Finite Group Databasebeta

← #14418640

Subgroups — #14418640

Order 1920.

696 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11752789C211yesmaximal, characteristic
2#11752355C211yesmaximal, characteristic
3C3×#65691C211yesmaximal, characteristic
4#11752014C211yesmaximal, characteristic
5#11752012C211yesmaximal, characteristic
6C2×#91169C211yesmaximal, characteristic
7#11751988C211yesmaximal, characteristic
8#11751823C211yesmaximal, characteristic, complemented
9#11751789C211yesmaximal, characteristic, complemented
10#11751785C211yesmaximal, characteristic, complemented
11#11751785C211yesmaximal, characteristic, complemented
12#11751784C211yesmaximal, characteristic, complemented
13#11751780C211yesmaximal, characteristic, complemented
14#11751778C211yesmaximal, characteristic, complemented
15#11751774C211yesmaximal, characteristic, complemented
16#10617262—31nomaximal, Hall
17C2×#6762C2211yescharacteristic
18C22×C3⋊#170C2211yescharacteristic
19C2×C3×#704C2211yescharacteristic
20C22×C3×#170C2211yescharacteristic
21#91767C411yescharacteristic
22C5×#878C411yescharacteristic
23C3×#4023C2211yescharacteristic
24#91576C411yescharacteristic
25#91575C411yescharacteristic
26#91528C2211yescharacteristic
27C2×Dic3×Dic5C2211yescharacteristic
28#91335C411yescharacteristic
29#91334C411yescharacteristic
30#91333C2211yescharacteristic
31Dic5×Dic6C2211yescharacteristic
32#91329C411yescharacteristic
33#91328C411yescharacteristic
34#91323C2211yescharacteristic
35#91323C2211yescharacteristic
36#91233C2211yescharacteristic
37#91233C2211yescharacteristic
38C4⋊(C3⋊#170)C2211yescharacteristic
39C4⋊(C3⋊#170)C2211yescharacteristic
40C3×#4002C2211yescharacteristic
41C3×#4002C2211yescharacteristic
42C3×C4×#170C2211yescharacteristic
43C3×C4×#170C2211yescharacteristic
44#91177C2211yescharacteristic
45#91177C2211yescharacteristic
46#91177C2211yescharacteristic
47#91177C2211yescharacteristic
48Dic3⋊#170C2211yescharacteristic
49Dic3⋊#170C2211yescharacteristic
50Dic3⋊#170C2211yescharacteristic
51Dic3⋊#170C2211yescharacteristic
52#91169C2211yescharacteristic
53#91169C2211yescharacteristic
54#91169C2211yescharacteristic
55#91169C2211yescharacteristic
56#91169C2211yescharacteristic
57#91169C2211yescharacteristic
58Dic3×#170C2211yescharacteristic
59Dic3×#170C2211yescharacteristic
60#78543—51nomaximal, Hall
61#65956—31no—
62#65788—31no—
63#65788—31no—
64#65758—31no—
65C2×#4002—31no—
66#65709—31no—
67#65709—31no—
68#65707—31no—
69#65707—31no—
70#65707—31no—
71#65707—31no—
72#65696—31no—
73#65696—31no—
74#65691—31no—
75#65691D311yescharacteristic
76C22×Dic15C2×C411yescharacteristic
77C2×Dic30C2×C411yescharacteristic
78C22×C5×Dic3C2×C411yescharacteristic
79C2×C5×Dic6C2×C411yescharacteristic
80C22×C3×Dic5C2311yescharacteristic
81C2×#1204C2×C411yescharacteristic
82C2×#1204C2×C411yescharacteristic
83#6762—22no—
84C2×C3⋊#170—22no—
85C2×C3⋊#170C2311yescharacteristic
86C2×C3⋊#170C2311yescharacteristic
87C2×C3⋊#170C2311yescharacteristic
88C2×C3⋊#170C2311yescharacteristic
89C3×#704—22no—
90C2×C3×#170Q814yescomplemented
91C2×C3×#170C2311yescharacteristic
92C2×C3×#170C2311yescharacteristic
93C2×C3×#170C2311yescharacteristic
94C2×C3×#170C2311yescharacteristic
95C3×C5×C22⋊C4C2×C411yescharacteristic
96#6719C2×C411yescharacteristic
97C4⋊Dic15C2×C411yescharacteristic
98#6712C2×C411yescharacteristic
99#6712C2×C411yescharacteristic
100C5×#217C2×C411yescharacteristic
101C5×C4⋊Dic3C2×C411yescharacteristic
102C5×Dic3⋊C4C2×C411yescharacteristic
103C5×Dic3⋊C4C2×C411yescharacteristic
104C3×C22⋊Dic5C2×C411yescharacteristic
105C3×#683C2×C411yescharacteristic
106C3×#683C2×C411yescharacteristic
107C3×C4×Dic5C2311yescharacteristic
108C3×C4×Dic5C2311yescharacteristic
109Dic3⋊Dic5C2×C411yescharacteristic
110Dic3⋊Dic5C2×C411yescharacteristic
111#6669C2×C411yescharacteristic
112#6669C2×C411yescharacteristic
113#6668C2×C411yescharacteristic
114#6668C2×C411yescharacteristic
115#6668C2×C411yescharacteristic
116#6668C2×C411yescharacteristic
117Dic3×Dic5C2×C411yescharacteristic
118Dic3×Dic5C2×C411yescharacteristic
119Dic3×Dic5C2311yescharacteristic
120Dic3×Dic5C2311yescharacteristic
121Dic3×Dic5C2311yescharacteristic
122Dic3×Dic5C2311yescharacteristic
123#5517—51no—
124C3×#405—51no—
125#5119—51no—
126#5118—51no—
127#5101—51no—
128C2×#815—51no—
129#4999—51no—
130#4701—51no—
131#4701—51no—
132#4701—51no—
133#4701—51no—
134#4679—51no—
135#4679—51no—
136#4679—51no—
137#4679—51no—
138C2×#704—31no—
139C2×#704C2×D311yescharacteristic
140C22×#170—31no—
141C22×#170C2×D311yescharacteristic
142C5×C22⋊Q8—31no—
143Q8×Dic5—31no—
144#4090—31no—
145#4070—31no—
146C2×C4×Dic5—31no—
147#4035—31no—
148#4035—31no—
149#4034—31no—
150#4033—31no—
151#4033—31no—
152#4033—31no—
153#4024—31no—
154#4024—31no—
155#4023—31no—
156#4023C2×D311yescharacteristic
157#4004—31no—
158#4004—31no—
159#4004—31no—
160#4004—31no—
161#4004—31no—
162#4004—31no—
163#4002—31no—
164#4002—31no—
165#4002—31no—
166#4002—31no—
167#4002—31no—
168#4002—31no—
169#4002C2×D311yescharacteristic
170#4002C2×D311yescharacteristic
171C4⋊#170—31no—
172C4⋊#170—31no—
173C4⋊#170—31no—
174C4⋊#170—31no—
175C4⋊#170—31no—
176C4⋊#170—31no—
177C4×#170—31no—
178C4×#170—31no—
179C4×#170C2×D311yescharacteristic
180C4×#170C2×D311yescharacteristic
181#2961—151noSylow
182C23×C3×C5C22×C411yescharacteristic
183C2×C3×C4×C5C22×C411yescharacteristic
184C2×C3×C4×C5C22×C411yescharacteristic
185C2×Dic15—41no—
186C2×Dic15C22×C411yescharacteristic
187C2×Dic15C22×C411yescharacteristic
188C2×Dic15C22×C411yescharacteristic
189C2×Dic15C22×C411yescharacteristic
190Dic30—41no—
191C2×C5×Dic3—41no—
192C2×C5×Dic3C22×C411yescharacteristic
193C2×C5×Dic3C22×C411yescharacteristic
194C2×C5×Dic3C22×C411yescharacteristic
195C2×C5×Dic3C22×C411yescharacteristic
196C5×Dic6—41no—
197C2×C3×Dic5C22×C411yescharacteristic
198C2×C3×Dic5C22×C411yescharacteristic
199C2×C3×Dic5C22×C411yescharacteristic
200C2×C3×Dic5C2×Q812yes—
201C2×C3×Dic5#4112yes—
202C2×C3×Dic5C2411yescharacteristic
203#1204—41no—
204#1204—41no—
205C3⋊#170—21no—
206C3⋊#170—21no—
207C3⋊#170—21no—
208C3⋊#170—21no—
209C3×#170—21no—
210C3×#170—21no—
211C3×#170C2×Q812yes—
212C3×#170C2×Q812yes—
213C2×C3×M16—51no—
214C22×C3×C8—51no—
215C3×#117—51no—
216#926—51no—
217C2×C4×Dic3—51no—
218C2×#208—51no—
219C22×C3⋊C8—51no—
220#890—51no—
221#890—51no—
222#888—51no—
223#888—51no—
224#878—51no—
225#878—51no—
226#878—51no—
227#878AGL(1,𝔽5)11yescharacteristic
228C4×Dic6—51no—
229C4×Dic6—51no—
230C4×Dic6—51no—
231C4×Dic6—51no—
232C3×#97—51no—
233C3×#97—51no—
234C3×C4×C8—51no—
235C3×C4×C8—51no—
236#815—51no—
237#815—51no—
238#815—51no—
239#815—51no—
240#815—51no—
241#815—51no—
242Dic3⋊C8—51no—
243Dic3⋊C8—51no—
244Dic3⋊C8—51no—
245Dic3⋊C8—51no—
246Dic3⋊C8—51no—
247Dic3⋊C8—51no—
248Dic3⋊C8—51no—
249Dic3⋊C8—51no—
250C8×Dic3—51no—
251C8×Dic3—51no—
252C4⋊(C3⋊C8)—51no—
253C4⋊(C3⋊C8)—51no—
254C4⋊(C3⋊C8)—51no—
255C4⋊(C3⋊C8)—51no—
256C2×C5×Q8—31no—
257C22×C4×C5—31no—
258C22×Dic5—31no—
259C22×Dic5C22×D311yescharacteristic
260C2×Dic10—31no—
261C2×Dic10—31no—
262C2×Dic10—31no—
263#704—22no—
264#704—62no—
265C2×#170—31no—
266C2×#170—31no—
267C2×#170—31no—
268C2×#170—31no—
269C2×#170—62no—
270C2×#170Dic614yescomplemented
271C2×#170C22×D311yescharacteristic
272C2×#170C22×D311yescharacteristic
273C2×#170C22×D311yescharacteristic
274C2×#170C22×D311yescharacteristic
275C5×C4⋊C4—31no—
276C5×C4⋊C4—31no—
277C5×C4⋊C4—31no—
278C5×C22⋊C4—31no—
279C5×C22⋊C4C4×D311yescharacteristic
280C22⋊Dic5—31no—
281C22⋊Dic5C4×D311yescharacteristic
282C4⋊Dic5—31no—
283C4⋊Dic5—31no—
284C4⋊Dic5—31no—
285#683—31no—
286#683—31no—
287#683—31no—
288#683—31no—
289#683—31no—
290#683—31no—
291#683—31no—
292#683—31no—
293#683C4×D311yescharacteristic
294#683C4×D311yescharacteristic
295C4×Dic5—31no—
296C4×Dic5—31no—
297C4×Dic5—31no—
298C4×Dic5—31no—
299C4×Dic5—31no—
300C4×Dic5—31no—
301C4×Dic5C22×D311yescharacteristic
302C4×Dic5C22×D311yescharacteristic
303#533—151no—
304#446—151no—
305#446—151no—
306#446—151no—
307#446—151no—
308#446—151no—
309#446—151no—
310#446—151no—
311#446—151no—
312#424—151no—
313#424—151no—
314#423—151no—
315#405—51no—
316#405—151no—
317C2×#97—151no—
318C22×C3×C5—21no—
319C22×C3×C5—21no—
320C22×C3×C5C4×Q812yes—
321C22×C3×C5C23×C411yescharacteristic
322C3×C4×C5—21no—
323C3×C4×C5—21no—
324Dic15—21no—
325Dic15—21no—
326Dic15—21no—
327Dic15—21no—
328C3×Dic5—21no—
329C3×Dic5—21no—
330C3×Dic5C4×Q812yes—
331C3×Dic5C22×Q811yescharacteristic
332C3×Dic5C2×#4111yescharacteristic
333C5×Dic3—21no—
334C5×Dic3—21no—
335C5×Dic3—21no—
336C5×Dic3—21no—
337C22×C3×C4—51no—
338C22×Dic3—51no—
339C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
340C2×Dic6—51no—
341C2×Dic6—51no—
342C2×Dic6—51no—
343C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
344C3×M16—102no—
345C2×C3×C8—51no—
346C2×C3×C8—51no—
347C2×C3×C8—54no—
348C2×C3×C8—51no—
349C2×C3×C8—51no—
350C3×C4⋊C4—51no—
351C3×C4⋊C4—51no—
352C3×C22⋊C4—51no—
353C3×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
354C3×C42—51no—
355C3×C42—51no—
356#217—51no—
357#217C2×AGL(1,𝔽5)11yescharacteristic
358C4⋊Dic3—51no—
359C4⋊Dic3—51no—
360C4⋊Dic3—51no—
361C4⋊Dic3—51no—
362C4⋊Dic3—51no—
363C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
364Dic3⋊C4—51no—
365Dic3⋊C4—51no—
366Dic3⋊C4—51no—
367Dic3⋊C4—51no—
368Dic3⋊C4—51no—
369Dic3⋊C4—51no—
370Dic3⋊C4C2×AGL(1,𝔽5)11yescharacteristic
371Dic3⋊C4C2×AGL(1,𝔽5)11yescharacteristic
372C4×Dic3—51no—
373C4×Dic3—51no—
374C4×Dic3—51no—
375C4×Dic3—51no—
376C4×Dic3—51no—
377C4×Dic3—51no—
378#208—102no—
379C2×C3⋊C8—51no—
380C2×C3⋊C8—51no—
381C2×C3⋊C8—51no—
382C2×C3⋊C8—51no—
383C2×C3⋊C8—102no—
384C23×C5C2×C4×D311yescharacteristic
385C5×Q8—121no—
386C2×C4×C5—31no—
387C2×C4×C5—31no—
388C2×C4×C5—31no—
389C2×C4×C5—31no—
390C2×C4×C5—121no—
391C2×C4×C5C2×C4×D311yescharacteristic
392C2×C4×C5C2×C4×D311yescharacteristic
393C2×Dic5—31no—
394C2×Dic5—31no—
395C2×Dic5—31no—
396C2×Dic5—31no—
397C2×Dic5—121no—
398C2×Dic5C2×Dic612yes—
399C2×Dic5C2×C4×D311yescharacteristic
400C2×Dic5C2×C4×D311yescharacteristic
401C2×Dic5C2×C4×D311yescharacteristic
402C2×Dic5#23512yes—
403C2×Dic5C23×D311yescharacteristic
404Dic10—121no—
405Dic10—121no—
406Dic10—121no—
407#170—21no—
408#170—21no—
409#170—61no—
410#170—61no—
411#170—61no—
412#170—61no—
413#170C2×Dic612yes—
414#170C2×Dic612yes—
415C2×M16—51no—
416C2×M16—151no—
417C22×C8—51no—
418C22×C8—151no—
419C4⋊Q8—151no—
420C4⋊Q8—151no—
421#125—151no—
422#125—151no—
423C22⋊Q8—31no—
424C22⋊Q8—151no—
425C22⋊Q8—151no—
426C22⋊Q8—151no—
427C4×Q8—151no—
428C4×Q8—151no—
429C4×Q8—151no—
430C4×Q8—151no—
431#117—51no—
432#117—151no—
433C2×C42—151no—
434C4⋊C8—151no—
435C4⋊C8—151no—
436C4⋊C8—151no—
437C4⋊C8—151no—
438C4⋊C8—151no—
439C4⋊C8—151no—
440C4⋊C8—151no—
441C4⋊C8—151no—
442C4⋊C8—151no—
443C4⋊C8—151no—
444C4⋊C8—151no—
445C4⋊C8—151no—
446#97—51no—
447#97—51no—
448#97—151no—
449#97—151no—
450#97—151no—
451#97—151no—
452#97—151no—
453#97—151no—
454C4×C8—51no—
455C4×C8—51no—
456C4×C8—151no—
457C4×C8—151no—
458C2×C3×C5—22no—
459C2×C3×C5C2×C4×Q811yescharacteristic
460C2×C3×C5#56811yescharacteristic
461C2×C3×C5#56811yescharacteristic
462C23×C3C22×AGL(1,𝔽5)11yescharacteristic
463C2×C3×C4—52no—
464C2×C3×C4—52no—
465C2×C3×C4—51no—
466C2×C3×C4—51no—
467C2×C3×C4—51no—
468C2×C3×C4—51no—
469C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
470C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
471C2×Dic3—41no—
472C2×Dic3—51no—
473C2×Dic3—51no—
474C2×Dic3—51no—
475C2×Dic3—51no—
476C2×Dic3—201no—
477C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
478C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
479C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
480C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
481Dic6—41no—
482Dic6—201no—
483Dic6—201no—
484Dic6—201no—
485C3×C8—52no—
486C3×C8—52no—
487C3×C8—101no—
488C3×C8—101no—
489C3⋊C8—101no—
490C3⋊C8—101no—
491C3⋊C8—101no—
492C3⋊C8—101no—
493C22×C5—21no—
494C22×C5—21no—
495C22×C5C4×Dic612yes—
496C22×C5C22×C4×D311yescharacteristic
497C4×C5—21no—
498C4×C5—21no—
499C4×C5—61no—
500C4×C5—61no—
501C4×C5—61no—
502C4×C5—61no—
503Dic5—21no—
504Dic5—21no—
505Dic5—61no—
506Dic5—61no—
507Dic5—61no—
508Dic5—61no—
509Dic5C4×Dic612yes—
510Dic5C22×Dic611yescharacteristic
511Dic5C2×#23511yescharacteristic
512C2×Q8—31no—
513C2×Q8—151no—
514C2×Q8—151no—
515C2×Q8—151no—
516C22×C4—31no—
517C22×C4—51no—
518C22×C4—151no—
519M16—102no—
520M16—302no—
521C2×C8—51no—
522C2×C8—51no—
523C2×C8—54no—
524C2×C8—51no—
525C2×C8—51no—
526C2×C8—151no—
527C2×C8—151no—
528C2×C8—151no—
529C2×C8—151no—
530C2×C8—302no—
531C4⋊C4—31no—
532C4⋊C4—31no—
533C4⋊C4—31no—
534C4⋊C4—51no—
535C4⋊C4—51no—
536C4⋊C4—151no—
537C4⋊C4—151no—
538C4⋊C4—151no—
539C4⋊C4—151no—
540C4⋊C4—151no—
541C4⋊C4—151no—
542C4⋊C4—151no—
543C4⋊C4—151no—
544C4⋊C4—151no—
545C4⋊C4—151no—
546C4⋊C4—151no—
547C22⋊C4—31no—
548C22⋊C4—51no—
549C22⋊C4—151no—
550C22⋊C4D3×AGL(1,𝔽5)11yescharacteristic
551C42—51no—
552C42—51no—
553C42—151no—
554C42—151no—
555C42—151no—
556C42—151no—
557C42—151no—
558C42—151no—
559C3×C5#296111yescharacteristic, complemented, Hall
560C22×C3—21no—
561C22×C3—21no—
562C22×C3Q8×AGL(1,𝔽5)12yes—
563C22×C3C23×AGL(1,𝔽5)11yescharacteristic
564C3×C4—21no—
565C3×C4—21no—
566C3×C4—52no—
567C3×C4—51no—
568C3×C4—51no—
569C3×C4—101no—
570C3×C4—101no—
571Dic3—21no—
572Dic3—21no—
573Dic3—21no—
574Dic3—21no—
575Dic3—101no—
576Dic3—101no—
577Dic3—101no—
578Dic3—101no—
579C2×C5—22no—
580C2×C5C2×C4×Dic611yescharacteristic
581C2×C5#575011yescharacteristic
582C2×C5#575011yescharacteristic
583C23C2×D3×AGL(1,𝔽5)11yescharacteristic
584Q8—121no—
585Q8—601no—
586Q8—601no—
587Q8—601no—
588C2×C4—31no—
589C2×C4—31no—
590C2×C4—31no—
591C2×C4—31no—
592C2×C4—52no—
593C2×C4—52no—
594C2×C4—51no—
595C2×C4—51no—
596C2×C4—51no—
597C2×C4—51no—
598C2×C4—121no—
599C2×C4—151no—
600C2×C4—151no—
601C2×C4—151no—
602C2×C4—151no—
603C2×C4—601no—
604C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
605C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
606C8—52no—
607C8—52no—
608C8—101no—
609C8—101no—
610C8—301no—
611C8—301no—
612C8—301no—
613C8—301no—
614C2×C3—22no—
615C2×C3#6625611yescharacteristic
616C2×C3C2×Q8×AGL(1,𝔽5)11yescharacteristic
617C2×C3#6626411yescharacteristic
618C5#7854311yescharacteristic, complemented, Sylow
619C22—21no—
620C22—21no—
621C22AGL(1,𝔽5)×Dic612yes—
622C22C22×D3×AGL(1,𝔽5)11yescharacteristic
623C4—21no—
624C4—21no—
625C4—51no—
626C4—51no—
627C4—52no—
628C4—61no—
629C4—61no—
630C4—61no—
631C4—61no—
632C4—101no—
633C4—101no—
634C4—301no—
635C4—301no—
636C4—301no—
637C4—301no—
638C3#1061726211yescharacteristic, complemented, Sylow
639C2—22no—
640C2C2×AGL(1,𝔽5)×Dic611yescharacteristic
641C2#1175684111yescharacteristic
642C2#1175684211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC3×C5×C22⋊C4C2×C4
prime radicalC23×C3×C5C22×C4
cosocleC2×C3×Dic5C24
hypercenterC22⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2961
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
8. #11751823C2semidirect
9. #11751789C2semidirect
10. #11751785C2semidirect
11. #11751785C2semidirect
12. #11751784C2semidirect
13. #11751780C2semidirect
14. #11751778C2semidirect
15. #11751774C2semidirect
90. C2×C3×#170Q8semidirect
270. C2×#170Dic6semidirect
559. C3×C5#2961semidirect
618. C5#78543semidirect
638. C3#10617262semidirect

Maximal subgroups

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

subgroupindex
#117527892
#117523552
C3×#656912
#117520142
#117520122
C2×#911692
#117519882
#117518232
#117517892
#117517852
#117517842
#117517802
#117517782
#117517742
#106172623
#785435