Finite Group Databasebeta

← #34329621

Subgroups — #34329621

Order 20224.

794 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098863C211yesmaximal, characteristic, complemented
2C2×#19098624C211yesmaximal, characteristic, complemented
3#23769230C211yesmaximal, characteristic, complemented
4#23764836C211yesmaximal, characteristic, complemented
5#23764831C211yesmaximal, characteristic, complemented
6#23764579C211yesmaximal, characteristic
7#23764579C211yesmaximal, characteristic
8#23764357C211yesmaximal, characteristic, complemented
9#23764158C211yesmaximal, characteristic, complemented
10#23764135C211yesmaximal, characteristic, complemented
11#23764130C211yesmaximal, characteristic, complemented
12#23764100C211yesmaximal, characteristic, complemented
13#23764086C211yesmaximal, characteristic, complemented
14#23763400C211yesmaximal, characteristic, complemented
15C79×#2324C211yesmaximal, characteristic
16C2×#14579153C2211yescharacteristic
17#19098863—24no—
18#19098624—24no—
19C2×#14579094C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×C4⋊(C79⋊C8)C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25C2×#14579082C2211yescharacteristic
26C2×#14579082C2211yescharacteristic
27C2×#14579082C2211yescharacteristic
28C2×#14579082C2211yescharacteristic
29C2×C22⋊(C79⋊C8)C2211yescharacteristic
30C2×C22⋊(C79⋊C8)C2211yescharacteristic
31C2×#14579072C2211yescharacteristic
32C2×C4×C79⋊C8C2211yescharacteristic
33#19098496—21no—
34#19098496—21no—
35#19098486—21no—
36#19098486—21no—
37#19098461C2211yescharacteristic
38#19098461C2211yescharacteristic
39#19098433C2211yescharacteristic
40#19098423C2211yescharacteristic
41#19098423C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50C2×#121×C79C2211yescharacteristic
51C2×C4×D4×C79C2211yescharacteristic
52C2×#117×C79C2211yescharacteristic
53#390×C79—21no—
54#390×C79C2211yescharacteristic
55#389×C79C2211yescharacteristic
56#388×C79—21no—
57#386×C79—21no—
58#385×C79—21no—
59#380×C79C2211yescharacteristic
60C4×C4⋊C4×C79C2211yescharacteristic
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65#14579153—22no—
66#14579153—22no—
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79—21no—
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C2×C22⋊Dic79—21no—
78C2×C22⋊Dic79—21no—
79#14579094—22no—
80#14579094—22no—
81#14579094—22no—
82#14579092—22no—
83C4⋊(C79⋊C8)—22no—
84C4⋊(C79⋊C8)D414yescomplemented
85#14579082—22no—
86#14579082—22no—
87#14579082—22no—
88#14579082—22no—
89C22⋊(C79⋊C8)—22no—
90C22⋊(C79⋊C8)—22no—
91#14579072—22no—
92C4×C79⋊C8—22no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C22×D4×C79—21no—
98C22×D4×C79C2311yescharacteristic
99C23×C4×C79—21no—
100C23×C4×C79C2311yescharacteristic
101#121×C79—22no—
102#121×C79—22no—
103C4×D4×C79—22no—
104C4×D4×C79—22no—
105#117×C79—22no—
106#117×C79—22no—
107C2×C4⋊C4×C79—21no—
108C2×C4⋊C4×C79D411yescharacteristic
109C2×C4⋊C4×C79D411yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C22⋊C4×C79—21no—
114C2×C22⋊C4×C79—21no—
115C2×C22⋊C4×C79—21no—
116C2×C22⋊C4×C79—21no—
117C2×C22⋊C4×C79—21no—
118C2×C22⋊C4×C79D411yescharacteristic
119C2×C22⋊C4×C79D411yescharacteristic
120C2×C42×C79—21no—
121C2×C42×C79C2311yescharacteristic
122C2×C42×C79C2311yescharacteristic
123#95×C79—21no—
124#95×C79—21no—
125#95×C79—21no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—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⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8C2×D412yes—
141C2×C79⋊C8C2×D412yes—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—81no—
151C4⋊Dic79—81no—
152Dic79⋊C4—42no—
153Dic79⋊C4—42no—
154Dic79⋊C4—81no—
155Dic79⋊C4—81no—
156C22⋊Dic79—81no—
157C22⋊Dic79—81no—
158C24×C79—21no—
159C24×C79C2×D411yescharacteristic
160C2×D4×C79—24no—
161C2×D4×C79—21no—
162C2×D4×C79—21no—
163C2×D4×C79—21no—
164C2×D4×C79—21no—
165C2×D4×C79—21no—
166C2×D4×C79—81no—
167C2×D4×C79—81no—
168C2×D4×C79#4112yes—
169C22×C4×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79—21no—
173C22×C4×C79—21no—
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79—21no—
177C22×C4×C79—21no—
178C22×C4×C79—42no—
179C22×C4×C79—81no—
180C22×C4×C79—81no—
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2×D411yescharacteristic
184C22×C4×C79C2×D411yescharacteristic
185C22×C4×C79C2×D411yescharacteristic
186C22×C4×C79C2411yescharacteristic
187C4⋊C4×C79—22no—
188C4⋊C4×C79—22no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—81no—
195C4⋊C4×C79#4112yes—
196C22⋊C4×C79—24no—
197C22⋊C4×C79—22no—
198C22⋊C4×C79—22no—
199C22⋊C4×C79—42no—
200C22⋊C4×C79—81no—
201C22⋊C4×C79—81no—
202C22⋊C4×C79—81no—
203C42×C79—21no—
204C42×C79—21no—
205C42×C79—81no—
206C42×C79C2×D412yes—
207C42×C79#4112yes—
208C23×C79—21no—
209C23×C79—22no—
210C23×C79—21no—
211C23×C79—21no—
212C23×C79—21no—
213C23×C79—21no—
214C23×C79—21no—
215C23×C79—21no—
216C23×C79—41no—
217C23×C79—41no—
218C23×C79—41no—
219C23×C79—41no—
220C23×C79—41no—
221C23×C79—41no—
222C23×C79C22×D411yescharacteristic
223D4×C79—22no—
224D4×C79—22no—
225D4×C79—41no—
226D4×C79—41no—
227D4×C79—84no—
228C2×C4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—21no—
234C2×C4×C79—21no—
235C2×C4×C79—21no—
236C2×C4×C79—22no—
237C2×C4×C79—21no—
238C2×C4×C79—21no—
239C2×C4×C79—21no—
240C2×C4×C79—21no—
241C2×C4×C79—21no—
242C2×C4×C79—41no—
243C2×C4×C79—41no—
244C2×C4×C79—41no—
245C2×C4×C79—41no—
246C2×C4×C79—41no—
247C2×C4×C79—41no—
248C2×C4×C79—41no—
249C2×C4×C79—41no—
250C2×C4×C79—41no—
251C2×C4×C79—41no—
252C2×C4×C79—41no—
253C2×C4×C79—41no—
254C2×C4×C79—84no—
255C2×C4×C79#12112yes—
256C2×C4×C79#12112yes—
257C2×C4×C79C22×D411yescharacteristic
258C2×C4×C79C2×#4111yescharacteristic
259C2×C4×C79C2×#4111yescharacteristic
260C2×C4×C79C2×#4111yescharacteristic
261C2×C4×C792−1+411yescharacteristic
262C2×C4×C792−1+411yescharacteristic
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—41no—
272C2×Dic79—41no—
273C2×Dic79—41no—
274C2×Dic79—41no—
275C79⋊C8—22no—
276C79⋊C8—22no—
277C79⋊C8—41no—
278C79⋊C8—41no—
279C22×C79—41no—
280C22×C79—42no—
281C22×C79—41no—
282C22×C79—42no—
283C22×C79—41no—
284C22×C79—42no—
285C22×C79—41no—
286C22×C79—41no—
287C22×C79—41no—
288C22×C79—41no—
289C22×C79—81no—
290C22×C79—81no—
291C22×C79—81no—
292C22×C79—81no—
293C22×C79C2×#12111yescharacteristic
294C22×C79#52511yescharacteristic
295C22×C79#53611yescharacteristic
296C22×C79#57611yescharacteristic
297C22×C79#57611yescharacteristic
298C22×C79#57711yescharacteristic
299C22×C79#57811yescharacteristic
300C4×C79—22no—
301C4×C79—22no—
302C4×C79—42no—
303C4×C79—41no—
304C4×C79—41no—
305C4×C79—41no—
306C4×C79—81no—
307C4×C79—81no—
308C4×C79#54711yescharacteristic
309C4×C79#54811yescharacteristic
310C4×C79#55211yescharacteristic
311C4×C79#55611yescharacteristic
312Dic79—81no—
313Dic79—81no—
314Dic79—81no—
315Dic79—81no—
316#24713—791nomaximal, Sylow
317C2×C79—81no—
318C2×C79—82no—
319C2×C79#241511yescharacteristic
320C2×C79#306111yescharacteristic
321C2×C79#306211yescharacteristic
322C2×C79#312011yescharacteristic
323C2×C79#312211yescharacteristic
324C2×C79#324111yescharacteristic
325C2×C79#324311yescharacteristic
326C2×#478—791no—
327C2×#433—791no—
328#2629—791no—
329#2324D7911yescharacteristic
330#2077—791no—
331#2076—791no—
332#2033—791no—
333#2033—791no—
334#1989—791no—
335#1947—791no—
336#1943—791no—
337#1942—791no—
338#1936—791no—
339#1934—791no—
340#1802—791no—
341C79#2471311yescharacteristic, complemented, Sylow
342C2×#125—791no—
343C2×#121C2×D7911yescharacteristic
344C2×C4×D4C2×D7911yescharacteristic
345C2×#117C2×D7911yescharacteristic
346#478—1584no—
347#433—1584no—
348C2×#107—791no—
349C2×#107—791no—
350C2×#107—791no—
351C2×#106—791no—
352C2×C4⋊C8—791no—
353C2×C4⋊C8—791no—
354C2×#102—791no—
355C2×#102—791no—
356C2×#102—791no—
357C2×#102—791no—
358C2×C22⋊C8—791no—
359C2×C22⋊C8—791no—
360C2×#97—791no—
361C2×C4×C8—791no—
362#398—1581no—
363#398—1581no—
364#396—1581no—
365#396—1581no—
366#390—21no—
367#390C2×D7911yescharacteristic
368#389—791no—
369#389—791no—
370#389C2×D7911yescharacteristic
371#388—21no—
372#386—21no—
373#385—21no—
374#384—791no—
375#380—791no—
376#380—791no—
377#380C2×D7911yescharacteristic
378C4×C4⋊C4C2×D7911yescharacteristic
379#340—791no—
380#340—791no—
381#340—791no—
382#340—791no—
383#340—791no—
384#340—791no—
385#340—791no—
386#340—791no—
387C22×D4—21no—
388C22×D4C22×D7911yescharacteristic
389C23×C4—21no—
390C23×C4C22×D7911yescharacteristic
391C22×C8—791no—
392C22×C8—791no—
393C22×C8—791no—
394C22×C8—791no—
395#125—1582no—
396#125—1582no—
397#121—22no—
398#121—22no—
399C4×D4—22no—
400C4×D4—22no—
401#117—22no—
402#117—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—791no—
405C2×C4⋊C4—791no—
406C2×C4⋊C4—791no—
407C2×C4⋊C4—791no—
408C2×C4⋊C4—1581no—
409C2×C4⋊C4—1581no—
410C2×C4⋊C4—1581no—
411C2×C4⋊C4—1581no—
412C2×C4⋊C4—1581no—
413C2×C4⋊C4—1581no—
414C2×C4⋊C4C22⋊D7911yescharacteristic
415C2×C4⋊C4C22⋊D7911yescharacteristic
416C2×C4⋊C4C22×D7911yescharacteristic
417C2×C4⋊C4C22×D7911yescharacteristic
418C2×C4⋊C4C22×D7911yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1581no—
425C2×C22⋊C4—1581no—
426C2×C22⋊C4C22⋊D7911yescharacteristic
427C2×C22⋊C4C22⋊D7911yescharacteristic
428C2×C42—21no—
429C2×C42C22×D7911yescharacteristic
430C2×C42C22×D7911yescharacteristic
431#107—1582no—
432#107—1582no—
433#107—1582no—
434#106—1582no—
435C4⋊C8—794no—
436C4⋊C8—1582no—
437#102—1582no—
438#102—1582no—
439#102—1582no—
440#102—1582no—
441C22⋊C8—1582no—
442C22⋊C8—1582no—
443#97—1582no—
444C4×C8—1582no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—1581no—
449#95—1581no—
450#95—1581no—
451#95—1581no—
452C24—21no—
453C24C2×C22⋊D7911yescharacteristic
454C2×D4—24no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—81no—
461C2×D4—81no—
462C2×D4#1192962412yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—1581no—
476C22×C4—1581no—
477C22×C4—1581no—
478C22×C4—1581no—
479C22×C4C2×C22⋊D7911yescharacteristic
480C22×C4C2×C22⋊D7911yescharacteristic
481C22×C4C2×C22⋊D7911yescharacteristic
482C22×C4C2×C22⋊D7911yescharacteristic
483C22×C4C2×C22⋊D7911yescharacteristic
484C22×C4C23×D7911yescharacteristic
485C2×C8—792no—
486C2×C8—792no—
487C2×C8—1581no—
488C2×C8—1581no—
489C2×C8—1581no—
490C2×C8—1581no—
491C2×C8—1581no—
492C2×C8—1581no—
493C2×C8—1581no—
494C2×C8—1581no—
495C2×C8—1581no—
496C2×C8—1581no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—81no—
505C4⋊C4—3162no—
506C4⋊C4—3162no—
507C4⋊C4—6321no—
508C4⋊C4—6321no—
509C4⋊C4—6321no—
510C4⋊C4—6321no—
511C4⋊C4—1581no—
512C4⋊C4—1581no—
513C4⋊C4—1581no—
514C4⋊C4—1581no—
515C4⋊C4—1581no—
516C4⋊C4—1581no—
517C4⋊C4—1581no—
518C4⋊C4—1581no—
519C4⋊C4#1192962412yes—
520C22⋊C4—24no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—6321no—
528C22⋊C4—6321no—
529C42—21no—
530C42—21no—
531C42—81no—
532C42D4×D7912yes—
533C42#1192962312yes—
534C23—21no—
535C23—22no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C22×C22⋊D7911yescharacteristic
549D4—22no—
550D4—22no—
551D4—41no—
552D4—41no—
553D4—84no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—84no—
581C2×C4—3161no—
582C2×C4—3161no—
583C2×C4—3161no—
584C2×C4—3161no—
585C2×C4—3161no—
586C2×C4—3161no—
587C2×C4—3161no—
588C2×C4—3161no—
589C2×C4—3161no—
590C2×C4—3161no—
591C2×C4—3161no—
592C2×C4—3161no—
593C2×C4#1457913312yes—
594C2×C4#1457913312yes—
595C2×C4C2×D4×D7911yescharacteristic
596C2×C4C2×#1192962311yescharacteristic
597C2×C4C2×#1192962411yescharacteristic
598C2×C4C2×#1192962411yescharacteristic
599C2×C4#1457920811yescharacteristic
600C2×C4#1457920811yescharacteristic
601C8—3161no—
602C8—3161no—
603C8—1582no—
604C8—1582no—
605C22—41no—
606C22—42no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—41no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22C2×#1457913311yescharacteristic
620C22#1909907411yescharacteristic
621C22#1909913411yescharacteristic
622C22#1909936611yescharacteristic
623C22#1909936611yescharacteristic
624C22#1909937411yescharacteristic
625C22#1909937811yescharacteristic
626C4—22no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—6321no—
635C4—6321no—
636C4—6321no—
637C4—6321no—
638C4#1909921811yescharacteristic
639C4#1909922211yescharacteristic
640C4#1909925311yescharacteristic
641C4#1909928311yescharacteristic
642C2—81no—
643C2—82no—
644C2#2376682911yescharacteristic
645C2#2377306711yescharacteristic
646C2#2377307611yescharacteristic
647C2#2377361411yescharacteristic
648C2#2377364311yescharacteristic
649C2#2377464711yescharacteristic
650C2#2377467711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2324C2
socleC23×C79C22×D4
prime radicalC2×#121×C79C22
hypercenter#2324D79
hypocenterC79#24713
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
1. C2×#19098863C2semidirect
2. C2×#19098624C2semidirect
3. #23769230C2semidirect
4. #23764836C2semidirect
5. #23764831C2semidirect
8. #23764357C2semidirect
9. #23764158C2semidirect
10. #23764135C2semidirect
11. #23764130C2semidirect
12. #23764100C2semidirect
13. #23764086C2semidirect
14. #23763400C2semidirect
84. C4⋊(C79⋊C8)D4semidirect
341. C79#24713semidirect

Maximal subgroups

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

subgroupindex
C2×#190988632
C2×#190986242
#237692302
#237648362
#237648312
#237645792
#237643572
#237641582
#237641352
#237641302
#237641002
#237640862
#237634002
C79×#23242
#2471379