Finite Group Databasebeta

← #34340064

Subgroups — #34340064

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×#19098969C211yesmaximal, characteristic, complemented
2C2×#19098863C211yesmaximal, characteristic
3#23766494C211yesmaximal, characteristic, complemented
4#23764882C211yesmaximal, characteristic, complemented
5#23764882C211yesmaximal, characteristic, complemented
6#23764788C211yesmaximal, characteristic
7#23764774C211yesmaximal, characteristic
8#23764353C211yesmaximal, characteristic, complemented
9#23764166C211yesmaximal, characteristic, complemented
10#23764166C211yesmaximal, characteristic, complemented
11#23764124C211yesmaximal, characteristic, complemented
12#23764024C211yesmaximal, characteristic
13#23764017C211yesmaximal, characteristic
14#23763225C211yesmaximal, characteristic
15C79×#2563C211yesmaximal, characteristic
16C2×#14579153C2211yescharacteristic
17#19098969—24no—
18#19098863—24no—
19C2×#14579094C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×C4⋊(C79⋊C8)C2211yescharacteristic
26C2×C4⋊(C79⋊C8)C2211yescharacteristic
27C2×C4⋊(C79⋊C8)C2211yescharacteristic
28C2×#14579082C2211yescharacteristic
29C2×#14579082C2211yescharacteristic
30C2×#14579082C2211yescharacteristic
31C2×#14579082C2211yescharacteristic
32C2×#14579072C2211yescharacteristic
33#19098461C2211yescharacteristic
34#19098461C2211yescharacteristic
35#19098458—21no—
36#19098445—21no—
37#19098444—21no—
38C4×C4⋊Dic79C2211yescharacteristic
39C4×C4⋊Dic79C2211yescharacteristic
40C4×C4⋊Dic79C2211yescharacteristic
41#19098414—21no—
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098318C2211yescharacteristic
49#19098318C2211yescharacteristic
50C2×C4⋊Q8×C79C2211yescharacteristic
51C2×C4×D4×C79C2211yescharacteristic
52#393×C79—21no—
53#393×C79—21no—
54#390×C79C2211yescharacteristic
55#390×C79C2211yescharacteristic
56#382×C79C2211yescharacteristic
57#382×C79C2211yescharacteristic
58#380×C79—21no—
59#380×C79—21no—
60C4×C4⋊C4×C79C2211yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65#14579153—22no—
66#14579153D414yes—
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×Dic79⋊C4—21no—
72C2×Dic79⋊C4—21no—
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×C4×Dic79—21no—
76C2×C4×Dic79—21no—
77C2×C4×Dic79—21no—
78#14579094—22no—
79#14579094—22no—
80#14579094—22no—
81#14579092—22no—
82#14579092—22no—
83#14579092—22no—
84C4⋊(C79⋊C8)—22no—
85C4⋊(C79⋊C8)—22no—
86C4⋊(C79⋊C8)—22no—
87#14579082—22no—
88#14579082—22no—
89#14579082—22no—
90#14579082Q814yescomplemented
91#14579072—22no—
92C79⋊#95—21no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C22×Q8×C79—21no—
98C22×D4×C79C2311yescharacteristic
99C23×C4×C79—21no—
100C4⋊Q8×C79—22no—
101C4⋊Q8×C79—22no—
102C4×D4×C79—22no—
103C4×D4×C79—22no—
104C2×C4⋊C4×C79—21no—
105C2×C4⋊C4×C79D411yescharacteristic
106C2×C4⋊C4×C79D411yescharacteristic
107C2×C4⋊C4×C79C2311yescharacteristic
108C2×C4⋊C4×C79C2311yescharacteristic
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C22⋊C4×C79—21no—
111C2×C22⋊C4×C79—21no—
112C2×C22⋊C4×C79—21no—
113C2×C22⋊C4×C79—21no—
114C2×C22⋊C4×C79—21no—
115C2×C42×C79C2311yescharacteristic
116C2×C42×C79C2311yescharacteristic
117C2×C42×C79C2311yescharacteristic
118#95×C79—21no—
119#95×C79—21no—
120#95×C79—21no—
121#95×C79—21no—
122#95×C79D411yescharacteristic
123#95×C79D411yescharacteristic
124C22×Dic79—21no—
125C22×Dic79—21no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C2×C79⋊C8—21no—
129C2×C79⋊C8—21no—
130C2×C79⋊C8—21no—
131C2×C79⋊C8—21no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8—21no—
136C2×C79⋊C8—21no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8C2×Q812yes—
139C2×C79⋊C8#4112yes—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79C2×D412yes—
146C4⋊Dic79C2×D412yes—
147C4⋊Dic79C2×Q812yes—
148Dic79⋊C4—24no—
149Dic79⋊C4—24no—
150Dic79⋊C4—42no—
151Dic79⋊C4—81no—
152C4×Dic79—42no—
153C4×Dic79—81no—
154C4×Dic79—81no—
155C24×C79—21no—
156C2×Q8×C79—42no—
157C2×Q8×C79—81no—
158C2×Q8×C79—81no—
159C2×D4×C79—21no—
160C2×D4×C79—21no—
161C2×D4×C79—21no—
162C2×D4×C79—21no—
163C2×D4×C79—21no—
164C2×D4×C79C2×Q812yes—
165C22×C4×C79—21no—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79—42no—
169C22×C4×C79—42no—
170C22×C4×C79—42no—
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2411yescharacteristic
178C4⋊C4×C79—21no—
179C4⋊C4×C79—21no—
180C4⋊C4×C79—21no—
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—41no—
185C4⋊C4×C79—41no—
186C4⋊C4×C79—42no—
187C22⋊C4×C79—42no—
188C22⋊C4×C79—81no—
189C22⋊C4×C79—81no—
190C22⋊C4×C79—81no—
191C22⋊C4×C79—81no—
192C42×C79—21no—
193C42×C79—21no—
194C42×C79—21no—
195C42×C79—21no—
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
206Q8×C79—84no—
207D4×C79—22no—
208D4×C79—22no—
209D4×C79—41no—
210D4×C79—41no—
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—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
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—41no—
236C2×C4×C79—84no—
237C2×C4×C79#12112yes—
238C2×C4×C79#12112yes—
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—22no—
246C2×Dic79—22no—
247C2×Dic79—22no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C79⋊C8—22no—
258C79⋊C8—22no—
259C79⋊C8—41no—
260C79⋊C8—41no—
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—41no—
278C4×C79—41no—
279C4×C79—41no—
280C4×C79—41no—
281C4×C79—81no—
282C4×C79—81no—
283C4×C79—81no—
284C4×C79#53311yescharacteristic
285C4×C79#54111yescharacteristic
286C4×C79#54811yescharacteristic
287C4×C79D4×Q811yescharacteristic
288Dic79—42no—
289Dic79—42no—
290Dic79—42no—
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×#478—791no—
303#2563D7911yescharacteristic
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⋊Q8C2×D7911yescharacteristic
318C2×#125—791no—
319C2×C4×D4C2×D7911yescharacteristic
320#501—1584no—
321#478—1584no—
322C2×#107—791no—
323C2×#107—791no—
324C2×#107—791no—
325C2×#106—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×#97—791no—
336#393—21no—
337#393—21no—
338#390C2×D7911yescharacteristic
339#390C2×D7911yescharacteristic
340#389—791no—
341#389—791no—
342#389—1581no—
343#387—1581no—
344#387—1581no—
345#382C2×D7911yescharacteristic
346#382C2×D7911yescharacteristic
347#380—21no—
348#380—21no—
349C4×C4⋊C4—791no—
350C4×C4⋊C4—791no—
351C4×C4⋊C4—791no—
352C4×C4⋊C4—1581no—
353C4×C4⋊C4C2×D7911yescharacteristic
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—21no—
363C22×D4C22×D7911yescharacteristic
364C23×C4—21no—
365C22×C8—791no—
366C22×C8—791no—
367C22×C8—791no—
368C22×C8—791no—
369C4⋊Q8—22no—
370C4⋊Q8—22no—
371#125—794no—
372#125—1582no—
373C4×D4—22no—
374C4×D4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—791no—
377C2×C4⋊C4—791no—
378C2×C4⋊C4—791no—
379C2×C4⋊C4—791no—
380C2×C4⋊C4—1581no—
381C2×C4⋊C4—1581no—
382C2×C4⋊C4—1581no—
383C2×C4⋊C4—1581no—
384C2×C4⋊C4C22⋊D7911yescharacteristic
385C2×C4⋊C4C22⋊D7911yescharacteristic
386C2×C4⋊C4C22×D7911yescharacteristic
387C2×C4⋊C4C22×D7911yescharacteristic
388C2×C4⋊C4C22×D7911yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C42—1581no—
395C2×C42—1581no—
396C2×C42—1581no—
397C2×C42C22×D7911yescharacteristic
398C2×C42C22×D7911yescharacteristic
399C2×C42C22×D7911yescharacteristic
400#107—1582no—
401#107—1582no—
402#107—1582no—
403#106—1582no—
404#106—1582no—
405#106—1582no—
406C4⋊C8—1582no—
407C4⋊C8—1582no—
408C4⋊C8—1582no—
409#102—794no—
410#102—1582no—
411#102—1582no—
412#102—1582no—
413#97—1582no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—1581no—
419#95—1581no—
420#95—1581no—
421#95—1581no—
422#95—1581no—
423#95C22⋊D7911yescharacteristic
424#95C22⋊D7911yescharacteristic
425C24—21no—
426C2×Q8—42no—
427C2×Q8—81no—
428C2×Q8—81no—
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—1581no—
442C22×C4—1581no—
443C22×C4—1581no—
444C22×C4—1581no—
445C22×C4C2×C22⋊D7911yescharacteristic
446C22×C4C2×C22⋊D7911yescharacteristic
447C22×C4C2×C22⋊D7911yescharacteristic
448C22×C4C2×C22⋊D7911yescharacteristic
449C22×C4C2×C22⋊D7911yescharacteristic
450C22×C4C2×C22⋊D7911yescharacteristic
451C22×C4C23×D7911yescharacteristic
452C2×C8—792no—
453C2×C8—792no—
454C2×C8—1581no—
455C2×C8—1581no—
456C2×C8—1581no—
457C2×C8—1581no—
458C2×C8—1581no—
459C2×C8—1581no—
460C2×C8—1581no—
461C2×C8—1581no—
462C2×C8—1581no—
463C2×C8—1581no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—42no—
473C4⋊C4—3162no—
474C4⋊C4—792no—
475C4⋊C4—792no—
476C4⋊C4—792no—
477C4⋊C4—6321no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1584no—
484C4⋊C4—1584no—
485C22⋊C4—42no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—81no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—3162no—
495C42—6321no—
496C42—6321no—
497C42D4×D7912yes—
498C42#1192962312yes—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C22×C22⋊D7911yescharacteristic
507Q8—84no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
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—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—3161no—
547C2×C4—1582no—
548C2×C4—1582no—
549C2×C4—1582no—
550C2×C4#1457913212yes—
551C2×C4#1457913212yes—
552C2×C4C2×D4×D7911yescharacteristic
553C2×C4C2×Q8×D7911yescharacteristic
554C2×C4#41×D7911yescharacteristic
555C2×C4C2×#1192962311yescharacteristic
556C2×C4#1457920811yescharacteristic
557C2×C4#1457920911yescharacteristic
558C8—3161no—
559C8—3161no—
560C8—1582no—
561C8—1582no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C2×#1457913211yescharacteristic
570C22#1909921211yescharacteristic
571C22Q8×C22⋊D7911yescharacteristic
572C22#1909936111yescharacteristic
573C22#1909936111yescharacteristic
574C22#1909936611yescharacteristic
575C22#1909937411yescharacteristic
576C4—22no—
577C4—22no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—3162no—
586C4—3162no—
587C4—3162no—
588C4—6321no—
589C4#1909911511yescharacteristic
590C4#1909917311yescharacteristic
591C4#1909922311yescharacteristic
592C4#1909922611yescharacteristic
593C2—82no—
594C2#2376699511yescharacteristic
595C2#2377289011yescharacteristic
596C2#2377293511yescharacteristic
597C2#2377590011yescharacteristic
598C2#2377591511yescharacteristic
599C2#2377634511yescharacteristic
600C2#2377635411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2563C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
hypercenter#2563D79
hypocenterC79#25466
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2563

Complements

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

NH ≅ G/Ntype
1. C2×#19098969C2semidirect
3. #23766494C2semidirect
4. #23764882C2semidirect
5. #23764882C2semidirect
8. #23764353C2semidirect
9. #23764166C2semidirect
10. #23764166C2semidirect
11. #23764124C2semidirect
90. #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×#190989692
C2×#190988632
#237664942
#237648822
#237647882
#237647742
#237643532
#237641662
#237641242
#237640242
#237640172
#237632252
C79×#25632
#2546679