Finite Group Databasebeta

← #34259667

Subgroups — #34259667

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#23773440C211yesmaximal, characteristic
2#23772501C211yesmaximal, characteristic
3#23772487C211yesmaximal, characteristic
4#23762310C211yesmaximal, characteristic, complemented
5#23762298C211yesmaximal, characteristic, complemented
6#23762273C211yesmaximal, characteristic, complemented
7#23762273C211yesmaximal, characteristic, complemented
8#23762252C211yesmaximal, characteristic, complemented
9#23762252C211yesmaximal, characteristic, complemented
10#23762220C211yesmaximal, characteristic, complemented
11C8×Dic316C211yesmaximal, characteristic, complemented
12#23762180C211yesmaximal, characteristic
13#23761874C211yesmaximal, characteristic
14#23761796C211yesmaximal, characteristic
15C79×#1454C211yesmaximal, characteristic
16#19099120C2211yescharacteristic
17C2×C4×Dic158C2211yescharacteristic
18#19098899C411yescharacteristic
19#19098899C411yescharacteristic
20#19098859C411yescharacteristic
21#19098859C411yescharacteristic
22#19098774C411yescharacteristic
23#19098774C411yescharacteristic
24#19098721C411yescharacteristic
25#19098721C411yescharacteristic
26#19098693C2211yescharacteristic
27#19098693C2211yescharacteristic
28#19098693C2211yescharacteristic
29#19098693C2211yescharacteristic
30#19098692—21no—
31#19098692—21no—
32#19098690—21no—
33C8×Dic158C2211yescharacteristic
34C8×Dic158C2211yescharacteristic
35C8×Dic158C2211yescharacteristic
36C8×Dic158C2211yescharacteristic
37#19098687—21no—
38C4×Dic316C2211yescharacteristic
39C4×Dic316C2211yescharacteristic
40C4×Dic316C2211yescharacteristic
41C4×Dic316C2211yescharacteristic
42#19098621—21no—
43#19098621—21no—
44#19098620C2211yescharacteristic
45#19098617—21no—
46#19098616C2211yescharacteristic
47#19098613—21no—
48#19098316C2211yescharacteristic
49#19098316C2211yescharacteristic
50#19098314C2211yescharacteristic
51#19098314C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098295C2211yescharacteristic
55#19098295C2211yescharacteristic
56#19098295C2211yescharacteristic
57#19098295C2211yescharacteristic
58#19098295C2211yescharacteristic
59#19098295C2211yescharacteristic
60#431×C79C2211yescharacteristic
61#431×C79C2211yescharacteristic
62#431×C79C2211yescharacteristic
63#322×C79C2211yescharacteristic
64#322×C79C2211yescharacteristic
65#322×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C22×Dic158C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×Dic316C2×C411yescharacteristic
70C2×Dic316C2×C411yescharacteristic
71C2×Dic316C2×C411yescharacteristic
72#14579161C2×C411yescharacteristic
73#14579161C2×C411yescharacteristic
74#14579153—21no—
75#14579140—21no—
76#14579140C2×C411yescharacteristic
77#14579140C2×C411yescharacteristic
78C4×Dic158—21no—
79C4×Dic158—21no—
80C4×Dic158—21no—
81C4×Dic158C2×C411yescharacteristic
82C4×Dic158C2×C411yescharacteristic
83C4×Dic158C2311yescharacteristic
84C4×Dic158C2311yescharacteristic
85C4×Dic158C2311yescharacteristic
86C4×Dic158C2311yescharacteristic
87#14579120—21no—
88C2×C4⋊Dic79C2×C411yescharacteristic
89C2×Dic79⋊C4—21no—
90C2×C4×Dic79—21no—
91#14579095C2×C411yescharacteristic
92#14579095C2×C411yescharacteristic
93#14579095C2×C411yescharacteristic
94#14579095C2×C411yescharacteristic
95C4⋊(C79⋊C8)C2311yescharacteristic
96C4⋊(C79⋊C8)C2311yescharacteristic
97C4⋊(C79⋊C8)C2311yescharacteristic
98C4⋊(C79⋊C8)C2311yescharacteristic
99Dic79⋊C8—21no—
100Dic79⋊C8—21no—
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106Dic79⋊C8—21no—
107#14579086C2×C411yescharacteristic
108#14579086C2×C411yescharacteristic
109#14579086C2×C411yescharacteristic
110#14579086C2×C411yescharacteristic
111#14579086C2×C411yescharacteristic
112#14579086C2×C411yescharacteristic
113#14579086C2×C411yescharacteristic
114#14579086C2×C411yescharacteristic
115C22⋊(C79⋊C8)—21no—
116C22⋊(C79⋊C8)—21no—
117#14579072—21no—
118#14579071—21no—
119#14579071—21no—
120#14579071—21no—
121#14579071—21no—
122C4×C79⋊C8—21no—
123C8×Dic79—21no—
124C8×Dic79—21no—
125C8×Dic79—21no—
126C8×Dic79—21no—
127C2×C42×C79C2311yescharacteristic
128C4⋊C8×C79—21no—
129C4⋊C8×C79C2×C411yescharacteristic
130C4⋊C8×C79C2×C411yescharacteristic
131C4⋊C8×C79D411yescharacteristic
132C4⋊C8×C79D411yescharacteristic
133C22⋊C8×C79—21no—
134C22⋊C8×C79C2×C411yescharacteristic
135C22⋊C8×C79C2×C411yescharacteristic
136C22⋊C8×C79D411yescharacteristic
137C22⋊C8×C79D411yescharacteristic
138C4×C8×C79C2311yescharacteristic
139C4×C8×C79C2311yescharacteristic
140C4×C8×C79C2311yescharacteristic
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—22no—
145C2×Dic158—21no—
146C2×Dic158—21no—
147C2×Dic158—41no—
148C2×Dic158C22×C411yescharacteristic
149C2×Dic158C22×C411yescharacteristic
150C2×Dic158C22×C411yescharacteristic
151C2×Dic158C22×C411yescharacteristic
152C22×Dic79—21no—
153C22×Dic79—21no—
154Dic316—22no—
155Dic316—22no—
156Dic316—41no—
157Dic316—41no—
158C2×C79⋊C8—21no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8—21no—
162C4⋊Dic79—21no—
163C4⋊Dic79—21no—
164C4⋊Dic79—21no—
165C4⋊Dic79—21no—
166C4⋊Dic79C22×C411yescharacteristic
167C4⋊Dic79C22×C411yescharacteristic
168C4⋊Dic79C22×C411yescharacteristic
169C4⋊Dic79C22×C411yescharacteristic
170Dic79⋊C4—21no—
171Dic79⋊C4—21no—
172Dic79⋊C4—21no—
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178C22⋊Dic79—21no—
179C22⋊Dic79—21no—
180C4×Dic79—21no—
181C4×Dic79—21no—
182C4×Dic79—21no—
183C4×Dic79—21no—
184C4×Dic79—21no—
185C4×Dic79—21no—
186C22×C4×C79C22×C411yescharacteristic
187C22×C4×C79C2×D411yescharacteristic
188C22×C4×C79#4111yescharacteristic
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79C22×C411yescharacteristic
192C2×C8×C79C22×C411yescharacteristic
193C2×C8×C79C22×C411yescharacteristic
194C2×C8×C79C22×C411yescharacteristic
195C2×C8×C79C2×D411yescharacteristic
196C2×C8×C79C2×D411yescharacteristic
197C2×C8×C79C2×D411yescharacteristic
198C2×C8×C79C2×D411yescharacteristic
199C42×C79C22×C411yescharacteristic
200C42×C79C2×D411yescharacteristic
201C42×C79#4111yescharacteristic
202C42×C79C2411yescharacteristic
203C23×C79C4×D411yescharacteristic
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—41no—
209C2×C4×C79C4×D411yescharacteristic
210C2×C4×C79C4×D411yescharacteristic
211C2×C4×C79C4×D411yescharacteristic
212C2×C4×C79C23×C411yescharacteristic
213C2×C4×C79C22×D411yescharacteristic
214C2×C4×C79C2×#4111yescharacteristic
215C2×Dic79—21no—
216C2×Dic79—21no—
217C2×Dic79—21no—
218C2×Dic79—21no—
219C2×Dic79—21no—
220C2×Dic79—21no—
221C2×Dic79—21no—
222C2×Dic79—21no—
223C2×Dic79—42no—
224C2×Dic79—42no—
225Dic158—21no—
226Dic158—21no—
227Dic158—21no—
228Dic158—21no—
229Dic158—21no—
230Dic158—21no—
231Dic158—21no—
232Dic158—42no—
233Dic158—42no—
234Dic158#13112yes—
235C8×C79—21no—
236C8×C79—21no—
237C8×C79—21no—
238C8×C79—21no—
239C8×C79—21no—
240C8×C79—21no—
241C8×C79—21no—
242C8×C79—21no—
243C8×C79—41no—
244C8×C79—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—21no—
255C4×C79—21no—
256C4×C79—21no—
257C4×C79C2×#13111yescharacteristic
258C4×C79#56811yescharacteristic
259C4×C79#57811yescharacteristic
260C4×C79#57811yescharacteristic
261Dic79—22no—
262Dic79—41no—
263Dic79—41no—
264Dic79—41no—
265Dic79—41no—
266Dic79—41no—
267Dic79—41no—
268Dic79—41no—
269#18032—791nomaximal, Sylow
270C2×C79—41no—
271C2×C79#293311yescharacteristic
272C2×C79#295011yescharacteristic
273C2×C79#295511yescharacteristic
274#3104—791no—
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#1454D7911yescharacteristic
289C79#1803211yescharacteristic, complemented, Sylow
290#533—791no—
291C2×C4×Q8—791no—
292#484—791no—
293#484—791no—
294#477—791no—
295#477—791no—
296#462—791no—
297#462—791no—
298#451—791no—
299#451—791no—
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×Q16—791no—
313C4×Q16—791no—
314C4×Q16—791no—
315C4×Q16—791no—
316#433—1581no—
317#433—1581no—
318#432—791no—
319#432—1581no—
320#431—791no—
321#431—1581no—
322#431C2×D7911yescharacteristic
323#431C2×D7911yescharacteristic
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#322C2×D7911yescharacteristic
338#322C2×D7911yescharacteristic
339#322C2×D7911yescharacteristic
340C82C2×D7911yescharacteristic
341C22×Q8—791no—
342C2×Q16—791no—
343C2×Q16—791no—
344C2×Q16—791no—
345C2×Q16—791no—
346C4⋊Q8—791no—
347C4⋊Q8—791no—
348#125—1581no—
349C22⋊Q8—791no—
350C22⋊Q8—791no—
351C22⋊Q8—1581no—
352C4×Q8—791no—
353C4×Q8—791no—
354C4×Q8—791no—
355C4×Q8—791no—
356C4×Q8—791no—
357C4×Q8—791no—
358C4×Q8—1581no—
359C4×Q8—1581no—
360C4×Q8—1581no—
361#117—1581no—
362C2×C4⋊C4—791no—
363C2×C4⋊C4—1581no—
364C2×C42—1581no—
365C2×C42C22×D7911yescharacteristic
366#107—791no—
367#107—791no—
368#107—791no—
369#107—791no—
370C4⋊C8—21no—
371C4⋊C8—791no—
372C4⋊C8—791no—
373C4⋊C8—791no—
374C4⋊C8—791no—
375C4⋊C8—1581no—
376C4⋊C8—1581no—
377C4⋊C8—1581no—
378C4⋊C8—1581no—
379C4⋊C8—1581no—
380C4⋊C8—1581no—
381C4⋊C8—1581no—
382C4⋊C8—1581no—
383C4⋊C8C4×D7911yescharacteristic
384C4⋊C8C4×D7911yescharacteristic
385C4⋊C8D31611yescharacteristic
386C4⋊C8D31611yescharacteristic
387#103—791no—
388#103—791no—
389#103—791no—
390#103—791no—
391#103—791no—
392#103—791no—
393#103—791no—
394#103—791no—
395C22⋊C8—21no—
396C22⋊C8—1581no—
397C22⋊C8—1581no—
398C22⋊C8C4×D7911yescharacteristic
399C22⋊C8C4×D7911yescharacteristic
400C22⋊C8D31611yescharacteristic
401C22⋊C8D31611yescharacteristic
402#97—1581no—
403#97—1581no—
404#97—1581no—
405#97—1581no—
406#97—1581no—
407C4×C8—1581no—
408C4×C8—1581no—
409C4×C8—1581no—
410C4×C8—1581no—
411C4×C8—1581no—
412C4×C8C22×D7911yescharacteristic
413C4×C8C22×D7911yescharacteristic
414C4×C8C22×D7911yescharacteristic
415C4×C8C22×D7911yescharacteristic
416C4×C8C22×D7911yescharacteristic
417C4×C8C22×D7911yescharacteristic
418C2×Q8—3161no—
419C2×Q8—791no—
420C2×Q8—791no—
421C2×Q8—791no—
422C2×Q8—791no—
423C2×Q8—1582no—
424C2×Q8—1581no—
425C2×Q8—1581no—
426C22×C4—1581no—
427C22×C4—1581no—
428C22×C4C2×C4×D7911yescharacteristic
429C22×C4C2×D31611yescharacteristic
430C22×C4#1192962411yescharacteristic
431Q16—3161no—
432Q16—3161no—
433Q16—1582no—
434Q16—1582no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—1581no—
438C2×C8—1581no—
439C2×C8—1581no—
440C2×C8—1581no—
441C2×C8C2×C4×D7911yescharacteristic
442C2×C8C2×C4×D7911yescharacteristic
443C2×C8C2×C4×D7911yescharacteristic
444C2×C8C2×C4×D7911yescharacteristic
445C2×C8C2×D31611yescharacteristic
446C2×C8C2×D31611yescharacteristic
447C2×C8C2×D31611yescharacteristic
448C2×C8C2×D31611yescharacteristic
449C4⋊C4—791no—
450C4⋊C4—791no—
451C4⋊C4—791no—
452C4⋊C4—791no—
453C4⋊C4—1581no—
454C4⋊C4—1581no—
455C4⋊C4—1581no—
456C4⋊C4—1581no—
457C4⋊C4—1581no—
458C4⋊C4—1581no—
459C4⋊C4—1581no—
460C4⋊C4—1581no—
461C4⋊C4—1581no—
462C4⋊C4—1581no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
465C22⋊C4—1581no—
466C22⋊C4—1581no—
467C42—1581no—
468C42—1581no—
469C42—1581no—
470C42—1581no—
471C42—1581no—
472C42—1581no—
473C42C2×C4×D7911yescharacteristic
474C42C2×D31611yescharacteristic
475C42#1192962411yescharacteristic
476C42C23×D7911yescharacteristic
477C23C4×D31611yescharacteristic
478Q8—3162no—
479Q8—3162no—
480Q8—792no—
481Q8—1581no—
482Q8—1581no—
483Q8—1581no—
484Q8—1581no—
485Q8—1581no—
486Q8—1581no—
487Q8—1581no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—41no—
493C2×C4—3162no—
494C2×C4—3162no—
495C2×C4—1581no—
496C2×C4—1581no—
497C2×C4—1581no—
498C2×C4—1581no—
499C2×C4—1581no—
500C2×C4—1581no—
501C2×C4—1581no—
502C2×C4—1581no—
503C2×C4C4×D31611yescharacteristic
504C2×C4C4×D31611yescharacteristic
505C2×C4C4×D31611yescharacteristic
506C2×C4C22×C4×D7911yescharacteristic
507C2×C4C22×D31611yescharacteristic
508C2×C4C2×#1192962411yescharacteristic
509C8—21no—
510C8—21no—
511C8—21no—
512C8—21no—
513C8—21no—
514C8—21no—
515C8—21no—
516C8—21no—
517C8—41no—
518C8—41no—
519C8—3161no—
520C8—3161no—
521C8—3161no—
522C8—3161no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22C2×C4×D31611yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—3161no—
532C4—3161no—
533C4—3161no—
534C4—3161no—
535C4—3161no—
536C4—3161no—
537C4—3161no—
538C4—1582no—
539C4#131×D7911yescharacteristic
540C4#1909932811yescharacteristic
541C4#1909938011yescharacteristic
542C4#1909938011yescharacteristic
543C2—41no—
544C2#2377212811yescharacteristic
545C2#2377226911yescharacteristic
546C2#2377230411yescharacteristic
subgroupG/H
centerC2×C4C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1454C2
socleC22×C79C2×C4×D4
prime radicalC23×C79C4×D4
cosocleC42×C79C24
hypercenter#1454D79
hypocenterC79#18032
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1454

Complements

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

NH ≅ G/Ntype
4. #23762310C2semidirect
5. #23762298C2semidirect
6. #23762273C2semidirect
7. #23762273C2semidirect
8. #23762252C2semidirect
9. #23762252C2semidirect
10. #23762220C2semidirect
11. C8×Dic316C2semidirect
289. C79#18032semidirect

Maximal subgroups

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

subgroupindex
#237734402
#237725012
#237724872
#237623102
#237622982
#237622732
#237622522
#237622202
C8×Dic3162
#237621802
#237618742
#237617962
C79×#14542
#1803279