Finite Group Databasebeta

← #35152788

Subgroups — #35152788

Order 20224.

1950 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#23776695C211yesmaximal, characteristic
2#23772382C218yesmaximal, complemented
3#23772360C218yesmaximal, complemented
4#23772340C211yesmaximal, characteristic, complemented
5#23772337C211yesmaximal, characteristic, complemented
6#23772118C211yesmaximal, characteristic, complemented
7#23772106C212yesmaximal, complemented
8#23772101C212yesmaximal, complemented
9#23772040C212yesmaximal, complemented
10#23772032C211yesmaximal, characteristic, complemented
11#131×Dic79C212yesmaximal, complemented
12C2×#19098515C211yesmaximal, characteristic, complemented
13C79×#2932C211yesmaximal, characteristic, complemented
14C2×#14579168C2212yes—
15C22×#11929607C2211yescharacteristic
16C22×#11929607C2211yescharacteristic
17#19099120C2211yescharacteristic
18#19099118C412yes—
19#19099111C412yes—
20#19099095C2214yes—
21#19099094C414yes—
22#19099092C414yes—
23#19099073C414yes—
24#19099070C2212yes—
25#41×Dic79C2212yes—
26C4×#11929623C2214yes—
27C2×C42×D79C2211yescharacteristic
28#19098692C2218yescomplemented
29#19098691C2218yescomplemented
30#19098639C2218yes—
31#19098638C2218yes—
32#19098635C2218yescomplemented
33#19098634C2218yes—
34#19098631C2218yes—
35#19098621C2218yescomplemented
36#19098619C2214yes—
37#19098615C2214yes—
38#19098613C2218yescomplemented
39#19098591C2212yes—
40#19098589C2212yes—
41#19098587C2211yescharacteristic
42#19098530C2212yes—
43#19098526C2212yes—
44#19098521C2212yes—
45#19098521C2214yescomplemented
46#19098518C2212yes—
47#19098518C2214yescomplemented
48#19098515C2214yes—
49#19098515C2214yes—
50M16×Dic79C2212yes—
51M16×Dic79C2214yescomplemented
52C2×#14579072C2211yescharacteristic
53C2×C8×Dic79C2211yescharacteristic
54C2×C8×Dic79C2211yescharacteristic
55C2×C8×Dic79C2214yescomplemented
56C2×#131×C79C2212yes—
57#532×C79C2211yescharacteristic
58#435×C79C2218yescomplemented
59#423×C79C2211yescharacteristic
60#408×C79C2212yes—
61C2×#97×C79C2211yescharacteristic
62C2×#11929623C2×C414yes—
63C22×C4×D79C2×C412yes—
64C22×C4×D79C2311yescharacteristic
65#14579168—28no—
66C2×#11929608C2314yes—
67C2×#11929608C2312yes—
68C2×#11929607—24no—
69C2×#11929607—24no—
70C2×#11929607C2314yes—
71C2×#11929607C2314yes—
72C22×C79⋊C8C2314yes—
73C22×C79⋊C8C2311yescharacteristic
74C22×C79⋊C8C2311yescharacteristic
75#14579161C2×C412yes—
76#14579154C2×C418yes—
77#14579153C2×C412yes—
78#14579152C2×C414yes—
79#14579148C2×C414yes—
80#14579143C2×C418yes—
81#14579142C2×C414yes—
82#14579141C2×C414yes—
83#14579140C2×C414yes—
84#14579137C2×C414yes—
85#14579132C2×C414yes—
86#14579132C2×C414yes—
87#14579129C2×C414yes—
88Q8×Dic79C2×C414yes—
89C4×Dic158C2314yes—
90#14579126C2×C414yes—
91D4×Dic79C2×C414yes—
92D4×Dic79C2×C418yes—
93D4×Dic79C2×C414yes—
94C4×C22⋊D79C2318yes—
95(C22⋊D79)⋊C4C2×C418yes—
96#14579120C2312yes—
97#14579120C2314yes—
98#14579119C2×C414yes—
99#14579117C2×C418yes—
100C2×C4×Dic79C2×C412yes—
101C2×C4×Dic79C2311yescharacteristic
102C2×C4×Dic79C2314yes—
103C42×D79C2×C412yes—
104C42×D79C2×C412yes—
105C42×D79C2312yes—
106C42×D79C2312yes—
107C4⋊(C79⋊C8)C2314yes—
108Dic79⋊C8C2318yes—
109Dic79⋊C8C2318yes—
110C22⋊(C79⋊C8)C2318yes—
111(C2×D79)⋊C8C2318yes—
112#14579072C2314yes—
113#14579071C2314yes—
114#14579071C2314yes—
115#14579071C2314yes—
116C8×Dic79C2314yes—
117C8×Dic79C2314yes—
118C8×Dic79C2312yes—
119C8×Dic79C2312yes—
120C8×Dic79C2312yes—
121C8×Dic79C2312yes—
122C8×Dic79C2314yes—
123C2×#41×C79C2312yes—
124#131×C79—28no—
125C2×M16×C79C2312yes—
126C2×M16×C79C2314yes—
127C22×C8×C79C2314yes—
128C22×C8×C79C2311yescharacteristic
129C22×C8×C79C2311yescharacteristic
130C4⋊Q8×C79C2×C411yescharacteristic
131#127×C79C2×C411yescharacteristic
132#124×C79C2×C412yes—
133#121×C79C2×C414yes—
134C4×D4×C79C2314yes—
135C2×C42×C79C2311yescharacteristic
136C4⋊C8×C79C2314yes—
137C22⋊C8×C79C2318yes—
138#97×C79C2314yes—
139C23×D79C22×C411yescharacteristic
140#11929623—48no—
141C2×Dic158C22×C414yes—
142C2×C22⋊D79C22×C418yes—
143C22×Dic79C22×C418yes—
144C22×Dic79C22×C411yescharacteristic
145C22×Dic79C22×C412yes—
146C2×C4×D79—22no—
147C2×C4×D79—22no—
148C2×C4×D79—44no—
149C2×C4×D79C22×C412yes—
150C2×C4×D79C22×C412yes—
151C2×C4×D79C22×C412yes—
152C2×C4×D79C22×C412yes—
153C2×C4×D79C22×C412yes—
154C2×C4×D79C2412yes—
155#11929608—28no—
156#11929608—24no—
157#11929607—28no—
158#11929607—28no—
159C2×C79⋊C8—28no—
160C2×C79⋊C8—22no—
161C2×C79⋊C8#4114yescomplemented
162C2×C79⋊C8C2414yes—
163C2×C79⋊C8C2412yes—
164C2×C79⋊C8C2412yes—
165C4⋊Dic79C22×C414yes—
166C4⋊Dic79C22×C414yes—
167C4⋊Dic79C22×C418yes—
168Dic79⋊C4C22×C418yes—
169Dic79⋊C4C22×C418yes—
170C22⋊Dic79C22×C418yes—
171C22⋊Dic79C22×C414yes—
172C22⋊Dic79C22×C418yes—
173(C2×D79)⋊C4C22×C418yes—
174C4×Dic79C22×C412yes—
175C4×Dic79C22×C412yes—
176C4×Dic79C22×C412yes—
177C4×Dic79C22×C412yes—
178C4×Dic79C22×C412yes—
179C4×Dic79C22×C418yes—
180C4×Dic79C2411yescharacteristic
181C4×Dic79C2411yescharacteristic
182C4×Dic79C2414yes—
183#41×C79—24no—
184#41×C79—24no—
185C2×Q8×C79C22×C412yes—
186C2×D4×C79C22×C414yes—
187C2×D4×C79C22×C412yes—
188C2×D4×C79C22×C414yes—
189C22×C4×C79C22×C412yes—
190C22×C4×C79C2414yes—
191C22×C4×C79C2411yescharacteristic
192M16×C79—28no—
193M16×C79—24no—
194C2×C8×C79—28no—
195C2×C8×C79—22no—
196C2×C8×C79#4114yescomplemented
197C2×C8×C79C2412yes—
198C2×C8×C79C2414yes—
199C2×C8×C79C2412yes—
200C4⋊C4×C79C22×C414yes—
201C22⋊C4×C79C22×C418yes—
202C42×C79C22×C411yescharacteristic
203C42×C79C22×C411yescharacteristic
204C42×C79C2412yes—
205C23×C79C23×C414yes—
206C23×C79C23×C411yescharacteristic
207C22×D79—22no—
208C22×D79—42no—
209C22×D79C23×C412yes—
210Q8×C79—24no—
211D4×C79—28no—
212D4×C79—24no—
213D4×C79—44no—
214C2×C4×C79—24no—
215C2×C4×C79—24no—
216C2×C4×C79—42no—
217C2×C4×C79C23×C414yes—
218C2×C4×C79C23×C412yes—
219C2×C4×C79C23×C412yes—
220C2×C4×C79C23×C411yescharacteristic
221C2×C4×C79C2×#4112yes—
222C2×C4×C79C2×#4112yes—
223C2×C4×C79C2511yescharacteristic
224C22⋊D79—48no—
225C2×Dic79—22no—
226C2×Dic79—24no—
227C2×Dic79—216no—
228C2×Dic79C23×C412yes—
229C2×Dic79C23×C412yes—
230C2×Dic79C23×C418yes—
231C2×Dic79C23×C412yes—
232C4×D79—24no—
233C4×D79—22no—
234C4×D79—22no—
235C4×D79—24no—
236C4×D79—24no—
237C4×D79—42no—
238C4×D79—42no—
239Dic158—44no—
240C8×C79—22no—
241C8×C79—24no—
242C8×C79C2×#4114yes—
243C79⋊C8—22no—
244C79⋊C8—24no—
245C79⋊C8C2×#4114yes—
246C22×C79—22no—
247C22×C79—24no—
248C22×C79—28no—
249C22×C79C4×#4112yes—
250C22×C79C24×C411yescharacteristic
251C2×D79—22no—
252C2×D79—22no—
253C2×D79—22no—
254C2×D79—44no—
255C4×C79—24no—
256C4×C79—22no—
257C4×C79—22no—
258C4×C79C4×#4112yes—
259C4×C79C22×#4111yescharacteristic
260C4×C79C22×#4111yescharacteristic
261Dic79—22no—
262Dic79—28no—
263Dic79—22no—
264Dic79#56814yes—
265#60205—791nomaximal, Sylow
266C2×C79—22no—
267C2×C79—44no—
268C2×C79C2×C4×#4111yescharacteristic
269C2×C79C2×#56812yes—
270D79—42no—
271#3444—791no—
272#2967—798no—
273#2965—798no—
274#2961—791no—
275#2961—791no—
276#2932—791no—
277#2932D7911yescharacteristic, complemented
278#2930—792no—
279#2930—792no—
280#2922—792no—
281#2920—791no—
282C4×#131—792no—
283C2×C4×M16—791no—
284C79#6020511yescharacteristic, complemented, Sylow
285C2×#131—792no—
286C2×#131C2×D7912yes—
287C22×M16—791no—
288C22×M16—791no—
289#533—792no—
290#533—791no—
291#532—792no—
292#532C2×D7911yescharacteristic
293#529—794no—
294#529—794no—
295#529—794no—
296#525—792no—
297#525—794no—
298C4×#41—792no—
299C4×#41—794no—
300C22×C42—791no—
301#446—798no—
302#446—798no—
303#436—798no—
304#436—798no—
305#435—798no—
306#435—798no—
307#435—798no—
308#435C2×D7918yescomplemented
309#433—798no—
310#432—794no—
311#431—794no—
312#431—798no—
313#424—792no—
314#424—792no—
315#423—791no—
316#423C2×D7911yescharacteristic
317#408—792no—
318#408C2×D7912yes—
319#407—792no—
320#405—792no—
321#405—794no—
322#405—792no—
323#405—794no—
324C4×M16—792no—
325C4×M16—794no—
326C4×M16—794no—
327C4×M16—794no—
328C2×#97—791no—
329C2×#97C2×D7911yescharacteristic
330C2×C4×C8—791no—
331C2×C4×C8—791no—
332C2×C4×C8—794no—
333C2×#41—794no—
334C2×#41C22×D7912yes—
335C23×C4—792no—
336C23×C4—791no—
337#131—28no—
338#131—1588no—
339C2×M16—792no—
340C2×M16—794no—
341C2×M16—794no—
342C2×M16—794no—
343C2×M16—1584no—
344C2×M16—1584no—
345C2×M16C22×D7912yes—
346C2×M16C22×D7914yes—
347C22×C8—791no—
348C22×C8—791no—
349C22×C8—794no—
350C22×C8C22×D7911yescharacteristic
351C22×C8C22×D7914yes—
352C22×C8C22×D7911yescharacteristic
353C4⋊Q8—792no—
354C4⋊Q8C4×D7911yescharacteristic
355#127C4×D7911yescharacteristic
356#126—798no—
357#125—794no—
358#125—792no—
359#124—794no—
360#124C4×D7912yes—
361#123—794no—
362#123—798no—
363C22⋊Q8—794no—
364C22⋊Q8—794no—
365C22⋊Q8—794no—
366#121—794no—
367#121—794no—
368#121C4×D7914yes—
369#120—794no—
370C4×Q8—794no—
371C4×Q8—794no—
372C4×Q8—794no—
373C4×D4—798no—
374C4×D4—794no—
375C4×D4—794no—
376C4×D4—798no—
377C4×D4—798no—
378C4×D4C22×D7914yes—
379#117—792no—
380#117—794no—
381#117—794no—
382#117—798no—
383C2×C42—792no—
384C2×C42—794no—
385C2×C42—791no—
386C2×C42—792no—
387C2×C42—792no—
388C2×C42—792no—
389C2×C42—792no—
390C2×C42C22×D7911yescharacteristic
391C4⋊C8—798no—
392C4⋊C8—798no—
393C4⋊C8—794no—
394C4⋊C8C22×D7914yes—
395C22⋊C8—798no—
396C22⋊C8—798no—
397C22⋊C8C22×D7918yes—
398#97—794no—
399#97—794no—
400#97—794no—
401#97—794no—
402#97C22×D7914yes—
403C4×C8—792no—
404C4×C8—792no—
405C4×C8—794no—
406C4×C8—792no—
407C4×C8—792no—
408C4×C8—794no—
409C4×C8—794no—
410C24—791no—
411#41—24no—
412#41—24no—
413#41—3168no—
414C2×Q8—794no—
415C2×Q8C2×C4×D7912yes—
416C2×D4—798no—
417C2×D4C2×C4×D7914yes—
418C2×D4C2×C4×D7912yes—
419C2×D4C2×C4×D7914yes—
420C22×C4—3164no—
421C22×C4—792no—
422C22×C4—792no—
423C22×C4—792no—
424C22×C4—791no—
425C22×C4—798no—
426C22×C4—792no—
427C22×C4—792no—
428C22×C4—792no—
429C22×C4—792no—
430C22×C4—1582no—
431C22×C4—1582no—
432C22×C4C2×C4×D7912yes—
433C22×C4C23×D7914yes—
434C22×C4C23×D7911yescharacteristic
435M16—24no—
436M16—28no—
437M16—1588no—
438M16—1588no—
439M16—1584no—
440M16—1588no—
441C2×C8—22no—
442C2×C8—28no—
443C2×C8—794no—
444C2×C8—792no—
445C2×C8—794no—
446C2×C8—792no—
447C2×C8—1588no—
448C2×C8—1582no—
449C2×C8#1192962314yescomplemented
450C2×C8C23×D7914yes—
451C2×C8C23×D7912yes—
452C2×C8C23×D7912yes—
453C4⋊C4—798no—
454C4⋊C4—798no—
455C4⋊C4—794no—
456C4⋊C4—794no—
457C4⋊C4—798no—
458C4⋊C4C2×C4×D7914yes—
459C22⋊C4—794no—
460C22⋊C4—798no—
461C22⋊C4—798no—
462C22⋊C4—798no—
463C22⋊C4C2×C4×D7918yes—
464C42—791no—
465C42—791no—
466C42—794no—
467C42—792no—
468C42—792no—
469C42—792no—
470C42—792no—
471C42—792no—
472C42—798no—
473C42C2×C4×D7911yescharacteristic
474C42C2×C4×D7911yescharacteristic
475C42C23×D7912yes—
476C23—3162no—
477C23—792no—
478C23—1582no—
479C23C22×C4×D7914yes—
480C23C22×C4×D7911yescharacteristic
481Q8—24no—
482Q8—3164no—
483D4—28no—
484D4—24no—
485D4—44no—
486D4—3168no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—3162no—
491C2×C4—3162no—
492C2×C4—792no—
493C2×C4—792no—
494C2×C4—792no—
495C2×C4—798no—
496C2×C4—1584no—
497C2×C4—1584no—
498C2×C4—1584no—
499C2×C4—15816no—
500C2×C4—1582no—
501C2×C4—1584no—
502C2×C4—1582no—
503C2×C4—1582no—
504C2×C4C22×C4×D7912yes—
505C2×C4C22×C4×D7912yes—
506C2×C4C22×C4×D7914yes—
507C2×C4C22×C4×D7911yescharacteristic
508C2×C4#41×D7912yes—
509C2×C4C2×#1192962312yes—
510C2×C4C24×D7911yescharacteristic
511C8—22no—
512C8—24no—
513C8—794no—
514C8—1584no—
515C8—1582no—
516C8C2×#1192962314yes—
517C22—22no—
518C22—28no—
519C22—24no—
520C22—3164no—
521C22—1582no—
522C22—1582no—
523C22—1582no—
524C22C4×#1192962312yes—
525C22C23×C4×D7911yescharacteristic
526C4—22no—
527C4—22no—
528C4—24no—
529C4—794no—
530C4—1582no—
531C4—1588no—
532C4—1582no—
533C4C4×#1192962312yes—
534C4C2×#41×D7911yescharacteristic
535C4C22×#1192962311yescharacteristic
536C2—22no—
537C2—44no—
538C2—3162no—
539C2C2×C4×#1192962311yescharacteristic
540C2#568×D7912yes—
subgroupG/H
center, FrattiniC2×C4C24×D79
commutator, socleC22×C79C24×C4
FittingC79×#2932C2
prime radical#23776695C2
cosocleC2×C4×C79C25
hypercenter#2932D79
hypocenterC79#60205
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #2932

Complements

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

NH ≅ G/Ntype
2. #23772382C2semidirect
3. #23772360C2semidirect
4. #23772340C2semidirect
5. #23772337C2semidirect
6. #23772118C2semidirect
7. #23772106C2semidirect
8. #23772101C2semidirect
9. #23772040C2semidirect
10. #23772032C2semidirect
11. #131×Dic79C2semidirect
12. C2×#19098515C2semidirect
13. C79×#2932C2semidirect
28. #19098692C22semidirect
29. #19098691C22semidirect
32. #19098635C22semidirect
35. #19098621C22semidirect
38. #19098613C22semidirect
45. #19098521C22semidirect
47. #19098518C22semidirect
51. M16×Dic79C22semidirect
55. C2×C8×Dic79C22semidirect
58. #435×C79C22semidirect
161. C2×C79⋊C8#41semidirect
196. C2×C8×C79#41semidirect
277. #2932D79semidirect
284. C79#60205semidirect
308. #435C2×D79semidirect
449. C2×C8#11929623semidirect

Maximal subgroups

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

subgroupindex
#237766952
#237723822
#237723602
#237723402
#237723372
#237721182
#237721062
#237721012
#237720402
#237720322
#131×Dic792
C2×#190985152
C79×#29322
#6020579