Finite Group Databasebeta

← #34340163

Subgroups — #34340163

Order 20224.

774 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098860C211yesmaximal, characteristic
2C2×Q8×C79⋊C8C211yesmaximal, characteristic, complemented
3#23768758C211yesmaximal, characteristic
4#23764901C211yesmaximal, characteristic, complemented
5#23764896C211yesmaximal, characteristic, complemented
6#23764785C211yesmaximal, characteristic
7#23764771C211yesmaximal, characteristic
8#23764366C211yesmaximal, characteristic, complemented
9#23764168C211yesmaximal, characteristic, complemented
10#23764130C211yesmaximal, characteristic, complemented
11#23764125C211yesmaximal, characteristic, complemented
12#23764082C211yesmaximal, characteristic
13#23764082C211yesmaximal, characteristic
14C4×#14579081C211yesmaximal, characteristic
15C79×#2584C211yesmaximal, characteristic, complemented
16C2×C4×D316C2211yescharacteristic
17#19098860—24no—
18Q8×C79⋊C8—24no—
19C2×#14579094C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×C4⋊(C79⋊C8)C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25C2×C4⋊(C79⋊C8)C2211yescharacteristic
26C2×#14579081C2211yescharacteristic
27C2×#14579081C2211yescharacteristic
28C2×#14579081C2211yescharacteristic
29C2×#14579081C2211yescharacteristic
30C2×C4×C79⋊C8C2211yescharacteristic
31C2×C4×C79⋊C8C2211yescharacteristic
32C2×C4×C79⋊C8C2211yescharacteristic
33#19098498—21no—
34#19098474—21no—
35#19098465C2211yescharacteristic
36#19098465C2211yescharacteristic
37#19098461C2211yescharacteristic
38#19098461C2211yescharacteristic
39#19098434—21no—
40#19098434—21no—
41#19098429C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50C2×#125×C79C2211yescharacteristic
51C2×C4×Q8×C79C2211yescharacteristic
52#398×C79—21no—
53#395×C79—21no—
54#389×C79—21no—
55#389×C79—21no—
56#389×C79C2211yescharacteristic
57#389×C79C2211yescharacteristic
58#389×C79C2211yescharacteristic
59#389×C79C2211yescharacteristic
60C4×C4⋊C4×C79C2211yescharacteristic
61C22×D316C2311yescharacteristic
62C22×C4×D79—21no—
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C4×D316—22no—
68C4×D316—22no—
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×Dic79⋊C4—21no—
73C2×Dic79⋊C4—21no—
74C2×(C2×D79)⋊C4—21no—
75C2×(C2×D79)⋊C4—21no—
76C2×(C2×D79)⋊C4—21no—
77C2×(C2×D79)⋊C4—21no—
78C2×(C2×D79)⋊C4—21no—
79#14579094—22no—
80#14579094—22no—
81#14579092—22no—
82#14579092—22no—
83C4⋊(C79⋊C8)—22no—
84C4⋊(C79⋊C8)—22no—
85C4⋊(C79⋊C8)D414yes—
86#14579081—22no—
87#14579081—22no—
88#14579081—22no—
89#14579081Q814yescomplemented
90C4×C79⋊C8—22no—
91C4×C79⋊C8—22no—
92C4×C79⋊C8—22no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C22×Q8×C79D411yescharacteristic
98#125×C79—22no—
99#125×C79—22no—
100C4×Q8×C79—22no—
101C4×Q8×C79—22no—
102C2×C4⋊C4×C79—21no—
103C2×C4⋊C4×C79—21no—
104C2×C4⋊C4×C79—21no—
105C2×C4⋊C4×C79—21no—
106C2×C4⋊C4×C79—21no—
107C2×C4⋊C4×C79—21no—
108C2×C4⋊C4×C79D411yescharacteristic
109C2×C4⋊C4×C79D411yescharacteristic
110C2×C4⋊C4×C79D411yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C42×C79—21no—
116C2×C42×C79C2311yescharacteristic
117C2×C42×C79C2311yescharacteristic
118C2×C42×C79C2311yescharacteristic
119#95×C79—21no—
120#95×C79—21no—
121#95×C79—21no—
122#95×C79—21no—
123#95×C79—21no—
124C23×D79—21no—
125C2×D316—21no—
126C2×D316—21no—
127C2×D316—21no—
128C2×D316—21no—
129C2×D316—21no—
130C2×D316C2×Q812yes—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C2×C4×D79—42no—
135C2×C4×D79—81no—
136C2×C4×D79—81no—
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⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8C2×D412yes—
147C2×C79⋊C8C2×D412yes—
148C2×C79⋊C8C2×Q812yes—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155Dic79⋊C4—81no—
156Dic79⋊C4—81no—
157(C2×D79)⋊C4—42no—
158(C2×D79)⋊C4—42no—
159(C2×D79)⋊C4—42no—
160(C2×D79)⋊C4—81no—
161(C2×D79)⋊C4—81no—
162C2×Q8×C79—22no—
163C2×Q8×C79—22no—
164C2×Q8×C79—22no—
165C22×C4×C79—21no—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79—21no—
169C22×C4×C79C2×D411yescharacteristic
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2411yescharacteristic
176C4⋊C4×C79—22no—
177C4⋊C4×C79—22no—
178C4⋊C4×C79—22no—
179C4⋊C4×C79—21no—
180C4⋊C4×C79—21no—
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—42no—
186C4⋊C4×C79—42no—
187C4⋊C4×C79—42no—
188C4⋊C4×C79—42no—
189C4⋊C4×C79—81no—
190C4⋊C4×C79—81no—
191C4⋊C4×C79C2×Q812yes—
192C4⋊C4×C79#4112yes—
193C42×C79—21no—
194C42×C79—21no—
195C42×C79—21no—
196C42×C79—21no—
197C42×C79—81no—
198C42×C79C2×D412yes—
199C42×C79#4112yes—
200C23×C79C22×D411yescharacteristic
201C22×D79—22no—
202C22×D79—41no—
203C22×D79—41no—
204C22×D79—41no—
205C22×D79—41no—
206C22×D79—41no—
207C22×D79—41no—
208Q8×C79—28no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—22no—
220C2×C4×C79—22no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—41no—
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#12112yes—
236C2×C4×C79#12112yes—
237C2×C4×C79#13512yes—
238C2×C4×C79#13512yes—
239C2×C4×C79C22×D411yescharacteristic
240C2×C4×C79C22×Q811yescharacteristic
241C2×C4×C79C2×#4111yescharacteristic
242C2×C4×C79C2×#4111yescharacteristic
243C2×C4×C792−1+411yescharacteristic
244C2×C4×C792−1+411yescharacteristic
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—41no—
254D316—22no—
255D316—22no—
256D316—41no—
257D316—41no—
258C4×D79—84no—
259C79⋊C8—22no—
260C79⋊C8—22no—
261C79⋊C8—22no—
262C79⋊C8—41no—
263C22×C79C2×#12111yescharacteristic
264C22×C79#54811yescharacteristic
265C22×C79D4×Q811yescharacteristic
266C22×C79C2×#13511yescharacteristic
267C22×C79C2×#13511yescharacteristic
268C22×C79#57611yescharacteristic
269C22×C79#57711yescharacteristic
270C2×D79—42no—
271C2×D79—42no—
272C2×D79—42no—
273C2×D79—81no—
274C2×D79—81no—
275C2×D79—81no—
276C2×D79—81no—
277C4×C79—22no—
278C4×C79—22no—
279C4×C79—22no—
280C4×C79—22no—
281C4×C79—42no—
282C4×C79—42no—
283C4×C79—41no—
284C4×C79—41no—
285C4×C79—81no—
286C4×C79—81no—
287C4×C79#53311yescharacteristic
288C4×C79#54111yescharacteristic
289C4×C79#54711yescharacteristic
290C4×C79D4×Q811yescharacteristic
291Dic79—81no—
292Dic79—81no—
293Dic79—81no—
294#25473—791nomaximal, Sylow
295C2×C79#242911yescharacteristic
296C2×C79#305011yescharacteristic
297C2×C79#306211yescharacteristic
298C2×C79#336111yescharacteristic
299C2×C79#336511yescharacteristic
300C2×C79#338011yescharacteristic
301C2×C79#338111yescharacteristic
302D79—82no—
303C2×#478—791no—
304C2×C8×Q8—791no—
305#2584D7911yescharacteristic, complemented
306#2580—791no—
307#2088—791no—
308#2087—791no—
309#2067—791no—
310#2065—791no—
311#1991—791no—
312#1949—791no—
313#1942—791no—
314#1941—791no—
315#1934—791no—
316#1934—791no—
317C4×#102—791no—
318C79#2547311yescharacteristic, complemented, Sylow
319C2×#125C2×D7911yescharacteristic
320C2×C4×Q8C2×D7911yescharacteristic
321C2×C4×D4—791no—
322#478—1584no—
323C8×Q8—1584no—
324C2×#107—791no—
325C2×#107—791no—
326C2×#106—791no—
327C2×#106—791no—
328C2×C4⋊C8—791no—
329C2×C4⋊C8—791no—
330C2×C4⋊C8—791no—
331C2×#102—791no—
332C2×#102—791no—
333C2×#102—791no—
334C2×#102—791no—
335C2×C4×C8—791no—
336C2×C4×C8—791no—
337C2×C4×C8—791no—
338#399—1581no—
339#398—21no—
340#395—21no—
341#393—1581no—
342#390—791no—
343#390—791no—
344#389—21no—
345#389—21no—
346#389—791no—
347#389—791no—
348#389C2×D7911yescharacteristic
349#389C2×D7911yescharacteristic
350#389C2×D7911yescharacteristic
351#389C2×D7911yescharacteristic
352#385—1581no—
353#385—1581no—
354#382—791no—
355C4×C4⋊C4C2×D7911yescharacteristic
356#340—791no—
357#340—791no—
358#340—791no—
359#340—791no—
360#340—791no—
361#340—791no—
362#340—791no—
363#340—791no—
364C22×Q8C22⋊D7911yescharacteristic
365C22×D4—791no—
366C23×C4—1581no—
367C22×C8—791no—
368C22×C8—791no—
369C22×C8—791no—
370C22×C8—791no—
371#125—22no—
372#125—22no—
373C4×Q8—22no—
374C4×Q8—22no—
375C4×D4—1582no—
376C4×D4—1582no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—791no—
385C2×C4⋊C4—791no—
386C2×C4⋊C4—1581no—
387C2×C4⋊C4—1581no—
388C2×C4⋊C4C22⋊D7911yescharacteristic
389C2×C4⋊C4C22⋊D7911yescharacteristic
390C2×C4⋊C4C22⋊D7911yescharacteristic
391C2×C4⋊C4C22×D7911yescharacteristic
392C2×C4⋊C4C22×D7911yescharacteristic
393C2×C4⋊C4C22×D7911yescharacteristic
394C2×C4⋊C4C22×D7911yescharacteristic
395C2×C22⋊C4—1581no—
396C2×C22⋊C4—1581no—
397C2×C22⋊C4—1581no—
398C2×C22⋊C4—1581no—
399C2×C22⋊C4—1581no—
400C2×C42—21no—
401C2×C42C22×D7911yescharacteristic
402C2×C42C22×D7911yescharacteristic
403C2×C42C22×D7911yescharacteristic
404#107—1582no—
405#107—1582no—
406#106—1582no—
407#106—1582no—
408C4⋊C8—794no—
409C4⋊C8—1582no—
410C4⋊C8—1582no—
411#102—794no—
412#102—1582no—
413#102—1582no—
414#102—1582no—
415C4×C8—1582no—
416C4×C8—1582no—
417C4×C8—1582no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—1581no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427C24—1581no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×D4—792no—
432C2×D4—1581no—
433C2×D4—1581no—
434C2×D4—1581no—
435C2×D4—1581no—
436C2×D4—1581no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—3162no—
442C22×C4—6321no—
443C22×C4—6321no—
444C22×C4—1581no—
445C22×C4—1581no—
446C22×C4—1581no—
447C22×C4C2×C22⋊D7911yescharacteristic
448C22×C4C2×C22⋊D7911yescharacteristic
449C22×C4C2×C22⋊D7911yescharacteristic
450C22×C4C2×C22⋊D7911yescharacteristic
451C22×C4C2×C22⋊D7911yescharacteristic
452C22×C4C2×C22⋊D7911yescharacteristic
453C22×C4C23×D7911yescharacteristic
454C2×C8—792no—
455C2×C8—792no—
456C2×C8—792no—
457C2×C8—1581no—
458C2×C8—1581no—
459C2×C8—1581no—
460C2×C8—1581no—
461C2×C8—1581no—
462C2×C8—1581no—
463C2×C8—1581no—
464C2×C8—1581no—
465C2×C8—1581no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—6321no—
482C4⋊C4—6321no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4—1581no—
487C4⋊C4—1581no—
488C4⋊C4—1581no—
489C4⋊C4Q8×D7912yes—
490C4⋊C4#1192962512yes—
491C22⋊C4—3162no—
492C22⋊C4—3162no—
493C22⋊C4—3162no—
494C22⋊C4—6321no—
495C22⋊C4—6321no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—81no—
501C42D4×D7912yes—
502C42#1192962312yes—
503C23—3161no—
504C23—3161no—
505C23—3161no—
506C23—3161no—
507C23—3161no—
508C23—3161no—
509C23—1582no—
510C23C22×C22⋊D7911yescharacteristic
511Q8—28no—
512D4—3161no—
513D4—3161no—
514D4—1582no—
515D4—1582no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—3161no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—3161no—
547C2×C4—3161no—
548C2×C4—3161no—
549C2×C4—3161no—
550C2×C4—3161no—
551C2×C4—6324no—
552C2×C4#1457913312yes—
553C2×C4#1457913312yes—
554C2×C4#1457918112yes—
555C2×C4#1457918112yes—
556C2×C4C2×D4×D7911yescharacteristic
557C2×C4C2×Q8×D7911yescharacteristic
558C2×C4C2×#1192962311yescharacteristic
559C2×C4C2×#1192962511yescharacteristic
560C2×C4#1457920911yescharacteristic
561C2×C4#1457920911yescharacteristic
562C8—3161no—
563C8—1582no—
564C8—1582no—
565C8—1582no—
566C22—3162no—
567C22—3162no—
568C22—3162no—
569C22—6321no—
570C22—6321no—
571C22—6321no—
572C22—6321no—
573C22C2×#1457913311yescharacteristic
574C22#1909922411yescharacteristic
575C22Q8×C22⋊D7911yescharacteristic
576C22C2×#1457918111yescharacteristic
577C22C2×#1457918111yescharacteristic
578C22#1909936611yescharacteristic
579C22#1909937111yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—6321no—
591C4—6321no—
592C4—6321no—
593C4#1909911811yescharacteristic
594C4#1909917411yescharacteristic
595C4#1909921611yescharacteristic
596C4#1909922811yescharacteristic
597C2—6322no—
598C2#2376700211yescharacteristic
599C2#2377297811yescharacteristic
600C2#2377308211yescharacteristic
601C2#2377606011yescharacteristic
602C2#2377609611yescharacteristic
603C2#2377626311yescharacteristic
604C2#2377627111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2584C2
socleC23×C79C22×D4
prime radicalC22×D316C23
hypercenter#2584D79
hypocenterC79#25473
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2584

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C79⋊C8C2semidirect
4. #23764901C2semidirect
5. #23764896C2semidirect
8. #23764366C2semidirect
9. #23764168C2semidirect
10. #23764130C2semidirect
11. #23764125C2semidirect
15. C79×#2584C2semidirect
89. #14579081Q8semidirect
305. #2584D79semidirect
318. C79#25473semidirect

Maximal subgroups

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

subgroupindex
C2×#190988602
C2×Q8×C79⋊C82
#237687582
#237649012
#237648962
#237647852
#237647712
#237643662
#237641682
#237641302
#237641252
#237640822
C4×#145790812
C79×#25842
#2547379