Finite Group Databasebeta

← #14319731

Subgroups — #14319731

Order 1920.

1560 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×C3×C4×C22⋊D5C211yesmaximal, characteristic
2C2×Dic3×C22⋊D5C212yesmaximal
3C4×#6719C211yesmaximal, characteristic, complemented
4C2×C5×C3⋊#95C211yesmaximal, characteristic, complemented
5Dic3×C22⋊Dic5C212yesmaximal, complemented
6#11748884C212yesmaximal, complemented
7D5×C3⋊#95C211yesmaximal, characteristic, complemented
8#11748702C211yesmaximal, characteristic, complemented
9#11748638C211yesmaximal, characteristic, complemented
10#11748637C212yesmaximal, complemented
11#11748635C211yesmaximal, characteristic, complemented
12#10605053—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C23×C5×Dic3C2212yes—
15C22×C3×C22⋊D5C2211yescharacteristic
16C22×C3×C4×D5C2211yescharacteristic
17C22×D5×Dic3C2212yes—
18C2×#6719C2212yes—
19C2×C4×Dic15C2212yes—
20C2×C3×C22⋊Dic5C2211yescharacteristic
21C3×C4×C22⋊D5C418yescomplemented
22C2×C3×#685C2211yescharacteristic
23C2×C3×#683C2211yescharacteristic
24C2×C3×C4×Dic5C2211yescharacteristic
25Dic3×C22⋊D5C4116yescomplemented
26C2×#6668C2212yes—
27C2×#6665C2212yes—
28C2×Dic3×Dic5C2212yes—
29C2×Dic3×Dic5C2212yes—
30C2×Dic3×Dic5C2212yes—
31#91115C2211yescharacteristic
32#91115C2211yescharacteristic
33C5×C3⋊#95C2211yescharacteristic
34C5×C3⋊#95C2212yes—
35C5×C3⋊#95C2211yescharacteristic
36#90994C2212yes—
37#90994C2212yes—
38#82317—51nomaximal, Hall
39C2×C4×C22⋊D5—32no—
40C2×C4×C22⋊D5D311yescharacteristic
41C2×C5×#95—31no—
42C4×C22⋊Dic5—31no—
43C4×C22⋊Dic5—32no—
44#64955—32no—
45#64955—31no—
46D5×#95—31no—
47#64938—32no—
48#64938—31no—
49#64937—31no—
50C24×C3×C5C2311yescharacteristic
51C23×C3×D5C2311yescharacteristic
52C22×C3×C4×C5—42no—
53C22×C3×C4×C5C2×C414yescomplemented
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×Dic15C2312yes—
57C22×Dic15C2312yes—
58C22×C5×Dic3—42no—
59C22×C5×Dic3—42no—
60C22×C5×Dic3C2×C418yescomplemented
61C22×C5×Dic3C2312yes—
62C22×C5×Dic3C2312yes—
63C2×C3×C22⋊D5C2×C418yes—
64C2×C3×C22⋊D5C2×C414yes—
65C22×C3×Dic5C2311yescharacteristic
66C22×C3×Dic5C2311yescharacteristic
67C22×C3×Dic5C2311yescharacteristic
68C2×C3×C4×D5—42no—
69C2×C3×C4×D5C2×C414yescomplemented
70C2×D5×Dic3—42no—
71C2×D5×Dic3—42no—
72C2×D5×Dic3C2×C418yescomplemented
73#6719C2×C418yescomplemented
74C4×Dic15—42no—
75C3×C22⋊Dic5C2×C414yescomplemented
76C3×#685C2×C414yescomplemented
77C3×#683C2×C414yescomplemented
78C3×C4×Dic5C2×C414yescomplemented
79#6668C2×C418yescomplemented
80#6665C2×C418yescomplemented
81Dic3×Dic5—42no—
82Dic3×Dic5—42no—
83Dic3×Dic5C2×C418yescomplemented
84C2×C3×C4×D4—51no—
85C2×D4×Dic3—52no—
86C4×#217—51no—
87C4×#217—51no—
88C2×C3⋊#95—51no—
89C2×C3⋊#95D511yescharacteristic, complemented
90Dic3×C4⋊C4—52no—
91Dic3×C22⋊C4—52no—
92Dic3×C22⋊C4—52no—
93C4×C4⋊Dic3—51no—
94#4932—51no—
95C23×C4×C5—32no—
96C23×C4×C5C2×D311yescharacteristic
97C22×C22⋊D5C2×D311yescharacteristic
98C22×C4×D5—32no—
99C22×C4×D5C2×D311yescharacteristic
100C2×C22⋊Dic5—32no—
101C2×C22⋊Dic5C2×D311yescharacteristic
102C4×C22⋊D5—316no—
103C4×C22⋊D5Dic318yescomplemented
104C2×#685—32no—
105C2×#685C2×D311yescharacteristic
106C2×#683—32no—
107C2×#683C2×D311yescharacteristic
108C2×C4×Dic5—32no—
109C2×C4×Dic5—32no—
110C2×C4×Dic5—32no—
111C2×C4×Dic5—32no—
112C2×C4×Dic5C2×D311yescharacteristic
113C5×#95—31no—
114C5×#95—32no—
115C5×#95—31no—
116#3963—31no—
117#3963—31no—
118#3963—32no—
119#3963—32no—
120#2275—151noSylow
121C23×C3×C5—21no—
122C23×C3×C5—22no—
123C23×C3×C5—21no—
124C23×C3×C5C22×C414yes—
125C23×C3×C5C22×C412yes—
126C23×C3×C5C2411yescharacteristic
127C22×C3×D5—21no—
128C22×C3×D5—22no—
129C22×C3×D5—21no—
130C22×C3×D5C22×C414yes—
131C22×C3×D5C22×C412yes—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—44no—
135C2×C3×C4×C5C22×C412yes—
136C2×C3×C4×C5C22×C412yes—
137C2×Dic15—22no—
138C2×Dic15—22no—
139C2×Dic15—22no—
140C2×Dic15—22no—
141C2×Dic15C22×C414yes—
142C2×Dic15C22×C414yes—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—48no—
148C2×C5×Dic3C22×C414yes—
149C2×C5×Dic3C22×C414yes—
150C3×C22⋊D5C42116yescomplemented
151C2×C3×Dic5—21no—
152C2×C3×Dic5—22no—
153C2×C3×Dic5—21no—
154C2×C3×Dic5C22×C412yes—
155C2×C3×Dic5C22×C412yes—
156C2×C3×Dic5C22×C414yes—
157C2×C3×Dic5C22×C412yes—
158C3×C4×D5—44no—
159D5×Dic3—48no—
160C22×C3×D4—51no—
161C23×C3×C4—51no—
162C23×C3×C4C2×D511yescharacteristic
163C23×Dic3—52no—
164C23×Dic3C2×D512yes—
165C3×C4×D4—58no—
166C2×C3×C4⋊C4—51no—
167C2×C3×C22⋊C4—51no—
168C2×C3×C22⋊C4—51no—
169C2×C3×C42—51no—
170C2×#217—52no—
171C2×#217—52no—
172D4×Dic3—516no—
173C2×C4⋊Dic3—52no—
174C2×C4×Dic3—52no—
175C2×C4×Dic3—52no—
176C2×C4×Dic3—52no—
177C2×C4×Dic3—52no—
178C3⋊#95—51no—
179C3⋊#95—51no—
180C3⋊#95—52no—
181C3⋊#95—52no—
182C3⋊#95C2×D511yescharacteristic
183C3⋊#95C2×D511yescharacteristic
184C3⋊#95C2×D512yes—
185C24×C5C22×D311yescharacteristic
186C23×D5C22×D311yescharacteristic
187C22×C4×C5—32no—
188C22×C4×C5—32no—
189C22×C4×C5—38no—
190C22×C4×C5—42no—
191C22×C4×C5—122no—
192C22×C4×C5—122no—
193C22×C4×C5C2×Dic314yescomplemented
194C22×C4×C5C22×D311yescharacteristic
195C22×C4×C5C22×D311yescharacteristic
196C2×C22⋊D5C4×D318yes—
197C2×C22⋊D5C2×Dic314yes—
198C22×Dic5—32no—
199C22×Dic5—32no—
200C22×Dic5C22×D311yescharacteristic
201C22×Dic5C22×D311yescharacteristic
202C22×Dic5C22×D311yescharacteristic
203C2×C4×D5—38no—
204C2×C4×D5—42no—
205C2×C4×D5—122no—
206C2×C4×D5—122no—
207C2×C4×D5C2×Dic314yescomplemented
208C22⋊Dic5—38no—
209C22⋊Dic5C2×Dic314yescomplemented
210#685—38no—
211#685C2×Dic314yescomplemented
212#683—38no—
213#683C2×Dic314yescomplemented
214C4×Dic5—38no—
215C4×Dic5—122no—
216C4×Dic5—122no—
217C4×Dic5—122no—
218C4×Dic5C2×Dic314yescomplemented
219C2×C4×D4—51no—
220C2×C4×D4—152no—
221C4×C4⋊C4—151no—
222C4×C4⋊C4—152no—
223C4×C22⋊C4—152no—
224C4×C22⋊C4—151no—
225C4×C22⋊C4—152no—
226C4×C22⋊C4—151no—
227#376—151no—
228C2×#95—31no—
229C2×#95—151no—
230C22×C3×C5—24no—
231C22×C3×C5—22no—
232C22×C3×C5—24no—
233C22×C3×C5—22no—
234C22×C3×C5C2×C4214yes—
235C22×C3×C5C23×C412yes—
236C22×C3×C5C23×C411yescharacteristic
237C22×C3×C5H(5,𝔽2)12yes—
238C22×C3×C5H(5,𝔽2)11yescharacteristic
239C22×C3×C52−1+411yescharacteristic
240C2×C3×D5—24no—
241C2×C3×D5—22no—
242C2×C3×D5—24no—
243C2×C3×D5—22no—
244C2×C3×D5C2×C4214yes—
245C3×C4×C5—22no—
246C3×C4×C5—22no—
247Dic15—24no—
248Dic15—24no—
249C3×Dic5—22no—
250C3×Dic5—22no—
251C3×Dic5C2×C4214yes—
252C5×Dic3—24no—
253C5×Dic3—24no—
254C24×C3—51no—
255C24×C3C22×D511yescharacteristic
256C2×C3×D4—58no—
257C2×C3×D4—54no—
258C22×C3×C4—42no—
259C22×C3×C4—51no—
260C22×C3×C4—51no—
261C22×C3×C4—51no—
262C22×C3×C4—54no—
263C22×C3×C4—202no—
264C22×C3×C4C4×D514yescomplemented
265C22×C3×C4C22×D511yescharacteristic
266C22×C3×C4C22×D511yescharacteristic
267C22×Dic3—42no—
268C22×Dic3—42no—
269C22×Dic3—52no—
270C22×Dic3—52no—
271C22×Dic3—58no—
272C22×Dic3—202no—
273C22×Dic3—202no—
274C22×Dic3C4×D518yescomplemented
275C22×Dic3C22×D512yes—
276C22×Dic3C22×D512yes—
277C3×C4⋊C4—54no—
278C3×C22⋊C4—54no—
279C3×C22⋊C4—54no—
280C3×C42—54no—
281#217—58no—
282#217—58no—
283C4⋊Dic3—58no—
284C4×Dic3—58no—
285C4×Dic3—202no—
286C4×Dic3—202no—
287C4×Dic3—202no—
288C23×C5—22no—
289C23×C5—21no—
290C23×C5—21no—
291C23×C5C2×C4×D314yes—
292C23×C5C22×Dic312yes—
293C23×C5C23×D311yescharacteristic
294C22×D5—22no—
295C22×D5—21no—
296C22×D5—21no—
297C22×D5C2×C4×D314yes—
298C22×D5C22×Dic312yes—
299C2×C4×C5—22no—
300C2×C4×C5—22no—
301C2×C4×C5—34no—
302C2×C4×C5—34no—
303C2×C4×C5—44no—
304C2×C4×C5—62no—
305C2×C4×C5—62no—
306C2×C4×C5—62no—
307C2×C4×C5—62no—
308C2×C4×C5—128no—
309C2×C4×C5C22×Dic312yes—
310C2×C4×C5C22×Dic312yes—
311C22⋊D5C4×Dic3116yescomplemented
312C2×Dic5—22no—
313C2×Dic5—21no—
314C2×Dic5—21no—
315C2×Dic5—34no—
316C2×Dic5—34no—
317C2×Dic5—62no—
318C2×Dic5—62no—
319C2×Dic5—62no—
320C2×Dic5—62no—
321C2×Dic5C2×C4×D314yes—
322C2×Dic5C22×Dic312yes—
323C2×Dic5C22×Dic312yes—
324C2×Dic5C22×Dic312yes—
325C4×D5—44no—
326C4×D5—128no—
327C22×D4—51no—
328C23×C4—32no—
329C23×C4—51no—
330C23×C4—152no—
331C23×C4D3×D511yescharacteristic
332C4×D4—58no—
333C4×D4—1516no—
334C2×C4⋊C4—51no—
335C2×C4⋊C4—152no—
336C2×C22⋊C4—51no—
337C2×C22⋊C4—51no—
338C2×C22⋊C4—152no—
339C2×C22⋊C4—152no—
340C2×C42—51no—
341C2×C42—152no—
342C2×C42—152no—
343C2×C42—152no—
344C2×C42—152no—
345#95—31no—
346#95—32no—
347#95—31no—
348#95—152no—
349#95—152no—
350#95—151no—
351#95—151no—
352C2×C3×C5—24no—
353C2×C3×C5C22×C4211yescharacteristic
354C2×C3×C5#51811yescharacteristic
355C2×C3×C5#51812yes—
356C2×C3×C5#52012yes—
357C2×C3×C5#52011yescharacteristic
358C3×D5—24no—
359C23×C3—21no—
360C23×C3—22no—
361C23×C3—21no—
362C23×C3—54no—
363C23×C3—52no—
364C23×C3—101no—
365C23×C3—102no—
366C23×C3—101no—
367C23×C3C2×C4×D514yes—
368C23×C3C2×C4×D512yes—
369C23×C3C23×D511yescharacteristic
370C3×D4—516no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—44no—
374C2×C3×C4—54no—
375C2×C3×C4—52no—
376C2×C3×C4—52no—
377C2×C3×C4—52no—
378C2×C3×C4—101no—
379C2×C3×C4—102no—
380C2×C3×C4—101no—
381C2×C3×C4—204no—
382C2×C3×C4C2×C4×D512yes—
383C2×C3×C4C2×C4×D512yes—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—48no—
389C2×Dic3—54no—
390C2×Dic3—54no—
391C2×Dic3—102no—
392C2×Dic3—102no—
393C2×Dic3—102no—
394C2×Dic3—102no—
395C2×Dic3—208no—
396C2×Dic3C2×C4×D514yes—
397C2×Dic3C2×C4×D514yes—
398C22×C5—22no—
399C22×C5—24no—
400C22×C5—24no—
401C22×C5—22no—
402C22×C5C2×C4×Dic314yes—
403C22×C5C22×C4×D312yes—
404C22×C5#100412yes—
405C22×C5#100911yescharacteristic
406C22×C5#101011yescharacteristic
407C22×C5C23×Dic311yescharacteristic
408C2×D5—22no—
409C2×D5—24no—
410C2×D5—24no—
411C2×D5—22no—
412C2×D5C2×C4×Dic314yes—
413C4×C5—22no—
414C4×C5—22no—
415C4×C5—64no—
416C4×C5—64no—
417Dic5—22no—
418Dic5—22no—
419Dic5—64no—
420Dic5—64no—
421Dic5C2×C4×Dic314yes—
422C24—51no—
423C24C2×D3×D511yescharacteristic
424C2×D4—58no—
425C2×D4—54no—
426C22×C4—32no—
427C22×C4—32no—
428C22×C4—38no—
429C22×C4—42no—
430C22×C4—51no—
431C22×C4—51no—
432C22×C4—51no—
433C22×C4—54no—
434C22×C4—122no—
435C22×C4—122no—
436C22×C4—158no—
437C22×C4—152no—
438C22×C4—152no—
439C22×C4—202no—
440C22×C4—602no—
441C22×C4—602no—
442C22×C4D5×Dic314yescomplemented
443C22×C4C2×D3×D511yescharacteristic
444C22×C4C2×D3×D511yescharacteristic
445C4⋊C4—54no—
446C4⋊C4—158no—
447C22⋊C4—54no—
448C22⋊C4—54no—
449C22⋊C4—158no—
450C22⋊C4—158no—
451C42—54no—
452C42—158no—
453C42—602no—
454C42—602no—
455C42—602no—
456C3×C5#227511yescharacteristic, complemented, Hall
457C22×C3—24no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—24no—
461C22×C3—54no—
462C22×C3—104no—
463C22×C3—102no—
464C22×C3—102no—
465C22×C3—104no—
466C22×C3C42×D514yes—
467C22×C3C22×C4×D512yes—
468C22×C3C22×C4×D511yescharacteristic
469C22×C3#414411yescharacteristic
470C22×C3#414412yes—
471C22×C3#414711yescharacteristic
472C3×C4—22no—
473C3×C4—22no—
474C3×C4—54no—
475C3×C4—102no—
476C3×C4—102no—
477Dic3—24no—
478Dic3—24no—
479Dic3—104no—
480Dic3—104no—
481C2×C5—24no—
482C2×C5#554512yes—
483C2×C5#554612yes—
484C2×C5C22×C4×Dic311yescharacteristic
485C2×C5#579811yescharacteristic
486C2×C5#582711yescharacteristic
487D5—24no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—54no—
492C23—52no—
493C23—101no—
494C23—101no—
495C23—102no—
496C23C4×D3×D514yes—
497C23C2×D5×Dic312yes—
498C23C22×D3×D511yescharacteristic
499D4—516no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—34no—
503C2×C4—34no—
504C2×C4—44no—
505C2×C4—54no—
506C2×C4—52no—
507C2×C4—52no—
508C2×C4—52no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—101no—
514C2×C4—101no—
515C2×C4—102no—
516C2×C4—128no—
517C2×C4—154no—
518C2×C4—154no—
519C2×C4—204no—
520C2×C4—302no—
521C2×C4—302no—
522C2×C4—302no—
523C2×C4—302no—
524C2×C4—608no—
525C2×C4C2×D5×Dic312yes—
526C2×C4C2×D5×Dic312yes—
527C2×C3—24no—
528C2×C3—104no—
529C2×C3C2×C42×D511yescharacteristic
530C2×C3#6582012yes—
531C2×C3#6582011yescharacteristic
532C2×C3#6583811yescharacteristic
533C2×C3#6583812yes—
534C5#8231711yescharacteristic, complemented, Sylow
535C22—22no—
536C22—24no—
537C22—24no—
538C22—22no—
539C22—54no—
540C22—102no—
541C22—104no—
542C22—104no—
543C22—102no—
544C22C4×D5×Dic314yes—
545C22#9200011yescharacteristic
546C22C2×C4×D3×D512yes—
547C22#9201811yescharacteristic
548C22C22×D5×Dic311yescharacteristic
549C22#9204912yes—
550C4—22no—
551C4—22no—
552C4—54no—
553C4—64no—
554C4—64no—
555C4—102no—
556C4—102no—
557C4—304no—
558C4—304no—
559C3#1060505311yescharacteristic, complemented, Sylow
560C2—24no—
561C2—104no—
562C2C2×C4×D5×Dic311yescharacteristic
563C2#1175428412yes—
564C2#1175449911yescharacteristic
565C2#1175462712yes—
566C2#1175540611yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC2×C3×C5C22×C42
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C3×C22⋊D5C22
hypercenterC23×C4D3×D5
hypocenterC3×C5#2275
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
3. C4×#6719C2semidirect
4. C2×C5×C3⋊#95C2semidirect
5. Dic3×C22⋊Dic5C2semidirect
6. #11748884C2semidirect
7. D5×C3⋊#95C2semidirect
8. #11748702C2semidirect
9. #11748638C2semidirect
10. #11748637C2semidirect
11. #11748635C2semidirect
21. C3×C4×C22⋊D5C4semidirect
25. Dic3×C22⋊D5C4semidirect
53. C22×C3×C4×C5C2×C4semidirect
60. C22×C5×Dic3C2×C4semidirect
69. C2×C3×C4×D5C2×C4semidirect
72. C2×D5×Dic3C2×C4semidirect
73. #6719C2×C4semidirect
75. C3×C22⋊Dic5C2×C4semidirect
76. C3×#685C2×C4semidirect
77. C3×#683C2×C4semidirect
78. C3×C4×Dic5C2×C4semidirect
79. #6668C2×C4semidirect
80. #6665C2×C4semidirect
83. Dic3×Dic5C2×C4semidirect
89. C2×C3⋊#95D5semidirect
103. C4×C22⋊D5Dic3semidirect
150. C3×C22⋊D5C42semidirect
193. C22×C4×C5C2×Dic3semidirect
207. C2×C4×D5C2×Dic3semidirect
209. C22⋊Dic5C2×Dic3semidirect
211. #685C2×Dic3semidirect
213. #683C2×Dic3semidirect
218. C4×Dic5C2×Dic3semidirect
264. C22×C3×C4C4×D5semidirect
274. C22×Dic3C4×D5semidirect
311. C22⋊D5C4×Dic3semidirect
442. C22×C4D5×Dic3semidirect
456. C3×C5#2275semidirect
534. C5#82317semidirect
559. C3#10605053semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊D52
C2×Dic3×C22⋊D52
C4×#67192
C2×C5×C3⋊#952
Dic3×C22⋊Dic52
#117488842
D5×C3⋊#952
#117487022
#117486382
#117486372
#117486352
#106050533
#823175