Finite Group Databasebeta

← C2×#21425750 #28741426

Subgroups — C2×#21425750 #28741426

Order 15104.

1214 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837386C211yesmaximal, characteristic
2C2×#16837376C211yesmaximal, characteristic
3#21425750C218yesmaximal, direct product factor
4C2×#16837192C214yesmaximal, complemented
5C2×C59×#536C211yesmaximal, characteristic
6C22×#11745664—21no—
7C2×#14273686—21no—
8#16837386—22no—
9#16837386C2214yes—
10C2×#14273646—21no—
11C2×#14273644C2212yes—
12#16837376—22no—
13#16837376C2214yes—
14C2×#14273642—21no—
15C2×#14273642C2212yes—
16#16837192C22116yescomplemented
17C22×#41×C59C2211yescharacteristic
18C23×Q8×C59C411yescharacteristic, complemented
19C59×#536—22no—
20C59×#536—22no—
21C59×#536C414yescomplemented
22C59×#536C2214yes—
23C2×C4⋊Q8×C59—21no—
24C2×#124×C59—21no—
25C2×#123×C59C412yescomplemented
26C2×#123×C59C2212yes—
27C2×C22⋊Q8×C59—21no—
28C2×C22⋊Q8×C59—21no—
29C2×#117×C59C411yescharacteristic, complemented
30C2×#11745664—22no—
31C2×#11745664—24no—
32C2×#11745664—41no—
33C2×#11745664—42no—
34C22×C59⋊C8—41no—
35#14273686—24no—
36#14273686—42no—
37C2×C4⋊Dic59—41no—
38C2×C22⋊Dic59—22no—
39C2×C4×Dic59—41no—
40#14273646—24no—
41#14273644—22no—
42#14273644C2314yes—
43#14273642—22no—
44#14273642—24no—
45#14273642C2314yes—
46C2×#41×C59—41no—
47C2×#41×C59—41no—
48C2×#41×C59—42no—
49C2×#41×C59C2×C412yes—
50C2×#41×C59D412yes—
51C2×#41×C59D412yes—
52C2×#41×C59C2312yes—
53C22×Q8×C59—21no—
54C22×Q8×C59—21no—
55C22×Q8×C59—22no—
56C22×Q8×C59—42no—
57C22×Q8×C59—41no—
58C22×Q8×C59—41no—
59C22×Q8×C59C2×C414yescomplemented
60C22×Q8×C59D411yescharacteristic
61C22×D4×C59C2×C411yescharacteristic
62C22×D4×C59D411yescharacteristic
63C22×D4×C59C2311yescharacteristic
64C23×C4×C59C2×C411yescharacteristic
65C23×C4×C59C2×C412yes—
66C23×C4×C59D411yescharacteristic
67C23×C4×C59D411yescharacteristic
68C4⋊Q8×C59—24no—
69C4⋊Q8×C59—24no—
70#124×C59—24no—
71#124×C59—24no—
72#123×C59—24no—
73#123×C59—22no—
74#123×C59—22no—
75#123×C59C2×C414yes—
76#123×C59C2×C418yescomplemented
77#123×C59C2314yes—
78C22⋊Q8×C59—24no—
79C22⋊Q8×C59—24no—
80C22⋊Q8×C59—24no—
81C22⋊Q8×C59—24no—
82#117×C59—22no—
83#117×C59C2×C414yescomplemented
84C2×C4⋊C4×C59—21no—
85C2×C4⋊C4×C59—22no—
86C2×C4⋊C4×C59—22no—
87C2×C22⋊C4×C59—22no—
88C2×C22⋊C4×C59—22no—
89C2×C22⋊C4×C59C2×C412yes—
90C2×C42×C59—21no—
91C24×C59C22⋊C411yescharacteristic
92C24×C59C22×C411yescharacteristic
93C24×C59C2×D411yescharacteristic
94#41×C59—42no—
95#41×C59—42no—
96#41×C59—42no—
97#41×C59—42no—
98#41×C59—42no—
99#41×C59—42no—
100#41×C59—42no—
101#41×C59—42no—
102C2×Q8×C59—24no—
103C2×Q8×C59—24no—
104C2×Q8×C59—41no—
105C2×Q8×C59—41no—
106C2×Q8×C59—42no—
107C2×Q8×C59—42no—
108C2×Q8×C59—44no—
109C2×Q8×C59—44no—
110C2×Q8×C59—44no—
111C2×Q8×C59—44no—
112C2×Q8×C59—42no—
113C2×Q8×C59—41no—
114C2×Q8×C59—41no—
115C2×Q8×C59C22⋊C412yes—
116C2×Q8×C59C2×D412yes—
117C2×D4×C59—41no—
118C2×D4×C59—41no—
119C2×D4×C59—41no—
120C2×D4×C59—41no—
121C2×D4×C59—41no—
122C2×D4×C59—41no—
123C2×D4×C59C22⋊C412yes—
124C2×D4×C59C22⋊C412yes—
125C2×D4×C59C22×C411yescharacteristic
126C2×D4×C59C22×C412yes—
127C2×D4×C59C2×D411yescharacteristic
128C2×D4×C59C2×D412yes—
129C2×D4×C59C2×D411yescharacteristic
130C2×D4×C59C2411yescharacteristic
131C22×C4×C59—22no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—22no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—22no—
140C22×C4×C59—22no—
141C22×C4×C59—22no—
142C22×C4×C59—22no—
143C22×C4×C59—42no—
144C22×C4×C59—42no—
145C22×C4×C59C22⋊C411yescharacteristic
146C22×C4×C59C22⋊C411yescharacteristic
147C22×C4×C59C22⋊C412yes—
148C22×C4×C59C22⋊C412yes—
149C22×C4×C59C22⋊C412yes—
150C22×C4×C59C22⋊C411yescharacteristic
151C22×C4×C59C22×C412yes—
152C22×C4×C59C22×C414yes—
153C22×C4×C59C2×D412yes—
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D412yes—
156C22×Dic59—42no—
157C4⋊C4×C59—24no—
158C4⋊C4×C59—24no—
159C4⋊C4×C59—28no—
160C4⋊C4×C59—24no—
161C22⋊C4×C59—24no—
162C22⋊C4×C59—28no—
163C22⋊C4×C59—22no—
164C22⋊C4×C59—24no—
165C22⋊C4×C59C22×C414yes—
166C42×C59—24no—
167C22⋊Dic59—24no—
168C22⋊Dic59—42no—
169C4⋊Dic59—44no—
170C4×Dic59—44no—
171#11745664—24no—
172#11745664—42no—
173#11745664—44no—
174C2×C59⋊C8—42no—
175C2×C59⋊C8—44no—
176C23×C59—21no—
177C23×C59—21no—
178C23×C59—21no—
179C23×C59—21no—
180C23×C59—21no—
181C23×C59—21no—
182C23×C59—42no—
183C23×C59—42no—
184C23×C59—42no—
185C23×C59C2×C22⋊C411yescharacteristic
186C23×C59C2×C22⋊C411yescharacteristic
187C23×C59C2×C22⋊C412yes—
188C23×C59C2×C22⋊C411yescharacteristic
189C23×C59C23×C411yescharacteristic
190C23×C59C22×D411yescharacteristic
191Q8×C59—42no—
192Q8×C59—42no—
193Q8×C59—48no—
194Q8×C59—44no—
195Q8×C59—44no—
196D4×C59—42no—
197D4×C59—42no—
198D4×C59—42no—
199D4×C59—41no—
200D4×C59—41no—
201D4×C59—41no—
202D4×C59—42no—
203D4×C59—41no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—22no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—24no—
223C2×C4×C59—24no—
224C2×C4×C59—44no—
225C2×C4×C59—44no—
226C2×C4×C59—42no—
227C2×C4×C59—42no—
228C2×C4×C59—44no—
229C2×C4×C59#9912yes—
230C2×C4×C59#9911yescharacteristic
231C2×C4×C59#9911yescharacteristic
232C2×C4×C59C2×C22⋊C412yes—
233C2×C4×C59C2×C22⋊C412yes—
234C2×C4×C59C2×C22⋊C411yescharacteristic
235C2×C4×C59C2×C22⋊C412yes—
236C2×C4×C59C22×D411yescharacteristic
237C2×Dic59—44no—
238C2×Dic59—84no—
239C59⋊C8—44no—
240C2×#2123—591nomaximal, Sylow
241C22×C59—22no—
242C22×C59—21no—
243C22×C59—21no—
244C22×C59—21no—
245C22×C59—22no—
246C22×C59—21no—
247C22×C59—41no—
248C22×C59—42no—
249C22×C59—41no—
250C22×C59—41no—
251C22×C59—41no—
252C22×C59—42no—
253C22×C59—42no—
254C22×C59—42no—
255C22×C59—42no—
256C22×C59—81no—
257C22×C59—81no—
258C22×C59C2×#9911yescharacteristic
259C22×C59C2×#9911yescharacteristic
260C22×C59C22×C22⋊C411yescharacteristic
261C4×C59—21no—
262C4×C59—21no—
263C4×C59—22no—
264C4×C59—22no—
265C4×C59—44no—
266C4×C59—44no—
267C4×C59C2×#9912yes—
268C4×C59C2×#9912yes—
269Dic59—84no—
270C2×#536D5911yescharacteristic
271C2×#413—591no—
272C2×#410—591no—
273#2123—598no—
274C2×#352—594no—
275C2×C59—21no—
276C2×C59—21no—
277C2×C59—41no—
278C2×C59—41no—
279C2×C59—41no—
280C2×C59—41no—
281C2×C59—42no—
282C2×C59#212312yescomplemented
283C2×C59C22×#9911yescharacteristic
284C22×#41C2×D5911yescharacteristic
285C23×Q8Dic5911yescharacteristic, complemented
286C22×M16—1181no—
287#536—22no—
288#536—22no—
289#536Dic5914yescomplemented
290#536C2×D5914yes—
291C2×C4⋊Q8—21no—
292C2×#124—21no—
293C2×#123Dic5912yescomplemented
294C2×#123C2×D5912yes—
295C2×C22⋊Q8—21no—
296C2×C22⋊Q8—21no—
297C2×#117—1181no—
298C2×#117Dic5911yescharacteristic, complemented
299#413—594no—
300#413—1182no—
301C2×#101—1181no—
302C2×#100—592no—
303#410—594no—
304#410—1182no—
305C2×#99—592no—
306C2×#99—1181no—
307#352—5916no—
308C59C2×#212311yescharacteristic, complemented, Sylow
309C2×#41—41no—
310C2×#41—41no—
311C2×#41—42no—
312C2×#41C2×Dic5912yes—
313C2×#41C22⋊D5912yes—
314C2×#41C22⋊D5912yes—
315C2×#41C22×D5912yes—
316C22×Q8—21no—
317C22×Q8—22no—
318C22×Q8—21no—
319C22×Q8—42no—
320C22×Q8—41no—
321C22×Q8—41no—
322C22×Q8C2×Dic5914yescomplemented
323C22×Q8C22⋊D5911yescharacteristic
324C22×D4C2×Dic5911yescharacteristic
325C22×D4C22⋊D5911yescharacteristic
326C22×D4C22×D5911yescharacteristic
327C23×C4C2×Dic5911yescharacteristic
328C23×C4C2×Dic5912yes—
329C23×C4C22⋊D5911yescharacteristic
330C23×C4C22⋊D5911yescharacteristic
331C2×M16—1182no—
332C2×M16—1184no—
333C2×M16—2361no—
334C2×M16—2362no—
335C22×C8—2361no—
336C4⋊Q8—24no—
337C4⋊Q8—24no—
338#124—24no—
339#124—24no—
340#123—24no—
341#123—22no—
342#123—22no—
343#123C2×Dic5914yes—
344#123C2×Dic5918yescomplemented
345#123C22×D5914yes—
346C22⋊Q8—24no—
347C22⋊Q8—24no—
348C22⋊Q8—24no—
349C22⋊Q8—24no—
350#117—22no—
351#117—1184no—
352#117—2362no—
353#117C2×Dic5914yescomplemented
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—2361no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—1182no—
361C2×C22⋊C4C2×Dic5912yes—
362C2×C42—21no—
363C2×C42—2361no—
364#101—1184no—
365#100—594no—
366#100—1182no—
367#99—594no—
368#99—1182no—
369#99—1184no—
370C24C22⋊Dic5911yescharacteristic
371C24C22×Dic5911yescharacteristic
372C24C2×C22⋊D5911yescharacteristic
373#41—42no—
374#41—42no—
375#41—42no—
376#41—42no—
377#41—42no—
378#41—42no—
379#41—42no—
380#41—42no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—41no—
393C2×Q8—41no—
394C2×Q8C22⋊Dic5912yes—
395C2×Q8C2×C22⋊D5912yes—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4C22⋊Dic5912yes—
403C2×D4C22⋊Dic5912yes—
404C2×D4C22×Dic5911yescharacteristic
405C2×D4C22×Dic5912yes—
406C2×D4C2×C22⋊D5911yescharacteristic
407C2×D4C2×C22⋊D5912yes—
408C2×D4C2×C22⋊D5911yescharacteristic
409C2×D4C23×D5911yescharacteristic
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—2362no—
425C22×C4C22⋊Dic5911yescharacteristic
426C22×C4C22⋊Dic5911yescharacteristic
427C22×C4C22⋊Dic5912yes—
428C22×C4C22⋊Dic5912yes—
429C22×C4C22⋊Dic5912yes—
430C22×C4C22⋊Dic5911yescharacteristic
431C22×C4C22×Dic5912yes—
432C22×C4C22×Dic5914yes—
433C22×C4C2×C22⋊D5912yes—
434C22×C4C2×C22⋊D5911yescharacteristic
435C22×C4C2×C22⋊D5912yes—
436M16—1184no—
437M16—2362no—
438M16—2364no—
439C2×C8—2362no—
440C2×C8—2364no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—28no—
444C4⋊C4—24no—
445C4⋊C4—2364no—
446C22⋊C4—24no—
447C22⋊C4—28no—
448C22⋊C4—22no—
449C22⋊C4—24no—
450C22⋊C4—1184no—
451C22⋊C4—2362no—
452C22⋊C4C22×Dic5914yes—
453C42—24no—
454C42—2364no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C2×C22⋊Dic5911yescharacteristic
465C23C2×C22⋊Dic5911yescharacteristic
466C23C2×C22⋊Dic5912yes—
467C23C2×C22⋊Dic5911yescharacteristic
468C23C23×Dic5911yescharacteristic
469C23C22×C22⋊D5911yescharacteristic
470Q8—42no—
471Q8—42no—
472Q8—48no—
473Q8—44no—
474Q8—44no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—41no—
479D4—41no—
480D4—42no—
481D4—41no—
482D4—41no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—4724no—
509C2×C4—2364no—
510C2×C4#1427364212yes—
511C2×C4#1427364211yescharacteristic
512C2×C4#1427364211yescharacteristic
513C2×C4C2×C22⋊Dic5912yes—
514C2×C4C2×C22⋊Dic5912yes—
515C2×C4C2×C22⋊Dic5911yescharacteristic
516C2×C4C2×C22⋊Dic5912yes—
517C2×C4C22×C22⋊D5911yescharacteristic
518C8—2364no—
519C22—22no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—22no—
524C22—21no—
525C22—41no—
526C22—42no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22C2×#1427364211yescharacteristic
537C22C2×#1427364211yescharacteristic
538C22C22×C22⋊Dic5911yescharacteristic
539C4—21no—
540C4—22no—
541C4—21no—
542C4—22no—
543C4—44no—
544C4—44no—
545C4—4724no—
546C4C2×#1427364212yes—
547C4C2×#1427364212yes—
548C2—21no—
549C2—21no—
550C2—41no—
551C2—41no—
552C2—41no—
553C2—41no—
554C2—42no—
555C2#2142575012yesdirect product factor
556C2C22×#1427364211yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC23×C59C23×C4
FrattiniC2×D4C23×D59
FittingC2×C59×#536C2
socleC22×C59C2×#99
prime radicalC22×#41×C59C22
cosocleC2×D4×C59C24
hypercenterC2×#536D59
hypocenterC59C2×#2123
LCSG ▷ C23×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
3. #21425750C2direct (N × H)
4. C2×#16837192C2semidirect
16. #16837192C22semidirect
18. C23×Q8×C59C4semidirect
21. C59×#536C4semidirect
25. C2×#123×C59C4semidirect
29. C2×#117×C59C4semidirect
59. C22×Q8×C59C2×C4semidirect
76. #123×C59C2×C4semidirect
83. #117×C59C2×C4semidirect
282. C2×C59#2123semidirect
285. C23×Q8Dic59semidirect
289. #536Dic59semidirect
293. C2×#123Dic59semidirect
298. C2×#117Dic59semidirect
308. C59C2×#2123semidirect
322. C22×Q8C2×Dic59semidirect
344. #123C2×Dic59semidirect
353. #117C2×Dic59semidirect
555. C2#21425750direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168373862
C2×#168373762
#214257502
C2×#168371922
C2×C59×#5362
C2×#212359