Finite Group Databasebeta

← #35152813

Subgroups — #35152813

Order 20224.

1962 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23776697C211yesmaximal, characteristic
2#23772413C218yesmaximal, complemented
3#23772393C218yesmaximal, complemented
4#23772341C211yesmaximal, characteristic, complemented
5#23772119C211yesmaximal, characteristic, complemented
6#23772117C211yesmaximal, characteristic, complemented
7#23772096C212yesmaximal, complemented
8#23772093C212yesmaximal, complemented
9#23772050C212yesmaximal, complemented
10#23772032C211yesmaximal, characteristic, complemented
11#23771827C212yesmaximal, complemented
12C2×#97×D79C211yesmaximal, characteristic, complemented
13C79×#2961C211yesmaximal, characteristic, complemented
14C2×#14579169C2212yes—
15C2×#14579167C2212yes—
16C22×C8×D79C2211yescharacteristic
17C22×C8×D79C2211yescharacteristic
18#19099118C412yes—
19#19099114C412yes—
20#19099113C2211yescharacteristic
21#19099100C2214yes—
22#19099099C414yes—
23#19099098C414yes—
24#19099076C414yes—
25#19099075C2212yes—
26C4×#11929625C2214yes—
27#19099031C2212yes—
28C2×C42×D79C2211yescharacteristic
29#19098694C2218yescomplemented
30#19098636C2218yes—
31#19098636C2218yescomplemented
32#19098633C2218yes—
33#19098633C2218yescomplemented
34C8×D316C2218yes—
35#19098628C2218yes—
36#19098623C2218yescomplemented
37#19098619C2214yes—
38#19098619C2218yescomplemented
39#19098615C2214yes—
40#19098590C2212yes—
41#19098587C2211yescharacteristic
42#19098531C2212yes—
43#19098529C2212yes—
44#19098526C2212yes—
45#19098520C2214yescomplemented
46#19098519C2212yes—
47#19098517C2214yescomplemented
48M16×Dic79C2212yes—
49C4×#11929607C2214yescomplemented
50C2×#14579072C2211yescharacteristic
51C2×#14579071C2211yescharacteristic
52C2×#14579071C2211yescharacteristic
53#97×D79C2214yes—
54#97×D79C2214yes—
55#97×D79C2214yescomplemented
56#533×C79C2211yescharacteristic
57#446×C79C2218yescomplemented
58#424×C79C2212yes—
59#423×C79C2211yescharacteristic
60#405×C79C2212yes—
61C2×#97×C79C2211yescharacteristic
62C2×#11929625C2×C414yes—
63C2×#11929624C2312yes—
64C22×C4×D79C2×C412yes—
65C22×C4×D79C2311yescharacteristic
66#14579169—28no—
67#14579167—28no—
68C2×#11929608C2312yes—
69C2×#11929607C2314yes—
70C2×#11929607C2314yes—
71C22×C79⋊C8C2311yescharacteristic
72C22×C79⋊C8C2311yescharacteristic
73C2×C8×D79—24no—
74C2×C8×D79—24no—
75C2×C8×D79C2314yes—
76C2×C8×D79C2314yes—
77C2×C8×D79C2314yes—
78C2×C8×D79C2314yes—
79#14579161C2×C411yescharacteristic
80#14579158C2×C411yescharacteristic
81#14579156C2×C418yes—
82#14579152C2×C414yes—
83#14579150C2×C414yes—
84#14579149C2×C412yes—
85#14579146C2×C418yes—
86#14579145C2×C414yes—
87#14579141C2×C414yes—
88#14579137C2×C414yes—
89#14579136C2×C414yes—
90#14579136C2×C414yes—
91#14579135C2×C414yes—
92#14579131C2×C414yes—
93Q8×Dic79C2×C414yes—
94#14579126C2×C414yes—
95C4×D316C2318yes—
96C4×D316C2314yes—
97#14579123C2×C414yes—
98#14579123C2×C418yes—
99#14579123C2×C414yes—
100(C22⋊D79)⋊C4C2×C418yes—
101#14579119C2×C414yes—
102#14579119C2×C418yes—
103#14579118C2314yes—
104C2×C4×Dic79C2×C412yes—
105C2×C4×Dic79C2311yescharacteristic
106C42×D79C2×C412yes—
107C42×D79C2×C412yes—
108C42×D79C2312yes—
109C42×D79C2312yes—
110C42×D79C2314yes—
111C4⋊(C79⋊C8)C2318yes—
112C4⋊(C79⋊C8)C2314yes—
113(C2×D79)⋊C8C2318yes—
114(C2×D79)⋊C8C2318yes—
115(C2×D79)⋊C8C2318yes—
116#14579072C2314yes—
117#14579072C2314yes—
118#14579071C2312yes—
119#14579071C2312yes—
120#14579071C2312yes—
121#14579071C2312yes—
122#14579071C2314yes—
123C4×C79⋊C8C2314yes—
124C8×Dic79C2314yes—
125C2×M16×C79C2312yes—
126C22×C8×C79C2311yescharacteristic
127C22×C8×C79C2311yescharacteristic
128C4⋊Q8×C79C2×C412yes—
129#125×C79C2×C412yes—
130C22⋊Q8×C79C2×C414yes—
131C4×Q8×C79C2314yes—
132#117×C79C2312yes—
133C2×C42×C79C2311yescharacteristic
134C4⋊C8×C79C2318yes—
135C4⋊C8×C79C2314yes—
136#97×C79C2314yes—
137#97×C79C2314yes—
138C4×C8×C79C2314yes—
139C23×D79C22×C411yescharacteristic
140#11929625—48no—
141#11929624—24no—
142#11929624—24no—
143C2×Dic158C22×C412yes—
144C2×D316C22×C412yes—
145C2×D316C22×C418yes—
146C2×D316C22×C414yes—
147C2×C22⋊D79C22×C414yes—
148C22×Dic79C22×C411yescharacteristic
149C22×Dic79C22×C412yes—
150C2×C4×D79—22no—
151C2×C4×D79—22no—
152C2×C4×D79—44no—
153C2×C4×D79C22×C412yes—
154C2×C4×D79C22×C412yes—
155C2×C4×D79C22×C412yes—
156C2×C4×D79C22×C418yes—
157C2×C4×D79C22×C412yes—
158C2×C4×D79C22×C412yes—
159C2×C4×D79C2414yes—
160C2×C4×D79C2412yes—
161#11929608—24no—
162#11929607—28no—
163#11929607—28no—
164C2×C79⋊C8—22no—
165C2×C79⋊C8#4114yescomplemented
166C2×C79⋊C8C2414yes—
167C2×C79⋊C8C2412yes—
168C2×C79⋊C8C2412yes—
169C8×D79—28no—
170C8×D79—28no—
171C8×D79—28no—
172C8×D79—28no—
173C4⋊Dic79C22×C414yes—
174C4⋊Dic79C22×C418yes—
175Dic79⋊C4C22×C418yes—
176(C2×D79)⋊C4C22×C418yes—
177(C2×D79)⋊C4C22×C418yes—
178(C2×D79)⋊C4C22×C418yes—
179(C2×D79)⋊C4C22×C418yes—
180C4×Dic79C22×C412yes—
181C4×Dic79C22×C412yes—
182C4×Dic79C22×C412yes—
183C4×Dic79C22×C412yes—
184C4×Dic79C22×C412yes—
185C4×Dic79C22×C418yes—
186C4×Dic79C2411yescharacteristic
187C4×Dic79C2411yescharacteristic
188C2×Q8×C79C22×C414yes—
189C22×C4×C79C22×C412yes—
190C22×C4×C79C2411yescharacteristic
191M16×C79—24no—
192C2×C8×C79—22no—
193C2×C8×C79#4114yescomplemented
194C2×C8×C79C2414yes—
195C2×C8×C79C2412yes—
196C2×C8×C79C2412yes—
197C4⋊C4×C79C22×C414yes—
198C4⋊C4×C79C22×C418yes—
199C4⋊C4×C79C22×C414yes—
200C22⋊C4×C79C22×C414yes—
201C42×C79C22×C411yescharacteristic
202C42×C79C22×C411yescharacteristic
203C42×C79C2414yes—
204C42×C79C2412yes—
205C23×C79C23×C411yescharacteristic
206C22×D79—22no—
207C22×D79—24no—
208C22×D79C23×C414yes—
209C22×D79C23×C412yes—
210Q8×C79—44no—
211C2×C4×C79—42no—
212C2×C4×C79C23×C412yes—
213C2×C4×C79C23×C412yes—
214C2×C4×C79C23×C418yes—
215C2×C4×C79C23×C411yescharacteristic
216C2×C4×C79C2×#4112yes—
217C2×C4×C79C2×#4112yes—
218C2×C4×C79C2511yescharacteristic
219C22⋊D79—28no—
220C2×Dic79—22no—
221C2×Dic79—42no—
222C2×Dic79C23×C412yes—
223C2×Dic79C23×C412yes—
224C2×Dic79C23×C412yes—
225C2×Dic79C23×C414yes—
226D316—216no—
227D316—24no—
228D316—44no—
229C4×D79—24no—
230C4×D79—24no—
231C4×D79—24no—
232C4×D79—22no—
233C4×D79—22no—
234C4×D79—24no—
235C4×D79—24no—
236C4×D79—42no—
237C4×D79—42no—
238C4×D79—48no—
239Dic158—24no—
240C8×C79—22no—
241C8×C79—24no—
242C8×C79C2×#4114yes—
243C79⋊C8—22no—
244C79⋊C8—24no—
245C79⋊C8C2×#4114yes—
246C22×C79—22no—
247C22×C79C4×#4112yes—
248C22×C79C24×C411yescharacteristic
249C2×D79—28no—
250C2×D79—22no—
251C2×D79—22no—
252C2×D79—24no—
253C2×D79—28no—
254C2×D79#56814yes—
255C4×C79—22no—
256C4×C79—22no—
257C4×C79—28no—
258C4×C79C4×#4112yes—
259C4×C79C22×#4111yescharacteristic
260C4×C79C22×#4111yescharacteristic
261Dic79—24no—
262Dic79—22no—
263Dic79—22no—
264Dic79—22no—
265#60207—791nomaximal, Sylow
266C2×C79—22no—
267C2×C79C2×C4×#4111yescharacteristic
268C2×C79C2×#56812yes—
269D79—24no—
270D79—44no—
271#3444—791no—
272#2970—798no—
273#2968—798no—
274#2961—791no—
275#2961D7911yescharacteristic, complemented
276#2932—791no—
277#2932—791no—
278#2929—792no—
279#2929—792no—
280#2923—792no—
281#2920—791no—
282#2875—792no—
283C22×#97—791no—
284C79#6020711yescharacteristic, complemented, Sylow
285C2×#131—792no—
286C2×#131—792no—
287C23×C8—791no—
288C23×C8—791no—
289#533—792no—
290#533C2×D7911yescharacteristic
291#532—792no—
292#532—791no—
293#529—794no—
294#529—794no—
295#529—794no—
296#525—794no—
297#525—792no—
298C4×#41—794no—
299C4×#41—792no—
300C22×C42—791no—
301#446—798no—
302#446C2×D7918yescomplemented
303#435—798no—
304#435—798no—
305#435—798no—
306#435—798no—
307C8×D4—798no—
308C8×D4—798no—
309#433—798no—
310#432—794no—
311#432—798no—
312#431—794no—
313#424—792no—
314#424C2×D7912yes—
315#423—791no—
316#423C2×D7911yescharacteristic
317#408—792no—
318#408—792no—
319#407—792no—
320#405—792no—
321#405—794no—
322#405—794no—
323#405C2×D7912yes—
324C4×M16—792no—
325C4×M16—794no—
326C2×#97—791no—
327C2×#97—791no—
328C2×#97—791no—
329C2×#97—794no—
330C2×#97—794no—
331C2×#97—794no—
332C2×#97C2×D7911yescharacteristic
333C2×#41—794no—
334C2×#41—792no—
335C23×C4—792no—
336C23×C4—791no—
337#131—1588no—
338#131—1588no—
339C2×M16—792no—
340C2×M16—794no—
341C2×M16—794no—
342C2×M16C22×D7912yes—
343C22×C8—791no—
344C22×C8—791no—
345C22×C8—794no—
346C22×C8—794no—
347C22×C8—794no—
348C22×C8—794no—
349C22×C8—1584no—
350C22×C8—1584no—
351C22×C8C22×D7911yescharacteristic
352C22×C8C22×D7911yescharacteristic
353C4⋊Q8—791no—
354C4⋊Q8C4×D7912yes—
355#127—791no—
356#126—798no—
357#125—794no—
358#125C4×D7912yes—
359#124—794no—
360#124—792no—
361#123—798no—
362#123—794no—
363C22⋊Q8—794no—
364C22⋊Q8—794no—
365C22⋊Q8C4×D7914yes—
366#121—794no—
367#121—794no—
368#121—794no—
369#120—794no—
370C4×Q8—794no—
371C4×Q8—794no—
372C4×Q8C22×D7914yes—
373C4×D4—798no—
374C4×D4—794no—
375C4×D4—794no—
376C4×D4—798no—
377C4×D4—798no—
378C4×D4—794no—
379#117—794no—
380#117—798no—
381#117—794no—
382#117C22×D7912yes—
383C2×C42—792no—
384C2×C42—791no—
385C2×C42—792no—
386C2×C42—792no—
387C2×C42—794no—
388C2×C42—792no—
389C2×C42—792no—
390C2×C42C22×D7911yescharacteristic
391C4⋊C8—794no—
392C4⋊C8—798no—
393C4⋊C8C22×D7914yes—
394C4⋊C8C22×D7918yes—
395C22⋊C8—798no—
396C22⋊C8—798no—
397C22⋊C8—798no—
398#97—792no—
399#97—792no—
400#97—792no—
401#97—792no—
402#97—794no—
403#97—794no—
404#97—794no—
405#97C22×D7914yes—
406#97C22×D7914yes—
407C4×C8—794no—
408C4×C8—794no—
409C4×C8C22×D7914yes—
410C24—791no—
411#41—3168no—
412#41—1584no—
413#41—1584no—
414C2×Q8—792no—
415C2×Q8C2×C4×D7914yes—
416C2×D4—794no—
417C2×D4—794no—
418C2×D4—798no—
419C2×D4—792no—
420C22×C4—3164no—
421C22×C4—792no—
422C22×C4—792no—
423C22×C4—792no—
424C22×C4—798no—
425C22×C4—791no—
426C22×C4—792no—
427C22×C4—794no—
428C22×C4—792no—
429C22×C4—792no—
430C22×C4—792no—
431C22×C4—1582no—
432C22×C4—1582no—
433C22×C4C2×C4×D7912yes—
434C22×C4C23×D7911yescharacteristic
435M16—24no—
436M16—1588no—
437M16—1584no—
438M16—1588no—
439C2×C8—22no—
440C2×C8—794no—
441C2×C8—792no—
442C2×C8—794no—
443C2×C8—792no—
444C2×C8—1588no—
445C2×C8—1588no—
446C2×C8—1582no—
447C2×C8—1588no—
448C2×C8—1588no—
449C2×C8#1192962514yescomplemented
450C2×C8C23×D7914yes—
451C2×C8C23×D7912yes—
452C2×C8C23×D7912yes—
453C4⋊C4—798no—
454C4⋊C4—794no—
455C4⋊C4—798no—
456C4⋊C4C2×C4×D7914yes—
457C4⋊C4C2×C4×D7918yes—
458C4⋊C4C2×C4×D7914yes—
459C22⋊C4—798no—
460C22⋊C4—798no—
461C22⋊C4—798no—
462C22⋊C4—798no—
463C22⋊C4C2×C4×D7914yes—
464C42—791no—
465C42—791no—
466C42—792no—
467C42—792no—
468C42—792no—
469C42—792no—
470C42—792no—
471C42—798no—
472C42C2×C4×D7911yescharacteristic
473C42C2×C4×D7911yescharacteristic
474C42C23×D7914yes—
475C42C23×D7912yes—
476C23—792no—
477C23—794no—
478C23—1584no—
479C23—1582no—
480C23C22×C4×D7911yescharacteristic
481Q8—44no—
482Q8—1584no—
483D4—3164no—
484D4—15816no—
485D4—1588no—
486D4—1584no—
487C2×C4—42no—
488C2×C4—3162no—
489C2×C4—3162no—
490C2×C4—3168no—
491C2×C4—3162no—
492C2×C4—792no—
493C2×C4—792no—
494C2×C4—792no—
495C2×C4—794no—
496C2×C4—1584no—
497C2×C4—1584no—
498C2×C4—1582no—
499C2×C4—1584no—
500C2×C4—1582no—
501C2×C4—1584no—
502C2×C4—1582no—
503C2×C4—1584no—
504C2×C4C22×C4×D7912yes—
505C2×C4C22×C4×D7912yes—
506C2×C4C22×C4×D7918yes—
507C2×C4C22×C4×D7911yescharacteristic
508C2×C4#41×D7912yes—
509C2×C4C2×#1192962512yes—
510C2×C4C24×D7911yescharacteristic
511C8—22no—
512C8—24no—
513C8—794no—
514C8—1582no—
515C8—1584no—
516C8C2×#1192962514yes—
517C22—22no—
518C22—794no—
519C22—1588no—
520C22—1588no—
521C22—1584no—
522C22—1582no—
523C22—1582no—
524C22C4×#1192962512yes—
525C22C23×C4×D7911yescharacteristic
526C4—22no—
527C4—22no—
528C4—28no—
529C4—1582no—
530C4—1582no—
531C4—1582no—
532C4—1584no—
533C4C4×#1192962512yes—
534C4C2×#41×D7911yescharacteristic
535C4C22×#1192962511yescharacteristic
536C2—22no—
537C2—3164no—
538C2—1584no—
539C2C2×C4×#1192962511yescharacteristic
540C2#568×D7912yes—
subgroupG/H
center, FrattiniC2×C4C24×D79
commutator, socleC22×C79C24×C4
FittingC79×#2961C2
prime radical#23776697C2
cosocleC2×C4×C79C25
hypercenter#2961D79
hypocenterC79#60207
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #2961

Complements

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

NH ≅ G/Ntype
2. #23772413C2semidirect
3. #23772393C2semidirect
4. #23772341C2semidirect
5. #23772119C2semidirect
6. #23772117C2semidirect
7. #23772096C2semidirect
8. #23772093C2semidirect
9. #23772050C2semidirect
10. #23772032C2semidirect
11. #23771827C2semidirect
12. C2×#97×D79C2semidirect
13. C79×#2961C2semidirect
29. #19098694C22semidirect
31. #19098636C22semidirect
33. #19098633C22semidirect
36. #19098623C22semidirect
38. #19098619C22semidirect
45. #19098520C22semidirect
47. #19098517C22semidirect
49. C4×#11929607C22semidirect
55. #97×D79C22semidirect
57. #446×C79C22semidirect
165. C2×C79⋊C8#41semidirect
193. C2×C8×C79#41semidirect
275. #2961D79semidirect
284. C79#60207semidirect
302. #446C2×D79semidirect
449. C2×C8#11929625semidirect

Maximal subgroups

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

subgroupindex
#237766972
#237724132
#237723932
#237723412
#237721192
#237721172
#237720962
#237720932
#237720502
#237720322
#237718272
C2×#97×D792
C79×#29612
#6020779