Finite Group Databasebeta

← #34259663

Subgroups — #34259663

Order 20224.

566 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773439C211yesmaximal, characteristic
2#23772501C211yesmaximal, characteristic
3#23762307C211yesmaximal, characteristic, complemented
4#23762295C211yesmaximal, characteristic, complemented
5#23762271C211yesmaximal, characteristic, complemented
6#23762269C211yesmaximal, characteristic, complemented
7#23762250C211yesmaximal, characteristic, complemented
8#23762248C211yesmaximal, characteristic, complemented
9#23762217C211yesmaximal, characteristic, complemented
10C8×#11929614C211yesmaximal, characteristic, complemented
11#23762181C211yesmaximal, characteristic
12#23761872C211yesmaximal, characteristic
13#23761795C211yesmaximal, characteristic
14#23761700C211yesmaximal, characteristic
15C79×#2984C211yesmaximal, characteristic
16#19099120C2211yescharacteristic
17#19098898C411yescharacteristic
18#19098898C411yescharacteristic
19#19098858C411yescharacteristic
20#19098858C411yescharacteristic
21#19098773C411yescharacteristic
22#19098773C411yescharacteristic
23#19098719C411yescharacteristic
24#19098719C411yescharacteristic
25#19098693C2211yescharacteristic
26#19098693C2211yescharacteristic
27#19098690—21no—
28C8×Dic158C2211yescharacteristic
29C8×Dic158C2211yescharacteristic
30#19098687—21no—
31C4×#11929614C2211yescharacteristic
32C4×#11929614C2211yescharacteristic
33C4×#11929614C2211yescharacteristic
34C4×#11929614C2211yescharacteristic
35#19098621—21no—
36#19098621—21no—
37#19098616C2211yescharacteristic
38#19098616C2211yescharacteristic
39#19098616C2211yescharacteristic
40#19098315C2211yescharacteristic
41#19098315C2211yescharacteristic
42#19098313C2211yescharacteristic
43#19098313C2211yescharacteristic
44Q8⋊(C79⋊C8)C2211yescharacteristic
45Q8⋊(C79⋊C8)C2211yescharacteristic
46Q8⋊(C79⋊C8)C2211yescharacteristic
47Q8⋊(C79⋊C8)C2211yescharacteristic
48#19098293C2211yescharacteristic
49#19098293C2211yescharacteristic
50#19098293C2211yescharacteristic
51#19098293C2211yescharacteristic
52#19098285C2211yescharacteristic
53#19098285C2211yescharacteristic
54#19098284C2211yescharacteristic
55C8×C79⋊C8C2211yescharacteristic
56C2×C4×Q8×C79C2211yescharacteristic
57#446×C79—21no—
58#446×C79—21no—
59#446×C79C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C8×Q8×C79C2211yescharacteristic
62C8×Q8×C79C2211yescharacteristic
63#432×C79—21no—
64#432×C79C2211yescharacteristic
65#431×C79—21no—
66#431×C79C2211yescharacteristic
67C2×#11929614C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929614C2×C411yescharacteristic
70C2×#11929614C2×C411yescharacteristic
71#14579161C2×C411yescharacteristic
72#14579161C2×C411yescharacteristic
73#14579153—21no—
74#14579140—21no—
75#14579140C2×C411yescharacteristic
76#14579140C2×C411yescharacteristic
77C4×Dic158—21no—
78C4×Dic158C2311yescharacteristic
79C4×Dic158C2311yescharacteristic
80#14579120—21no—
81#14579094C2×C411yescharacteristic
82#14579094C2×C411yescharacteristic
83#14579094C2×C411yescharacteristic
84#14579094C2×C411yescharacteristic
85C4⋊(C79⋊C8)—21no—
86C4⋊(C79⋊C8)C2×C411yescharacteristic
87C4⋊(C79⋊C8)C2×C411yescharacteristic
88C4⋊(C79⋊C8)C2311yescharacteristic
89C4⋊(C79⋊C8)C2311yescharacteristic
90Dic79⋊C8—21no—
91Dic79⋊C8—21no—
92Dic79⋊C8—21no—
93Dic79⋊C8—21no—
94#14579085C2×C411yescharacteristic
95#14579085C2×C411yescharacteristic
96#14579085C2×C411yescharacteristic
97#14579085C2×C411yescharacteristic
98#14579084C2×C411yescharacteristic
99#14579084C2×C411yescharacteristic
100#14579084C2×C411yescharacteristic
101#14579084C2×C411yescharacteristic
102C22⋊(C79⋊C8)—21no—
103C22⋊(C79⋊C8)—21no—
104C22⋊(C79⋊C8)C2×C411yescharacteristic
105C22⋊(C79⋊C8)C2×C411yescharacteristic
106#14579071—21no—
107#14579071—21no—
108C4×C79⋊C8—21no—
109C4×C79⋊C8C2311yescharacteristic
110C4×C79⋊C8C2311yescharacteristic
111C4×C79⋊C8C2311yescharacteristic
112C4×C79⋊C8C2311yescharacteristic
113C8×Dic79—21no—
114C8×Dic79—21no—
115C22×Q8×C79C2×C411yescharacteristic
116C4×Q8×C79—21no—
117C4×Q8×C79—21no—
118C4×Q8×C79C2×C411yescharacteristic
119C4×Q8×C79C2×C411yescharacteristic
120C4×Q8×C79C2311yescharacteristic
121C4×Q8×C79C2311yescharacteristic
122C2×C4⋊C4×C79—21no—
123C2×C4⋊C4×C79C2×C411yescharacteristic
124C2×C42×C79—21no—
125C2×C42×C79C2311yescharacteristic
126C4⋊C8×C79—21no—
127C4⋊C8×C79—21no—
128C4⋊C8×C79—21no—
129C4⋊C8×C79—21no—
130C4⋊C8×C79D411yescharacteristic
131C4⋊C8×C79D411yescharacteristic
132C4⋊C8×C79C2311yescharacteristic
133C4⋊C8×C79C2311yescharacteristic
134C22⋊C8×C79—21no—
135C22⋊C8×C79D411yescharacteristic
136C22⋊C8×C79D411yescharacteristic
137#97×C79—21no—
138#97×C79—21no—
139#97×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158C22×C411yescharacteristic
146C2×Dic158C22×C411yescharacteristic
147#11929614—22no—
148#11929614—22no—
149#11929614—41no—
150#11929614—41no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8—21no—
155C2×C79⋊C8C22×C411yescharacteristic
156C2×C79⋊C8C22×C411yescharacteristic
157C2×C79⋊C8C22×C411yescharacteristic
158C2×C79⋊C8C22×C411yescharacteristic
159C4⋊Dic79—21no—
160C4⋊Dic79—21no—
161C4⋊Dic79—21no—
162C4⋊Dic79C22×C411yescharacteristic
163C4⋊Dic79C22×C411yescharacteristic
164Dic79⋊C4—21no—
165Dic79⋊C4—21no—
166Dic79⋊C4—21no—
167Dic79⋊C4—21no—
168C22⋊Dic79—21no—
169C22⋊Dic79—21no—
170C4×Dic79—21no—
171C4×Dic79—21no—
172C2×Q8×C79—21no—
173C2×Q8×C79—22no—
174C2×Q8×C79—41no—
175C2×Q8×C79C22×C411yescharacteristic
176C2×Q8×C79C22×C411yescharacteristic
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79C22×C411yescharacteristic
180C22×C4×C79C2×D411yescharacteristic
181C22×C4×C79#4111yescharacteristic
182C2×C8×C79—21no—
183C2×C8×C79—21no—
184C2×C8×C79C2×D411yescharacteristic
185C2×C8×C79C2×D411yescharacteristic
186C2×C8×C79C2×D411yescharacteristic
187C2×C8×C79C2×D411yescharacteristic
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79C22×C411yescharacteristic
194C4⋊C4×C79C22×C411yescharacteristic
195C42×C79—21no—
196C42×C79—21no—
197C42×C79—21no—
198C42×C79—21no—
199C42×C79C22×C411yescharacteristic
200C42×C79C2×D411yescharacteristic
201C42×C79#4111yescharacteristic
202C42×C79C2411yescharacteristic
203C23×C79C4×D411yescharacteristic
204Q8×C79—21no—
205Q8×C79—21no—
206Q8×C79—21no—
207Q8×C79—42no—
208Q8×C79#13112yes—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—41no—
218C2×C4×C79—42no—
219C2×C4×C79—42no—
220C2×C4×C79C4×D411yescharacteristic
221C2×C4×C79C4×D411yescharacteristic
222C2×C4×C79C4×D411yescharacteristic
223C2×C4×C79C23×C411yescharacteristic
224C2×C4×C79C22×D411yescharacteristic
225C2×C4×C79C2×#4111yescharacteristic
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230Dic158—21no—
231Dic158—21no—
232Dic158—21no—
233Dic158—21no—
234Dic158—42no—
235C8×C79—21no—
236C8×C79—21no—
237C8×C79—21no—
238C8×C79—21no—
239C8×C79—41no—
240C8×C79—41no—
241C79⋊C8—21no—
242C79⋊C8—21no—
243C79⋊C8—21no—
244C79⋊C8—21no—
245C79⋊C8—41no—
246C79⋊C8—41no—
247C79⋊C8—41no—
248C79⋊C8—41no—
249C22×C79—21no—
250C22×C79—21no—
251C22×C79—21no—
252C22×C79C2×C4×D411yescharacteristic
253C4×C79—21no—
254C4×C79—22no—
255C4×C79—21no—
256C4×C79—21no—
257C4×C79—21no—
258C4×C79—41no—
259C4×C79—41no—
260C4×C79—41no—
261C4×C79C2×#13111yescharacteristic
262C4×C79#56811yescharacteristic
263C4×C79#57811yescharacteristic
264C4×C79#57811yescharacteristic
265Dic79—41no—
266Dic79—41no—
267Dic79—41no—
268Dic79—41no—
269#18032—791nomaximal, Sylow
270C2×C79—41no—
271C2×C79#293311yescharacteristic
272C2×C79#295011yescharacteristic
273C2×C79#295511yescharacteristic
274#3104—791no—
275#2986—791no—
276#2984D7911yescharacteristic
277#1593—791no—
278#1591—791no—
279#1587—791no—
280#1587—791no—
281#1584—791no—
282#1584—791no—
283#1579—791no—
284C8×Q16—791no—
285#1572—791no—
286#1497—791no—
287#1479—791no—
288#1454—791no—
289C79#1803211yescharacteristic, complemented, Sylow
290#533—791no—
291C2×C4×Q8C2×D7911yescharacteristic
292#484—791no—
293#484—791no—
294#477—791no—
295#477—791no—
296#462—791no—
297#462—791no—
298#451—791no—
299#451—791no—
300#446—21no—
301#446—21no—
302#446—791no—
303#446—791no—
304#446—1581no—
305#446C2×D7911yescharacteristic
306#446C2×D7911yescharacteristic
307C8×Q8—791no—
308C8×Q8—791no—
309C8×Q8—1581no—
310C8×Q8C2×D7911yescharacteristic
311C8×Q8C2×D7911yescharacteristic
312C4×Q16—791no—
313C4×Q16—791no—
314C4×Q16—791no—
315C4×Q16—791no—
316#433—1581no—
317#433—1581no—
318#432—21no—
319#432C2×D7911yescharacteristic
320#431—21no—
321#431—791no—
322#431—791no—
323#431—791no—
324#431C2×D7911yescharacteristic
325#335—791no—
326#335—791no—
327#334—791no—
328#334—791no—
329Q8⋊C8—791no—
330Q8⋊C8—791no—
331Q8⋊C8—791no—
332Q8⋊C8—791no—
333Q8⋊C8—791no—
334Q8⋊C8—791no—
335Q8⋊C8—791no—
336Q8⋊C8—791no—
337#322—791no—
338#322—791no—
339#322—791no—
340C82—791no—
341C22×Q8C4×D7911yescharacteristic
342C2×Q16—791no—
343C2×Q16—791no—
344C2×Q16—791no—
345C2×Q16—791no—
346C4⋊Q8—791no—
347C4⋊Q8—791no—
348#125—1581no—
349C22⋊Q8—791no—
350C22⋊Q8—791no—
351C22⋊Q8—1581no—
352C4×Q8—21no—
353C4×Q8—21no—
354C4×Q8—791no—
355C4×Q8—791no—
356C4×Q8—1581no—
357C4×Q8C4×D7911yescharacteristic
358C4×Q8C4×D7911yescharacteristic
359C4×Q8C22×D7911yescharacteristic
360C4×Q8C22×D7911yescharacteristic
361#117—1581no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4C4×D7911yescharacteristic
364C2×C42—21no—
365C2×C42C22×D7911yescharacteristic
366#107—791no—
367#107—791no—
368#107—791no—
369#107—791no—
370C4⋊C8—21no—
371C4⋊C8—21no—
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—791no—
375C4⋊C8—791no—
376C4⋊C8—791no—
377C4⋊C8—791no—
378C4⋊C8—1581no—
379C4⋊C8—1581no—
380C4⋊C8—1581no—
381C4⋊C8—1581no—
382C4⋊C8—1581no—
383C4⋊C8C22⋊D7911yescharacteristic
384C4⋊C8C22⋊D7911yescharacteristic
385C4⋊C8C22×D7911yescharacteristic
386C4⋊C8C22×D7911yescharacteristic
387#103—791no—
388#103—791no—
389#103—791no—
390#103—791no—
391#103—791no—
392#103—791no—
393#103—791no—
394#103—791no—
395C22⋊C8—21no—
396C22⋊C8—791no—
397C22⋊C8—791no—
398C22⋊C8—1581no—
399C22⋊C8—1581no—
400C22⋊C8C22⋊D7911yescharacteristic
401C22⋊C8C22⋊D7911yescharacteristic
402#97—21no—
403#97—21no—
404#97—21no—
405#97—1581no—
406#97—1581no—
407C4×C8—21no—
408C4×C8—21no—
409C4×C8—791no—
410C4×C8—791no—
411C4×C8—791no—
412C4×C8—791no—
413C4×C8—1581no—
414C4×C8—1581no—
415C4×C8—1581no—
416C4×C8C22×D7911yescharacteristic
417C4×C8C22×D7911yescharacteristic
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—41no—
421C2×Q8—791no—
422C2×Q8—791no—
423C2×Q8—1581no—
424C2×Q8C2×C4×D7911yescharacteristic
425C2×Q8C2×C4×D7911yescharacteristic
426C22×C4—21no—
427C22×C4—21no—
428C22×C4C2×C4×D7911yescharacteristic
429C22×C4C2×C22⋊D7911yescharacteristic
430C22×C4#1192962411yescharacteristic
431Q16—3161no—
432Q16—3161no—
433Q16—1582no—
434Q16—1582no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—791no—
438C2×C8—791no—
439C2×C8—791no—
440C2×C8—791no—
441C2×C8—1581no—
442C2×C8—1581no—
443C2×C8—1581no—
444C2×C8—1581no—
445C2×C8C2×C22⋊D7911yescharacteristic
446C2×C8C2×C22⋊D7911yescharacteristic
447C2×C8C2×C22⋊D7911yescharacteristic
448C2×C8C2×C22⋊D7911yescharacteristic
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—791no—
455C4⋊C4—791no—
456C4⋊C4—1581no—
457C4⋊C4—1581no—
458C4⋊C4—1581no—
459C4⋊C4—1581no—
460C4⋊C4—1581no—
461C4⋊C4—1581no—
462C4⋊C4—1581no—
463C4⋊C4C2×C4×D7911yescharacteristic
464C4⋊C4C2×C4×D7911yescharacteristic
465C22⋊C4—1581no—
466C22⋊C4—1581no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—1581no—
472C42—1581no—
473C42C2×C4×D7911yescharacteristic
474C42C2×C22⋊D7911yescharacteristic
475C42#1192962411yescharacteristic
476C42C23×D7911yescharacteristic
477C23C4×C22⋊D7911yescharacteristic
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—42no—
482Q8—3162no—
483Q8—1581no—
484Q8—1581no—
485Q8—1581no—
486Q8—1581no—
487Q8#1457916912yes—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—1581no—
500C2×C4—1581no—
501C2×C4—1581no—
502C2×C4—1581no—
503C2×C4C4×C22⋊D7911yescharacteristic
504C2×C4C4×C22⋊D7911yescharacteristic
505C2×C4C4×C22⋊D7911yescharacteristic
506C2×C4C22×C4×D7911yescharacteristic
507C2×C4C22×C22⋊D7911yescharacteristic
508C2×C4C2×#1192962411yescharacteristic
509C8—21no—
510C8—21no—
511C8—21no—
512C8—21no—
513C8—41no—
514C8—41no—
515C8—3161no—
516C8—3161no—
517C8—3161no—
518C8—3161no—
519C8—1581no—
520C8—1581no—
521C8—1581no—
522C8—1581no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22C2×C4×C22⋊D7911yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—22no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—3161no—
536C4—3161no—
537C4—3161no—
538C4—3161no—
539C4C2×#1457916911yescharacteristic
540C4#1909932911yescharacteristic
541C4#1909937811yescharacteristic
542C4#1909937811yescharacteristic
543C2—41no—
544C2#2377212511yescharacteristic
545C2#2377226611yescharacteristic
546C2#2377230111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#2984C2
socleC22×C79C2×C4×D4
prime radicalC23×C79C4×D4
cosocleC42×C79C24
hypercenter#2984D79
hypocenterC79#18032
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2984

Complements

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

NH ≅ G/Ntype
3. #23762307C2semidirect
4. #23762295C2semidirect
5. #23762271C2semidirect
6. #23762269C2semidirect
7. #23762250C2semidirect
8. #23762248C2semidirect
9. #23762217C2semidirect
10. C8×#11929614C2semidirect
289. C79#18032semidirect

Maximal subgroups

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

subgroupindex
#237734392
#237725012
#237623072
#237622952
#237622712
#237622692
#237622502
#237622482
#237622172
C8×#119296142
#237621812
#237618722
#237617952
#237617002
C79×#29842
#1803279