Finite Group Databasebeta

← #34293148

Subgroups — #34293148

Order 20224.

894 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098617C211yesmaximal, characteristic
2#23765802C211yesmaximal, characteristic
3#117×Dic79C211yesmaximal, characteristic
4#23764162C211yesmaximal, characteristic, complemented
5#23764160C211yesmaximal, characteristic, complemented
6#23764157C211yesmaximal, characteristic, complemented
7#23764155C211yesmaximal, characteristic, complemented
8#23763601C211yesmaximal, characteristic
9#23763599C211yesmaximal, characteristic
10#23763597C211yesmaximal, characteristic
11#23763300C211yesmaximal, characteristic, complemented
12#23763299C211yesmaximal, characteristic, complemented
13#23763296C211yesmaximal, characteristic, complemented
14#23763296C211yesmaximal, characteristic, complemented
15C79×#1854C211yesmaximal, characteristic
16C2×#14579120C2211yescharacteristic
17C2×#14579117—21no—
18C22×C4×Dic79C2211yescharacteristic
19#19098617C418yescomplemented
20C2×#14579095C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579093C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×#14579092C2211yescharacteristic
26C2×Dic79⋊C8C2211yescharacteristic
27C2×Dic79⋊C8C2211yescharacteristic
28C2×C22⋊(C79⋊C8)C2211yescharacteristic
29C2×#14579071C2211yescharacteristic
30C2×#14579071C2211yescharacteristic
31#19098432C2211yescharacteristic
32#19098432C2211yescharacteristic
33#19098427—21no—
34#19098423C2211yescharacteristic
35#19098421—21no—
36C4⋊C4×Dic79C2211yescharacteristic
37C4⋊C4×Dic79C2211yescharacteristic
38C4⋊C4×Dic79C2211yescharacteristic
39C4⋊C4×Dic79C2211yescharacteristic
40C4⋊C4×Dic79C2211yescharacteristic
41C4⋊C4×Dic79C2211yescharacteristic
42#19098414—21no—
43C22⋊C4×Dic79—21no—
44C22⋊C4×Dic79—21no—
45#19098409—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—
66#14579117—44no—
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×C22⋊Dic79—21no—
76C2×C22⋊Dic79—21no—
77C2×C4×Dic79—21no—
78C2×C4×Dic79—21no—
79C2×C4×Dic79—21no—
80C2×C4×Dic79—21no—
81C2×C4×Dic79—21no—
82C2×C4×Dic79—21no—
83C2×C4×Dic79—21no—
84C2×C4×Dic79C2×C414yes—
85C2×C4×Dic79D411yescharacteristic
86C2×C4×Dic79D411yescharacteristic
87C2×C4×Dic79Q814yes—
88C2×C4×Dic79C2311yescharacteristic
89C2×C4×Dic79C2311yescharacteristic
90#14579095—22no—
91#14579094—22no—
92#14579094—22no—
93#14579093—22no—
94#14579092—22no—
95#14579092—22no—
96Dic79⋊C8C2×C414yes—
97Dic79⋊C8C2×C414yes—
98C22⋊(C79⋊C8)C2×C414yes—
99#14579071C2×C414yes—
100#14579071C2×C414yes—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C79⋊#95—21no—
108C23×C4×C79C2311yescharacteristic
109C22×C8×C79C2311yescharacteristic
110C22×C8×C79C2311yescharacteristic
111#117×C79—22no—
112#117×C79—22no—
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C22⋊C4×C79—21no—
118C2×C22⋊C4×C79—21no—
119C2×C42×C79—21no—
120#107×C79—22no—
121#106×C79—22no—
122C22⋊C8×C79C2×C414yes—
123#95×C79—21no—
124C22×Dic79—22no—
125C22×Dic79—22no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—21no—
130C22×Dic79—41no—
131C22×Dic79—41no—
132C22×Dic79C4⋊C414yes—
133C22×Dic79C4⋊C414yes—
134C22×Dic79C2×D411yescharacteristic
135C22×Dic79C2×D411yescharacteristic
136C22×Dic79C2×D411yescharacteristic
137C22×Dic79C2×D411yescharacteristic
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8C22×C412yes—
143C2×C79⋊C8C22×C412yes—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79#4112yes—
151C4⋊Dic79#4112yes—
152Dic79⋊C4—42no—
153Dic79⋊C4—42no—
154Dic79⋊C4—42no—
155Dic79⋊C4—42no—
156C22⋊Dic79—24no—
157C22⋊Dic79—42no—
158C4×Dic79—24no—
159C4×Dic79—42no—
160C4×Dic79—42no—
161C4×Dic79—42no—
162C4×Dic79—42no—
163C4×Dic79—42no—
164C4×Dic79—42no—
165C4×Dic79C4⋊C414yes—
166C4×Dic79C4⋊C414yes—
167C4×Dic79C22×C412yes—
168C4×Dic79C22×C412yes—
169C4×Dic79C2×Q812yes—
170C4×Dic79C2×Q812yes—
171C24×C79C2×D411yescharacteristic
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—21no—
179C22×C4×C79—21no—
180C22×C4×C79C22×C412yes—
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2×Q812yes—
183C22×C4×C79C2411yescharacteristic
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79C22×C412yes—
189C2×C8×C79C22×C412yes—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79#4112yes—
197C4⋊C4×C79#4112yes—
198C22⋊C4×C79—24no—
199C22⋊C4×C79—42no—
200C42×C79—24no—
201C23×C79—21no—
202C23×C79—21no—
203C23×C79—21no—
204C23×C79—21no—
205C23×C79—21no—
206C23×C79—21no—
207C23×C79C2×C4⋊C412yes—
208C23×C79C22×D411yescharacteristic
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—22no—
212C2×C4×C79—22no—
213C2×C4×C79—22no—
214C2×C4×C79—22no—
215C2×C4×C79—22no—
216C2×C4×C79—22no—
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—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79C2×C4⋊C412yes—
228C2×C4×C79C23×C411yescharacteristic
229C2×C4×C79C22×Q811yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79—21no—
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—28no—
241C2×Dic79—22no—
242C2×Dic79—22no—
243C2×Dic79—44no—
244C2×Dic79—41no—
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×Dic79C2×C4⋊C412yes—
255C2×Dic79C2×C4⋊C412yes—
256C2×Dic79C2×C4⋊C412yes—
257C2×Dic79C2×C4⋊C412yes—
258C2×Dic79AGL(1,ℤ/8)12yes—
259C2×Dic79#13712yes—
260C8×C79—22no—
261C8×C79—22no—
262C79⋊C8—22no—
263C79⋊C8—22no—
264C22×C79—22no—
265C22×C79—22no—
266C22×C79—22no—
267C22×C79—41no—
268C22×C79—41no—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79C22×C4⋊C411yescharacteristic
272C22×C79#52511yescharacteristic
273C22×C79#52511yescharacteristic
274C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C79C2×#13711yescharacteristic
276C22×C79#57511yescharacteristic
277C22×C79#57511yescharacteristic
278C4×C79—22no—
279C4×C79—42no—
280C4×C79—42no—
281C4×C79—81no—
282C4×C79—81no—
283C4×C79C4×#4111yescharacteristic
284C4×C79C4×#4111yescharacteristic
285C4×C79#53311yescharacteristic
286C4×C79#53311yescharacteristic
287Dic79—22no—
288Dic79—22no—
289Dic79—22no—
290Dic79—42no—
291Dic79—42no—
292Dic79—81no—
293Dic79—81no—
294Dic79#42814yes—
295#21364—791nomaximal, Sylow
296C2×C79—42no—
297C2×C79#229611yescharacteristic
298C2×C79C2×#42811yescharacteristic
299C2×C79#291111yescharacteristic
300C2×C79#311511yescharacteristic
301C2×C79#311511yescharacteristic
302C2×C79#311611yescharacteristic
303C2×C79#311611yescharacteristic
304C2×#432—791no—
305#2285—791no—
306C4×#117—791no—
307#1948—791no—
308#1948—791no—
309#1947—791no—
310#1947—791no—
311#1854—791no—
312#1854—791no—
313#1854—791no—
314#1854D7911yescharacteristic
315#1776—791no—
316#1776—791no—
317#1776—791no—
318#1776—791no—
319C79#2136411yescharacteristic, complemented, Sylow
320C2×#117—791no—
321C2×#117—1581no—
322C2×#117C2×D7911yescharacteristic
323C22×C42—791no—
324#432—798no—
325C2×#107—791no—
326C2×#107—791no—
327C2×#107—791no—
328C2×#107C2×D7911yescharacteristic
329C2×#106—791no—
330C2×#106—791no—
331C2×#106—791no—
332C2×#106C2×D7911yescharacteristic
333C2×C4⋊C8—791no—
334C2×C4⋊C8—791no—
335C2×C22⋊C8—791no—
336C2×C22⋊C8C2×D7911yescharacteristic
337C2×#97—791no—
338C2×#97—791no—
339#384—791no—
340#384—791no—
341#382—1581no—
342#380—791no—
343#380—1581no—
344#380C2×D7911yescharacteristic
345C4×C4⋊C4—791no—
346C4×C4⋊C4—791no—
347C4×C4⋊C4—791no—
348C4×C4⋊C4—791no—
349C4×C4⋊C4—791no—
350C4×C4⋊C4—791no—
351C4×C4⋊C4—1581no—
352C4×C22⋊C4—1581no—
353C4×C22⋊C4—1581no—
354#376—1581no—
355C43—1581no—
356#340—791no—
357#340—791no—
358#340—791no—
359#340—791no—
360#340—791no—
361#340—791no—
362#340C2×D7911yescharacteristic
363#340C2×D7911yescharacteristic
364C23×C4—791no—
365C23×C4—791no—
366C23×C4C22×D7911yescharacteristic
367C22×C8—791no—
368C22×C8—791no—
369C22×C8C22×D7911yescharacteristic
370C22×C8C22×D7911yescharacteristic
371#117—22no—
372#117—22no—
373#117—3164no—
374#117—1582no—
375#117—1582no—
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×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1581no—
391C2×C22⋊C4—1581no—
392C2×C42—21no—
393C2×C42—794no—
394C2×C42—791no—
395C2×C42—791no—
396C2×C42—794no—
397C2×C42—791no—
398C2×C42—791no—
399C2×C42—1581no—
400C2×C42—1581no—
401C2×C42—1581no—
402C2×C42—1581no—
403C2×C42—1581no—
404C2×C42—1581no—
405C2×C42—1581no—
406#107—22no—
407#107—1582no—
408#107—1582no—
409#107—1582no—
410#106—22no—
411#106—1582no—
412#106—1582no—
413#106—1582no—
414C4⋊C8—794no—
415C4⋊C8—794no—
416C22⋊C8—794no—
417C22⋊C8C4×D7914yes—
418#97—794no—
419#97—794no—
420#95—21no—
421#95—1581no—
422#95—1581no—
423#95—1581no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427#95—1581no—
428C24D4×D7911yescharacteristic
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—3161no—
438C22×C4—3161no—
439C22×C4—794no—
440C22×C4—791no—
441C22×C4—794no—
442C22×C4—791no—
443C22×C4—791no—
444C22×C4—791no—
445C22×C4—1581no—
446C22×C4—1581no—
447C22×C4—1581no—
448C22×C4—1581no—
449C22×C4—1582no—
450C22×C4—1582no—
451C22×C4C2×C4×D7912yes—
452C22×C4D4×D7911yescharacteristic
453C22×C4Q8×D7912yes—
454C22×C4C23×D7911yescharacteristic
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—792no—
460C2×C8—792no—
461C2×C8—1581no—
462C2×C8—1581no—
463C2×C8—1581no—
464C2×C8—1581no—
465C2×C8C2×C4×D7912yes—
466C2×C8C2×C4×D7912yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—3162no—
474C4⋊C4—3162no—
475C4⋊C4—3162no—
476C4⋊C4—3162no—
477C4⋊C4—792no—
478C4⋊C4—792no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4#1192962312yes—
486C4⋊C4#1192962312yes—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—3162no—
490C22⋊C4—1584no—
491C42—24no—
492C42—3162no—
493C42—3162no—
494C42—3162no—
495C42—3162no—
496C42—3162no—
497C42—3162no—
498C42—792no—
499C42—792no—
500C42—794no—
501C42—794no—
502C42—792no—
503C42—792no—
504C42—1584no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C4⋊C4×D7912yes—
512C23C2×D4×D7911yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—3161no—
532C2×C4—3161no—
533C2×C4—3161no—
534C2×C4—3161no—
535C2×C4—3161no—
536C2×C4—3161no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3164no—
542C2×C4—792no—
543C2×C4—792no—
544C2×C4—792no—
545C2×C4—792no—
546C2×C4—792no—
547C2×C4—792no—
548C2×C4—1582no—
549C2×C4—1582no—
550C2×C4—1588no—
551C2×C4—1581no—
552C2×C4—1581no—
553C2×C4—1581no—
554C2×C4—1581no—
555C2×C4—1581no—
556C2×C4—1581no—
557C2×C4C4⋊C4×D7912yes—
558C2×C4C22×C4×D7911yescharacteristic
559C2×C4C2×Q8×D7911yescharacteristic
560C2×C4#41×D7911yescharacteristic
561C2×C4#41×D7911yescharacteristic
562C2×C4C2×#1192962311yescharacteristic
563C2×C4C2×#1192962311yescharacteristic
564C8—22no—
565C8—22no—
566C8—1582no—
567C8—1582no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×C4⋊C4×D7911yescharacteristic
576C22#1909906911yescharacteristic
577C22#1909907111yescharacteristic
578C22AGL(1,ℤ/8)×D7911yescharacteristic
579C22#137×D7911yescharacteristic
580C22#1909936011yescharacteristic
581C22#1909936011yescharacteristic
582C4—22no—
583C4—42no—
584C4—42no—
585C4—81no—
586C4—81no—
587C4—3162no—
588C4—3162no—
589C4—794no—
590C4—6321no—
591C4—6321no—
592C4—1582no—
593C4—1582no—
594C4—1582no—
595C4C4×#1192962311yescharacteristic
596C4C4×#1192962311yescharacteristic
597C4#1909911511yescharacteristic
598C4#1909911511yescharacteristic
599C2—42no—
600C2#2376587911yescharacteristic
601C2#428×D7911yescharacteristic
602C2#2377198811yescharacteristic
603C2#2377354411yescharacteristic
604C2#2377354611yescharacteristic
605C2#2377355511yescharacteristic
606C2#2377355711yescharacteristic
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#21364
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. #23764162C2semidirect
5. #23764160C2semidirect
6. #23764157C2semidirect
7. #23764155C2semidirect
11. #23763300C2semidirect
12. #23763299C2semidirect
13. #23763296C2semidirect
14. #23763296C2semidirect
19. #19098617C4semidirect
319. C79#21364semidirect

Maximal subgroups

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

subgroupindex
C2×#190986172
#237658022
#117×Dic792
#237641622
#237641602
#237641572
#237641552
#237636012
#237635992
#237635972
#237633002
#237632992
#237632962
C79×#18542
#2136479