Finite Group Databasebeta

← #323×Dic53 #27274596

Subgroups — #323×Dic53 #27274596

Order 13568.

946 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×Dic53C211yesmaximal, characteristic
2C22⋊C8×Dic53C212yesmaximal, complemented
3C2×#16810347C211yesmaximal, characteristic
4#20272557C212yesmaximal, complemented
5C2×C53×#323C211yesmaximal, characteristic
6C24×Dic53C411yescharacteristic
7C22×C22⋊Dic53C411yescharacteristic
8C22×C4×Dic53C411yescharacteristic
9C22×C4×Dic53C2211yescharacteristic
10C2×C22⋊#88322C2212yes—
11C2×C8×Dic53—22no—
12C22⋊C4×Dic53—22no—
13C22⋊C4×Dic53—22no—
14C2×#13133124C411yescharacteristic
15C2×#13133124C2211yescharacteristic
16#16810377—22no—
17#16810347C414yescomplemented
18C22×C22⋊C4×C53C2211yescharacteristic
19C2×C22⋊C8×C53C2212yes—
20C53×#323C414yescomplemented
21C23×Dic53—21no—
22C23×Dic53—21no—
23C23×Dic53—22no—
24C23×Dic53—42no—
25C23×Dic53—42no—
26C23×Dic53C814yescomplemented
27C23×Dic53C2×C411yescharacteristic
28C23×Dic53C2×C411yescharacteristic
29C23×Dic53C2×C411yescharacteristic
30C23×Dic53C2×C411yescharacteristic
31C22×#88322—22no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53C814yescomplemented
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—42no—
38C2×C4×Dic53—42no—
39C2×C4×Dic53D412yes—
40C2×C4×Dic53D412yes—
41C22⋊#88322—22no—
42C22⋊#88322C2×C414yescomplemented
43C8×Dic53—44no—
44#13133124—22no—
45#13133124—22no—
46C25×C53C2×C411yescharacteristic
47C23×C4×C53C2×C411yescharacteristic
48C23×C4×C53C2311yescharacteristic
49C22×C8×C53—22no—
50C2×C22⋊C4×C53—22no—
51C2×C22⋊C4×C53—22no—
52C2×C22⋊C4×C53C2×C412yes—
53C2×C22⋊C4×C53C2×C412yes—
54C22⋊C8×C53—22no—
55C22⋊C8×C53C2×C414yescomplemented
56C24×C53—21no—
57C24×C53—21no—
58C24×C53—21no—
59C24×C53—21no—
60C24×C53—41no—
61C24×C53—41no—
62C24×C53—41no—
63C24×C53—41no—
64C24×C53C4212yes—
65C24×C53C2×C812yes—
66C24×C53C2×C812yes—
67C24×C53C22×C411yescharacteristic
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53C4212yes—
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D412yes—
78C22×C4×C53#4112yes—
79C22×Dic53—22no—
80C22×Dic53—22no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—24no—
86C22×Dic53—24no—
87C22×Dic53—41no—
88C22×Dic53—41no—
89C22×Dic53—42no—
90C22×Dic53—42no—
91C22×Dic53—48no—
92C22×Dic53—48no—
93C22×Dic53—42no—
94C22×Dic53—42no—
95C22×Dic53—41no—
96C22×Dic53—41no—
97C22×Dic53—41no—
98C22×Dic53—41no—
99C22×Dic53C22⋊C412yes—
100C22×Dic53C22⋊C411yescharacteristic
101C22×Dic53C22⋊C411yescharacteristic
102C22×Dic53C2×C812yes—
103C22×Dic53C2×C812yes—
104C22×Dic53C2×C812yes—
105C22×Dic53C2×C812yes—
106C22×Dic53M1612yes—
107C22×Dic53M1612yes—
108C2×C8×C53—24no—
109C2×C8×C53—42no—
110C2×C8×C53—42no—
111C22⋊C4×C53—24no—
112C22⋊C4×C53—24no—
113C22⋊Dic53—28no—
114C4×Dic53—24no—
115C4×Dic53—42no—
116C4×Dic53—42no—
117C4×Dic53—42no—
118C2×#88322—24no—
119C2×#88322—42no—
120C2×#88322—42no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—22no—
126C23×C53—22no—
127C23×C53—22no—
128C23×C53—22no—
129C23×C53—22no—
130C23×C53—22no—
131C23×C53—42no—
132C23×C53—42no—
133C23×C53—42no—
134C23×C53—42no—
135C23×C53—42no—
136C23×C53—42no—
137C23×C53—42no—
138C23×C53—42no—
139C23×C53—42no—
140C23×C53—42no—
141C23×C53—42no—
142C23×C53—42no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53C4×C814yescomplemented
148C23×C53C2×C4211yescharacteristic
149C23×C53C2×C22⋊C411yescharacteristic
150C23×C53#11711yescharacteristic
151C23×C53C22×C811yescharacteristic
152C23×C53C22×C811yescharacteristic
153C23×C53C2×M1611yescharacteristic
154C23×C53C2×M1611yescharacteristic
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—22no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53C4×D412yes—
166C2×C4×C53C4×D412yes—
167C2×Dic53—21no—
168C2×Dic53—22no—
169C2×Dic53—21no—
170C2×Dic53—22no—
171C2×Dic53—24no—
172C2×Dic53—24no—
173C2×Dic53—22no—
174C2×Dic53—22no—
175C2×Dic53—42no—
176C2×Dic53—42no—
177C2×Dic53—48no—
178C2×Dic53—42no—
179C2×Dic53—42no—
180C2×Dic53—41no—
181C2×Dic53—41no—
182C2×Dic53—41no—
183C2×Dic53—41no—
184C2×Dic53—48no—
185C2×Dic53—42no—
186C2×Dic53—42no—
187C2×Dic53C22⋊C812yes—
188C2×Dic53C22⋊C814yes—
189C2×Dic53C22⋊C812yes—
190C2×Dic53#9912yes—
191C2×Dic53#10012yes—
192C8×C53—44no—
193#88322—44no—
194C4×#323—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—24no—
208C22×C53—44no—
209C22×C53—44no—
210C22×C53—44no—
211C22×C53—44no—
212C22×C53—42no—
213C22×C53—42no—
214C22×C53—42no—
215C22×C53—42no—
216C22×C53—42no—
217C22×C53—42no—
218C22×C53C4×C22⋊C411yescharacteristic
219C22×C53C2×C4×C811yescharacteristic
220C22×C53C4×M1611yescharacteristic
221C22×C53C2×C22⋊C811yescharacteristic
222C22×C53C2×C22⋊C811yescharacteristic
223C22×C53C2×#9911yescharacteristic
224C22×C53#41011yescharacteristic
225C22×C53C2×#10011yescharacteristic
226C22×C53#41311yescharacteristic
227C22×C53#43111yescharacteristic
228C22×C53#43111yescharacteristic
229C4×C53—22no—
230C4×C53—22no—
231C4×C53—42no—
232Dic53—22no—
233Dic53—24no—
234Dic53—42no—
235Dic53—42no—
236Dic53#32314yesdirect product factor
237C2×C4×C22⋊C4—531no—
238C4×C22⋊C8—532no—
239C2×#323—531no—
240C2×#323D5311yescharacteristic
241#1280—532no—
242C2×C53—21no—
243C2×C53—21no—
244C2×C53—21no—
245C2×C53—21no—
246C2×C53—44no—
247C2×C53C2×#32311yescharacteristic
248C2×C53#145811yescharacteristic
249C2×C53#145911yescharacteristic
250C2×C53#146011yescharacteristic
251C2×C53C4×C22⋊C811yescharacteristic
252C2×C53C4×#9911yescharacteristic
253C2×C53C4×#10011yescharacteristic
254C24×C4—531no—
255C22×C22⋊C4—531no—
256C22×C22⋊C4C2×D5311yescharacteristic
257C22×C42—531no—
258C22×C42—531no—
259C2×C22⋊C8—532no—
260C2×C22⋊C8C2×D5312yes—
261C2×C4×C8—1062no—
262C4×C22⋊C4—1062no—
263C4×C22⋊C4—1062no—
264C2×#95—531no—
265C2×#95—531no—
266#336—1062no—
267#323—534no—
268#323Dic5314yesdirect product factor
269C53C4×#32311yescharacteristic, complemented, Sylow
270C25C4×D5311yescharacteristic
271C23×C4—531no—
272C23×C4—531no—
273C23×C4—531no—
274C23×C4—531no—
275C23×C4—534no—
276C23×C4—1061no—
277C23×C4—1061no—
278C23×C4—1062no—
279C23×C4—2122no—
280C23×C4—2122no—
281C23×C4C4×D5311yescharacteristic
282C23×C4C22×D5311yescharacteristic
283C22×C8—22no—
284C22×C8—1062no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—534no—
288C2×C22⋊C4—1062no—
289C2×C22⋊C4—1062no—
290C2×C22⋊C4C4×D5312yes—
291C2×C22⋊C4C2×Dic5312yes—
292C2×C42—532no—
293C2×C42—532no—
294C2×C42—1061no—
295C2×C42—1061no—
296C2×C42—2122no—
297C2×C42—2122no—
298C22⋊C8—22no—
299C22⋊C8—534no—
300C22⋊C8—1062no—
301C22⋊C8C2×Dic5314yescomplemented
302C4×C8—2124no—
303#95—1062no—
304#95—1062no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24C8×D5312yes—
314C24C8×D5312yes—
315C24C4×Dic5312yes—
316C24C2×C4×D5311yescharacteristic
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—531no—
324C22×C4—532no—
325C22×C4—531no—
326C22×C4—532no—
327C22×C4—532no—
328C22×C4—532no—
329C22×C4—532no—
330C22×C4—532no—
331C22×C4—532no—
332C22×C4—1062no—
333C22×C4—1062no—
334C22×C4—1064no—
335C22×C4—1064no—
336C22×C4—1061no—
337C22×C4—1061no—
338C22×C4—1061no—
339C22×C4—1061no—
340C22×C4—2122no—
341C22×C4—2122no—
342C22×C4—2121no—
343C22×C4—2121no—
344C22×C4—2128no—
345C22×C4—2122no—
346C22×C4—2122no—
347C22×C4—2128no—
348C22×C4—2121no—
349C22×C4—2121no—
350C22×C4—2121no—
351C22×C4—2121no—
352C22×C4C4×Dic5312yes—
353C22×C4C2×C4×D5311yescharacteristic
354C22×C4D4×D5312yes—
355C22×C4#1171994712yes—
356C22×C4C22×Dic5311yescharacteristic
357C2×C8—24no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—1064no—
361C2×C8—2122no—
362C2×C8—2122no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—1068no—
366C42—1064no—
367C42—2122no—
368C42—2122no—
369C42—2122no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23C8×Dic5314yescomplemented
397C23C2×C4×Dic5311yescharacteristic
398C23C22⋊C4×D5311yescharacteristic
399C23#1313317311yescharacteristic
400C23C2×C8×D5311yescharacteristic
401C23C2×C8×D5311yescharacteristic
402C23M16×D5311yescharacteristic
403C23M16×D5311yescharacteristic
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—532no—
415C2×C4—532no—
416C2×C4—532no—
417C2×C4—534no—
418C2×C4—532no—
419C2×C4—1064no—
420C2×C4—1064no—
421C2×C4—1062no—
422C2×C4—1061no—
423C2×C4—1061no—
424C2×C4—1062no—
425C2×C4—1062no—
426C2×C4—1062no—
427C2×C4—2128no—
428C2×C4—2122no—
429C2×C4—2122no—
430C2×C4—2128no—
431C2×C4—2122no—
432C2×C4—2122no—
433C2×C4—2121no—
434C2×C4—2121no—
435C2×C4—2121no—
436C2×C4—2121no—
437C2×C4—2122no—
438C2×C4—2122no—
439C2×C4#1313317712yes—
440C2×C4D4×Dic5312yes—
441C8—44no—
442C8—2124no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—24no—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—44no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22C22⋊C4×Dic5311yescharacteristic
467C22C2×C8×Dic5311yescharacteristic
468C22M16×Dic5311yescharacteristic
469C22C22⋊C8×D5311yescharacteristic
470C22C22⋊C8×D5311yescharacteristic
471C22#99×D5311yescharacteristic
472C22#1681059511yescharacteristic
473C22#100×D5311yescharacteristic
474C22#1681060411yescharacteristic
475C22#1681067311yescharacteristic
476C22#1681067311yescharacteristic
477C4—22no—
478C4—22no—
479C4—42no—
480C4—534no—
481C4—1062no—
482C4—1064no—
483C4—2122no—
484C4—2122no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—44no—
490C2#323×D5311yescharacteristic
491C2#2027302611yescharacteristic
492C2#2027303011yescharacteristic
493C2#2027303411yescharacteristic
494C2C22⋊C8×Dic5311yescharacteristic
495C2#99×Dic5311yescharacteristic
496C2#100×Dic5311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C53C2×C4×C8
FrattiniC23×C4C22×D53
FittingC2×C53×#323C2
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC23×C4×C53C23
hypercenterC2×#323D53
hypocenterC53C4×#323
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. C22⋊C8×Dic53C2semidirect
4. #20272557C2semidirect
17. #16810347C4semidirect
20. C53×#323C4semidirect
26. C23×Dic53C8semidirect
34. C2×C22⋊Dic53C8semidirect
42. C22⋊#88322C2×C4semidirect
55. C22⋊C8×C53C2×C4semidirect
147. C23×C53C4×C8semidirect
236. Dic53#323direct (N × H)
268. #323Dic53direct (N × H)
269. C53C4×#323semidirect
301. C22⋊C8C2×Dic53semidirect
396. C23C8×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×Dic532
C22⋊C8×Dic532
C2×#168103472
#202725572
C2×C53×#3232
C4×#32353