Finite Group Databasebeta

← #34356716

Subgroups — #34356716

Order 20224.

798 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766425C211yesmaximal, characteristic, complemented
2#23764717C211yesmaximal, characteristic, complemented
3#23764603C211yesmaximal, characteristic, complemented
4#23764538C211yesmaximal, characteristic, complemented
5#23764403C211yesmaximal, characteristic
6#23764353C211yesmaximal, characteristic, complemented
7#23764100C211yesmaximal, characteristic, complemented
8#23764017C211yesmaximal, characteristic, complemented
9#23764015C211yesmaximal, characteristic, complemented
10#23763992C211yesmaximal, characteristic, complemented
11#23763758C211yesmaximal, characteristic, complemented
12#23763723C211yesmaximal, characteristic
13#23763492C211yesmaximal, characteristic, complemented
14#23763233C211yesmaximal, characteristic, complemented
15C79×#2491C211yesmaximal, characteristic
16C22×#11929612C2211yescharacteristic
17C2×#14579140C2211yescharacteristic
18C2×C4×Dic158C2211yescharacteristic
19C2×#14579120C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×C4⋊(C79⋊C8)C2211yescharacteristic
22C2×#14579084C2211yescharacteristic
23C2×#14579084C2211yescharacteristic
24C2×#14579084C2211yescharacteristic
25C2×#14579082C2211yescharacteristic
26C2×#14579082C2211yescharacteristic
27C2×#14579082C2211yescharacteristic
28C2×C22⋊(C79⋊C8)C2211yescharacteristic
29C2×C22⋊(C79⋊C8)C2211yescharacteristic
30C2×#14579072C2211yescharacteristic
31#19098469C2211yescharacteristic
32#19098461C2211yescharacteristic
33#19098457—21no—
34#19098451—21no—
35#19098445—21no—
36C4×C4⋊Dic79C2211yescharacteristic
37#19098414—21no—
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×C4⋊Q8×C79C2211yescharacteristic
49C2×#124×C79—21no—
50C2×#121×C79C2211yescharacteristic
51#394×C79—21no—
52#393×C79—21no—
53#390×C79C2211yescharacteristic
54#390×C79C2211yescharacteristic
55#388×C79—21no—
56#380×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58C4×C22⋊C4×C79C2211yescharacteristic
59C22×Dic158C2311yescharacteristic
60C2×#11929612—22no—
61C2×#11929612—41no—
62C2×#11929612—41no—
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67#14579140—22no—
68#14579140—22no—
69C4×Dic158—22no—
70C4×Dic158—41no—
71#14579120—22no—
72#14579120—41no—
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×C22⋊Dic79—21no—
80C2×C22⋊Dic79—21no—
81C2×C4×Dic79—21no—
82C2×C4×Dic79—21no—
83C2×C4×Dic79—21no—
84#14579094—22no—
85C4⋊(C79⋊C8)—41no—
86#14579084—22no—
87#14579084—22no—
88#14579084—41no—
89#14579082—22no—
90#14579082—22no—
91#14579082—41no—
92C22⋊(C79⋊C8)—41no—
93C22⋊(C79⋊C8)—41no—
94#14579072—41no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C22×Q8×C79—21no—
100C22×D4×C79—21no—
101C22×D4×C79C2311yescharacteristic
102C23×C4×C79C2311yescharacteristic
103C4⋊Q8×C79—22no—
104C4⋊Q8×C79—41no—
105#124×C79—44no—
106#121×C79—22no—
107#121×C79—41no—
108C2×C4⋊C4×C79—21no—
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C22⋊C4×C79—21no—
113C2×C22⋊C4×C79—21no—
114C2×C22⋊C4×C79—21no—
115C2×C22⋊C4×C79—21no—
116C2×C22⋊C4×C79—21no—
117C2×C22⋊C4×C79—21no—
118C2×C22⋊C4×C79D411yescharacteristic
119C2×C22⋊C4×C79D411yescharacteristic
120C2×C42×C79—21no—
121C2×C42×C79C2311yescharacteristic
122C2×C42×C79C2311yescharacteristic
123#95×C79—21no—
124#95×C79—21no—
125#95×C79D411yescharacteristic
126#95×C79D411yescharacteristic
127C2×Dic158—21no—
128C2×Dic158—21no—
129C2×Dic158—21no—
130C2×Dic158—21no—
131C2×Dic158—21no—
132C2×Dic158#4112yes—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C22×Dic79—21no—
137#11929612—44no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
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—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79#4112yes—
155C4⋊Dic79#4112yes—
156Dic79⋊C4—24no—
157Dic79⋊C4—42no—
158Dic79⋊C4—42no—
159C22⋊Dic79—24no—
160C22⋊Dic79—42no—
161C4×Dic79—42no—
162C4×Dic79—42no—
163C4×Dic79—42no—
164C24×C79—21no—
165C24×C79C2×D411yescharacteristic
166C2×Q8×C79—42no—
167C2×Q8×C79—42no—
168C2×Q8×C79—81no—
169C2×D4×C79—21no—
170C2×D4×C79—21no—
171C2×D4×C79—21no—
172C2×D4×C79—21no—
173C2×D4×C79—21no—
174C2×D4×C79—21no—
175C2×D4×C79—42no—
176C2×D4×C79—42no—
177C2×D4×C79—81no—
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—21no—
185C22×C4×C79—21no—
186C22×C4×C79—21no—
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—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79—42no—
199C4⋊C4×C79#4112yes—
200C22⋊C4×C79—22no—
201C22⋊C4×C79—41no—
202C22⋊C4×C79—42no—
203C22⋊C4×C79—42no—
204C22⋊C4×C79—42no—
205C22⋊C4×C79—42no—
206C22⋊C4×C79—42no—
207C22⋊C4×C79—81no—
208C42×C79—21no—
209C42×C79—21no—
210C42×C79—21no—
211C42×C79—21no—
212C42×C79—42no—
213C23×C79—21no—
214C23×C79—21no—
215C23×C79—21no—
216C23×C79—21no—
217C23×C79—21no—
218C23×C79—21no—
219C23×C79—21no—
220C23×C79—41no—
221C23×C79—41no—
222C23×C79—41no—
223C23×C79—41no—
224C23×C79—41no—
225C23×C79—41no—
226C23×C79—41no—
227C23×C79C22×D411yescharacteristic
228Q8×C79—84no—
229D4×C79—41no—
230D4×C79—41no—
231D4×C79—41no—
232D4×C79—41no—
233D4×C79—84no—
234C2×C4×C79—21no—
235C2×C4×C79—21no—
236C2×C4×C79—21no—
237C2×C4×C79—21no—
238C2×C4×C79—21no—
239C2×C4×C79—21no—
240C2×C4×C79—21no—
241C2×C4×C79—21no—
242C2×C4×C79—22no—
243C2×C4×C79—21no—
244C2×C4×C79—21no—
245C2×C4×C79—21no—
246C2×C4×C79—21no—
247C2×C4×C79—21no—
248C2×C4×C79—21no—
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×C79AGL(1,ℤ/8)12yes—
262C2×C4×C79C2×#4111yescharacteristic
263C2×C4×C79C2×#4111yescharacteristic
264C2×C4×C79C2×#4111yescharacteristic
265C2×C4×C79C2×#4111yescharacteristic
266C2×C4×C79H(5,𝔽2)11yescharacteristic
267C2×C4×C79H(5,𝔽2)11yescharacteristic
268C2×Dic79—22no—
269C2×Dic79—22no—
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—
279C2×Dic79—41no—
280Dic158—22no—
281Dic158—22no—
282Dic158—41no—
283Dic158—41no—
284C79⋊C8—41no—
285C79⋊C8—41no—
286C79⋊C8—41no—
287C79⋊C8—41no—
288C22×C79—41no—
289C22×C79—41no—
290C22×C79—41no—
291C22×C79—41no—
292C22×C79—41no—
293C22×C79—41no—
294C22×C79—41no—
295C22×C79—81no—
296C22×C79—81no—
297C22×C79—81no—
298C22×C79—81no—
299C22×C79—81no—
300C22×C79—81no—
301C22×C79—81no—
302C22×C79#52511yescharacteristic
303C22×C79#54611yescharacteristic
304C22×C79#54611yescharacteristic
305C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C79#57511yescharacteristic
307C22×C79#57711yescharacteristic
308C22×C79#57811yescharacteristic
309C4×C79—22no—
310C4×C79—41no—
311C4×C79—41no—
312C4×C79—42no—
313C4×C79—41no—
314C4×C79—41no—
315C4×C79—81no—
316C4×C79—81no—
317C4×C79#52911yescharacteristic
318C4×C79#53811yescharacteristic
319C4×C79#55511yescharacteristic
320C4×C79#55911yescharacteristic
321Dic79—42no—
322Dic79—42no—
323Dic79—42no—
324Dic79—81no—
325#26748—791nomaximal, Sylow
326C2×C79—81no—
327C2×C79—161no—
328C2×C79#245911yescharacteristic
329C2×C79#311511yescharacteristic
330C2×C79#312111yescharacteristic
331C2×C79#331011yescharacteristic
332C2×C79#331311yescharacteristic
333C2×C79#332111yescharacteristic
334C2×C79#333011yescharacteristic
335#2491D7911yescharacteristic
336#2362—791no—
337#2055—791no—
338#2036—791no—
339#2024—791no—
340#2000—791no—
341#1988—791no—
342#1936—791no—
343#1925—791no—
344#1924—791no—
345#1921—791no—
346#1880—791no—
347#1875—791no—
348#1825—791no—
349#1763—791no—
350C79#2674811yescharacteristic, complemented, Sylow
351C22×SD16—791no—
352C2×C4⋊Q8C2×D7911yescharacteristic
353C2×#124—21no—
354C2×C22⋊Q8—791no—
355C2×#121C2×D7911yescharacteristic
356C2×C4×Q8—791no—
357C2×#117—791no—
358C2×#107—791no—
359C2×C4⋊C8—791no—
360C2×#103—791no—
361C2×#103—791no—
362C2×#103—791no—
363C2×#102—791no—
364C2×#102—791no—
365C2×#102—791no—
366C2×C22⋊C8—791no—
367C2×C22⋊C8—791no—
368C2×#97—791no—
369#394—21no—
370#393—21no—
371#391—791no—
372#390C2×D7911yescharacteristic
373#390C2×D7911yescharacteristic
374#389—791no—
375#388—21no—
376#388—1581no—
377#388—1581no—
378#387—1581no—
379#380C2×D7911yescharacteristic
380C4×C4⋊C4—791no—
381C4×C4⋊C4—1581no—
382C4×C4⋊C4C2×D7911yescharacteristic
383C4×C22⋊C4C2×D7911yescharacteristic
384#340—791no—
385#340—791no—
386#340—791no—
387#340—791no—
388#340—791no—
389#340—791no—
390#340—791no—
391#340—791no—
392#336—791no—
393#336—791no—
394C22×Q8—21no—
395C22×Q8—791no—
396C22×D4—21no—
397C22×D4C22×D7911yescharacteristic
398C23×C4C22×D7911yescharacteristic
399C2×SD16—3161no—
400C2×SD16—3161no—
401C2×SD16—1582no—
402C22×C8—791no—
403C22×C8—791no—
404C22×C8—791no—
405C22×C8—791no—
406C4⋊Q8—22no—
407C4⋊Q8—41no—
408#124—44no—
409C22⋊Q8—1582no—
410C22⋊Q8—1582no—
411#121—22no—
412#121—41no—
413C4×Q8—3161no—
414C4×Q8—1582no—
415#117—3161no—
416#117—1582no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—791no—
419C2×C4⋊C4—791no—
420C2×C4⋊C4—791no—
421C2×C4⋊C4—1581no—
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—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—1581no—
434C2×C22⋊C4—1581no—
435C2×C22⋊C4C22⋊D7911yescharacteristic
436C2×C22⋊C4C22⋊D7911yescharacteristic
437C2×C42—21no—
438C2×C42—1581no—
439C2×C42—1581no—
440C2×C42—1581no—
441C2×C42C22×D7911yescharacteristic
442C2×C42C22×D7911yescharacteristic
443#107—1582no—
444C4⋊C8—3161no—
445#103—3161no—
446#103—1582no—
447#103—1582no—
448#102—3161no—
449#102—1582no—
450#102—1582no—
451C22⋊C8—3161no—
452C22⋊C8—3161no—
453#97—3161no—
454#95—21no—
455#95—21no—
456#95—1581no—
457#95—1581no—
458#95—1581no—
459#95—1581no—
460#95C22⋊D7911yescharacteristic
461#95C22⋊D7911yescharacteristic
462C24—21no—
463C24C2×C22⋊D7911yescharacteristic
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—81no—
467C2×Q8—792no—
468C2×Q8—1581no—
469C2×Q8—1581no—
470C2×Q8—1581no—
471C2×Q8—1581no—
472C2×Q8—1581no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—81no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—1581no—
492C22×C4—1581no—
493C22×C4—1581no—
494C22×C4—1581no—
495C22×C4C2×C22⋊D7911yescharacteristic
496C22×C4C2×C22⋊D7911yescharacteristic
497C22×C4C2×C22⋊D7911yescharacteristic
498C22×C4C2×C22⋊D7911yescharacteristic
499C22×C4C2×C22⋊D7911yescharacteristic
500C22×C4C23×D7911yescharacteristic
501SD16—3164no—
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—
512C2×C8—1581no—
513C2×C8—1581no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—3162no—
521C4⋊C4—3162no—
522C4⋊C4—792no—
523C4⋊C4—792no—
524C4⋊C4—1584no—
525C4⋊C4—1581no—
526C4⋊C4—1581no—
527C4⋊C4—1581no—
528C4⋊C4—1581no—
529C4⋊C4#1192962312yes—
530C22⋊C4—22no—
531C22⋊C4—41no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—81no—
538C22⋊C4—3162no—
539C22⋊C4—1584no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—42no—
545C42—3162no—
546C42—3162no—
547C42—3162no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C22×C22⋊D7911yescharacteristic
563Q8—84no—
564Q8—3161no—
565Q8—3161no—
566Q8—1582no—
567Q8—1582no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—84no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
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—3161no—
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—1582no—
611C2×C4—1582no—
612C2×C4#1457918712yes—
613C2×C4#41×D7911yescharacteristic
614C2×C4#41×D7911yescharacteristic
615C2×C4#41×D7911yescharacteristic
616C2×C4C2×#1192962311yescharacteristic
617C2×C4#1457920611yescharacteristic
618C2×C4#1457920711yescharacteristic
619C8—3161no—
620C8—3161no—
621C8—3161no—
622C8—3161no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22#1909907011yescharacteristic
638C22#1909920111yescharacteristic
639C22#1909920711yescharacteristic
640C22C2×#1457918711yescharacteristic
641C22#1909936111yescharacteristic
642C22#1909937411yescharacteristic
643C22#1909937811yescharacteristic
644C4—22no—
645C4—41no—
646C4—41no—
647C4—42no—
648C4—41no—
649C4—41no—
650C4—81no—
651C4—81no—
652C4—3162no—
653C4—3162no—
654C4—3162no—
655C4—6321no—
656C4#1909909211yescharacteristic
657C4#1909914711yescharacteristic
658C4#1909927511yescharacteristic
659C4#1909929511yescharacteristic
660C2—81no—
661C2—161no—
662C2#2376733311yescharacteristic
663C2#2377354511yescharacteristic
664C2#2377362311yescharacteristic
665C2#2377542111yescharacteristic
666C2#2377546611yescharacteristic
667C2#2377558011yescharacteristic
668C2#2377569911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2491C2
socleC23×C79C22×D4
prime radicalC2×#121×C79C22
hypercenter#2491D79
hypocenterC79#26748
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2491

Complements

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

NH ≅ G/Ntype
1. #23766425C2semidirect
2. #23764717C2semidirect
3. #23764603C2semidirect
4. #23764538C2semidirect
6. #23764353C2semidirect
7. #23764100C2semidirect
8. #23764017C2semidirect
9. #23764015C2semidirect
10. #23763992C2semidirect
11. #23763758C2semidirect
13. #23763492C2semidirect
14. #23763233C2semidirect
350. C79#26748semidirect

Maximal subgroups

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

subgroupindex
#237664252
#237647172
#237646032
#237645382
#237644032
#237643532
#237641002
#237640172
#237640152
#237639922
#237637582
#237637232
#237634922
#237632332
C79×#24912
#2674879