Finite Group Databasebeta

← #34340072

Subgroups — #34340072

Order 20224.

756 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098973C211yesmaximal, characteristic, complemented
2#23768573C211yesmaximal, characteristic
3#23766500C211yesmaximal, characteristic, complemented
4#23764885C211yesmaximal, characteristic, complemented
5#23764884C211yesmaximal, characteristic, complemented
6#23764784C211yesmaximal, characteristic
7#23764770C211yesmaximal, characteristic
8#23764351C211yesmaximal, characteristic, complemented
9#23764169C211yesmaximal, characteristic, complemented
10#23764165C211yesmaximal, characteristic, complemented
11#23764121C211yesmaximal, characteristic, complemented
12#23764030C211yesmaximal, characteristic
13#23764023C211yesmaximal, characteristic
14#23763225C211yesmaximal, characteristic
15C2×#478×C79C211yesmaximal, characteristic
16C2×#14579161C2211yescharacteristic
17#19098973—24no—
18C2×#14579094C2211yescharacteristic
19C2×#14579094C2211yescharacteristic
20C2×#14579092C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×Dic79⋊C8C2211yescharacteristic
23C2×Dic79⋊C8C2211yescharacteristic
24C2×#14579082C2211yescharacteristic
25C2×#14579082C2211yescharacteristic
26C2×#14579072C2211yescharacteristic
27#19098475—21no—
28#19098475—21no—
29#19098466C2211yescharacteristic
30#19098466C2211yescharacteristic
31#19098462—21no—
32#19098462C2211yescharacteristic
33#19098462C2211yescharacteristic
34#19098449—21no—
35#19098449—21no—
36#19098428C2211yescharacteristic
37#19098428C2211yescharacteristic
38#19098423—21no—
39#19098423—21no—
40C4⋊C4×Dic79—21no—
41C4⋊C4×Dic79C2211yescharacteristic
42C4⋊C4×Dic79C2211yescharacteristic
43C4×C4⋊Dic79C2211yescharacteristic
44C4×C4⋊Dic79C2211yescharacteristic
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098317C2211yescharacteristic
52#19098317C2211yescharacteristic
53C2×#125×C79C2211yescharacteristic
54C2×C4×D4×C79C2211yescharacteristic
55#478×C79—24no—
56C2×#107×C79C2211yescharacteristic
57C2×#106×C79C2211yescharacteristic
58C2×C4⋊C8×C79C2211yescharacteristic
59C2×#102×C79C2211yescharacteristic
60C2×#102×C79C2211yescharacteristic
61C22×Dic158—21no—
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64#14579161—22no—
65#14579161—22no—
66C2×C4⋊Dic79—21no—
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79—21no—
69C2×C4⋊Dic79D411yescharacteristic
70C2×C4⋊Dic79D411yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C22⋊Dic79—21no—
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×Dic79C2311yescharacteristic
83C2×C4×Dic79C2311yescharacteristic
84#14579094—22no—
85#14579094—22no—
86#14579092—22no—
87#14579092—22no—
88Dic79⋊C8—22no—
89Dic79⋊C8—22no—
90#14579082—22no—
91#14579082Q814yescomplemented
92#14579072—22no—
93C79⋊#95—21no—
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⋊#95D411yescharacteristic
103C79⋊#95D411yescharacteristic
104C22×D4×C79C2311yescharacteristic
105C23×C4×C79—21no—
106C22×C8×C79C2311yescharacteristic
107C22×C8×C79C2311yescharacteristic
108#125×C79—22no—
109#125×C79D414yes—
110C4×D4×C79—22no—
111C4×D4×C79—22no—
112C2×C4⋊C4×C79—21no—
113C2×C4⋊C4×C79—21no—
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C22⋊C4×C79—21no—
118C2×C42×C79C2311yescharacteristic
119#107×C79—22no—
120#106×C79—22no—
121C4⋊C8×C79—22no—
122#102×C79—22no—
123#102×C79—22no—
124C2×Dic158—42no—
125C2×Dic158—81no—
126C2×Dic158—81no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—21no—
130C22×Dic79—21no—
131C22×Dic79C2×D411yescharacteristic
132C22×Dic79C2×D411yescharacteristic
133C22×Dic79C2×D411yescharacteristic
134C22×Dic79C2×D411yescharacteristic
135C2×C79⋊C8—21no—
136C2×C79⋊C8—21no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8C2×Q812yes—
140C2×C79⋊C8#4112yes—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—41no—
149C4⋊Dic79—41no—
150C4⋊Dic79—42no—
151C4⋊Dic79—42no—
152C4⋊Dic79—81no—
153C4⋊Dic79C2×Q812yes—
154C22⋊Dic79—81no—
155C22⋊Dic79—81no—
156C22⋊Dic79—81no—
157C22⋊Dic79—81no—
158C4×Dic79—21no—
159C4×Dic79—21no—
160C4×Dic79—21no—
161C4×Dic79—21no—
162C4×Dic79—42no—
163C4×Dic79—81no—
164C4×Dic79—81no—
165C24×C79—21no—
166C2×D4×C79—21no—
167C2×D4×C79—21no—
168C2×D4×C79—21no—
169C2×D4×C79—21no—
170C2×D4×C79—21no—
171C2×D4×C79C2×Q812yes—
172C22×C4×C79—21no—
173C22×C4×C79—21no—
174C22×C4×C79—21no—
175C22×C4×C79—42no—
176C22×C4×C79—42no—
177C22×C4×C79—42no—
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2411yescharacteristic
181C2×C8×C79—21no—
182C2×C8×C79—21no—
183C2×C8×C79—21no—
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C4⋊C4×C79—24no—
188C4⋊C4×C79—24no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79C2×D412yes—
194C4⋊C4×C79C2×D412yes—
195C22⋊C4×C79—42no—
196C42×C79C2×D412yes—
197C42×C79#4112yes—
198C23×C79—22no—
199C23×C79—41no—
200C23×C79—41no—
201C23×C79—41no—
202C23×C79—41no—
203C23×C79—41no—
204C23×C79—41no—
205C23×C79C22×D411yescharacteristic
206D4×C79—22no—
207D4×C79—22no—
208D4×C79—41no—
209D4×C79—41no—
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—41no—
217C2×C4×C79—41no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—84no—
224C2×C4×C79#12112yes—
225C2×C4×C79#12112yes—
226C2×C4×C79C22×D411yescharacteristic
227C2×C4×C79C22×Q811yescharacteristic
228C2×C4×C79C2×#4111yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C2×C4×C792−1+411yescharacteristic
231C2×C4×C792−1+411yescharacteristic
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—22no—
237C2×Dic79—21no—
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—21no—
244C2×Dic79—21no—
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—
254C2×Dic79—41no—
255C2×Dic79—41no—
256Dic158—84no—
257C8×C79—41no—
258C8×C79—41no—
259C79⋊C8—22no—
260C79⋊C8—22no—
261C22×C79—42no—
262C22×C79—42no—
263C22×C79—42no—
264C22×C79—81no—
265C22×C79—81no—
266C22×C79—81no—
267C22×C79—81no—
268C22×C79C2×#12111yescharacteristic
269C22×C79#54711yescharacteristic
270C22×C79D4×Q811yescharacteristic
271C22×C79#57511yescharacteristic
272C22×C79#57511yescharacteristic
273C22×C79#57611yescharacteristic
274C22×C79#57711yescharacteristic
275C4×C79—22no—
276C4×C79—22no—
277C4×C79—42no—
278C4×C79—42no—
279C4×C79—81no—
280C4×C79#53311yescharacteristic
281C4×C79#54111yescharacteristic
282C4×C79#54811yescharacteristic
283C4×C79D4×Q811yescharacteristic
284Dic79—41no—
285Dic79—41no—
286Dic79—41no—
287Dic79—41no—
288Dic79—42no—
289Dic79—81no—
290Dic79—81no—
291Dic79—81no—
292#25466—791nomaximal, Sylow
293C2×C79—82no—
294C2×C79#242911yescharacteristic
295C2×C79#304011yescharacteristic
296C2×C79#304511yescharacteristic
297C2×C79#334711yescharacteristic
298C2×C79#334811yescharacteristic
299C2×C79#338811yescharacteristic
300C2×C79#338911yescharacteristic
301C2×#501—791no—
302C2×#478D7911yescharacteristic
303#2563—791no—
304#2370—791no—
305#2085—791no—
306#2085—791no—
307#2067—791no—
308#2065—791no—
309#1988—791no—
310#1949—791no—
311#1949—791no—
312#1940—791no—
313#1926—791no—
314#1925—791no—
315#1762—791no—
316C79#2546611yescharacteristic, complemented, Sylow
317C2×C4⋊Q8—791no—
318C2×#125C2×D7911yescharacteristic
319C2×C4×D4C2×D7911yescharacteristic
320#501—1584no—
321#478—24no—
322C2×#107—791no—
323C2×#107—791no—
324C2×#107C2×D7911yescharacteristic
325C2×#106—791no—
326C2×#106—791no—
327C2×#106C2×D7911yescharacteristic
328C2×C4⋊C8—791no—
329C2×C4⋊C8—791no—
330C2×C4⋊C8C2×D7911yescharacteristic
331C2×#102—791no—
332C2×#102—791no—
333C2×#102C2×D7911yescharacteristic
334C2×#102C2×D7911yescharacteristic
335C2×#97—791no—
336#393—1581no—
337#393—1581no—
338#390—791no—
339#390—791no—
340#389—791no—
341#389—791no—
342#389—1581no—
343#387—1581no—
344#387—1581no—
345#382—791no—
346#382—791no—
347#380—1581no—
348#380—1581no—
349C4×C4⋊C4—791no—
350C4×C4⋊C4—791no—
351C4×C4⋊C4—791no—
352C4×C4⋊C4—791no—
353C4×C4⋊C4—1581no—
354#340—791no—
355#340—791no—
356#340—791no—
357#340—791no—
358#340—791no—
359#340—791no—
360#336—791no—
361#336—791no—
362C22×Q8—1581no—
363C22×D4C22×D7911yescharacteristic
364C23×C4—21no—
365C22×C8—791no—
366C22×C8—791no—
367C22×C8C22×D7911yescharacteristic
368C22×C8C22×D7911yescharacteristic
369C4⋊Q8—1582no—
370C4⋊Q8—1582no—
371#125—22no—
372#125C22⋊D7914yes—
373C4×D4—22no—
374C4×D4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—791no—
378C2×C4⋊C4—791no—
379C2×C4⋊C4—791no—
380C2×C4⋊C4—791no—
381C2×C4⋊C4—791no—
382C2×C4⋊C4—791no—
383C2×C4⋊C4—1581no—
384C2×C4⋊C4—1581no—
385C2×C4⋊C4—1581no—
386C2×C4⋊C4C22×D7911yescharacteristic
387C2×C4⋊C4C22×D7911yescharacteristic
388C2×C4⋊C4C22×D7911yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1581no—
391C2×C22⋊C4—1581no—
392C2×C22⋊C4—1581no—
393C2×C22⋊C4—1581no—
394C2×C42—791no—
395C2×C42—791no—
396C2×C42—1581no—
397C2×C42—1581no—
398C2×C42—1581no—
399C2×C42C22×D7911yescharacteristic
400#107—22no—
401#107—1582no—
402#107—1582no—
403#106—22no—
404#106—1582no—
405#106—1582no—
406C4⋊C8—22no—
407C4⋊C8—1582no—
408C4⋊C8—1582no—
409#102—22no—
410#102—22no—
411#102—794no—
412#102—1582no—
413#97—1582no—
414#95—791no—
415#95—791no—
416#95—1581no—
417#95—1581no—
418#95—1581no—
419#95—1581no—
420#95—1581no—
421#95—1581no—
422#95—1581no—
423#95—1581no—
424#95—1581no—
425C24—21no—
426C2×Q8—3162no—
427C2×Q8—6321no—
428C2×Q8—6321no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4Q8×D7912yes—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—791no—
442C22×C4—791no—
443C22×C4—791no—
444C22×C4—791no—
445C22×C4—1581no—
446C22×C4—1581no—
447C22×C4—1581no—
448C22×C4—1581no—
449C22×C4D4×D7911yescharacteristic
450C22×C4D4×D7911yescharacteristic
451C22×C4C23×D7911yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—792no—
459C2×C8—792no—
460C2×C8—1581no—
461C2×C8—1581no—
462C2×C8—1581no—
463C2×C8—1581no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—3161no—
471C4⋊C4—3161no—
472C4⋊C4—3162no—
473C4⋊C4—3162no—
474C4⋊C4—792no—
475C4⋊C4—6321no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4C2×C22⋊D7912yes—
484C4⋊C4C2×C22⋊D7912yes—
485C22⋊C4—42no—
486C22⋊C4—6321no—
487C22⋊C4—6321no—
488C22⋊C4—6321no—
489C22⋊C4—6321no—
490C42—3162no—
491C42—6321no—
492C42—6321no—
493C42—1581no—
494C42—1581no—
495C42—1581no—
496C42—1581no—
497C42C2×C22⋊D7912yes—
498C42#1192962312yes—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C2×D4×D7911yescharacteristic
507Q8—6324no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—84no—
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—3161no—
537C2×C4—1581no—
538C2×C4—1581no—
539C2×C4—1581no—
540C2×C4—1581no—
541C2×C4—1582no—
542C2×C4—1581no—
543C2×C4—1581no—
544C2×C4—1581no—
545C2×C4—1581no—
546C2×C4—1581no—
547C2×C4—1581no—
548C2×C4—1581no—
549C2×C4—1581no—
550C2×C4#1457913312yes—
551C2×C4#1457913312yes—
552C2×C4C22×C22⋊D7911yescharacteristic
553C2×C4C2×Q8×D7911yescharacteristic
554C2×C4#41×D7911yescharacteristic
555C2×C4C2×#1192962311yescharacteristic
556C2×C4#1457920811yescharacteristic
557C2×C4#1457920911yescharacteristic
558C8—41no—
559C8—41no—
560C8—1582no—
561C8—1582no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C2×#1457913311yescharacteristic
570C22#1909921511yescharacteristic
571C22#1909922611yescharacteristic
572C22#1909935911yescharacteristic
573C22#1909935911yescharacteristic
574C22#1909936511yescharacteristic
575C22#1909937011yescharacteristic
576C4—22no—
577C4—22no—
578C4—42no—
579C4—42no—
580C4—81no—
581C4—3162no—
582C4—3161no—
583C4—3161no—
584C4—3161no—
585C4—3161no—
586C4—6321no—
587C4—6321no—
588C4—6321no—
589C4#1909911511yescharacteristic
590C4#1909917311yescharacteristic
591C4#1909922011yescharacteristic
592C4Q8×C22⋊D7911yescharacteristic
593C2—82no—
594C2#2376700111yescharacteristic
595C2#2377289211yescharacteristic
596C2#2377293711yescharacteristic
597C2#2377589711yescharacteristic
598C2#2377591211yescharacteristic
599C2#2377634211yescharacteristic
600C2#2377635711yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC2×#478×C79C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
hypercenterC2×#478D79
hypocenterC79#25466
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. C2×#19098973C2semidirect
3. #23766500C2semidirect
4. #23764885C2semidirect
5. #23764884C2semidirect
8. #23764351C2semidirect
9. #23764169C2semidirect
10. #23764165C2semidirect
11. #23764121C2semidirect
91. #14579082Q8semidirect
316. C79#25466semidirect

Maximal subgroups

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

subgroupindex
C2×#190989732
#237685732
#237665002
#237648852
#237648842
#237647842
#237647702
#237643512
#237641692
#237641652
#237641212
#237640302
#237640232
#237632252
C2×#478×C792
#2546679