Finite Group Databasebeta

← #34307723

Subgroups — #34307723

Order 20224.

740 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14579183C211yesmaximal, characteristic
2#23772065C211yesmaximal, characteristic, complemented
3#23771965C211yesmaximal, characteristic, complemented
4#23764111C212yesmaximal, complemented
5#23764000C212yesmaximal, complemented
6#23763726C212yesmaximal
7#23763538C211yesmaximal, characteristic, complemented
8#23763421C211yesmaximal, characteristic, complemented
9#23763421C211yesmaximal, characteristic, complemented
10#23763244C212yesmaximal, complemented
11C79×#1770C211yesmaximal, characteristic, complemented
12C2×#14579183C2211yescharacteristic
13C2×#14579167C2212yes—
14C4×#11929624C2211yescharacteristic
15C4×#11929624C2211yescharacteristic
16C4×Dic316C411yescharacteristic, complemented
17C4×Dic316C411yescharacteristic, complemented
18C4×#11929613C411yescharacteristic, complemented
19C4×#11929613C411yescharacteristic, complemented
20C4×#11929613C411yescharacteristic, complemented
21C4×#11929613C411yescharacteristic, complemented
22C4×D632C411yescharacteristic, complemented
23C4×D632C411yescharacteristic, complemented
24#19098623—22no—
25#19098619—21no—
26#19098615—21no—
27#19098601C2211yescharacteristic
28#19098589C2212yes—
29#19098588—22no—
30#19098587C2211yescharacteristic
31#19098587C2211yescharacteristic
32#19098585—21no—
33C4⋊C8×D79—21no—
34C2×#14579089C2212yes—
35C2×#14579089C2212yes—
36#19098560C2211yescharacteristic
37#19098560C2211yescharacteristic
38#19098529C2212yes—
39#19098518C2212yes—
40#19098328C2212yes—
41#19098328C2212yes—
42#19098323C2212yes—
43#19098323C2212yes—
44C2×C4⋊C8×C79C2211yescharacteristic
45C2×#97×C79C2212yes—
46C2×C4×C8×C79C2211yescharacteristic
47C4×C4⋊C4×C79C2211yescharacteristic
48#336×C79C2212yes—
49C2×#11929624C2311yescharacteristic
50C2×#11929624C2311yescharacteristic
51#14579183—22no—
52#14579183—22no—
53C2×Dic316C2×C411yescharacteristic
54C2×#11929613C2×C411yescharacteristic
55C2×#11929613C2×C411yescharacteristic
56C2×D632C2×C411yescharacteristic
57#14579167—24no—
58#14579167—42no—
59C2×#11929608C2312yes—
60C2×#11929608C2312yes—
61C2×#11929607—22no—
62C2×C8×D79—22no—
63C4×Dic158C2×C411yescharacteristic
64C4×Dic158C2×C411yescharacteristic
65C4×Dic158C2×C411yescharacteristic
66C4×Dic158C2×C411yescharacteristic
67C4×D316C2×C411yescharacteristic
68C4×D316C2×C411yescharacteristic
69C4×D316C2×C411yescharacteristic
70C4×D316C2×C411yescharacteristic
71C4×C22⋊D79—21no—
72C4×C22⋊D79—21no—
73C4×C22⋊D79—21no—
74C4×C22⋊D79—21no—
75#14579120C2311yescharacteristic
76#14579120C2311yescharacteristic
77#14579118—21no—
78#14579118—21no—
79C42×D79—21no—
80C42×D79—21no—
81#14579095C2×C411yescharacteristic
82#14579095C2×C411yescharacteristic
83#14579093C2×C411yescharacteristic
84#14579093C2×C411yescharacteristic
85C4⋊(C79⋊C8)—21no—
86C4⋊(C79⋊C8)—21no—
87C4⋊(C79⋊C8)—21no—
88C4⋊(C79⋊C8)—21no—
89Dic79⋊C8—22no—
90Dic79⋊C8—22no—
91#14579089—24no—
92#14579089—42no—
93#14579086C2×C411yescharacteristic
94#14579086C2×C411yescharacteristic
95#14579086C2×C411yescharacteristic
96#14579086C2×C411yescharacteristic
97#14579083C2×C411yescharacteristic
98#14579083C2×C411yescharacteristic
99#14579083C2×C411yescharacteristic
100#14579083C2×C411yescharacteristic
101(C2×D79)⋊C8—22no—
102(C2×D79)⋊C8—22no—
103#14579071—22no—
104C8×Dic79—22no—
105C22×C8×C79C2311yescharacteristic
106C22×C8×C79C2311yescharacteristic
107C22×C8×C79C2312yes—
108C2×C4⋊C4×C79—21no—
109C2×C42×C79C2312yes—
110C2×C42×C79C2311yescharacteristic
111C4⋊C8×C79D411yescharacteristic
112C4⋊C8×C79D411yescharacteristic
113C4⋊C8×C79D412yes—
114#97×C79—42no—
115C4×C8×C79C2×C411yescharacteristic
116C4×C8×C79C2×C411yescharacteristic
117C4×C8×C79C2×C411yescharacteristic
118C4×C8×C79C2×C411yescharacteristic
119#95×C79—21no—
120#11929624—21no—
121#11929624—21no—
122#11929624—21no—
123#11929624—21no—
124#11929624—21no—
125#11929624—21no—
126#11929624#4112yes—
127#11929624#4112yes—
128C2×Dic158C22×C411yescharacteristic
129C2×Dic158C22×C411yescharacteristic
130C2×D316C22×C411yescharacteristic
131C2×D316C22×C411yescharacteristic
132C2×C22⋊D79—21no—
133C2×C22⋊D79—21no—
134C2×C4×D79—21no—
135C2×C4×D79—21no—
136C2×C4×D79—21no—
137C2×C4×D79—21no—
138C2×C4×D79—21no—
139C2×C4×D79—21no—
140Dic316—22no—
141#11929613—22no—
142#11929613—22no—
143D632—22no—
144#11929608—22no—
145#11929608—22no—
146#11929608—22no—
147#11929608—22no—
148#11929607—44no—
149C2×C79⋊C8—22no—
150C2×C79⋊C8—22no—
151C8×D79—44no—
152C4⋊Dic79C22×C411yescharacteristic
153C4⋊Dic79C22×C411yescharacteristic
154C4⋊Dic79C22×C411yescharacteristic
155C4⋊Dic79C22×C411yescharacteristic
156Dic79⋊C4—21no—
157Dic79⋊C4—21no—
158Dic79⋊C4—21no—
159Dic79⋊C4—21no—
160C22⋊Dic79—21no—
161C22⋊Dic79—21no—
162(C2×D79)⋊C4—21no—
163(C2×D79)⋊C4—21no—
164(C2×D79)⋊C4—21no—
165(C2×D79)⋊C4—21no—
166C4×Dic79—21no—
167C4×Dic79—21no—
168C4×Dic79—21no—
169C4×Dic79—21no—
170C4×Dic79—21no—
171C4×Dic79—21no—
172C22×C4×C79—22no—
173C22×C4×C79#4111yescharacteristic
174C22×C4×C79#4111yescharacteristic
175C22×C4×C79C2411yescharacteristic
176C2×C8×C79—22no—
177C2×C8×C79—21no—
178C2×C8×C79—21no—
179C2×C8×C79—21no—
180C2×C8×C79—22no—
181C2×C8×C79—21no—
182C2×C8×C79C22×C411yescharacteristic
183C2×C8×C79C22×C411yescharacteristic
184C2×C8×C79C22×C411yescharacteristic
185C2×C8×C79C22×C411yescharacteristic
186C2×C8×C79C2×D412yes—
187C2×C8×C79C2×D412yes—
188C4⋊C4×C79—42no—
189C42×C79—22no—
190C42×C79—22no—
191C42×C79C22×C411yescharacteristic
192C42×C79C22×C411yescharacteristic
193C42×C79C2×D411yescharacteristic
194C42×C79C2×D411yescharacteristic
195C23×C79C2×#4111yescharacteristic
196C22×D79—21no—
197C22×D79—21no—
198C2×C4×C79—22no—
199C2×C4×C79—41no—
200C2×C4×C79—42no—
201C2×C4×C79—42no—
202C2×C4×C79—42no—
203C2×C4×C79C4×D411yescharacteristic
204C2×C4×C79C4×D411yescharacteristic
205C2×C4×C79C4×D411yescharacteristic
206C2×C4×C79C4×D411yescharacteristic
207C2×C4×C79C23×C411yescharacteristic
208C2×C4×C79C22×D411yescharacteristic
209C2×C4×C79C2×#4111yescharacteristic
210C2×C4×C79C2×#4111yescharacteristic
211C2×C4×C79H(5,𝔽2)11yescharacteristic
212C2×C4×C792−1+411yescharacteristic
213C22⋊D79—22no—
214C22⋊D79—22no—
215C22⋊D79—41no—
216C22⋊D79—41no—
217C2×Dic79—21no—
218C2×Dic79—21no—
219C2×Dic79—21no—
220C2×Dic79—21no—
221C2×Dic79—21no—
222C2×Dic79—21no—
223D316—21no—
224D316—21no—
225D316—21no—
226D316#11712yes—
227C4×D79—22no—
228C4×D79—22no—
229C4×D79—41no—
230C4×D79—41no—
231C4×D79—42no—
232C4×D79—42no—
233C4×D79—81no—
234C4×D79—81no—
235Dic158—21no—
236Dic158—21no—
237Dic158—21no—
238Dic158#11712yes—
239C8×C79—21no—
240C8×C79—21no—
241C8×C79—21no—
242C8×C79—21no—
243C8×C79—22no—
244C8×C79—22no—
245C79⋊C8—42no—
246C79⋊C8—42no—
247C22×C79—21no—
248C22×C79—21no—
249C22×C79C2×C4×D411yescharacteristic
250C22×C79#55211yescharacteristic
251C22×C79#55311yescharacteristic
252C2×D79—22no—
253C2×D79—41no—
254C2×D79—41no—
255C2×D79—41no—
256C2×D79—41no—
257C2×D79—41no—
258C4×C79—21no—
259C4×C79—21no—
260C4×C79—21no—
261C4×C79—82no—
262C4×C79#44412yes—
263C4×C79C2×#11711yescharacteristic
264C4×C79#52011yescharacteristic
265C4×C79#54611yescharacteristic
266C4×C79#54811yescharacteristic
267Dic79—22no—
268Dic79—41no—
269Dic79—41no—
270Dic79—41no—
271Dic79—41no—
272Dic79—41no—
273#22782—791nomaximal, Sylow
274C2×C79—21no—
275C2×C79—21no—
276C2×C79#235211yescharacteristic
277C2×C79C2×#44411yescharacteristic
278C2×C79#295711yescharacteristic
279D79—42no—
280D79—81no—
281C4×#135—791no—
282#2925—791no—
283#2906—791no—
284#1938—792no—
285#1922—792no—
286#1876—792no—
287#1837—791no—
288#1806—791no—
289#1806—791no—
290#1770D7911yescharacteristic, complemented
291#1765—792no—
292C79#2278211yescharacteristic, complemented, Sylow
293C2×#135—791no—
294C2×#131—792no—
295C4×#41—791no—
296C4×#41—791no—
297C4×Q16—791no—
298C4×Q16—791no—
299C4×SD16—791no—
300C4×SD16—791no—
301C4×SD16—791no—
302C4×SD16—791no—
303C4×D8—791no—
304C4×D8—791no—
305#433—1582no—
306#432—1581no—
307#431—1581no—
308#427—791no—
309#424—792no—
310#424—1582no—
311#423—791no—
312#423—791no—
313#423—1581no—
314C2×C4⋊C8—1581no—
315C2×C4⋊C8C2×D7911yescharacteristic
316C2×#104—792no—
317C2×#104—792no—
318#416—791no—
319#416—791no—
320#408—792no—
321#405—792no—
322C2×#97C2×D7912yes—
323C2×C4×C8C2×D7911yescharacteristic
324C4×C4⋊C4C2×D7911yescharacteristic
325#341—792no—
326#341—792no—
327#339—792no—
328#339—792no—
329#336C2×D7912yes—
330C2×#41—791no—
331C2×#41—791no—
332#135—1582no—
333#135—1582no—
334C2×Q16—791no—
335C2×SD16—791no—
336C2×SD16—791no—
337C2×D8—791no—
338#131—3162no—
339#131—1584no—
340C2×M16—792no—
341C2×M16—792no—
342C2×M16—1582no—
343C22×C8—1582no—
344C22×C8C22×D7911yescharacteristic
345C22×C8C22×D7911yescharacteristic
346C22×C8C22×D7912yes—
347C4×Q8—791no—
348C4×Q8—791no—
349C4×Q8—791no—
350C4×Q8—791no—
351C4×D4—791no—
352C4×D4—791no—
353C4×D4—791no—
354C4×D4—791no—
355C4×D4—1581no—
356C4×D4—1581no—
357C4×D4—1581no—
358C4×D4—1581no—
359#117—791no—
360#117—791no—
361#117—1581no—
362#117—1581no—
363C2×C4⋊C4—21no—
364C2×C42—1581no—
365C2×C42—1581no—
366C2×C42C22×D7912yes—
367C2×C42C22×D7911yescharacteristic
368#107—791no—
369#107—791no—
370#106—791no—
371#106—791no—
372C4⋊C8—1582no—
373C4⋊C8—1582no—
374C4⋊C8—1581no—
375C4⋊C8—1581no—
376C4⋊C8—1581no—
377C4⋊C8—1581no—
378C4⋊C8D31611yescharacteristic
379C4⋊C8D31611yescharacteristic
380C4⋊C8D31612yes—
381#104—3162no—
382#104—1584no—
383#103—791no—
384#103—791no—
385#103—791no—
386#103—791no—
387#102—791no—
388#102—791no—
389#102—791no—
390#102—791no—
391C22⋊C8—1582no—
392C22⋊C8—1582no—
393#97—42no—
394#97—1582no—
395C4×C8—1582no—
396C4×C8C4×D7911yescharacteristic
397C4×C8C4×D7911yescharacteristic
398C4×C8C4×D7911yescharacteristic
399C4×C8C4×D7911yescharacteristic
400#95—21no—
401#41—792no—
402#41—792no—
403#41—1581no—
404#41—1581no—
405#41—1581no—
406#41—1581no—
407#41—1581no—
408#41—1581no—
409C2×Q8—791no—
410C2×Q8—791no—
411C2×D4—791no—
412C2×D4—791no—
413C2×D4—1581no—
414C2×D4—1581no—
415C22×C4—22no—
416C22×C4—1581no—
417C22×C4—1581no—
418C22×C4—1581no—
419C22×C4—1581no—
420C22×C4—1581no—
421C22×C4—1581no—
422C22×C4#1192962411yescharacteristic
423C22×C4#1192962411yescharacteristic
424C22×C4C23×D7911yescharacteristic
425Q16—1582no—
426SD16—1582no—
427SD16—1582no—
428D8—1582no—
429M16—3164no—
430M16—1582no—
431M16—1582no—
432M16—1582no—
433M16—1582no—
434C2×C8—22no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—21no—
440C2×C8—3164no—
441C2×C8—1582no—
442C2×C8—1582no—
443C2×C8C2×C4×D7911yescharacteristic
444C2×C8C2×C4×D7911yescharacteristic
445C2×C8C2×C4×D7911yescharacteristic
446C2×C8C2×C4×D7911yescharacteristic
447C2×C8C2×D31612yes—
448C2×C8C2×D31612yes—
449C4⋊C4—42no—
450C4⋊C4—791no—
451C4⋊C4—791no—
452C4⋊C4—791no—
453C4⋊C4—791no—
454C4⋊C4—1581no—
455C4⋊C4—1581no—
456C4⋊C4—1581no—
457C4⋊C4—1581no—
458C22⋊C4—1581no—
459C22⋊C4—1581no—
460C22⋊C4—1581no—
461C22⋊C4—1581no—
462C22⋊C4—1581no—
463C22⋊C4—1581no—
464C42—22no—
465C42—22no—
466C42—1581no—
467C42—1581no—
468C42—1581no—
469C42—1581no—
470C42—1581no—
471C42—1581no—
472C42C2×C4×D7911yescharacteristic
473C42C2×C4×D7911yescharacteristic
474C42C2×D31611yescharacteristic
475C42C2×D31611yescharacteristic
476C23—1581no—
477C23—1581no—
478C23C2×#1192962411yescharacteristic
479Q8—792no—
480Q8—1581no—
481Q8—1581no—
482Q8—1581no—
483D4—3161no—
484D4—3161no—
485D4—792no—
486D4—1582no—
487D4—1582no—
488D4—1581no—
489D4—1581no—
490D4—1581no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—3162no—
497C2×C4—3162no—
498C2×C4—3161no—
499C2×C4—3161no—
500C2×C4—6321no—
501C2×C4—6321no—
502C2×C4—1581no—
503C2×C4—1581no—
504C2×C4—1581no—
505C2×C4—1581no—
506C2×C4—1581no—
507C2×C4—1581no—
508C2×C4—1582no—
509C2×C4—1582no—
510C2×C4C4×D31611yescharacteristic
511C2×C4C4×D31611yescharacteristic
512C2×C4C4×D31611yescharacteristic
513C2×C4C4×D31611yescharacteristic
514C2×C4C22×C4×D7911yescharacteristic
515C2×C4C22×D31611yescharacteristic
516C2×C4#41×D7911yescharacteristic
517C2×C4#41×D7911yescharacteristic
518C2×C4#1457920611yescharacteristic
519C2×C4#1457920911yescharacteristic
520C8—22no—
521C8—22no—
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—3162no—
527C8—3162no—
528C22—21no—
529C22—21no—
530C22—3161no—
531C22—3161no—
532C22—3161no—
533C22—3161no—
534C22—3161no—
535C22—1582no—
536C22C2×C4×D31611yescharacteristic
537C22#1909925411yescharacteristic
538C22#1909926311yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—82no—
543C4—3161no—
544C4—3161no—
545C4—3161no—
546C4—3161no—
547C4—3161no—
548C4—1582no—
549C4#1909868612yes—
550C4#117×D7911yescharacteristic
551C4#1909904411yescharacteristic
552C4#1909920811yescharacteristic
553C4#1909922511yescharacteristic
554C2—21no—
555C2—21no—
556C2—3162no—
557C2—6321no—
558C2#2376633211yescharacteristic
559C2C2×#1909868611yescharacteristic
560C2#2377231811yescharacteristic
subgroupG/H
centerC2×C4C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1770C2
socleC22×C79C2×C4×D4
prime radicalC2×#14579183C22
cosocleC22×C4×C79C24
hypercenter#1770D79
hypocenterC79#22782
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ #1770

Complements

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

NH ≅ G/Ntype
2. #23772065C2semidirect
3. #23771965C2semidirect
4. #23764111C2semidirect
5. #23764000C2semidirect
7. #23763538C2semidirect
8. #23763421C2semidirect
9. #23763421C2semidirect
10. #23763244C2semidirect
11. C79×#1770C2semidirect
16. C4×Dic316C4semidirect
17. C4×Dic316C4semidirect
18. C4×#11929613C4semidirect
19. C4×#11929613C4semidirect
20. C4×#11929613C4semidirect
21. C4×#11929613C4semidirect
22. C4×D632C4semidirect
23. C4×D632C4semidirect
290. #1770D79semidirect
292. C79#22782semidirect

Maximal subgroups

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

subgroupindex
C4×#145791832
#237720652
#237719652
#237641112
#237640002
#237637262
#237635382
#237634212
#237632442
C79×#17702
#2278279