Finite Group Databasebeta

← C2×#20290402 #28317959

Subgroups — C2×#20290402 #28317959

Order 13568.

2574 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D53⋊M16C211yesmaximal, characteristic
2C22×#13133296C211yesmaximal, characteristic
3#20290402C218yesmaximal, direct product factor
4C2×#16811496C214yesmaximal, complemented
5C22×C4⋊C4×D53C211yesmaximal, characteristic
6C22×#11719936—21no—
7C2×D53⋊M16—22no—
8C2×D53⋊M16—24no—
9C2×D53⋊M16C418yescomplemented
10C2×D53⋊M16C2214yes—
11C22×#11719940—21no—
12C22×D53⋊C8C2212yes—
13C2×#13133296—22no—
14C2×#13133296—24no—
15C2×#13133296C418yescomplemented
16C2×#13133296C2214yes—
17C22×C4⋊#6157C2212yes—
18C22×C22⋊#6157—21no—
19C22×C4×#6157—21no—
20#16811496C22116yescomplemented
21C23×C4×D53—21no—
22C23×C4×D53C2211yescharacteristic
23C22×C4⋊Dic53C411yescharacteristic, complemented
24C22×#11719919—21no—
25C2×C4⋊C4×D53—21no—
26C2×C4⋊C4×D53—22no—
27C2×C4⋊C4×D53—22no—
28C2×C4⋊C4×D53—21no—
29C2×C4⋊C4×D53C412yescomplemented
30C2×C4⋊C4×D53C414yescomplemented
31C2×C4⋊C4×D53C2212yes—
32C2×C4⋊C4×D53C2214yes—
33C22×C4⋊C4×C53C411yescharacteristic, complemented
34C23×#6157—22no—
35C2×#11719936—28no—
36C2×#11719936—24no—
37D53⋊M16—216no—
38D53⋊M16—28no—
39D53⋊M16C2×C4116yescomplemented
40C2×#11719940—28no—
41C2×#11719940—24no—
42C2×D53⋊C8—24no—
43C2×D53⋊C8—22no—
44C2×D53⋊C8C2×C418yescomplemented
45C2×D53⋊C8C2314yes—
46C22×#88324—22no—
47#13133296—216no—
48#13133296—28no—
49#13133296C2×C4116yescomplemented
50C2×C4⋊#6157—22no—
51C2×C4⋊#6157—24no—
52C2×C4⋊#6157C2×C418yescomplemented
53C2×C4⋊#6157C2314yes—
54C2×C22⋊#6157—28no—
55C2×C22⋊#6157—24no—
56C2×C4×#6157—28no—
57C2×C4×#6157—24no—
58C24×D53D411yescharacteristic
59C23×Dic53—21no—
60C23×Dic53D411yescharacteristic
61C22×C4×D53—24no—
62C22×C4×D53—22no—
63C22×C4×D53—22no—
64C22×C4×D53—24no—
65C22×C4×D53—48no—
66C22×C4×D53C2×C412yes—
67C22×C4×D53C2×C411yescharacteristic
68C22×C4×D53C2×C414yes—
69C22×C4×D53C2×C414yes—
70C22×C4×D53D411yescharacteristic
71C22×C4×D53D412yes—
72C22×C4×D53D412yes—
73C22×C4×D53D411yescharacteristic
74C22×C4×D53D414yes—
75C22×C4×D53Q814yes—
76C22×C4×D53C2311yescharacteristic
77C22×C4×D53C2312yes—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—22no—
80C2×C4⋊Dic53C2×C412yes—
81C2×C4⋊Dic53C2×C414yescomplemented
82C2×#11719919—22no—
83C2×#11719919—24no—
84C2×#11719919—22no—
85C2×#11719919—24no—
86C4⋊C4×D53—24no—
87C4⋊C4×D53—24no—
88C4⋊C4×D53—24no—
89C4⋊C4×D53—22no—
90C4⋊C4×D53—24no—
91C4⋊C4×D53—24no—
92C4⋊C4×D53—22no—
93C4⋊C4×D53C2×C414yes—
94C4⋊C4×D53C2×C418yescomplemented
95C4⋊C4×D53C2314yes—
96C23×C4×C53—21no—
97C23×C4×C53C2×C411yescharacteristic
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—22no—
100C2×C4⋊C4×C53C2×C412yes—
101C2×C4⋊C4×C53C2×C414yescomplemented
102C24×C53C22⋊C411yescharacteristic
103C23×D53—24no—
104C23×D53—24no—
105C23×D53C22⋊C411yescharacteristic
106C23×D53C22⋊C412yes—
107C23×D53C22⋊C414yes—
108C23×D53C4⋊C414yes—
109C23×D53C2×D412yes—
110C23×D53C2×D411yescharacteristic
111C22×#6157—28no—
112C22×#6157—24no—
113C22×#6157—44no—
114C22×#6157—44no—
115C22×C4×C53—22no—
116C22×C4×C53—24no—
117C22×C4×C53—44no—
118C22×C4×C53C4214yes—
119C22×C4×C53C22⋊C412yes—
120C22×C4×C53C22⋊C411yescharacteristic
121C22×C4×C53C4⋊C414yes—
122C22×C4×C53C22×C411yescharacteristic
123C22×C4×C53C22×C412yes—
124C22×Dic53—22no—
125C22×Dic53—24no—
126C22×Dic53—44no—
127C22×Dic53C22⋊C412yes—
128C22×Dic53C22⋊C411yescharacteristic
129C22×Dic53C22⋊C414yes—
130C22×Dic53C4⋊C414yes—
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D412yes—
133C2×C4×D53—24no—
134C2×C4×D53—28no—
135C2×C4×D53—24no—
136C2×C4×D53—48no—
137C2×C4×D53—416no—
138C2×C4×D53—416no—
139C2×C4×D53—48no—
140C2×C4×D53C4214yes—
141C2×C4×D53C4218yes—
142C2×C4×D53C22⋊C412yes—
143C2×C4×D53C22⋊C412yes—
144C2×C4×D53C22⋊C412yes—
145C2×C4×D53C22⋊C414yes—
146C2×C4×D53C22⋊C418yes—
147C2×C4×D53C22⋊C418yes—
148C2×C4×D53C22⋊C414yes—
149C2×C4×D53C4⋊C418yes—
150C2×C4×D53C4⋊C414yes—
151C2×C4×D53C4⋊C414yes—
152C2×C4×D53C4⋊C418yes—
153C2×C4×D53C4⋊C418yes—
154C2×C4×D53C4⋊C414yes—
155C2×C4×D53C22×C411yescharacteristic
156C2×C4×D53C22×C412yes—
157C2×C4×D53C22×C412yes—
158C2×C4×D53C22×C414yes—
159C2×C4×D53C22×C412yes—
160C2×C4×D53C22×C414yes—
161C2×C4×D53C2×D412yes—
162C2×C4×D53C2×D411yescharacteristic
163C2×C4×D53C2×D411yescharacteristic
164C2×C4×D53C2×D412yes—
165C2×C4×D53C2×D412yes—
166C2×C4×D53C2×D414yes—
167C2×C4×D53C2×Q814yes—
168C2×C4×D53C2×Q812yes—
169C2×C4×D53C2411yescharacteristic
170C22⋊#6157—216no—
171#11719940—216no—
172C2×#88324—28no—
173C2×#88324—24no—
174C4⋊#6157—28no—
175C4⋊#6157—24no—
176C4⋊#6157C22×C418yescomplemented
177C4×#6157—216no—
178#11719936—216no—
179D53⋊C8—28no—
180D53⋊C8—24no—
181D53⋊C8C22×C418yescomplemented
182C4⋊C4×C53—22no—
183C4⋊C4×C53—24no—
184C4⋊C4×C53C22×C414yes—
185C4⋊Dic53—22no—
186C4⋊Dic53—24no—
187C4⋊Dic53C22×C414yes—
188#11719919—24no—
189#11719919—28no—
190#11719919—24no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53#9514yes—
194C23×C53C2×C22⋊C412yes—
195C23×C53C2×C22⋊C411yescharacteristic
196C22×D53—22no—
197C22×D53—22no—
198C22×D53—24no—
199C22×D53—24no—
200C22×D53—216no—
201C22×D53—24no—
202C22×D53—24no—
203C22×D53—216no—
204C22×D53#9514yes—
205C22×D53#9518yes—
206C22×D53C2×C22⋊C414yes—
207C22×D53C2×C22⋊C412yes—
208C22×D53C2×C22⋊C412yes—
209C22×D53C2×C22⋊C411yescharacteristic
210C22×D53C2×C4⋊C414yes—
211C22×D53C2×C4⋊C412yes—
212C22×D53AGL(1,ℤ/8)14yes—
213C22×D53#13714yes—
214C22×D53C22×D411yescharacteristic
215C2×#6157—28no—
216C2×#6157—44no—
217C2×#6157—44no—
218C2×#6157—416no—
219C2×C4×C53—24no—
220C2×C4×C53—48no—
221C2×C4×C53—44no—
222C2×C4×C53#9518yes—
223C2×C4×C53#9514yes—
224C2×C4×C53C2×C4214yes—
225C2×C4×C53C2×C4212yes—
226C2×C4×C53C2×C22⋊C412yes—
227C2×C4×C53C2×C22⋊C411yescharacteristic
228C2×C4×C53C2×C4⋊C414yes—
229C2×C4×C53C2×C4⋊C412yes—
230C2×C4×C53C23×C411yescharacteristic
231C2×Dic53—24no—
232C2×Dic53—48no—
233C2×Dic53—44no—
234C2×Dic53#9518yes—
235C2×Dic53#9514yes—
236C2×Dic53C2×C22⋊C412yes—
237C2×Dic53C2×C22⋊C414yes—
238C2×Dic53C2×C22⋊C411yescharacteristic
239C2×Dic53C2×C22⋊C412yes—
240C2×Dic53C2×C4⋊C414yes—
241C2×Dic53C2×C4⋊C412yes—
242C2×Dic53C22×D411yescharacteristic
243C4×D53—416no—
244C4×D53—48no—
245C4×D53—48no—
246C4×D53#9518yes—
247C4×D53#9518yes—
248C4×D53#9518yes—
249C4×D53C2×C4214yes—
250C4×D53C2×C4212yes—
251C4×D53C2×C22⋊C414yes—
252C4×D53C2×C22⋊C412yes—
253C4×D53C2×C22⋊C414yes—
254C4×D53C2×C22⋊C412yes—
255C4×D53C2×C4⋊C414yes—
256C4×D53C2×C4⋊C412yes—
257C4×D53C2×C4⋊C414yes—
258C4×D53C2×C4⋊C412yes—
259C4×D53C2×C4⋊C412yes—
260C4×D53C2×C4⋊C414yes—
261C4×D53C23×C411yescharacteristic
262C4×D53C23×C411yescharacteristic
263C4×D53C22×D411yescharacteristic
264C4×D53C22×Q811yescharacteristic
265#88324—28no—
266C2×#1736—531nomaximal, Sylow
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—28no—
270C22×C53C2×#9514yes—
271C22×C53C2×#9512yes—
272C22×C53#41812yes—
273C22×C53#41912yes—
274C22×C53C22×C22⋊C411yescharacteristic
275C2×D53—22no—
276C2×D53—22no—
277C2×D53—28no—
278C2×D53—28no—
279C2×D53—28no—
280C2×D53—216no—
281C2×D53C2×#9514yes—
282C2×D53C2×#9512yes—
283C2×D53#41718yes—
284C2×D53#41812yes—
285C2×D53#41912yes—
286C2×D53#42818yes—
287C2×D53C22×C22⋊C411yescharacteristic
288C2×D53C22×C4⋊C411yescharacteristic
289C2×D53C2×AGL(1,ℤ/8)12yes—
290C2×D53C2×#13712yes—
291#6157—48no—
292C4×C53—44no—
293C4×C53C2×#9514yes—
294C4×C53C2×#9512yes—
295C4×C53C22×C4211yescharacteristic
296C4×C53C22×C4⋊C411yescharacteristic
297Dic53—44no—
298Dic53C2×#9514yes—
299Dic53C2×#9512yes—
300Dic53C22×C22⋊C411yescharacteristic
301Dic53C22×C4⋊C411yescharacteristic
302C23×M16—531no—
303C22×#117—531no—
304C23×C4⋊C4—531no—
305#1736—538no—
306C2×#340—534no—
307C2×C53—24no—
308C2×C53#173614yescomplemented
309C2×C53C22×#9511yescharacteristic
310C2×C53C2×#41811yescharacteristic
311C2×C53C2×#41911yescharacteristic
312D53—24no—
313D53#173614yescomplemented
314D53C2×#41712yes—
315D53C2×#42812yes—
316C24×C4—531no—
317C24×C4—1061no—
318C22×M16—538no—
319C22×M16—534no—
320C22×M16—1061no—
321C22×M16—1064no—
322C22×M16—1062no—
323C22×M16—1061no—
324C23×C8—532no—
325C2×#117—538no—
326C2×#117—534no—
327C2×#117—1064no—
328C2×#117—1062no—
329C22×C4⋊C4—532no—
330C22×C4⋊C4—531no—
331C22×C4⋊C4—532no—
332C22×C4⋊C4—532no—
333C22×C4⋊C4—534no—
334C22×C4⋊C4—534no—
335C22×C4⋊C4—1062no—
336C22×C4⋊C4—1062no—
337C22×C4⋊C4—1061no—
338C22×C4⋊C4—1061no—
339C22×C4⋊C4—1061no—
340C22×C4⋊C4#615711yescharacteristic, complemented
341C22×C22⋊C4—1061no—
342C22×C42—1061no—
343#340—5316no—
344C53C2×#173611yescharacteristic, complemented, Sylow
345C25—531no—
346C23×C4—21no—
347C23×C4—531no—
348C23×C4—534no—
349C23×C4—534no—
350C23×C4—534no—
351C23×C4—534no—
352C23×C4—532no—
353C23×C4—532no—
354C23×C4—531no—
355C23×C4—531no—
356C23×C4—531no—
357C23×C4—532no—
358C23×C4—532no—
359C23×C4—531no—
360C23×C4—1064no—
361C23×C4—1064no—
362C23×C4—1062no—
363C23×C4—1062no—
364C23×C4—1062no—
365C23×C4—1061no—
366C23×C4—2128no—
367C23×C4C2×#615711yescharacteristic
368C2×M16—5316no—
369C2×M16—1068no—
370C2×M16—1064no—
371C2×M16—10616no—
372C2×M16—1068no—
373C2×M16—1068no—
374C2×M16—1064no—
375C22×C8—534no—
376C22×C8—538no—
377C22×C8—1062no—
378C22×C8—1062no—
379C22×C8—1064no—
380#117—5316no—
381#117—10616no—
382#117—1068no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—534no—
386C2×C4⋊C4—538no—
387C2×C4⋊C4—532no—
388C2×C4⋊C4—534no—
389C2×C4⋊C4—534no—
390C2×C4⋊C4—534no—
391C2×C4⋊C4—538no—
392C2×C4⋊C4—1064no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1064no—
396C2×C4⋊C4—1064no—
397C2×C4⋊C4—1064no—
398C2×C4⋊C4—1064no—
399C2×C4⋊C4—1064no—
400C2×C4⋊C4—1064no—
401C2×C4⋊C4—1062no—
402C2×C4⋊C4—1062no—
403C2×C4⋊C4—1062no—
404C2×C4⋊C4—1064no—
405C2×C4⋊C4—1062no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4C2×#615712yes—
408C2×C4⋊C4C2×#615714yescomplemented
409C2×C22⋊C4—1068no—
410C2×C22⋊C4—1064no—
411C2×C42—1068no—
412C2×C42—1064no—
413C24—531no—
414C24—534no—
415C24—534no—
416C24—532no—
417C24—532no—
418C24—531no—
419C24—1064no—
420C24—1064no—
421C24C22⋊#615711yescharacteristic
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—44no—
425C22×C4—534no—
426C22×C4—534no—
427C22×C4—532no—
428C22×C4—532no—
429C22×C4—531no—
430C22×C4—531no—
431C22×C4—534no—
432C22×C4—538no—
433C22×C4—534no—
434C22×C4—538no—
435C22×C4—534no—
436C22×C4—538no—
437C22×C4—538no—
438C22×C4—534no—
439C22×C4—538no—
440C22×C4—534no—
441C22×C4—532no—
442C22×C4—534no—
443C22×C4—532no—
444C22×C4—534no—
445C22×C4—532no—
446C22×C4—532no—
447C22×C4—532no—
448C22×C4—532no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—532no—
452C22×C4—538no—
453C22×C4—534no—
454C22×C4—532no—
455C22×C4—534no—
456C22×C4—532no—
457C22×C4—534no—
458C22×C4—531no—
459C22×C4—532no—
460C22×C4—531no—
461C22×C4—1064no—
462C22×C4—1064no—
463C22×C4—1068no—
464C22×C4—1068no—
465C22×C4—1064no—
466C22×C4—1064no—
467C22×C4—1062no—
468C22×C4—2128no—
469C22×C4—21216no—
470C22×C4—21216no—
471C22×C4—2128no—
472C22×C4—2124no—
473C22×C4—2124no—
474C22×C4—2124no—
475C22×C4C4×#615714yes—
476C22×C4C4⋊#615714yes—
477C22×C4C22⋊#615712yes—
478C22×C4C22⋊#615711yescharacteristic
479C22×C4C22×#615712yes—
480C22×C4C22×#615711yescharacteristic
481M16—10616no—
482M16—10616no—
483C2×C8—538no—
484C2×C8—1068no—
485C2×C8—1064no—
486C2×C8—1068no—
487C2×C8—1064no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—538no—
491C4⋊C4—534no—
492C4⋊C4—1064no—
493C4⋊C4—1068no—
494C4⋊C4—1064no—
495C4⋊C4—1068no—
496C4⋊C4—1064no—
497C4⋊C4—1062no—
498C4⋊C4—1064no—
499C4⋊C4C22×#615714yes—
500C22⋊C4—10616no—
501C42—10616no—
502C23—22no—
503C23—22no—
504C23—532no—
505C23—532no—
506C23—534no—
507C23—534no—
508C23—531no—
509C23—531no—
510C23—538no—
511C23—534no—
512C23—534no—
513C23—534no—
514C23—532no—
515C23—1064no—
516C23—1064no—
517C23—1064no—
518C23—1064no—
519C23—10616no—
520C23—10616no—
521C23—1062no—
522C23—1062no—
523C23(C2×#6157)⋊C414yes—
524C23C2×C22⋊#615712yes—
525C23C2×C22⋊#615711yescharacteristic
526C2×C4—24no—
527C2×C4—48no—
528C2×C4—44no—
529C2×C4—538no—
530C2×C4—534no—
531C2×C4—532no—
532C2×C4—534no—
533C2×C4—534no—
534C2×C4—532no—
535C2×C4—531no—
536C2×C4—532no—
537C2×C4—531no—
538C2×C4—538no—
539C2×C4—538no—
540C2×C4—538no—
541C2×C4—534no—
542C2×C4—532no—
543C2×C4—532no—
544C2×C4—534no—
545C2×C4—532no—
546C2×C4—534no—
547C2×C4—532no—
548C2×C4—534no—
549C2×C4—532no—
550C2×C4—534no—
551C2×C4—531no—
552C2×C4—531no—
553C2×C4—532no—
554C2×C4—534no—
555C2×C4—531no—
556C2×C4—531no—
557C2×C4—1068no—
558C2×C4—1064no—
559C2×C4—2128no—
560C2×C4—2128no—
561C2×C4—21216no—
562C2×C4—2128no—
563C2×C4—2124no—
564C2×C4—2124no—
565C2×C4—21216no—
566C2×C4—2124no—
567C2×C4(C2×#6157)⋊C418yes—
568C2×C4(C2×#6157)⋊C414yes—
569C2×C4C2×C4×#615714yes—
570C2×C4C2×C4×#615712yes—
571C2×C4C2×C22⋊#615711yescharacteristic
572C2×C4C2×C22⋊#615712yes—
573C2×C4C2×C4⋊#615714yes—
574C2×C4C2×C4⋊#615712yes—
575C2×C4C23×#615711yescharacteristic
576C8—1068no—
577C22—22no—
578C22—22no—
579C22—28no—
580C22—532no—
581C22—532no—
582C22—532no—
583C22—532no—
584C22—531no—
585C22—531no—
586C22—532no—
587C22—538no—
588C22—538no—
589C22—534no—
590C22—1068no—
591C22—1068no—
592C22—10616no—
593C22—1062no—
594C22—1062no—
595C22—1068no—
596C22C2×(C2×#6157)⋊C414yes—
597C22C2×(C2×#6157)⋊C412yes—
598C22#1681161512yes—
599C22#1681161612yes—
600C22C22×C22⋊#615711yescharacteristic
601C4—44no—
602C4—531no—
603C4—531no—
604C4—532no—
605C4—534no—
606C4—2128no—
607C4—2124no—
608C4C2×(C2×#6157)⋊C414yes—
609C4C2×(C2×#6157)⋊C412yes—
610C4C22×C4×#615711yescharacteristic
611C4C22×C4⋊#615711yescharacteristic
612C2—24no—
613C2—532no—
614C2—532no—
615C2—534no—
616C2—1064no—
617C2#2029040214yesdirect product factor
618C2C22×(C2×#6157)⋊C411yescharacteristic
619C2C2×#1681161511yescharacteristic
620C2C2×#1681161611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC4×C53C22×C42
FrattiniC2×C4C23×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×D53D4
cosocleC2×C4×D53C24
hypercenterC22×C4⋊C4#6157
hypocenterC53C2×#1736
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20290402C2direct (N × H)
4. C2×#16811496C2semidirect
9. C2×D53⋊M16C4semidirect
15. C2×#13133296C4semidirect
20. #16811496C22semidirect
23. C22×C4⋊Dic53C4semidirect
29. C2×C4⋊C4×D53C4semidirect
30. C2×C4⋊C4×D53C4semidirect
33. C22×C4⋊C4×C53C4semidirect
39. D53⋊M16C2×C4semidirect
44. C2×D53⋊C8C2×C4semidirect
49. #13133296C2×C4semidirect
52. C2×C4⋊#6157C2×C4semidirect
81. C2×C4⋊Dic53C2×C4semidirect
94. C4⋊C4×D53C2×C4semidirect
101. C2×C4⋊C4×C53C2×C4semidirect
176. C4⋊#6157C22×C4semidirect
181. D53⋊C8C22×C4semidirect
308. C2×C53#1736semidirect
313. D53#1736semidirect
340. C22×C4⋊C4#6157semidirect
344. C53C2×#1736semidirect
408. C2×C4⋊C4C2×#6157semidirect
617. C2#20290402direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D53⋊M162
C22×#131332962
#202904022
C2×#168114962
C22×C4⋊C4×D532
C2×#173653