Finite Group Databasebeta

← #34304937

Subgroups — #34304937

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×#19099270C211yesmaximal, characteristic
2C2×#19098692C211yesmaximal, characteristic, complemented
3#23766352C211yesmaximal, characteristic, complemented
4#23764372C211yesmaximal, characteristic, complemented
5#23764367C211yesmaximal, characteristic, complemented
6#23764363C211yesmaximal, characteristic, complemented
7#23764356C211yesmaximal, characteristic, complemented
8#23764354C211yesmaximal, characteristic, complemented
9#23764351C211yesmaximal, characteristic, complemented
10#23763408C211yesmaximal, characteristic
11#23763398C211yesmaximal, characteristic
12#23763227C211yesmaximal, characteristic
13#23763225C211yesmaximal, characteristic
14#102×Dic79C211yesmaximal, characteristic
15C79×#1804C211yesmaximal, characteristic
16#19099270—24no—
17#19099270C418yescomplemented
18C2×#14579161C2211yescharacteristic
19C2×#14579152C2211yescharacteristic
20C2×#14579152C2211yescharacteristic
21C2×#14579140—21no—
22C2×#14579138—21no—
23C2×#14579138—21no—
24C2×C4×Dic158C2211yescharacteristic
25C2×D4×Dic79C2211yescharacteristic
26C2×D4×Dic79C2211yescharacteristic
27C22×C4⋊Dic79—21no—
28#19098692—24no—
29C2×Dic79⋊C8C2211yescharacteristic
30C2×Dic79⋊C8C2211yescharacteristic
31C2×#14579082C2211yescharacteristic
32C2×#14579082C2211yescharacteristic
33C2×#14579072C2211yescharacteristic
34C2×#14579071C2211yescharacteristic
35C2×C8×Dic79C2211yescharacteristic
36#19098467—21no—
37#19098448—21no—
38#19098445—21no—
39#19098432C2211yescharacteristic
40#19098428C2211yescharacteristic
41C4⋊C4×Dic79C2211yescharacteristic
42C4⋊C4×Dic79C2211yescharacteristic
43C4×C4⋊Dic79C2211yescharacteristic
44#19098414—21no—
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098325C2211yescharacteristic
53#19098325C2211yescharacteristic
54#19098325C2211yescharacteristic
55C2×C4×D4×C79C2211yescharacteristic
56C2×C4⋊C8×C79C2211yescharacteristic
57C2×#102×C79C2211yescharacteristic
58C2×#102×C79C2211yescharacteristic
59#382×C79C2211yescharacteristic
60#340×C79C2211yescharacteristic
61#340×C79C2211yescharacteristic
62C22×Dic158—21no—
63C22×Dic158D411yescharacteristic
64C23×Dic79—21no—
65C23×Dic79—21no—
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68#14579161—22no—
69#14579161C2×C414yes—
70#14579152—22no—
71#14579152—22no—
72#14579152C2×C414yes—
73#14579152C2×C414yes—
74#14579140—28no—
75#14579138—28no—
76#14579138—28no—
77C4×Dic158C2×C414yes—
78C4×Dic158D414yes—
79D4×Dic79—22no—
80D4×Dic79—22no—
81D4×Dic79C2×C414yes—
82D4×Dic79C2×C414yes—
83C2×C4⋊Dic79—21no—
84C2×C4⋊Dic79—21no—
85C2×C4⋊Dic79—21no—
86C2×C4⋊Dic79—28no—
87C2×C4⋊Dic79D411yescharacteristic
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—
96C2×Dic79⋊C4D411yescharacteristic
97C2×Dic79⋊C4D411yescharacteristic
98C2×C22⋊Dic79—21no—
99C2×C22⋊Dic79—21no—
100C2×C4×Dic79—21no—
101C2×C4×Dic79—21no—
102C2×C4×Dic79—21no—
103C2×C4×Dic79C2311yescharacteristic
104C2×C4×Dic79C2311yescharacteristic
105Dic79⋊C8—22no—
106Dic79⋊C8—22no—
107#14579082—22no—
108#14579082Q814yescomplemented
109#14579072—22no—
110#14579071—22no—
111C8×Dic79—22no—
112C79⋊#95—21no—
113C79⋊#95—21no—
114C79⋊#95—21no—
115C79⋊#95—21no—
116C79⋊#95—21no—
117C22×D4×C79C2311yescharacteristic
118C23×C4×C79—21no—
119C22×C8×C79C2311yescharacteristic
120C22×C8×C79C2311yescharacteristic
121C4×D4×C79—22no—
122C4×D4×C79C2×C414yes—
123C2×C4⋊C4×C79C2311yescharacteristic
124C2×C4⋊C4×C79C2311yescharacteristic
125C2×C4⋊C4×C79C2311yescharacteristic
126C2×C22⋊C4×C79—21no—
127C2×C42×C79C2311yescharacteristic
128C4⋊C8×C79—22no—
129#102×C79—22no—
130#102×C79—22no—
131#95×C79—21no—
132#95×C79—21no—
133C2×Dic158—24no—
134C2×Dic158—41no—
135C2×Dic158—41no—
136C2×Dic158—81no—
137C2×Dic158—81no—
138C2×Dic158C22⋊C414yes—
139C22×Dic79—24no—
140C22×Dic79—24no—
141C22×Dic79—21no—
142C22×Dic79—21no—
143C22×Dic79—21no—
144C22×Dic79—21no—
145C22×Dic79—81no—
146C22×Dic79—81no—
147C22×Dic79—81no—
148C22×Dic79—81no—
149C22×Dic79C2×D411yescharacteristic
150C22×Dic79C2×D411yescharacteristic
151C22×Dic79C2×D411yescharacteristic
152C22×Dic79C2×D411yescharacteristic
153C2×C79⋊C8—21no—
154C2×C79⋊C8—21no—
155C2×C79⋊C8—21no—
156C2×C79⋊C8—21no—
157C2×C79⋊C8—21no—
158C2×C79⋊C8C2×Q812yes—
159C4⋊Dic79—24no—
160C4⋊Dic79—24no—
161C4⋊Dic79—24no—
162C4⋊Dic79—21no—
163C4⋊Dic79—21no—
164C4⋊Dic79—21no—
165C4⋊Dic79—21no—
166C4⋊Dic79—21no—
167C4⋊Dic79C22⋊C414yes—
168C4⋊Dic79C22×C412yes—
169C4⋊Dic79C22×C412yes—
170C4⋊Dic79C2×Q812yes—
171Dic79⋊C4—24no—
172Dic79⋊C4—24no—
173Dic79⋊C4—42no—
174Dic79⋊C4—81no—
175Dic79⋊C4C22⋊C414yes—
176Dic79⋊C4C22⋊C414yes—
177C22⋊Dic79—24no—
178C22⋊Dic79—24no—
179C4×Dic79—42no—
180C4×Dic79—81no—
181C4×Dic79—81no—
182C4×Dic79C22×C412yes—
183C4×Dic79C22×C412yes—
184C4×Dic79C2×D412yes—
185C4×Dic79C2×D412yes—
186C24×C79—21no—
187C2×D4×C79—21no—
188C2×D4×C79—21no—
189C2×D4×C79—21no—
190C2×D4×C79C22×C412yes—
191C2×D4×C79C2×Q812yes—
192C2×D4×C79#4112yes—
193C22×C4×C79—21no—
194C22×C4×C79—24no—
195C22×C4×C79—21no—
196C22×C4×C79—21no—
197C22×C4×C79—42no—
198C22×C4×C79—42no—
199C22×C4×C79C2×D411yescharacteristic
200C22×C4×C79C2×D411yescharacteristic
201C22×C4×C79C2411yescharacteristic
202C2×C8×C79—21no—
203C2×C8×C79—21no—
204C2×C8×C79—21no—
205C2×C8×C79—21no—
206C2×C8×C79—21no—
207C2×C8×C79—21no—
208C4⋊C4×C79—21no—
209C4⋊C4×C79—21no—
210C4⋊C4×C79—21no—
211C4⋊C4×C79—21no—
212C4⋊C4×C79—21no—
213C4⋊C4×C79C22×C412yes—
214C22⋊C4×C79—24no—
215C42×C79C22×C412yes—
216C42×C79C2×D412yes—
217C23×C79—22no—
218C23×C79—22no—
219C23×C79—22no—
220C23×C79—41no—
221C23×C79—41no—
222C23×C79—41no—
223C23×C79—41no—
224C23×C79C22×D411yescharacteristic
225D4×C79—22no—
226D4×C79—22no—
227D4×C79—22no—
228D4×C79C4×Q814yes—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—22no—
234C2×C4×C79—48no—
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×C79C2×C22⋊C412yes—
244C2×C4×C79C2×C22⋊C412yes—
245C2×C4×C79C23×C411yescharacteristic
246C2×C4×C79C22×D411yescharacteristic
247C2×C4×C79C22×Q811yescharacteristic
248C2×C4×C79C2×#4111yescharacteristic
249C2×C4×C792−1+411yescharacteristic
250C2×C4×C792−1+411yescharacteristic
251C2×Dic79—21no—
252C2×Dic79—21no—
253C2×Dic79—21no—
254C2×Dic79—21no—
255C2×Dic79—21no—
256C2×Dic79—21no—
257C2×Dic79—21no—
258C2×Dic79—21no—
259C2×Dic79—22no—
260C2×Dic79—22no—
261C2×Dic79—22no—
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—
271C2×Dic79—84no—
272C2×Dic79—84no—
273C2×Dic79C2×C22⋊C412yes—
274C2×Dic79C2×C22⋊C412yes—
275C2×Dic79C2×C22⋊C412yes—
276C2×Dic79C2×C22⋊C412yes—
277Dic158—44no—
278Dic158—84no—
279C8×C79—41no—
280C8×C79—41no—
281C79⋊C8—22no—
282C79⋊C8—41no—
283C22×C79—24no—
284C22×C79—42no—
285C22×C79—42no—
286C22×C79—42no—
287C22×C79—42no—
288C22×C79—42no—
289C22×C79—42no—
290C22×C79C22×C22⋊C411yescharacteristic
291C22×C79#54811yescharacteristic
292C22×C79D4×Q811yescharacteristic
293C22×C79#57511yescharacteristic
294C22×C79#57511yescharacteristic
295C22×C79#57611yescharacteristic
296C22×C79#57711yescharacteristic
297C4×C79—22no—
298C4×C79—22no—
299C4×C79—42no—
300C4×C79—81no—
301C4×C79—81no—
302C4×C79C2×C4×Q811yescharacteristic
303C4×C79#51911yescharacteristic
304C4×C79#54811yescharacteristic
305C4×C79D4×Q811yescharacteristic
306Dic79—22no—
307Dic79—22no—
308Dic79—22no—
309Dic79—22no—
310Dic79—42no—
311Dic79—42no—
312Dic79—42no—
313Dic79—81no—
314#22526—791nomaximal, Sylow
315C2×C79—44no—
316C2×C79Q8×C22⋊C411yescharacteristic
317C2×C79#289611yescharacteristic
318C2×C79#289711yescharacteristic
319C2×C79#337011yescharacteristic
320C2×C79#337611yescharacteristic
321C2×C79#338811yescharacteristic
322C2×C79#338911yescharacteristic
323C2×#554—791no—
324C2×#446—791no—
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#1804D7911yescharacteristic
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×Q8—791no—
348C2×C4×D4—791no—
349C2×C4×D4—791no—
350C2×C4×D4C2×D7911yescharacteristic
351C22×C4⋊C4—1581no—
352#446—1584no—
353C2×C4⋊C8—791no—
354C2×C4⋊C8—791no—
355C2×C4⋊C8C2×D7911yescharacteristic
356C2×#102—791no—
357C2×#102—791no—
358C2×#102C2×D7911yescharacteristic
359C2×#102C2×D7911yescharacteristic
360C2×#97—791no—
361C2×#97—791no—
362C2×C4×C8—791no—
363#391—1581no—
364#387—1581no—
365#387—1581no—
366#384—791no—
367#382—791no—
368#382C2×D7911yescharacteristic
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#340C2×D7911yescharacteristic
384#340C2×D7911yescharacteristic
385C22×Q8—791no—
386C22×Q8—1581no—
387C22×D4C22×D7911yescharacteristic
388C23×C4—21no—
389C23×C4—1581no—
390C23×C4—1581no—
391C22×C8—791no—
392C22×C8—791no—
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×Q8—794no—
405C4×Q8—794no—
406C4×D4—22no—
407C4×D4—794no—
408C4×D4—794no—
409C4×D4—1582no—
410C4×D4—1582no—
411C4×D4C4×D7914yes—
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—1588no—
421C2×C4⋊C4—1581no—
422C2×C4⋊C4—1581no—
423C2×C4⋊C4—1581no—
424C2×C4⋊C4—1581no—
425C2×C4⋊C4—1581no—
426C2×C4⋊C4—1581no—
427C2×C4⋊C4C22×D7911yescharacteristic
428C2×C4⋊C4C22×D7911yescharacteristic
429C2×C4⋊C4C22×D7911yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1581no—
432C2×C22⋊C4—1581no—
433C2×C42—791no—
434C2×C42—791no—
435C2×C42—1581no—
436C2×C42—1581no—
437C2×C42—1581no—
438C2×C42C22×D7911yescharacteristic
439C4⋊C8—22no—
440C4⋊C8—1582no—
441C4⋊C8—1582no—
442#102—22no—
443#102—22no—
444#102—794no—
445#102—1582no—
446#97—1582no—
447#97—1582no—
448C4×C8—1582no—
449#95—21no—
450#95—21no—
451#95—1581no—
452#95—1581no—
453#95—1581no—
454#95—1581no—
455#95—1581no—
456C24—21no—
457C2×Q8—3161no—
458C2×Q8—3161no—
459C2×Q8—794no—
460C2×Q8—6321no—
461C2×Q8—6321no—
462C2×Q8—1584no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4C2×C4×D7912yes—
467C2×D4Q8×D7912yes—
468C2×D4#1192962312yes—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—791no—
476C22×C4—791no—
477C22×C4—791no—
478C22×C4—791no—
479C22×C4—6321no—
480C22×C4—6321no—
481C22×C4—6321no—
482C22×C4—6321no—
483C22×C4—1581no—
484C22×C4—1581no—
485C22×C4—1581no—
486C22×C4—1581no—
487C22×C4—1584no—
488C22×C4—1584no—
489C22×C4D4×D7911yescharacteristic
490C22×C4D4×D7911yescharacteristic
491C22×C4C23×D7911yescharacteristic
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—792no—
499C2×C8—1581no—
500C2×C8—1581no—
501C2×C8—1581no—
502C2×C8—1581no—
503C2×C8—1581no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—3162no—
510C4⋊C4—794no—
511C4⋊C4—794no—
512C4⋊C4—794no—
513C4⋊C4—792no—
514C4⋊C4—792no—
515C4⋊C4—792no—
516C4⋊C4—6321no—
517C4⋊C4—1584no—
518C4⋊C4—1584no—
519C4⋊C4—1584no—
520C4⋊C4—1584no—
521C4⋊C4—1581no—
522C4⋊C4—1581no—
523C4⋊C4—1581no—
524C4⋊C4—1584no—
525C4⋊C4—1581no—
526C4⋊C4—1581no—
527C4⋊C4C2×C4×D7912yes—
528C22⋊C4—24no—
529C22⋊C4—1584no—
530C22⋊C4—1584no—
531C42—3162no—
532C42—792no—
533C42—792no—
534C42—792no—
535C42—792no—
536C42—6321no—
537C42—6321no—
538C42C2×C4×D7912yes—
539C42D4×D7912yes—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C2×D4×D7911yescharacteristic
548Q8—3164no—
549Q8—6324no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4#1457912614yes—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—48no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
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—792no—
578C2×C4—792no—
579C2×C4—792no—
580C2×C4—792no—
581C2×C4—6324no—
582C2×C4—6324no—
583C2×C4—1582no—
584C2×C4—1582no—
585C2×C4—1582no—
586C2×C4—1581no—
587C2×C4—1581no—
588C2×C4—1581no—
589C2×C4—1581no—
590C2×C4—1581no—
591C2×C4—1581no—
592C2×C4—1581no—
593C2×C4—1581no—
594C2×C4C22⋊C4×D7912yes—
595C2×C4C22⋊C4×D7912yes—
596C2×C4C22×C4×D7911yescharacteristic
597C2×C4C2×D4×D7911yescharacteristic
598C2×C4C2×Q8×D7911yescharacteristic
599C2×C4C2×#1192962311yescharacteristic
600C2×C4#1457920811yescharacteristic
601C2×C4#1457920811yescharacteristic
602C8—41no—
603C8—41no—
604C8—3161no—
605C8—1582no—
606C22—24no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22C2×C22⋊C4×D7911yescharacteristic
614C22#1909921911yescharacteristic
615C22#1909922611yescharacteristic
616C22#1909935911yescharacteristic
617C22#1909935911yescharacteristic
618C22#1909936511yescharacteristic
619C22#1909937011yescharacteristic
620C4—22no—
621C4—22no—
622C4—42no—
623C4—81no—
624C4—81no—
625C4—3162no—
626C4—3162no—
627C4—3162no—
628C4—6321no—
629C4—1582no—
630C4—1582no—
631C4—1582no—
632C4—1582no—
633C4C2×#1457912611yescharacteristic
634C4#1909904011yescharacteristic
635C4#1909921911yescharacteristic
636C4#1909922611yescharacteristic
637C2—44no—
638C2#2376623211yescharacteristic
639C2#2377191411yescharacteristic
640C2#2377192211yescharacteristic
641C2#2377615011yescharacteristic
642C2#2377622411yescharacteristic
643C2#2377634211yescharacteristic
644C2#2377635711yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1804C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenter#1804D79
hypocenterC79#22526
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×#19098692C2semidirect
3. #23766352C2semidirect
4. #23764372C2semidirect
5. #23764367C2semidirect
6. #23764363C2semidirect
7. #23764356C2semidirect
8. #23764354C2semidirect
9. #23764351C2semidirect
17. #19099270C4semidirect
108. #14579082Q8semidirect
338. C79#22526semidirect

Maximal subgroups

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

subgroupindex
C2×#190992702
C2×#190986922
#237663522
#237643722
#237643672
#237643632
#237643562
#237643542
#237643512
#237634082
#237633982
#237632272
#237632252
#102×Dic792
C79×#18042
#2252679