Finite Group Databasebeta

← #28651820

Subgroups — #28651820

Order 15104.

932 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837692C211yesmaximal, characteristic, complemented
2C2×C4×#11745669C211yesmaximal, characteristic
3#21427946C211yesmaximal, characteristic, complemented
4#21425620C211yesmaximal, characteristic, complemented
5#21425578C211yesmaximal, characteristic, complemented
6#21425467C211yesmaximal, characteristic, complemented
7#21425424C211yesmaximal, characteristic, complemented
8#21424821C211yesmaximal, characteristic, complemented
9#21424655C211yesmaximal, characteristic, complemented
10#21424444C211yesmaximal, characteristic
11#21424444C211yesmaximal, characteristic
12#21424357C211yesmaximal, characteristic, complemented
13#21424356C211yesmaximal, characteristic, complemented
14#21424343C211yesmaximal, characteristic, complemented
15C59×#2349C211yesmaximal, characteristic, complemented
16C22×#11745669C2211yescharacteristic
17C2×#14273727C2211yescharacteristic
18C2×#14273705—21no—
19C2×C4×D236C2211yescharacteristic
20#16837692—24no—
21C4×#11745669—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273648C2211yescharacteristic
29C2×#14273648C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273647C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×#14273647C2211yescharacteristic
34C2×C4×C59⋊C8C2211yescharacteristic
35#16837324—21no—
36#16837315—21no—
37#16837304C2211yescharacteristic
38#16837304C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837300C2211yescharacteristic
41#16837276—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C2×C4×D4×C59C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#385—21no—
56C59×#385—21no—
57C59×#382C2211yescharacteristic
58C59×#380—21no—
59C4×C4⋊C4×C59C2211yescharacteristic
60C2×#95×C59—21no—
61C22×Dic118—21no—
62C22×D236C2311yescharacteristic
63C22×C4×D59—21no—
64C2×#11745669—22no—
65C2×#11745669—22no—
66C2×#11745669Q814yescomplemented
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71#14273727—22no—
72#14273727—41no—
73#14273705—44no—
74C4×D236—22no—
75C4×D236—41no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87#14273660—22no—
88#14273660—41no—
89#14273658—22no—
90#14273658—22no—
91#14273658—22no—
92C4⋊(C59⋊C8)—22no—
93#14273648—22no—
94#14273648—22no—
95#14273648D414yes—
96#14273647—22no—
97#14273647—22no—
98#14273647—41no—
99C4×C59⋊C8—22no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C22×D4×C59C2311yescharacteristic
103C23×C4×C59—21no—
104C23×C4×C59—21no—
105C23×C4×C59—21no—
106C4×D4×C59—22no—
107C4×D4×C59—22no—
108C4×D4×C59—22no—
109C4×D4×C59—22no—
110C4×D4×C59—41no—
111C4×D4×C59—41no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C4⋊C4×C59D411yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124C2×C42×C59C2311yescharacteristic
125#95×C59—21no—
126#95×C59—21no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59D411yescharacteristic
130#95×C59D411yescharacteristic
131C24×C59—21no—
132C23×D59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59C2×D412yes—
137C2×D4×C59C2×Q812yes—
138C2×D4×C59#4112yes—
139C22×C4×C59—21no—
140C22×C4×C59—24no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—42no—
144C22×C4×C59—42no—
145C22×C4×C59—42no—
146C22×C4×C59—42no—
147C22×C4×C59—81no—
148C22×C4×C59—81no—
149C22×C4×C59—81no—
150C22×C4×C59—81no—
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2411yescharacteristic
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C2×D236—21no—
162C2×D236—21no—
163C2×D236—21no—
164C2×D236—21no—
165C2×D236—21no—
166C2×D236C2×Q812yes—
167C2×C4×D59—42no—
168C2×C4×D59—42no—
169C2×C4×D59—81no—
170C2×Dic118—42no—
171C2×Dic118—42no—
172C2×Dic118—81no—
173C4⋊C4×C59—22no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59—81no—
181C4⋊C4×C59—81no—
182C4⋊C4×C59#4112yes—
183C22⋊C4×C59—24no—
184C22⋊C4×C59—42no—
185C22⋊C4×C59—42no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—21no—
189C42×C59—21no—
190C42×C59—21no—
191C42×C59—21no—
192#11745669—28no—
193(C2×D59)⋊C4—42no—
194(C2×D59)⋊C4—42no—
195(C2×D59)⋊C4—42no—
196(C2×D59)⋊C4—42no—
197(C2×D59)⋊C4—81no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—21no—
203C4⋊Dic59—42no—
204C4⋊Dic59C2×D412yes—
205Dic59⋊C4—42no—
206Dic59⋊C4—42no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8C2×D412yes—
218C2×C59⋊C8C2×Q812yes—
219C23×C59—22no—
220C23×C59—22no—
221C23×C59—22no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59C22×D411yescharacteristic
227C22×D59—22no—
228C22×D59—41no—
229C22×D59—41no—
230C22×D59—41no—
231C22×D59—41no—
232C22×D59—41no—
233C22×D59—41no—
234D4×C59—22no—
235D4×C59—22no—
236D4×C59—22no—
237D4×C59C22⋊Q814yescomplemented
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—22no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—21no—
254C2×C4×C59—21no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59—84no—
264C2×C4×C59—84no—
265C2×C4×C59—84no—
266C2×C4×C59#13512yes—
267C2×C4×C59AGL(1,ℤ/8)12yes—
268C2×C4×C59C22×D411yescharacteristic
269C2×C4×C59C22×Q811yescharacteristic
270C2×C4×C59C2×#4111yescharacteristic
271C2×C4×C59C2×#4111yescharacteristic
272C2×C4×C59H(5,𝔽2)11yescharacteristic
273C2×C4×C592−1+411yescharacteristic
274C2×Dic59—22no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283D236—22no—
284D236—22no—
285D236—41no—
286D236—41no—
287C4×D59—84no—
288Dic118—84no—
289C59⋊C8—22no—
290C59⋊C8—22no—
291C59⋊C8—41no—
292C59⋊C8—41no—
293#26380—591nomaximal, Sylow
294C22×C59—24no—
295C22×C59—42no—
296C22×C59—42no—
297C22×C59—42no—
298C22×C59—42no—
299C22×C59—42no—
300C22×C59—42no—
301C22×C59#52511yescharacteristic
302C22×C59D4211yescharacteristic
303C22×C59D4×Q811yescharacteristic
304C22×C59C2×#13511yescharacteristic
305C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C59#57611yescharacteristic
307C22×C59#57711yescharacteristic
308C2×D59—42no—
309C2×D59—42no—
310C2×D59—42no—
311C2×D59—81no—
312C2×D59—81no—
313C2×D59—81no—
314C2×D59—81no—
315C4×C59—22no—
316C4×C59—22no—
317C4×C59—41no—
318C4×C59—41no—
319C4×C59—42no—
320C4×C59—41no—
321C4×C59—41no—
322C4×C59—81no—
323C4×C59—81no—
324C4×C59C2×C22⋊Q811yescharacteristic
325C4×C59#53911yescharacteristic
326C4×C59#54711yescharacteristic
327C4×C59#55411yescharacteristic
328Dic59—42no—
329Dic59—81no—
330Dic59—81no—
331C2×#476—591no—
332C2×C4×D8—591no—
333#2452—591no—
334#2349D5911yescharacteristic, complemented
335#2092—591no—
336#2085—591no—
337#2065—591no—
338#2057—591no—
339#1941—591no—
340#1916—591no—
341#1879—591no—
342#1879—591no—
343#1864—591no—
344#1864—591no—
345#1862—591no—
346C2×C59—44no—
347C2×C59#244911yescharacteristic
348C2×C59#310611yescharacteristic
349C2×C59#311811yescharacteristic
350C2×C59#326411yescharacteristic
351C2×C59#328211yescharacteristic
352C2×C59#338011yescharacteristic
353C2×C59#338411yescharacteristic
354D59—82no—
355C22×D8—591no—
356C2×C4⋊Q8—591no—
357C2×C22⋊Q8—1181no—
358C2×C4×D4—591no—
359C2×C4×D4C2×D5911yescharacteristic
360C2×C4×D4C2×D5911yescharacteristic
361C2×C4×D4C2×D5911yescharacteristic
362#476—1184no—
363C4×D8—1184no—
364C2×#107—591no—
365C2×#107—591no—
366C2×#106—591no—
367C2×#106—591no—
368C2×#106—591no—
369C2×C4⋊C8—591no—
370C2×#102—591no—
371C2×#102—591no—
372C2×#102—591no—
373C2×#102—591no—
374C2×#102—591no—
375C2×#102—591no—
376C2×C4×C8—591no—
377#396—1181no—
378#393—1181no—
379#390—591no—
380#390—591no—
381#389—591no—
382#389—591no—
383#389C2×D5911yescharacteristic
384#389C2×D5911yescharacteristic
385#385—21no—
386#385—21no—
387#385—1181no—
388#382C2×D5911yescharacteristic
389#380—21no—
390C4×C4⋊C4C2×D5911yescharacteristic
391C2×#95—21no—
392#340—591no—
393#340—591no—
394#340—591no—
395#340—591no—
396#336—591no—
397#336—591no—
398#336—591no—
399#336—591no—
400C59#2638011yescharacteristic, complemented, Sylow
401C22×Q8—1181no—
402C22×D4—591no—
403C22×D4C22×D5911yescharacteristic
404C23×C4—21no—
405C23×C4—21no—
406C23×C4—21no—
407C23×C4—1181no—
408C2×D8—594no—
409C2×D8—1182no—
410C2×D8—1182no—
411C22×C8—591no—
412C22×C8—591no—
413C22×C8—591no—
414C22×C8—591no—
415C4⋊Q8—1182no—
416C4⋊Q8—2361no—
417C22⋊Q8—2364no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—41no—
423C4×D4—41no—
424C4×D4—1182no—
425C4×D4—2361no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—591no—
429C2×C4⋊C4—591no—
430C2×C4⋊C4—591no—
431C2×C4⋊C4—1181no—
432C2×C4⋊C4—1181no—
433C2×C4⋊C4—1181no—
434C2×C4⋊C4C22⋊D5911yescharacteristic
435C2×C4⋊C4C22⋊D5911yescharacteristic
436C2×C4⋊C4C22×D5911yescharacteristic
437C2×C4⋊C4C22×D5911yescharacteristic
438C2×C4⋊C4C22×D5911yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1181no—
443C2×C22⋊C4—1181no—
444C2×C22⋊C4—1181no—
445C2×C22⋊C4—1181no—
446C2×C22⋊C4—1181no—
447C2×C42C22×D5911yescharacteristic
448C2×C42C22×D5911yescharacteristic
449C2×C42C22×D5911yescharacteristic
450#107—1182no—
451#107—2361no—
452#106—1182no—
453#106—1182no—
454#106—1182no—
455C4⋊C8—1182no—
456#102—594no—
457#102—1182no—
458#102—1182no—
459#102—1182no—
460#102—1182no—
461#102—2361no—
462C4×C8—1182no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—1181no—
468#95—1181no—
469#95C22⋊D5911yescharacteristic
470#95C22⋊D5911yescharacteristic
471C24—21no—
472C24—1181no—
473C2×Q8—4721no—
474C2×Q8—2362no—
475C2×Q8—2362no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—592no—
480C2×D4—1181no—
481C2×D4—1181no—
482C2×D4—1181no—
483C2×D4—1181no—
484C2×D4—1181no—
485C2×D4#1174568512yes—
486C2×D4D4×D5912yes—
487C2×D4Q8×D5912yes—
488C22×C4—21no—
489C22×C4—24no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—1181no—
501C22×C4—1181no—
502C22×C4—1181no—
503C22×C4—4721no—
504C22×C4—2362no—
505C22×C4—2362no—
506C22×C4C2×C22⋊D5911yescharacteristic
507C22×C4C2×C22⋊D5911yescharacteristic
508C22×C4C2×C22⋊D5911yescharacteristic
509C22×C4C2×C22⋊D5911yescharacteristic
510C22×C4C2×C22⋊D5911yescharacteristic
511C22×C4C2×C22⋊D5911yescharacteristic
512C22×C4C23×D5911yescharacteristic
513D8—1188no—
514C2×C8—592no—
515C2×C8—592no—
516C2×C8—1181no—
517C2×C8—1181no—
518C2×C8—1181no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—1181no—
522C2×C8—1181no—
523C2×C8—1181no—
524C2×C8—1181no—
525C2×C8—1181no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—81no—
534C4⋊C4—81no—
535C4⋊C4—592no—
536C4⋊C4—1181no—
537C4⋊C4—1181no—
538C4⋊C4—1181no—
539C4⋊C4—1181no—
540C4⋊C4—1181no—
541C4⋊C4—2362no—
542C4⋊C4—2362no—
543C4⋊C4—2362no—
544C4⋊C4#1174568512yes—
545C22⋊C4—24no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—4721no—
549C22⋊C4—2362no—
550C22⋊C4—2362no—
551C22⋊C4—2362no—
552C22⋊C4—2362no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—1182no—
567C23—2361no—
568C23—2361no—
569C23—2361no—
570C23—2361no—
571C23—2361no—
572C23—2361no—
573C23C22×C22⋊D5911yescharacteristic
574Q8—4724no—
575D4—22no—
576D4—22no—
577D4—22no—
578D4—1182no—
579D4—1182no—
580D4—2361no—
581D4—2361no—
582D4#1427370514yescomplemented
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—1182no—
612C2×C4—4724no—
613C2×C4—2361no—
614C2×C4—2361no—
615C2×C4—2361no—
616C2×C4—2361no—
617C2×C4—2361no—
618C2×C4—2361no—
619C2×C4—2361no—
620C2×C4—2361no—
621C2×C4#1427374712yes—
622C2×C4#1427375212yes—
623C2×C4C2×D4×D5911yescharacteristic
624C2×C4C2×Q8×D5911yescharacteristic
625C2×C4C2×#1174568511yescharacteristic
626C2×C4C2×#1174568511yescharacteristic
627C2×C4#1427377211yescharacteristic
628C2×C4#1427377511yescharacteristic
629C8—1182no—
630C8—1182no—
631C8—2361no—
632C8—2361no—
633C22—24no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—4721no—
641C22—4721no—
642C22—4721no—
643C22—4721no—
644C22—2362no—
645C22—2362no—
646C22—2362no—
647C22#1683791311yescharacteristic
648C22D4×C22⋊D5911yescharacteristic
649C22Q8×C22⋊D5911yescharacteristic
650C22C2×#1427374711yescharacteristic
651C22C2×#1427375211yescharacteristic
652C22#1683820511yescharacteristic
653C22#1683821311yescharacteristic
654C4—22no—
655C4—22no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—42no—
660C4—41no—
661C4—81no—
662C4—81no—
663C4—4721no—
664C4—4721no—
665C4—2362no—
666C4C2×#1427370511yescharacteristic
667C4#1683799911yescharacteristic
668C4#1683805311yescharacteristic
669C4#1683810711yescharacteristic
670C2—44no—
671C2—4722no—
672C2#2142790511yescharacteristic
673C2#2143415611yescharacteristic
674C2#2143428011yescharacteristic
675C2#2143557711yescharacteristic
676C2#2143579911yescharacteristic
677C2#2143695911yescharacteristic
678C2#2143699511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2349C2
socleC23×C59C22×D4
prime radicalC22×#11745669C22
hypercenter#2349D59
hypocenterC59#26380
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
1. C2×#16837692C2semidirect
3. #21427946C2semidirect
4. #21425620C2semidirect
5. #21425578C2semidirect
6. #21425467C2semidirect
7. #21425424C2semidirect
8. #21424821C2semidirect
9. #21424655C2semidirect
12. #21424357C2semidirect
13. #21424356C2semidirect
14. #21424343C2semidirect
15. C59×#2349C2semidirect
66. C2×#11745669Q8semidirect
237. D4×C59C22⋊Q8semidirect
334. #2349D59semidirect
400. C59#26380semidirect
582. D4#14273705semidirect

Maximal subgroups

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

subgroupindex
C2×#168376922
C2×C4×#117456692
#214279462
#214256202
#214255782
#214254672
#214254242
#214248212
#214246552
#214244442
#214243572
#214243562
#214243432
C59×#23492
#2638059