313321211yescharacteristic
525C4—42no—
526C4—42no—
527C4—82no—
528C4—2122no—
529C4#1681054512yes—
530C4#1681054512yes—
531C4C2×#1313319111yescharacteristic
532C4C2×#1313319211yescharacteristic
533C4C2×#1313319311yescharacteristic
534C4C2×#1313319511yescharacteristic
535C2—21no—
536C2—21no—
537C2—22no—
538C2—22no—
539C2—4242no—
540C2C2×#1681054511yescharacteristic
541C2#2028444011yescharacteristic
542C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#428C2
socleC22×C53C2×#121
prime radicalC22×#11719945C22
hypercenterC2×#428D53
hypocenterC53#23604
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#16810671C2semidirect
2. #20283279C2semidirect
6. #20276079C2semidirect
7. #20276072C2semidirect
8. C2×#16810392C2semidirect
9. C2×C53×#428C2semidirect
265. C2×#428D53semidirect
310. C53#23604semidirect

Maximal subgroups

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

subgroupindex
C2×#168106712
#202832792
#202832322
C2×#168106222
C2×#168106052
#202760792
#202760722
C2×#168103922
C2×C53×#4282
#2360453