Finite Group Databasebeta

← #34293180

Subgroups — #34293180

Order 20224.

826 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766009C211yesmaximal, characteristic
2#23765756C211yesmaximal, characteristic
3#23764170C211yesmaximal, characteristic, complemented
4#23764170C211yesmaximal, characteristic, complemented
5#23764164C211yesmaximal, characteristic, complemented
6#23764159C211yesmaximal, characteristic, complemented
7#23763601C211yesmaximal, characteristic
8#23763601C211yesmaximal, characteristic
9#23763595C211yesmaximal, characteristic
10#23763590C211yesmaximal, characteristic
11#23763297C211yesmaximal, characteristic, complemented
12#23763297C211yesmaximal, characteristic, complemented
13C4×#14579095C211yesmaximal, characteristic, complemented
14C4×#14579093C211yesmaximal, characteristic, complemented
15C2×#433×C79C211yesmaximal, characteristic
16C2×#14579120C2211yescharacteristic
17C22×C4⋊Dic79C2211yescharacteristic
18C2×#14579095C2211yescharacteristic
19C2×#14579095C2211yescharacteristic
20C2×#14579095C2211yescharacteristic
21C2×#14579095C2211yescharacteristic
22C2×#14579093C2211yescharacteristic
23C2×#14579093C2211yescharacteristic
24C2×#14579093C2211yescharacteristic
25C2×#14579093C2211yescharacteristic
26#19098456—21no—
27#19098456—21no—
28#19098449—21no—
29#19098449—21no—
30#19098433C2211yescharacteristic
31#19098429C2211yescharacteristic
32#19098423C2211yescharacteristic
33#19098423C2211yescharacteristic
34C4⋊C4×Dic79—21no—
35C4⋊C4×Dic79—21no—
36C4×C4⋊Dic79C2211yescharacteristic
37C4×C4⋊Dic79C2211yescharacteristic
38C4×C4⋊Dic79C2211yescharacteristic
39C4×C4⋊Dic79C2211yescharacteristic
40C4×C4⋊Dic79C2211yescharacteristic
41C4×C4⋊Dic79C2211yescharacteristic
42C22⋊C4×Dic79—21no—
43C22⋊C4×Dic79—21no—
44#19098326C2211yescharacteristic
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098326C2211yescharacteristic
50#19098326C2211yescharacteristic
51#19098326C2211yescharacteristic
52C2×#117×C79C2211yescharacteristic
53#433×C79C418yescomplemented
54C2×C4⋊C8×C79C2211yescharacteristic
55C2×C4⋊C8×C79C2211yescharacteristic
56C2×C22⋊C8×C79C2211yescharacteristic
57C2×C22⋊C8×C79C2211yescharacteristic
58C2×#97×C79C2211yescharacteristic
59C2×C4×C8×C79C2211yescharacteristic
60C23×Dic79—21no—
61#14579120—22no—
62#14579120—22no—
63C2×C4⋊Dic79—21no—
64C2×C4⋊Dic79—22no—
65C2×C4⋊Dic79—22no—
66C2×C4⋊Dic79—21no—
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C22⋊Dic79—21no—
77C2×C22⋊Dic79—21no—
78C2×C22⋊Dic79—21no—
79C2×C4×Dic79—21no—
80C2×C4×Dic79—21no—
81C2×C4×Dic79—21no—
82C2×C4×Dic79—21no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79—21no—
85C2×C4×Dic79—21no—
86#14579095—22no—
87#14579095—22no—
88#14579095—22no—
89#14579095—22no—
90#14579093—22no—
91#14579093—22no—
92#14579093—22no—
93#14579093—22no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C23×C4×C79C2311yescharacteristic
105C22×C8×C79C2311yescharacteristic
106C22×C8×C79C2311yescharacteristic
107C22×C8×C79C2311yescharacteristic
108C22×C8×C79C2311yescharacteristic
109#117×C79C2×C414yes—
110#117×C79Q814yes—
111C2×C4⋊C4×C79D411yescharacteristic
112C2×C4⋊C4×C79D411yescharacteristic
113C2×C22⋊C4×C79D411yescharacteristic
114C2×C22⋊C4×C79D411yescharacteristic
115C2×C42×C79C2311yescharacteristic
116C2×C42×C79C2311yescharacteristic
117C4⋊C8×C79C2×C414yescomplemented
118C4⋊C8×C79C2×C414yescomplemented
119C22⋊C8×C79C2×C414yes—
120C22⋊C8×C79C2×C414yes—
121#97×C79C2×C414yescomplemented
122C4×C8×C79C2×C414yescomplemented
123C22×Dic79—24no—
124C22×Dic79—21no—
125C22×Dic79—21no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—21no—
130C22×Dic79—21no—
131C22×Dic79—21no—
132C22×Dic79—81no—
133C22×Dic79—81no—
134C4⋊Dic79—24no—
135C4⋊Dic79—21no—
136C4⋊Dic79—21no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—81no—
148C4⋊Dic79—81no—
149C4⋊Dic79#4112yes—
150C4⋊Dic79#4112yes—
151C4⋊Dic79#4112yes—
152C4⋊Dic79#4112yes—
153C22⋊Dic79—24no—
154C22⋊Dic79—81no—
155C22⋊Dic79—81no—
156C4×Dic79—24no—
157C4×Dic79—81no—
158C4×Dic79—81no—
159C4×Dic79—81no—
160C4×Dic79—81no—
161C4×Dic79—81no—
162C4×Dic79—81no—
163C24×C79C2×D411yescharacteristic
164C22×C4×C79—21no—
165C22×C4×C79—21no—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79C22×C412yes—
169C22×C4×C79C2×D411yescharacteristic
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2×Q812yes—
175C22×C4×C79C2411yescharacteristic
176C2×C8×C79—21no—
177C2×C8×C79—21no—
178C2×C8×C79—21no—
179C2×C8×C79—21no—
180C2×C8×C79—21no—
181C2×C8×C79—21no—
182C2×C8×C79—21no—
183C2×C8×C79—21no—
184C2×C8×C79C22×C412yes—
185C2×C8×C79C22×C412yes—
186C2×C8×C79C22×C412yes—
187C2×C8×C79C22×C412yes—
188C4⋊C4×C79C4⋊C414yes—
189C4⋊C4×C79C4⋊C414yes—
190C22⋊C4×C79C4⋊C414yes—
191C22⋊C4×C79C4⋊C414yes—
192C42×C79C22×C412yes—
193C42×C79C22×C412yes—
194C42×C79C2×Q812yes—
195C42×C79C2×Q812yes—
196C23×C79—21no—
197C23×C79—21no—
198C23×C79—21no—
199C23×C79—21no—
200C23×C79—21no—
201C23×C79—21no—
202C23×C79C2×C4⋊C412yes—
203C23×C79C22×D411yescharacteristic
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—22no—
215C2×C4×C79—22no—
216C2×C4×C79—22no—
217C2×C4×C79—22no—
218C2×C4×C79C2×C4⋊C412yes—
219C2×C4×C79C2×C4⋊C412yes—
220C2×C4×C79C2×C4⋊C412yes—
221C2×C4×C79C2×C4⋊C412yes—
222C2×C4×C79C2×C4⋊C412yes—
223C2×C4×C79C23×C411yescharacteristic
224C2×C4×C79C22×Q811yescharacteristic
225C2×C4×C79C2×#4111yescharacteristic
226C2×C4×C79C2×#4111yescharacteristic
227C2×C4×C79C2×#4111yescharacteristic
228C2×C4×C79C2×#4111yescharacteristic
229C2×Dic79—22no—
230C2×Dic79—22no—
231C2×Dic79—22no—
232C2×Dic79—22no—
233C2×Dic79—41no—
234C2×Dic79—41no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—84no—
254C8×C79—22no—
255C8×C79—22no—
256C8×C79—22no—
257C8×C79—22no—
258C22×C79—22no—
259C22×C79—22no—
260C22×C79—22no—
261C22×C79—41no—
262C22×C79—41no—
263C22×C79—41no—
264C22×C79—41no—
265C22×C79C22×C4⋊C411yescharacteristic
266C22×C79#52511yescharacteristic
267C22×C79#52511yescharacteristic
268C22×C79#57611yescharacteristic
269C22×C79#57711yescharacteristic
270C22×C79#57711yescharacteristic
271C22×C79#57811yescharacteristic
272C4×C79—22no—
273C4×C79—22no—
274C4×C79—22no—
275C4×C79—22no—
276C4×C79—22no—
277C4×C79C4×#4111yescharacteristic
278C4×C79C4×#4111yescharacteristic
279C4×C79#53311yescharacteristic
280C4×C79#53311yescharacteristic
281Dic79—42no—
282Dic79—42no—
283Dic79—42no—
284Dic79—42no—
285Dic79—81no—
286Dic79—81no—
287Dic79—81no—
288Dic79—81no—
289#21367—791nomaximal, Sylow
290C2×C79—42no—
291C2×C79#229611yescharacteristic
292C2×C79#291211yescharacteristic
293C2×C79#291311yescharacteristic
294C2×C79#311811yescharacteristic
295C2×C79#311911yescharacteristic
296C2×C79#312011yescharacteristic
297C2×C79#312111yescharacteristic
298C2×#433D7911yescharacteristic
299#2314—791no—
300#2274—791no—
301#1949—791no—
302#1949—791no—
303#1948—791no—
304#1947—791no—
305#1854—791no—
306#1854—791no—
307#1853—791no—
308#1852—791no—
309#1776—791no—
310#1776—791no—
311C4×#107—791no—
312C4×#106—791no—
313C79#2136711yescharacteristic, complemented, Sylow
314C2×#117—791no—
315C2×#117C2×D7911yescharacteristic
316C22×C4⋊C4—791no—
317#433Dic7918yescomplemented
318C2×#107—791no—
319C2×#107—791no—
320C2×#107—791no—
321C2×#107—791no—
322C2×#106—791no—
323C2×#106—791no—
324C2×#106—791no—
325C2×#106—791no—
326C2×C4⋊C8C2×D7911yescharacteristic
327C2×C4⋊C8C2×D7911yescharacteristic
328C2×C22⋊C8C2×D7911yescharacteristic
329C2×C22⋊C8C2×D7911yescharacteristic
330C2×#97C2×D7911yescharacteristic
331C2×C4×C8C2×D7911yescharacteristic
332#388—1581no—
333#388—1581no—
334#387—1581no—
335#387—1581no—
336#384—791no—
337#382—791no—
338#380—791no—
339#380—791no—
340C4×C4⋊C4—791no—
341C4×C4⋊C4—791no—
342C4×C4⋊C4—791no—
343C4×C4⋊C4—791no—
344C4×C4⋊C4—791no—
345C4×C4⋊C4—791no—
346C4×C4⋊C4—1581no—
347C4×C4⋊C4—1581no—
348C4×C22⋊C4—1581no—
349C4×C22⋊C4—1581no—
350#340—791no—
351#340—791no—
352#340—791no—
353#340—791no—
354#340—791no—
355#340—791no—
356#340—791no—
357#340—791no—
358C23×C4—1581no—
359C23×C4C22×D7911yescharacteristic
360C22×C8C22×D7911yescharacteristic
361C22×C8C22×D7911yescharacteristic
362C22×C8C22×D7911yescharacteristic
363C22×C8C22×D7911yescharacteristic
364#117—1582no—
365#117—1582no—
366#117Dic15814yes—
367#117C2×Dic7914yes—
368C2×C4⋊C4—791no—
369C2×C4⋊C4—791no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—791no—
372C2×C4⋊C4—791no—
373C2×C4⋊C4—791no—
374C2×C4⋊C4—791no—
375C2×C4⋊C4—791no—
376C2×C4⋊C4—1582no—
377C2×C4⋊C4—1582no—
378C2×C4⋊C4—1581no—
379C2×C4⋊C4—1581no—
380C2×C4⋊C4—1581no—
381C2×C4⋊C4D31611yescharacteristic
382C2×C4⋊C4D31611yescharacteristic
383C2×C22⋊C4—1581no—
384C2×C22⋊C4—1581no—
385C2×C22⋊C4—1581no—
386C2×C22⋊C4D31611yescharacteristic
387C2×C22⋊C4D31611yescharacteristic
388C2×C42—1581no—
389C2×C42—1581no—
390C2×C42—1581no—
391C2×C42—1581no—
392C2×C42—1581no—
393C2×C42—1581no—
394C2×C42—1581no—
395C2×C42C22×D7911yescharacteristic
396C2×C42C22×D7911yescharacteristic
397#107—1582no—
398#107—1582no—
399#107—1582no—
400#107—1582no—
401#106—1582no—
402#106—1582no—
403#106—1582no—
404#106—1582no—
405C4⋊C8C2×Dic7914yescomplemented
406C4⋊C8C2×Dic7914yescomplemented
407C22⋊C8C2×Dic7914yes—
408C22⋊C8C2×Dic7914yes—
409#97C2×Dic7914yescomplemented
410C4×C8C2×Dic7914yescomplemented
411#95—1581no—
412#95—1581no—
413#95—1581no—
414#95—1581no—
415#95—1581no—
416#95—1581no—
417#95—1581no—
418#95—1581no—
419#95—1581no—
420#95—1581no—
421C24C2×D31611yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—6321no—
427C22×C4—6321no—
428C22×C4—1581no—
429C22×C4—1581no—
430C22×C4—1581no—
431C22×C4—1581no—
432C22×C4—1581no—
433C22×C4—1581no—
434C22×C4—1581no—
435C22×C4—1581no—
436C22×C4—1584no—
437C22×C4C22×Dic7912yes—
438C22×C4C2×D31611yescharacteristic
439C22×C4C2×D31611yescharacteristic
440C22×C4C2×D31611yescharacteristic
441C22×C4C2×D31611yescharacteristic
442C22×C4C2×D31611yescharacteristic
443C22×C4C2×Dic15812yes—
444C22×C4C23×D7911yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8C22×Dic7912yes—
454C2×C8C22×Dic7912yes—
455C2×C8C22×Dic7912yes—
456C2×C8C22×Dic7912yes—
457C4⋊C4—792no—
458C4⋊C4—792no—
459C4⋊C4—792no—
460C4⋊C4—792no—
461C4⋊C4—6321no—
462C4⋊C4—6321no—
463C4⋊C4—1584no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4C4⋊Dic7914yes—
477C4⋊C4C4⋊Dic7914yes—
478C22⋊C4—6321no—
479C22⋊C4—6321no—
480C22⋊C4—1584no—
481C22⋊C4C4⋊Dic7914yes—
482C22⋊C4C4⋊Dic7914yes—
483C42—6321no—
484C42—6321no—
485C42—6321no—
486C42—6321no—
487C42—6321no—
488C42—6321no—
489C42—1584no—
490C42C22×Dic7912yes—
491C42C22×Dic7912yes—
492C42C2×Dic15812yes—
493C42C2×Dic15812yes—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C2×C4⋊Dic7912yes—
501C23C22×D31611yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—3161no—
517C2×C4—3161no—
518C2×C4—3161no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—3161no—
523C2×C4—3161no—
524C2×C4—3161no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—3161no—
531C2×C4—3161no—
532C2×C4—3161no—
533C2×C4—3161no—
534C2×C4—3161no—
535C2×C4—3161no—
536C2×C4—6324no—
537C2×C4—1582no—
538C2×C4—1582no—
539C2×C4—1582no—
540C2×C4—1582no—
541C2×C4C2×C4⋊Dic7912yes—
542C2×C4C2×C4⋊Dic7912yes—
543C2×C4C2×C4⋊Dic7912yes—
544C2×C4C2×C4⋊Dic7912yes—
545C2×C4C2×C4⋊Dic7912yes—
546C2×C4C23×Dic7911yescharacteristic
547C2×C4C22×Dic15811yescharacteristic
548C2×C4#41×D7911yescharacteristic
549C2×C4#41×D7911yescharacteristic
550C2×C4#41×D7911yescharacteristic
551C2×C4#41×D7911yescharacteristic
552C8—22no—
553C8—22no—
554C8—22no—
555C8—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C22×C4⋊Dic7911yescharacteristic
564C22#1909907511yescharacteristic
565C22#1909907511yescharacteristic
566C22#1909936811yescharacteristic
567C22#1909937511yescharacteristic
568C22#1909937511yescharacteristic
569C22#1909938011yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—3162no—
576C4—3162no—
577C4—3162no—
578C4—3162no—
579C4—6321no—
580C4—6321no—
581C4—6321no—
582C4—6321no—
583C4#41×Dic7911yescharacteristic
584C4#41×Dic7911yescharacteristic
585C4#1909911611yescharacteristic
586C4#1909911611yescharacteristic
587C2—42no—
588C2#2376588411yescharacteristic
589C2#2377199511yescharacteristic
590C2#2377200011yescharacteristic
591C2#2377358611yescharacteristic
592C2#2377360111yescharacteristic
593C2#2377361611yescharacteristic
594C2#2377363111yescharacteristic
subgroupG/H
centerC23C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#433×C79C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
cosocleC22×C4×C79C24
hypercenterC2×#433D79
hypocenterC79#21367
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
3. #23764170C2semidirect
4. #23764170C2semidirect
5. #23764164C2semidirect
6. #23764159C2semidirect
11. #23763297C2semidirect
12. #23763297C2semidirect
13. C4×#14579095C2semidirect
14. C4×#14579093C2semidirect
53. #433×C79C4semidirect
117. C4⋊C8×C79C2×C4semidirect
118. C4⋊C8×C79C2×C4semidirect
121. #97×C79C2×C4semidirect
122. C4×C8×C79C2×C4semidirect
313. C79#21367semidirect
317. #433Dic79semidirect
405. C4⋊C8C2×Dic79semidirect
406. C4⋊C8C2×Dic79semidirect
409. #97C2×Dic79semidirect
410. C4×C8C2×Dic79semidirect

Maximal subgroups

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

subgroupindex
#237660092
#237657562
#237641702
#237641642
#237641592
#237636012
#237635952
#237635902
#237632972
C4×#145790952
C4×#145790932
C2×#433×C792
#2136779