Finite Group Databasebeta

← #34293118

Subgroups — #34293118

Order 20224.

870 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098613C211yesmaximal, characteristic
2#23765795C211yesmaximal, characteristic
3#117×Dic79C211yesmaximal, characteristic
4#23764171C211yesmaximal, characteristic, complemented
5#23764169C211yesmaximal, characteristic, complemented
6#23764167C211yesmaximal, characteristic, complemented
7#23764165C211yesmaximal, characteristic, complemented
8#23763601C211yesmaximal, characteristic
9#23763599C211yesmaximal, characteristic
10#23763597C211yesmaximal, characteristic
11#107×Dic79C211yesmaximal, characteristic, complemented
12#23763292C211yesmaximal, characteristic, complemented
13#106×Dic79C211yesmaximal, characteristic, complemented
14#23763288C211yesmaximal, characteristic, complemented
15C79×#1854C211yesmaximal, characteristic
16C2×#14579120C2211yescharacteristic
17C22×C4×Dic79C2211yescharacteristic
18#19098613C418yescomplemented
19C2×#14579095C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579093C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×Dic79⋊C8C2211yescharacteristic
26C2×Dic79⋊C8C2211yescharacteristic
27C2×C22⋊(C79⋊C8)C2211yescharacteristic
28C2×C8×Dic79C2211yescharacteristic
29C2×C8×Dic79C2211yescharacteristic
30#19098430—21no—
31#19098428C2211yescharacteristic
32#19098428C2211yescharacteristic
33#19098424—21no—
34#19098423C2211yescharacteristic
35C4⋊C4×Dic79C2211yescharacteristic
36C4⋊C4×Dic79C2211yescharacteristic
37C4⋊C4×Dic79C2211yescharacteristic
38C4⋊C4×Dic79C2211yescharacteristic
39C4⋊C4×Dic79C2211yescharacteristic
40C4⋊C4×Dic79C2211yescharacteristic
41C22⋊C4×Dic79—21no—
42C22⋊C4×Dic79—21no—
43C22⋊C4×Dic79—21no—
44#19098409—21no—
45#19098408—21no—
46C42×Dic79—21no—
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098325C2211yescharacteristic
53C2×#117×C79C2211yescharacteristic
54C2×#107×C79C2211yescharacteristic
55C2×#106×C79C2211yescharacteristic
56C2×C22⋊C8×C79C2211yescharacteristic
57#380×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C23×Dic79D411yescharacteristic
61C23×Dic79D411yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64#14579120—22no—
65#14579120—22no—
66C2×C4⋊Dic79C2311yescharacteristic
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C22⋊Dic79—21no—
71C2×C22⋊Dic79—21no—
72C2×C4×Dic79—21no—
73C2×C4×Dic79—21no—
74C2×C4×Dic79—21no—
75C2×C4×Dic79—21no—
76C2×C4×Dic79—21no—
77C2×C4×Dic79—21no—
78C2×C4×Dic79—21no—
79C2×C4×Dic79C2×C414yes—
80C2×C4×Dic79D411yescharacteristic
81C2×C4×Dic79D411yescharacteristic
82C2×C4×Dic79Q814yes—
83C2×C4×Dic79C2311yescharacteristic
84C2×C4×Dic79C2311yescharacteristic
85#14579095—22no—
86#14579094—22no—
87#14579094—22no—
88#14579093—22no—
89#14579092—22no—
90#14579092—22no—
91Dic79⋊C8C2×C414yes—
92Dic79⋊C8C2×C414yes—
93C22⋊(C79⋊C8)C2×C414yes—
94C8×Dic79C2×C414yes—
95C8×Dic79C2×C414yes—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C23×C4×C79C2311yescharacteristic
108C22×C8×C79C2311yescharacteristic
109C22×C8×C79C2311yescharacteristic
110#117×C79—22no—
111#117×C79—22no—
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C22⋊C4×C79—21no—
117C2×C22⋊C4×C79—21no—
118C2×C42×C79—21no—
119#107×C79—22no—
120#106×C79—22no—
121C22⋊C8×C79C2×C414yes—
122#95×C79—21no—
123C22×Dic79—22no—
124C22×Dic79—22no—
125C22×Dic79—21no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—41no—
130C22×Dic79—41no—
131C22×Dic79C4⋊C414yes—
132C22×Dic79C4⋊C414yes—
133C22×Dic79C2×D411yescharacteristic
134C22×Dic79C2×D411yescharacteristic
135C22×Dic79C2×D411yescharacteristic
136C22×Dic79C2×D411yescharacteristic
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8C22×C412yes—
142C2×C79⋊C8C22×C412yes—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79#4112yes—
150C4⋊Dic79#4112yes—
151C22⋊Dic79—24no—
152C22⋊Dic79—42no—
153C4×Dic79—24no—
154C4×Dic79—42no—
155C4×Dic79—42no—
156C4×Dic79—42no—
157C4×Dic79—42no—
158C4×Dic79—42no—
159C4×Dic79—42no—
160C4×Dic79C4⋊C414yes—
161C4×Dic79C4⋊C414yes—
162C4×Dic79C22×C412yes—
163C4×Dic79C22×C412yes—
164C4×Dic79C2×Q812yes—
165C4×Dic79C2×Q812yes—
166C24×C79C2×D411yescharacteristic
167C22×C4×C79—21no—
168C22×C4×C79—21no—
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×C79C22×C412yes—
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×Q812yes—
178C22×C4×C79C2411yescharacteristic
179C2×C8×C79—21no—
180C2×C8×C79—21no—
181C2×C8×C79—21no—
182C2×C8×C79—21no—
183C2×C8×C79C22×C412yes—
184C2×C8×C79C22×C412yes—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79#4112yes—
192C4⋊C4×C79#4112yes—
193C22⋊C4×C79—24no—
194C22⋊C4×C79—42no—
195C42×C79—24no—
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—22no—
207C2×C4×C79—22no—
208C2×C4×C79—22no—
209C2×C4×C79—22no—
210C2×C4×C79—22no—
211C2×C4×C79—22no—
212C2×C4×C79—41no—
213C2×C4×C79—41no—
214C2×C4×C79—41no—
215C2×C4×C79—41no—
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×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—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—28no—
236C2×Dic79—22no—
237C2×Dic79—22no—
238C2×Dic79—44no—
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×Dic79C2×C4⋊C412yes—
250C2×Dic79C2×C4⋊C412yes—
251C2×Dic79C2×C4⋊C412yes—
252C2×Dic79C2×C4⋊C412yes—
253C2×Dic79#13512yes—
254C2×Dic79#13512yes—
255C8×C79—22no—
256C8×C79—22no—
257C79⋊C8—22no—
258C79⋊C8—22no—
259C22×C79—22no—
260C22×C79—22no—
261C22×C79—22no—
262C22×C79—41no—
263C22×C79—41no—
264C22×C79—41no—
265C22×C79—41no—
266C22×C79C22×C4⋊C411yescharacteristic
267C22×C79#52511yescharacteristic
268C22×C79#52511yescharacteristic
269C22×C79C2×#13511yescharacteristic
270C22×C79C2×#13511yescharacteristic
271C22×C79#57511yescharacteristic
272C22×C79#57511yescharacteristic
273C4×C79—22no—
274C4×C79—42no—
275C4×C79—42no—
276C4×C79—81no—
277C4×C79—81no—
278C4×C79C4×#4111yescharacteristic
279C4×C79C4×#4111yescharacteristic
280C4×C79#53311yescharacteristic
281C4×C79#53311yescharacteristic
282Dic79—22no—
283Dic79—22no—
284Dic79—22no—
285Dic79—42no—
286Dic79—42no—
287Dic79—81no—
288Dic79—81no—
289Dic79#42714yes—
290#21360—791nomaximal, Sylow
291C2×C79—42no—
292C2×C79#229611yescharacteristic
293C2×C79C2×#42711yescharacteristic
294C2×C79#291111yescharacteristic
295C2×C79#310211yescharacteristic
296C2×C79#310211yescharacteristic
297C2×C79#310211yescharacteristic
298C2×C79#310211yescharacteristic
299C2×#431—791no—
300#2283—791no—
301C4×#117—791no—
302#1949—791no—
303#1949—791no—
304#1949—791no—
305#1949—791no—
306#1854—791no—
307#1854—791no—
308#1854—791no—
309#1854D7911yescharacteristic
310C4×#107—791no—
311C4×#107—791no—
312C4×#106—791no—
313C4×#106—791no—
314C79#2136011yescharacteristic, complemented, Sylow
315C2×#117—791no—
316C2×#117C2×D7911yescharacteristic
317C22×C42—791no—
318#431—798no—
319C2×#107—791no—
320C2×#107—791no—
321C2×#107—791no—
322C2×#107C2×D7911yescharacteristic
323C2×#106—791no—
324C2×#106—791no—
325C2×#106—791no—
326C2×#106C2×D7911yescharacteristic
327C2×C4⋊C8—791no—
328C2×C4⋊C8—791no—
329C2×C22⋊C8—791no—
330C2×C22⋊C8C2×D7911yescharacteristic
331C2×C4×C8—791no—
332C2×C4×C8—791no—
333#383—1581no—
334#382—791no—
335#382—791no—
336#381—1581no—
337#380—791no—
338#380C2×D7911yescharacteristic
339C4×C4⋊C4—791no—
340C4×C4⋊C4—791no—
341C4×C4⋊C4—791no—
342C4×C4⋊C4—791no—
343C4×C4⋊C4—791no—
344C4×C4⋊C4—791no—
345C4×C22⋊C4—1581no—
346C4×C22⋊C4—1581no—
347C4×C22⋊C4—1581no—
348#376—1581no—
349#376—1581no—
350C43—1581no—
351#340—791no—
352#340—791no—
353#340—791no—
354#340—791no—
355#340—791no—
356#340—791no—
357#340C2×D7911yescharacteristic
358#340C2×D7911yescharacteristic
359C23×C4—791no—
360C23×C4—791no—
361C23×C4C22×D7911yescharacteristic
362C22×C8—791no—
363C22×C8—791no—
364C22×C8C22×D7911yescharacteristic
365C22×C8C22×D7911yescharacteristic
366#117—22no—
367#117—22no—
368#117—1582no—
369#117—1582no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—791no—
372C2×C4⋊C4—791no—
373C2×C4⋊C4—791no—
374C2×C4⋊C4C22×D7911yescharacteristic
375C2×C4⋊C4C22×D7911yescharacteristic
376C2×C4⋊C4C22×D7911yescharacteristic
377C2×C4⋊C4C22×D7911yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1581no—
381C2×C22⋊C4—1581no—
382C2×C42—21no—
383C2×C42—794no—
384C2×C42—791no—
385C2×C42—791no—
386C2×C42—794no—
387C2×C42—791no—
388C2×C42—791no—
389C2×C42—1581no—
390C2×C42—1581no—
391C2×C42—1581no—
392C2×C42—1581no—
393C2×C42—1581no—
394C2×C42—1581no—
395C2×C42—1581no—
396#107—22no—
397#107—1582no—
398#107—1582no—
399#107—1582no—
400#106—22no—
401#106—1582no—
402#106—1582no—
403#106—1582no—
404C4⋊C8—794no—
405C4⋊C8—794no—
406C22⋊C8—794no—
407C22⋊C8C4×D7914yes—
408C4×C8—794no—
409C4×C8—794no—
410#95—21no—
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—
421#95—1581no—
422C24D4×D7911yescharacteristic
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—3161no—
432C22×C4—3161no—
433C22×C4—794no—
434C22×C4—791no—
435C22×C4—794no—
436C22×C4—791no—
437C22×C4—791no—
438C22×C4—791no—
439C22×C4—1581no—
440C22×C4—1581no—
441C22×C4—1581no—
442C22×C4—1581no—
443C22×C4—1582no—
444C22×C4—1582no—
445C22×C4C2×C4×D7912yes—
446C22×C4D4×D7911yescharacteristic
447C22×C4Q8×D7912yes—
448C22×C4C23×D7911yescharacteristic
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—792no—
454C2×C8—792no—
455C2×C8—1581no—
456C2×C8—1581no—
457C2×C8—1581no—
458C2×C8—1581no—
459C2×C8C2×C4×D7912yes—
460C2×C8C2×C4×D7912yes—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—792no—
468C4⋊C4—792no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4#1192962312yes—
476C4⋊C4#1192962312yes—
477C22⋊C4—24no—
478C22⋊C4—42no—
479C22⋊C4—3162no—
480C22⋊C4—1584no—
481C42—24no—
482C42—3162no—
483C42—3162no—
484C42—3162no—
485C42—3162no—
486C42—3162no—
487C42—3162no—
488C42—792no—
489C42—792no—
490C42—794no—
491C42—794no—
492C42—792no—
493C42—792no—
494C42—1584no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23C4⋊C4×D7912yes—
502C23C2×D4×D7911yescharacteristic
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
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—3164no—
532C2×C4—792no—
533C2×C4—792no—
534C2×C4—792no—
535C2×C4—792no—
536C2×C4—792no—
537C2×C4—792no—
538C2×C4—1582no—
539C2×C4—1582no—
540C2×C4—1588no—
541C2×C4—1581no—
542C2×C4—1581no—
543C2×C4—1581no—
544C2×C4—1581no—
545C2×C4—1581no—
546C2×C4—1581no—
547C2×C4C4⋊C4×D7912yes—
548C2×C4C22×C4×D7911yescharacteristic
549C2×C4C2×Q8×D7911yescharacteristic
550C2×C4#41×D7911yescharacteristic
551C2×C4#41×D7911yescharacteristic
552C2×C4C2×#1192962311yescharacteristic
553C2×C4C2×#1192962311yescharacteristic
554C8—22no—
555C8—22no—
556C8—1582no—
557C8—1582no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22C2×C4⋊C4×D7911yescharacteristic
566C22#1909906911yescharacteristic
567C22#1909907111yescharacteristic
568C22#135×D7911yescharacteristic
569C22#135×D7911yescharacteristic
570C22#1909935911yescharacteristic
571C22#1909936211yescharacteristic
572C4—22no—
573C4—42no—
574C4—42no—
575C4—81no—
576C4—81no—
577C4—3162no—
578C4—3162no—
579C4—794no—
580C4—6321no—
581C4—6321no—
582C4—1582no—
583C4—1582no—
584C4—1582no—
585C4C4×#1192962311yescharacteristic
586C4C4×#1192962311yescharacteristic
587C4#1909911511yescharacteristic
588C4#1909911511yescharacteristic
589C2—42no—
590C2#2376587911yescharacteristic
591C2#427×D7911yescharacteristic
592C2#2377198711yescharacteristic
593C2#2377341111yescharacteristic
594C2#2377341311yescharacteristic
595C2#2377341911yescharacteristic
596C2#2377342111yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1854C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
cosocleC22×C4×C79C24
hypercenter#1854D79
hypocenterC79#21360
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
4. #23764171C2semidirect
5. #23764169C2semidirect
6. #23764167C2semidirect
7. #23764165C2semidirect
11. #107×Dic79C2semidirect
12. #23763292C2semidirect
13. #106×Dic79C2semidirect
14. #23763288C2semidirect
18. #19098613C4semidirect
314. C79#21360semidirect

Maximal subgroups

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

subgroupindex
C2×#190986132
#237657952
#117×Dic792
#237641712
#237641692
#237641672
#237641652
#237636012
#237635992
#237635972
#107×Dic792
#237632922
#106×Dic792
#237632882
C79×#18542
#2136079