Finite Group Databasebeta

← #34258380

Subgroups — #34258380

Order 20224.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773465C211yesmaximal, characteristic
2#23772509C211yesmaximal, characteristic
3#23772495C211yesmaximal, characteristic, complemented
4#23762274C211yesmaximal, characteristic, complemented
5#23762260C211yesmaximal, characteristic, complemented
6#23762253C211yesmaximal, characteristic, complemented
7#23762239C211yesmaximal, characteristic, complemented
8#23762216C211yesmaximal, characteristic, complemented
9#23762215C211yesmaximal, characteristic, complemented
10#23762204C211yesmaximal, characteristic, complemented
11#23762197C211yesmaximal, characteristic, complemented
12#23762183C211yesmaximal, characteristic, complemented
13#23761835C211yesmaximal, characteristic
14#23761700C211yesmaximal, characteristic, complemented
15C79×#1481C211yesmaximal, characteristic, complemented
16C4×#11929624C2211yescharacteristic
17#19098901C411yescharacteristic
18#19098901C411yescharacteristic
19#19098845C411yescharacteristic
20#19098840C411yescharacteristic
21#19098770C411yescharacteristic
22#19098765C411yescharacteristic
23#19098723C411yescharacteristic
24#19098723C411yescharacteristic
25#19098694—21no—
26#19098694—21no—
27#19098694C2211yescharacteristic
28#19098694C2211yescharacteristic
29Q8×C79⋊C8C2211yescharacteristic
30Q8×C79⋊C8C2211yescharacteristic
31#19098687C2211yescharacteristic
32#19098687C2211yescharacteristic
33C4×#11929614C2211yescharacteristic
34C4×#11929614C2211yescharacteristic
35C4×#11929611C2211yescharacteristic
36C4×#11929611C2211yescharacteristic
37#19098628C2211yescharacteristic
38#19098628C2211yescharacteristic
39C8×C22⋊D79—21no—
40C8×C22⋊D79—21no—
41#19098624—21no—
42#19098624—21no—
43#19098620C2211yescharacteristic
44#19098616C2211yescharacteristic
45#19098616C2211yescharacteristic
46#19098615—21no—
47C4⋊C8×D79—21no—
48#19098315C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098313C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098292C2211yescharacteristic
55#19098292C2211yescharacteristic
56#19098285C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#533×C79C2211yescharacteristic
61#431×C79C2211yescharacteristic
62C2×C4⋊C8×C79C2211yescharacteristic
63Q8⋊C8×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65Q8⋊C8×C79C2211yescharacteristic
66Q8⋊C8×C79C2211yescharacteristic
67C2×#11929624C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929614C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71C2×#11929611C2×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)—21no—
89C4⋊(C79⋊C8)—21no—
90C4⋊(C79⋊C8)—21no—
91C4⋊(C79⋊C8)C2×C411yescharacteristic
92C4⋊(C79⋊C8)C2×C411yescharacteristic
93C4⋊(C79⋊C8)D411yescharacteristic
94C4⋊(C79⋊C8)D411yescharacteristic
95C4⋊(C79⋊C8)C2311yescharacteristic
96C4⋊(C79⋊C8)C2311yescharacteristic
97Dic79⋊C8—21no—
98Dic79⋊C8—21no—
99#14579085C2×C411yescharacteristic
100#14579085C2×C411yescharacteristic
101#14579085C2×C411yescharacteristic
102#14579085C2×C411yescharacteristic
103#14579084C2×C411yescharacteristic
104#14579084C2×C411yescharacteristic
105#14579081C2×C411yescharacteristic
106#14579081C2×C411yescharacteristic
107C22⋊(C79⋊C8)—21no—
108C22⋊(C79⋊C8)C2×C411yescharacteristic
109C22⋊(C79⋊C8)C2×C411yescharacteristic
110C22⋊(C79⋊C8)D411yescharacteristic
111C22⋊(C79⋊C8)D411yescharacteristic
112(C2×D79)⋊C8—21no—
113(C2×D79)⋊C8—21no—
114#14579072—21no—
115#14579072—21no—
116#14579072—21no—
117C4×C79⋊C8—21no—
118C4×C79⋊C8—21no—
119C4×C79⋊C8C2311yescharacteristic
120C4×C79⋊C8C2311yescharacteristic
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C8×Dic79—21no—
124C8×Dic79—21no—
125C22×C8×C79—21no—
126C4⋊Q8×C79C2×C411yescharacteristic
127C4⋊Q8×C79C2×C411yescharacteristic
128#125×C79—21no—
129C22⋊Q8×C79—21no—
130C22⋊Q8×C79C2×C411yescharacteristic
131C22⋊Q8×C79C2×C411yescharacteristic
132C4×Q8×C79—21no—
133C4×Q8×C79C2311yescharacteristic
134C4×Q8×C79C2311yescharacteristic
135#117×C79—21no—
136C2×C42×C79C2311yescharacteristic
137C4⋊C8×C79—21no—
138C4⋊C8×C79—21no—
139C4⋊C8×C79C2311yescharacteristic
140C4⋊C8×C79C2311yescharacteristic
141C22⋊C8×C79—21no—
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#11929614—22no—
154#11929611—22no—
155#11929611—22no—
156C2×C79⋊C8—21no—
157C2×C79⋊C8—21no—
158C2×C79⋊C8C22×C411yescharacteristic
159C2×C79⋊C8C22×C411yescharacteristic
160C2×C79⋊C8C22×C411yescharacteristic
161C2×C79⋊C8C22×C411yescharacteristic
162C2×C79⋊C8C2×D411yescharacteristic
163C2×C79⋊C8C2×D411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C2×C79⋊C8C2×D411yescharacteristic
166C8×D79—81no—
167C8×D79—81no—
168C4⋊Dic79C22×C411yescharacteristic
169C4⋊Dic79C22×C411yescharacteristic
170Dic79⋊C4—21no—
171Dic79⋊C4—21no—
172C22⋊Dic79—21no—
173(C2×D79)⋊C4—21no—
174(C2×D79)⋊C4—21no—
175C4×Dic79—21no—
176C4×Dic79—21no—
177C4×Dic79—21no—
178C2×Q8×C79—21no—
179C2×Q8×C79C22×C411yescharacteristic
180C2×Q8×C79C22×C411yescharacteristic
181C22×C4×C79C22×C411yescharacteristic
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79#4111yescharacteristic
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79—81no—
189C4⋊C4×C79—21no—
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×C79C22×C411yescharacteristic
197C4⋊C4×C79C22×C411yescharacteristic
198C22⋊C4×C79—21no—
199C22⋊C4×C79—21no—
200C42×C79—21no—
201C42×C79—21no—
202C42×C79C22×C411yescharacteristic
203C42×C79C2×D411yescharacteristic
204C42×C79#4111yescharacteristic
205C42×C79C2411yescharacteristic
206C23×C79C4×D411yescharacteristic
207C22×D79—21no—
208Q8×C79—21no—
209Q8×C79—21no—
210Q8×C79—21no—
211Q8×C79—21no—
212Q8×C79—81no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—41no—
222C2×C4×C79C4×D411yescharacteristic
223C2×C4×C79C4×D411yescharacteristic
224C2×C4×C79C4×D411yescharacteristic
225C2×C4×C79C23×C411yescharacteristic
226C2×C4×C79C22×D411yescharacteristic
227C2×C4×C79C2×#4111yescharacteristic
228C22⋊D79—24no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232D316—21no—
233D316#13112yes—
234C4×D79—24no—
235C4×D79—41no—
236C4×D79—41no—
237C4×D79—81no—
238Dic158—21no—
239Dic158#13112yes—
240C8×C79—41no—
241C8×C79—41no—
242C8×C79—41no—
243C8×C79—41no—
244C79⋊C8—21no—
245C79⋊C8—21no—
246C79⋊C8—21no—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C22×C79—21no—
255C22×C79—21no—
256C22×C79—21no—
257C22×C79C2×C4×D411yescharacteristic
258C2×D79—22no—
259C2×D79—41no—
260C2×D79—41no—
261C4×C79—21no—
262C4×C79—21no—
263C4×C79—21no—
264C4×C79—21no—
265C4×C79—41no—
266C4×C79—41no—
267C4×C79—41no—
268C4×C79—41no—
269C4×C79C2×#13111yescharacteristic
270C4×C79C2×#13111yescharacteristic
271C4×C79#57711yescharacteristic
272C4×C79#57811yescharacteristic
273Dic79—22no—
274Dic79—41no—
275Dic79—41no—
276#17921—791nomaximal, Sylow
277C2×C79—41no—
278C2×C79#292911yescharacteristic
279C2×C79#294711yescharacteristic
280C2×C79#295711yescharacteristic
281D79—42no—
282#3107—791no—
283#2988—791no—
284#2985—791no—
285#1587—791no—
286#1585—791no—
287#1584—791no—
288#1582—791no—
289#1579—791no—
290SD16⋊C8—791no—
291C8×Q16—791no—
292C8×SD16—791no—
293#1573—791no—
294#1488—791no—
295#1481D7911yescharacteristic, complemented
296#1454—791no—
297C79#1792111yescharacteristic, complemented, Sylow
298#533C2×D7911yescharacteristic
299C4×#41—791no—
300#485—791no—
301#485—791no—
302#475—791no—
303#474—791no—
304#462—791no—
305#461—791no—
306#452—791no—
307#452—791no—
308#446—791no—
309#446—791no—
310#446—1581no—
311#446—1581no—
312C8×Q8—791no—
313C8×Q8—791no—
314C8×Q8—791no—
315C8×Q8—791no—
316C4×Q16—791no—
317C4×Q16—791no—
318C4×SD16—791no—
319C4×SD16—791no—
320C8×D4—791no—
321C8×D4—791no—
322C8×D4—1581no—
323C8×D4—1581no—
324#433—1581no—
325#433—1581no—
326#432—791no—
327#431—791no—
328#431—791no—
329#431—1581no—
330#431C2×D7911yescharacteristic
331C2×C4⋊C8—1581no—
332C2×C4⋊C8C2×D7911yescharacteristic
333#335—791no—
334#335—791no—
335#334—791no—
336#334—791no—
337Q8⋊C8—791no—
338Q8⋊C8—791no—
339Q8⋊C8C2×D7911yescharacteristic
340Q8⋊C8C2×D7911yescharacteristic
341Q8⋊C8C2×D7911yescharacteristic
342Q8⋊C8C2×D7911yescharacteristic
343D4⋊C8—791no—
344D4⋊C8—791no—
345#322—791no—
346#322—791no—
347#322—791no—
348C82—791no—
349C2×#41—791no—
350C2×Q16—791no—
351C2×Q16—791no—
352C2×SD16—791no—
353C2×SD16—791no—
354C22×C8—21no—
355C22×C8—1581no—
356C22×C8—1581no—
357C4⋊Q8C4×D7911yescharacteristic
358C4⋊Q8C4×D7911yescharacteristic
359#125—21no—
360C22⋊Q8—21no—
361C22⋊Q8C4×D7911yescharacteristic
362C22⋊Q8C4×D7911yescharacteristic
363C4×Q8—21no—
364C4×Q8—791no—
365C4×Q8—791no—
366C4×Q8C22×D7911yescharacteristic
367C4×Q8C22×D7911yescharacteristic
368C4×D4—791no—
369C4×D4—791no—
370C4×D4—1581no—
371C4×D4—1581no—
372#117—21no—
373#117—791no—
374#117—1581no—
375C2×C42—1581no—
376C2×C42C22×D7911yescharacteristic
377#107—791no—
378#107—791no—
379#106—791no—
380#106—791no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—791no—
384C4⋊C8—791no—
385C4⋊C8—791no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—791no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8—1581no—
395C4⋊C8C22×D7911yescharacteristic
396C4⋊C8C22×D7911yescharacteristic
397#103—791no—
398#103—791no—
399#103—791no—
400#103—791no—
401#103—791no—
402#103—791no—
403#102—791no—
404#102—791no—
405C22⋊C8—21no—
406C22⋊C8—791no—
407C22⋊C8—791no—
408C22⋊C8—791no—
409C22⋊C8—791no—
410C22⋊C8—1581no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413#97—1581no—
414#97—1581no—
415#97—1581no—
416C4×C8—791no—
417C4×C8—791no—
418C4×C8—791no—
419C4×C8—791no—
420C4×C8—1581no—
421C4×C8—1581no—
422C4×C8—1581no—
423C4×C8—1581no—
424C4×C8C22×D7911yescharacteristic
425C4×C8C22×D7911yescharacteristic
426#41—1582no—
427#41—1582no—
428C2×Q8—21no—
429C2×Q8—791no—
430C2×Q8C2×C4×D7911yescharacteristic
431C2×Q8C2×C4×D7911yescharacteristic
432C2×D4—791no—
433C2×D4—1581no—
434C22×C4—1581no—
435C22×C4—1581no—
436C22×C4—1581no—
437C22×C4C2×C4×D7911yescharacteristic
438C22×C4D4×D7911yescharacteristic
439C22×C4#1192962311yescharacteristic
440Q16—1582no—
441Q16—1582no—
442SD16—1582no—
443SD16—1582no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—81no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—791no—
452C2×C8—791no—
453C2×C8—791no—
454C2×C8—791no—
455C2×C8—791no—
456C2×C8—791no—
457C2×C8—6321no—
458C2×C8—6321no—
459C2×C8—1581no—
460C2×C8—1581no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—791no—
469C4⋊C4—791no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4C2×C4×D7911yescharacteristic
473C4⋊C4C2×C4×D7911yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—1581no—
477C22⋊C4—1581no—
478C22⋊C4—1581no—
479C42—21no—
480C42—21no—
481C42—1581no—
482C42—1581no—
483C42—1581no—
484C42C2×C4×D7911yescharacteristic
485C42D4×D7911yescharacteristic
486C42#1192962311yescharacteristic
487C42C23×D7911yescharacteristic
488C23—1581no—
489C23(C22⋊D79)⋊C411yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—81no—
495Q8—792no—
496Q8—1581no—
497D4—792no—
498D4—1584no—
499D4—1581no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—3161no—
510C2×C4—3161no—
511C2×C4—6321no—
512C2×C4—1584no—
513C2×C4—1581no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4(C22⋊D79)⋊C411yescharacteristic
517C2×C4(C22⋊D79)⋊C411yescharacteristic
518C2×C4(C22⋊D79)⋊C411yescharacteristic
519C2×C4C22×C4×D7911yescharacteristic
520C2×C4C2×D4×D7911yescharacteristic
521C2×C4C2×#1192962311yescharacteristic
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—3161no—
527C8—3161no—
528C8—1581no—
529C8—1581no—
530C8—1581no—
531C8—1581no—
532C8—1581no—
533C8—1581no—
534C8—1581no—
535C8—1581no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—3161no—
540C22—3161no—
541C22—1582no—
542C22C2×(C22⋊D79)⋊C411yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—3161no—
552C4—3161no—
553C4—1582no—
554C4#131×D7911yescharacteristic
555C4#131×D7911yescharacteristic
556C4#1909936911yescharacteristic
557C4#1909937611yescharacteristic
558C2—41no—
559C2—3162no—
560C2#2377209711yescharacteristic
561C2#2377223711yescharacteristic
562C2#2377231311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1481C2
socleC22×C79C2×C4×D4
prime radicalC2×#11929624C2×C4
cosocleC42×C79C24
hypercenter#1481D79
hypocenterC79#17921
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
3. #23772495C2semidirect
4. #23762274C2semidirect
5. #23762260C2semidirect
6. #23762253C2semidirect
7. #23762239C2semidirect
8. #23762216C2semidirect
9. #23762215C2semidirect
10. #23762204C2semidirect
11. #23762197C2semidirect
12. #23762183C2semidirect
14. #23761700C2semidirect
15. C79×#1481C2semidirect
295. #1481D79semidirect
297. C79#17921semidirect

Maximal subgroups

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

subgroupindex
#237734652
#237725092
#237724952
#237622742
#237622602
#237622532
#237622392
#237622162
#237622152
#237622042
#237621972
#237621832
#237618352
#237617002
C79×#14812
#1792179