Finite Group Databasebeta

← #34347063

Subgroups — #34347063

Order 20224.

826 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098966C211yesmaximal, characteristic, complemented
2C2×#19098603C211yesmaximal, characteristic
3#23767091C211yesmaximal, characteristic
4#23764881C211yesmaximal, characteristic, complemented
5#23764872C211yesmaximal, characteristic, complemented
6#23764730C211yesmaximal, characteristic, complemented
7#23764728C211yesmaximal, characteristic, complemented
8#23764135C211yesmaximal, characteristic, complemented
9#23764125C211yesmaximal, characteristic, complemented
10#23763597C211yesmaximal, characteristic
11#23763588C211yesmaximal, characteristic
12#23763566C211yesmaximal, characteristic, complemented
13#23763431C211yesmaximal, characteristic
14#23763184C211yesmaximal, characteristic
15C79×#2440C211yesmaximal, characteristic
16C22×#11929608C2211yescharacteristic
17C2×#14579161C2211yescharacteristic
18C2×#14579120C2211yescharacteristic
19#19098966—24no—
20#19098603—24no—
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579094C2211yescharacteristic
24C2×#14579094C2211yescharacteristic
25C2×#14579094C2211yescharacteristic
26C2×#14579094C2211yescharacteristic
27C2×#14579092C2211yescharacteristic
28C2×#14579092C2211yescharacteristic
29C2×#14579092C2211yescharacteristic
30C2×#14579092C2211yescharacteristic
31C2×C22⋊(C79⋊C8)C2211yescharacteristic
32C2×#14579072C2211yescharacteristic
33C2×C4×C79⋊C8C2211yescharacteristic
34#19098486—21no—
35#19098475—21no—
36#19098467—21no—
37#19098461C2211yescharacteristic
38#19098461C2211yescharacteristic
39#19098450—21no—
40#19098429C2211yescharacteristic
41#19098423C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098318C2211yescharacteristic
49#19098318C2211yescharacteristic
50C2×C4⋊Q8×C79C2211yescharacteristic
51C2×C22⋊Q8×C79—21no—
52C22×C4⋊C4×C79C2211yescharacteristic
53#398×C79—21no—
54#395×C79—21no—
55#389×C79C2211yescharacteristic
56#389×C79C2211yescharacteristic
57#385×C79—21no—
58#384×C79C2211yescharacteristic
59#380×C79C2211yescharacteristic
60C4×C22⋊C4×C79C2211yescharacteristic
61C22×Dic158—21no—
62C2×#11929608—22no—
63C2×#11929608—22no—
64C2×#11929608Q814yescomplemented
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68C22×C79⋊C8C2311yescharacteristic
69#14579161—22no—
70#14579161—41no—
71#14579120—22no—
72#14579120—41no—
73C2×C4⋊Dic79—21no—
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×C4⋊Dic79C2311yescharacteristic
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×C22⋊Dic79—21no—
81C2×C22⋊Dic79—21no—
82C2×C4×Dic79—21no—
83#14579094—22no—
84#14579094—22no—
85#14579094—22no—
86#14579094—22no—
87#14579094—41no—
88#14579094D414yes—
89#14579092—22no—
90#14579092—22no—
91#14579092—22no—
92#14579092—41no—
93C22⋊(C79⋊C8)Q814yescomplemented
94#14579072—22no—
95C4×C79⋊C8—22no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C22×Q8×C79—21no—
102C23×C4×C79—21no—
103C23×C4×C79C2311yescharacteristic
104C4⋊Q8×C79—22no—
105C4⋊Q8×C79—41no—
106C22⋊Q8×C79—44no—
107C2×C4⋊C4×C79—22no—
108C2×C4⋊C4×C79—21no—
109C2×C4⋊C4×C79—21no—
110C2×C4⋊C4×C79—21no—
111C2×C4⋊C4×C79—21no—
112C2×C4⋊C4×C79—21no—
113C2×C4⋊C4×C79—21no—
114C2×C4⋊C4×C79—41no—
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C4⋊C4×C79C2311yescharacteristic
119C2×C22⋊C4×C79—21no—
120C2×C22⋊C4×C79D411yescharacteristic
121C2×C22⋊C4×C79D411yescharacteristic
122C2×C42×C79C2311yescharacteristic
123C2×C42×C79C2311yescharacteristic
124#95×C79—21no—
125#95×C79—21no—
126#95×C79—21no—
127#95×C79D411yescharacteristic
128#95×C79D411yescharacteristic
129C2×Dic158—42no—
130C2×Dic158—42no—
131C2×Dic158—81no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136#11929608—28no—
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⋊C8C2×D412yes—
144C2×C79⋊C8C2×Q812yes—
145C2×C79⋊C8C2×Q812yes—
146C2×C79⋊C8C2×Q812yes—
147C2×C79⋊C8C2×Q812yes—
148C2×C79⋊C8#4112yes—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—42no—
157C4⋊Dic79C2×D412yes—
158Dic79⋊C4—42no—
159Dic79⋊C4—81no—
160C22⋊Dic79—42no—
161C22⋊Dic79—81no—
162C4×Dic79—42no—
163C24×C79C2×D411yescharacteristic
164C2×Q8×C79—42no—
165C2×Q8×C79—42no—
166C2×Q8×C79—81no—
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—21no—
175C22×C4×C79—42no—
176C22×C4×C79—42no—
177C22×C4×C79—81no—
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2×D411yescharacteristic
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2×Q812yes—
184C22×C4×C79C2×Q812yes—
185C22×C4×C79C2411yescharacteristic
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—42no—
194C4⋊C4×C79—42no—
195C4⋊C4×C79—42no—
196C4⋊C4×C79—42no—
197C4⋊C4×C79—42no—
198C4⋊C4×C79—81no—
199C4⋊C4×C79C2×D412yes—
200C22⋊C4×C79—22no—
201C22⋊C4×C79—22no—
202C22⋊C4×C79—42no—
203C42×C79—21no—
204C42×C79—21no—
205C42×C79—21no—
206C42×C79—21no—
207C23×C79—21no—
208C23×C79—21no—
209C23×C79—21no—
210C23×C79—21no—
211C23×C79—21no—
212C23×C79—21no—
213C23×C79C4⋊Q812yes—
214C23×C79C22×D411yescharacteristic
215Q8×C79—84no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—22no—
223C2×C4×C79—22no—
224C2×C4×C79—22no—
225C2×C4×C79—22no—
226C2×C4×C79—22no—
227C2×C4×C79—21no—
228C2×C4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—41no—
234C2×C4×C79—41no—
235C2×C4×C79—41no—
236C2×C4×C79—41no—
237C2×C4×C79—41no—
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—84no—
245C2×C4×C79C4⋊Q812yes—
246C2×C4×C79AGL(1,ℤ/8)12yes—
247C2×C4×C79#13712yes—
248C2×C4×C79C22×D411yescharacteristic
249C2×C4×C79C22×Q811yescharacteristic
250C2×C4×C79C22×Q811yescharacteristic
251C2×C4×C79C2×#4111yescharacteristic
252C2×C4×C79H(5,𝔽2)11yescharacteristic
253C2×C4×C792−1+411yescharacteristic
254C2×Dic79—22no—
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×Dic79—41no—
265C2×Dic79—41no—
266Dic158—84no—
267C79⋊C8—22no—
268C79⋊C8—22no—
269C79⋊C8—22no—
270C79⋊C8C22⋊Q814yescomplemented
271C22×C79—22no—
272C22×C79—22no—
273C22×C79—22no—
274C22×C79—41no—
275C22×C79—41no—
276C22×C79—41no—
277C22×C79—41no—
278C22×C79C2×C4⋊Q811yescharacteristic
279C22×C79D4211yescharacteristic
280C22×C79#54711yescharacteristic
281C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C79C2×#13711yescharacteristic
283C22×C79#57611yescharacteristic
284C22×C79#57811yescharacteristic
285C4×C79—22no—
286C4×C79—22no—
287C4×C79—22no—
288C4×C79—41no—
289C4×C79—41no—
290C4×C79—42no—
291C4×C79—81no—
292C4×C79—81no—
293C4×C79—81no—
294C4×C79C2×C22⋊Q811yescharacteristic
295C4×C79#54411yescharacteristic
296C4×C79D4×Q811yescharacteristic
297C4×C79#55411yescharacteristic
298Dic79—42no—
299Dic79—81no—
300Dic79—81no—
301Dic79—81no—
302#26074—791nomaximal, Sylow
303C2×C79—42no—
304C2×C79#244011yescharacteristic
305C2×C79#316411yescharacteristic
306C2×C79#316511yescharacteristic
307C2×C79#326411yescharacteristic
308C2×C79#327011yescharacteristic
309C2×C79#335311yescharacteristic
310C2×C79#335711yescharacteristic
311C2×#500—791no—
312C2×#428—791no—
313#2440D7911yescharacteristic
314#2439—791no—
315#2085—791no—
316#2083—791no—
317#2058—791no—
318#2057—791no—
319#1943—791no—
320#1941—791no—
321#1854—791no—
322#1852—791no—
323#1847—791no—
324#1809—791no—
325#1753—791no—
326C79#2607411yescharacteristic, complemented, Sylow
327C22×M16—791no—
328C2×C4⋊Q8—791no—
329C2×C4⋊Q8C2×D7911yescharacteristic
330C2×C22⋊Q8—21no—
331C2×#117—791no—
332C22×C4⋊C4C2×D7911yescharacteristic
333#500—1584no—
334#428—1584no—
335C2×#107—791no—
336C2×#107—791no—
337C2×#107—791no—
338C2×#107—791no—
339C2×#107—791no—
340C2×#107—791no—
341C2×#106—791no—
342C2×#106—791no—
343C2×#106—791no—
344C2×#106—791no—
345C2×C22⋊C8—791no—
346C2×#97—791no—
347C2×C4×C8—791no—
348#398—21no—
349#396—1581no—
350#395—21no—
351#393—1581no—
352#391—1581no—
353#389—791no—
354#389—791no—
355#389C2×D7911yescharacteristic
356#389C2×D7911yescharacteristic
357#387—1581no—
358#385—21no—
359#384C2×D7911yescharacteristic
360#382—791no—
361#380—791no—
362#380C2×D7911yescharacteristic
363C4×C22⋊C4C2×D7911yescharacteristic
364#340—791no—
365#340—791no—
366#340—791no—
367#340—791no—
368#340—791no—
369#340—791no—
370#336—791no—
371#336—791no—
372C22×Q8—21no—
373C22×Q8—1581no—
374C23×C4—21no—
375C23×C4C22×D7911yescharacteristic
376C2×M16—794no—
377C2×M16—1582no—
378C2×M16—1582no—
379C22×C8—791no—
380C22×C8—791no—
381C22×C8—791no—
382C22×C8—791no—
383C4⋊Q8—22no—
384C4⋊Q8—41no—
385C4⋊Q8—3161no—
386C4⋊Q8—1582no—
387C22⋊Q8—44no—
388#117—3161no—
389#117—1582no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—41no—
398C2×C4⋊C4—791no—
399C2×C4⋊C4—791no—
400C2×C4⋊C4—791no—
401C2×C4⋊C4—791no—
402C2×C4⋊C4—1581no—
403C2×C4⋊C4—1581no—
404C2×C4⋊C4—1581no—
405C2×C4⋊C4C22×D7911yescharacteristic
406C2×C4⋊C4C22×D7911yescharacteristic
407C2×C4⋊C4C22×D7911yescharacteristic
408C2×C4⋊C4C22×D7911yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1581no—
411C2×C22⋊C4—1581no—
412C2×C22⋊C4C22⋊D7911yescharacteristic
413C2×C22⋊C4C22⋊D7911yescharacteristic
414C2×C42—1581no—
415C2×C42C22×D7911yescharacteristic
416C2×C42C22×D7911yescharacteristic
417#107—3161no—
418#107—794no—
419#107—1582no—
420#107—1582no—
421#107—1582no—
422#107—1582no—
423#106—3161no—
424#106—1582no—
425#106—1582no—
426#106—1582no—
427C22⋊C8—794no—
428#97—1582no—
429C4×C8—1582no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1581no—
434#95—1581no—
435#95—1581no—
436#95—1581no—
437#95—1581no—
438#95C22⋊D7911yescharacteristic
439#95C22⋊D7911yescharacteristic
440C24C2×C22⋊D7911yescharacteristic
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—81no—
444C2×Q8—3162no—
445C2×Q8—3162no—
446C2×Q8—6321no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—81no—
458C22×C4—1581no—
459C22×C4—1581no—
460C22×C4—1581no—
461C22×C4—1581no—
462C22×C4C2×C22⋊D7911yescharacteristic
463C22×C4C2×C22⋊D7911yescharacteristic
464C22×C4C2×C22⋊D7911yescharacteristic
465C22×C4C2×C22⋊D7911yescharacteristic
466C22×C4C2×C22⋊D7911yescharacteristic
467C22×C4Q8×D7912yes—
468C22×C4Q8×D7912yes—
469C22×C4C23×D7911yescharacteristic
470M16—1588no—
471C2×C8—792no—
472C2×C8—792no—
473C2×C8—792no—
474C2×C8—792no—
475C2×C8—792no—
476C2×C8—792no—
477C2×C8—1581no—
478C2×C8—1581no—
479C2×C8—1581no—
480C2×C8—1581no—
481C2×C8—1581no—
482C2×C8—1581no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—3162no—
497C4⋊C4—3162no—
498C4⋊C4—792no—
499C4⋊C4—6321no—
500C4⋊C4—1581no—
501C4⋊C4—1581no—
502C4⋊C4—1581no—
503C4⋊C4—1581no—
504C4⋊C4—1581no—
505C4⋊C4—1581no—
506C4⋊C4—1581no—
507C4⋊C4D4×D7912yes—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—42no—
511C22⋊C4—3162no—
512C22⋊C4—6321no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—3162no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23#1457915912yes—
525C23C22×C22⋊D7911yescharacteristic
526Q8—84no—
527Q8—6324no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—84no—
557C2×C4—3161no—
558C2×C4—3161no—
559C2×C4—3161no—
560C2×C4—3161no—
561C2×C4—3161no—
562C2×C4—3161no—
563C2×C4—3161no—
564C2×C4—3161no—
565C2×C4—3161no—
566C2×C4—3161no—
567C2×C4—3161no—
568C2×C4—1582no—
569C2×C4#1457915912yes—
570C2×C4#1457918712yes—
571C2×C4#1457919212yes—
572C2×C4C2×D4×D7911yescharacteristic
573C2×C4C2×Q8×D7911yescharacteristic
574C2×C4C2×Q8×D7911yescharacteristic
575C2×C4#41×D7911yescharacteristic
576C2×C4#1457920611yescharacteristic
577C2×C4#1457920811yescharacteristic
578C8—794no—
579C8—1582no—
580C8—1582no—
581C8—1582no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22C2×#1457915911yescharacteristic
590C22D4×C22⋊D7911yescharacteristic
591C22#1909921211yescharacteristic
592C22C2×#1457918711yescharacteristic
593C22C2×#1457919211yescharacteristic
594C22#1909936611yescharacteristic
595C22#1909937811yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—41no—
600C4—42no—
601C4—41no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—3162no—
606C4—6321no—
607C4—6321no—
608C4—6321no—
609C4C22⋊Q8×D7911yescharacteristic
610C4#1909919211yescharacteristic
611C4#1909922611yescharacteristic
612C4#1909926711yescharacteristic
613C2—42no—
614C2#2376709911yescharacteristic
615C2#2377393411yescharacteristic
616C2#2377394211yescharacteristic
617C2#2377487611yescharacteristic
618C2#2377497111yescharacteristic
619C2#2377598411yescharacteristic
620C2#2377602611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2440C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#2440D79
hypocenterC79#26074
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
1. C2×#19098966C2semidirect
4. #23764881C2semidirect
5. #23764872C2semidirect
6. #23764730C2semidirect
7. #23764728C2semidirect
8. #23764135C2semidirect
9. #23764125C2semidirect
12. #23763566C2semidirect
64. C2×#11929608Q8semidirect
93. C22⋊(C79⋊C8)Q8semidirect
270. C79⋊C8C22⋊Q8semidirect
326. C79#26074semidirect

Maximal subgroups

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

subgroupindex
C2×#190989662
C2×#190986032
#237670912
#237648812
#237648722
#237647302
#237647282
#237641352
#237641252
#237635972
#237635882
#237635662
#237634312
#237631842
C79×#24402
#2607479