Finite Group Databasebeta

← #34258222

Subgroups — #34258222

Order 20224.

626 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773412C211yesmaximal, characteristic
2C8×#11929624C211yesmaximal, characteristic, complemented
3#23762276C211yesmaximal, characteristic, complemented
4#23762269C211yesmaximal, characteristic, complemented
5#23762262C211yesmaximal, characteristic, complemented
6#23762255C211yesmaximal, characteristic, complemented
7#23762222C211yesmaximal, characteristic, complemented
8#23762217C211yesmaximal, characteristic, complemented
9#23762212C211yesmaximal, characteristic, complemented
10#23762210C211yesmaximal, characteristic, complemented
11#23762181C211yesmaximal, characteristic, complemented
12#23761832C211yesmaximal, characteristic, complemented
13#23761808C211yesmaximal, characteristic, complemented
14#23761692C211yesmaximal, characteristic, complemented
15C79×#2966C211yesmaximal, characteristic, complemented
16C4×#11929624C2211yescharacteristic
17#19098901C411yescharacteristic
18#19098884C411yescharacteristic
19#19098867C411yescharacteristic
20#19098860C411yescharacteristic
21#19098782C411yescharacteristic
22#19098775C411yescharacteristic
23#19098723C411yescharacteristic
24#19098706C411yescharacteristic
25C8×Dic158C2211yescharacteristic
26C8×Dic158C2211yescharacteristic
27C4×#11929614C2211yescharacteristic
28C4×#11929611C2211yescharacteristic
29C4×#11929612C2211yescharacteristic
30C4×#11929609C2211yescharacteristic
31C8×D316C2211yescharacteristic
32C8×D316C2211yescharacteristic
33C8×C22⋊D79—21no—
34C8×C22⋊D79—21no—
35#19098616C2211yescharacteristic
36#19098616C2211yescharacteristic
37#19098616C2211yescharacteristic
38#19098614—21no—
39C4×C8×D79—21no—
40#19098315C2211yescharacteristic
41#19098315C2211yescharacteristic
42#19098313C2211yescharacteristic
43#19098313C2211yescharacteristic
44Q8⋊(C79⋊C8)C2211yescharacteristic
45Q8⋊(C79⋊C8)C2211yescharacteristic
46#19098293C2211yescharacteristic
47#19098293C2211yescharacteristic
48D4⋊(C79⋊C8)C2211yescharacteristic
49D4⋊(C79⋊C8)C2211yescharacteristic
50#19098290C2211yescharacteristic
51#19098290C2211yescharacteristic
52#19098285C2211yescharacteristic
53#19098285C2211yescharacteristic
54#19098285C2211yescharacteristic
55#19098285C2211yescharacteristic
56#529×C79C2211yescharacteristic
57#446×C79C2211yescharacteristic
58C8×Q8×C79C2211yescharacteristic
59#436×C79C2211yescharacteristic
60#435×C79—21no—
61C8×D4×C79—21no—
62C8×D4×C79C2211yescharacteristic
63#433×C79—21no—
64#432×C79C2211yescharacteristic
65#405×C79—21no—
66C2×C4×C8×C79C2211yescharacteristic
67C2×#11929624C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929612C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71C2×#11929609C2×C411yescharacteristic
72C2×C8×D79—21no—
73C2×C8×D79—21no—
74C4×Dic158C2×C411yescharacteristic
75C4×Dic158C2311yescharacteristic
76C4×D316C2×C411yescharacteristic
77C4×D316C2311yescharacteristic
78C4×C22⋊D79—21no—
79C4×C22⋊D79—21no—
80#14579120C2×C411yescharacteristic
81#14579118—21no—
82C42×D79—21no—
83#14579094C2×C411yescharacteristic
84#14579094C2×C411yescharacteristic
85#14579092C2×C411yescharacteristic
86#14579092C2×C411yescharacteristic
87C4⋊(C79⋊C8)—21no—
88C4⋊(C79⋊C8)C2×C411yescharacteristic
89C4⋊(C79⋊C8)C2×C411yescharacteristic
90C4⋊(C79⋊C8)C2311yescharacteristic
91C4⋊(C79⋊C8)C2311yescharacteristic
92Dic79⋊C8—21no—
93Dic79⋊C8—21no—
94#14579085C2×C411yescharacteristic
95#14579085C2×C411yescharacteristic
96#14579084C2×C411yescharacteristic
97#14579084C2×C411yescharacteristic
98#14579082C2×C411yescharacteristic
99#14579082C2×C411yescharacteristic
100#14579081C2×C411yescharacteristic
101#14579081C2×C411yescharacteristic
102C22⋊(C79⋊C8)—21no—
103C22⋊(C79⋊C8)—21no—
104C22⋊(C79⋊C8)C2×C411yescharacteristic
105C22⋊(C79⋊C8)C2×C411yescharacteristic
106(C2×D79)⋊C8—21no—
107(C2×D79)⋊C8—21no—
108C4×C79⋊C8—21no—
109C4×C79⋊C8C2311yescharacteristic
110C4×C79⋊C8C2311yescharacteristic
111C4×C79⋊C8C2311yescharacteristic
112C4×C79⋊C8C2311yescharacteristic
113C8×Dic79—21no—
114C8×Dic79—21no—
115C2×M16×C79—21no—
116C22×C8×C79—21no—
117C22×C8×C79D411yescharacteristic
118C22×C8×C79D411yescharacteristic
119#126×C79—21no—
120#125×C79C2×C411yescharacteristic
121#124×C79C2×C411yescharacteristic
122#123×C79—21no—
123C22⋊Q8×C79C2×C411yescharacteristic
124#121×C79C2×C411yescharacteristic
125C4×Q8×C79C2311yescharacteristic
126C4×D4×C79—21no—
127C4×D4×C79C2311yescharacteristic
128#117×C79—21no—
129C2×C42×C79C2311yescharacteristic
130C4⋊C8×C79—21no—
131C4⋊C8×C79—21no—
132C4⋊C8×C79C2311yescharacteristic
133C4⋊C8×C79C2311yescharacteristic
134C22⋊C8×C79—21no—
135C22⋊C8×C79—21no—
136C22⋊C8×C79—21no—
137#97×C79—21no—
138#97×C79—21no—
139C4×C8×C79—21no—
140C4×C8×C79D411yescharacteristic
141C4×C8×C79D411yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929624—22no—
145#11929624—22no—
146C2×Dic158C22×C411yescharacteristic
147C2×D316C22×C411yescharacteristic
148C2×C22⋊D79—21no—
149C2×C4×D79—21no—
150C2×C4×D79—21no—
151C2×C4×D79—21no—
152#11929614—22no—
153#11929612—22no—
154#11929611—22no—
155#11929609—22no—
156C2×C79⋊C8—21no—
157C2×C79⋊C8—21no—
158C2×C79⋊C8—21no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8C22×C411yescharacteristic
161C2×C79⋊C8C22×C411yescharacteristic
162C2×C79⋊C8C22×C411yescharacteristic
163C2×C79⋊C8C22×C411yescharacteristic
164C8×D79—42no—
165C8×D79—42no—
166C4⋊Dic79C22×C411yescharacteristic
167C4⋊Dic79C22×C411yescharacteristic
168Dic79⋊C4—21no—
169Dic79⋊C4—21no—
170C22⋊Dic79—21no—
171(C2×D79)⋊C4—21no—
172(C2×D79)⋊C4—21no—
173C4×Dic79—21no—
174C4×Dic79—21no—
175C4×Dic79—21no—
176C2×Q8×C79C22×C411yescharacteristic
177C2×D4×C79—21no—
178C2×D4×C79C22×C411yescharacteristic
179C22×C4×C79—21no—
180C22×C4×C79C22×C411yescharacteristic
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79#4111yescharacteristic
183M16×C79—42no—
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79—41no—
187C2×C8×C79—41no—
188C2×C8×C79—42no—
189C2×C8×C79C2×D411yescharacteristic
190C2×C8×C79C2×D411yescharacteristic
191C2×C8×C79C2×D411yescharacteristic
192C2×C8×C79C2×D411yescharacteristic
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79C22×C411yescharacteristic
198C4⋊C4×C79C22×C411yescharacteristic
199C22⋊C4×C79—21no—
200C22⋊C4×C79—21no—
201C22⋊C4×C79—21no—
202C22⋊C4×C79—21no—
203C22⋊C4×C79—21no—
204C42×C79—21no—
205C42×C79C22×C411yescharacteristic
206C42×C79C2×D411yescharacteristic
207C42×C79#4111yescharacteristic
208C42×C79C2411yescharacteristic
209C23×C79—21no—
210C23×C79C4×D411yescharacteristic
211C22×D79—21no—
212Q8×C79—21no—
213Q8×C79—21no—
214D4×C79—21no—
215D4×C79—21no—
216D4×C79—42no—
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—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79C4×D411yescharacteristic
228C2×C4×C79C4×D411yescharacteristic
229C2×C4×C79C4×D411yescharacteristic
230C2×C4×C79C23×C411yescharacteristic
231C2×C4×C79C22×D411yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C22⋊D79—24no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237D316—21no—
238D316#13112yes—
239C4×D79—24no—
240C4×D79—42no—
241C4×D79—41no—
242C4×D79—41no—
243Dic158—21no—
244Dic158#13112yes—
245C8×C79—21no—
246C8×C79—21no—
247C8×C79—21no—
248C8×C79—21no—
249C8×C79—41no—
250C8×C79—41no—
251C79⋊C8—21no—
252C79⋊C8—21no—
253C79⋊C8—21no—
254C79⋊C8—21no—
255C79⋊C8—41no—
256C79⋊C8—41no—
257C79⋊C8—41no—
258C79⋊C8—41no—
259C22×C79—21no—
260C22×C79—21no—
261C22×C79—21no—
262C22×C79—41no—
263C22×C79—41no—
264C22×C79—41no—
265C22×C79C2×C4×D411yescharacteristic
266C2×D79—22no—
267C2×D79—41no—
268C2×D79—41no—
269C4×C79—21no—
270C4×C79—21no—
271C4×C79—21no—
272C4×C79—21no—
273C4×C79—41no—
274C4×C79—41no—
275C4×C79—41no—
276C4×C79C2×#13111yescharacteristic
277C4×C79C2×#13111yescharacteristic
278C4×C79#57511yescharacteristic
279C4×C79#57511yescharacteristic
280Dic79—22no—
281Dic79—41no—
282Dic79—41no—
283#17909—791nomaximal, Sylow
284C2×C79—41no—
285C2×C79—81no—
286C2×C79#292911yescharacteristic
287C2×C79#294311yescharacteristic
288C2×C79#295511yescharacteristic
289D79—42no—
290#3102—791no—
291#2966D7911yescharacteristic, complemented
292C8×#41—791no—
293#1588—791no—
294#1587—791no—
295#1586—791no—
296#1585—791no—
297#1580—791no—
298#1579—791no—
299SD16⋊C8—791no—
300SD16⋊C8—791no—
301#1572—791no—
302#1488—791no—
303#1482—791no—
304#1452—791no—
305C79#1790911yescharacteristic, complemented, Sylow
306#529C2×D7911yescharacteristic
307C4×#41—791no—
308#485—791no—
309#482—791no—
310#479—791no—
311#478—791no—
312#464—791no—
313#463—791no—
314#452—791no—
315#449—791no—
316#446C2×D7911yescharacteristic
317C8×Q8—791no—
318C8×Q8—791no—
319C8×Q8C2×D7911yescharacteristic
320C4×Q16—791no—
321C4×SD16—791no—
322C4×SD16—791no—
323C4×D8—791no—
324#436C2×D7911yescharacteristic
325#435—21no—
326C8×D4—21no—
327C8×D4—791no—
328C8×D4—791no—
329C8×D4—1581no—
330C8×D4—1581no—
331C8×D4C2×D7911yescharacteristic
332#433—21no—
333#432C2×D7911yescharacteristic
334#431—791no—
335#431—791no—
336#431—791no—
337#431—1581no—
338#405—21no—
339C2×C4×C8—1581no—
340C2×C4×C8C2×D7911yescharacteristic
341#335—791no—
342#335—791no—
343#334—791no—
344#334—791no—
345Q8⋊C8—791no—
346Q8⋊C8—791no—
347Q8⋊C8—791no—
348Q8⋊C8—791no—
349D4⋊C8—791no—
350D4⋊C8—791no—
351D4⋊C8—791no—
352D4⋊C8—791no—
353#322—791no—
354#322—791no—
355#322—791no—
356#322—791no—
357C2×#41—791no—
358C2×Q16—791no—
359C2×SD16—791no—
360C2×SD16—791no—
361C2×D8—791no—
362C2×M16—21no—
363C22×C8—21no—
364C22×C8—1581no—
365C22×C8—1581no—
366C22×C8C22⋊D7911yescharacteristic
367C22×C8C22⋊D7911yescharacteristic
368#126—21no—
369#125C4×D7911yescharacteristic
370#124C4×D7911yescharacteristic
371#123—21no—
372C22⋊Q8C4×D7911yescharacteristic
373#121C4×D7911yescharacteristic
374C4×Q8—791no—
375C4×Q8—791no—
376C4×Q8C22×D7911yescharacteristic
377C4×D4—21no—
378C4×D4—791no—
379C4×D4—791no—
380C4×D4—1581no—
381C4×D4—1581no—
382C4×D4C22×D7911yescharacteristic
383#117—21no—
384#117—791no—
385#117—1581no—
386C2×C42—1581no—
387C2×C42C22×D7911yescharacteristic
388#107—791no—
389#107—791no—
390#106—791no—
391#106—791no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8C22×D7911yescharacteristic
402C4⋊C8C22×D7911yescharacteristic
403#103—791no—
404#103—791no—
405#103—791no—
406#103—791no—
407#102—791no—
408#102—791no—
409#102—791no—
410#102—791no—
411C22⋊C8—21no—
412C22⋊C8—21no—
413C22⋊C8—21no—
414C22⋊C8—791no—
415C22⋊C8—791no—
416C22⋊C8—1581no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419C22⋊C8—1581no—
420#97—21no—
421#97—21no—
422C4×C8—21no—
423C4×C8—791no—
424C4×C8—791no—
425C4×C8—791no—
426C4×C8—791no—
427C4×C8—1581no—
428C4×C8—1581no—
429C4×C8—1581no—
430C4×C8C22⋊D7911yescharacteristic
431C4×C8C22⋊D7911yescharacteristic
432C4×C8C22×D7911yescharacteristic
433C4×C8C22×D7911yescharacteristic
434#41—1582no—
435#41—1582no—
436C2×Q8—791no—
437C2×Q8C2×C4×D7911yescharacteristic
438C2×D4—21no—
439C2×D4—791no—
440C2×D4—1581no—
441C2×D4C2×C4×D7911yescharacteristic
442C22×C4—21no—
443C22×C4—1581no—
444C22×C4—1581no—
445C22×C4—1581no—
446C22×C4C2×C4×D7911yescharacteristic
447C22×C4C2×C22⋊D7911yescharacteristic
448C22×C4#1192962411yescharacteristic
449Q16—1582no—
450SD16—1582no—
451SD16—1582no—
452D8—1582no—
453M16—42no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—42no—
459C2×C8—3162no—
460C2×C8—3162no—
461C2×C8—791no—
462C2×C8—791no—
463C2×C8—791no—
464C2×C8—791no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8—1581no—
468C2×C8—1581no—
469C2×C8C2×C22⋊D7911yescharacteristic
470C2×C8C2×C22⋊D7911yescharacteristic
471C2×C8C2×C22⋊D7911yescharacteristic
472C2×C8C2×C22⋊D7911yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—791no—
478C4⋊C4—791no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4C2×C4×D7911yescharacteristic
482C4⋊C4C2×C4×D7911yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—1581no—
489C22⋊C4—1581no—
490C22⋊C4—1581no—
491C42—21no—
492C42—1581no—
493C42—1581no—
494C42—1581no—
495C42C2×C4×D7911yescharacteristic
496C42C2×C22⋊D7911yescharacteristic
497C42#1192962411yescharacteristic
498C42C23×D7911yescharacteristic
499C23—21no—
500C23—1581no—
501C23C4×C22⋊D7911yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—792no—
505Q8—1581no—
506D4—21no—
507D4—21no—
508D4—42no—
509D4—792no—
510D4—1584no—
511D4—1581no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—3162no—
523C2×C4—3161no—
524C2×C4—3161no—
525C2×C4—1584no—
526C2×C4—1581no—
527C2×C4—1581no—
528C2×C4—1581no—
529C2×C4C4×C22⋊D7911yescharacteristic
530C2×C4C4×C22⋊D7911yescharacteristic
531C2×C4C4×C22⋊D7911yescharacteristic
532C2×C4C22×C4×D7911yescharacteristic
533C2×C4C22×C22⋊D7911yescharacteristic
534C2×C4C2×#1192962411yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—3161no—
542C8—3161no—
543C8—3161no—
544C8—3161no—
545C8—1581no—
546C8—1581no—
547C8—1581no—
548C8—1581no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—3161no—
556C22—3161no—
557C22—1582no—
558C22C2×C4×C22⋊D7911yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—3161no—
567C4—3161no—
568C4—1582no—
569C4#131×D7911yescharacteristic
570C4#131×D7911yescharacteristic
571C4#1909936111yescharacteristic
572C4#1909936111yescharacteristic
573C2—41no—
574C2—81no—
575C2—3162no—
576C2#2377209211yescharacteristic
577C2#2377219811yescharacteristic
578C2#2377230111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#2966C2
socleC22×C79C2×C4×D4
prime radical#19098706C4
cosocleC42×C79C24
hypercenter#2966D79
hypocenterC79#17909
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #2966

Complements

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

NH ≅ G/Ntype
2. C8×#11929624C2semidirect
3. #23762276C2semidirect
4. #23762269C2semidirect
5. #23762262C2semidirect
6. #23762255C2semidirect
7. #23762222C2semidirect
8. #23762217C2semidirect
9. #23762212C2semidirect
10. #23762210C2semidirect
11. #23762181C2semidirect
12. #23761832C2semidirect
13. #23761808C2semidirect
14. #23761692C2semidirect
15. C79×#2966C2semidirect
291. #2966D79semidirect
305. C79#17909semidirect

Maximal subgroups

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

subgroupindex
#237734122
C8×#119296242
#237622762
#237622692
#237622622
#237622552
#237622222
#237622172
#237622122
#237622102
#237621812
#237618322
#237618082
#237616922
C79×#29662
#1790979