Finite Group Databasebeta

← C2×#11752497 #14480171

Subgroups — C2×#11752497 #14480171

Order 1920.

2336 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
1C2×C4×Dic30C211yesmaximal, characteristic, complemented
2C2×C4×C5×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×C42×D5C211yesmaximal, characteristic
4C2×#91468C211yesmaximal, characteristic, complemented
5C2×C4×#1204C212yesmaximal, complemented
6C2×#91378C212yesmaximal, complemented
7C2×#91376C211yesmaximal, characteristic, complemented
8C2×#91360C211yesmaximal, characteristic
9C2×#91359C211yesmaximal, characteristic
10C2×#91357C212yesmaximal
11C2×#91349C212yesmaximal
12#11752497C2116yesmaximal, direct product factor
13C2×#65912—31nomaximal, Hall
14C22×Dic30C2211yescharacteristic
15C22×C5×Dic6C2211yescharacteristic
16C22×C3×C4×D5C2211yescharacteristic
17C22×C3×C4×D5C2211yescharacteristic
18C22×C3×C4×D5C2211yescharacteristic
19C22×#1204C2212yes—
20C2×C3×C42×C5C2211yescharacteristic
21C2×C4⋊Dic15C2211yescharacteristic
22C2×#6712C2212yes—
23C2×C4×Dic15C2212yes—
24C4×Dic30C2218yescomplemented
25C2×C5×C4⋊Dic3C2211yescharacteristic
26C2×C5×Dic3⋊C4C2212yes—
27C2×C4×C5×Dic3C2212yes—
28C4×C5×Dic6C2218yescomplemented
29C2×C3×C4×Dic5C2211yescharacteristic
30C2×C3×C4×Dic5C2211yescharacteristic
31C2×C3×C4×Dic5C2211yescharacteristic
32C3×C42×D5C2218yes—
33C2×Dic3⋊Dic5C2212yes—
34C2×#6669C2212yes—
35C2×#6668C2212yes—
36C2×#6668C2212yes—
37C2×#6668C2212yes—
38C2×#6665C2212yes—
39C2×#6665C2212yes—
40#91468C2218yescomplemented
41C4×#1204C22116yescomplemented
42#91378C22116yescomplemented
43#91376C2218yescomplemented
44#91360C2218yes—
45#91359C2218yes—
46#91357C22116yes—
47#91349C22116yes—
48C2×#5470—51nomaximal, Hall
49C2×C4×C5×Q8—31no—
50C2×#4092—31no—
51C2×#4090—31no—
52#65912—316no—
53C2×#4043—31no—
54C2×#4042—32no—
55C2×#4035—32no—
56C2×#4034—31no—
57C2×#4018—32no—
58C2×C42×D5D311yescharacteristic
59C2×C4×Dic10—32no—
60C2×C4×Dic10—31no—
61C23×C3×D5C2311yescharacteristic
62C22×C3×C4×C5C2311yescharacteristic
63C22×C3×C4×C5C2311yescharacteristic
64C22×C3×C4×C5C2311yescharacteristic
65C22×Dic15C2312yes—
66C22×Dic15C2312yes—
67C2×Dic30—24no—
68C2×Dic30C2314yes—
69C22×C5×Dic3C2312yes—
70C22×C5×Dic3C2312yes—
71C2×C5×Dic6—24no—
72C2×C5×Dic6C2314yes—
73C22×C3×Dic5C2311yescharacteristic
74C22×C3×Dic5C2311yescharacteristic
75C22×C3×Dic5C2311yescharacteristic
76C2×C3×C4×D5—24no—
77C2×C3×C4×D5—24no—
78C2×C3×C4×D5Q818yescomplemented
79C2×C3×C4×D5C2314yes—
80C2×C3×C4×D5C2314yes—
81C2×C3×C4×D5C2314yes—
82C2×#1204—28no—
83C2×#1204C2318yes—
84C3×C42×C5C2314yes—
85C4⋊Dic15C2314yes—
86#6712C2318yes—
87C4×Dic15C2318yes—
88C5×C4⋊Dic3C2314yes—
89C5×Dic3⋊C4C2318yes—
90C4×C5×Dic3C2318yes—
91C3×C4×Dic5C2314yes—
92C3×C4×Dic5C2314yes—
93C3×C4×Dic5C2314yes—
94Dic3⋊Dic5C2318yes—
95#6669C2318yes—
96#6668C2318yes—
97#6668C2318yes—
98#6668C2318yes—
99#6665C2318yes—
100#6665C2318yes—
101C22×C3×C42—51no—
102C2×#926—52no—
103C2×#924—51no—
104C2×#924—51no—
105C2×#924—52no—
106#5470—516no—
107C2×#870—52no—
108C2×#869—51no—
109C2×#869—51no—
110C2×C4×Dic6—51no—
111C2×C4×Dic6—52no—
112C2×C4×Dic6D511yescharacteristic, complemented
113C22×C5×Q8—31no—
114C22×C4×D5C2×D311yescharacteristic
115C22×C4×D5C2×D311yescharacteristic
116C22×C4×D5C2×D311yescharacteristic
117C22×Dic10—31no—
118C22×Dic10—32no—
119C4×C5×Q8—38no—
120C2×C5×C4⋊C4—31no—
121C2×C5×C4⋊C4—32no—
122C2×C42×C5—32no—
123C2×C42×C5C2×D311yescharacteristic
124#4092—38no—
125#4090—38no—
126C2×#685—32no—
127C2×#685—32no—
128C2×C4⋊Dic5—31no—
129C2×C4⋊Dic5—32no—
130C2×#683—32no—
131C2×#683—32no—
132C2×#683—32no—
133C2×#683—32no—
134C2×#683—32no—
135C2×C4×Dic5—32no—
136C2×C4×Dic5C2×D311yescharacteristic
137C2×C4×Dic5C2×D311yescharacteristic
138C2×C4×Dic5C2×D311yescharacteristic
139#4043—38no—
140#4042—316no—
141#4035—316no—
142#4034—38no—
143#4018—316no—
144C42×D5C2×D318yes—
145C4×Dic10—38no—
146C4×Dic10—316no—
147C2×#533—151noSylow
148C23×C3×C5C2411yescharacteristic
149C22×C3×D5—22no—
150C22×C3×D5—22no—
151C22×C3×D5C2×Q814yes—
152C22×C3×D5C2412yes—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5C2×Q814yes—
155C2×C3×C4×C5#4114yes—
156C2×C3×C4×C5C2412yes—
157C2×C3×C4×C5C2412yes—
158C2×C3×C4×C5C2412yes—
159C2×Dic15—24no—
160C2×Dic15—24no—
161C2×Dic15C2414yes—
162C2×Dic15C2414yes—
163Dic30—28no—
164C2×C5×Dic3—24no—
165C2×C5×Dic3—24no—
166C2×C5×Dic3C2414yes—
167C2×C5×Dic3C2414yes—
168C5×Dic6—28no—
169C2×C3×Dic5—22no—
170C2×C3×Dic5C2×Q814yes—
171C2×C3×Dic5#4114yes—
172C2×C3×Dic5C2412yes—
173C2×C3×Dic5C2412yes—
174C2×C3×Dic5C2412yes—
175C3×C4×D5—28no—
176C3×C4×D5—28no—
177C3×C4×D5C2×Q8116yescomplemented
178#1204—216no—
179C23×C3×C4—51no—
180C23×C3×C4—51no—
181C23×C3×C4—51no—
182C22×Dic6—51no—
183C22×Dic6—52no—
184C22×Dic6C2×D511yescharacteristic
185C2×C3×C42—51no—
186C2×C3×C42—51no—
187C2×C3×C42—51no—
188C2×C3×C42—58no—
189C2×C3×C42C2×D511yescharacteristic
190C2×#217—52no—
191C2×#217—52no—
192#926—516no—
193C2×C4⋊Dic3—51no—
194C2×C4⋊Dic3—52no—
195C2×C4⋊Dic3—52no—
196C2×C4⋊Dic3—52no—
197C2×C4⋊Dic3C2×D511yescharacteristic
198#924—58no—
199#924—58no—
200#924—516no—
201C2×Dic3⋊C4—52no—
202C2×Dic3⋊C4—52no—
203C2×Dic3⋊C4—52no—
204C2×Dic3⋊C4C2×D512yes—
205C2×C4×Dic3—52no—
206C2×C4×Dic3C2×D512yes—
207#870—516no—
208#869—58no—
209#869—58no—
210C4×Dic6—58no—
211C4×Dic6—516no—
212C4×Dic6C2×D518yescomplemented
213C23×D5C22×D311yescharacteristic
214C2×C5×Q8—34no—
215C2×C5×Q8—64no—
216C22×C4×C5—32no—
217C22×C4×C5—32no—
218C22×C4×C5C22×D311yescharacteristic
219C22×C4×C5C22×D311yescharacteristic
220C22×C4×C5C22×D311yescharacteristic
221C22×Dic5—32no—
222C22×Dic5—32no—
223C22×Dic5C22×D311yescharacteristic
224C22×Dic5C22×D311yescharacteristic
225C22×Dic5C22×D311yescharacteristic
226C2×C4×D5—24no—
227C2×C4×D5—24no—
228C2×C4×D5Dic618yescomplemented
229C2×C4×D5C22×D314yes—
230C2×C4×D5C22×D314yes—
231C2×C4×D5C22×D314yes—
232C2×Dic10—34no—
233C2×Dic10—38no—
234C2×Dic10—64no—
235C2×Dic10—68no—
236C5×C4⋊C4—38no—
237C5×C4⋊C4—34no—
238C42×C5—38no—
239C42×C5C22×D314yes—
240#685—38no—
241#685—38no—
242C4⋊Dic5—34no—
243C4⋊Dic5—38no—
244#683—38no—
245#683—38no—
246#683—38no—
247#683—38no—
248#683—38no—
249C4×Dic5—38no—
250C4×Dic5C22×D314yes—
251C4×Dic5C22×D314yes—
252C4×Dic5C22×D314yes—
253#533—1516no—
254C2×C4⋊Q8—151no—
255C2×C4⋊Q8—151no—
256C2×#125—152no—
257C2×C22⋊Q8—152no—
258C2×C22⋊Q8—151no—
259C2×C22⋊Q8—151no—
260C2×C4×Q8—31no—
261C2×C4×Q8—152no—
262C2×C4×Q8—151no—
263C2×#117—152no—
264C22×C42—51no—
265C22×C3×C5C22×Q812yes—
266C22×C3×C5C2×#4112yes—
267C22×C3×C5C2×#4112yes—
268C22×C3×C5C2511yescharacteristic
269C2×C3×D5—28no—
270C2×C3×D5—22no—
271C2×C3×D5—22no—
272C2×C3×D5C22×Q814yes—
273C3×C4×C5—22no—
274C3×C4×C5C22×Q814yes—
275C3×C4×C5C2×#4114yes—
276Dic15—24no—
277Dic15—24no—
278C3×Dic5—22no—
279C3×Dic5C22×Q814yes—
280C3×Dic5C2×#4114yes—
281C5×Dic3—24no—
282C5×Dic3—24no—
283C24×C3—51no—
284C22×C3×C4—51no—
285C22×C3×C4—51no—
286C22×C3×C4—58no—
287C22×C3×C4—54no—
288C22×C3×C4—51no—
289C22×C3×C4—54no—
290C22×C3×C4—54no—
291C22×C3×C4—104no—
292C22×C3×C4—104no—
293C22×C3×C4C22×D511yescharacteristic
294C22×C3×C4C22×D511yescharacteristic
295C22×C3×C4C22×D511yescharacteristic
296C22×Dic3—52no—
297C22×Dic3—52no—
298C22×Dic3C22×D512yes—
299C22×Dic3C22×D512yes—
300C2×Dic6—24no—
301C2×Dic6—54no—
302C2×Dic6—58no—
303C2×Dic6—108no—
304C2×Dic6—104no—
305C2×Dic6C22×D514yes—
306C3×C42—54no—
307C3×C42—54no—
308C3×C42—54no—
309C3×C42C22×D514yes—
310#217—58no—
311#217—58no—
312C4⋊Dic3—54no—
313C4⋊Dic3—58no—
314C4⋊Dic3—58no—
315C4⋊Dic3—58no—
316C4⋊Dic3C22×D514yes—
317Dic3⋊C4—58no—
318Dic3⋊C4—58no—
319Dic3⋊C4—58no—
320Dic3⋊C4C22×D518yes—
321C4×Dic3—58no—
322C4×Dic3C22×D518yes—
323C23×C5C23×D311yescharacteristic
324C22×D5—22no—
325C22×D5—22no—
326C22×D5C2×Dic614yes—
327C22×D5C23×D312yes—
328C5×Q8—68no—
329C2×C4×C5—22no—
330C2×C4×C5—34no—
331C2×C4×C5—34no—
332C2×C4×C5—64no—
333C2×C4×C5—64no—
334C2×C4×C5C2×Dic614yes—
335C2×C4×C5#23514yes—
336C2×C4×C5C23×D312yes—
337C2×C4×C5C23×D312yes—
338C2×C4×C5C23×D312yes—
339C2×Dic5—22no—
340C2×Dic5—34no—
341C2×Dic5—34no—
342C2×Dic5—64no—
343C2×Dic5—64no—
344C2×Dic5C2×Dic614yes—
345C2×Dic5#23514yes—
346C2×Dic5C23×D312yes—
347C2×Dic5C23×D312yes—
348C2×Dic5C23×D312yes—
349C4×D5—28no—
350C4×D5—28no—
351C4×D5C2×Dic6116yescomplemented
352Dic10—68no—
353Dic10—616no—
354C22×Q8—31no—
355C22×Q8—152no—
356C22×Q8—151no—
357C23×C4—51no—
358C23×C4—51no—
359C23×C4—51no—
360C4⋊Q8—158no—
361C4⋊Q8—158no—
362#125—1516no—
363C22⋊Q8—1516no—
364C22⋊Q8—158no—
365C22⋊Q8—158no—
366C4×Q8—38no—
367C4×Q8—1516no—
368C4×Q8—158no—
369#117—1516no—
370C2×C4⋊C4—32no—
371C2×C4⋊C4—31no—
372C2×C4⋊C4—152no—
373C2×C4⋊C4—152no—
374C2×C4⋊C4—152no—
375C2×C4⋊C4—152no—
376C2×C4⋊C4—152no—
377C2×C4⋊C4—152no—
378C2×C4⋊C4—151no—
379C2×C22⋊C4—152no—
380C2×C22⋊C4—152no—
381C2×C42—32no—
382C2×C42—51no—
383C2×C42—51no—
384C2×C42—51no—
385C2×C42—58no—
386C2×C42—152no—
387C2×C42D3×D511yescharacteristic
388C2×C3×C5#53314yescomplemented
389C2×C3×C5C23×Q811yescharacteristic
390C2×C3×C5C22×#4111yescharacteristic
391C2×C3×C5C22×#4111yescharacteristic
392C3×D5—24no—
393C23×C3—54no—
394C23×C3—52no—
395C23×C3—102no—
396C23×C3—102no—
397C23×C3C23×D511yescharacteristic
398C2×C3×C4—22no—
399C2×C3×C4—54no—
400C2×C3×C4—54no—
401C2×C3×C4—52no—
402C2×C3×C4—52no—
403C2×C3×C4—516no—
404C2×C3×C4—52no—
405C2×C3×C4—108no—
406C2×C3×C4—102no—
407C2×C3×C4—108no—
408C2×C3×C4#70914yes—
409C2×C3×C4Q8×D514yes—
410C2×C3×C4C23×D512yes—
411C2×C3×C4C23×D512yes—
412C2×C3×C4C23×D512yes—
413C2×Dic3—24no—
414C2×Dic3—24no—
415C2×Dic3—54no—
416C2×Dic3—54no—
417C2×Dic3—104no—
418C2×Dic3—104no—
419C2×Dic3C23×D514yes—
420C2×Dic3C23×D514yes—
421Dic6—28no—
422Dic6—1016no—
423Dic6—108no—
424C22×C5C22×Dic612yes—
425C22×C5C2×#23512yes—
426C22×C5C2×#23512yes—
427C22×C5C24×D311yescharacteristic
428C2×D5—22no—
429C2×D5—22no—
430C2×D5—28no—
431C2×D5C22×Dic614yes—
432C4×C5—22no—
433C4×C5—64no—
434C4×C5—64no—
435C4×C5C22×Dic614yes—
436C4×C5C2×#23514yes—
437Dic5—22no—
438Dic5—64no—
439Dic5—64no—
440Dic5C22×Dic614yes—
441Dic5C2×#23514yes—
442C24—51no—
443C2×Q8—34no—
444C2×Q8—64no—
445C2×Q8—158no—
446C2×Q8—154no—
447C2×Q8—308no—
448C2×Q8—304no—
449C22×C4—32no—
450C22×C4—32no—
451C22×C4—51no—
452C22×C4—51no—
453C22×C4—58no—
454C22×C4—54no—
455C22×C4—51no—
456C22×C4—54no—
457C22×C4—54no—
458C22×C4—104no—
459C22×C4—104no—
460C22×C4—152no—
461C22×C4—152no—
462C22×C4C2×D3×D511yescharacteristic
463C22×C4C2×D3×D511yescharacteristic
464C22×C4C2×D3×D511yescharacteristic
465C4⋊C4—38no—
466C4⋊C4—34no—
467C4⋊C4—158no—
468C4⋊C4—158no—
469C4⋊C4—158no—
470C4⋊C4—158no—
471C4⋊C4—158no—
472C4⋊C4—158no—
473C4⋊C4—154no—
474C22⋊C4—158no—
475C22⋊C4—158no—
476C42—38no—
477C42—54no—
478C42—54no—
479C42—54no—
480C42—158no—
481C42C2×D3×D514yes—
482C3×C5C2×#53311yescharacteristic, complemented, Hall
483C22×C3—54no—
484C22×C3—108no—
485C22×C3—102no—
486C22×C3—102no—
487C22×C3C2×#70912yes—
488C22×C3C2×Q8×D512yes—
489C22×C3D5×#4112yes—
490C22×C3C24×D511yescharacteristic
491C3×C4—22no—
492C3×C4—54no—
493C3×C4—54no—
494C3×C4—102no—
495C3×C4C2×#70914yes—
496C3×C4C2×Q8×D514yes—
497Dic3—24no—
498Dic3—24no—
499Dic3—104no—
500Dic3—104no—
501C2×C5#547014yescomplemented
502C2×C5C23×Dic611yescharacteristic
503C2×C5C22×#23511yescharacteristic
504C2×C5C22×#23511yescharacteristic
505D5—24no—
506C23—54no—
507C23—52no—
508C23—102no—
509C23—102no—
510C23C22×D3×D511yescharacteristic
511Q8—68no—
512Q8—3016no—
513Q8—308no—
514C2×C4—22no—
515C2×C4—34no—
516C2×C4—34no—
517C2×C4—54no—
518C2×C4—54no—
519C2×C4—52no—
520C2×C4—52no—
521C2×C4—516no—
522C2×C4—52no—
523C2×C4—64no—
524C2×C4—64no—
525C2×C4—108no—
526C2×C4—102no—
527C2×C4—108no—
528C2×C4—154no—
529C2×C4—154no—
530C2×C4—304no—
531C2×C4—304no—
532C2×C4D5×Dic614yes—
533C2×C4#677114yes—
534C2×C4C22×D3×D512yes—
535C2×C4C22×D3×D512yes—
536C2×C4C22×D3×D512yes—
537C2×C3—104no—
538C2×C3#6591214yescomplemented
539C2×C3C22×#70911yescharacteristic
540C2×C3C22×Q8×D511yescharacteristic
541C2×C3C2×D5×#4111yescharacteristic
542C5C2×#547011yescharacteristic, complemented, Sylow
543C22—54no—
544C22—108no—
545C22—102no—
546C22—102no—
547C22C2×D5×Dic612yes—
548C22C2×#677112yes—
549C22D5×#23512yes—
550C22C23×D3×D511yescharacteristic
551C4—22no—
552C4—54no—
553C4—54no—
554C4—64no—
555C4—64no—
556C4—102no—
557C4—304no—
558C4—304no—
559C4C2×D5×Dic614yes—
560C4C2×#677114yes—
561C3C2×#6591211yescharacteristic, complemented, Sylow
562C2—104no—
563C2#1175249714yesdirect product factor
564C2C22×D5×Dic611yescharacteristic
565C2C22×#677111yescharacteristic
566C2C2×D5×#23511yescharacteristic
subgroupG/H
centerC22×C4C2×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC2×C3×C42×C5C22
socleC23×C3×C5C24
prime radicalC23×C3×D5C23
hypercenterC2×C42D3×D5
hypocenterC3×C5C2×#533
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic30C2semidirect
2. C2×C4×C5×Dic6C2semidirect
4. C2×#91468C2semidirect
5. C2×C4×#1204C2semidirect
6. C2×#91378C2semidirect
7. C2×#91376C2semidirect
12. #11752497C2direct (N × H)
24. C4×Dic30C22semidirect
28. C4×C5×Dic6C22semidirect
40. #91468C22semidirect
41. C4×#1204C22semidirect
42. #91378C22semidirect
43. #91376C22semidirect
78. C2×C3×C4×D5Q8semidirect
112. C2×C4×Dic6D5semidirect
177. C3×C4×D5C2×Q8semidirect
212. C4×Dic6C2×D5semidirect
228. C2×C4×D5Dic6semidirect
351. C4×D5C2×Dic6semidirect
388. C2×C3×C5#533semidirect
482. C3×C5C2×#533semidirect
501. C2×C5#5470semidirect
538. C2×C3#65912semidirect
542. C5C2×#5470semidirect
561. C3C2×#65912semidirect
563. C2#11752497direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×Dic302
C2×C4×C5×Dic62
C2×C3×C42×D52
C2×#914682
C2×C4×#12042
C2×#913782
C2×#913762
C2×#913602
C2×#913592
C2×#913572
C2×#913492
#117524972
C2×#659123
C2×#54705