Finite Group Databasebeta

← #27484913

Subgroups — #27484913

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133191C211yesmaximal, characteristic
2#20276206C212yesmaximal, complemented
3#20276100C212yesmaximal, complemented
4#20275283C212yesmaximal, complemented
5#20275032C211yesmaximal, characteristic
6#20275032C211yesmaximal, characteristic
7#20274961C212yesmaximal, complemented
8#20274903C211yesmaximal, characteristic, complemented
9#20274903C211yesmaximal, characteristic, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C2×C53×#389C211yesmaximal, characteristic, complemented
12C2×#13133191C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14C2×C4×C22⋊D53—21no—
15C2×#13133150C2212yes—
16C2×#13133150C2212yes—
17C2×C4⋊#88322C2212yes—
18C2×#13133137C2212yes—
19C2×#13133137C2212yes—
20C2×C22⋊#88322C2211yescharacteristic
21C2×C22⋊#88322C2211yescharacteristic
22#16810525C2211yescharacteristic
23#16810521C2211yescharacteristic
24#16810514—21no—
25#16810502—21no—
26#16810483C2211yescharacteristic
27C4×C4⋊Dic53C2211yescharacteristic
28C4×#11719921—21no—
29#16810385C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C22×C4⋊C4×C53—21no—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C22×C42×C53C2211yescharacteristic
38C53×#389—21no—
39C53×#389—21no—
40C53×#389C2212yes—
41C53×#389C2212yes—
42C2×#95×C53—21no—
43C22×D212C2311yescharacteristic
44C22×C22⋊D53—21no—
45C22×C4×D53—21no—
46C22×#88322C2312yes—
47C22×#88322C2312yes—
48#13133191—22no—
49#13133191Q814yes—
50C4×D212—42no—
51C4×C22⋊D53—44no—
52C2×C4⋊Dic53C2311yescharacteristic
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2311yescharacteristic
55C2×#11719919—21no—
56C2×C22⋊Dic53—21no—
57C2×#11719921—21no—
58C2×#11719921—21no—
59C2×#11719921—21no—
60C2×#11719921—21no—
61C2×C4×Dic53—21no—
62#13133150—24no—
63#13133150—42no—
64C4⋊#88322—42no—
65#13133137—24no—
66#13133137—42no—
67C22⋊#88322—22no—
68C22⋊#88322—22no—
69#13133124—21no—
70#13133124—21no—
71#13133124—21no—
72C23×C4×C53—21no—
73C23×C4×C53—21no—
74C23×C4×C53D411yescharacteristic
75C23×C4×C53D411yescharacteristic
76C23×C4×C53C2311yescharacteristic
77C2×C4⋊C4×C53—24no—
78C2×C4⋊C4×C53—21no—
79C2×C4⋊C4×C53—22no—
80C2×C4⋊C4×C53—22no—
81C2×C4⋊C4×C53—21no—
82C2×C4⋊C4×C53—42no—
83C2×C4⋊C4×C53—44no—
84C2×C4⋊C4×C53C2312yes—
85C2×C4⋊C4×C53C2312yes—
86C2×C42×C53—42no—
87C2×C42×C53D411yescharacteristic
88C2×C42×C53D411yescharacteristic
89C2×C42×C53C2311yescharacteristic
90C2×C42×C53C2311yescharacteristic
91#95×C53—22no—
92#95×C53—22no—
93C24×C53C2×D411yescharacteristic
94C23×D53—21no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—24no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53—41no—
105C22×C4×C53—42no—
106C22×C4×C53—42no—
107C22×C4×C53—42no—
108C22×C4×C53—42no—
109C22×C4×C53—42no—
110C22×C4×C53D814yes—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×Q812yes—
117C22×C4×C53#4112yes—
118C22×C4×C53C2411yescharacteristic
119C2×C22⋊D53—24no—
120C2×C22⋊D53—42no—
121C2×C22⋊D53—42no—
122C22×Dic53—21no—
123C22×Dic53—21no—
124C22×Dic53—21no—
125C2×D212—22no—
126C2×D212—22no—
127C2×D212—21no—
128C2×D212C2×Q812yes—
129C2×C4×D53—42no—
130C2×C4×D53—42no—
131C2×C4×D53—42no—
132C4⋊C4×C53—24no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—22no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—42no—
137C4⋊C4×C53—44no—
138C4⋊C4×C53—44no—
139C4⋊C4×C53#4114yes—
140C42×C53—22no—
141C42×C53—22no—
142C42×C53—41no—
143C42×C53—41no—
144C22⋊Dic53—42no—
145#11719921—24no—
146#11719921—42no—
147#11719921—42no—
148#11719921—42no—
149C4⋊Dic53—21no—
150C4⋊Dic53—22no—
151C4⋊Dic53—22no—
152C4⋊Dic53C2×Q812yes—
153#11719919—42no—
154C4×Dic53—42no—
155C2×#88322—22no—
156C2×#88322—22no—
157C2×#88322—22no—
158C2×#88322—22no—
159C2×#88322—22no—
160C2×#88322—22no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53C22⋊Q812yes—
166C23×C53C2×D812yes—
167C23×C53#13712yes—
168C23×C53C22×D411yescharacteristic
169C22×D53—22no—
170C22×D53—41no—
171C22×D53—41no—
172C22×D53—41no—
173C22×D53—41no—
174C22×D53—41no—
175C22×D53—41no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—24no—
183C2×C4×C53—24no—
184C2×C4×C53—24no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—44no—
192C2×C4×C53—48no—
193C2×C4×C53—44no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—84no—
200C2×C4×C53C22⋊Q812yes—
201C2×C4×C53C2×D812yes—
202C2×C4×C53C2×D812yes—
203C2×C4×C53C22×Q811yescharacteristic
204C2×C4×C53C2×#4112yes—
205C2×C4×C53C2×#4111yescharacteristic
206C2×C4×C53H(5,𝔽2)12yes—
207C22⋊D53—48no—
208C2×Dic53—22no—
209C2×Dic53—41no—
210C2×Dic53—41no—
211C2×Dic53—41no—
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53—41no—
216C2×Dic53—41no—
217D212—24no—
218D212—42no—
219C4×D53—84no—
220#88322—42no—
221#88322—42no—
222#29234—531nomaximal, Sylow
223C22×C53—22no—
224C22×C53—22no—
225C22×C53—22no—
226C22×C53—22no—
227C22×C53—22no—
228C22×C53—22no—
229C22×C53#47414yes—
230C22×C53C2×C22⋊Q811yescharacteristic
231C22×C53#52511yescharacteristic
232C22×C53#53411yescharacteristic
233C22×C53C22×D811yescharacteristic
234C22×C53C2×#13711yescharacteristic
235C22×C53#57511yescharacteristic
236C22×C53#57511yescharacteristic
237C2×D53—42no—
238C2×D53—42no—
239C2×D53—42no—
240C2×D53—81no—
241C2×D53—81no—
242C2×D53—81no—
243C2×D53—81no—
244C4×C53—22no—
245C4×C53—22no—
246C4×C53—22no—
247C4×C53—44no—
248C4×C53—41no—
249C4×C53—41no—
250C4×C53—82no—
251C4×C53#55312yes—
252C4×C53#55412yes—
253Dic53—42no—
254Dic53—81no—
255Dic53—81no—
256C4×#121—531no—
257C2×#389D5311yescharacteristic, complemented
258#2086—532no—
259#2067—532no—
260#1918—532no—
261#1874—531no—
262#1874—531no—
263#1862—532no—
264#1852—531no—
265#1852—531no—
266#1797—531no—
267C2×C53—24no—
268C2×C53#257911yescharacteristic
269C2×C53C2×#47411yescharacteristic
270C2×C53#308011yescharacteristic
271C2×C53#310011yescharacteristic
272C2×C53#311711yescharacteristic
273C2×C53#318611yescharacteristic
274C2×C53#319611yescharacteristic
275D53—82no—
276C2×#121—531no—
277C2×C4×D4—531no—
278C2×C4×D4—1061no—
279C22×C4⋊C4—21no—
280C22×C4⋊C4C2×D5311yescharacteristic
281C22×C4⋊C4C2×D5311yescharacteristic
282C22×C42C2×D5311yescharacteristic
283C2×#107—532no—
284C2×#107—532no—
285C2×C4⋊C8—532no—
286C2×#102—532no—
287C2×#102—532no—
288C2×C22⋊C8—531no—
289C2×C22⋊C8—531no—
290#390—531no—
291#389—21no—
292#389—21no—
293#389—531no—
294#389C2×D5312yes—
295#389C2×D5312yes—
296#388—1061no—
297#386—1061no—
298#380—531no—
299C4×C4⋊C4—531no—
300C4×C22⋊C4—1061no—
301C2×#95—21no—
302#340—532no—
303#340—532no—
304#340—532no—
305#340—532no—
306#336—532no—
307C53#2923411yescharacteristic, complemented, Sylow
308C22×D4—531no—
309C22×D4—1061no—
310C23×C4—21no—
311C23×C4—21no—
312C23×C4—1061no—
313C23×C4C22⋊D5311yescharacteristic
314C23×C4C22⋊D5311yescharacteristic
315C23×C4C22×D5311yescharacteristic
316C22×C8—532no—
317C22×C8—532no—
318#121—534no—
319#121—1062no—
320C4×D4—2122no—
321C4×D4—2124no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—531no—
330C2×C4⋊C4—531no—
331C2×C4⋊C4—531no—
332C2×C4⋊C4—1061no—
333C2×C4⋊C4C22×D5312yes—
334C2×C4⋊C4C22×D5312yes—
335C2×C22⋊C4—1061no—
336C2×C22⋊C4—1061no—
337C2×C22⋊C4—1061no—
338C2×C22⋊C4—1061no—
339C2×C22⋊C4—1061no—
340C2×C42—42no—
341C2×C42—1061no—
342C2×C42C22⋊D5311yescharacteristic
343C2×C42C22⋊D5311yescharacteristic
344C2×C42C22×D5311yescharacteristic
345C2×C42C22×D5311yescharacteristic
346#107—1064no—
347#107—2122no—
348C4⋊C8—2122no—
349#102—1064no—
350#102—2122no—
351C22⋊C8—1062no—
352C22⋊C8—1062no—
353#95—22no—
354#95—22no—
355#95—1061no—
356#95—1061no—
357#95—1061no—
358C24—1061no—
359C24C2×C22⋊D5311yescharacteristic
360C2×D4—532no—
361C2×D4—1064no—
362C2×D4—1062no—
363C2×D4—1061no—
364C2×D4—1062no—
365C2×D4—2122no—
366C2×D4—2122no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—1061no—
383C22×C4—1061no—
384C22×C4—1061no—
385C22×C4—2122no—
386C22×C4—2122no—
387C22×C4—2122no—
388C22×C4#1171992214yes—
389C22×C4Q8×D5312yes—
390C22×C4#1171994912yes—
391C22×C4C2×C22⋊D5311yescharacteristic
392C22×C4C2×C22⋊D5311yescharacteristic
393C22×C4C2×C22⋊D5311yescharacteristic
394C22×C4C2×C22⋊D5311yescharacteristic
395C22×C4C2×C22⋊D5311yescharacteristic
396C22×C4C23×D5311yescharacteristic
397C2×C8—1062no—
398C2×C8—1062no—
399C2×C8—1062no—
400C2×C8—1062no—
401C2×C8—1062no—
402C2×C8—1062no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—532no—
411C4⋊C4—1061no—
412C4⋊C4—1062no—
413C4⋊C4—1062no—
414C4⋊C4—2122no—
415C4⋊C4#1171994514yes—
416C22⋊C4—1064no—
417C22⋊C4—2122no—
418C22⋊C4—2122no—
419C22⋊C4—2122no—
420C22⋊C4—2122no—
421C42—22no—
422C42—22no—
423C42—41no—
424C42—41no—
425C42—2122no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—1062no—
431C23—2121no—
432C23—2121no—
433C23—2121no—
434C23—2121no—
435C23—2121no—
436C23—2121no—
437C23#1313319312yes—
438C23C2×#1171992212yes—
439C23#1313324812yes—
440C23C22×C22⋊D5311yescharacteristic
441D4—1064no—
442D4—2128no—
443D4—2122no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—44no—
460C2×C4—48no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—84no—
468C2×C4—1062no—
469C2×C4—4244no—
470C2×C4—2121no—
471C2×C4—2121no—
472C2×C4—2121no—
473C2×C4—2121no—
474C2×C4—2121no—
475C2×C4—2121no—
476C2×C4—2121no—
477C2×C4—2121no—
478C2×C4#1313319312yes—
479C2×C4C2×#1171992212yes—
480C2×C4C2×#1171992212yes—
481C2×C4C2×Q8×D5311yescharacteristic
482C2×C4C2×#1171994512yes—
483C2×C4C2×#1171994911yescharacteristic
484C2×C4#1313326212yes—
485C8—2122no—
486C8—2122no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—4241no—
494C22—4241no—
495C22—4241no—
496C22—4241no—
497C22—2122no—
498C22—2122no—
499C22—2122no—
500C22#1681090014yes—
501C22C2×#1313319311yescharacteristic
502C22#1681113411yescharacteristic
503C22#1681118211yescharacteristic
504C22C22×#1171992211yescharacteristic
505C22C2×#1313324811yescharacteristic
506C22#1681142111yescharacteristic
507C22#1681142111yescharacteristic
508C4—22no—
509C4—22no—
510C4—22no—
511C4—44no—
512C4—41no—
513C4—41no—
514C4—82no—
515C4—4241no—
516C4—4241no—
517C4—2122no—
518C4#1681132112yes—
519C4#1681132812yes—
520C2—24no—
521C2—4242no—
522C2#2028006211yescharacteristic
523C2C2×#1681090011yescharacteristic
524C2#2028453311yescharacteristic
525C2#2028471511yescharacteristic
526C2#2028488611yescharacteristic
527C2#2028546311yescharacteristic
528C2#2028554511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#389C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC2×#389D53
hypocenterC53#29234
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. #20276206C2semidirect
3. #20276100C2semidirect
4. #20275283C2semidirect
7. #20274961C2semidirect
8. #20274903C2semidirect
9. #20274903C2semidirect
10. #20274695C2semidirect
11. C2×C53×#389C2semidirect
257. C2×#389D53semidirect
307. C53#29234semidirect

Maximal subgroups

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

subgroupindex
C4×#131331912
#202762062
#202761002
#202752832
#202750322
#202749612
#202749032
#202746952
C2×C53×#3892
#2923453