Finite Group Databasebeta

← #34297816

Subgroups — #34297816

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×#19099194C211yesmaximal, characteristic
2C2×#19098621C211yesmaximal, characteristic
3#23766007C211yesmaximal, characteristic
4#23764371C211yesmaximal, characteristic, complemented
5#23764367C211yesmaximal, characteristic, complemented
6#23764363C211yesmaximal, characteristic, complemented
7#23764359C211yesmaximal, characteristic, complemented
8#23764356C211yesmaximal, characteristic, complemented
9#23764354C211yesmaximal, characteristic, complemented
10#23764350C211yesmaximal, characteristic, complemented
11#23764346C211yesmaximal, characteristic, complemented
12#23763496C211yesmaximal, characteristic
13#23763492C211yesmaximal, characteristic
14#23763490C211yesmaximal, characteristic
15C79×#1823C211yesmaximal, characteristic
16#19099194—24no—
17#19099194C418yescomplemented
18C2×#14579160—21no—
19C2×#14579160C2211yescharacteristic
20C2×#14579160C2211yescharacteristic
21C2×#14579152—21no—
22C2×#14579152C2211yescharacteristic
23C2×#14579152C2211yescharacteristic
24C2×#14579140—21no—
25C2×#14579140—21no—
26C2×#14579120C2211yescharacteristic
27C2×#14579120C2211yescharacteristic
28#19098621—24no—
29C2×Dic79⋊C8C2211yescharacteristic
30C2×Dic79⋊C8C2211yescharacteristic
31C2×C22⋊(C79⋊C8)C2211yescharacteristic
32C2×#14579071C2211yescharacteristic
33C2×C8×Dic79C2211yescharacteristic
34#19098457—21no—
35#19098451—21no—
36#19098448—21no—
37#19098446—21no—
38#19098432C2211yescharacteristic
39#19098428C2211yescharacteristic
40#19098423C2211yescharacteristic
41C4⋊C4×Dic79C2211yescharacteristic
42C4⋊C4×Dic79C2211yescharacteristic
43#19098326C2211yescharacteristic
44#19098326C2211yescharacteristic
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098325C2211yescharacteristic
53#19098325C2211yescharacteristic
54#19098325C2211yescharacteristic
55C22×C4⋊C4×C79C2211yescharacteristic
56C2×C22⋊C8×C79C2211yescharacteristic
57#380×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60#340×C79C2211yescharacteristic
61#340×C79C2211yescharacteristic
62C22×Dic158—21no—
63C22×Dic158—21no—
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66#14579160—28no—
67#14579160—22no—
68#14579160—22no—
69#14579160C2×C414yes—
70#14579160C2×C414yes—
71#14579152—28no—
72#14579152—22no—
73#14579152—22no—
74#14579152C2×C414yes—
75#14579152C2×C414yes—
76#14579140—28no—
77#14579140—28no—
78#14579120—22no—
79#14579120C2×C414yes—
80#14579120C2×C414yes—
81#14579120D414yes—
82C2×C4⋊Dic79—21no—
83C2×C4⋊Dic79—21no—
84C2×C4⋊Dic79D411yescharacteristic
85C2×C4⋊Dic79D411yescharacteristic
86C2×C4⋊Dic79C2311yescharacteristic
87C2×C4⋊Dic79C2311yescharacteristic
88C2×C4⋊Dic79C2311yescharacteristic
89C2×C4⋊Dic79C2311yescharacteristic
90C2×Dic79⋊C4—21no—
91C2×Dic79⋊C4—21no—
92C2×Dic79⋊C4—21no—
93C2×Dic79⋊C4—21no—
94C2×Dic79⋊C4—21no—
95C2×Dic79⋊C4—21no—
96C2×C22⋊Dic79—21no—
97C2×C22⋊Dic79—21no—
98C2×C22⋊Dic79D411yescharacteristic
99C2×C22⋊Dic79D411yescharacteristic
100C2×C4×Dic79—21no—
101C2×C4×Dic79—21no—
102C2×C4×Dic79—21no—
103C2×C4×Dic79C2311yescharacteristic
104C2×C4×Dic79C2311yescharacteristic
105Dic79⋊C8—22no—
106Dic79⋊C8—22no—
107C22⋊(C79⋊C8)—22no—
108#14579071—22no—
109C8×Dic79—22no—
110C79⋊#95—21no—
111C79⋊#95—21no—
112C79⋊#95—21no—
113C79⋊#95—21no—
114C79⋊#95—21no—
115C23×C4×C79—21no—
116C23×C4×C79C2311yescharacteristic
117C22×C8×C79C2311yescharacteristic
118C22×C8×C79C2311yescharacteristic
119C2×C4⋊C4×C79—22no—
120C2×C4⋊C4×C79—21no—
121C2×C4⋊C4×C79C2×C414yes—
122C2×C4⋊C4×C79C2311yescharacteristic
123C2×C4⋊C4×C79C2311yescharacteristic
124C2×C4⋊C4×C79C2311yescharacteristic
125C2×C4⋊C4×C79C2311yescharacteristic
126C2×C22⋊C4×C79—21no—
127C22⋊C8×C79—22no—
128#95×C79—21no—
129C2×Dic158—24no—
130C2×Dic158—24no—
131C2×Dic158—81no—
132C2×Dic158—81no—
133C2×Dic158—81no—
134C2×Dic158—81no—
135C22×Dic79—21no—
136C22×Dic79—21no—
137C22×Dic79—21no—
138C22×Dic79—21no—
139C22×Dic79C2×D411yescharacteristic
140C22×Dic79C2×D411yescharacteristic
141C22×Dic79C2×D411yescharacteristic
142C22×Dic79C2×D411yescharacteristic
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8—21no—
149C4⋊Dic79—24no—
150C4⋊Dic79—24no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79C22⋊C414yes—
158C4⋊Dic79C22⋊C414yes—
159C4⋊Dic79C22×C412yes—
160C4⋊Dic79C22×C412yes—
161Dic79⋊C4—24no—
162Dic79⋊C4—24no—
163Dic79⋊C4—24no—
164Dic79⋊C4—24no—
165Dic79⋊C4—81no—
166Dic79⋊C4—81no—
167C22⋊Dic79—24no—
168C22⋊Dic79—81no—
169C22⋊Dic79C22⋊C414yes—
170C22⋊Dic79C22⋊C414yes—
171C4×Dic79—24no—
172C4×Dic79—81no—
173C4×Dic79—81no—
174C4×Dic79C22×C412yes—
175C4×Dic79C22×C412yes—
176C4×Dic79C2×D412yes—
177C4×Dic79C2×D412yes—
178C24×C79C2×D411yescharacteristic
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—24no—
186C22×C4×C79—21no—
187C22×C4×C79—21no—
188C22×C4×C79—81no—
189C22×C4×C79—81no—
190C22×C4×C79C22×C412yes—
191C22×C4×C79C2×D411yescharacteristic
192C22×C4×C79C2×D412yes—
193C22×C4×C79C2411yescharacteristic
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—21no—
199C2×C8×C79—21no—
200C4⋊C4×C79—24no—
201C4⋊C4×C79—21no—
202C4⋊C4×C79—21no—
203C4⋊C4×C79—21no—
204C4⋊C4×C79—21no—
205C4⋊C4×C79—21no—
206C4⋊C4×C79—21no—
207C4⋊C4×C79C22×C412yes—
208C4⋊C4×C79C22×C412yes—
209C22⋊C4×C79—81no—
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
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—22no—
221C2×C4×C79—22no—
222C2×C4×C79—22no—
223C2×C4×C79—22no—
224C2×C4×C79—22no—
225C2×C4×C79—22no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79—41no—
233C2×C4×C79—41no—
234C2×C4×C79—41no—
235C2×C4×C79—41no—
236C2×C4×C79—84no—
237C2×C4×C79C2×C22⋊C412yes—
238C2×C4×C79C23×C411yescharacteristic
239C2×C4×C79C22×D411yescharacteristic
240C2×C4×C792−1+411yescharacteristic
241C2×C4×C792−1+411yescharacteristic
242C2×C4×C792−1+411yescharacteristic
243C2×C4×C792−1+411yescharacteristic
244C2×Dic79—21no—
245C2×Dic79—21no—
246C2×Dic79—21no—
247C2×Dic79—21no—
248C2×Dic79—21no—
249C2×Dic79—21no—
250C2×Dic79—21no—
251C2×Dic79—21no—
252C2×Dic79—22no—
253C2×Dic79—22no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79—41no—
262C2×Dic79—41no—
263C2×Dic79—41no—
264C2×Dic79C2×C22⋊C412yes—
265C2×Dic79C2×C22⋊C412yes—
266C2×Dic79C2×C22⋊C412yes—
267C2×Dic79C2×C22⋊C412yes—
268Dic158—84no—
269Dic158—84no—
270C8×C79—41no—
271C8×C79—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—42no—
290C4×C79—42no—
291C4×C79—81no—
292C4×C79—81no—
293C4×C79#51911yescharacteristic
294C4×C79#51911yescharacteristic
295C4×C79#53611yescharacteristic
296C4×C79#53611yescharacteristic
297Dic79—22no—
298Dic79—22no—
299Dic79—22no—
300Dic79—22no—
301Dic79—42no—
302Dic79—42no—
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×#544—791no—
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#1823D7911yescharacteristic
329C79#2184311yescharacteristic, complemented, Sylow
330#544—798no—
331#544—1584no—
332C2×C4⋊Q8—791no—
333C2×C4⋊Q8—791no—
334C2×C4⋊Q8—1581no—
335C2×#125—791no—
336C2×#125—791no—
337C2×#125—1581no—
338C2×C22⋊Q8—1581no—
339C2×C22⋊Q8—1581no—
340C2×#117—791no—
341C2×#117—791no—
342C22×C4⋊C4C2×D7911yescharacteristic
343#433—1584no—
344C2×C4⋊C8—791no—
345C2×C4⋊C8—791no—
346C2×C22⋊C8—791no—
347C2×C22⋊C8C2×D7911yescharacteristic
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#380C2×D7911yescharacteristic
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#340C2×D7911yescharacteristic
373#340C2×D7911yescharacteristic
374#340C2×D7911yescharacteristic
375#340C2×D7911yescharacteristic
376C22×Q8—1581no—
377C22×Q8—1581no—
378C23×C4—21no—
379C23×C4C22×D7911yescharacteristic
380C22×C8—791no—
381C22×C8—791no—
382C22×C8C22×D7911yescharacteristic
383C22×C8C22×D7911yescharacteristic
384C4⋊Q8—794no—
385C4⋊Q8—794no—
386C4⋊Q8—1582no—
387C4⋊Q8—1582no—
388C4⋊Q8—1588no—
389#125—794no—
390#125—794no—
391#125—1582no—
392#125—1582no—
393#125—1588no—
394C22⋊Q8—1588no—
395C22⋊Q8—1588no—
396#117—794no—
397#117—794no—
398#117—794no—
399#117—1582no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—791no—
403C2×C4⋊C4—791no—
404C2×C4⋊C4—791no—
405C2×C4⋊C4—791no—
406C2×C4⋊C4—791no—
407C2×C4⋊C4—791no—
408C2×C4⋊C4—1581no—
409C2×C4⋊C4—1581no—
410C2×C4⋊C4—1581no—
411C2×C4⋊C4—1581no—
412C2×C4⋊C4—1581no—
413C2×C4⋊C4—1581no—
414C2×C4⋊C4—1581no—
415C2×C4⋊C4—1581no—
416C2×C4⋊C4C4×D7914yes—
417C2×C4⋊C4C22×D7911yescharacteristic
418C2×C4⋊C4C22×D7911yescharacteristic
419C2×C4⋊C4C22×D7911yescharacteristic
420C2×C4⋊C4C22×D7911yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—791no—
423C2×C22⋊C4—791no—
424C2×C22⋊C4—1581no—
425C2×C22⋊C4—1581no—
426C2×C42—791no—
427C2×C42—791no—
428C2×C42—1581no—
429C2×C42—1581no—
430C2×C42—1581no—
431C4⋊C8—1582no—
432C4⋊C8—1582no—
433C22⋊C8—22no—
434C22⋊C8—1582no—
435#97—1582no—
436C4×C8—1582no—
437#95—21no—
438#95—1581no—
439#95—1581no—
440#95—1581no—
441#95—1581no—
442#95—1581no—
443C24D4×D7911yescharacteristic
444C2×Q8—6321no—
445C2×Q8—6321no—
446C2×Q8—6321no—
447C2×Q8—6321no—
448C2×Q8—1584no—
449C2×Q8—1584no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—791no—
462C22×C4—791no—
463C22×C4—791no—
464C22×C4—791no—
465C22×C4—1581no—
466C22×C4—1581no—
467C22×C4—1581no—
468C22×C4—1581no—
469C22×C4C2×C4×D7912yes—
470C22×C4D4×D7911yescharacteristic
471C22×C4D4×D7912yes—
472C22×C4C23×D7911yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—1581no—
480C2×C8—1581no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8—1581no—
484C2×C8—1581no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—794no—
493C4⋊C4—794no—
494C4⋊C4—792no—
495C4⋊C4—792no—
496C4⋊C4—6321no—
497C4⋊C4—6321no—
498C4⋊C4—1584no—
499C4⋊C4—1584no—
500C4⋊C4—1584no—
501C4⋊C4—1584no—
502C4⋊C4—1584no—
503C4⋊C4—1584no—
504C4⋊C4—1581no—
505C4⋊C4—1581no—
506C4⋊C4—1581no—
507C4⋊C4—1581no—
508C4⋊C4—1581no—
509C4⋊C4—1581no—
510C4⋊C4C2×C4×D7912yes—
511C4⋊C4C2×C4×D7912yes—
512C22⋊C4—81no—
513C22⋊C4—794no—
514C22⋊C4—794no—
515C22⋊C4—6321no—
516C22⋊C4—1584no—
517C42—792no—
518C42—792no—
519C42—792no—
520C42—792no—
521C42—6321no—
522C42—6321no—
523C42—1584no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C22⋊C4×D7912yes—
531C23C2×D4×D7911yescharacteristic
532Q8—6324no—
533Q8—6324no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—84no—
553C2×C4—3161no—
554C2×C4—3161no—
555C2×C4—3161no—
556C2×C4—3161no—
557C2×C4—3161no—
558C2×C4—3161no—
559C2×C4—3161no—
560C2×C4—3161no—
561C2×C4—3161no—
562C2×C4—3161no—
563C2×C4—792no—
564C2×C4—792no—
565C2×C4—792no—
566C2×C4—792no—
567C2×C4—1582no—
568C2×C4—1582no—
569C2×C4—1581no—
570C2×C4—1581no—
571C2×C4—1581no—
572C2×C4—1581no—
573C2×C4—1581no—
574C2×C4—1581no—
575C2×C4—1581no—
576C2×C4—1581no—
577C2×C4C22⋊C4×D7912yes—
578C2×C4C22×C4×D7911yescharacteristic
579C2×C4C2×D4×D7911yescharacteristic
580C2×C4#1457920811yescharacteristic
581C2×C4#1457920811yescharacteristic
582C2×C4#1457920811yescharacteristic
583C2×C4#1457920811yescharacteristic
584C8—41no—
585C8—41no—
586C8—3161no—
587C8—3161no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C2×C22⋊C4×D7911yescharacteristic
596C22#1909913511yescharacteristic
597C22#1909913511yescharacteristic
598C22#1909936411yescharacteristic
599C22#1909936911yescharacteristic
600C22#1909937211yescharacteristic
601C22#1909937911yescharacteristic
602C4—22no—
603C4—42no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—3162no—
608C4—3162no—
609C4—6321no—
610C4—6321no—
611C4—1582no—
612C4—1582no—
613C4—1582no—
614C4—1582no—
615C4#1909904011yescharacteristic
616C4#1909904011yescharacteristic
617C4#1909913511yescharacteristic
618C4#1909913511yescharacteristic
619C2—42no—
620C2#2376598511yescharacteristic
621C2#2377192111yescharacteristic
622C2#2377193211yescharacteristic
623C2#2377464911yescharacteristic
624C2#2377466411yescharacteristic
625C2#2377467911yescharacteristic
626C2#2377470211yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1823C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
cosocleC22×C4×C79C24
hypercenter#1823D79
hypocenterC79#21843
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
4. #23764371C2semidirect
5. #23764367C2semidirect
6. #23764363C2semidirect
7. #23764359C2semidirect
8. #23764356C2semidirect
9. #23764354C2semidirect
10. #23764350C2semidirect
11. #23764346C2semidirect
17. #19099194C4semidirect
329. C79#21843semidirect

Maximal subgroups

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

subgroupindex
C2×#190991942
C2×#190986212
#237660072
#237643712
#237643672
#237643632
#237643592
#237643562
#237643542
#237643502
#237643462
#237634962
#237634922
#237634902
C79×#18232
#2184379