Finite Group Databasebeta

← #34259639

Subgroups — #34259639

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#23773427C211yesmaximal, characteristic
2#23772490C211yesmaximal, characteristic, complemented
3#23772467C211yesmaximal, characteristic
4#23762299C211yesmaximal, characteristic, complemented
5#23762293C211yesmaximal, characteristic, complemented
6#23762281C211yesmaximal, characteristic, complemented
7#23762254C211yesmaximal, characteristic, complemented
8#23762246C211yesmaximal, characteristic, complemented
9#23762233C211yesmaximal, characteristic, complemented
10#23762209C211yesmaximal, characteristic, complemented
11#23762203C211yesmaximal, characteristic, complemented
12#23762181C211yesmaximal, characteristic, complemented
13#23761796C211yesmaximal, characteristic, complemented
14#23761700C211yesmaximal, characteristic, complemented
15C79×#1496C211yesmaximal, characteristic
16C2×C4×Dic158C2211yescharacteristic
17#19098879C411yescharacteristic
18#19098879C411yescharacteristic
19#19098845C411yescharacteristic
20#19098845C411yescharacteristic
21#19098760C411yescharacteristic
22#19098760C411yescharacteristic
23#19098701C411yescharacteristic
24#19098701C411yescharacteristic
25#19098691—21no—
26#19098691—21no—
27#19098690C2211yescharacteristic
28#19098690C2211yescharacteristic
29#19098687C2211yescharacteristic
30#19098687C2211yescharacteristic
31C4×#11929612C2211yescharacteristic
32C4×#11929612C2211yescharacteristic
33C4×#11929612C2211yescharacteristic
34C4×#11929612C2211yescharacteristic
35#19098641—21no—
36#19098641C2211yescharacteristic
37#19098641C2211yescharacteristic
38D4×C79⋊C8—21no—
39D4×C79⋊C8C2211yescharacteristic
40D4×C79⋊C8C2211yescharacteristic
41#19098617—21no—
42#19098616C2211yescharacteristic
43#19098616C2211yescharacteristic
44#19098616C2211yescharacteristic
45#19098613—21no—
46#19098530—21no—
47#19098530—21no—
48#19098315C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098313C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098295C2211yescharacteristic
55#19098295C2211yescharacteristic
56#19098285C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#532×C79C2211yescharacteristic
61#432×C79C2211yescharacteristic
62#431×C79C2211yescharacteristic
63D4⋊C8×C79C2211yescharacteristic
64D4⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66D4⋊C8×C79C2211yescharacteristic
67C22×Dic158C2×C411yescharacteristic
68C2×#11929612C2×C411yescharacteristic
69C2×#11929612C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C2×#11929612C2×C411yescharacteristic
72C2×#11929608—21no—
73C2×#11929608—21no—
74C22×C79⋊C8—21no—
75C22×C79⋊C8—21no—
76C4×Dic158—21no—
77C4×Dic158—21no—
78C4×Dic158C2×C411yescharacteristic
79C4×Dic158C2×C411yescharacteristic
80C4×Dic158C2311yescharacteristic
81C4×Dic158C2311yescharacteristic
82C2×C4⋊Dic79C2×C411yescharacteristic
83C2×Dic79⋊C4—21no—
84C2×C4×Dic79—21no—
85#14579092C2×C411yescharacteristic
86#14579092C2×C411yescharacteristic
87#14579092C2×C411yescharacteristic
88#14579092C2×C411yescharacteristic
89C4⋊(C79⋊C8)C2×C411yescharacteristic
90C4⋊(C79⋊C8)C2×C411yescharacteristic
91C4⋊(C79⋊C8)D411yescharacteristic
92C4⋊(C79⋊C8)D411yescharacteristic
93C4⋊(C79⋊C8)C2311yescharacteristic
94C4⋊(C79⋊C8)C2311yescharacteristic
95Dic79⋊C8—21no—
96Dic79⋊C8—21no—
97Dic79⋊C8—21no—
98Dic79⋊C8—21no—
99#14579084C2×C411yescharacteristic
100#14579084C2×C411yescharacteristic
101#14579084C2×C411yescharacteristic
102#14579084C2×C411yescharacteristic
103#14579082C2×C411yescharacteristic
104#14579082C2×C411yescharacteristic
105#14579082C2×C411yescharacteristic
106#14579082C2×C411yescharacteristic
107C22⋊(C79⋊C8)—21no—
108C22⋊(C79⋊C8)—21no—
109C22⋊(C79⋊C8)—21no—
110C22⋊(C79⋊C8)—21no—
111C22⋊(C79⋊C8)—21no—
112C22⋊(C79⋊C8)C2×C411yescharacteristic
113C22⋊(C79⋊C8)C2×C411yescharacteristic
114C22⋊(C79⋊C8)D411yescharacteristic
115C22⋊(C79⋊C8)D411yescharacteristic
116#14579071—21no—
117#14579071—21no—
118C4×C79⋊C8—21no—
119C4×C79⋊C8C2311yescharacteristic
120C4×C79⋊C8C2311yescharacteristic
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C8×Dic79—21no—
124C8×Dic79—21no—
125C2×#41×C79—21no—
126C4⋊Q8×C79C2×C411yescharacteristic
127#127×C79C2×C411yescharacteristic
128#124×C79—21no—
129#121×C79—21no—
130#121×C79C2×C411yescharacteristic
131#121×C79C2×C411yescharacteristic
132C4×D4×C79—21no—
133C4×D4×C79C2311yescharacteristic
134C4×D4×C79C2311yescharacteristic
135C2×C42×C79C2311yescharacteristic
136C4⋊C8×C79—21no—
137C4⋊C8×C79C2311yescharacteristic
138C4⋊C8×C79C2311yescharacteristic
139C22⋊C8×C79—21no—
140C22⋊C8×C79—21no—
141#97×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—22no—
145C2×Dic158—21no—
146C2×Dic158—41no—
147C2×Dic158C22×C411yescharacteristic
148C2×Dic158C22×C411yescharacteristic
149C22×Dic79—21no—
150C22×Dic79—21no—
151#11929612—22no—
152#11929612—22no—
153#11929612—41no—
154#11929612—41no—
155#11929608—81no—
156#11929608—81no—
157C2×C79⋊C8—21no—
158C2×C79⋊C8—21no—
159C2×C79⋊C8—81no—
160C2×C79⋊C8—81no—
161C2×C79⋊C8C22×C411yescharacteristic
162C2×C79⋊C8C22×C411yescharacteristic
163C2×C79⋊C8C22×C411yescharacteristic
164C2×C79⋊C8C22×C411yescharacteristic
165C2×C79⋊C8C2×D411yescharacteristic
166C2×C79⋊C8C2×D411yescharacteristic
167C2×C79⋊C8C2×D411yescharacteristic
168C2×C79⋊C8C2×D411yescharacteristic
169C4⋊Dic79—21no—
170C4⋊Dic79C22×C411yescharacteristic
171C4⋊Dic79C22×C411yescharacteristic
172Dic79⋊C4—21no—
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176C4×Dic79—21no—
177C4×Dic79—21no—
178C4×Dic79—21no—
179C4×Dic79—21no—
180#41×C79—81no—
181#41×C79—81no—
182C2×Q8×C79—21no—
183C2×D4×C79—21no—
184C2×D4×C79—21no—
185C2×D4×C79—21no—
186C2×D4×C79—21no—
187C2×D4×C79C22×C411yescharacteristic
188C2×D4×C79C22×C411yescharacteristic
189C22×C4×C79—21no—
190C22×C4×C79—21no—
191C22×C4×C79C22×C411yescharacteristic
192C22×C4×C79C2×D411yescharacteristic
193C22×C4×C79#4111yescharacteristic
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79C22×C411yescharacteristic
200C4⋊C4×C79C22×C411yescharacteristic
201C22⋊C4×C79—21no—
202C22⋊C4×C79—21no—
203C22⋊C4×C79—21no—
204C22⋊C4×C79—21no—
205C42×C79C22×C411yescharacteristic
206C42×C79C2×D411yescharacteristic
207C42×C79#4111yescharacteristic
208C42×C79C2411yescharacteristic
209C23×C79—21no—
210C23×C79—21no—
211C23×C79C4×D411yescharacteristic
212Q8×C79—81no—
213D4×C79—21no—
214D4×C79—21no—
215D4×C79—21no—
216D4×C79—21no—
217D4×C79—42no—
218D4×C79—81no—
219D4×C79—81no—
220D4×C79—81no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79C4×D411yescharacteristic
233C2×C4×C79C4×D411yescharacteristic
234C2×C4×C79C4×D411yescharacteristic
235C2×C4×C79C23×C411yescharacteristic
236C2×C4×C79C22×D411yescharacteristic
237C2×C4×C79C2×#4111yescharacteristic
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—42no—
243C2×Dic79—42no—
244Dic158—21no—
245Dic158—21no—
246Dic158—21no—
247Dic158—42no—
248Dic158#13112yes—
249C8×C79—41no—
250C8×C79—41no—
251C8×C79—41no—
252C8×C79—41no—
253C79⋊C8—21no—
254C79⋊C8—21no—
255C79⋊C8—21no—
256C79⋊C8—21no—
257C79⋊C8—21no—
258C79⋊C8—21no—
259C79⋊C8—21no—
260C79⋊C8—21no—
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—21no—
275C4×C79—21no—
276C4×C79—21no—
277C4×C79—41no—
278C4×C79—41no—
279C4×C79C2×#13111yescharacteristic
280C4×C79#56811yescharacteristic
281C4×C79#57711yescharacteristic
282C4×C79#57711yescharacteristic
283Dic79—22no—
284Dic79—41no—
285Dic79—41no—
286Dic79—41no—
287#18030—791nomaximal, Sylow
288C2×C79—41no—
289C2×C79—81no—
290C2×C79—81no—
291C2×C79#293311yescharacteristic
292C2×C79#294911yescharacteristic
293C2×C79#295511yescharacteristic
294#3103—791no—
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#1496D7911yescharacteristic
307#1479—791no—
308#1454—791no—
309C79#1803011yescharacteristic, complemented, Sylow
310#532C2×D7911yescharacteristic
311C2×C4×Q8—791no—
312#481—791no—
313#481—791no—
314#475—791no—
315#475—791no—
316#460—791no—
317#460—791no—
318#448—791no—
319#448—791no—
320#446—791no—
321#446—791no—
322#446—1581no—
323#446—1581no—
324C8×Q8—791no—
325C8×Q8—791no—
326C4×SD16—791no—
327C4×SD16—791no—
328C4×SD16—791no—
329C4×SD16—791no—
330#436—791no—
331#436—791no—
332#436—1581no—
333C8×D4—791no—
334C8×D4—791no—
335C8×D4—1581no—
336#432—1581no—
337#432C2×D7911yescharacteristic
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⋊C8C2×D7911yescharacteristic
354D4⋊C8C2×D7911yescharacteristic
355D4⋊C8C2×D7911yescharacteristic
356D4⋊C8C2×D7911yescharacteristic
357#322—791no—
358#322—791no—
359#322—791no—
360C82—791no—
361C2×#41—21no—
362C22×Q8—791no—
363C2×SD16—791no—
364C2×SD16—791no—
365C2×SD16—791no—
366C2×SD16—791no—
367C2×M16—1581no—
368C2×M16—1581no—
369C22×C8—1581no—
370C22×C8—1581no—
371C4⋊Q8C4×D7911yescharacteristic
372#127C4×D7911yescharacteristic
373#124—21no—
374#121—21no—
375#121C4×D7911yescharacteristic
376#121C4×D7911yescharacteristic
377C4×Q8—791no—
378C4×Q8—791no—
379C4×Q8—791no—
380C4×Q8—791no—
381C4×Q8—1581no—
382C4×Q8—1581no—
383C4×D4—21no—
384C4×D4C22×D7911yescharacteristic
385C4×D4C22×D7911yescharacteristic
386C2×C4⋊C4—791no—
387C2×C4⋊C4—1581no—
388C2×C42—1581no—
389C2×C42C22×D7911yescharacteristic
390#106—791no—
391#106—791no—
392#106—791no—
393#106—791no—
394C4⋊C8—21no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—791no—
399C4⋊C8—791no—
400C4⋊C8—791no—
401C4⋊C8—1581no—
402C4⋊C8—1581no—
403C4⋊C8—1581no—
404C4⋊C8—1581no—
405C4⋊C8C22×D7911yescharacteristic
406C4⋊C8C22×D7911yescharacteristic
407#103—791no—
408#103—791no—
409#103—791no—
410#103—791no—
411#102—791no—
412#102—791no—
413#102—791no—
414#102—791no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—791no—
418C22⋊C8—791no—
419C22⋊C8—791no—
420C22⋊C8—791no—
421C22⋊C8—1581no—
422C22⋊C8—1581no—
423C22⋊C8—1581no—
424C22⋊C8—1581no—
425C22⋊C8—1581no—
426#97—21no—
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—3161no—
442C2×Q8—791no—
443C2×Q8—791no—
444C2×Q8—1582no—
445C2×Q8—1581no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C2×C4×D7911yescharacteristic
451C2×D4C2×C4×D7911yescharacteristic
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—1581no—
455C22×C4—1581no—
456C22×C4C2×C4×D7911yescharacteristic
457C22×C4D4×D7911yescharacteristic
458C22×C4#1192962311yescharacteristic
459SD16—3161no—
460SD16—3161no—
461SD16—1582no—
462SD16—1582no—
463M16—6321no—
464M16—6321no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—791no—
470C2×C8—791no—
471C2×C8—791no—
472C2×C8—791no—
473C2×C8—791no—
474C2×C8—791no—
475C2×C8—791no—
476C2×C8—791no—
477C2×C8—6321no—
478C2×C8—6321no—
479C2×C8—1581no—
480C2×C8—1581no—
481C4⋊C4—21no—
482C4⋊C4—791no—
483C4⋊C4—791no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4—1581no—
487C4⋊C4—1581no—
488C4⋊C4—1581no—
489C4⋊C4C2×C4×D7911yescharacteristic
490C4⋊C4C2×C4×D7911yescharacteristic
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C42—1581no—
496C42—1581no—
497C42—1581no—
498C42—1581no—
499C42C2×C4×D7911yescharacteristic
500C42D4×D7911yescharacteristic
501C42#1192962311yescharacteristic
502C42C23×D7911yescharacteristic
503C23—21no—
504C23—21no—
505C23(C22⋊D79)⋊C411yescharacteristic
506Q8—81no—
507Q8—3162no—
508Q8—792no—
509Q8—1581no—
510Q8—1581no—
511Q8—1581no—
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—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—3162no—
532C2×C4—3162no—
533C2×C4—1581no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4—1581no—
537C2×C4(C22⋊D79)⋊C411yescharacteristic
538C2×C4(C22⋊D79)⋊C411yescharacteristic
539C2×C4(C22⋊D79)⋊C411yescharacteristic
540C2×C4C22×C4×D7911yescharacteristic
541C2×C4C2×D4×D7911yescharacteristic
542C2×C4C2×#1192962311yescharacteristic
543C8—41no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—3161no—
548C8—3161no—
549C8—1581no—
550C8—1581no—
551C8—1581no—
552C8—1581no—
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×(C22⋊D79)⋊C411yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—41no—
572C4—41no—
573C4—3161no—
574C4—3161no—
575C4—3161no—
576C4—1582no—
577C4#131×D7911yescharacteristic
578C4#1909932811yescharacteristic
579C4#1909937311yescharacteristic
580C4#1909937311yescharacteristic
581C2—41no—
582C2—81no—
583C2—81no—
584C2#2377213011yescharacteristic
585C2#2377226211yescharacteristic
586C2#2377229911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1496C2
socleC22×C79C2×C4×D4
prime radical#532×C79C22
cosocleC42×C79C24
hypercenter#1496D79
hypocenterC79#18030
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1496

Complements

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

NH ≅ G/Ntype
2. #23772490C2semidirect
4. #23762299C2semidirect
5. #23762293C2semidirect
6. #23762281C2semidirect
7. #23762254C2semidirect
8. #23762246C2semidirect
9. #23762233C2semidirect
10. #23762209C2semidirect
11. #23762203C2semidirect
12. #23762181C2semidirect
13. #23761796C2semidirect
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
#237734272
#237724902
#237724672
#237622992
#237622932
#237622812
#237622542
#237622462
#237622332
#237622092
#237622032
#237621812
#237617962
#237617002
C79×#14962
#1803079