Finite Group Databasebeta

← #34259012

Subgroups — #34259012

Order 20224.

590 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773983C211yesmaximal, characteristic
2#23772331C211yesmaximal, characteristic
3#23772329C211yesmaximal, characteristic
4#23762274C211yesmaximal, characteristic, complemented
5#23762273C211yesmaximal, characteristic, complemented
6#23762272C211yesmaximal, characteristic, complemented
7#23762271C211yesmaximal, characteristic, complemented
8#23762270C211yesmaximal, characteristic, complemented
9#23762269C211yesmaximal, characteristic, complemented
10#23762268C211yesmaximal, characteristic, complemented
11#23762154C211yesmaximal, characteristic
12#23761853C211yesmaximal, characteristic
13#23761853C211yesmaximal, characteristic
14#322×D79C211yesmaximal, characteristic
15C79×#1587C211yesmaximal, characteristic, complemented
16#19099117C2211yescharacteristic
17#19099117C2211yescharacteristic
18#19098967C411yescharacteristic
19#19098941C411yescharacteristic
20#19098912C411yescharacteristic
21#19098912C411yescharacteristic
22#19098805C411yescharacteristic
23#19098803C411yescharacteristic
24#19098803C411yescharacteristic
25#19098801C411yescharacteristic
26#19098694C2211yescharacteristic
27#19098693C2211yescharacteristic
28#19098692—21no—
29#19098691—21no—
30#19098690C2211yescharacteristic
31Q8×C79⋊C8C2211yescharacteristic
32C8×Dic158—21no—
33C8×Dic158C2211yescharacteristic
34#19098687—21no—
35#19098687C2211yescharacteristic
36C4×Dic316C2211yescharacteristic
37C4×#11929614C2211yescharacteristic
38C4×#11929614C2211yescharacteristic
39#19098588—21no—
40#19098588—21no—
41#19098585C2211yescharacteristic
42#19098585C2211yescharacteristic
43#19098520—21no—
44#19098517—21no—
45C4×C8×D79C2211yescharacteristic
46C4×C8×D79C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098314C2211yescharacteristic
49#19098313C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52#19098295C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54Q8⋊(C79⋊C8)C2211yescharacteristic
55#19098293C2211yescharacteristic
56#19098293C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59#19098284C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C8×Q8×C79C2211yescharacteristic
62C4×Q16×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65Q8⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×Dic316C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929614C2×C411yescharacteristic
70C2×#11929607—21no—
71C2×#11929607—21no—
72C2×C8×D79—21no—
73C2×C8×D79C2×C411yescharacteristic
74C2×C8×D79C2×C411yescharacteristic
75C2×C8×D79D411yescharacteristic
76C2×C8×D79D411yescharacteristic
77#14579160C2×C411yescharacteristic
78#14579160C2×C411yescharacteristic
79#14579159C2×C411yescharacteristic
80#14579159C2×C411yescharacteristic
81#14579151—21no—
82#14579151—21no—
83#14579141C2×C411yescharacteristic
84#14579141C2×C411yescharacteristic
85#14579139—21no—
86#14579139—21no—
87#14579137C2×C411yescharacteristic
88#14579137C2×C411yescharacteristic
89C4×Dic158—21no—
90C4×Dic158—21no—
91C4×Dic158C2311yescharacteristic
92C4×Dic158C2311yescharacteristic
93#14579118—21no—
94#14579118—21no—
95C42×D79C2311yescharacteristic
96#14579095C2×C411yescharacteristic
97#14579094C2×C411yescharacteristic
98#14579094C2×C411yescharacteristic
99C4⋊(C79⋊C8)—21no—
100C4⋊(C79⋊C8)—21no—
101C4⋊(C79⋊C8)C2311yescharacteristic
102C4⋊(C79⋊C8)C2311yescharacteristic
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106Dic79⋊C8—21no—
107Dic79⋊C8—21no—
108Dic79⋊C8—21no—
109#14579086C2×C411yescharacteristic
110#14579086C2×C411yescharacteristic
111#14579085C2×C411yescharacteristic
112#14579085C2×C411yescharacteristic
113#14579084C2×C411yescharacteristic
114#14579084C2×C411yescharacteristic
115#14579072—21no—
116#14579071—21no—
117#14579071—21no—
118C4×C79⋊C8—21no—
119C4×C79⋊C8C2311yescharacteristic
120C4×C79⋊C8C2311yescharacteristic
121C4×C79⋊C8C2311yescharacteristic
122C8×Dic79—21no—
123C8×Dic79—21no—
124C8×Dic79—21no—
125C8×Dic79C2×C411yescharacteristic
126C8×Dic79C2×C411yescharacteristic
127C8×Dic79D411yescharacteristic
128C8×Dic79D411yescharacteristic
129C2×Q16×C79C2×C411yescharacteristic
130C4×Q8×C79C2311yescharacteristic
131C4×Q8×C79C2311yescharacteristic
132#107×C79C2×C411yescharacteristic
133C4⋊C8×C79—21no—
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#103×C79C2×C411yescharacteristic
139#97×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158—21no—
146C2×Dic158C22×C411yescharacteristic
147C2×Dic158C22×C411yescharacteristic
148C2×C4×D79C22×C411yescharacteristic
149C2×C4×D79C2×D411yescharacteristic
150C2×C4×D79#4111yescharacteristic
151Dic316—41no—
152#11929614—41no—
153#11929614—41no—
154#11929607—42no—
155#11929607—42no—
156C2×C79⋊C8—21no—
157C2×C79⋊C8—21no—
158C2×C79⋊C8—21no—
159C2×C79⋊C8C22×C411yescharacteristic
160C2×C79⋊C8C22×C411yescharacteristic
161C2×C79⋊C8C2×D411yescharacteristic
162C2×C79⋊C8C2×D411yescharacteristic
163C8×D79—24no—
164C8×D79—22no—
165C8×D79—22no—
166C8×D79—41no—
167C8×D79—41no—
168C4⋊Dic79—21no—
169C4⋊Dic79—21no—
170C4⋊Dic79C22×C411yescharacteristic
171C4⋊Dic79C22×C411yescharacteristic
172Dic79⋊C4—21no—
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179Dic79⋊C4—21no—
180Dic79⋊C4—21no—
181Dic79⋊C4—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185(C2×D79)⋊C4—21no—
186C4×Dic79—21no—
187C4×Dic79—21no—
188C4×Dic79C22×C411yescharacteristic
189C4×Dic79C2×D411yescharacteristic
190C4×Dic79#4111yescharacteristic
191C2×Q8×C79C22×C411yescharacteristic
192C2×Q8×C79C22×C411yescharacteristic
193Q16×C79—41no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79C22×C411yescharacteristic
198C2×C8×C79C22×C411yescharacteristic
199C2×C8×C79C2×D411yescharacteristic
200C2×C8×C79C2×D411yescharacteristic
201C4⋊C4×C79—21no—
202C4⋊C4×C79—21no—
203C4⋊C4×C79C22×C411yescharacteristic
204C4⋊C4×C79C22×C411yescharacteristic
205C42×C79—21no—
206C42×C79—21no—
207C42×C79C2411yescharacteristic
208C22×D79C4×D411yescharacteristic
209Q8×C79—21no—
210Q8×C79—21no—
211Q8×C79—21no—
212Q8×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79C23×C411yescharacteristic
218C2×C4×C79C22×D411yescharacteristic
219C2×C4×C79C2×#4111yescharacteristic
220C2×Dic79—21no—
221C2×Dic79—21no—
222C2×Dic79—21no—
223C2×Dic79—21no—
224C2×Dic79C4×D411yescharacteristic
225C2×Dic79C4×D411yescharacteristic
226C2×Dic79C4×D411yescharacteristic
227C4×D79—21no—
228C4×D79—21no—
229C4×D79—22no—
230C4×D79#13712yes—
231C4×D79#13712yes—
232Dic158—21no—
233Dic158—21no—
234Dic158—21no—
235Dic158—21no—
236Dic158—42no—
237Dic158—42no—
238C8×C79—22no—
239C8×C79—21no—
240C8×C79—21no—
241C8×C79—21no—
242C8×C79—21no—
243C8×C79—41no—
244C8×C79—41no—
245C79⋊C8—22no—
246C79⋊C8—21no—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—41no—
251C79⋊C8—41no—
252C22×C79C2×C4×D411yescharacteristic
253C2×D79—21no—
254C2×D79—21no—
255C2×D79#44112yes—
256C4×C79—21no—
257C4×C79—41no—
258C4×C79—41no—
259C4×C79—41no—
260C4×C79—41no—
261C4×C79#56811yescharacteristic
262C4×C79#56811yescharacteristic
263C4×C79C2×#13711yescharacteristic
264C4×C79C2×#13711yescharacteristic
265Dic79—21no—
266Dic79—41no—
267Dic79—41no—
268Dic79—41no—
269Dic79—41no—
270Dic79#44112yes—
271Dic79#44312yes—
272#17980—791nomaximal, Sylow
273C2×C79#293211yescharacteristic
274C2×C79C2×#44111yescharacteristic
275C2×C79C2×#44311yescharacteristic
276D79—22no—
277#3170—791no—
278#2960—791no—
279#2960—791no—
280#1587—791no—
281#1587—791no—
282#1587—791no—
283#1587—791no—
284#1587—791no—
285#1587—791no—
286#1587—791no—
287#1587D7911yescharacteristic, complemented
288#1566—791no—
289#1493—791no—
290#1493—791no—
291C2×#322—791no—
292C79#1798011yescharacteristic, complemented, Sylow
293#533—791no—
294#533—791no—
295#500—791no—
296#494—791no—
297#487—791no—
298#487—791no—
299#467—791no—
300#467—791no—
301#467—791no—
302#467—791no—
303#446—791no—
304#446—791no—
305#446—791no—
306#446—1581no—
307#446—1581no—
308#446C2×D7911yescharacteristic
309C8×Q8—791no—
310C8×Q8—791no—
311C8×Q8—791no—
312C8×Q8—1581no—
313C8×Q8—1581no—
314C8×Q8C2×D7911yescharacteristic
315C4×Q16—791no—
316C4×Q16—791no—
317C4×Q16—791no—
318C4×Q16C2×D7911yescharacteristic
319#424—1581no—
320#424—1581no—
321#423—791no—
322#423—791no—
323#405—1581no—
324#405—1581no—
325C2×C4×C8—791no—
326C2×C4×C8—791no—
327C2×C4×C8—791no—
328#334—791no—
329#334—791no—
330#334—791no—
331#334C2×D7911yescharacteristic
332Q8⋊C8—791no—
333Q8⋊C8—791no—
334Q8⋊C8—791no—
335Q8⋊C8—791no—
336Q8⋊C8—791no—
337Q8⋊C8—791no—
338Q8⋊C8C2×D7911yescharacteristic
339Q8⋊C8C2×D7911yescharacteristic
340#322—791no—
341#322—791no—
342#322—791no—
343#322C2×D7911yescharacteristic
344C2×Q16—791no—
345C2×Q16—791no—
346C2×Q16—791no—
347C2×Q16C4×D7911yescharacteristic
348C2×M16—1581no—
349C2×M16—1581no—
350C22×C8—791no—
351C22×C8—791no—
352C22×C8—791no—
353C22×C8—791no—
354C22×C8—1581no—
355C4⋊Q8—791no—
356C4⋊Q8—791no—
357C4⋊Q8—791no—
358C4⋊Q8—791no—
359#125—1581no—
360#125—1581no—
361C22⋊Q8—791no—
362C22⋊Q8—791no—
363C22⋊Q8—791no—
364C22⋊Q8—791no—
365C22⋊Q8—1581no—
366C22⋊Q8—1581no—
367C4×Q8—791no—
368C4×Q8—791no—
369C4×Q8—1581no—
370C4×Q8—1581no—
371C4×Q8C22×D7911yescharacteristic
372C4×Q8C22×D7911yescharacteristic
373#117—1581no—
374#117—1581no—
375C2×C42—791no—
376#107—791no—
377#107—791no—
378#107—791no—
379#107C4×D7911yescharacteristic
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8—791no—
383C4⋊C8—791no—
384C4⋊C8—1581no—
385C4⋊C8—1581no—
386C4⋊C8—1581no—
387C4⋊C8—1581no—
388C4⋊C8—1581no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8C22×D7911yescharacteristic
393C4⋊C8C22×D7911yescharacteristic
394#103—791no—
395#103—791no—
396#103—791no—
397#103—791no—
398#103—791no—
399#103—791no—
400#103C4×D7911yescharacteristic
401#103C4×D7911yescharacteristic
402#97—21no—
403#97—1581no—
404#97—1581no—
405#97—1581no—
406C4×C8—21no—
407C4×C8—791no—
408C4×C8—791no—
409C4×C8—791no—
410C4×C8—791no—
411C4×C8—791no—
412C4×C8—791no—
413C4×C8—791no—
414C4×C8—1581no—
415C4×C8—1581no—
416C4×C8—1581no—
417C4×C8—1581no—
418C4×C8C22×D7911yescharacteristic
419C4×C8C22×D7911yescharacteristic
420C4×C8C22×D7911yescharacteristic
421C2×Q8—791no—
422C2×Q8—791no—
423C2×Q8—1581no—
424C2×Q8—1581no—
425C2×Q8C2×C4×D7911yescharacteristic
426C2×Q8C2×C4×D7911yescharacteristic
427C22×C4—791no—
428C22×C4—791no—
429C22×C4—791no—
430Q16—41no—
431Q16—3161no—
432Q16—3161no—
433Q16—3161no—
434M16—3162no—
435M16—3162no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—3161no—
440C2×C8—3161no—
441C2×C8—791no—
442C2×C8—791no—
443C2×C8—791no—
444C2×C8—791no—
445C2×C8—1582no—
446C2×C8—1582no—
447C2×C8—1584no—
448C2×C8—1581no—
449C2×C8—1581no—
450C2×C8—1581no—
451C2×C8C2×C4×D7911yescharacteristic
452C2×C8C2×C4×D7911yescharacteristic
453C2×C8D4×D7911yescharacteristic
454C2×C8D4×D7911yescharacteristic
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—791no—
458C4⋊C4—791no—
459C4⋊C4—1581no—
460C4⋊C4—1581no—
461C4⋊C4—1581no—
462C4⋊C4—1581no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4C2×C4×D7911yescharacteristic
472C4⋊C4C2×C4×D7911yescharacteristic
473C22⋊C4—1581no—
474C22⋊C4—1581no—
475C22⋊C4—1581no—
476C22⋊C4—1581no—
477C42—21no—
478C42—21no—
479C42—791no—
480C42—791no—
481C42—791no—
482C42—1581no—
483C42—1581no—
484C42C23×D7911yescharacteristic
485C23—791no—
486Q8—21no—
487Q8—21no—
488Q8—21no—
489Q8—21no—
490Q8—3162no—
491Q8—3162no—
492Q8—1581no—
493Q8—1581no—
494Q8—1581no—
495Q8—1581no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—791no—
501C2×C4—791no—
502C2×C4—792no—
503C2×C4—792no—
504C2×C4—791no—
505C2×C4—1582no—
506C2×C4—1581no—
507C2×C4—1581no—
508C2×C4—1581no—
509C2×C4—1581no—
510C2×C4—1581no—
511C2×C4—1581no—
512C2×C4C22×C4×D7911yescharacteristic
513C2×C4C2×D4×D7911yescharacteristic
514C2×C4#41×D7911yescharacteristic
515C8—22no—
516C8—21no—
517C8—21no—
518C8—21no—
519C8—21no—
520C8—41no—
521C8—41no—
522C8—3161no—
523C8—3161no—
524C8—1582no—
525C8—1581no—
526C8—1581no—
527C8—1581no—
528C8—1581no—
529C22—792no—
530C22—1581no—
531C22—1581no—
532C22C4×D4×D7911yescharacteristic
533C4—21no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—3161no—
539C4—3161no—
540C4—3161no—
541C4—3161no—
542C4—792no—
543C4—792no—
544C4—1581no—
545C4#1909932811yescharacteristic
546C4#1909932911yescharacteristic
547C4#137×D7911yescharacteristic
548C4#137×D7911yescharacteristic
549C2—1582no—
550C2#2377211411yescharacteristic
551C2#441×D7911yescharacteristic
552C2#443×D7911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1587C2
socleC22×C79C2×C4×D4
prime radicalC22×D79C4×D4
cosocleC42×C79C24
hypercenter#1587D79
hypocenterC79#17980
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1587

Complements

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

NH ≅ G/Ntype
4. #23762274C2semidirect
5. #23762273C2semidirect
6. #23762272C2semidirect
7. #23762271C2semidirect
8. #23762270C2semidirect
9. #23762269C2semidirect
10. #23762268C2semidirect
15. C79×#1587C2semidirect
287. #1587D79semidirect
292. C79#17980semidirect

Maximal subgroups

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

subgroupindex
#237739832
#237723312
#237723292
#237622742
#237622732
#237622722
#237622712
#237622702
#237622692
#237622682
#237621542
#237618532
#322×D792
C79×#15872
#1798079