Finite Group Databasebeta

← #34259359

Subgroups — #34259359

Order 20224.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773480C211yesmaximal, characteristic
2#23772398C211yesmaximal, characteristic
3Q8×#11929608C211yesmaximal, characteristic, complemented
4#23762305C211yesmaximal, characteristic, complemented
5#23762294C211yesmaximal, characteristic, complemented
6#23762275C211yesmaximal, characteristic, complemented
7#23762274C211yesmaximal, characteristic, complemented
8#23762247C211yesmaximal, characteristic, complemented
9#23762239C211yesmaximal, characteristic, complemented
10#23762216C211yesmaximal, characteristic, complemented
11#23762197C211yesmaximal, characteristic, complemented
12#23762188C211yesmaximal, characteristic, complemented
13#23761866C211yesmaximal, characteristic
14#23761695C211yesmaximal, characteristic, complemented
15C79×#1479C211yesmaximal, characteristic, complemented
16#19099097C2211yescharacteristic
17#19098893C411yescharacteristic
18#19098876C411yescharacteristic
19#19098870C411yescharacteristic
20#19098870C411yescharacteristic
21#19098785C411yescharacteristic
22#19098785C411yescharacteristic
23#19098719C411yescharacteristic
24#19098702C411yescharacteristic
25#19098694—21no—
26#19098694—21no—
27#19098694C2211yescharacteristic
28#19098694C2211yescharacteristic
29#19098690C2211yescharacteristic
30Q8×C79⋊C8C2211yescharacteristic
31Q8×C79⋊C8C2211yescharacteristic
32#19098687C2211yescharacteristic
33C4×#11929614C2211yescharacteristic
34C4×#11929614C2211yescharacteristic
35C4×#11929611C2211yescharacteristic
36C4×#11929611C2211yescharacteristic
37#19098640C2211yescharacteristic
38#19098638—21no—
39#19098637—21no—
40#19098628C2211yescharacteristic
41#19098622—21no—
42#19098616C2211yescharacteristic
43#19098587—21no—
44#19098587C2211yescharacteristic
45#19098517—21no—
46C4×#11929608—21no—
47C4×#11929608C2211yescharacteristic
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
60C2×C4×Q8×C79C2211yescharacteristic
61#431×C79C2211yescharacteristic
62#431×C79C2211yescharacteristic
63Q8⋊C8×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65Q8⋊C8×C79C2211yescharacteristic
66Q8⋊C8×C79C2211yescharacteristic
67C2×#11929614C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929611C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71C2×#11929608—21no—
72C2×#11929608D411yescharacteristic
73C2×#11929608D411yescharacteristic
74C2×#11929607—21no—
75C2×C8×D79—21no—
76#14579155—21no—
77#14579153C2×C411yescharacteristic
78#14579149C2×C411yescharacteristic
79#14579144—21no—
80#14579140C2×C411yescharacteristic
81#14579135C2×C411yescharacteristic
82C4×Dic158C2311yescharacteristic
83C4×D316C2311yescharacteristic
84C4×C22⋊D79—21no—
85#14579118—21no—
86#14579094C2×C411yescharacteristic
87#14579094C2×C411yescharacteristic
88#14579092C2×C411yescharacteristic
89#14579092C2×C411yescharacteristic
90C4⋊(C79⋊C8)—21no—
91C4⋊(C79⋊C8)—21no—
92C4⋊(C79⋊C8)—21no—
93C4⋊(C79⋊C8)—21no—
94C4⋊(C79⋊C8)—21no—
95C4⋊(C79⋊C8)C2×C411yescharacteristic
96C4⋊(C79⋊C8)C2×C411yescharacteristic
97C4⋊(C79⋊C8)C2311yescharacteristic
98C4⋊(C79⋊C8)C2311yescharacteristic
99Dic79⋊C8—21no—
100Dic79⋊C8—21no—
101#14579085C2×C411yescharacteristic
102#14579085C2×C411yescharacteristic
103#14579085C2×C411yescharacteristic
104#14579085C2×C411yescharacteristic
105#14579084C2×C411yescharacteristic
106#14579084C2×C411yescharacteristic
107#14579081C2×C411yescharacteristic
108#14579081C2×C411yescharacteristic
109C22⋊(C79⋊C8)C2×C411yescharacteristic
110C22⋊(C79⋊C8)C2×C411yescharacteristic
111(C2×D79)⋊C8—21no—
112(C2×D79)⋊C8—21no—
113#14579072—21no—
114#14579072—21no—
115#14579072D411yescharacteristic
116#14579072D411yescharacteristic
117#14579071—21no—
118C4×C79⋊C8—21no—
119C4×C79⋊C8—21no—
120C4×C79⋊C8C2311yescharacteristic
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C8×Dic79—21no—
125C22×Q8×C79C2×C411yescharacteristic
126C4×Q8×C79—21no—
127C4×Q8×C79—21no—
128C4×Q8×C79C2×C411yescharacteristic
129C4×Q8×C79C2×C411yescharacteristic
130C4×Q8×C79C2311yescharacteristic
131C4×Q8×C79C2311yescharacteristic
132C2×C4⋊C4×C79—21no—
133C2×C4⋊C4×C79C2×C411yescharacteristic
134C2×C42×C79—21no—
135C2×C42×C79C2311yescharacteristic
136C4⋊C8×C79—21no—
137C4⋊C8×C79C2311yescharacteristic
138C4⋊C8×C79C2311yescharacteristic
139C22⋊C8×C79—21no—
140C22⋊C8×C79—21no—
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158C22×C411yescharacteristic
145C2×D316C22×C411yescharacteristic
146C2×C22⋊D79—21no—
147C2×C4×D79—21no—
148#11929614—22no—
149#11929614—41no—
150#11929611—22no—
151#11929611—41no—
152#11929608—22no—
153#11929608—41no—
154#11929608—42no—
155#11929607—42no—
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—42no—
167C4⋊Dic79C22×C411yescharacteristic
168C4⋊Dic79C22×C411yescharacteristic
169Dic79⋊C4—21no—
170Dic79⋊C4—21no—
171Dic79⋊C4—21no—
172Dic79⋊C4—21no—
173C22⋊Dic79—21no—
174(C2×D79)⋊C4—21no—
175(C2×D79)⋊C4—21no—
176(C2×D79)⋊C4—21no—
177(C2×D79)⋊C4—21no—
178C4×Dic79—21no—
179C2×Q8×C79—21no—
180C2×Q8×C79—22no—
181C2×Q8×C79—41no—
182C2×Q8×C79C22×C411yescharacteristic
183C2×Q8×C79C22×C411yescharacteristic
184C22×C4×C79—21no—
185C22×C4×C79—21no—
186C22×C4×C79C22×C411yescharacteristic
187C22×C4×C79C2×D411yescharacteristic
188C22×C4×C79#4111yescharacteristic
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79C22×C411yescharacteristic
199C4⋊C4×C79C22×C411yescharacteristic
200C42×C79—21no—
201C42×C79—21no—
202C42×C79—21no—
203C42×C79—21no—
204C42×C79C22×C411yescharacteristic
205C42×C79C2×D411yescharacteristic
206C42×C79#4111yescharacteristic
207C42×C79C2411yescharacteristic
208C23×C79C4×D411yescharacteristic
209C22×D79—21no—
210Q8×C79—21no—
211Q8×C79—21no—
212Q8×C79—21no—
213Q8×C79—42no—
214Q8×C79#13112yes—
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—21no—
222C2×C4×C79—21no—
223C2×C4×C79—41no—
224C2×C4×C79—42no—
225C2×C4×C79—81no—
226C2×C4×C79C4×D411yescharacteristic
227C2×C4×C79C4×D411yescharacteristic
228C2×C4×C79C4×D411yescharacteristic
229C2×C4×C79C23×C411yescharacteristic
230C2×C4×C79C22×D411yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C22⋊D79—81no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236D316—21no—
237D316—21no—
238C4×D79—41no—
239C4×D79—41no—
240Dic158—21no—
241Dic158—21no—
242C8×C79—41no—
243C8×C79—41no—
244C8×C79—41no—
245C8×C79—41no—
246C79⋊C8—21no—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—21no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C79⋊C8#13712yes—
256C22×C79—21no—
257C22×C79—21no—
258C22×C79—21no—
259C22×C79C2×C4×D411yescharacteristic
260C2×D79—41no—
261C2×D79—41no—
262C2×D79—41no—
263C4×C79—21no—
264C4×C79—22no—
265C4×C79—21no—
266C4×C79—21no—
267C4×C79—21no—
268C4×C79—41no—
269C4×C79—41no—
270C4×C79—41no—
271C4×C79C2×#13111yescharacteristic
272C4×C79#56811yescharacteristic
273C4×C79C2×#13711yescharacteristic
274C4×C79#57511yescharacteristic
275Dic79—41no—
276Dic79—41no—
277Dic79—41no—
278#18010—791nomaximal, Sylow
279C2×C79—41no—
280C2×C79#293311yescharacteristic
281C2×C79C4×#13711yescharacteristic
282C2×C79#295711yescharacteristic
283D79—81no—
284#3108—791no—
285#2969—791no—
286Q8×M16—791no—
287#1592—791no—
288#1591—791no—
289#1588—791no—
290#1587—791no—
291#1584—791no—
292#1582—791no—
293#1579—791no—
294C8×SD16—791no—
295#1574—791no—
296#1495—791no—
297#1479D7911yescharacteristic, complemented
298#1453—791no—
299C79#1801011yescharacteristic, complemented, Sylow
300#529—791no—
301C2×C4×Q8C2×D7911yescharacteristic
302#483—791no—
303#480—791no—
304#479—791no—
305#479—791no—
306#464—791no—
307#464—791no—
308#451—791no—
309#448—791no—
310#446—791no—
311#446—791no—
312#446—791no—
313#446—1581no—
314#446—1581no—
315C8×Q8—791no—
316C8×Q8—791no—
317C8×Q8—791no—
318C4×Q16—791no—
319C4×Q16—791no—
320C4×SD16—791no—
321C4×SD16—791no—
322#436—791no—
323#436—1581no—
324#435—1581no—
325C8×D4—791no—
326#433—1581no—
327#431—791no—
328#431C2×D7911yescharacteristic
329#431C2×D7911yescharacteristic
330#423—791no—
331#423—1581no—
332#405—1581no—
333C4×M16—791no—
334C4×M16—1581no—
335#335—791no—
336#335—791no—
337#334—791no—
338#334—791no—
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8C2×D7911yescharacteristic
342Q8⋊C8C2×D7911yescharacteristic
343Q8⋊C8C2×D7911yescharacteristic
344Q8⋊C8C2×D7911yescharacteristic
345D4⋊C8—791no—
346D4⋊C8—791no—
347#322—791no—
348#322—791no—
349#322—791no—
350C82—791no—
351C22×Q8C4×D7911yescharacteristic
352C2×Q16—791no—
353C2×Q16—791no—
354C2×SD16—791no—
355C2×SD16—791no—
356C2×M16—791no—
357C2×M16—791no—
358C2×M16—1581no—
359C2×M16—1581no—
360C22×C8—1581no—
361#126—1581no—
362#125—791no—
363#124—791no—
364#123—1581no—
365C22⋊Q8—791no—
366#121—791no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8—791no—
370C4×Q8C4×D7911yescharacteristic
371C4×Q8C4×D7911yescharacteristic
372C4×Q8C22×D7911yescharacteristic
373C4×Q8C22×D7911yescharacteristic
374C4×D4—791no—
375C4×D4—1581no—
376#117—1581no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C4×D7911yescharacteristic
379C2×C42—21no—
380C2×C42C22×D7911yescharacteristic
381#107—791no—
382#107—791no—
383#106—791no—
384#106—791no—
385C4⋊C8—21no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—791no—
389C4⋊C8—791no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8—1581no—
395C4⋊C8—1581no—
396C4⋊C8—1581no—
397C4⋊C8C22×D7911yescharacteristic
398C4⋊C8C22×D7911yescharacteristic
399#103—791no—
400#103—791no—
401#103—791no—
402#103—791no—
403#103—791no—
404#103—791no—
405#102—791no—
406#102—791no—
407C22⋊C8—21no—
408C22⋊C8—21no—
409C22⋊C8—791no—
410C22⋊C8—791no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413#97—791no—
414#97—791no—
415#97—1581no—
416#97—1581no—
417#97—1581no—
418C4×C8—21no—
419C4×C8—791no—
420C4×C8—791no—
421C4×C8—791no—
422C4×C8—791no—
423C4×C8—1581no—
424C4×C8—1581no—
425C4×C8—1581no—
426C4×C8C22×D7911yescharacteristic
427C4×C8C22×D7911yescharacteristic
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—41no—
431C2×Q8—791no—
432C2×Q8C2×C4×D7911yescharacteristic
433C2×Q8C2×C4×D7911yescharacteristic
434C2×D4—791no—
435C2×D4—1581no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—1581no—
439C22×C4C2×C4×D7911yescharacteristic
440C22×C4D4×D7911yescharacteristic
441C22×C4#1192962311yescharacteristic
442Q16—3161no—
443Q16—1582no—
444SD16—3161no—
445SD16—1582no—
446M16—3162no—
447M16—3162no—
448M16—3161no—
449M16—1582no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—3162no—
455C2×C8—791no—
456C2×C8—791no—
457C2×C8—791no—
458C2×C8—791no—
459C2×C8—791no—
460C2×C8—791no—
461C2×C8—791no—
462C2×C8—791no—
463C2×C8—1581no—
464C2×C8—1581no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—791no—
471C4⋊C4—791no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4C2×C4×D7911yescharacteristic
477C4⋊C4C2×C4×D7911yescharacteristic
478C22⋊C4—1581no—
479C22⋊C4—1581no—
480C22⋊C4—1581no—
481C22⋊C4—1581no—
482C22⋊C4—1581no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—1581no—
488C42C2×C4×D7911yescharacteristic
489C42D4×D7911yescharacteristic
490C42#1192962311yescharacteristic
491C42C23×D7911yescharacteristic
492C23—1581no—
493C23(C22⋊D79)⋊C411yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—42no—
498Q8—1581no—
499Q8—1581no—
500Q8#1457916712yes—
501D4—6321no—
502D4—1581no—
503D4—1581no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—81no—
515C2×C4—3161no—
516C2×C4—3161no—
517C2×C4—1581no—
518C2×C4—1581no—
519C2×C4—1581no—
520C2×C4(C22⋊D79)⋊C411yescharacteristic
521C2×C4(C22⋊D79)⋊C411yescharacteristic
522C2×C4(C22⋊D79)⋊C411yescharacteristic
523C2×C4C22×C4×D7911yescharacteristic
524C2×C4C2×D4×D7911yescharacteristic
525C2×C4C2×#1192962311yescharacteristic
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—3161no—
531C8—3161no—
532C8—792no—
533C8—1581no—
534C8—1581no—
535C8—1581no—
536C8—1581no—
537C8—1581no—
538C8—1581no—
539C8—1581no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—3161no—
544C22—3161no—
545C22—3161no—
546C22C2×(C22⋊D79)⋊C411yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—3161no—
556C4—3161no—
557C4—3161no—
558C4C2×#1457916711yescharacteristic
559C4#1909932911yescharacteristic
560C4#137×D7911yescharacteristic
561C4#1909936211yescharacteristic
562C2—41no—
563C2—6321no—
564C2#2377212111yescharacteristic
565C2#2377221511yescharacteristic
566C2#2377231311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1479C2
socleC22×C79C2×C4×D4
prime radical#14579135C2×C4
cosocleC42×C79C24
hypercenter#1479D79
hypocenterC79#18010
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
3. Q8×#11929608C2semidirect
4. #23762305C2semidirect
5. #23762294C2semidirect
6. #23762275C2semidirect
7. #23762274C2semidirect
8. #23762247C2semidirect
9. #23762239C2semidirect
10. #23762216C2semidirect
11. #23762197C2semidirect
12. #23762188C2semidirect
14. #23761695C2semidirect
15. C79×#1479C2semidirect
297. #1479D79semidirect
299. C79#18010semidirect

Maximal subgroups

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

subgroupindex
#237734802
#237723982
Q8×#119296082
#237623052
#237622942
#237622752
#237622742
#237622472
#237622392
#237622162
#237621972
#237621882
#237618662
#237616952
C79×#14792
#1801079