Finite Group Databasebeta

← #34181010

Subgroups — #34181010

Order 20224.

698 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#121×Dic79C211yesmaximal, characteristic
2#23763378C211yesmaximal, characteristic, complemented
3C2×#19098297C211yesmaximal, characteristic
4#23761626C211yesmaximal, characteristic, complemented
5#23761280C211yesmaximal, characteristic, complemented
6#23761279C211yesmaximal, characteristic, complemented
7C2×#327×C79C211yesmaximal, characteristic
8C2×D4×Dic79—21no—
9C2×D4×Dic79C411yescharacteristic, complemented
10C22×C4×Dic79C2211yescharacteristic
11C2×Dic79⋊C8—21no—
12C2×C22⋊(C79⋊C8)C2211yescharacteristic
13#19098466C411yescharacteristic, complemented
14#19098462C411yescharacteristic, complemented
15#19098456—21no—
16#19098443—21no—
17#19098423C2211yescharacteristic
18C4⋊C4×Dic79C411yescharacteristic, complemented
19C22⋊C4×Dic79—21no—
20C2×C79⋊#95—21no—
21C2×C79⋊#95—21no—
22C2×C79⋊#95C2211yescharacteristic
23#19098317—21no—
24#19098297—22no—
25C2×#121×C79C2211yescharacteristic
26C2×C22⋊C8×C79C2211yescharacteristic
27C2×#95×C79C2211yescharacteristic
28#327×C79—22no—
29C23×Dic79—21no—
30C23×Dic79—21no—
31C23×Dic79D411yescharacteristic
32C23×Dic79D411yescharacteristic
33C22×C79⋊C8—21no—
34D4×Dic79—42no—
35D4×Dic79—42no—
36D4×Dic79—42no—
37C2×C4⋊Dic79—21no—
38C2×C4⋊Dic79C2×C411yescharacteristic
39C2×C4⋊Dic79C2×C411yescharacteristic
40C2×C22⋊Dic79—21no—
41C2×C22⋊Dic79—21no—
42C2×C22⋊Dic79—21no—
43C2×C22⋊Dic79—21no—
44C2×C4×Dic79—21no—
45C2×C4×Dic79—42no—
46C2×C4×Dic79C2×C411yescharacteristic
47C2×C4×Dic79C2×C411yescharacteristic
48C2×C4×Dic79D411yescharacteristic
49C2×C4×Dic79D411yescharacteristic
50Dic79⋊C8—42no—
51C22⋊(C79⋊C8)—21no—
52C22⋊(C79⋊C8)—21no—
53C79⋊#95—24no—
54C79⋊#95—21no—
55C79⋊#95—21no—
56C79⋊#95—21no—
57C79⋊#95—21no—
58C79⋊#95—21no—
59C79⋊#95—42no—
60C22×D4×C79—21no—
61C22×D4×C79C2×C411yescharacteristic
62C23×C4×C79—21no—
63C23×C4×C79C2311yescharacteristic
64C22×C8×C79—21no—
65#121×C79—21no—
66#121×C79—21no—
67#121×C79D412yes—
68#121×C79Q812yes—
69C2×C4⋊C4×C79C2×C411yescharacteristic
70C2×C22⋊C4×C79—21no—
71C22⋊C8×C79—21no—
72C22⋊C8×C79—21no—
73#95×C79—21no—
74#95×C79—21no—
75C22×Dic79—22no—
76C22×Dic79—22no—
77C22×Dic79—22no—
78C22×Dic79—21no—
79C22×Dic79—21no—
80C22×Dic79—21no—
81C22×Dic79—21no—
82C22×Dic79—21no—
83C22×Dic79—21no—
84C22×Dic79—41no—
85C22×Dic79—41no—
86C22×Dic79—41no—
87C22×Dic79—41no—
88C22×Dic79—41no—
89C22×Dic79—42no—
90C22×Dic79—42no—
91C22×Dic79—42no—
92C22×Dic79—41no—
93C22×Dic79—41no—
94C22×Dic79—41no—
95C22×Dic79C22⋊C411yescharacteristic
96C22×Dic79C22⋊C411yescharacteristic
97C22×Dic79C22⋊C411yescharacteristic
98C22×Dic79C22⋊C411yescharacteristic
99C2×C79⋊C8—41no—
100C2×C79⋊C8—41no—
101C2×C79⋊C8—41no—
102C4⋊Dic79—41no—
103C4⋊Dic79—41no—
104C4⋊Dic79—42no—
105C22⋊Dic79—42no—
106C22⋊Dic79—42no—
107C22⋊Dic79—42no—
108C22⋊Dic79—41no—
109C22⋊Dic79—41no—
110C4×Dic79—22no—
111C4×Dic79—22no—
112C4×Dic79—41no—
113C4×Dic79—41no—
114C4×Dic79—42no—
115C24×C79—21no—
116C24×C79C2×D411yescharacteristic
117C2×D4×C79—22no—
118C2×D4×C79—22no—
119C2×D4×C79—21no—
120C2×D4×C79—41no—
121C2×D4×C79—41no—
122C2×D4×C79—41no—
123C2×D4×C79—41no—
124C2×D4×C79—41no—
125C2×D4×C79—41no—
126C2×D4×C79C4⋊C412yes—
127C22×C4×C79—21no—
128C22×C4×C79—22no—
129C22×C4×C79—41no—
130C22×C4×C79—41no—
131C22×C4×C79—41no—
132C22×C4×C79—41no—
133C22×C4×C79—41no—
134C22×C4×C79—41no—
135C22×C4×C79—41no—
136C22×C4×C79—41no—
137C22×C4×C79C22×C411yescharacteristic
138C22×C4×C79C2×D411yescharacteristic
139C22×C4×C79C2×D411yescharacteristic
140C22×C4×C79C2×Q811yescharacteristic
141C22×C4×C79#4111yescharacteristic
142C22×C4×C79#4111yescharacteristic
143C2×C8×C79—41no—
144C2×C8×C79—41no—
145C2×C8×C79—41no—
146C4⋊C4×C79—21no—
147C4⋊C4×C79C4⋊C412yes—
148C22⋊C4×C79—22no—
149C22⋊C4×C79—22no—
150C23×C79—21no—
151C23×C79—21no—
152C23×C79—21no—
153C23×C79—21no—
154C23×C79—22no—
155C23×C79—41no—
156C23×C79—41no—
157C23×C79—41no—
158C23×C79—41no—
159C23×C79—41no—
160C23×C79—41no—
161C23×C79C2×C22⋊C411yescharacteristic
162C23×C79#12111yescharacteristic
163C23×C79C22⋊Q811yescharacteristic
164C23×C79#13511yescharacteristic
165C23×C79#13511yescharacteristic
166C23×C79AGL(1,ℤ/8)11yescharacteristic
167C23×C79AGL(1,ℤ/8)11yescharacteristic
168D4×C79—42no—
169D4×C79—42no—
170D4×C79—42no—
171D4×C79—42no—
172D4×C79—42no—
173D4×C79—42no—
174C2×C4×C79—21no—
175C2×C4×C79—21no—
176C2×C4×C79—21no—
177C2×C4×C79—21no—
178C2×C4×C79—22no—
179C2×C4×C79—41no—
180C2×C4×C79—41no—
181C2×C4×C79—41no—
182C2×C4×C79—41no—
183C2×C4×C79—41no—
184C2×C4×C79—41no—
185C2×C4×C79—42no—
186C2×C4×C79—42no—
187C2×C4×C79—42no—
188C2×C4×C79—81no—
189C2×C4×C79—81no—
190C2×C4×C79—81no—
191C2×C4×C79—81no—
192C2×C4×C79C2×C4⋊C411yescharacteristic
193C2×C4×C79#11711yescharacteristic
194C2×C4×C79#12111yescharacteristic
195C2×C4×C79C22⋊Q811yescharacteristic
196C2×Dic79—21no—
197C2×Dic79—21no—
198C2×Dic79—21no—
199C2×Dic79—21no—
200C2×Dic79—21no—
201C2×Dic79—21no—
202C2×Dic79—21no—
203C2×Dic79—21no—
204C2×Dic79—21no—
205C2×Dic79—21no—
206C2×Dic79—42no—
207C2×Dic79—42no—
208C2×Dic79—42no—
209C2×Dic79—42no—
210C2×Dic79—42no—
211C2×Dic79—42no—
212C2×Dic79—42no—
213C2×Dic79—41no—
214C2×Dic79—41no—
215C2×Dic79—41no—
216C2×Dic79—41no—
217C2×Dic79—41no—
218C2×Dic79—41no—
219C2×Dic79—84no—
220C2×Dic79—81no—
221C2×Dic79—81no—
222C2×Dic79—81no—
223C2×Dic79—81no—
224C2×Dic79#9912yes—
225C2×Dic79#10412yes—
226C8×C79—81no—
227C79⋊C8—81no—
228C22×C79—21no—
229C22×C79—21no—
230C22×C79—21no—
231C22×C79—21no—
232C22×C79—21no—
233C22×C79—21no—
234C22×C79—21no—
235C22×C79—21no—
236C22×C79—21no—
237C22×C79—21no—
238C22×C79—21no—
239C22×C79—21no—
240C22×C79—42no—
241C22×C79—42no—
242C22×C79—42no—
243C22×C79—81no—
244C22×C79—81no—
245C22×C79—81no—
246C22×C79—81no—
247C22×C79#38011yescharacteristic
248C22×C79C2×#9911yescharacteristic
249C22×C79#41011yescharacteristic
250C22×C79#41611yescharacteristic
251C22×C79#41811yescharacteristic
252C22×C79C2×#10411yescharacteristic
253C22×C79#42111yescharacteristic
254C22×C79#46311yescharacteristic
255C22×C79#46311yescharacteristic
256C22×C79#47811yescharacteristic
257C22×C79#47811yescharacteristic
258C4×C79—21no—
259C4×C79—21no—
260C4×C79—21no—
261C4×C79—21no—
262C4×C79—42no—
263C4×C79—82no—
264Dic79—22no—
265Dic79—22no—
266Dic79—41no—
267Dic79—41no—
268Dic79—82no—
269Dic79—81no—
270Dic79—81no—
271#8575—791nomaximal, Sylow
272C2×C79—21no—
273C2×C79—21no—
274C2×C79—21no—
275C2×C79—21no—
276C2×C79—82no—
277C2×C79#150111yescharacteristic
278C2×C79#150511yescharacteristic
279C2×C79#151011yescharacteristic
280C2×C79#152111yescharacteristic
281C2×C79#179911yescharacteristic
282C2×C79#180411yescharacteristic
283C2×C79#180611yescharacteristic
284C4×#121—791no—
285#1797—791no—
286C2×#327—791no—
287C2×#327D7911yescharacteristic
288#1434—791no—
289#1294—791no—
290#1294—791no—
291C79#857511yescharacteristic, complemented, Sylow
292C2×#121C2×D7911yescharacteristic
293C2×C4×D4—791no—
294C2×C4×D4—1581no—
295C22×C42—791no—
296C2×C4⋊C8—1581no—
297C2×C22⋊C8—791no—
298C2×C22⋊C8C2×D7911yescharacteristic
299#390—791no—
300#389—791no—
301#388—1581no—
302#386—1581no—
303#380—791no—
304C4×C4⋊C4—791no—
305C4×C22⋊C4—1581no—
306C2×#95—791no—
307C2×#95—1581no—
308C2×#95—1581no—
309C2×#95C2×D7911yescharacteristic
310#336—1581no—
311#327—22no—
312#327—1582no—
313C22×D4—21no—
314C22×D4C4×D7911yescharacteristic
315C23×C4—21no—
316C23×C4—791no—
317C23×C4—791no—
318C23×C4—1581no—
319C23×C4—1581no—
320C23×C4C22×D7911yescharacteristic
321C22×C8—21no—
322C22×C8—1581no—
323#121—21no—
324#121—21no—
325#121Dic15812yes—
326#121C22⋊D7912yes—
327C4×D4—3162no—
328C4×D4—3162no—
329C4×D4—3162no—
330C2×C4⋊C4—791no—
331C2×C4⋊C4—791no—
332C2×C4⋊C4—1581no—
333C2×C4⋊C4C4×D7911yescharacteristic
334C2×C22⋊C4—21no—
335C2×C22⋊C4—1581no—
336C2×C22⋊C4—1581no—
337C2×C22⋊C4—1581no—
338C2×C22⋊C4—1581no—
339C2×C42—3162no—
340C2×C42—791no—
341C2×C42—791no—
342C2×C42—791no—
343C2×C42—791no—
344C2×C42—1581no—
345C4⋊C8—3162no—
346C22⋊C8—21no—
347C22⋊C8—21no—
348C22⋊C8—1581no—
349C22⋊C8—1581no—
350#95—21no—
351#95—21no—
352#95—3162no—
353#95—1581no—
354#95—1581no—
355#95—1584no—
356#95—1581no—
357#95—1581no—
358#95—1581no—
359C24—21no—
360C24D4×D7911yescharacteristic
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4Dic79⋊C412yes—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—3161no—
382C22×C4—3161no—
383C22×C4—3161no—
384C22×C4—3161no—
385C22×C4—3161no—
386C22×C4—3161no—
387C22×C4—3161no—
388C22×C4—3161no—
389C22×C4—3162no—
390C22×C4—3162no—
391C22×C4—3162no—
392C22×C4—791no—
393C22×C4—791no—
394C22×C4—791no—
395C22×C4—791no—
396C22×C4—1582no—
397C22×C4—1581no—
398C22×C4—1581no—
399C22×C4—1582no—
400C22×C4—1582no—
401C22×C4—1581no—
402C22×C4—1581no—
403C22×C4—1581no—
404C22×C4—1581no—
405C22×C4C2×C4×D7911yescharacteristic
406C22×C4D4×D7911yescharacteristic
407C22×C4C2×C22⋊D7911yescharacteristic
408C22×C4C2×Dic15811yescharacteristic
409C22×C4#1192962311yescharacteristic
410C22×C4#1192962311yescharacteristic
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—3161no—
415C2×C8—3161no—
416C2×C8—3161no—
417C4⋊C4—21no—
418C4⋊C4—3161no—
419C4⋊C4—3161no—
420C4⋊C4—3162no—
421C4⋊C4Dic79⋊C412yes—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—3161no—
425C22⋊C4—3161no—
426C22⋊C4—3162no—
427C22⋊C4—3162no—
428C22⋊C4—3162no—
429C42—3161no—
430C42—3161no—
431C42—3162no—
432C42—1582no—
433C42—1582no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—22no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23C22⋊C4×D7911yescharacteristic
446C23#1457913311yescharacteristic
447C23#1457913811yescharacteristic
448C23#1457918011yescharacteristic
449C23#1457918211yescharacteristic
450C23#1457918511yescharacteristic
451C23#1457918811yescharacteristic
452D4—42no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—3162no—
477C2×C4—3162no—
478C2×C4—3162no—
479C2×C4—3162no—
480C2×C4—3162no—
481C2×C4—3162no—
482C2×C4—3162no—
483C2×C4—3161no—
484C2×C4—3161no—
485C2×C4—3161no—
486C2×C4—3161no—
487C2×C4—3161no—
488C2×C4—3161no—
489C2×C4—792no—
490C2×C4—792no—
491C2×C4—6321no—
492C2×C4—6321no—
493C2×C4—6321no—
494C2×C4—6321no—
495C2×C4—6324no—
496C2×C4—1581no—
497C2×C4—1581no—
498C2×C4—1581no—
499C2×C4—1581no—
500C2×C4—1581no—
501C2×C4—1581no—
502C2×C4—1581no—
503C2×C4—1581no—
504C2×C4—1581no—
505C2×C4—1581no—
506C2×C4C2×Dic79⋊C411yescharacteristic
507C2×C4#1457911711yescharacteristic
508C2×C4#1457913311yescharacteristic
509C2×C4#1457913811yescharacteristic
510C8—81no—
511C8—6321no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22#1909842111yescharacteristic
532C22#99×D7911yescharacteristic
533C22#1909853511yescharacteristic
534C22#1909855611yescharacteristic
535C22#1909856611yescharacteristic
536C22#104×D7911yescharacteristic
537C22#1909857811yescharacteristic
538C22#1909877611yescharacteristic
539C22#1909878011yescharacteristic
540C22#1909886111yescharacteristic
541C22#1909886511yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—42no—
547C4—82no—
548C4—3161no—
549C4—3161no—
550C4—6322no—
551C4—6321no—
552C4—6321no—
553C4—1582no—
554C4—1582no—
555C2—21no—
556C2—21no—
557C2—21no—
558C2—21no—
559C2—82no—
560C2#2376188811yescharacteristic
561C2#2376190611yescharacteristic
562C2#2376192711yescharacteristic
563C2#2376197411yescharacteristic
564C2#2376338611yescharacteristic
565C2#2376340811yescharacteristic
566C2#2376341811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D79
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#327×C79C2
socleC23×C79C2×C22⋊C4
prime radicalC2×#121×C79C22
cosocleC23×C4×C79C23
hypercenterC2×#327D79
hypocenterC79#8575
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. #23763378C2semidirect
4. #23761626C2semidirect
5. #23761280C2semidirect
6. #23761279C2semidirect
9. C2×D4×Dic79C4semidirect
13. #19098466C4semidirect
14. #19098462C4semidirect
18. C4⋊C4×Dic79C4semidirect
291. C79#8575semidirect

Maximal subgroups

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

subgroupindex
#121×Dic792
#237633782
C2×#190982972
#237616262
#237612802
#237612792
C2×#327×C792
#857579