Finite Group Databasebeta

← #34259665

Subgroups — #34259665

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#23772502C211yesmaximal, characteristic
2#23772489C211yesmaximal, characteristic
3#23762309C211yesmaximal, characteristic, complemented
4#23762297C211yesmaximal, characteristic, complemented
5#23762272C211yesmaximal, characteristic, complemented
6#23762272C211yesmaximal, characteristic, complemented
7#23762251C211yesmaximal, characteristic, complemented
8#23762251C211yesmaximal, characteristic, complemented
9#23762219C211yesmaximal, characteristic, complemented
10Q16×C79⋊C8C211yesmaximal, characteristic, complemented
11#23762180C211yesmaximal, characteristic
12#23761873C211yesmaximal, characteristic
13#23761795C211yesmaximal, characteristic
14#23761700C211yesmaximal, characteristic
15C79×#3104C211yesmaximal, characteristic
16#19098694—21no—
17#19098694—21no—
18#19098694—21no—
19#19098694C2211yescharacteristic
20#19098694C2211yescharacteristic
21#19098694C2211yescharacteristic
22#19098694C2211yescharacteristic
23Q8×C79⋊C8—21no—
24Q8×C79⋊C8C2211yescharacteristic
25Q8×C79⋊C8C2211yescharacteristic
26Q8×C79⋊C8C2211yescharacteristic
27Q8×C79⋊C8C2211yescharacteristic
28#19098624—21no—
29#19098624—21no—
30#19098620—21no—
31#19098620C2211yescharacteristic
32#19098616—21no—
33#19098616C2211yescharacteristic
34#19098616C2211yescharacteristic
35#19098616C2211yescharacteristic
36#19098316C2211yescharacteristic
37#19098316C2211yescharacteristic
38#19098314C2211yescharacteristic
39#19098314C2211yescharacteristic
40Q8⋊(C79⋊C8)C2211yescharacteristic
41Q8⋊(C79⋊C8)C2211yescharacteristic
42Q8⋊(C79⋊C8)C2211yescharacteristic
43Q8⋊(C79⋊C8)C2211yescharacteristic
44Q8⋊(C79⋊C8)C2211yescharacteristic
45Q8⋊(C79⋊C8)C2211yescharacteristic
46Q8⋊(C79⋊C8)C2211yescharacteristic
47Q8⋊(C79⋊C8)C2211yescharacteristic
48#19098285C2211yescharacteristic
49#19098285C2211yescharacteristic
50#19098284C2211yescharacteristic
51C8×C79⋊C8C2211yescharacteristic
52#533×C79C2211yescharacteristic
53C2×C4×Q8×C79C2211yescharacteristic
54#484×C79C411yescharacteristic
55#484×C79C411yescharacteristic
56#477×C79C411yescharacteristic
57#477×C79C411yescharacteristic
58#462×C79C411yescharacteristic
59#462×C79C411yescharacteristic
60#451×C79C411yescharacteristic
61#451×C79C411yescharacteristic
62C4×Q16×C79C2211yescharacteristic
63C4×Q16×C79C2211yescharacteristic
64C4×Q16×C79C2211yescharacteristic
65C4×Q16×C79C2211yescharacteristic
66#431×C79C2211yescharacteristic
67C4⋊(C79⋊C8)—21no—
68C4⋊(C79⋊C8)—21no—
69C4⋊(C79⋊C8)—21no—
70C4⋊(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)D411yescharacteristic
86C22⋊(C79⋊C8)D411yescharacteristic
87#14579072—21no—
88#14579072—21no—
89#14579072—21no—
90#14579072—21no—
91#14579072—21no—
92C4×C79⋊C8—21no—
93C4×C79⋊C8—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
101C22×Q8×C79C2×C411yescharacteristic
102C2×Q16×C79C2×C411yescharacteristic
103C2×Q16×C79C2×C411yescharacteristic
104C2×Q16×C79C2×C411yescharacteristic
105C2×Q16×C79C2×C411yescharacteristic
106C4⋊Q8×C79C2×C411yescharacteristic
107C4⋊Q8×C79C2×C411yescharacteristic
108#125×C79—21no—
109C22⋊Q8×C79—21no—
110C22⋊Q8×C79C2×C411yescharacteristic
111C22⋊Q8×C79C2×C411yescharacteristic
112C4×Q8×C79—21no—
113C4×Q8×C79—21no—
114C4×Q8×C79—21no—
115C4×Q8×C79C2×C411yescharacteristic
116C4×Q8×C79C2×C411yescharacteristic
117C4×Q8×C79C2311yescharacteristic
118C4×Q8×C79C2311yescharacteristic
119C4×Q8×C79C2311yescharacteristic
120C4×Q8×C79C2311yescharacteristic
121#117×C79—21no—
122C2×C4⋊C4×C79—21no—
123C2×C4⋊C4×C79C2×C411yescharacteristic
124C2×C42×C79—21no—
125C2×C42×C79C2311yescharacteristic
126#107×C79C2×C411yescharacteristic
127#107×C79C2×C411yescharacteristic
128#107×C79C2×C411yescharacteristic
129#107×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#103×C79C2×C411yescharacteristic
137#103×C79C2×C411yescharacteristic
138#103×C79C2×C411yescharacteristic
139#103×C79C2×C411yescharacteristic
140C22⋊C8×C79C2×C411yescharacteristic
141C22⋊C8×C79C2×C411yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8—21no—
149C2×C79⋊C8—21no—
150C2×C79⋊C8C2×D411yescharacteristic
151C2×C79⋊C8C2×D411yescharacteristic
152C2×C79⋊C8C2×D411yescharacteristic
153C2×C79⋊C8C2×D411yescharacteristic
154C2×Q8×C79—21no—
155C2×Q8×C79—21no—
156C2×Q8×C79—22no—
157C2×Q8×C79—41no—
158C2×Q8×C79C22×C411yescharacteristic
159C2×Q8×C79C22×C411yescharacteristic
160C2×Q8×C79C22×C411yescharacteristic
161C2×Q8×C79C22×C411yescharacteristic
162C22×C4×C79—21no—
163C22×C4×C79—21no—
164C22×C4×C79C22×C411yescharacteristic
165C22×C4×C79C2×D411yescharacteristic
166C22×C4×C79#4111yescharacteristic
167Q16×C79—22no—
168Q16×C79—22no—
169Q16×C79—41no—
170Q16×C79—41no—
171C2×C8×C79C22×C411yescharacteristic
172C2×C8×C79C22×C411yescharacteristic
173C2×C8×C79C22×C411yescharacteristic
174C2×C8×C79C22×C411yescharacteristic
175C4⋊C4×C79—21no—
176C4⋊C4×C79—21no—
177C4⋊C4×C79—21no—
178C4⋊C4×C79—21no—
179C4⋊C4×C79—21no—
180C4⋊C4×C79—21no—
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79C22×C411yescharacteristic
188C4⋊C4×C79C22×C411yescharacteristic
189C4⋊C4×C79C22×C411yescharacteristic
190C4⋊C4×C79C22×C411yescharacteristic
191C22⋊C4×C79—21no—
192C22⋊C4×C79—21no—
193C42×C79—21no—
194C42×C79—21no—
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—21no—
208Q8×C79—21no—
209Q8×C79—21no—
210Q8×C79—21no—
211Q8×C79—42no—
212Q8×C79—42no—
213Q8×C79#13112yes—
214C2×C4×C79—21no—
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—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—41no—
227C2×C4×C79—42no—
228C2×C4×C79—42no—
229C2×C4×C79C4×D411yescharacteristic
230C2×C4×C79C4×D411yescharacteristic
231C2×C4×C79C4×D411yescharacteristic
232C2×C4×C79C23×C411yescharacteristic
233C2×C4×C79C22×D411yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C8×C79—21no—
236C8×C79—21no—
237C8×C79—21no—
238C8×C79—21no—
239C79⋊C8—21no—
240C79⋊C8—21no—
241C79⋊C8—21no—
242C79⋊C8—21no—
243C79⋊C8—41no—
244C79⋊C8—41no—
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×C79—41no—
262C4×C79—41no—
263C4×C79—41no—
264C4×C79—41no—
265C4×C79C2×#13111yescharacteristic
266C4×C79#56811yescharacteristic
267C4×C79#57811yescharacteristic
268C4×C79#57811yescharacteristic
269#18032—791nomaximal, Sylow
270C2×C79—41no—
271C2×C79#293311yescharacteristic
272C2×C79#295011yescharacteristic
273C2×C79#295511yescharacteristic
274#3104D7911yescharacteristic
275#2986—791no—
276#2984—791no—
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#533C2×D7911yescharacteristic
291C2×C4×Q8C2×D7911yescharacteristic
292#484Dic7911yescharacteristic
293#484Dic7911yescharacteristic
294#477Dic7911yescharacteristic
295#477Dic7911yescharacteristic
296#462Dic7911yescharacteristic
297#462Dic7911yescharacteristic
298#451Dic7911yescharacteristic
299#451Dic7911yescharacteristic
300#446—791no—
301#446—791no—
302#446—791no—
303#446—791no—
304#446—1581no—
305#446—1581no—
306#446—1581no—
307C8×Q8—791no—
308C8×Q8—791no—
309C8×Q8—791no—
310C8×Q8—791no—
311C8×Q8—1581no—
312C4×Q16C2×D7911yescharacteristic
313C4×Q16C2×D7911yescharacteristic
314C4×Q16C2×D7911yescharacteristic
315C4×Q16C2×D7911yescharacteristic
316#433—1581no—
317#433—1581no—
318#432—791no—
319#432—1581no—
320#431—791no—
321#431—791no—
322#431—791no—
323#431—1581no—
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×Q8C2×Dic7911yescharacteristic
342C2×Q16C2×Dic7911yescharacteristic
343C2×Q16C2×Dic7911yescharacteristic
344C2×Q16C2×Dic7911yescharacteristic
345C2×Q16C2×Dic7911yescharacteristic
346C4⋊Q8C2×Dic7911yescharacteristic
347C4⋊Q8C2×Dic7911yescharacteristic
348#125—21no—
349C22⋊Q8—21no—
350C22⋊Q8C2×Dic7911yescharacteristic
351C22⋊Q8C2×Dic7911yescharacteristic
352C4×Q8—21no—
353C4×Q8—21no—
354C4×Q8—21no—
355C4×Q8C2×Dic7911yescharacteristic
356C4×Q8C2×Dic7911yescharacteristic
357C4×Q8C22×D7911yescharacteristic
358C4×Q8C22×D7911yescharacteristic
359C4×Q8C22×D7911yescharacteristic
360C4×Q8C22×D7911yescharacteristic
361#117—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4C2×Dic7911yescharacteristic
364C2×C42—21no—
365C2×C42C22×D7911yescharacteristic
366#107C2×Dic7911yescharacteristic
367#107C2×Dic7911yescharacteristic
368#107C2×Dic7911yescharacteristic
369#107C2×Dic7911yescharacteristic
370C4⋊C8—791no—
371C4⋊C8—791no—
372C4⋊C8—791no—
373C4⋊C8—791no—
374C4⋊C8—791no—
375C4⋊C8—791no—
376C4⋊C8—1581no—
377C4⋊C8—1581no—
378C4⋊C8—1581no—
379C4⋊C8—1581no—
380C4⋊C8—1581no—
381C4⋊C8—1581no—
382C4⋊C8—1581no—
383C4⋊C8—1581no—
384C4⋊C8—1581no—
385C4⋊C8C2×Dic7911yescharacteristic
386C4⋊C8C2×Dic7911yescharacteristic
387#103C2×Dic7911yescharacteristic
388#103C2×Dic7911yescharacteristic
389#103C2×Dic7911yescharacteristic
390#103C2×Dic7911yescharacteristic
391#103C2×Dic7911yescharacteristic
392#103C2×Dic7911yescharacteristic
393#103C2×Dic7911yescharacteristic
394#103C2×Dic7911yescharacteristic
395C22⋊C8—791no—
396C22⋊C8—791no—
397C22⋊C8—1581no—
398C22⋊C8—1581no—
399C22⋊C8—1581no—
400C22⋊C8C2×Dic7911yescharacteristic
401C22⋊C8C2×Dic7911yescharacteristic
402#97—1581no—
403#97—1581no—
404#97—1581no—
405#97—1581no—
406#97—1581no—
407C4×C8—791no—
408C4×C8—791no—
409C4×C8—791no—
410C4×C8—791no—
411C4×C8—1581no—
412C4×C8—1581no—
413C4×C8—1581no—
414C4×C8—1581no—
415C4×C8—1581no—
416C4×C8C22×D7911yescharacteristic
417C4×C8C22×D7911yescharacteristic
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—41no—
422C2×Q8C22×Dic7911yescharacteristic
423C2×Q8C22×Dic7911yescharacteristic
424C2×Q8C22×Dic7911yescharacteristic
425C2×Q8C22×Dic7911yescharacteristic
426C22×C4—21no—
427C22×C4—21no—
428C22×C4C22×Dic7911yescharacteristic
429C22×C4D4×D7911yescharacteristic
430C22×C4#1192962311yescharacteristic
431Q16—22no—
432Q16—22no—
433Q16—41no—
434Q16—41no—
435C2×C8—791no—
436C2×C8—791no—
437C2×C8—791no—
438C2×C8—791no—
439C2×C8—1581no—
440C2×C8—1581no—
441C2×C8—1581no—
442C2×C8—1581no—
443C2×C8—1581no—
444C2×C8—1581no—
445C2×C8C22×Dic7911yescharacteristic
446C2×C8C22×Dic7911yescharacteristic
447C2×C8C22×Dic7911yescharacteristic
448C2×C8C22×Dic7911yescharacteristic
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4C22×Dic7911yescharacteristic
462C4⋊C4C22×Dic7911yescharacteristic
463C4⋊C4C22×Dic7911yescharacteristic
464C4⋊C4C22×Dic7911yescharacteristic
465C22⋊C4—21no—
466C22⋊C4—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42C22×Dic7911yescharacteristic
474C42D4×D7911yescharacteristic
475C42#1192962311yescharacteristic
476C42C23×D7911yescharacteristic
477C23D4×Dic7911yescharacteristic
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—21no—
484Q8—21no—
485Q8—42no—
486Q8—42no—
487Q8#1457916812yes—
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—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4D4×Dic7911yescharacteristic
504C2×C4D4×Dic7911yescharacteristic
505C2×C4D4×Dic7911yescharacteristic
506C2×C4C23×Dic7911yescharacteristic
507C2×C4C2×D4×D7911yescharacteristic
508C2×C4C2×#1192962311yescharacteristic
509C8—21no—
510C8—21no—
511C8—21no—
512C8—21no—
513C8—3161no—
514C8—3161no—
515C8—3161no—
516C8—3161no—
517C8—3161no—
518C8—3161no—
519C8—1581no—
520C8—1581no—
521C8—1581no—
522C8—1581no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22C2×D4×Dic7911yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—22no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4C2×#1457916811yescharacteristic
540C4#1909933011yescharacteristic
541C4#1909937611yescharacteristic
542C4#1909937611yescharacteristic
543C2—41no—
544C2#2377212711yescharacteristic
545C2#2377226811yescharacteristic
546C2#2377230311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#3104C2
socleC22×C79C2×C4×D4
prime radicalC23×C79C4×D4
cosocleC42×C79C24
hypercenter#3104D79
hypocenterC79#18032
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
3. #23762309C2semidirect
4. #23762297C2semidirect
5. #23762272C2semidirect
6. #23762272C2semidirect
7. #23762251C2semidirect
8. #23762251C2semidirect
9. #23762219C2semidirect
10. Q16×C79⋊C8C2semidirect
289. C79#18032semidirect

Maximal subgroups

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

subgroupindex
#237725022
#237724892
#237623092
#237622972
#237622722
#237622512
#237622192
Q16×C79⋊C82
#237621802
#237618732
#237617952
#237617002
C79×#31042
#1803279