Finite Group Databasebeta

← #34356399

Subgroups — #34356399

Order 20224.

790 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23767752C211yesmaximal, characteristic
2#23764715C211yesmaximal, characteristic, complemented
3#23764600C211yesmaximal, characteristic, complemented
4#23764551C211yesmaximal, characteristic, complemented
5#23764401C211yesmaximal, characteristic
6#23764357C211yesmaximal, characteristic, complemented
7#23764082C211yesmaximal, characteristic, complemented
8#23764020C211yesmaximal, characteristic, complemented
9#23764008C211yesmaximal, characteristic, complemented
10#23763989C211yesmaximal, characteristic, complemented
11#23763755C211yesmaximal, characteristic, complemented
12#23763717C211yesmaximal, characteristic
13#23763486C211yesmaximal, characteristic, complemented
14C4×#14579085C211yesmaximal, characteristic, complemented
15C79×#2360C211yesmaximal, characteristic, complemented
16C22×#11929611C2211yescharacteristic
17C2×#14579153C2211yescharacteristic
18C2×#14579135C2211yescharacteristic
19C2×#14579094C2211yescharacteristic
20C2×C4⋊(C79⋊C8)C2211yescharacteristic
21C2×#14579085C2211yescharacteristic
22C2×#14579085C2211yescharacteristic
23C2×#14579085C2211yescharacteristic
24C2×#14579081C2211yescharacteristic
25C2×#14579081C2211yescharacteristic
26C2×#14579081C2211yescharacteristic
27C2×C22⋊(C79⋊C8)C2211yescharacteristic
28C2×C22⋊(C79⋊C8)C2211yescharacteristic
29C2×C4×C79⋊C8C2211yescharacteristic
30#19098492—21no—
31#19098478—21no—
32#19098465C2211yescharacteristic
33#19098465C2211yescharacteristic
34#19098463—21no—
35#19098452—21no—
36#19098423C2211yescharacteristic
37C4×C4⋊Dic79C2211yescharacteristic
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×C22⋊Q8×C79C2211yescharacteristic
49C2×C4×Q8×C79C2211yescharacteristic
50C22×C4⋊C4×C79C2211yescharacteristic
51#391×C79C2211yescharacteristic
52#389×C79—21no—
53#389×C79C2211yescharacteristic
54#387×C79—21no—
55#385×C79—21no—
56C4×C4⋊C4×C79C2211yescharacteristic
57C4×C22⋊C4×C79—21no—
58C4×C22⋊C4×C79C2211yescharacteristic
59C22×D316C2311yescharacteristic
60C22×C22⋊D79—21no—
61C2×#11929611—22no—
62C2×#11929611—41no—
63C2×#11929611—41no—
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68#14579153—22no—
69#14579153—41no—
70#14579135—22no—
71#14579135—41no—
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C2×C22⋊Dic79—21no—
78C2×(C2×D79)⋊C4—21no—
79C2×(C2×D79)⋊C4—21no—
80C2×(C2×D79)⋊C4—21no—
81C2×(C2×D79)⋊C4—21no—
82C2×(C2×D79)⋊C4—21no—
83C2×C4×Dic79—21no—
84#14579094—22no—
85C4⋊(C79⋊C8)—41no—
86#14579085—22no—
87#14579085—22no—
88#14579085—41no—
89#14579081—22no—
90#14579081—22no—
91#14579081—41no—
92C22⋊(C79⋊C8)—41no—
93C22⋊(C79⋊C8)—41no—
94C4×C79⋊C8—41no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C22×Q8×C79C2311yescharacteristic
98C23×C4×C79—21no—
99C23×C4×C79C2311yescharacteristic
100C22⋊Q8×C79—22no—
101C22⋊Q8×C79—22no—
102C4×Q8×C79—22no—
103C4×Q8×C79—41no—
104C2×C4⋊C4×C79—21no—
105C2×C4⋊C4×C79—22no—
106C2×C4⋊C4×C79—21no—
107C2×C4⋊C4×C79—21no—
108C2×C4⋊C4×C79—21no—
109C2×C4⋊C4×C79—41no—
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C22⋊C4×C79—21no—
114C2×C22⋊C4×C79D411yescharacteristic
115C2×C22⋊C4×C79D411yescharacteristic
116C2×C42×C79—21no—
117C2×C42×C79—21no—
118C2×C42×C79C2311yescharacteristic
119C2×C42×C79C2311yescharacteristic
120#95×C79—21no—
121#95×C79—21no—
122#95×C79—21no—
123#95×C79—21no—
124#95×C79D411yescharacteristic
125#95×C79D411yescharacteristic
126C23×D79—21no—
127C2×D316—21no—
128C2×D316—21no—
129C2×D316—21no—
130C2×D316—21no—
131C2×D316—21no—
132C2×D316—21no—
133C2×C22⋊D79—42no—
134C2×C22⋊D79—42no—
135C2×C22⋊D79—81no—
136C22×Dic79—21no—
137C22×Dic79—21no—
138C22×Dic79—21no—
139#11929611—44no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8—21no—
149C2×C79⋊C8—21no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79#4112yes—
158Dic79⋊C4—42no—
159Dic79⋊C4—42no—
160C22⋊Dic79—81no—
161(C2×D79)⋊C4—42no—
162(C2×D79)⋊C4—42no—
163(C2×D79)⋊C4—81no—
164(C2×D79)⋊C4—81no—
165(C2×D79)⋊C4—81no—
166C4×Dic79—81no—
167C24×C79C2×D411yescharacteristic
168C2×Q8×C79—21no—
169C2×Q8×C79—21no—
170C2×Q8×C79—21no—
171C2×Q8×C79—21no—
172C2×Q8×C79—21no—
173C2×Q8×C79#4112yes—
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79—21no—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79—21no—
182C22×C4×C79—21no—
183C22×C4×C79—21no—
184C22×C4×C79—42no—
185C22×C4×C79—42no—
186C22×C4×C79—81no—
187C22×C4×C79C2×D411yescharacteristic
188C22×C4×C79C2×D411yescharacteristic
189C22×C4×C79C2×D411yescharacteristic
190C22×C4×C79C2×D411yescharacteristic
191C22×C4×C79C2×D411yescharacteristic
192C22×C4×C79C2411yescharacteristic
193C4⋊C4×C79—24no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79—42no—
199C4⋊C4×C79—42no—
200C4⋊C4×C79—81no—
201C4⋊C4×C79#4112yes—
202C4⋊C4×C79#4112yes—
203C22⋊C4×C79—24no—
204C22⋊C4×C79—22no—
205C22⋊C4×C79—22no—
206C42×C79—21no—
207C42×C79—21no—
208C42×C79—21no—
209C42×C79—21no—
210C42×C79—42no—
211C42×C79—81no—
212C23×C79—21no—
213C23×C79—21no—
214C23×C79—21no—
215C23×C79—21no—
216C23×C79—21no—
217C23×C79—21no—
218C23×C79#13512yes—
219C23×C79C22×D411yescharacteristic
220C22×D79—41no—
221C22×D79—41no—
222C22×D79—41no—
223C22×D79—41no—
224C22×D79—41no—
225C22×D79—41no—
226C22×D79—41no—
227Q8×C79—22no—
228Q8×C79—22no—
229Q8×C79—41no—
230Q8×C79—41no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—22no—
234C2×C4×C79—22no—
235C2×C4×C79—21no—
236C2×C4×C79—21no—
237C2×C4×C79—21no—
238C2×C4×C79—21no—
239C2×C4×C79—21no—
240C2×C4×C79—22no—
241C2×C4×C79—21no—
242C2×C4×C79—21no—
243C2×C4×C79—21no—
244C2×C4×C79—21no—
245C2×C4×C79—21no—
246C2×C4×C79—21no—
247C2×C4×C79—21no—
248C2×C4×C79—41no—
249C2×C4×C79—41no—
250C2×C4×C79—41no—
251C2×C4×C79—41no—
252C2×C4×C79—41no—
253C2×C4×C79—41no—
254C2×C4×C79—41no—
255C2×C4×C79—41no—
256C2×C4×C79—41no—
257C2×C4×C79—41no—
258C2×C4×C79—41no—
259C2×C4×C79—41no—
260C2×C4×C79—41no—
261C2×C4×C79—84no—
262C2×C4×C79#13712yes—
263C2×C4×C79C2×#4111yescharacteristic
264C2×C4×C79C2×#4111yescharacteristic
265C2×C4×C79C2×#4111yescharacteristic
266C2×C4×C79C2×#4111yescharacteristic
267C2×C4×C79H(5,𝔽2)11yescharacteristic
268C2×C4×C79H(5,𝔽2)11yescharacteristic
269C22⋊D79—84no—
270C2×Dic79—22no—
271C2×Dic79—41no—
272C2×Dic79—41no—
273C2×Dic79—41no—
274C2×Dic79—41no—
275C2×Dic79—41no—
276C2×Dic79—41no—
277C2×Dic79—41no—
278C2×Dic79—41no—
279D316—41no—
280D316—41no—
281D316—41no—
282D316—41no—
283C79⋊C8—41no—
284C79⋊C8—41no—
285C79⋊C8—41no—
286C79⋊C8—41no—
287C22×C79—22no—
288C22×C79—22no—
289C22×C79—22no—
290C22×C79—41no—
291C22×C79—41no—
292C22×C79—41no—
293C22×C79—41no—
294C22×C79#52511yescharacteristic
295C22×C79#54611yescharacteristic
296C22×C79#54611yescharacteristic
297C22×C79C2×#13511yescharacteristic
298C22×C79C2×#13711yescharacteristic
299C22×C79#57611yescharacteristic
300C22×C79#57711yescharacteristic
301C2×D79—81no—
302C2×D79—81no—
303C2×D79—81no—
304C2×D79—81no—
305C2×D79—81no—
306C2×D79—81no—
307C2×D79—81no—
308C4×C79—22no—
309C4×C79—41no—
310C4×C79—42no—
311C4×C79—41no—
312C4×C79—42no—
313C4×C79—41no—
314C4×C79—41no—
315C4×C79—42no—
316C4×C79—81no—
317C4×C79#52911yescharacteristic
318C4×C79#53811yescharacteristic
319C4×C79#55511yescharacteristic
320C4×C79#55911yescharacteristic
321Dic79—42no—
322Dic79—81no—
323Dic79—81no—
324#26727—791nomaximal, Sylow
325C2×C79—42no—
326C2×C79#245911yescharacteristic
327C2×C79#310811yescharacteristic
328C2×C79#311911yescharacteristic
329C2×C79#329611yescharacteristic
330C2×C79#329711yescharacteristic
331C2×C79#332011yescharacteristic
332C2×C79#333311yescharacteristic
333D79—161no—
334#2490—791no—
335#2360D7911yescharacteristic, complemented
336#2055—791no—
337#2036—791no—
338#2026—791no—
339#2000—791no—
340#1989—791no—
341#1934—791no—
342#1925—791no—
343#1923—791no—
344#1921—791no—
345#1880—791no—
346#1874—791no—
347#1823—791no—
348C4×#103—791no—
349C79#2672711yescharacteristic, complemented, Sylow
350C22×SD16—791no—
351C2×#125—791no—
352C2×C22⋊Q8C2×D7911yescharacteristic
353C2×#121—791no—
354C2×C4×Q8C2×D7911yescharacteristic
355C22×C4⋊C4C2×D7911yescharacteristic
356C2×#107—791no—
357C2×C4⋊C8—791no—
358C2×#103—791no—
359C2×#103—791no—
360C2×#103—791no—
361C2×#102—791no—
362C2×#102—791no—
363C2×#102—791no—
364C2×C22⋊C8—791no—
365C2×C22⋊C8—791no—
366C2×C4×C8—791no—
367#397—1581no—
368#394—1581no—
369#391C2×D7911yescharacteristic
370#390—791no—
371#390—791no—
372#390—1581no—
373#389—21no—
374#389C2×D7911yescharacteristic
375#388—1581no—
376#387—21no—
377#385—21no—
378#380—791no—
379C4×C4⋊C4—791no—
380C4×C4⋊C4C2×D7911yescharacteristic
381C4×C22⋊C4—21no—
382C4×C22⋊C4C2×D7911yescharacteristic
383#340—791no—
384#340—791no—
385#340—791no—
386#340—791no—
387#340—791no—
388#340—791no—
389#340—791no—
390#340—791no—
391#336—791no—
392#336—791no—
393C22×Q8C22×D7911yescharacteristic
394C22×D4—791no—
395C22×D4—1581no—
396C23×C4—21no—
397C23×C4C22×D7911yescharacteristic
398C2×SD16—3161no—
399C2×SD16—3161no—
400C2×SD16—1582no—
401C22×C8—791no—
402C22×C8—791no—
403C22×C8—791no—
404C22×C8—791no—
405#125—3161no—
406#125—1582no—
407C22⋊Q8—22no—
408C22⋊Q8—22no—
409#121—3161no—
410#121—1582no—
411C4×Q8—22no—
412C4×Q8—41no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—22no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—41no—
419C2×C4⋊C4—791no—
420C2×C4⋊C4—791no—
421C2×C4⋊C4—791no—
422C2×C4⋊C4—1581no—
423C2×C4⋊C4—1581no—
424C2×C4⋊C4C22×D7911yescharacteristic
425C2×C4⋊C4C22×D7911yescharacteristic
426C2×C4⋊C4C22×D7911yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—1581no—
429C2×C22⋊C4—1581no—
430C2×C22⋊C4—1581no—
431C2×C22⋊C4—1581no—
432C2×C22⋊C4—1581no—
433C2×C22⋊C4—1581no—
434C2×C22⋊C4C22⋊D7911yescharacteristic
435C2×C22⋊C4C22⋊D7911yescharacteristic
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—1581no—
439C2×C42C22×D7911yescharacteristic
440C2×C42C22×D7911yescharacteristic
441#107—1582no—
442C4⋊C8—3161no—
443#103—3161no—
444#103—1582no—
445#103—1582no—
446#102—3161no—
447#102—1582no—
448#102—1582no—
449C22⋊C8—3161no—
450C22⋊C8—3161no—
451C4×C8—3161no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—1581no—
457#95—1581no—
458#95C22⋊D7911yescharacteristic
459#95C22⋊D7911yescharacteristic
460C24—1581no—
461C24C2×C22⋊D7911yescharacteristic
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8#1192962412yes—
468C2×D4—3162no—
469C2×D4—3162no—
470C2×D4—6321no—
471C2×D4—1581no—
472C2×D4—1581no—
473C2×D4—1581no—
474C2×D4—1581no—
475C2×D4—1581no—
476C2×D4—1581no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—81no—
490C22×C4—1581no—
491C22×C4—1581no—
492C22×C4—1581no—
493C22×C4C2×C22⋊D7911yescharacteristic
494C22×C4C2×C22⋊D7911yescharacteristic
495C22×C4C2×C22⋊D7911yescharacteristic
496C22×C4C2×C22⋊D7911yescharacteristic
497C22×C4C2×C22⋊D7911yescharacteristic
498C22×C4C23×D7911yescharacteristic
499SD16—3164no—
500C2×C8—1581no—
501C2×C8—1581no—
502C2×C8—1581no—
503C2×C8—1581no—
504C2×C8—1581no—
505C2×C8—1581no—
506C2×C8—1581no—
507C2×C8—1581no—
508C2×C8—1581no—
509C2×C8—1581no—
510C2×C8—1581no—
511C2×C8—1581no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—81no—
520C4⋊C4—3162no—
521C4⋊C4—3162no—
522C4⋊C4—792no—
523C4⋊C4—1581no—
524C4⋊C4—1581no—
525C4⋊C4—1581no—
526C4⋊C4—1581no—
527C4⋊C4—1581no—
528C4⋊C4#1192962312yes—
529C4⋊C4#1192962412yes—
530C22⋊C4—24no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—3162no—
534C22⋊C4—3162no—
535C22⋊C4—6321no—
536C22⋊C4—6321no—
537C22⋊C4—6321no—
538C22⋊C4—6321no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—42no—
544C42—81no—
545C42—6321no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—3161no—
553C23—3161no—
554C23—3161no—
555C23—3161no—
556C23—3161no—
557C23—3161no—
558C23—3161no—
559C23#1457918112yes—
560C23C22×C22⋊D7911yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—41no—
564Q8—41no—
565D4—3161no—
566D4—3161no—
567D4—3161no—
568D4—3161no—
569D4—6324no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—84no—
601C2×C4—3161no—
602C2×C4—3161no—
603C2×C4—3161no—
604C2×C4—3161no—
605C2×C4—3161no—
606C2×C4—3161no—
607C2×C4—3161no—
608C2×C4—3161no—
609C2×C4—1582no—
610C2×C4#1457919212yes—
611C2×C4#41×D7911yescharacteristic
612C2×C4C2×#1192962311yescharacteristic
613C2×C4C2×#1192962411yescharacteristic
614C2×C4C2×#1192962411yescharacteristic
615C2×C4#1457920611yescharacteristic
616C2×C4#1457920711yescharacteristic
617C8—3161no—
618C8—3161no—
619C8—3161no—
620C8—3161no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—6321no—
629C22—6321no—
630C22—6321no—
631C22—6321no—
632C22—6321no—
633C22—6321no—
634C22—6321no—
635C22#1909907411yescharacteristic
636C22#1909920111yescharacteristic
637C22#1909920711yescharacteristic
638C22C2×#1457918111yescharacteristic
639C22C2×#1457919211yescharacteristic
640C22#1909936611yescharacteristic
641C22#1909937111yescharacteristic
642C4—22no—
643C4—41no—
644C4—42no—
645C4—41no—
646C4—42no—
647C4—42no—
648C4—41no—
649C4—41no—
650C4—81no—
651C4—3162no—
652C4—6321no—
653C4—6321no—
654C4#1909909311yescharacteristic
655C4#1909914611yescharacteristic
656C4#1909927811yescharacteristic
657C4#1909929611yescharacteristic
658C2—42no—
659C2—12641no—
660C2#2376733111yescharacteristic
661C2#2377348011yescharacteristic
662C2#2377359911yescharacteristic
663C2#2377522911yescharacteristic
664C2#2377525011yescharacteristic
665C2#2377557111yescharacteristic
666C2#2377573211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2360C2
socleC23×C79C22×D4
prime radicalC2×#14579135C22
hypercenter#2360D79
hypocenterC79#26727
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #23764715C2semidirect
3. #23764600C2semidirect
4. #23764551C2semidirect
6. #23764357C2semidirect
7. #23764082C2semidirect
8. #23764020C2semidirect
9. #23764008C2semidirect
10. #23763989C2semidirect
11. #23763755C2semidirect
13. #23763486C2semidirect
14. C4×#14579085C2semidirect
15. C79×#2360C2semidirect
335. #2360D79semidirect
349. C79#26727semidirect

Maximal subgroups

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

subgroupindex
#237677522
#237647152
#237646002
#237645512
#237644012
#237643572
#237640822
#237640202
#237640082
#237639892
#237637552
#237637172
#237634862
C4×#145790852
C79×#23602
#2672779