Finite Group Databasebeta

← #3166×D79 #35157920

Subgroups — #3166×D79 #35157920

Order 20224.

1902 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D79C211yesmaximal, characteristic
2C2×#125×D79C211yesmaximal, characteristic
3#23773953C211yesmaximal, characteristic, complemented
4#23773952C211yesmaximal, characteristic, complemented
5#23773951C214yesmaximal, complemented
6#23773950C214yesmaximal, complemented
7#23773949C211yesmaximal, characteristic, complemented
8#23773948C214yesmaximal, complemented
9#501×D79C214yesmaximal, complemented
10#499×D79C214yesmaximal, complemented
11#428×D79C214yesmaximal
12C4×M16×D79C211yesmaximal, characteristic
13C79×#3166C211yesmaximal, characteristic, complemented
14C2×M16×D79C2212yes—
15#19099120C2211yescharacteristic
16#19099119C2212yes—
17#19099118C2212yes—
18#19099117—22no—
19#19099116C2212yes—
20#19099115—22no—
21C4⋊Q8×D79C2212yes—
22C2×#14579153C2211yescharacteristic
23C2×#14579152C2212yes—
24C2×#14579151—22no—
25#125×D79—22no—
26#125×D79—22no—
27#125×D79C2212yes—
28#125×D79C2212yes—
29#125×D79C2212yes—
30#125×D79C2212yes—
31#125×D79C2212yes—
32C22⋊Q8×D79—22no—
33C4×Q8×D79—22no—
34#117×D79C2212yes—
35C2×C4⋊C4×D79—22no—
36C2×C4⋊C4×D79C2212yes—
37C2×C42×D79C2211yescharacteristic
38#19098973C2214yescomplemented
39#19098972C2214yescomplemented
40#19098971C2218yescomplemented
41#19098970C2218yescomplemented
42#19098969C2214yescomplemented
43#19098965C2214yescomplemented
44#19098964C2218yescomplemented
45#19098963C2218yescomplemented
46#19098962C2214yescomplemented
47#19098962C2214yescomplemented
48#19098606C2214yes—
49#19098605C2214yes—
50#19098604C2218yes—
51#19098603C2214yes—
52#19098602C2218yes—
53#107×D79C2218yes—
54#106×D79C2218yes—
55C4×#11929608C2211yescharacteristic
56#19098515C2212yes—
57M16×Dic79C2212yes—
58C4×#11929607C2212yes—
59#97×D79C2212yes—
60C4×C8×D79C2212yes—
61#533×C79C2211yescharacteristic
62C2×#125×C79C2211yescharacteristic
63#501×C79C2214yescomplemented
64#499×C79C2214yescomplemented
65#428×C79C2214yes—
66C4×M16×C79C2211yescharacteristic
67C2×Q8×D79—22no—
68C22×C4×D79—22no—
69C22×C4×D79D412yes—
70C22×C4×D79C2311yescharacteristic
71C2×#11929608C2312yes—
72C2×#11929607C2314yes—
73M16×D79Q8116yescomplemented
74C2×C8×D79C2314yes—
75#14579161C2312yes—
76#14579160—22no—
77#14579160C2314yes—
78#14579159—22no—
79#14579153—22no—
80#14579153C2312yes—
81#14579153C2312yes—
82#14579153C2312yes—
83#14579152—22no—
84#14579152—22no—
85#14579152C2314yes—
86#14579152C2314yes—
87#14579152C2314yes—
88#14579151—24no—
89#14579151—24no—
90#14579151—24no—
91#14579151—24no—
92#14579151—24no—
93#14579141—22no—
94#14579140—22no—
95#14579139—24no—
96#14579138—24no—
97#14579137—22no—
98Q8×Dic79—22no—
99C4×Dic158—22no—
100C4×Dic158—24no—
101#14579126—22no—
102#14579126—24no—
103#14579120C2312yes—
104#14579119C2314yes—
105#14579118—22no—
106#14579117—22no—
107C2×C4⋊Dic79—22no—
108C2×C4⋊Dic79C2312yes—
109C2×Dic79⋊C4—22no—
110C2×Dic79⋊C4—22no—
111C2×Dic79⋊C4—22no—
112C4⋊C4×D79—24no—
113C4⋊C4×D79—24no—
114C4⋊C4×D79—24no—
115C4⋊C4×D79—24no—
116C4⋊C4×D79—22no—
117C4⋊C4×D79—22no—
118C4⋊C4×D79—24no—
119C4⋊C4×D79—22no—
120C4⋊C4×D79C2314yes—
121C4⋊C4×D79C2314yes—
122C4⋊C4×D79C2314yes—
123C22⋊C4×D79—22no—
124C2×C4×Dic79D412yes—
125C2×C4×Dic79C2311yescharacteristic
126C42×D79—22no—
127C42×D79D412yes—
128C42×D79D412yes—
129C42×D79C2311yescharacteristic
130C42×D79C2311yescharacteristic
131C42×D79C2311yescharacteristic
132C42×D79C2311yescharacteristic
133#14579095C2318yes—
134#14579094C2318yes—
135#14579094C2318yes—
136#14579093C2318yes—
137#14579092C2318yes—
138#14579092C2318yes—
139#14579072C2312yes—
140#14579071C2314yes—
141C4×C79⋊C8C2312yes—
142C8×Dic79C2314yes—
143C2×M16×C79C2312yes—
144C4⋊Q8×C79C2312yes—
145#125×C79—22no—
146#125×C79C2312yes—
147#125×C79C2312yes—
148#125×C79C2312yes—
149C22⋊Q8×C79—22no—
150C4×Q8×C79—22no—
151#117×C79C2312yes—
152C2×C4⋊C4×C79—22no—
153C2×C4⋊C4×C79C2312yes—
154C2×C42×C79C2311yescharacteristic
155#107×C79C2318yes—
156#106×C79C2318yes—
157#97×C79C2312yes—
158C4×C8×C79C2312yes—
159C23×D79C2×D411yescharacteristic
160C2×Dic158—24no—
161C2×Dic158—22no—
162Q8×D79—48no—
163C22×Dic79—22no—
164C22×Dic79C2×D412yes—
165C22×Dic79C2×D411yescharacteristic
166C2×C4×D79—24no—
167C2×C4×D79—24no—
168C2×C4×D79—24no—
169C2×C4×D79—44no—
170C2×C4×D79—48no—
171C2×C4×D79C2×D412yes—
172C2×C4×D79C2×D412yes—
173C2×C4×D79C2×D412yes—
174C2×C4×D79C2×D412yes—
175C2×C4×D79C2×D411yescharacteristic
176C2×C4×D79C2×D411yescharacteristic
177C2×C4×D79C2×Q818yes—
178C2×C4×D79C2411yescharacteristic
179C2×C4×D79C2411yescharacteristic
180#11929608C2×Q818yescomplemented
181#11929607C2×Q8116yescomplemented
182C2×C79⋊C8C2414yes—
183C8×D79C2×Q8116yes—
184C4⋊Dic79—24no—
185C4⋊Dic79—24no—
186C4⋊Dic79—24no—
187C4⋊Dic79—22no—
188C4⋊Dic79—22no—
189C4⋊Dic79C2414yes—
190C4⋊Dic79C2414yes—
191Dic79⋊C4—24no—
192Dic79⋊C4—24no—
193Dic79⋊C4—24no—
194Dic79⋊C4—24no—
195Dic79⋊C4—24no—
196Dic79⋊C4—24no—
197Dic79⋊C4—24no—
198Dic79⋊C4—24no—
199Dic79⋊C4—24no—
200Dic79⋊C4—24no—
201C22⋊Dic79—22no—
202(C2×D79)⋊C4—24no—
203C4×Dic79—24no—
204C4×Dic79—22no—
205C4×Dic79C2×D412yes—
206C4×Dic79C2×D412yes—
207C4×Dic79C2×D412yes—
208C4×Dic79C2×D412yes—
209C4×Dic79C2×D412yes—
210C4×Dic79C2411yescharacteristic
211C4×Dic79C2411yescharacteristic
212C2×Q8×C79—22no—
213C22×C4×C79—22no—
214C22×C4×C79C2×D412yes—
215C22×C4×C79C2411yescharacteristic
216M16×C79C2×Q818yescomplemented
217C2×C8×C79C2414yes—
218C4⋊C4×C79—24no—
219C4⋊C4×C79—24no—
220C4⋊C4×C79—22no—
221C4⋊C4×C79—24no—
222C4⋊C4×C79—22no—
223C4⋊C4×C79C2414yes—
224C4⋊C4×C79C2414yes—
225C22⋊C4×C79—22no—
226C42×C79—22no—
227C42×C79C2×D412yes—
228C42×C79C2411yescharacteristic
229C42×C79C2411yescharacteristic
230C23×C79C22×D411yescharacteristic
231C22×D79—24no—
232C22×D79C4⋊Q814yes—
233C22×D79C22×D411yescharacteristic
234C22×D79C22×D411yescharacteristic
235Q8×C79—44no—
236C2×C4×C79—24no—
237C2×C4×C79—24no—
238C2×C4×C79—42no—
239C2×C4×C79—44no—
240C2×C4×C79C22×D412yes—
241C2×C4×C79C22×D412yes—
242C2×C4×C79C22×D411yescharacteristic
243C2×C4×C79C22×Q814yes—
244C2×C4×C79C2511yescharacteristic
245C2×Dic79—24no—
246C2×Dic79—24no—
247C2×Dic79—42no—
248C2×Dic79—44no—
249C2×Dic79C4⋊Q814yes—
250C2×Dic79C22×D412yes—
251C2×Dic79C22×D412yes—
252C2×Dic79C22×D411yescharacteristic
253C2×Dic79C22×D411yescharacteristic
254C4×D79—24no—
255C4×D79—24no—
256C4×D79—42no—
257C4×D79—42no—
258C4×D79—48no—
259C4×D79—48no—
260C4×D79—48no—
261C4×D79C4⋊Q814yes—
262C4×D79C4⋊Q814yes—
263C4×D79C22×Q814yes—
264C4×D79C22×Q814yes—
265Dic158—48no—
266Dic158—44no—
267C8×C79C22×Q818yes—
268C79⋊C8C22×Q818yes—
269C22×C79—22no—
270C22×C79C2×C4⋊Q812yes—
271C22×C79C23×D411yescharacteristic
272C2×D79—22no—
273C2×D79—28no—
274C2×D79C2×C4⋊Q812yes—
275C2×D79C2×C4⋊Q812yes—
276C2×D79#57514yes—
277C4×C79—22no—
278C4×C79—22no—
279C4×C79—44no—
280C4×C79—44no—
281C4×C79C2×C4⋊Q812yes—
282C4×C79C23×Q812yes—
283Dic79—22no—
284Dic79—22no—
285Dic79—44no—
286Dic79—44no—
287Dic79C2×C4⋊Q812yes—
288Dic79C2×C4⋊Q812yes—
289C2×#3166—791nomaximal, Sylow
290C2×C79—22no—
291C2×C79C22×C4⋊Q811yescharacteristic
292C2×C79C2×#57512yes—
293D79—22no—
294D79#316614yesdirect product factor
295C2×#533—791no—
296C22×#125—791no—
297#3166—794no—
298#3166—794no—
299#3166—794no—
300#3166—791no—
301#3166—791no—
302#3166—791no—
303#3166D7911yescharacteristic, direct product factor
304C2×#501—794no—
305C2×#499—794no—
306C2×#428—794no—
307C2×C4×M16—791no—
308C79C2×#316611yescharacteristic, complemented, Sylow
309C22×M16—792no—
310#533—792no—
311#533—792no—
312#533—792no—
313#533—791no—
314#533—1582no—
315#533—1582no—
316#533C2×D7911yescharacteristic
317C2×C4⋊Q8—792no—
318C2×#125—792no—
319C2×#125—792no—
320C2×#125—792no—
321C2×#125—792no—
322C2×#125—792no—
323C2×#125—792no—
324C2×#125—791no—
325C2×#125—1582no—
326C2×#125—1582no—
327C2×#125—1582no—
328C2×#125C2×D7911yescharacteristic
329C2×C22⋊Q8—1582no—
330C2×C4×Q8—1582no—
331C2×#117—792no—
332C22×C4⋊C4—792no—
333C22×C4⋊C4—1582no—
334C22×C42—791no—
335#501—794no—
336#501—794no—
337#501—798no—
338#501—798no—
339#501—794no—
340#501C2×D7914yescomplemented
341#499—794no—
342#499—794no—
343#499—798no—
344#499—798no—
345#499—794no—
346#499C2×D7914yescomplemented
347#428—794no—
348#428—794no—
349#428—794no—
350#428—798no—
351#428—798no—
352#428C2×D7914yes—
353C2×#107—798no—
354C2×#106—798no—
355C4×M16—792no—
356C4×M16—791no—
357C4×M16—792no—
358C4×M16—792no—
359C4×M16C2×D7911yescharacteristic
360C2×#97—792no—
361C2×C4×C8—792no—
362C22×Q8—1582no—
363C23×C4—792no—
364C23×C4—791no—
365C23×C4—1582no—
366C2×M16—792no—
367C2×M16—7916no—
368C2×M16—794no—
369C2×M16C22×D7912yes—
370C22×C8—794no—
371C4⋊Q8—794no—
372C4⋊Q8—792no—
373C4⋊Q8—1582no—
374C4⋊Q8—1582no—
375C4⋊Q8C22×D7912yes—
376#125—22no—
377#125—794no—
378#125—794no—
379#125—794no—
380#125—792no—
381#125—792no—
382#125—792no—
383#125—1584no—
384#125—1584no—
385#125—1584no—
386#125—1584no—
387#125—1582no—
388#125—1582no—
389#125—1584no—
390#125—1582no—
391#125C22×D7912yes—
392#125C22×D7912yes—
393#125C22×D7912yes—
394C22⋊Q8—22no—
395C22⋊Q8—1584no—
396C22⋊Q8—1584no—
397C22⋊Q8—1582no—
398C22⋊Q8—1582no—
399C22⋊Q8—1582no—
400C4×Q8—22no—
401C4×Q8—1584no—
402C4×Q8—1584no—
403C4×Q8—1582no—
404C4×Q8—1582no—
405C4×Q8—1582no—
406#117—792no—
407#117—794no—
408#117—1582no—
409#117—1582no—
410#117C22×D7912yes—
411C2×C4⋊C4—22no—
412C2×C4⋊C4—792no—
413C2×C4⋊C4—794no—
414C2×C4⋊C4—794no—
415C2×C4⋊C4—794no—
416C2×C4⋊C4—1582no—
417C2×C4⋊C4—1582no—
418C2×C4⋊C4—1584no—
419C2×C4⋊C4—1584no—
420C2×C4⋊C4—1584no—
421C2×C4⋊C4—1584no—
422C2×C4⋊C4—1584no—
423C2×C4⋊C4—1582no—
424C2×C4⋊C4—1582no—
425C2×C4⋊C4—1582no—
426C2×C4⋊C4—1582no—
427C2×C4⋊C4—1582no—
428C2×C4⋊C4C22×D7912yes—
429C2×C22⋊C4—1582no—
430C2×C42—792no—
431C2×C42—791no—
432C2×C42—792no—
433C2×C42—792no—
434C2×C42—791no—
435C2×C42—791no—
436C2×C42—791no—
437C2×C42—791no—
438C2×C42—1582no—
439C2×C42C22×D7911yescharacteristic
440#107—798no—
441#107—798no—
442#107—798no—
443#107C22×D7918yes—
444#106—798no—
445#106—798no—
446#106—798no—
447#106C22×D7918yes—
448#97—794no—
449#97—792no—
450#97C22×D7912yes—
451C4×C8—794no—
452C4×C8—792no—
453C4×C8C22×D7912yes—
454C24—791no—
455C2×Q8—22no—
456C2×Q8—3168no—
457C2×Q8—1584no—
458C2×Q8—1582no—
459C22×C4—22no—
460C22×C4—3168no—
461C22×C4—3164no—
462C22×C4—792no—
463C22×C4—792no—
464C22×C4—792no—
465C22×C4—791no—
466C22×C4—798no—
467C22×C4—791no—
468C22×C4—791no—
469C22×C4—791no—
470C22×C4—791no—
471C22×C4—792no—
472C22×C4—792no—
473C22×C4—1584no—
474C22×C4—1584no—
475C22×C4—1584no—
476C22×C4—1582no—
477C22×C4D4×D7912yes—
478C22×C4C23×D7911yescharacteristic
479M16—798no—
480M16—7916no—
481M16Q8×D7918yescomplemented
482C2×C8—794no—
483C2×C8—7916no—
484C2×C8C23×D7914yes—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—794no—
491C4⋊C4—794no—
492C4⋊C4—1584no—
493C4⋊C4—1584no—
494C4⋊C4—1584no—
495C4⋊C4—1584no—
496C4⋊C4—1584no—
497C4⋊C4—1584no—
498C4⋊C4—1584no—
499C4⋊C4—1584no—
500C4⋊C4—1584no—
501C4⋊C4—1584no—
502C4⋊C4—1584no—
503C4⋊C4—1584no—
504C4⋊C4—1584no—
505C4⋊C4—1582no—
506C4⋊C4—1582no—
507C4⋊C4C23×D7914yes—
508C4⋊C4C23×D7914yes—
509C22⋊C4—22no—
510C22⋊C4—1584no—
511C22⋊C4—1582no—
512C42—22no—
513C42—791no—
514C42—791no—
515C42—792no—
516C42—792no—
517C42—792no—
518C42—792no—
519C42—792no—
520C42—1584no—
521C42—1582no—
522C42D4×D7912yes—
523C42C23×D7911yescharacteristic
524C42C23×D7911yescharacteristic
525C23—791no—
526C23—794no—
527C23—791no—
528C23—1584no—
529C23C2×D4×D7911yescharacteristic
530Q8—44no—
531Q8—3164no—
532Q8—3168no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—3162no—
538C2×C4—3162no—
539C2×C4—3168no—
540C2×C4—3168no—
541C2×C4—3168no—
542C2×C4—3164no—
543C2×C4—3162no—
544C2×C4—794no—
545C2×C4—791no—
546C2×C4—791no—
547C2×C4—794no—
548C2×C4—794no—
549C2×C4—794no—
550C2×C4—794no—
551C2×C4—792no—
552C2×C4—792no—
553C2×C4—1584no—
554C2×C4—1584no—
555C2×C4—1584no—
556C2×C4—1584no—
557C2×C4C2×D4×D7912yes—
558C2×C4C2×D4×D7912yes—
559C2×C4C2×D4×D7911yescharacteristic
560C2×C4C2×Q8×D7914yes—
561C2×C4C24×D7911yescharacteristic
562C8—798no—
563C8C2×Q8×D7918yes—
564C22—22no—
565C22—794no—
566C22—792no—
567C22—792no—
568C22—1588no—
569C22—1582no—
570C22C4⋊Q8×D7912yes—
571C22C22×D4×D7911yescharacteristic
572C4—22no—
573C4—22no—
574C4—44no—
575C4—44no—
576C4—3164no—
577C4—3164no—
578C4—792no—
579C4—792no—
580C4—1582no—
581C4—1582no—
582C4C4⋊Q8×D7912yes—
583C4C22×Q8×D7912yes—
584C2—22no—
585C2—794no—
586C2—1582no—
587C2C2×C4⋊Q8×D7911yescharacteristic
588C2#575×D7912yes—
subgroupG/H
centerC22C22×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC79×#3166C2
socleC22×C79C23×D4
prime radicalC23×D79C2×D4
hypercenter#3166D79
hypocenterC79C2×#3166
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3166

Complements

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

NH ≅ G/Ntype
3. #23773953C2semidirect
4. #23773952C2semidirect
5. #23773951C2semidirect
6. #23773950C2semidirect
7. #23773949C2semidirect
8. #23773948C2semidirect
9. #501×D79C2semidirect
10. #499×D79C2semidirect
13. C79×#3166C2semidirect
38. #19098973C22semidirect
39. #19098972C22semidirect
40. #19098971C22semidirect
41. #19098970C22semidirect
42. #19098969C22semidirect
43. #19098965C22semidirect
44. #19098964C22semidirect
45. #19098963C22semidirect
46. #19098962C22semidirect
47. #19098962C22semidirect
63. #501×C79C22semidirect
64. #499×C79C22semidirect
73. M16×D79Q8semidirect
180. #11929608C2×Q8semidirect
181. #11929607C2×Q8semidirect
216. M16×C79C2×Q8semidirect
294. D79#3166direct (N × H)
303. #3166D79direct (N × H)
308. C79C2×#3166semidirect
340. #501C2×D79semidirect
346. #499C2×D79semidirect
481. M16Q8×D79semidirect

Maximal subgroups

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

subgroupindex
#533×D792
C2×#125×D792
#237739532
#237739522
#237739512
#237739502
#237739492
#237739482
#501×D792
#499×D792
#428×D792
C4×M16×D792
C79×#31662
C2×#316679