Finite Group Databasebeta

← #34297820

Subgroups — #34297820

Order 20224.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098624C211yesmaximal, characteristic
2#23766009C211yesmaximal, characteristic
3#23764374C211yesmaximal, characteristic, complemented
4#23764370C211yesmaximal, characteristic, complemented
5#23764366C211yesmaximal, characteristic, complemented
6#23764362C211yesmaximal, characteristic, complemented
7#23764358C211yesmaximal, characteristic, complemented
8#23764358C211yesmaximal, characteristic, complemented
9#23764353C211yesmaximal, characteristic, complemented
10#23764349C211yesmaximal, characteristic, complemented
11#23763495C211yesmaximal, characteristic
12#23763495C211yesmaximal, characteristic
13#23763490C211yesmaximal, characteristic
14#23763486C211yesmaximal, characteristic
15C2×#544×C79C211yesmaximal, characteristic
16#19098624—24no—
17C2×C4⋊(C79⋊C8)C2211yescharacteristic
18C2×C4⋊(C79⋊C8)C2211yescharacteristic
19C2×C22⋊(C79⋊C8)C2211yescharacteristic
20C2×C22⋊(C79⋊C8)C2211yescharacteristic
21C2×#14579072C2211yescharacteristic
22C2×C4×C79⋊C8C2211yescharacteristic
23#19098456—21no—
24#19098456—21no—
25#19098449—21no—
26#19098449—21no—
27#19098433C2211yescharacteristic
28#19098429C2211yescharacteristic
29#19098423C2211yescharacteristic
30#19098423C2211yescharacteristic
31C4×C4⋊Dic79C2211yescharacteristic
32C4×C4⋊Dic79C2211yescharacteristic
33#19098325C2211yescharacteristic
34#19098325C2211yescharacteristic
35#19098325C2211yescharacteristic
36#19098325C2211yescharacteristic
37#19098325C2211yescharacteristic
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#544×C79—24no—
50#544×C79C418yescomplemented
51C2×C4⋊Q8×C79—21no—
52C2×C4⋊Q8×C79C2211yescharacteristic
53C2×C4⋊Q8×C79C2211yescharacteristic
54C2×#125×C79—21no—
55C2×#125×C79C2211yescharacteristic
56C2×#125×C79C2211yescharacteristic
57C2×C22⋊Q8×C79—21no—
58C2×C22⋊Q8×C79—21no—
59C2×#117×C79C2211yescharacteristic
60C2×#117×C79C2211yescharacteristic
61C22×C4⋊C4×C79C2211yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C2×C4⋊Dic79—21no—
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C22⋊Dic79—21no—
73C2×C22⋊Dic79—21no—
74C2×C4×Dic79—21no—
75C2×C4×Dic79—21no—
76C4⋊(C79⋊C8)—22no—
77C4⋊(C79⋊C8)—22no—
78C22⋊(C79⋊C8)—22no—
79C22⋊(C79⋊C8)—22no—
80#14579072—22no—
81C4×C79⋊C8—22no—
82C79⋊#95—21no—
83C79⋊#95—21no—
84C79⋊#95—21no—
85C79⋊#95—21no—
86C79⋊#95—21no—
87C79⋊#95—21no—
88C22×Q8×C79—21no—
89C22×Q8×C79—21no—
90C23×C4×C79—21no—
91C23×C4×C79C2311yescharacteristic
92C4⋊Q8×C79—28no—
93C4⋊Q8×C79—22no—
94C4⋊Q8×C79—22no—
95C4⋊Q8×C79C2×C414yes—
96C4⋊Q8×C79C2×C414yes—
97#125×C79—28no—
98#125×C79—22no—
99#125×C79—22no—
100#125×C79C2×C414yes—
101#125×C79C2×C414yes—
102C22⋊Q8×C79—28no—
103C22⋊Q8×C79—28no—
104#117×C79—22no—
105#117×C79C2×C414yes—
106#117×C79C2×C414yes—
107#117×C79D414yes—
108C2×C4⋊C4×C79—21no—
109C2×C4⋊C4×C79—22no—
110C2×C4⋊C4×C79—21no—
111C2×C4⋊C4×C79—21no—
112C2×C4⋊C4×C79—21no—
113C2×C4⋊C4×C79—21no—
114C2×C4⋊C4×C79—21no—
115C2×C4⋊C4×C79—21no—
116C2×C4⋊C4×C79C2×C414yes—
117C2×C4⋊C4×C79D411yescharacteristic
118C2×C4⋊C4×C79D411yescharacteristic
119C2×C4⋊C4×C79C2311yescharacteristic
120C2×C4⋊C4×C79C2311yescharacteristic
121C2×C4⋊C4×C79C2311yescharacteristic
122C2×C4⋊C4×C79C2311yescharacteristic
123C2×C22⋊C4×C79—21no—
124C2×C22⋊C4×C79D411yescharacteristic
125C2×C22⋊C4×C79D411yescharacteristic
126C2×C42×C79—21no—
127C2×C42×C79C2311yescharacteristic
128C2×C42×C79C2311yescharacteristic
129C22×Dic79—21no—
130C22×Dic79—21no—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8—21no—
136C2×C79⋊C8—21no—
137C2×C79⋊C8—21no—
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—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—81no—
154C4⋊Dic79—81no—
155C22⋊Dic79—81no—
156C22⋊Dic79—81no—
157C4×Dic79—81no—
158C4×Dic79—81no—
159C24×C79C2×D411yescharacteristic
160C2×Q8×C79—24no—
161C2×Q8×C79—24no—
162C2×Q8×C79—81no—
163C2×Q8×C79—81no—
164C2×Q8×C79—81no—
165C2×Q8×C79—81no—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79—21no—
169C22×C4×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79—21no—
173C22×C4×C79—21no—
174C22×C4×C79—24no—
175C22×C4×C79—81no—
176C22×C4×C79—81no—
177C22×C4×C79C22×C412yes—
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2×D411yescharacteristic
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2×D412yes—
184C22×C4×C79C2411yescharacteristic
185C4⋊C4×C79—24no—
186C4⋊C4×C79—24no—
187C4⋊C4×C79—24no—
188C4⋊C4×C79—24no—
189C4⋊C4×C79—24no—
190C4⋊C4×C79—24no—
191C4⋊C4×C79—24no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79C22⋊C414yes—
197C4⋊C4×C79C22⋊C414yes—
198C4⋊C4×C79C22×C412yes—
199C4⋊C4×C79C22×C412yes—
200C4⋊C4×C79C22×C412yes—
201C4⋊C4×C79C22×C412yes—
202C22⋊C4×C79—24no—
203C22⋊C4×C79C22⋊C414yes—
204C22⋊C4×C79C22⋊C414yes—
205C42×C79—24no—
206C42×C79C22×C412yes—
207C42×C79C22×C412yes—
208C42×C79C2×D412yes—
209C42×C79C2×D412yes—
210C23×C79—21no—
211C23×C79—21no—
212C23×C79—21no—
213C23×C79—21no—
214C23×C79—21no—
215C23×C79—21no—
216C23×C79C2×C22⋊C412yes—
217C23×C79C22×D411yescharacteristic
218Q8×C79—84no—
219Q8×C79—84no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—22no—
231C2×C4×C79—22no—
232C2×C4×C79—22no—
233C2×C4×C79—22no—
234C2×C4×C79—22no—
235C2×C4×C79—22no—
236C2×C4×C79—22no—
237C2×C4×C79—22no—
238C2×C4×C79—41no—
239C2×C4×C79—41no—
240C2×C4×C79—41no—
241C2×C4×C79—41no—
242C2×C4×C79—41no—
243C2×C4×C79—41no—
244C2×C4×C79—41no—
245C2×C4×C79—41no—
246C2×C4×C79—84no—
247C2×C4×C79C2×C22⋊C412yes—
248C2×C4×C79C2×C22⋊C412yes—
249C2×C4×C79C2×C22⋊C412yes—
250C2×C4×C79C2×C22⋊C412yes—
251C2×C4×C79C2×C22⋊C412yes—
252C2×C4×C79C23×C411yescharacteristic
253C2×C4×C79C22×D411yescharacteristic
254C2×C4×C792−1+411yescharacteristic
255C2×C4×C792−1+411yescharacteristic
256C2×C4×C792−1+411yescharacteristic
257C2×C4×C792−1+411yescharacteristic
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79—41no—
262C2×Dic79—41no—
263C2×Dic79—41no—
264C2×Dic79—41no—
265C2×Dic79—41no—
266C2×Dic79—41no—
267C2×Dic79—41no—
268C2×Dic79—41no—
269C2×Dic79—41no—
270C79⋊C8—41no—
271C79⋊C8—41no—
272C79⋊C8—41no—
273C79⋊C8—41no—
274C22×C79—22no—
275C22×C79—22no—
276C22×C79—22no—
277C22×C79—41no—
278C22×C79—41no—
279C22×C79—41no—
280C22×C79—41no—
281C22×C79C22×C22⋊C411yescharacteristic
282C22×C79#53611yescharacteristic
283C22×C79#53611yescharacteristic
284C22×C79#57611yescharacteristic
285C22×C79#57711yescharacteristic
286C22×C79#57711yescharacteristic
287C22×C79#57811yescharacteristic
288C4×C79—22no—
289C4×C79—22no—
290C4×C79—22no—
291C4×C79—22no—
292C4×C79—22no—
293C4×C79—42no—
294C4×C79—42no—
295C4×C79—42no—
296C4×C79—42no—
297C4×C79#51911yescharacteristic
298C4×C79#51911yescharacteristic
299C4×C79#53611yescharacteristic
300C4×C79#53611yescharacteristic
301Dic79—81no—
302Dic79—81no—
303Dic79—81no—
304Dic79—81no—
305#21843—791nomaximal, Sylow
306C2×C79—42no—
307C2×C79#231011yescharacteristic
308C2×C79#289711yescharacteristic
309C2×C79#289811yescharacteristic
310C2×C79#324111yescharacteristic
311C2×C79#324211yescharacteristic
312C2×C79#324311yescharacteristic
313C2×C79#324411yescharacteristic
314C2×#544D7911yescharacteristic
315C2×#433—791no—
316#2314—791no—
317#1993—791no—
318#1992—791no—
319#1991—791no—
320#1990—791no—
321#1989—791no—
322#1989—791no—
323#1988—791no—
324#1987—791no—
325#1825—791no—
326#1825—791no—
327#1824—791no—
328#1823—791no—
329C79#2184311yescharacteristic, complemented, Sylow
330#544—24no—
331#544Dic7918yescomplemented
332C2×C4⋊Q8—21no—
333C2×C4⋊Q8C2×D7911yescharacteristic
334C2×C4⋊Q8C2×D7911yescharacteristic
335C2×#125—21no—
336C2×#125C2×D7911yescharacteristic
337C2×#125C2×D7911yescharacteristic
338C2×C22⋊Q8—21no—
339C2×C22⋊Q8—21no—
340C2×#117C2×D7911yescharacteristic
341C2×#117C2×D7911yescharacteristic
342C22×C4⋊C4C2×D7911yescharacteristic
343#433—1584no—
344C2×C4⋊C8—791no—
345C2×C4⋊C8—791no—
346C2×C22⋊C8—791no—
347C2×C22⋊C8—791no—
348C2×#97—791no—
349C2×C4×C8—791no—
350#388—1581no—
351#388—1581no—
352#387—1581no—
353#387—1581no—
354#384—791no—
355#382—791no—
356#380—791no—
357#380—791no—
358C4×C4⋊C4—791no—
359C4×C4⋊C4—791no—
360#340—791no—
361#340—791no—
362#340—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#340—791no—
368#340—791no—
369#340—791no—
370#340—791no—
371#340—791no—
372#340—791no—
373#340—791no—
374#340—791no—
375#340—791no—
376C22×Q8—21no—
377C22×Q8—21no—
378C23×C4—21no—
379C23×C4C22×D7911yescharacteristic
380C22×C8—791no—
381C22×C8—791no—
382C22×C8—791no—
383C22×C8—791no—
384C4⋊Q8—28no—
385C4⋊Q8—22no—
386C4⋊Q8—22no—
387C4⋊Q8C2×Dic7914yes—
388C4⋊Q8C2×Dic7914yes—
389#125—28no—
390#125—22no—
391#125—22no—
392#125C2×Dic7914yes—
393#125C2×Dic7914yes—
394C22⋊Q8—28no—
395C22⋊Q8—28no—
396#117—22no—
397#117C2×Dic7914yes—
398#117C2×Dic7914yes—
399#117C22⋊D7914yes—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—791no—
409C2×C4⋊C4—791no—
410C2×C4⋊C4—791no—
411C2×C4⋊C4—791no—
412C2×C4⋊C4—1581no—
413C2×C4⋊C4—1581no—
414C2×C4⋊C4C2×Dic7914yes—
415C2×C4⋊C4C22⋊D7911yescharacteristic
416C2×C4⋊C4C22⋊D7911yescharacteristic
417C2×C4⋊C4C22×D7911yescharacteristic
418C2×C4⋊C4C22×D7911yescharacteristic
419C2×C4⋊C4C22×D7911yescharacteristic
420C2×C4⋊C4C22×D7911yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—1581no—
423C2×C22⋊C4—1581no—
424C2×C22⋊C4C22⋊D7911yescharacteristic
425C2×C22⋊C4C22⋊D7911yescharacteristic
426C2×C42—21no—
427C2×C42—1581no—
428C2×C42—1581no—
429C2×C42C22×D7911yescharacteristic
430C2×C42C22×D7911yescharacteristic
431C4⋊C8—1582no—
432C4⋊C8—1582no—
433C22⋊C8—1582no—
434C22⋊C8—1582no—
435#97—1582no—
436C4×C8—1582no—
437#95—1581no—
438#95—1581no—
439#95—1581no—
440#95—1581no—
441#95—1581no—
442#95—1581no—
443C24C2×C22⋊D7911yescharacteristic
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—81no—
447C2×Q8—81no—
448C2×Q8—81no—
449C2×Q8—81no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—1581no—
462C22×C4—1581no—
463C22×C4—1581no—
464C22×C4—1581no—
465C22×C4C22×Dic7912yes—
466C22×C4C2×C22⋊D7911yescharacteristic
467C22×C4C2×C22⋊D7911yescharacteristic
468C22×C4C2×C22⋊D7911yescharacteristic
469C22×C4C2×C22⋊D7911yescharacteristic
470C22×C4C2×C22⋊D7911yescharacteristic
471C22×C4C2×C22⋊D7912yes—
472C22×C4C23×D7911yescharacteristic
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8—1581no—
478C2×C8—1581no—
479C2×C8—1581no—
480C2×C8—1581no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8—1581no—
484C2×C8—1581no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—6321no—
497C4⋊C4—6321no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C4⋊C4—1581no—
501C4⋊C4—1581no—
502C4⋊C4—1581no—
503C4⋊C4—1581no—
504C4⋊C4—1581no—
505C4⋊C4—1581no—
506C4⋊C4C22⋊Dic7914yes—
507C4⋊C4C22⋊Dic7914yes—
508C4⋊C4C22×Dic7912yes—
509C4⋊C4C22×Dic7912yes—
510C4⋊C4C22×Dic7912yes—
511C4⋊C4C22×Dic7912yes—
512C22⋊C4—24no—
513C22⋊C4—6321no—
514C22⋊C4—6321no—
515C22⋊C4C22⋊Dic7914yes—
516C22⋊C4C22⋊Dic7914yes—
517C42—24no—
518C42—6321no—
519C42—6321no—
520C42C22×Dic7912yes—
521C42C22×Dic7912yes—
522C42C2×C22⋊D7912yes—
523C42C2×C22⋊D7912yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C2×C22⋊Dic7912yes—
531C23C22×C22⋊D7911yescharacteristic
532Q8—84no—
533Q8—84no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4—3161no—
562C2×C4—3161no—
563C2×C4—3161no—
564C2×C4—3161no—
565C2×C4—3161no—
566C2×C4—3161no—
567C2×C4—3161no—
568C2×C4—3161no—
569C2×C4—3161no—
570C2×C4—3161no—
571C2×C4—3161no—
572C2×C4—3161no—
573C2×C4C2×C22⋊Dic7912yes—
574C2×C4C2×C22⋊Dic7912yes—
575C2×C4C2×C22⋊Dic7912yes—
576C2×C4C2×C22⋊Dic7912yes—
577C2×C4C2×C22⋊Dic7912yes—
578C2×C4C23×Dic7911yescharacteristic
579C2×C4C22×C22⋊D7911yescharacteristic
580C2×C4#1457920911yescharacteristic
581C2×C4#1457920911yescharacteristic
582C2×C4#1457920911yescharacteristic
583C2×C4#1457920911yescharacteristic
584C8—3161no—
585C8—3161no—
586C8—3161no—
587C8—3161no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C22×C22⋊Dic7911yescharacteristic
596C22#1909913711yescharacteristic
597C22#1909913711yescharacteristic
598C22#1909936611yescharacteristic
599C22#1909937111yescharacteristic
600C22#1909937411yescharacteristic
601C22#1909937811yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—6321no—
612C4—6321no—
613C4—6321no—
614C4—6321no—
615C4#1909904111yescharacteristic
616C4#1909904111yescharacteristic
617C4#1909913711yescharacteristic
618C4#1909913711yescharacteristic
619C2—42no—
620C2#2376598611yescharacteristic
621C2#2377192611yescharacteristic
622C2#2377193311yescharacteristic
623C2#2377465311yescharacteristic
624C2#2377466811yescharacteristic
625C2#2377468311yescharacteristic
626C2#2377469811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#544×C79C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
cosocleC22×C4×C79C24
hypercenterC2×#544D79
hypocenterC79#21843
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #23764374C2semidirect
4. #23764370C2semidirect
5. #23764366C2semidirect
6. #23764362C2semidirect
7. #23764358C2semidirect
8. #23764358C2semidirect
9. #23764353C2semidirect
10. #23764349C2semidirect
50. #544×C79C4semidirect
329. C79#21843semidirect
331. #544Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190986242
#237660092
#237643742
#237643702
#237643662
#237643622
#237643582
#237643532
#237643492
#237634952
#237634902
#237634862
C2×#544×C792
#2184379