Finite Group Databasebeta

← #35152804

Subgroups — #35152804

Order 20224.

1878 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23776792C211yesmaximal, characteristic
2#23772384C218yesmaximal, complemented
3#23772377C218yesmaximal, complemented
4#23772341C211yesmaximal, characteristic, complemented
5#23772117C211yesmaximal, characteristic, complemented
6#23772113C211yesmaximal, characteristic, complemented
7#23772102C212yesmaximal, complemented
8#23772096C212yesmaximal, complemented
9#23772050C212yesmaximal, complemented
10#23772037C211yesmaximal, characteristic, complemented
11#23771822C212yesmaximal, complemented
12C2×#19098517C211yesmaximal, characteristic, complemented
13C79×#2960C211yesmaximal, characteristic, complemented
14C2×#14579169C2212yes—
15C2×#14579167C2212yes—
16C22×#11929607C2211yescharacteristic
17C22×C8×D79C2211yescharacteristic
18#19099187C414yes—
19#19099176C412yes—
20#19099169C412yes—
21#19099162C414yes—
22#19099153C414yes—
23#19099113C2211yescharacteristic
24#19099100C2218yes—
25#19099075C2212yes—
26#19099031C2212yes—
27C2×#14579118C2211yescharacteristic
28#19098694C2218yescomplemented
29#19098642C2214yescomplemented
30#19098640C2218yes—
31#19098636C2218yes—
32#19098633C2218yes—
33#19098633C2218yescomplemented
34C8×D316C2218yes—
35C8×D316C2214yescomplemented
36#19098623C2218yes—
37#19098623C2218yescomplemented
38#19098619C2218yescomplemented
39#19098590C2212yes—
40#19098587C2211yescharacteristic
41#19098531C2212yes—
42#19098529C2212yes—
43#19098526C2212yes—
44#19098520C2214yescomplemented
45#19098519C2212yes—
46#19098517C2214yes—
47#19098517C2214yes—
48#19098517C2214yescomplemented
49#19098515C2214yescomplemented
50M16×Dic79C2212yes—
51C2×#14579072C2211yescharacteristic
52C2×#14579071C2211yescharacteristic
53C2×C8×Dic79C2211yescharacteristic
54C4×C8×D79C2214yescomplemented
55#533×C79C2211yescharacteristic
56#446×C79C2214yescomplemented
57C8×Q8×C79C2214yescomplemented
58#424×C79C2212yes—
59#423×C79C2211yescharacteristic
60#405×C79C2212yes—
61C2×C4×C8×C79C2211yescharacteristic
62C2×#11929624C2312yes—
63C22×C4×D79C2311yescharacteristic
64#14579169—28no—
65#14579167—28no—
66C2×#11929608C2312yes—
67C2×#11929607—24no—
68C2×#11929607C2314yes—
69C2×#11929607C2314yes—
70C2×#11929607C2314yes—
71C22×C79⋊C8C2311yescharacteristic
72C22×C79⋊C8C2311yescharacteristic
73C2×C8×D79—24no—
74C2×C8×D79C2312yes—
75C2×C8×D79C2314yes—
76C2×C8×D79C2314yes—
77C2×C8×D79C2312yes—
78#14579161C2×C411yescharacteristic
79#14579158C2×C411yescharacteristic
80#14579156C2×C418yes—
81#14579156C2×C418yes—
82#14579152C2×C418yes—
83#14579150C2×C418yes—
84#14579149C2×C412yes—
85#14579146C2×C418yes—
86#14579146C2×C418yes—
87#14579145C2×C414yes—
88#14579141C2×C414yes—
89#14579137C2×C418yes—
90#14579136C2×C418yes—
91#14579136C2×C414yes—
92#14579135C2×C414yes—
93#14579131C2×C414yes—
94#14579126C2×C414yes—
95C4×D316C2318yes—
96C4×D316C2314yes—
97#14579123C2×C414yes—
98(C22⋊D79)⋊C4C2×C418yes—
99#14579118C2×C412yes—
100#14579118C2×C412yes—
101#14579118C2314yes—
102#14579118C2314yes—
103C2×Dic79⋊C4C2×C412yes—
104C2×(C2×D79)⋊C4C2×C412yes—
105C2×C4×Dic79C2311yescharacteristic
106C42×D79C2314yes—
107C4⋊(C79⋊C8)C2318yes—
108C4⋊(C79⋊C8)C2314yes—
109(C2×D79)⋊C8C2318yes—
110(C2×D79)⋊C8C2318yes—
111(C2×D79)⋊C8C2318yes—
112#14579072C2314yes—
113#14579072C2314yes—
114#14579071C2314yes—
115#14579071C2314yes—
116C4×C79⋊C8C2314yes—
117C8×Dic79C2312yes—
118C8×Dic79C2314yes—
119C8×Dic79C2312yes—
120C2×M16×C79C2312yes—
121C22×C8×C79C2311yescharacteristic
122C22×C8×C79C2311yescharacteristic
123C4⋊Q8×C79C2×C412yes—
124#125×C79C2×C412yes—
125C22⋊Q8×C79C2×C414yes—
126C4×Q8×C79C2314yes—
127#117×C79C2312yes—
128C2×C42×C79C2311yescharacteristic
129C4⋊C8×C79C2314yes—
130C4⋊C8×C79C2318yes—
131#97×C79C2314yes—
132C4×C8×C79C2314yes—
133C4×C8×C79C2312yes—
134C4×C8×C79C2312yes—
135C23×D79C22×C411yescharacteristic
136#11929624—24no—
137#11929624—24no—
138C2×Dic158C22×C412yes—
139C2×D316C22×C412yes—
140C2×D316C22×C418yes—
141C2×D316C22×C414yes—
142C2×C22⋊D79C22×C414yes—
143C22×Dic79C22×C411yescharacteristic
144C22×Dic79C22×C412yes—
145C2×C4×D79—22no—
146C2×C4×D79—22no—
147C2×C4×D79C22×C412yes—
148C2×C4×D79C22×C418yes—
149C2×C4×D79C2414yes—
150C2×C4×D79C2412yes—
151#11929608—24no—
152#11929607—28no—
153#11929607—28no—
154#11929607—28no—
155C2×C79⋊C8—22no—
156C2×C79⋊C8#4114yes—
157C2×C79⋊C8C2414yes—
158C2×C79⋊C8C2412yes—
159C2×C79⋊C8C2412yes—
160C8×D79—24no—
161C8×D79—24no—
162C8×D79—28no—
163C8×D79—28no—
164C4⋊Dic79C22×C414yes—
165C4⋊Dic79C22×C418yes—
166Dic79⋊C4C22×C414yes—
167Dic79⋊C4C22×C414yes—
168Dic79⋊C4C22×C418yes—
169(C2×D79)⋊C4C22×C414yes—
170(C2×D79)⋊C4C22×C414yes—
171(C2×D79)⋊C4C22×C418yes—
172(C2×D79)⋊C4C22×C418yes—
173(C2×D79)⋊C4C22×C418yes—
174(C2×D79)⋊C4C22×C418yes—
175C4×Dic79C22×C418yes—
176C4×Dic79C22×C412yes—
177C4×Dic79C2412yes—
178C2×Q8×C79C22×C414yes—
179C22×C4×C79C22×C412yes—
180C22×C4×C79C2411yescharacteristic
181M16×C79—24no—
182C2×C8×C79—22no—
183C2×C8×C79#4114yes—
184C2×C8×C79C2412yes—
185C2×C8×C79C2414yes—
186C2×C8×C79C2411yescharacteristic
187C2×C8×C79C2411yescharacteristic
188C4⋊C4×C79C22×C414yes—
189C4⋊C4×C79C22×C418yes—
190C4⋊C4×C79C22×C414yes—
191C22⋊C4×C79C22×C414yes—
192C42×C79C22×C411yescharacteristic
193C42×C79C22×C411yescharacteristic
194C42×C79C2412yes—
195C42×C79C2414yes—
196C23×C79C23×C411yescharacteristic
197C22×D79—22no—
198C22×D79—42no—
199C22×D79C23×C412yes—
200C22×D79C23×C414yes—
201Q8×C79—44no—
202C2×C4×C79—42no—
203C2×C4×C79C23×C418yes—
204C2×C4×C79C23×C412yes—
205C2×C4×C79C23×C412yes—
206C2×C4×C79C23×C411yescharacteristic
207C2×C4×C79C2×#4112yes—
208C2×C4×C79C2×#4112yes—
209C2×C4×C79C2511yescharacteristic
210C22⋊D79—28no—
211C2×Dic79—22no—
212C2×Dic79—42no—
213C2×Dic79C23×C414yes—
214C2×Dic79C23×C414yes—
215C2×Dic79C23×C412yes—
216D316—24no—
217D316—44no—
218D316—48no—
219C4×D79—24no—
220C4×D79—22no—
221C4×D79—24no—
222C4×D79—24no—
223C4×D79—22no—
224C4×D79—48no—
225Dic158—24no—
226C8×C79—24no—
227C8×C79—22no—
228C8×C79C2×#4112yes—
229C8×C79C2×#4112yes—
230C79⋊C8—22no—
231C79⋊C8—24no—
232C79⋊C8C2×#4114yes—
233C22×C79—22no—
234C22×C79C4×#4112yes—
235C22×C79C24×C411yescharacteristic
236C2×D79—24no—
237C2×D79—22no—
238C2×D79—24no—
239C2×D79—28no—
240C2×D79—44no—
241C4×C79—22no—
242C4×C79—22no—
243C4×C79—28no—
244C4×C79C4×#4112yes—
245C4×C79C22×#4111yescharacteristic
246C4×C79C22×#4111yescharacteristic
247Dic79—24no—
248Dic79—22no—
249Dic79—24no—
250#60206—791nomaximal, Sylow
251C2×C79—22no—
252C2×C79C2×C4×#4111yescharacteristic
253C2×C79#357312yes—
254D79—44no—
255D79—42no—
256#3455—791no—
257#2967—798no—
258#2966—798no—
259#2961—791no—
260#2960D7911yescharacteristic, complemented
261#2932—791no—
262#2931—791no—
263#2930—792no—
264#2929—792no—
265#2923—792no—
266#2921—791no—
267C4×#131—792no—
268C2×#405—791no—
269C79#6020611yescharacteristic, complemented, Sylow
270C2×#131—792no—
271C2×#131—792no—
272C22×M16—791no—
273C23×C8—791no—
274#542—794no—
275#541—792no—
276#540—792no—
277#539—794no—
278#538—794no—
279#533C2×D7911yescharacteristic
280#532—791no—
281#529—798no—
282#525—792no—
283C4×#41—792no—
284C2×#117—791no—
285#446—798no—
286#446C2×D7914yescomplemented
287C8×Q8C2×D7914yescomplemented
288#436—798no—
289#436—794no—
290#435—798no—
291#435—798no—
292#435—798no—
293C8×D4—798no—
294C8×D4—794no—
295#433—798no—
296#433—798no—
297#432—798no—
298#424—792no—
299#424C2×D7912yes—
300#423—791no—
301#423C2×D7911yescharacteristic
302#408—792no—
303#408—792no—
304#407—792no—
305#405—792no—
306#405—794no—
307#405—794no—
308#405—794no—
309#405—794no—
310#405C2×D7912yes—
311C4×M16—792no—
312C4×M16—794no—
313C2×#97—791no—
314C2×#97—791no—
315C2×C4×C8—791no—
316C2×C4×C8—794no—
317C2×C4×C8C2×D7911yescharacteristic
318C2×#41—792no—
319C23×C4—791no—
320#131—1588no—
321#131—1588no—
322C2×M16—792no—
323C2×M16—794no—
324C2×M16—794no—
325C2×M16—794no—
326C2×M16—1584no—
327C2×M16C22×D7912yes—
328C22×C8—791no—
329C22×C8—791no—
330C22×C8—792no—
331C22×C8—794no—
332C22×C8—792no—
333C22×C8—794no—
334C22×C8—1584no—
335C22×C8C22×D7911yescharacteristic
336C22×C8C22×D7911yescharacteristic
337C4⋊Q8—791no—
338C4⋊Q8C4×D7912yes—
339#127—791no—
340#126—798no—
341#126—798no—
342#125—798no—
343#125C4×D7912yes—
344#124—798no—
345#124—792no—
346#123—798no—
347#123—798no—
348#123—794no—
349C22⋊Q8—798no—
350C22⋊Q8—794no—
351C22⋊Q8C4×D7914yes—
352#121—794no—
353#121—794no—
354#121—798no—
355#120—794no—
356C4×Q8—794no—
357C4×Q8C22×D7914yes—
358C4×D4—794no—
359C4×D4—798no—
360C4×D4—798no—
361C4×D4—794no—
362#117—792no—
363#117—792no—
364#117—794no—
365#117—794no—
366#117C22×D7912yes—
367C2×C4⋊C4—792no—
368C2×C22⋊C4—792no—
369C2×C42—791no—
370C2×C42—794no—
371C2×C42C22×D7911yescharacteristic
372C4⋊C8—794no—
373C4⋊C8—798no—
374C4⋊C8C22×D7914yes—
375C4⋊C8C22×D7918yes—
376C22⋊C8—798no—
377C22⋊C8—798no—
378C22⋊C8—798no—
379#97—794no—
380#97—794no—
381#97—794no—
382#97—794no—
383#97C22×D7914yes—
384C4×C8—792no—
385C4×C8—794no—
386C4×C8—792no—
387C4×C8—794no—
388C4×C8C22×D7912yes—
389C4×C8C22×D7914yes—
390C4×C8C22×D7912yes—
391C24—791no—
392#41—1584no—
393#41—1584no—
394C2×Q8—792no—
395C2×Q8C2×C4×D7914yes—
396C2×D4—794no—
397C2×D4—798no—
398C2×D4—794no—
399C2×D4—792no—
400C22×C4—792no—
401C22×C4—798no—
402C22×C4—791no—
403C22×C4—794no—
404C22×C4—792no—
405C22×C4—792no—
406C22×C4—1582no—
407C22×C4—1582no—
408C22×C4C2×C4×D7912yes—
409C22×C4C23×D7911yescharacteristic
410M16—24no—
411M16—1588no—
412M16—1588no—
413M16—1584no—
414M16—1588no—
415C2×C8—22no—
416C2×C8—794no—
417C2×C8—792no—
418C2×C8—794no—
419C2×C8—792no—
420C2×C8—1588no—
421C2×C8—1582no—
422C2×C8—1584no—
423C2×C8—1584no—
424C2×C8—1588no—
425C2×C8#1192962514yes—
426C2×C8C23×D7914yes—
427C2×C8C23×D7912yes—
428C2×C8C23×D7911yescharacteristic
429C2×C8C23×D7911yescharacteristic
430C4⋊C4—794no—
431C4⋊C4—794no—
432C4⋊C4—798no—
433C4⋊C4—794no—
434C4⋊C4—798no—
435C4⋊C4C2×C4×D7914yes—
436C4⋊C4C2×C4×D7918yes—
437C4⋊C4C2×C4×D7914yes—
438C22⋊C4—794no—
439C22⋊C4—794no—
440C22⋊C4—798no—
441C22⋊C4—798no—
442C22⋊C4—798no—
443C22⋊C4—798no—
444C22⋊C4C2×C4×D7914yes—
445C42—792no—
446C42—792no—
447C42—798no—
448C42C2×C4×D7911yescharacteristic
449C42C2×C4×D7911yescharacteristic
450C42C23×D7914yes—
451C42C23×D7912yes—
452C23—3162no—
453C23—792no—
454C23—794no—
455C23—1582no—
456C23C22×C4×D7911yescharacteristic
457Q8—44no—
458Q8—1584no—
459D4—3164no—
460D4—3168no—
461D4—1588no—
462D4—1584no—
463C2×C4—42no—
464C2×C4—3168no—
465C2×C4—3162no—
466C2×C4—794no—
467C2×C4—794no—
468C2×C4—792no—
469C2×C4—1582no—
470C2×C4—1584no—
471C2×C4—1584no—
472C2×C4—1582no—
473C2×C4—1582no—
474C2×C4—1584no—
475C2×C4C22×C4×D7918yes—
476C2×C4C22×C4×D7912yes—
477C2×C4C22×C4×D7912yes—
478C2×C4C22×C4×D7911yescharacteristic
479C2×C4#41×D7912yes—
480C2×C4C2×#1192962512yes—
481C2×C4C24×D7911yescharacteristic
482C8—22no—
483C8—24no—
484C8—794no—
485C8—1582no—
486C8—1584no—
487C8C2×#1192962512yes—
488C8C2×#1192962512yes—
489C22—22no—
490C22—3164no—
491C22—1588no—
492C22—1584no—
493C22—1582no—
494C22—1584no—
495C22C4×#1192962512yes—
496C22C23×C4×D7911yescharacteristic
497C4—22no—
498C4—22no—
499C4—28no—
500C4—1584no—
501C4—1582no—
502C4—1584no—
503C4C4×#1192962512yes—
504C4C2×#41×D7911yescharacteristic
505C4C22×#1192962511yescharacteristic
506C2—22no—
507C2—3162no—
508C2—3164no—
509C2C2×C4×#1192962511yescharacteristic
510C2#2377811112yes—
subgroupG/H
centerC2×C8C23×D79
commutator, socleC22×C79C24×C4
FrattiniC2×C4C24×D79
FittingC79×#2960C2
prime radical#23776792C2
cosocleC2×C4×C79C25
hypercenter#2960D79
hypocenterC79#60206
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C8 ◁ #2960

Complements

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

NH ≅ G/Ntype
2. #23772384C2semidirect
3. #23772377C2semidirect
4. #23772341C2semidirect
5. #23772117C2semidirect
6. #23772113C2semidirect
7. #23772102C2semidirect
8. #23772096C2semidirect
9. #23772050C2semidirect
10. #23772037C2semidirect
11. #23771822C2semidirect
12. C2×#19098517C2semidirect
13. C79×#2960C2semidirect
28. #19098694C22semidirect
29. #19098642C22semidirect
33. #19098633C22semidirect
35. C8×D316C22semidirect
37. #19098623C22semidirect
38. #19098619C22semidirect
44. #19098520C22semidirect
48. #19098517C22semidirect
49. #19098515C22semidirect
54. C4×C8×D79C22semidirect
56. #446×C79C22semidirect
57. C8×Q8×C79C22semidirect
260. #2960D79semidirect
269. C79#60206semidirect
286. #446C2×D79semidirect
287. C8×Q8C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237767922
#237723842
#237723772
#237723412
#237721172
#237721132
#237721022
#237720962
#237720502
#237720372
#237718222
C2×#190985172
C79×#29602
#6020679