Finite Group Databasebeta

← #34304942

Subgroups — #34304942

Order 20224.

1066 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099273C211yesmaximal, characteristic
2#23766357C211yesmaximal, characteristic, complemented
3#23764373C211yesmaximal, characteristic, complemented
4#23764369C211yesmaximal, characteristic, complemented
5#23764365C211yesmaximal, characteristic, complemented
6#23764355C211yesmaximal, characteristic, complemented
7#23764355C211yesmaximal, characteristic, complemented
8#23764351C211yesmaximal, characteristic, complemented
9#23763411C211yesmaximal, characteristic
10#23763411C211yesmaximal, characteristic
11#23763401C211yesmaximal, characteristic
12#23763226C211yesmaximal, characteristic
13#23763226C211yesmaximal, characteristic
14C4×#14579083C211yesmaximal, characteristic
15C2×#446×C79C211yesmaximal, characteristic, complemented
16#19099273—24no—
17#19099273C418yescomplemented
18C2×#14579161C2211yescharacteristic
19C2×#14579153C2211yescharacteristic
20C2×#14579153C2211yescharacteristic
21C2×#14579139—21no—
22C2×#14579139—21no—
23C2×#14579137—21no—
24C2×C4×D316C2211yescharacteristic
25C2×C4×D316C2211yescharacteristic
26C2×C4×D316C2211yescharacteristic
27C2×C4⋊C4×D79—21no—
28C2×#14579083C2211yescharacteristic
29C2×#14579083C2211yescharacteristic
30C2×#14579083C2211yescharacteristic
31C2×#14579083C2211yescharacteristic
32#19098470—21no—
33#19098449—21no—
34#19098449—21no—
35#19098433C2211yescharacteristic
36#19098429C2211yescharacteristic
37#19098429C2211yescharacteristic
38C4⋊C4×Dic79—21no—
39C4×C4⋊Dic79C2211yescharacteristic
40C4×C4⋊Dic79C2211yescharacteristic
41C4×C4⋊Dic79C2211yescharacteristic
42#19098326C2211yescharacteristic
43#19098326C2211yescharacteristic
44#19098326C2211yescharacteristic
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098326C2211yescharacteristic
50#19098326C2211yescharacteristic
51#19098326C2211yescharacteristic
52#19098326C2211yescharacteristic
53#19098326C2211yescharacteristic
54C2×C4×Q8×C79C2211yescharacteristic
55#446×C79—24no—
56C2×C4⋊C8×C79C2211yescharacteristic
57C2×C4⋊C8×C79C2211yescharacteristic
58C2×C4⋊C8×C79C2211yescharacteristic
59C2×#97×C79C2211yescharacteristic
60C2×#97×C79C2211yescharacteristic
61C2×C4×C8×C79C2211yescharacteristic
62C22×Dic158—21no—
63C22×D316C2311yescharacteristic
64C22×C4×D79—21no—
65C22×C4×D79—21no—
66C22×C4×D79—21no—
67#14579161—22no—
68#14579161C2×C414yes—
69#14579153—22no—
70#14579153—22no—
71#14579153C2×C414yes—
72#14579153C2×C414yes—
73#14579139—28no—
74#14579139—28no—
75#14579137—28no—
76C4×D316—22no—
77C4×D316—22no—
78C4×D316—22no—
79C4×D316C2×C414yes—
80C4×D316C2×C414yes—
81C4×D316C2×C414yes—
82C2×C4⋊Dic79—21no—
83C2×C4⋊Dic79—21no—
84C2×C4⋊Dic79—21no—
85C2×C4⋊Dic79C2311yescharacteristic
86C2×C4⋊Dic79C2311yescharacteristic
87C2×C4⋊Dic79C2311yescharacteristic
88C2×C4⋊Dic79C2311yescharacteristic
89C2×C4⋊Dic79C2311yescharacteristic
90C2×C4⋊Dic79C2311yescharacteristic
91C2×C4⋊Dic79C2311yescharacteristic
92C2×Dic79⋊C4—21no—
93C2×Dic79⋊C4—21no—
94C2×Dic79⋊C4—21no—
95C2×Dic79⋊C4—21no—
96C4⋊C4×D79—28no—
97C2×(C2×D79)⋊C4—21no—
98C2×(C2×D79)⋊C4—21no—
99C2×(C2×D79)⋊C4—21no—
100C2×C4×Dic79—21no—
101C2×C4×Dic79—21no—
102C2×C4×Dic79—21no—
103#14579083—22no—
104#14579083—22no—
105#14579083—22no—
106#14579083Q814yescomplemented
107C79⋊#95—21no—
108C79⋊#95—21no—
109C79⋊#95—21no—
110C79⋊#95—21no—
111C79⋊#95—21no—
112C79⋊#95—21no—
113C79⋊#95—21no—
114C22×Q8×C79D411yescharacteristic
115C22×C8×C79C2311yescharacteristic
116C22×C8×C79C2311yescharacteristic
117C22×C8×C79C2311yescharacteristic
118C22×C8×C79C2311yescharacteristic
119C4×Q8×C79C2×C414yes—
120C4×Q8×C79D414yes—
121C2×C4⋊C4×C79D411yescharacteristic
122C2×C4⋊C4×C79D411yescharacteristic
123C2×C4⋊C4×C79D411yescharacteristic
124C2×C42×C79C2311yescharacteristic
125C2×C42×C79C2311yescharacteristic
126C2×C42×C79C2311yescharacteristic
127C4⋊C8×C79—22no—
128C4⋊C8×C79—22no—
129C4⋊C8×C79—22no—
130#97×C79—22no—
131#97×C79—22no—
132C4×C8×C79—22no—
133C23×D79—21no—
134C2×Dic158—24no—
135C2×Dic158—81no—
136C2×Dic158—81no—
137C2×D316—21no—
138C2×D316—21no—
139C2×D316—21no—
140C2×D316C22×C412yes—
141C2×D316C2×Q812yes—
142C2×D316#4112yes—
143C22×Dic79—21no—
144C22×Dic79—21no—
145C22×Dic79—21no—
146C22×Dic79—21no—
147C22×Dic79—21no—
148C22×Dic79—21no—
149C22×Dic79—21no—
150C2×C4×D79—24no—
151C2×C4×D79—24no—
152C2×C4×D79—24no—
153C2×C4×D79—42no—
154C2×C4×D79—42no—
155C2×C4×D79—81no—
156C2×C4×D79—81no—
157C2×C4×D79—81no—
158C2×C4×D79—81no—
159C4⋊Dic79—24no—
160C4⋊Dic79—21no—
161C4⋊Dic79—21no—
162C4⋊Dic79—21no—
163C4⋊Dic79—21no—
164C4⋊Dic79—21no—
165C4⋊Dic79—21no—
166C4⋊Dic79—21no—
167C4⋊Dic79—21no—
168C4⋊Dic79—21no—
169C4⋊Dic79—21no—
170C4⋊Dic79—42no—
171C4⋊Dic79—81no—
172C4⋊Dic79C22×C412yes—
173C4⋊Dic79C22×C412yes—
174C4⋊Dic79C22×C412yes—
175C4⋊Dic79C2×Q812yes—
176Dic79⋊C4—24no—
177Dic79⋊C4—24no—
178Dic79⋊C4—24no—
179Dic79⋊C4—24no—
180(C2×D79)⋊C4—24no—
181(C2×D79)⋊C4—24no—
182(C2×D79)⋊C4—24no—
183C4×Dic79—42no—
184C4×Dic79—81no—
185C4×Dic79—81no—
186C2×Q8×C79—41no—
187C2×Q8×C79—41no—
188C2×Q8×C79C22⋊C414yes—
189C22×C4×C79C2×D411yescharacteristic
190C22×C4×C79C2×D411yescharacteristic
191C22×C4×C79C2×D411yescharacteristic
192C22×C4×C79C2×D411yescharacteristic
193C22×C4×C79C2×D411yescharacteristic
194C22×C4×C79C2×D411yescharacteristic
195C22×C4×C79C2411yescharacteristic
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—21no—
199C2×C8×C79—21no—
200C2×C8×C79—21no—
201C2×C8×C79—21no—
202C2×C8×C79—21no—
203C2×C8×C79—21no—
204C2×C8×C79—21no—
205C2×C8×C79—21no—
206C2×C8×C79—21no—
207C2×C8×C79C2×Q812yes—
208C4⋊C4×C79C22⋊C414yes—
209C4⋊C4×C79C22⋊C414yes—
210C4⋊C4×C79C22⋊C414yes—
211C42×C79C22×C412yes—
212C42×C79C22×C412yes—
213C42×C79C22×C412yes—
214C42×C79C2×D412yes—
215C42×C79C2×D412yes—
216C42×C79C2×D412yes—
217C23×C79C22×D411yescharacteristic
218C22×D79—22no—
219C22×D79—22no—
220C22×D79—22no—
221C22×D79—41no—
222C22×D79—41no—
223C22×D79—41no—
224C22×D79—41no—
225Q8×C79—44no—
226C2×C4×C79—21no—
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—21no—
234C2×C4×C79—21no—
235C2×C4×C79—21no—
236C2×C4×C79—21no—
237C2×C4×C79—21no—
238C2×C4×C79C2×C22⋊C412yes—
239C2×C4×C79C2×C22⋊C412yes—
240C2×C4×C79C2×C22⋊C412yes—
241C2×C4×C79C2×C22⋊C412yes—
242C2×C4×C79C2×C22⋊C412yes—
243C2×C4×C79C2×C22⋊C412yes—
244C2×C4×C79C23×C411yescharacteristic
245C2×C4×C79C22×D411yescharacteristic
246C2×C4×C79C22×Q811yescharacteristic
247C2×C4×C79C2×#4111yescharacteristic
248C2×C4×C792−1+411yescharacteristic
249C2×C4×C792−1+411yescharacteristic
250C2×Dic79—22no—
251C2×Dic79—22no—
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×Dic79—41no—
265C2×Dic79—41no—
266C2×Dic79—41no—
267C2×Dic79—41no—
268C2×Dic79—41no—
269C2×Dic79—41no—
270C2×Dic79—41no—
271D316—22no—
272D316—22no—
273D316—22no—
274D316C4×Q814yes—
275C4×D79—48no—
276C4×D79—84no—
277C4×D79—84no—
278Dic158—84no—
279C8×C79—22no—
280C8×C79—41no—
281C8×C79—41no—
282C8×C79—41no—
283C22×C79C22×C22⋊C411yescharacteristic
284C22×C79#54811yescharacteristic
285C22×C79D4×Q811yescharacteristic
286C22×C79#57511yescharacteristic
287C22×C79#57511yescharacteristic
288C22×C79#57611yescharacteristic
289C22×C79#57711yescharacteristic
290C2×D79—24no—
291C2×D79—42no—
292C2×D79—42no—
293C2×D79—42no—
294C2×D79—42no—
295C2×D79—42no—
296C2×D79—42no—
297C4×C79—22no—
298C4×C79—22no—
299C4×C79—22no—
300C4×C79—22no—
301C4×C79—22no—
302C4×C79—22no—
303C4×C79C2×C4×Q811yescharacteristic
304C4×C79#51911yescharacteristic
305C4×C79#54811yescharacteristic
306C4×C79D4×Q811yescharacteristic
307Dic79—42no—
308Dic79—42no—
309Dic79—42no—
310Dic79—42no—
311Dic79—81no—
312Dic79—81no—
313Dic79—81no—
314#22526—791nomaximal, Sylow
315C2×C79Q8×C22⋊C411yescharacteristic
316C2×C79#289611yescharacteristic
317C2×C79#289711yescharacteristic
318C2×C79#337011yescharacteristic
319C2×C79#337611yescharacteristic
320C2×C79#338811yescharacteristic
321C2×C79#338911yescharacteristic
322D79—44no—
323C2×#554—791no—
324C2×#446D7911yescharacteristic, complemented
325#2355—791no—
326#1993—791no—
327#1992—791no—
328#1991—791no—
329#1989—791no—
330#1989—791no—
331#1988—791no—
332#1804—791no—
333#1804—791no—
334#1802—791no—
335#1762—791no—
336#1762—791no—
337C4×#102—791no—
338C79#2252611yescharacteristic, complemented, Sylow
339#554—798no—
340#554—1584no—
341C2×C4⋊Q8—791no—
342C2×#125—791no—
343C2×#125—791no—
344C2×C22⋊Q8—1581no—
345C2×C22⋊Q8—1581no—
346C2×C22⋊Q8—1581no—
347C2×C4×Q8C2×D7911yescharacteristic
348C2×C4×D4—791no—
349C2×C4×D4—791no—
350C2×C4×D4—791no—
351C22×C4⋊C4—1581no—
352#446—24no—
353C2×C4⋊C8C2×D7911yescharacteristic
354C2×C4⋊C8C2×D7911yescharacteristic
355C2×C4⋊C8C2×D7911yescharacteristic
356C2×#102—791no—
357C2×#102—791no—
358C2×#102—791no—
359C2×#102—791no—
360C2×#97C2×D7911yescharacteristic
361C2×#97C2×D7911yescharacteristic
362C2×C4×C8C2×D7911yescharacteristic
363#391—1581no—
364#387—1581no—
365#387—1581no—
366#384—791no—
367#382—791no—
368#382—791no—
369C4×C4⋊C4—791no—
370C4×C4⋊C4—791no—
371C4×C4⋊C4—791no—
372C4×C4⋊C4—1581no—
373#340—791no—
374#340—791no—
375#340—791no—
376#340—791no—
377#340—791no—
378#340—791no—
379#340—791no—
380#340—791no—
381#340—791no—
382#340—791no—
383#340—791no—
384#340—791no—
385C22×Q8—1581no—
386C22×Q8D31611yescharacteristic
387C22×D4—791no—
388C23×C4—1581no—
389C23×C4—1581no—
390C23×C4—1581no—
391C22×C8C22×D7911yescharacteristic
392C22×C8C22×D7911yescharacteristic
393C22×C8C22×D7911yescharacteristic
394C22×C8C22×D7911yescharacteristic
395C4⋊Q8—794no—
396C4⋊Q8—1582no—
397#125—794no—
398#125—794no—
399#125—1582no—
400#125—1582no—
401C22⋊Q8—1588no—
402C22⋊Q8—1588no—
403C22⋊Q8—1588no—
404C4×Q8C4×D7914yes—
405C4×Q8C22⋊D7914yes—
406C4×D4—794no—
407C4×D4—794no—
408C4×D4—794no—
409C4×D4—1582no—
410C4×D4—1582no—
411C4×D4—1582no—
412C2×C4⋊C4—791no—
413C2×C4⋊C4—791no—
414C2×C4⋊C4—791no—
415C2×C4⋊C4—791no—
416C2×C4⋊C4—791no—
417C2×C4⋊C4—791no—
418C2×C4⋊C4—791no—
419C2×C4⋊C4—1581no—
420C2×C4⋊C4—1581no—
421C2×C4⋊C4—1581no—
422C2×C4⋊C4—1581no—
423C2×C4⋊C4—1581no—
424C2×C4⋊C4—1581no—
425C2×C4⋊C4—1581no—
426C2×C4⋊C4—1588no—
427C2×C4⋊C4D31611yescharacteristic
428C2×C4⋊C4D31611yescharacteristic
429C2×C4⋊C4D31611yescharacteristic
430C2×C22⋊C4—1581no—
431C2×C22⋊C4—1581no—
432C2×C22⋊C4—1581no—
433C2×C42—1581no—
434C2×C42—1581no—
435C2×C42—1581no—
436C2×C42C22×D7911yescharacteristic
437C2×C42C22×D7911yescharacteristic
438C2×C42C22×D7911yescharacteristic
439C4⋊C8—22no—
440C4⋊C8—22no—
441C4⋊C8—22no—
442#102—794no—
443#102—1582no—
444#102—1582no—
445#102—1582no—
446#97—22no—
447#97—22no—
448C4×C8—22no—
449#95—1581no—
450#95—1581no—
451#95—1581no—
452#95—1581no—
453#95—1581no—
454#95—1581no—
455#95—1581no—
456C24—1581no—
457C2×Q8—41no—
458C2×Q8—41no—
459C2×Q8—6321no—
460C2×Q8—6321no—
461C2×Q8—1584no—
462C2×Q8(C2×D79)⋊C414yes—
463C2×D4—792no—
464C2×D4—792no—
465C2×D4—792no—
466C2×D4—1581no—
467C2×D4—1581no—
468C2×D4—1581no—
469C22×C4—3162no—
470C22×C4—3162no—
471C22×C4—6321no—
472C22×C4—6321no—
473C22×C4—6321no—
474C22×C4—6321no—
475C22×C4—1581no—
476C22×C4—1581no—
477C22×C4—1581no—
478C22×C4—1581no—
479C22×C4—1581no—
480C22×C4—1581no—
481C22×C4—1581no—
482C22×C4—1584no—
483C22×C4—1584no—
484C22×C4—1584no—
485C22×C4C2×D31611yescharacteristic
486C22×C4C2×D31611yescharacteristic
487C22×C4C2×D31611yescharacteristic
488C22×C4C2×D31611yescharacteristic
489C22×C4C2×D31611yescharacteristic
490C22×C4C2×D31611yescharacteristic
491C22×C4C23×D7911yescharacteristic
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8Q8×D7912yes—
504C4⋊C4—3162no—
505C4⋊C4—792no—
506C4⋊C4—792no—
507C4⋊C4—792no—
508C4⋊C4—792no—
509C4⋊C4—6321no—
510C4⋊C4—1581no—
511C4⋊C4—1581no—
512C4⋊C4—1581no—
513C4⋊C4—1581no—
514C4⋊C4—1581no—
515C4⋊C4—1581no—
516C4⋊C4—1581no—
517C4⋊C4—1584no—
518C4⋊C4—1584no—
519C4⋊C4—1584no—
520C4⋊C4—1584no—
521C4⋊C4—1584no—
522C4⋊C4—1581no—
523C4⋊C4—1581no—
524C4⋊C4—1581no—
525C4⋊C4(C2×D79)⋊C414yes—
526C4⋊C4(C2×D79)⋊C414yes—
527C4⋊C4(C2×D79)⋊C414yes—
528C22⋊C4—1584no—
529C22⋊C4—1584no—
530C22⋊C4—1584no—
531C42—3162no—
532C42—6321no—
533C42—6321no—
534C42C2×C4×D7912yes—
535C42C2×C4×D7912yes—
536C42C2×C4×D7912yes—
537C42C2×C22⋊D7912yes—
538C42C2×C22⋊D7912yes—
539C42C2×C22⋊D7912yes—
540C23—3161no—
541C23—3161no—
542C23—3161no—
543C23—3161no—
544C23—1582no—
545C23—1582no—
546C23—1582no—
547C23C22×D31611yescharacteristic
548Q8—44no—
549Q8—6324no—
550D4—794no—
551D4—1582no—
552D4—1582no—
553D4—1582no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—3168no—
567C2×C4—3161no—
568C2×C4—3161no—
569C2×C4—3161no—
570C2×C4—3161no—
571C2×C4—3161no—
572C2×C4—3161no—
573C2×C4—3161no—
574C2×C4—3161no—
575C2×C4—3161no—
576C2×C4—3161no—
577C2×C4—3161no—
578C2×C4—3161no—
579C2×C4—3161no—
580C2×C4—3161no—
581C2×C4—3161no—
582C2×C4—3161no—
583C2×C4—3161no—
584C2×C4—6324no—
585C2×C4—6324no—
586C2×C4—1582no—
587C2×C4—1582no—
588C2×C4—1582no—
589C2×C4—1582no—
590C2×C4C2×(C2×D79)⋊C412yes—
591C2×C4C2×(C2×D79)⋊C412yes—
592C2×C4C2×(C2×D79)⋊C412yes—
593C2×C4C2×(C2×D79)⋊C412yes—
594C2×C4C2×(C2×D79)⋊C412yes—
595C2×C4C2×(C2×D79)⋊C412yes—
596C2×C4C22×C4×D7911yescharacteristic
597C2×C4C22×C22⋊D7911yescharacteristic
598C2×C4C2×Q8×D7911yescharacteristic
599C2×C4#41×D7911yescharacteristic
600C2×C4#1457920811yescharacteristic
601C2×C4#1457920911yescharacteristic
602C8—22no—
603C8—41no—
604C8—41no—
605C8—41no—
606C22—3162no—
607C22—3162no—
608C22—3162no—
609C22—3162no—
610C22—3162no—
611C22—3162no—
612C22—1584no—
613C22C22×(C2×D79)⋊C411yescharacteristic
614C22#1909922511yescharacteristic
615C22Q8×D31611yescharacteristic
616C22#1909936311yescharacteristic
617C22#1909936311yescharacteristic
618C22#1909936811yescharacteristic
619C22#1909937511yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—3162no—
627C4—3162no—
628C4—3162no—
629C4—3162no—
630C4—6321no—
631C4—6321no—
632C4—6321no—
633C4C4×Q8×D7911yescharacteristic
634C4#1909903911yescharacteristic
635C4#1909922011yescharacteristic
636C4Q8×C22⋊D7911yescharacteristic
637C2—3164no—
638C2Q8×(C2×D79)⋊C411yescharacteristic
639C2#2377192011yescharacteristic
640C2#2377192711yescharacteristic
641C2#2377615511yescharacteristic
642C2#2377622911yescharacteristic
643C2#2377634711yescharacteristic
644C2#2377636011yescharacteristic
subgroupG/H
centerC23C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#446×C79C2
socleC23×C79C22×D4
prime radicalC22×D316C23
cosocleC22×C4×C79C24
hypercenterC2×#446D79
hypocenterC79#22526
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
2. #23766357C2semidirect
3. #23764373C2semidirect
4. #23764369C2semidirect
5. #23764365C2semidirect
6. #23764355C2semidirect
7. #23764355C2semidirect
8. #23764351C2semidirect
15. C2×#446×C79C2semidirect
17. #19099273C4semidirect
106. #14579083Q8semidirect
324. C2×#446D79semidirect
338. C79#22526semidirect

Maximal subgroups

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

subgroupindex
C2×#190992732
#237663572
#237643732
#237643692
#237643652
#237643552
#237643512
#237634112
#237634012
#237632262
C4×#145790832
C2×#446×C792
#2252679