Finite Group Databasebeta

← C2×#21423814 #28586091

Subgroups — C2×#21423814 #28586091

Order 15104.

2574 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11745659C211yesmaximal, characteristic
2C22×#14273685C211yesmaximal, characteristic
3C22×C4⋊C4×D59C211yesmaximal, characteristic
4#21423814C218yesmaximal, direct product factor
5C2×#16837165C211yesmaximal, characteristic, complemented
6C2×#16837164C211yesmaximal, characteristic, complemented
7C2×#16837164C211yesmaximal, characteristic, complemented
8C2×C59×#340C211yesmaximal, characteristic, complemented
9C23×C4×D59—21no—
10C23×C4×D59C2211yescharacteristic
11C22×#11745659—24no—
12C22×#11745659—24no—
13C22×#11745659C418yescomplemented
14C22×#11745659C2214yes—
15C23×C59⋊C8C2211yescharacteristic
16C2×#14273685—24no—
17C2×#14273685—22no—
18C2×#14273685C418yescomplemented
19C2×#14273685C2214yes—
20C22×C4⋊Dic59C2211yescharacteristic
21C22×C4⋊Dic59C2211yescharacteristic
22C22×Dic59⋊C4—21no—
23C2×C4⋊C4×D59—22no—
24C2×C4⋊C4×D59—24no—
25C2×C4⋊C4×D59C418yescomplemented
26C2×C4⋊C4×D59C2214yes—
27C22×(C2×D59)⋊C4—21no—
28C22×C4×Dic59—21no—
29#16837165C2214yescomplemented
30#16837164C2214yescomplemented
31#16837164C2214yescomplemented
32C23×C8×C59C2211yescharacteristic
33C22×C4⋊C4×C59C2211yescharacteristic
34C22×C4⋊C4×C59C2211yescharacteristic
35C59×#340C2214yescomplemented
36C24×D59D411yescharacteristic
37C23×Dic59—21no—
38C23×Dic59—21no—
39C23×Dic59D411yescharacteristic
40C22×C4×D59—28no—
41C22×C4×D59—24no—
42C22×C4×D59—44no—
43C22×C4×D59—44no—
44C22×C4×D59C2×C414yes—
45C22×C4×D59C2×C414yes—
46C22×C4×D59C2×C414yes—
47C22×C4×D59D412yes—
48C22×C4×D59D414yes—
49C22×C4×D59D414yes—
50C22×C4×D59Q814yes—
51C22×C4×D59C2312yes—
52C2×#11745659—232no—
53C2×#11745659—28no—
54C2×#11745659—28no—
55C2×#11745659C2×C4116yescomplemented
56C22×C59⋊C8—22no—
57C22×C59⋊C8—22no—
58C22×C59⋊C8C2×C414yescomplemented
59C22×C59⋊C8C2312yes—
60#14273685—216no—
61#14273685—28no—
62#14273685C2×C4116yescomplemented
63C2×C4⋊Dic59—21no—
64C2×C4⋊Dic59—22no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—22no—
67C2×C4⋊Dic59C2×C414yes—
68C2×C4⋊Dic59C2×C414yescomplemented
69C2×C4⋊Dic59C2312yes—
70C2×C4⋊Dic59C2312yes—
71C2×Dic59⋊C4—28no—
72C2×Dic59⋊C4—24no—
73C4⋊C4×D59—216no—
74C4⋊C4×D59—28no—
75C4⋊C4×D59C2×C4116yescomplemented
76C2×(C2×D59)⋊C4—28no—
77C2×(C2×D59)⋊C4—24no—
78C2×C4×Dic59—28no—
79C2×C4×Dic59—24no—
80C23×C4×C59—21no—
81C23×C4×C59—21no—
82C23×C4×C59C2311yescharacteristic
83C22×C8×C59—22no—
84C22×C8×C59—22no—
85C22×C8×C59C2×C414yescomplemented
86C22×C8×C59C2312yes—
87C2×C4⋊C4×C59—21no—
88C2×C4⋊C4×C59—22no—
89C2×C4⋊C4×C59—21no—
90C2×C4⋊C4×C59—22no—
91C2×C4⋊C4×C59C2×C414yes—
92C2×C4⋊C4×C59C2×C414yescomplemented
93C2×C4⋊C4×C59C2312yes—
94C2×C4⋊C4×C59C2312yes—
95C24×C59C2×D411yescharacteristic
96C23×D59—22no—
97C23×D59—22no—
98C23×D59—22no—
99C23×D59—22no—
100C23×D59C22⋊C414yes—
101C23×D59C22⋊C414yes—
102C23×D59C4⋊C414yes—
103C23×D59C2×D412yes—
104C22×C4×C59—24no—
105C22×C4×C59—24no—
106C22×C4×C59—22no—
107C22×C4×C59—22no—
108C22×C4×C59—42no—
109C22×C4×C59—42no—
110C22×C4×C59—42no—
111C22×C4×C59—42no—
112C22×C4×C59C22×C412yes—
113C22×C4×C59C22×C412yes—
114C22×C4×C59C22×C412yes—
115C22×C4×C59C2×D411yescharacteristic
116C22×C4×C59C2×D412yes—
117C22×C4×C59C2×D412yes—
118C22×C4×C59C2×Q812yes—
119C22×C4×C59C2411yescharacteristic
120C22×Dic59—24no—
121C22×Dic59—24no—
122C22×Dic59—22no—
123C22×Dic59—22no—
124C22×Dic59—42no—
125C22×Dic59—42no—
126C22×Dic59—42no—
127C22×Dic59—42no—
128C22×Dic59C22⋊C414yes—
129C22×Dic59C22⋊C414yes—
130C22×Dic59C4⋊C414yes—
131C22×Dic59C2×D412yes—
132C2×C4×D59—216no—
133C2×C4×D59—48no—
134C2×C4×D59—48no—
135C2×C4×D59—432no—
136C2×C4×D59C42116yes—
137C2×C4×D59C22⋊C418yes—
138C2×C4×D59C22⋊C418yes—
139C2×C4×D59C22⋊C4116yes—
140C2×C4×D59C4⋊C418yes—
141C2×C4×D59C4⋊C4116yes—
142C2×C4×D59C4⋊C4116yes—
143C2×C4×D59C22×C414yes—
144C2×C4×D59C22×C414yes—
145C2×C4×D59C22×C414yes—
146C2×C4×D59C2×D414yes—
147C2×C4×D59C2×D414yes—
148C2×C4×D59C2×Q814yes—
149C2×C8×C59—28no—
150C2×C8×C59—22no—
151C2×C8×C59—22no—
152C2×C8×C59C22×C414yescomplemented
153C4⋊C4×C59—24no—
154C4⋊C4×C59—24no—
155C4⋊C4×C59—22no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59C22×C414yes—
158C4⋊C4×C59C22×C414yescomplemented
159(C2×D59)⋊C4—216no—
160C4⋊Dic59—24no—
161C4⋊Dic59—24no—
162C4⋊Dic59—22no—
163C4⋊Dic59—22no—
164C4⋊Dic59C22×C414yes—
165C4⋊Dic59C22×C414yescomplemented
166Dic59⋊C4—216no—
167C4×Dic59—216no—
168C2×C59⋊C8—28no—
169C2×C59⋊C8—22no—
170C2×C59⋊C8—22no—
171C2×C59⋊C8C22×C414yescomplemented
172#11745659—232no—
173C23×C59C2×C22⋊C412yes—
174C23×C59C2×C22⋊C412yes—
175C23×C59C2×C4⋊C412yes—
176C23×C59AGL(1,ℤ/8)12yes—
177C23×C59AGL(1,ℤ/8)12yes—
178C23×C59#13712yes—
179C23×C59#13712yes—
180C23×C59C22×D411yescharacteristic
181C22×D59—22no—
182C22×D59—22no—
183C22×D59—22no—
184C22×D59—22no—
185C22×D59—28no—
186C22×D59—28no—
187C22×D59—28no—
188C22×D59—28no—
189C22×D59—28no—
190C22×D59—28no—
191C22×D59#95116yes—
192C22×D59C2×C22⋊C414yes—
193C22×D59C2×C22⋊C414yes—
194C22×D59C2×C4⋊C414yes—
195C2×C4×C59—24no—
196C2×C4×C59—24no—
197C2×C4×C59—42no—
198C2×C4×C59—42no—
199C2×C4×C59—42no—
200C2×C4×C59—42no—
201C2×C4×C59—48no—
202C2×C4×C59—48no—
203C2×C4×C59C2×C4214yes—
204C2×C4×C59C2×C22⋊C414yes—
205C2×C4×C59C2×C22⋊C412yes—
206C2×C4×C59C2×C22⋊C412yes—
207C2×C4×C59C2×C4⋊C414yes—
208C2×C4×C59C2×C4⋊C414yes—
209C2×C4×C59C2×C4⋊C412yes—
210C2×C4×C59C23×C411yescharacteristic
211C2×C4×C59C23×C411yescharacteristic
212C2×C4×C59C23×C411yescharacteristic
213C2×C4×C59C22×D411yescharacteristic
214C2×C4×C59C22×D411yescharacteristic
215C2×C4×C59C22×Q811yescharacteristic
216C2×Dic59—24no—
217C2×Dic59—24no—
218C2×Dic59—42no—
219C2×Dic59—42no—
220C2×Dic59—42no—
221C2×Dic59—42no—
222C2×Dic59—48no—
223C2×Dic59—48no—
224C2×Dic59#95116yes—
225C2×Dic59C2×C22⋊C414yes—
226C2×Dic59C2×C22⋊C414yes—
227C2×Dic59C2×C4⋊C414yes—
228C4×D59—432no—
229C4×D59#95132yes—
230C4×D59C2×C4218yes—
231C4×D59C2×C22⋊C418yes—
232C4×D59C2×C4⋊C418yes—
233C4×D59C2×C4⋊C418yes—
234C8×C59—24no—
235C59⋊C8—24no—
236C2×#1736—591nomaximal, Sylow
237C22×C59C2×#9514yes—
238C22×C59#41714yes—
239C22×C59#41714yes—
240C22×C59#41814yes—
241C22×C59#41914yes—
242C22×C59#42814yes—
243C22×C59#42814yes—
244C22×C59C22×C22⋊C411yescharacteristic
245C22×C59C22×C22⋊C411yescharacteristic
246C22×C59C22×C4⋊C411yescharacteristic
247C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
249C22×C59C2×#13711yescharacteristic
250C22×C59C2×#13711yescharacteristic
251C2×D59—24no—
252C2×D59—24no—
253C2×D59—24no—
254C2×D59—24no—
255C2×D59—24no—
256C2×D59—24no—
257C2×D59—232no—
258C2×D59C2×#9518yes—
259C4×C59—44no—
260C4×C59—44no—
261C4×C59C2×#9514yes—
262C4×C59C22×C4211yescharacteristic
263C4×C59C22×C22⋊C411yescharacteristic
264C4×C59C22×C4⋊C411yescharacteristic
265C4×C59C22×C4⋊C411yescharacteristic
266Dic59—44no—
267Dic59—44no—
268Dic59C2×#9518yes—
269C23×M16—591no—
270C22×#117—591no—
271C23×C4⋊C4—591no—
272#1736—598no—
273C2×#340—591no—
274C2×#340—591no—
275C2×#340—591no—
276C2×#340D5911yescharacteristic, complemented
277C2×C59#173618yescomplemented
278C2×C59C22×#9511yescharacteristic
279C2×C59C2×#41811yescharacteristic
280C2×C59C2×#41911yescharacteristic
281C2×C59C2×#41711yescharacteristic
282C2×C59C2×#41711yescharacteristic
283C2×C59C2×#42811yescharacteristic
284C2×C59C2×#42811yescharacteristic
285D59—28no—
286C24×C4—591no—
287C24×C4—1181no—
288C22×M16—598no—
289C22×M16—594no—
290C22×M16—1184no—
291C22×M16—1184no—
292C23×C8—591no—
293C23×C8C2×D5911yescharacteristic
294C2×#117—598no—
295C2×#117—594no—
296C2×#117—1182no—
297C2×#117—1184no—
298C22×C4⋊C4—591no—
299C22×C4⋊C4—591no—
300C22×C4⋊C4—598no—
301C22×C4⋊C4—594no—
302C22×C4⋊C4—1182no—
303C22×C4⋊C4—1184no—
304C22×C4⋊C4—1181no—
305C22×C4⋊C4C2×D5911yescharacteristic
306C22×C4⋊C4C2×D5911yescharacteristic
307C22×C22⋊C4—1181no—
308C22×C42—1181no—
309#340—594no—
310#340—594no—
311#340—594no—
312#340C2×D5914yescomplemented
313C59C2×#173611yescharacteristic, complemented, Sylow
314C25—591no—
315C23×C4—21no—
316C23×C4—21no—
317C23×C4—591no—
318C23×C4—594no—
319C23×C4—594no—
320C23×C4—594no—
321C23×C4—592no—
322C23×C4—594no—
323C23×C4—594no—
324C23×C4—594no—
325C23×C4—592no—
326C23×C4—1188no—
327C23×C4—1184no—
328C23×C4—1181no—
329C23×C4—1181no—
330C23×C4—2364no—
331C23×C4—2364no—
332C23×C4C22×D5911yescharacteristic
333C2×M16—5916no—
334C2×M16—11832no—
335C2×M16—1188no—
336C2×M16—1188no—
337C22×C8—22no—
338C22×C8—22no—
339C22×C8—594no—
340C22×C8—592no—
341C22×C8—1182no—
342C22×C8—1182no—
343C22×C8C4×D5914yescomplemented
344C22×C8C22×D5912yes—
345#117—5916no—
346#117—11816no—
347#117—1188no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—594no—
353C2×C4⋊C4—594no—
354C2×C4⋊C4—592no—
355C2×C4⋊C4—592no—
356C2×C4⋊C4—5916no—
357C2×C4⋊C4—11816no—
358C2×C4⋊C4—1188no—
359C2×C4⋊C4—1188no—
360C2×C4⋊C4—1184no—
361C2×C4⋊C4—1181no—
362C2×C4⋊C4—1182no—
363C2×C4⋊C4—1181no—
364C2×C4⋊C4—1182no—
365C2×C4⋊C4C4×D5914yes—
366C2×C4⋊C4C4×D5914yescomplemented
367C2×C4⋊C4C22×D5912yes—
368C2×C4⋊C4C22×D5912yes—
369C2×C22⋊C4—1188no—
370C2×C22⋊C4—1184no—
371C2×C42—1188no—
372C2×C42—1184no—
373C24—594no—
374C24—594no—
375C24—594no—
376C24—592no—
377C24—1182no—
378C24—1182no—
379C24—1182no—
380C24—1182no—
381C24D4×D5911yescharacteristic
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—594no—
391C22×C4—594no—
392C22×C4—594no—
393C22×C4—592no—
394C22×C4—5916no—
395C22×C4—598no—
396C22×C4—598no—
397C22×C4—5916no—
398C22×C4—5916no—
399C22×C4—5916no—
400C22×C4—598no—
401C22×C4—594no—
402C22×C4—594no—
403C22×C4—594no—
404C22×C4—594no—
405C22×C4—594no—
406C22×C4—594no—
407C22×C4—11816no—
408C22×C4—1184no—
409C22×C4—1184no—
410C22×C4—1182no—
411C22×C4—1182no—
412C22×C4—23632no—
413C22×C4—2368no—
414C22×C4—2368no—
415C22×C4—2362no—
416C22×C4—2362no—
417C22×C4—2362no—
418C22×C4—2362no—
419C22×C4C2×C4×D5912yes—
420C22×C4C2×C4×D5912yes—
421C22×C4C2×C4×D5912yes—
422C22×C4D4×D5911yescharacteristic
423C22×C4D4×D5912yes—
424C22×C4D4×D5912yes—
425C22×C4Q8×D5912yes—
426C22×C4C23×D5911yescharacteristic
427M16—11832no—
428C2×C8—28no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—594no—
432C2×C8—1188no—
433C2×C8—1182no—
434C2×C8—1182no—
435C2×C8C2×C4×D5914yescomplemented
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—594no—
441C4⋊C4—594no—
442C4⋊C4—11816no—
443C4⋊C4—1184no—
444C4⋊C4—1184no—
445C4⋊C4—1182no—
446C4⋊C4—1182no—
447C4⋊C4C2×C4×D5914yes—
448C4⋊C4C2×C4×D5914yescomplemented
449C22⋊C4—11816no—
450C42—11816no—
451C23—5916no—
452C23—594no—
453C23—594no—
454C23—594no—
455C23—1188no—
456C23—1188no—
457C23—1188no—
458C23—1188no—
459C23—1182no—
460C23—1182no—
461C23—1182no—
462C23—1182no—
463C23—1188no—
464C23—1188no—
465C23C22⋊C4×D5912yes—
466C23C22⋊C4×D5912yes—
467C23C4⋊C4×D5912yes—
468C23#1427375112yes—
469C23#1427375412yes—
470C23#1427375612yes—
471C23#1427375912yes—
472C23C2×D4×D5911yescharacteristic
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—5916no—
482C2×C4—594no—
483C2×C4—594no—
484C2×C4—594no—
485C2×C4—5932no—
486C2×C4—598no—
487C2×C4—598no—
488C2×C4—598no—
489C2×C4—598no—
490C2×C4—1184no—
491C2×C4—1184no—
492C2×C4—23632no—
493C2×C4—2368no—
494C2×C4—2368no—
495C2×C4—2362no—
496C2×C4—2362no—
497C2×C4—2362no—
498C2×C4—2362no—
499C2×C4C42×D5914yes—
500C2×C4C22⋊C4×D5912yes—
501C2×C4C22⋊C4×D5912yes—
502C2×C4C22⋊C4×D5914yes—
503C2×C4C4⋊C4×D5912yes—
504C2×C4C4⋊C4×D5914yes—
505C2×C4C4⋊C4×D5914yes—
506C2×C4C22×C4×D5911yescharacteristic
507C2×C4C22×C4×D5911yescharacteristic
508C2×C4C22×C4×D5911yescharacteristic
509C2×C4C2×D4×D5911yescharacteristic
510C2×C4C2×D4×D5911yescharacteristic
511C2×C4C2×Q8×D5911yescharacteristic
512C8—24no—
513C8—1184no—
514C22—598no—
515C22—1184no—
516C22—1184no—
517C22—1184no—
518C22—1184no—
519C22—1184no—
520C22—1184no—
521C22—11832no—
522C22#95×D5914yes—
523C22#1683740014yes—
524C22#1683740214yes—
525C22#1683740514yes—
526C22#1683740914yes—
527C22#1683744114yes—
528C22#1683744314yes—
529C22C2×C22⋊C4×D5911yescharacteristic
530C22C2×C22⋊C4×D5911yescharacteristic
531C22C2×C4⋊C4×D5911yescharacteristic
532C22C2×#1427375111yescharacteristic
533C22C2×#1427375411yescharacteristic
534C22C2×#1427375611yescharacteristic
535C22C2×#1427375911yescharacteristic
536C4—44no—
537C4—44no—
538C4—598no—
539C4—2364no—
540C4—2364no—
541C4#95×D5914yes—
542C4C2×C42×D5911yescharacteristic
543C4C2×C22⋊C4×D5911yescharacteristic
544C4C2×C4⋊C4×D5911yescharacteristic
545C4C2×C4⋊C4×D5911yescharacteristic
546C2—1188no—
547C2#2142381418yesdirect product factor
548C2C2×#95×D5911yescharacteristic
549C2C2×#1683740511yescharacteristic
550C2C2×#1683740911yescharacteristic
551C2C2×#1683740011yescharacteristic
552C2C2×#1683740211yescharacteristic
553C2C2×#1683744111yescharacteristic
554C2C2×#1683744311yescharacteristic
subgroupG/H
centerC24D4×D59
commutatorC4×C59C22×C42
FrattiniC22×C4C23×D59
FittingC2×C59×#340C2
socleC24×C59C2×D4
prime radicalC24×D59D4
cosocleC22×C4×C59C24
hypercenterC2×#340D59
hypocenterC59C2×#1736
LCSG ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
4. #21423814C2direct (N × H)
5. C2×#16837165C2semidirect
6. C2×#16837164C2semidirect
7. C2×#16837164C2semidirect
8. C2×C59×#340C2semidirect
13. C22×#11745659C4semidirect
18. C2×#14273685C4semidirect
25. C2×C4⋊C4×D59C4semidirect
29. #16837165C22semidirect
30. #16837164C22semidirect
31. #16837164C22semidirect
35. C59×#340C22semidirect
55. C2×#11745659C2×C4semidirect
58. C22×C59⋊C8C2×C4semidirect
62. #14273685C2×C4semidirect
68. C2×C4⋊Dic59C2×C4semidirect
75. C4⋊C4×D59C2×C4semidirect
85. C22×C8×C59C2×C4semidirect
92. C2×C4⋊C4×C59C2×C4semidirect
152. C2×C8×C59C22×C4semidirect
158. C4⋊C4×C59C22×C4semidirect
165. C4⋊Dic59C22×C4semidirect
171. C2×C59⋊C8C22×C4semidirect
276. C2×#340D59semidirect
277. C2×C59#1736semidirect
312. #340C2×D59semidirect
313. C59C2×#1736semidirect
343. C22×C8C4×D59semidirect
366. C2×C4⋊C4C4×D59semidirect
435. C2×C8C2×C4×D59semidirect
448. C4⋊C4C2×C4×D59semidirect
547. C2#21423814direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117456592
C22×#142736852
C22×C4⋊C4×D592
#214238142
C2×#168371652
C2×#168371642
C2×C59×#3402
C2×#173659