Finite Group Databasebeta

← C2×#21425289 #28616142

Subgroups — C2×#21425289 #28616142

Order 15104.

1662 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273658C211yesmaximal, characteristic, complemented
2C22×#14273651C211yesmaximal, characteristic, complemented
3C22×#14273651C211yesmaximal, characteristic, complemented
4C22×C22⋊(C59⋊C8)C211yesmaximal, characteristic
5C2×#16837262C211yesmaximal, characteristic
6#21425289C218yesmaximal, direct product factor
7C2×#16837164C211yesmaximal, characteristic, complemented
8C22×C22⋊Q8×C59C211yesmaximal, characteristic
9C23×C59⋊C8C2211yescharacteristic
10C23×C59⋊C8C2211yescharacteristic
11C22×C4⋊Dic59C2211yescharacteristic
12C22×C4⋊Dic59C2211yescharacteristic
13C22×C22⋊Dic59—21no—
14C2×#14273658—24no—
15C2×#14273658C2214yescomplemented
16C2×#14273651—24no—
17C2×#14273651—24no—
18C2×#14273651C2214yescomplemented
19C2×#14273651C2214yescomplemented
20C2×C22⋊(C59⋊C8)—24no—
21C2×C22⋊(C59⋊C8)C2214yes—
22#16837262—22no—
23#16837262C2214yes—
24C2×C59⋊#95—21no—
25#16837164C2214yescomplemented
26C23×Q8×C59C2211yescharacteristic
27C24×C4×C59C2211yescharacteristic
28C2×C22⋊Q8×C59—24no—
29C2×C22⋊Q8×C59—22no—
30C2×C22⋊Q8×C59—24no—
31C2×C22⋊Q8×C59C2214yes—
32C22×C4⋊C4×C59—21no—
33C22×C4⋊C4×C59C2211yescharacteristic
34C22×C22⋊C4×C59—21no—
35C23×Dic59—21no—
36C23×Dic59—21no—
37C22×C59⋊C8—22no—
38C22×C59⋊C8—22no—
39C22×C59⋊C8—22no—
40C22×C59⋊C8—22no—
41C22×C59⋊C8—22no—
42C22×C59⋊C8D414yes—
43C22×C59⋊C8C2312yes—
44C22×C59⋊C8C2312yes—
45C2×C4⋊Dic59—22no—
46C2×C4⋊Dic59—22no—
47C2×C4⋊Dic59—22no—
48C2×C4⋊Dic59—22no—
49C2×C4⋊Dic59—21no—
50C2×C4⋊Dic59—21no—
51C2×C4⋊Dic59C2312yes—
52C2×C4⋊Dic59C2312yes—
53C2×C22⋊Dic59—24no—
54C2×C22⋊Dic59—44no—
55#14273658—28no—
56#14273651—28no—
57#14273651—28no—
58C22⋊(C59⋊C8)—28no—
59C59⋊#95—24no—
60C25×C59D411yescharacteristic
61C22×Q8×C59—22no—
62C22×Q8×C59—21no—
63C22×Q8×C59—22no—
64C22×Q8×C59—22no—
65C22×Q8×C59—22no—
66C22×Q8×C59—22no—
67C22×Q8×C59D414yes—
68C22×Q8×C59C2312yes—
69C23×C4×C59—21no—
70C23×C4×C59—21no—
71C23×C4×C59—22no—
72C23×C4×C59—22no—
73C23×C4×C59—22no—
74C23×C4×C59—22no—
75C23×C4×C59—22no—
76C23×C4×C59—22no—
77C23×C4×C59D411yescharacteristic
78C23×C4×C59D412yes—
79C23×C4×C59C2312yes—
80C23×C4×C59C2311yescharacteristic
81C22⋊Q8×C59—216no—
82C22⋊Q8×C59—28no—
83C22⋊Q8×C59—28no—
84C2×C4⋊C4×C59—28no—
85C2×C4⋊C4×C59—24no—
86C2×C4⋊C4×C59—21no—
87C2×C4⋊C4×C59—22no—
88C2×C4⋊C4×C59D414yes—
89C2×C4⋊C4×C59C2312yes—
90C2×C22⋊C4×C59—28no—
91C2×C22⋊C4×C59—24no—
92C24×C59—22no—
93C24×C59—22no—
94C24×C59—22no—
95C24×C59—22no—
96C24×C59—22no—
97C24×C59—22no—
98C24×C59SD1614yes—
99C24×C59C2×D412yes—
100C24×C59C2×D411yescharacteristic
101C2×Q8×C59—24no—
102C2×Q8×C59—22no—
103C2×Q8×C59—28no—
104C2×Q8×C59—28no—
105C2×Q8×C59—22no—
106C2×Q8×C59—22no—
107C2×Q8×C59—22no—
108C2×Q8×C59—22no—
109C2×Q8×C59—42no—
110C2×Q8×C59—44no—
111C2×Q8×C59—42no—
112C2×Q8×C59—44no—
113C2×Q8×C59C2×D414yes—
114C22×C4×C59—24no—
115C22×C4×C59—24no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—24no—
122C22×C4×C59—24no—
123C22×C4×C59—24no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59—22no—
130C22×C4×C59—42no—
131C22×C4×C59—42no—
132C22×C4×C59—42no—
133C22×C4×C59—42no—
134C22×C4×C59—44no—
135C22×C4×C59—44no—
136C22×C4×C59—44no—
137C22×C4×C59—44no—
138C22×C4×C59C2×D412yes—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D412yes—
141C22×C4×C59C2×D412yes—
142C22×C4×C59C2×D412yes—
143C22×C4×C59#4112yes—
144C22×C4×C59#4112yes—
145C22×C4×C59#4112yes—
146C22×C4×C59C2411yescharacteristic
147C22×Dic59—22no—
148C22×Dic59—22no—
149C22×Dic59—42no—
150C22×Dic59—42no—
151C22×Dic59—42no—
152C22×Dic59—42no—
153C22×Dic59—42no—
154C22×Dic59—42no—
155C4⋊C4×C59—216no—
156C4⋊C4×C59—24no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59C2×D414yes—
159C22⋊C4×C59—216no—
160C22⋊Dic59—48no—
161C4⋊Dic59—22no—
162C4⋊Dic59—22no—
163C4⋊Dic59—22no—
164C4⋊Dic59—22no—
165C4⋊Dic59—42no—
166C4⋊Dic59—42no—
167C2×C59⋊C8—28no—
168C2×C59⋊C8—22no—
169C2×C59⋊C8—22no—
170C2×C59⋊C8—22no—
171C2×C59⋊C8—22no—
172C2×C59⋊C8—22no—
173C2×C59⋊C8—44no—
174C2×C59⋊C8C2×D414yes—
175C23×C59—22no—
176C23×C59—22no—
177C23×C59—22no—
178C23×C59—22no—
179C23×C59—22no—
180C23×C59—22no—
181C23×C59—28no—
182C23×C59—28no—
183C23×C59—28no—
184C23×C59—44no—
185C23×C59—44no—
186C23×C59—44no—
187C23×C59—44no—
188C23×C59#12012yes—
189C23×C59#12112yes—
190C23×C59#12312yes—
191C23×C59C2×SD1614yes—
192C23×C59C2×SD1612yes—
193C23×C59#13512yes—
194C23×C59#13712yes—
195C23×C59#13712yes—
196C23×C59C22×D411yescharacteristic
197Q8×C59—24no—
198Q8×C59—24no—
199Q8×C59—42no—
200Q8×C59—42no—
201Q8×C59—48no—
202C2×C4×C59—24no—
203C2×C4×C59—24no—
204C2×C4×C59—22no—
205C2×C4×C59—22no—
206C2×C4×C59—22no—
207C2×C4×C59—42no—
208C2×C4×C59—42no—
209C2×C4×C59—42no—
210C2×C4×C59—42no—
211C2×C4×C59—44no—
212C2×C4×C59—48no—
213C2×C4×C59—48no—
214C2×C4×C59—42no—
215C2×C4×C59—42no—
216C2×C4×C59—42no—
217C2×C4×C59—42no—
218C2×C4×C59—48no—
219C2×C4×C59#12012yes—
220C2×C4×C59#12114yes—
221C2×C4×C59#12114yes—
222C2×C4×C59#12112yes—
223C2×C4×C59#12314yes—
224C2×C4×C59#12312yes—
225C2×C4×C59#12414yes—
226C2×C4×C59C22×D411yescharacteristic
227C2×C4×C59C22×D411yescharacteristic
228C2×C4×C59C22×D411yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×Dic59—42no—
233C2×Dic59—42no—
234C2×Dic59—42no—
235C2×Dic59—42no—
236C2×Dic59—42no—
237C2×Dic59—42no—
238C2×Dic59—84no—
239C2×Dic59—84no—
240C59⋊C8—24no—
241C59⋊C8—42no—
242C2×#2035—591nomaximal, Sylow
243C22×C59—24no—
244C22×C59—24no—
245C22×C59—24no—
246C22×C59—42no—
247C22×C59—42no—
248C22×C59—42no—
249C22×C59—42no—
250C22×C59—416no—
251C22×C59#39414yes—
252C22×C59#44814yes—
253C22×C59#45214yes—
254C22×C59#46514yes—
255C22×C59#47014yes—
256C22×C59#48314yes—
257C22×C59#48514yes—
258C22×C59C2×#12011yescharacteristic
259C22×C59C2×#12111yescharacteristic
260C22×C59C2×#12311yescharacteristic
261C22×C59C22×SD1611yescharacteristic
262C22×C59C2×#13511yescharacteristic
263C22×C59C2×#13711yescharacteristic
264C22×C59C2×#13711yescharacteristic
265C4×C59—42no—
266C4×C59—44no—
267C4×C59—44no—
268C4×C59#39414yes—
269C4×C59C2×#12111yescharacteristic
270C4×C59C2×#12111yescharacteristic
271C4×C59C2×#12311yescharacteristic
272C4×C59C2×#12411yescharacteristic
273Dic59—82no—
274Dic59—82no—
275C22×C22⋊Q8D5911yescharacteristic
276C22×#106—591no—
277C22×#103—591no—
278C22×#103—591no—
279C22×C22⋊C8—591no—
280C2×#380—591no—
281#2035—598no—
282C2×#340—591no—
283C2×C59—44no—
284C2×C59#203518yescomplemented
285C2×C59C2×#39411yescharacteristic
286C2×C59C2×#44811yescharacteristic
287C2×C59C2×#45211yescharacteristic
288C2×C59C2×#46511yescharacteristic
289C2×C59C2×#47011yescharacteristic
290C2×C59C2×#48311yescharacteristic
291C2×C59C2×#48511yescharacteristic
292C23×Q8C2×D5911yescharacteristic
293C24×C4C2×D5911yescharacteristic
294C23×C8—591no—
295C23×C8—591no—
296C2×C22⋊Q8—24no—
297C2×C22⋊Q8—22no—
298C2×C22⋊Q8—24no—
299C2×C22⋊Q8C2×D5914yes—
300C22×C4⋊C4—21no—
301C22×C4⋊C4—591no—
302C22×C4⋊C4—591no—
303C22×C4⋊C4C2×D5911yescharacteristic
304C22×C22⋊C4—21no—
305C22×C22⋊C4—1181no—
306C2×#106—594no—
307C2×#106—1184no—
308C2×#103—594no—
309C2×#103—594no—
310C2×#103—1184no—
311C2×#103—1184no—
312C2×C22⋊C8—594no—
313C2×C22⋊C8—1184no—
314#380—594no—
315#380—1182no—
316C2×#95—1181no—
317#340—594no—
318C59C2×#203511yescharacteristic, complemented, Sylow
319C25C22⋊D5911yescharacteristic
320C22×Q8—22no—
321C22×Q8—21no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8C22⋊D5914yes—
327C22×Q8C22×D5912yes—
328C23×C4—21no—
329C23×C4—21no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—22no—
335C23×C4—22no—
336C23×C4—1181no—
337C23×C4—1181no—
338C23×C4C22⋊D5912yes—
339C23×C4C22⋊D5911yescharacteristic
340C23×C4C22×D5912yes—
341C23×C4C22×D5911yescharacteristic
342C22×C8—594no—
343C22×C8—592no—
344C22×C8—592no—
345C22×C8—1182no—
346C22×C8—1182no—
347C22×C8—1182no—
348C22×C8—1182no—
349C22×C8—1182no—
350C22⋊Q8—216no—
351C22⋊Q8—28no—
352C22⋊Q8—28no—
353C2×C4⋊C4—28no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—592no—
358C2×C4⋊C4—592no—
359C2×C4⋊C4—1182no—
360C2×C4⋊C4—1182no—
361C2×C4⋊C4—1182no—
362C2×C4⋊C4—1182no—
363C2×C4⋊C4—1181no—
364C2×C4⋊C4—1181no—
365C2×C4⋊C4C22⋊D5914yes—
366C2×C4⋊C4C22×D5912yes—
367C2×C22⋊C4—28no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—1184no—
370C2×C22⋊C4—2364no—
371#106—1188no—
372#103—1188no—
373#103—1188no—
374C22⋊C8—1188no—
375#95—1184no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24#1174567114yes—
383C24C2×C22⋊D5912yes—
384C24C2×C22⋊D5911yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×C22⋊D5914yes—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—1182no—
423C22×C4—1182no—
424C22×C4—2362no—
425C22×C4—2362no—
426C22×C4—2362no—
427C22×C4—2362no—
428C22×C4—2362no—
429C22×C4—2362no—
430C22×C4D4×D5912yes—
431C22×C4#1174568712yes—
432C22×C4#1174568712yes—
433C22×C4#1174568712yes—
434C22×C4C2×C22⋊D5912yes—
435C22×C4C2×C22⋊D5911yescharacteristic
436C22×C4C2×C22⋊D5912yes—
437C22×C4C2×C22⋊D5912yes—
438C22×C4C23×D5911yescharacteristic
439C2×C8—594no—
440C2×C8—1188no—
441C2×C8—1182no—
442C2×C8—1182no—
443C2×C8—1182no—
444C2×C8—1182no—
445C2×C8—1182no—
446C2×C8—2364no—
447C4⋊C4—216no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—1182no—
451C4⋊C4—1182no—
452C4⋊C4—1182no—
453C4⋊C4—1182no—
454C4⋊C4—2362no—
455C4⋊C4—2362no—
456C4⋊C4C2×C22⋊D5914yes—
457C22⋊C4—216no—
458C22⋊C4—2368no—
459C23—22no—
460C23—28no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—28no—
467C23—28no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23#1427369612yes—
473C23#1427369912yes—
474C23#1427370812yes—
475C23C2×#1174567114yes—
476C23C2×#1174567112yes—
477C23#1427374712yes—
478C23#1427375712yes—
479C23#1427375812yes—
480C23C22×C22⋊D5911yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—42no—
484Q8—42no—
485Q8—48no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—48no—
496C2×C4—44no—
497C2×C4—48no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—48no—
503C2×C4—4724no—
504C2×C4—4724no—
505C2×C4—2362no—
506C2×C4—2362no—
507C2×C4—2362no—
508C2×C4—2362no—
509C2×C4—2362no—
510C2×C4—2362no—
511C2×C4#1427369612yes—
512C2×C4#1427369814yes—
513C2×C4#1427369914yes—
514C2×C4#1427369912yes—
515C2×C4#1427370814yes—
516C2×C4#1427370812yes—
517C2×C4#1427371414yes—
518C2×C4C2×D4×D5911yescharacteristic
519C2×C4C22×C22⋊D5911yescharacteristic
520C2×C4C22×C22⋊D5911yescharacteristic
521C2×C4C2×#1174568711yescharacteristic
522C2×C4C2×#1174568711yescharacteristic
523C2×C4C2×#1174568711yescharacteristic
524C8—1184no—
525C8—2362no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—416no—
533C22—42no—
534C22#1683731614yes—
535C22#1683754114yes—
536C22#1683756414yes—
537C22#1683763314yes—
538C22#1683765814yes—
539C22#1683773014yes—
540C22#1683774014yes—
541C22C2×#1427369611yescharacteristic
542C22C2×#1427369911yescharacteristic
543C22C2×#1427370811yescharacteristic
544C22C22×#1174567111yescharacteristic
545C22C2×#1427374711yescharacteristic
546C22C2×#1427375711yescharacteristic
547C22C2×#1427375811yescharacteristic
548C4—42no—
549C4—44no—
550C4—44no—
551C4—4722no—
552C4—4722no—
553C4#1683731614yes—
554C4C2×#1427369811yescharacteristic
555C4C2×#1427369911yescharacteristic
556C4C2×#1427370811yescharacteristic
557C4C2×#1427371411yescharacteristic
558C2—44no—
559C2#2142528918yesdirect product factor
560C2C2×#1683731611yescharacteristic
561C2C2×#1683754111yescharacteristic
562C2C2×#1683756411yescharacteristic
563C2C2×#1683763311yescharacteristic
564C2C2×#1683765811yescharacteristic
565C2C2×#1683773011yescharacteristic
566C2C2×#1683774011yescharacteristic
subgroupG/H
centerC24C2×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC22×C22⋊Q8×C59C2
socleC24×C59C2×D4
prime radicalC25×C59D4
hypercenterC22×C22⋊Q8D59
hypocenterC59C2×#2035
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14273658C2semidirect
2. C22×#14273651C2semidirect
3. C22×#14273651C2semidirect
6. #21425289C2direct (N × H)
7. C2×#16837164C2semidirect
15. C2×#14273658C22semidirect
18. C2×#14273651C22semidirect
19. C2×#14273651C22semidirect
25. #16837164C22semidirect
284. C2×C59#2035semidirect
318. C59C2×#2035semidirect
559. C2#21425289direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#142736582
C22×#142736512
C22×C22⋊(C59⋊C8)2
C2×#168372622
#214252892
C2×#168371642
C22×C22⋊Q8×C592
C2×#203559