Finite Group Databasebeta

← C2×#11755174 #14480151

Subgroups — C2×#11755174 #14480151

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×Q8×Dic15C211yesmaximal, characteristic, complemented
2C2×C5×#944C211yesmaximal, characteristic, complemented
3C2×C3×#4092C211yesmaximal, characteristic
4#11755174C2116yesmaximal, direct product factor
5C2×#91469C211yesmaximal, characteristic, complemented
6C2×#91422C212yesmaximal
7C2×#91394C212yesmaximal
8C2×C4×D5×Dic3C211yesmaximal, characteristic
9C2×#91359C211yesmaximal, characteristic
10C2×#91344C211yesmaximal, characteristic, complemented
11C2×#91343C212yesmaximal, complemented
12C2×#91325C212yesmaximal, complemented
13C2×#66062—31nomaximal, Hall
14C22×C3×C5×Q8C2211yescharacteristic
15C22×C5×Dic6C2211yescharacteristic
16C22×C3×C4×D5C2211yescharacteristic
17C22×#1204C2212yes—
18C22×D5×Dic3C2211yescharacteristic
19C22×D5×Dic3C2211yescharacteristic
20Q8×Dic15C2218yescomplemented
21C2×C4⋊Dic15C2212yes—
22C2×C4⋊Dic15C2211yescharacteristic
23C2×C4×Dic15C2211yescharacteristic
24C2×C4×Dic15C2212yes—
25C5×#944C2218yescomplemented
26C2×C5×Dic3⋊C4C2212yes—
27C2×C5×Dic3⋊C4C2212yes—
28C2×C4×C5×Dic3C2211yescharacteristic
29C3×#4092C2218yes—
30C2×C3×#685C2212yes—
31C2×C3×C4⋊Dic5C2211yescharacteristic
32C2×C3×#683C2212yes—
33C2×Dic3⋊Dic5C2212yes—
34C2×Dic3⋊Dic5C2212yes—
35C2×#6669C2212yes—
36C2×#6665C2212yes—
37C2×Dic3×Dic5C2211yescharacteristic
38C2×Dic3×Dic5C2211yescharacteristic
39C2×Dic3×Dic5C2212yes—
40#91469C2218yescomplemented
41#91422C22116yes—
42#91394C22116yes—
43C4×D5×Dic3C2218yes—
44#91359C2218yes—
45#91344C2218yescomplemented
46#91343C22116yescomplemented
47#91325C22116yescomplemented
48C2×#5622—51nomaximal, Hall
49C2×C5×C4⋊Q8—31no—
50C2×#4092—31no—
51C2×#4092D311yescharacteristic
52C2×Q8×Dic5—31no—
53C2×Q8×Dic5—31no—
54#66062—316no—
55C2×#4043—32no—
56C2×#4038—32no—
57C2×#4036—32no—
58C2×#4033—32no—
59C2×C42×D5—31no—
60C2×#4015—31no—
61C23×C3×D5C2311yescharacteristic
62C2×C3×C5×Q8—24no—
63C2×C3×C5×Q8C2314yes—
64C22×C3×C4×C5C2312yes—
65C22×C3×C4×C5C2311yescharacteristic
66C22×Dic15C2311yescharacteristic
67C22×Dic15C2312yes—
68C22×Dic15C2311yescharacteristic
69C22×C5×Dic3C2312yes—
70C22×C5×Dic3C2311yescharacteristic
71C22×C5×Dic3C2311yescharacteristic
72C2×C5×Dic6—24no—
73C2×C5×Dic6C2314yes—
74C22×C3×Dic5C2311yescharacteristic
75C22×C3×Dic5C2312yes—
76C2×C3×C4×D5—24no—
77C2×C3×C4×D5C2314yes—
78C2×#1204—28no—
79C2×#1204C2318yes—
80C2×D5×Dic3—24no—
81C2×D5×Dic3Q818yescomplemented
82C2×D5×Dic3C2314yes—
83C2×D5×Dic3C2314yes—
84C4⋊Dic15C2318yes—
85C4⋊Dic15C2314yes—
86C4×Dic15C2318yes—
87C4×Dic15C2314yes—
88C5×Dic3⋊C4C2318yes—
89C5×Dic3⋊C4C2318yes—
90C4×C5×Dic3C2314yes—
91C3×#685C2318yes—
92C3×C4⋊Dic5C2314yes—
93C3×#683C2318yes—
94Dic3⋊Dic5C2318yes—
95Dic3⋊Dic5C2318yes—
96#6669C2318yes—
97#6665C2318yes—
98Dic3×Dic5C2318yes—
99Dic3×Dic5C2314yes—
100Dic3×Dic5C2314yes—
101C2×C3×C22⋊Q8—51no—
102C2×Q8×Dic3—51no—
103C2×#944D511yescharacteristic, complemented
104C2×#924—51no—
105C22×C4×Dic3—51no—
106#5622—516no—
107C2×#889—52no—
108C2×#888—51no—
109C2×#887—51no—
110C2×#887—52no—
111C2×#878—52no—
112C2×#877—52no—
113C22×C5×Q8—31no—
114C22×C5×Q8C2×D311yescharacteristic
115C22×C4×D5—31no—
116C22×C4×D5—31no—
117C22×C4×D5C2×D311yescharacteristic
118C22×Dic10—32no—
119C5×C4⋊Q8—38no—
120C2×C5×C4⋊C4—32no—
121C2×C5×C4⋊C4—32no—
122C2×C42×C5—31no—
123#4092—38no—
124#4092C2×D318yes—
125Q8×Dic5—38no—
126Q8×Dic5—38no—
127C2×#685—32no—
128C2×#685C2×D312yes—
129C2×C4⋊Dic5—31no—
130C2×C4⋊Dic5—32no—
131C2×C4⋊Dic5—32no—
132C2×C4⋊Dic5—32no—
133C2×C4⋊Dic5C2×D311yescharacteristic
134C2×#683—32no—
135C2×#683C2×D312yes—
136C2×C4×Dic5—31no—
137C2×C4×Dic5—32no—
138C2×C4×Dic5—31no—
139C2×C4×Dic5—31no—
140C2×C4×Dic5—32no—
141#4043—316no—
142#4038—316no—
143#4036—316no—
144#4033—316no—
145C42×D5—38no—
146#4015—38no—
147C2×#533—151noSylow
148C23×C3×C5C2411yescharacteristic
149C22×C3×D5—22no—
150C22×C3×D5—22no—
151C22×C3×D5C2×Q814yes—
152C22×C3×D5C2412yes—
153C3×C5×Q8—28no—
154C2×C3×C4×C5—24no—
155C2×C3×C4×C5#4114yes—
156C2×C3×C4×C5C2414yes—
157C2×C3×C4×C5C2412yes—
158C2×Dic15—22no—
159C2×Dic15—24no—
160C2×Dic15C2×Q814yes—
161C2×Dic15C2412yes—
162C2×Dic15C2414yes—
163C2×Dic15C2412yes—
164C2×C5×Dic3—24no—
165C2×C5×Dic3C2×Q814yes—
166C2×C5×Dic3#4114yes—
167C2×C5×Dic3C2414yes—
168C2×C5×Dic3C2412yes—
169C2×C5×Dic3C2412yes—
170C5×Dic6—28no—
171C2×C3×Dic5—24no—
172C2×C3×Dic5—22no—
173C2×C3×Dic5C2412yes—
174C2×C3×Dic5C2414yes—
175C3×C4×D5—28no—
176#1204—216no—
177D5×Dic3—28no—
178D5×Dic3C2×Q8116yescomplemented
179C22×C3×Q8C2×D511yescharacteristic
180C23×C3×C4—51no—
181C23×Dic3—51no—
182C23×Dic3—51no—
183C22×Dic6—52no—
184C22×Dic6C2×D511yescharacteristic
185C3×C22⋊Q8—58no—
186C2×C3×C4⋊C4—51no—
187C2×C3×C4⋊C4—52no—
188C2×C3×C22⋊C4—52no—
189C2×#217—52no—
190Q8×Dic3—58no—
191#944C2×D518yescomplemented
192C2×C4⋊Dic3—52no—
193C2×C4⋊Dic3—51no—
194#924—58no—
195C2×Dic3⋊C4—52no—
196C2×Dic3⋊C4—52no—
197C2×Dic3⋊C4—52no—
198C2×Dic3⋊C4C2×D512yes—
199C2×Dic3⋊C4C2×D512yes—
200C2×C4×Dic3—52no—
201C2×C4×Dic3—51no—
202C2×C4×Dic3—52no—
203C2×C4×Dic3—51no—
204C2×C4×Dic3—51no—
205C2×C4×Dic3—58no—
206C2×C4×Dic3C2×D511yescharacteristic
207#889—516no—
208#888—58no—
209#887—58no—
210#887—516no—
211#878—516no—
212#877—516no—
213C23×D5C22×D311yescharacteristic
214C2×C5×Q8—24no—
215C2×C5×Q8—34no—
216C2×C5×Q8—64no—
217C2×C5×Q8C22×D314yes—
218C22×C4×C5—31no—
219C22×C4×C5—31no—
220C22×C4×C5—32no—
221C22×C4×C5C22×D312yes—
222C22×C4×C5C22×D311yescharacteristic
223C22×Dic5—31no—
224C22×Dic5—32no—
225C22×Dic5—31no—
226C22×Dic5C22×D312yes—
227C22×Dic5C22×D311yescharacteristic
228C2×C4×D5—24no—
229C2×C4×D5—38no—
230C2×C4×D5—34no—
231C2×C4×D5—34no—
232C2×C4×D5—64no—
233C2×C4×D5C22×D314yes—
234C2×Dic10—38no—
235C2×Dic10—68no—
236C5×C4⋊C4—38no—
237C5×C4⋊C4—38no—
238C42×C5—34no—
239#685—38no—
240#685C22×D318yes—
241C4⋊Dic5—34no—
242C4⋊Dic5—38no—
243C4⋊Dic5—38no—
244C4⋊Dic5—38no—
245C4⋊Dic5C22×D314yes—
246#683—38no—
247#683C22×D318yes—
248C4×Dic5—34no—
249C4×Dic5—38no—
250C4×Dic5—34no—
251C4×Dic5—34no—
252C4×Dic5—38no—
253#533—1516no—
254C2×C4⋊Q8—31no—
255C2×C4⋊Q8—151no—
256C2×#125—152no—
257C2×C22⋊Q8—51no—
258C2×C22⋊Q8—152no—
259C2×C22⋊Q8—151no—
260C2×C4×Q8—152no—
261C2×C4×Q8—151no—
262C2×C4×Q8—151no—
263C2×#117—152no—
264C22×C42—151no—
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—24no—
274C3×C4×C5C2×#4114yes—
275Dic15—22no—
276Dic15—24no—
277Dic15C22×Q814yes—
278C3×Dic5—24no—
279C3×Dic5—22no—
280C5×Dic3—24no—
281C5×Dic3C22×Q814yes—
282C5×Dic3C2×#4114yes—
283C24×C3—51no—
284C2×C3×Q8—24no—
285C2×C3×Q8C22×D514yes—
286C22×C3×C4—51no—
287C22×C3×C4—52no—
288C22×C3×C4—54no—
289C22×C3×C4—104no—
290C22×C3×C4C22×D512yes—
291C22×C3×C4C22×D511yescharacteristic
292C22×Dic3—51no—
293C22×Dic3—58no—
294C22×Dic3—54no—
295C22×Dic3—51no—
296C22×Dic3—52no—
297C22×Dic3—54no—
298C22×Dic3—104no—
299C22×Dic3C22×D512yes—
300C22×Dic3C22×D511yescharacteristic
301C22×Dic3C22×D511yescharacteristic
302C2×Dic6—24no—
303C2×Dic6—58no—
304C2×Dic6—108no—
305C2×Dic6C22×D514yes—
306C3×C4⋊C4—58no—
307C3×C4⋊C4—54no—
308C3×C22⋊C4—58no—
309#217—58no—
310C4⋊Dic3—58no—
311C4⋊Dic3—54no—
312Dic3⋊C4—58no—
313Dic3⋊C4—58no—
314Dic3⋊C4—58no—
315Dic3⋊C4C22×D518yes—
316Dic3⋊C4C22×D518yes—
317C4×Dic3—58no—
318C4×Dic3—54no—
319C4×Dic3—58no—
320C4×Dic3—54no—
321C4×Dic3—54no—
322C4×Dic3C22×D514yes—
323C23×C5C23×D311yescharacteristic
324C22×D5—22no—
325C22×D5—22no—
326C22×D5D3×Q814yes—
327C22×D5C23×D312yes—
328C5×Q8—28no—
329C5×Q8—68no—
330C2×C4×C5—24no—
331C2×C4×C5—34no—
332C2×C4×C5—34no—
333C2×C4×C5—32no—
334C2×C4×C5—32no—
335C2×C4×C5—34no—
336C2×C4×C5—64no—
337C2×C4×C5#23714yes—
338C2×C4×C5C23×D314yes—
339C2×C4×C5C23×D312yes—
340C2×Dic5—22no—
341C2×Dic5—24no—
342C2×Dic5—34no—
343C2×Dic5—32no—
344C2×Dic5—32no—
345C2×Dic5—34no—
346C2×Dic5—64no—
347C2×Dic5—62no—
348C2×Dic5C23×D312yes—
349C2×Dic5C23×D314yes—
350C4×D5—28no—
351C4×D5—316no—
352C4×D5—68no—
353Dic10—616no—
354C22×Q8—31no—
355C22×Q8—152no—
356C22×Q8D3×D511yescharacteristic
357C23×C4—51no—
358C23×C4—151no—
359C23×C4—151no—
360C4⋊Q8—38no—
361C4⋊Q8—158no—
362#125—1516no—
363C22⋊Q8—58no—
364C22⋊Q8—1516no—
365C22⋊Q8—158no—
366C4×Q8—1516no—
367C4×Q8—158no—
368C4×Q8—158no—
369#117—1516no—
370C2×C4⋊C4—32no—
371C2×C4⋊C4—32no—
372C2×C4⋊C4—52no—
373C2×C4⋊C4—51no—
374C2×C4⋊C4—152no—
375C2×C4⋊C4—152no—
376C2×C4⋊C4—152no—
377C2×C4⋊C4—152no—
378C2×C4⋊C4—151no—
379C2×C22⋊C4—52no—
380C2×C22⋊C4—152no—
381C2×C42—31no—
382C2×C42—158no—
383C2×C42—152no—
384C2×C42—151no—
385C2×C42—151no—
386C2×C42—152no—
387C2×C42—151no—
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
398C3×Q8—28no—
399C2×C3×C4—24no—
400C2×C3×C4—52no—
401C2×C3×C4—54no—
402C2×C3×C4—102no—
403C2×C3×C4—104no—
404C2×C3×C4—108no—
405C2×C3×C4#71114yes—
406C2×C3×C4C23×D514yes—
407C2×C3×C4C23×D512yes—
408C2×Dic3—24no—
409C2×Dic3—54no—
410C2×Dic3—52no—
411C2×Dic3—516no—
412C2×Dic3—52no—
413C2×Dic3—54no—
414C2×Dic3—104no—
415C2×Dic3—102no—
416C2×Dic3—108no—
417C2×Dic3Q8×D514yes—
418C2×Dic3#71314yes—
419C2×Dic3C23×D514yes—
420C2×Dic3C23×D512yes—
421C2×Dic3C23×D512yes—
422Dic6—28no—
423Dic6—1016no—
424C22×C5C2×#23712yes—
425C22×C5C2×D3×Q812yes—
426C22×C5D3×#4112yes—
427C22×C5C24×D311yescharacteristic
428C2×D5—22no—
429C2×D5—22no—
430C2×D5—28no—
431C2×D5C2×D3×Q814yes—
432C4×C5—24no—
433C4×C5—34no—
434C4×C5—34no—
435C4×C5—64no—
436C4×C5C2×#23714yes—
437Dic5—22no—
438Dic5—24no—
439Dic5—34no—
440Dic5—64no—
441Dic5—62no—
442C24—51no—
443C2×Q8—24no—
444C2×Q8—34no—
445C2×Q8—64no—
446C2×Q8—158no—
447C2×Q8—308no—
448C2×Q8C2×D3×D514yes—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—32no—
452C22×C4—51no—
453C22×C4—52no—
454C22×C4—54no—
455C22×C4—104no—
456C22×C4—154no—
457C22×C4—158no—
458C22×C4—154no—
459C22×C4—151no—
460C22×C4—151no—
461C22×C4—152no—
462C22×C4—304no—
463C22×C4C2×D3×D512yes—
464C22×C4C2×D3×D511yescharacteristic
465C4⋊C4—38no—
466C4⋊C4—38no—
467C4⋊C4—58no—
468C4⋊C4—54no—
469C4⋊C4—158no—
470C4⋊C4—158no—
471C4⋊C4—158no—
472C4⋊C4—158no—
473C4⋊C4—154no—
474C22⋊C4—58no—
475C22⋊C4—158no—
476C42—34no—
477C42—154no—
478C42—154no—
479C42—158no—
480C42—154no—
481C42—158no—
482C3×C5C2×#53311yescharacteristic, complemented, Hall
483C22×C3—54no—
484C22×C3—108no—
485C22×C3—102no—
486C22×C3—102no—
487C22×C3C2×#71112yes—
488C22×C3C2×Q8×D512yes—
489C22×C3C2×#71312yes—
490C22×C3C24×D511yescharacteristic
491C3×C4—24no—
492C3×C4—102no—
493C3×C4—104no—
494C3×C4C2×#71114yes—
495Dic3—24no—
496Dic3—54no—
497Dic3—104no—
498Dic3—102no—
499Dic3C2×Q8×D514yes—
500Dic3C2×#71314yes—
501C2×C5#562214yescomplemented
502C2×C5C22×#23711yescharacteristic
503C2×C5C22×D3×Q811yescharacteristic
504C2×C5C2×D3×#4111yescharacteristic
505D5—24no—
506C23—54no—
507C23—52no—
508C23—102no—
509C23—102no—
510C23C22×D3×D511yescharacteristic
511Q8—28no—
512Q8—68no—
513Q8—3016no—
514C2×C4—24no—
515C2×C4—34no—
516C2×C4—34no—
517C2×C4—32no—
518C2×C4—32no—
519C2×C4—34no—
520C2×C4—52no—
521C2×C4—54no—
522C2×C4—64no—
523C2×C4—104no—
524C2×C4—102no—
525C2×C4—108no—
526C2×C4—1516no—
527C2×C4—154no—
528C2×C4—152no—
529C2×C4—152no—
530C2×C4—154no—
531C2×C4—308no—
532C2×C4—302no—
533C2×C4—304no—
534C2×C4#678014yes—
535C2×C4C22×D3×D514yes—
536C2×C4C22×D3×D512yes—
537C2×C3—104no—
538C2×C3#6606214yescomplemented
539C2×C3C22×#71111yescharacteristic
540C2×C3C22×Q8×D511yescharacteristic
541C2×C3C22×#71311yescharacteristic
542C5C2×#562211yescharacteristic, complemented, Sylow
543C22—54no—
544C22—102no—
545C22—108no—
546C22—102no—
547C22D3×Q8×D512yes—
548C22D3×#71312yes—
549C22C2×#678012yes—
550C22C23×D3×D511yescharacteristic
551C4—24no—
552C4—34no—
553C4—34no—
554C4—64no—
555C4—102no—
556C4—104no—
557C4—154no—
558C4—302no—
559C4—304no—
560C4C2×#678014yes—
561C3C2×#6606211yescharacteristic, complemented, Sylow
562C2—104no—
563C2#1175517414yesdirect product factor
564C2C2×D3×Q8×D511yescharacteristic
565C2C2×D3×#71311yescharacteristic
566C2C22×#678011yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC22×C3×C5×Q8C22
socleC23×C3×C5C24
prime radicalC23×C3×D5C23
hypercenterC22×Q8D3×D5
hypocenterC3×C5C2×#533
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic15C2semidirect
2. C2×C5×#944C2semidirect
4. #11755174C2direct (N × H)
5. C2×#91469C2semidirect
10. C2×#91344C2semidirect
11. C2×#91343C2semidirect
12. C2×#91325C2semidirect
20. Q8×Dic15C22semidirect
25. C5×#944C22semidirect
40. #91469C22semidirect
45. #91344C22semidirect
46. #91343C22semidirect
47. #91325C22semidirect
81. C2×D5×Dic3Q8semidirect
103. C2×#944D5semidirect
178. D5×Dic3C2×Q8semidirect
191. #944C2×D5semidirect
388. C2×C3×C5#533semidirect
482. C3×C5C2×#533semidirect
501. C2×C5#5622semidirect
538. C2×C3#66062semidirect
542. C5C2×#5622semidirect
561. C3C2×#66062semidirect
563. C2#11755174direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×Dic152
C2×C5×#9442
C2×C3×#40922
#117551742
C2×#914692
C2×#914222
C2×#913942
C2×C4×D5×Dic32
C2×#913592
C2×#913442
C2×#913432
C2×#913252
C2×#660623
C2×#56225