Finite Group Databasebeta

← #34259635

Subgroups — #34259635

Order 20224.

606 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772489C211yesmaximal, characteristic, complemented
2#23772467C211yesmaximal, characteristic
3#23762303C211yesmaximal, characteristic, complemented
4#23762291C211yesmaximal, characteristic, complemented
5#23762279C211yesmaximal, characteristic, complemented
6#23762258C211yesmaximal, characteristic, complemented
7#23762244C211yesmaximal, characteristic, complemented
8#23762237C211yesmaximal, characteristic, complemented
9SD16⋊(C79⋊C8)C211yesmaximal, characteristic, complemented
10SD16×C79⋊C8C211yesmaximal, characteristic, complemented
11#23762180C211yesmaximal, characteristic, complemented
12#23761869C211yesmaximal, characteristic
13#23761795C211yesmaximal, characteristic, complemented
14#23761700C211yesmaximal, characteristic, complemented
15C79×#3103C211yesmaximal, characteristic
16#19098694—21no—
17#19098694—21no—
18#19098694C2211yescharacteristic
19#19098694C2211yescharacteristic
20Q8×C79⋊C8C2211yescharacteristic
21Q8×C79⋊C8C2211yescharacteristic
22#19098641—21no—
23#19098641C2211yescharacteristic
24#19098641C2211yescharacteristic
25D4×C79⋊C8—21no—
26D4×C79⋊C8C2211yescharacteristic
27D4×C79⋊C8C2211yescharacteristic
28#19098620—21no—
29#19098620C2211yescharacteristic
30#19098616—21no—
31#19098616C2211yescharacteristic
32#19098616C2211yescharacteristic
33#19098616C2211yescharacteristic
34#19098530—21no—
35#19098530—21no—
36#19098316C2211yescharacteristic
37#19098316C2211yescharacteristic
38#19098314C2211yescharacteristic
39#19098314C2211yescharacteristic
40Q8⋊(C79⋊C8)C2211yescharacteristic
41Q8⋊(C79⋊C8)C2211yescharacteristic
42Q8⋊(C79⋊C8)C2211yescharacteristic
43Q8⋊(C79⋊C8)C2211yescharacteristic
44D4⋊(C79⋊C8)C2211yescharacteristic
45D4⋊(C79⋊C8)C2211yescharacteristic
46D4⋊(C79⋊C8)C2211yescharacteristic
47D4⋊(C79⋊C8)C2211yescharacteristic
48#19098285C2211yescharacteristic
49#19098285C2211yescharacteristic
50#19098284C2211yescharacteristic
51C8×C79⋊C8C2211yescharacteristic
52#532×C79C2211yescharacteristic
53C2×C4×Q8×C79C2211yescharacteristic
54#481×C79C411yescharacteristic
55#481×C79C411yescharacteristic
56#475×C79C411yescharacteristic
57#475×C79C411yescharacteristic
58#460×C79C411yescharacteristic
59#460×C79C411yescharacteristic
60#448×C79C411yescharacteristic
61#448×C79C411yescharacteristic
62C4×SD16×C79C2211yescharacteristic
63C4×SD16×C79C2211yescharacteristic
64C4×SD16×C79C2211yescharacteristic
65C4×SD16×C79C2211yescharacteristic
66#431×C79C2211yescharacteristic
67C2×#11929608—21no—
68C2×#11929608—21no—
69C22×C79⋊C8—21no—
70C22×C79⋊C8—21no—
71C4⋊(C79⋊C8)—21no—
72C4⋊(C79⋊C8)—21no—
73C4⋊(C79⋊C8)—21no—
74C4⋊(C79⋊C8)—21no—
75C4⋊(C79⋊C8)—21no—
76C4⋊(C79⋊C8)D411yescharacteristic
77C4⋊(C79⋊C8)D411yescharacteristic
78C4⋊(C79⋊C8)C2311yescharacteristic
79C4⋊(C79⋊C8)C2311yescharacteristic
80C4⋊(C79⋊C8)C2311yescharacteristic
81C4⋊(C79⋊C8)C2311yescharacteristic
82C22⋊(C79⋊C8)—21no—
83C22⋊(C79⋊C8)—21no—
84C22⋊(C79⋊C8)—21no—
85C22⋊(C79⋊C8)—21no—
86C22⋊(C79⋊C8)—21no—
87C22⋊(C79⋊C8)—21no—
88C22⋊(C79⋊C8)—21no—
89C22⋊(C79⋊C8)D411yescharacteristic
90C22⋊(C79⋊C8)D411yescharacteristic
91#14579072—21no—
92#14579072—21no—
93#14579072—21no—
94C4×C79⋊C8—21no—
95C4×C79⋊C8—21no—
96C4×C79⋊C8—21no—
97C4×C79⋊C8C2311yescharacteristic
98C4×C79⋊C8C2311yescharacteristic
99C4×C79⋊C8C2311yescharacteristic
100C4×C79⋊C8C2311yescharacteristic
101C2×#41×C79—21no—
102C22×Q8×C79C2×C411yescharacteristic
103C2×SD16×C79C2×C411yescharacteristic
104C2×SD16×C79C2×C411yescharacteristic
105C2×SD16×C79C2×C411yescharacteristic
106C2×SD16×C79C2×C411yescharacteristic
107C4⋊Q8×C79C2×C411yescharacteristic
108#127×C79C2×C411yescharacteristic
109#124×C79—21no—
110#121×C79—21no—
111#121×C79C2×C411yescharacteristic
112#121×C79C2×C411yescharacteristic
113C4×Q8×C79—21no—
114C4×Q8×C79—21no—
115C4×Q8×C79C2×C411yescharacteristic
116C4×Q8×C79C2×C411yescharacteristic
117C4×Q8×C79C2311yescharacteristic
118C4×Q8×C79C2311yescharacteristic
119C4×D4×C79—21no—
120C4×D4×C79C2311yescharacteristic
121C4×D4×C79C2311yescharacteristic
122C2×C4⋊C4×C79—21no—
123C2×C4⋊C4×C79C2×C411yescharacteristic
124C2×C42×C79—21no—
125C2×C42×C79C2311yescharacteristic
126#106×C79C2×C411yescharacteristic
127#106×C79C2×C411yescharacteristic
128#106×C79C2×C411yescharacteristic
129#106×C79C2×C411yescharacteristic
130C4⋊C8×C79C2×C411yescharacteristic
131C4⋊C8×C79C2×C411yescharacteristic
132#103×C79C2×C411yescharacteristic
133#103×C79C2×C411yescharacteristic
134#103×C79C2×C411yescharacteristic
135#103×C79C2×C411yescharacteristic
136#102×C79C2×C411yescharacteristic
137#102×C79C2×C411yescharacteristic
138#102×C79C2×C411yescharacteristic
139#102×C79C2×C411yescharacteristic
140C22⋊C8×C79C2×C411yescharacteristic
141C22⋊C8×C79C2×C411yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929608—81no—
145#11929608—81no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8—21no—
149C2×C79⋊C8—21no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—81no—
153C2×C79⋊C8—81no—
154C2×C79⋊C8C2×D411yescharacteristic
155C2×C79⋊C8C2×D411yescharacteristic
156C2×C79⋊C8C2×D411yescharacteristic
157C2×C79⋊C8C2×D411yescharacteristic
158#41×C79—81no—
159#41×C79—81no—
160C2×Q8×C79—21no—
161C2×Q8×C79—21no—
162C2×Q8×C79—22no—
163C2×Q8×C79—41no—
164C2×Q8×C79C22×C411yescharacteristic
165C2×Q8×C79C22×C411yescharacteristic
166C2×D4×C79—21no—
167C2×D4×C79—21no—
168C2×D4×C79—21no—
169C2×D4×C79—21no—
170C2×D4×C79C22×C411yescharacteristic
171C2×D4×C79C22×C411yescharacteristic
172C22×C4×C79—21no—
173C22×C4×C79—21no—
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79C22×C411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79#4111yescharacteristic
179SD16×C79—22no—
180SD16×C79—22no—
181SD16×C79—41no—
182SD16×C79—41no—
183C2×C8×C79C22×C411yescharacteristic
184C2×C8×C79C22×C411yescharacteristic
185C2×C8×C79C22×C411yescharacteristic
186C2×C8×C79C22×C411yescharacteristic
187C4⋊C4×C79—21no—
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
195C4⋊C4×C79C22×C411yescharacteristic
196C4⋊C4×C79C22×C411yescharacteristic
197C22⋊C4×C79—21no—
198C22⋊C4×C79—21no—
199C22⋊C4×C79—21no—
200C22⋊C4×C79—21no—
201C42×C79—21no—
202C42×C79—21no—
203C42×C79—21no—
204C42×C79—21no—
205C42×C79C22×C411yescharacteristic
206C42×C79C2×D411yescharacteristic
207C42×C79#4111yescharacteristic
208C42×C79C2411yescharacteristic
209C23×C79—21no—
210C23×C79—21no—
211C23×C79C4×D411yescharacteristic
212Q8×C79—21no—
213Q8×C79—21no—
214Q8×C79—21no—
215Q8×C79—42no—
216Q8×C79—81no—
217Q8×C79#13112yes—
218D4×C79—21no—
219D4×C79—21no—
220D4×C79—21no—
221D4×C79—21no—
222D4×C79—42no—
223D4×C79—81no—
224D4×C79—81no—
225D4×C79—81no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—21no—
234C2×C4×C79—21no—
235C2×C4×C79—21no—
236C2×C4×C79—41no—
237C2×C4×C79—42no—
238C2×C4×C79—42no—
239C2×C4×C79—41no—
240C2×C4×C79—41no—
241C2×C4×C79—41no—
242C2×C4×C79—41no—
243C2×C4×C79C4×D411yescharacteristic
244C2×C4×C79C4×D411yescharacteristic
245C2×C4×C79C4×D411yescharacteristic
246C2×C4×C79C23×C411yescharacteristic
247C2×C4×C79C22×D411yescharacteristic
248C2×C4×C79C2×#4111yescharacteristic
249C8×C79—21no—
250C8×C79—21no—
251C8×C79—21no—
252C8×C79—21no—
253C79⋊C8—21no—
254C79⋊C8—21no—
255C79⋊C8—21no—
256C79⋊C8—21no—
257C79⋊C8—41no—
258C79⋊C8—41no—
259C79⋊C8—41no—
260C79⋊C8—41no—
261C79⋊C8—41no—
262C79⋊C8—41no—
263C22×C79—21no—
264C22×C79—21no—
265C22×C79—21no—
266C22×C79—41no—
267C22×C79—41no—
268C22×C79—41no—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79—41no—
272C22×C79C2×C4×D411yescharacteristic
273C4×C79—21no—
274C4×C79—22no—
275C4×C79—21no—
276C4×C79—21no—
277C4×C79—21no—
278C4×C79—41no—
279C4×C79—41no—
280C4×C79—41no—
281C4×C79—41no—
282C4×C79—41no—
283C4×C79C2×#13111yescharacteristic
284C4×C79#56811yescharacteristic
285C4×C79#57711yescharacteristic
286C4×C79#57711yescharacteristic
287#18030—791nomaximal, Sylow
288C2×C79—41no—
289C2×C79—81no—
290C2×C79—81no—
291C2×C79#293311yescharacteristic
292C2×C79#294911yescharacteristic
293C2×C79#295511yescharacteristic
294#3103D7911yescharacteristic
295#2984—791no—
296#2979—791no—
297#1592—791no—
298#1590—791no—
299#1588—791no—
300#1585—791no—
301#1583—791no—
302#1582—791no—
303SD16⋊C8—791no—
304C8×SD16—791no—
305#1572—791no—
306#1496—791no—
307#1479—791no—
308#1454—791no—
309C79#1803011yescharacteristic, complemented, Sylow
310#532C2×D7911yescharacteristic
311C2×C4×Q8C2×D7911yescharacteristic
312#481Dic7911yescharacteristic
313#481Dic7911yescharacteristic
314#475Dic7911yescharacteristic
315#475Dic7911yescharacteristic
316#460Dic7911yescharacteristic
317#460Dic7911yescharacteristic
318#448Dic7911yescharacteristic
319#448Dic7911yescharacteristic
320#446—791no—
321#446—791no—
322#446—1581no—
323#446—1581no—
324C8×Q8—791no—
325C8×Q8—791no—
326C4×SD16C2×D7911yescharacteristic
327C4×SD16C2×D7911yescharacteristic
328C4×SD16C2×D7911yescharacteristic
329C4×SD16C2×D7911yescharacteristic
330#436—791no—
331#436—791no—
332#436—1581no—
333C8×D4—791no—
334C8×D4—791no—
335C8×D4—1581no—
336#432—791no—
337#432—1581no—
338#431—791no—
339#431—791no—
340#431—791no—
341#431—1581no—
342#431C2×D7911yescharacteristic
343#408—1581no—
344#408—1581no—
345#335—791no—
346#335—791no—
347#334—791no—
348#334—791no—
349Q8⋊C8—791no—
350Q8⋊C8—791no—
351Q8⋊C8—791no—
352Q8⋊C8—791no—
353D4⋊C8—791no—
354D4⋊C8—791no—
355D4⋊C8—791no—
356D4⋊C8—791no—
357#322—791no—
358#322—791no—
359#322—791no—
360C82—791no—
361C2×#41—21no—
362C22×Q8C2×Dic7911yescharacteristic
363C2×SD16C2×Dic7911yescharacteristic
364C2×SD16C2×Dic7911yescharacteristic
365C2×SD16C2×Dic7911yescharacteristic
366C2×SD16C2×Dic7911yescharacteristic
367C2×M16—1581no—
368C2×M16—1581no—
369C22×C8—1581no—
370C22×C8—1581no—
371C4⋊Q8C2×Dic7911yescharacteristic
372#127C2×Dic7911yescharacteristic
373#124—21no—
374#121—21no—
375#121C2×Dic7911yescharacteristic
376#121C2×Dic7911yescharacteristic
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8C2×Dic7911yescharacteristic
380C4×Q8C2×Dic7911yescharacteristic
381C4×Q8C22×D7911yescharacteristic
382C4×Q8C22×D7911yescharacteristic
383C4×D4—21no—
384C4×D4C22×D7911yescharacteristic
385C4×D4C22×D7911yescharacteristic
386C2×C4⋊C4—21no—
387C2×C4⋊C4C2×Dic7911yescharacteristic
388C2×C42—21no—
389C2×C42C22×D7911yescharacteristic
390#106C2×Dic7911yescharacteristic
391#106C2×Dic7911yescharacteristic
392#106C2×Dic7911yescharacteristic
393#106C2×Dic7911yescharacteristic
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—791no—
399C4⋊C8—791no—
400C4⋊C8—1581no—
401C4⋊C8—1581no—
402C4⋊C8—1581no—
403C4⋊C8—1581no—
404C4⋊C8—1581no—
405C4⋊C8C2×Dic7911yescharacteristic
406C4⋊C8C2×Dic7911yescharacteristic
407#103C2×Dic7911yescharacteristic
408#103C2×Dic7911yescharacteristic
409#103C2×Dic7911yescharacteristic
410#103C2×Dic7911yescharacteristic
411#102C2×Dic7911yescharacteristic
412#102C2×Dic7911yescharacteristic
413#102C2×Dic7911yescharacteristic
414#102C2×Dic7911yescharacteristic
415C22⋊C8—791no—
416C22⋊C8—791no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419C22⋊C8—1581no—
420C22⋊C8—1581no—
421C22⋊C8—1581no—
422C22⋊C8—1581no—
423C22⋊C8—1581no—
424C22⋊C8C2×Dic7911yescharacteristic
425C22⋊C8C2×Dic7911yescharacteristic
426#97—1581no—
427#97—1581no—
428#97—1581no—
429C4×C8—791no—
430C4×C8—791no—
431C4×C8—791no—
432C4×C8—791no—
433C4×C8—1581no—
434C4×C8—1581no—
435C4×C8—1581no—
436C4×C8C22×D7911yescharacteristic
437C4×C8C22×D7911yescharacteristic
438#41—81no—
439#41—81no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—41no—
444C2×Q8C22×Dic7911yescharacteristic
445C2×Q8C22×Dic7911yescharacteristic
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C22×Dic7911yescharacteristic
451C2×D4C22×Dic7911yescharacteristic
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4C22×Dic7911yescharacteristic
457C22×C4D4×D7911yescharacteristic
458C22×C4#1192962311yescharacteristic
459SD16—22no—
460SD16—22no—
461SD16—41no—
462SD16—41no—
463M16—6321no—
464M16—6321no—
465C2×C8—791no—
466C2×C8—791no—
467C2×C8—791no—
468C2×C8—791no—
469C2×C8—6321no—
470C2×C8—6321no—
471C2×C8—1581no—
472C2×C8—1581no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8C22×Dic7911yescharacteristic
478C2×C8C22×Dic7911yescharacteristic
479C2×C8C22×Dic7911yescharacteristic
480C2×C8C22×Dic7911yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4C22×Dic7911yescharacteristic
488C4⋊C4C22×Dic7911yescharacteristic
489C4⋊C4C22×Dic7911yescharacteristic
490C4⋊C4C22×Dic7911yescharacteristic
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42C22×Dic7911yescharacteristic
500C42D4×D7911yescharacteristic
501C42#1192962311yescharacteristic
502C42C23×D7911yescharacteristic
503C23—21no—
504C23—21no—
505C23D4×Dic7911yescharacteristic
506Q8—21no—
507Q8—21no—
508Q8—21no—
509Q8—42no—
510Q8—81no—
511Q8#1457916812yes—
512D4—21no—
513D4—21no—
514D4—21no—
515D4—21no—
516D4—42no—
517D4—81no—
518D4—81no—
519D4—81no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4D4×Dic7911yescharacteristic
538C2×C4D4×Dic7911yescharacteristic
539C2×C4D4×Dic7911yescharacteristic
540C2×C4C23×Dic7911yescharacteristic
541C2×C4C2×D4×D7911yescharacteristic
542C2×C4C2×#1192962311yescharacteristic
543C8—21no—
544C8—21no—
545C8—21no—
546C8—21no—
547C8—3161no—
548C8—3161no—
549C8—3161no—
550C8—3161no—
551C8—3161no—
552C8—3161no—
553C8—1581no—
554C8—1581no—
555C8—1581no—
556C8—1581no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22C2×D4×Dic7911yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—22no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4C2×#1457916811yescharacteristic
578C4#1909933011yescharacteristic
579C4#1909936911yescharacteristic
580C4#1909936911yescharacteristic
581C2—41no—
582C2—81no—
583C2—81no—
584C2#2377212711yescharacteristic
585C2#2377226011yescharacteristic
586C2#2377230311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#3103C2
socleC22×C79C2×C4×D4
prime radical#532×C79C22
cosocleC42×C79C24
hypercenter#3103D79
hypocenterC79#18030
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
1. #23772489C2semidirect
3. #23762303C2semidirect
4. #23762291C2semidirect
5. #23762279C2semidirect
6. #23762258C2semidirect
7. #23762244C2semidirect
8. #23762237C2semidirect
9. SD16⋊(C79⋊C8)C2semidirect
10. SD16×C79⋊C8C2semidirect
11. #23762180C2semidirect
13. #23761795C2semidirect
14. #23761700C2semidirect
309. C79#18030semidirect

Maximal subgroups

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

subgroupindex
#237724892
#237724672
#237623032
#237622912
#237622792
#237622582
#237622442
#237622372
SD16⋊(C79⋊C8)2
SD16×C79⋊C82
#237621802
#237618692
#237617952
#237617002
C79×#31032
#1803079