Finite Group Databasebeta

← #35152668

Subgroups — #35152668

Order 20224.

1882 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D79C211yesmaximal, characteristic
2#23772352C218yesmaximal, complemented
3#23772345C218yesmaximal, complemented
4#23772341C211yesmaximal, characteristic, complemented
5#23772340C211yesmaximal, characteristic, complemented
6#23772337C211yesmaximal, characteristic, complemented
7#23772333C212yesmaximal, complemented
8#23772332C212yesmaximal, complemented
9#23771961C212yesmaximal
10#423×D79C211yesmaximal, characteristic
11#23771816C212yesmaximal
12C2×#19098515C211yesmaximal, characteristic
13C79×#2961C211yesmaximal, characteristic, complemented
14C22×#11929607C2211yescharacteristic
15C22×#11929607C2211yescharacteristic
16C2×M16×D79C2212yes—
17#19099120C2211yescharacteristic
18#19099119C412yes—
19#19099118C412yes—
20#19099117C2214yes—
21#19099116C2212yes—
22#19099115C414yes—
23C4⋊Q8×D79C412yes—
24#125×D79C412yes—
25C22⋊Q8×D79C414yes—
26C4×Q8×D79C2214yes—
27#117×D79C2212yes—
28C2×C42×D79C2211yescharacteristic
29#19098694C2218yescomplemented
30#19098692C2218yescomplemented
31#19098692C2218yescomplemented
32#19098691C2218yescomplemented
33#19098691C2218yescomplemented
34C8×Dic158C2218yescomplemented
35#19098687C2218yescomplemented
36#19098591C2212yes—
37#19098590C2212yes—
38#19098589C2212yes—
39#19098588C2218yes—
40#19098587C2211yescharacteristic
41#19098586C2212yes—
42#19098585C2214yes—
43#19098585C2218yes—
44C4⋊C8×D79C2214yes—
45#19098521C2212yes—
46#19098520C2214yes—
47#19098519C2212yes—
48#19098518C2212yes—
49#19098517C2214yes—
50#19098515C2214yes—
51#19098515C2214yes—
52#19098515C2214yes—
53C4×#11929607C2214yes—
54C2×#14579072C2211yescharacteristic
55C2×C8×Dic79C2211yescharacteristic
56C2×C8×Dic79C2211yescharacteristic
57#533×C79C2211yescharacteristic
58#446×C79C2218yescomplemented
59#424×C79C2212yes—
60#423×C79C2211yescharacteristic
61#405×C79C2212yes—
62C2×#97×C79C2211yescharacteristic
63C2×Q8×D79C2×C414yes—
64C22×C4×D79C2×C412yes—
65C22×C4×D79C2311yescharacteristic
66C2×#11929608C2312yes—
67C2×#11929607—24no—
68C2×#11929607Q818yescomplemented
69C2×#11929607C2314yes—
70C2×#11929607C2312yes—
71C2×#11929607C2312yes—
72C2×#11929607C2312yes—
73C2×#11929607C2312yes—
74M16×D79—28no—
75C22×C79⋊C8C2311yescharacteristic
76C22×C79⋊C8C2311yescharacteristic
77C2×C8×D79C2314yes—
78#14579161C2×C412yes—
79#14579160C2×C414yes—
80#14579160C2×C414yes—
81#14579159C2×C414yes—
82#14579153C2×C412yes—
83#14579152C2×C414yes—
84#14579151C2×C418yes—
85#14579141C2×C414yes—
86#14579140C2×C414yes—
87#14579139C2×C418yes—
88#14579138C2×C418yes—
89#14579137C2×C414yes—
90Q8×Dic79C2×C414yes—
91C4×Dic158C2318yes—
92C4×Dic158C2314yes—
93#14579126C2×C414yes—
94#14579126C2×C418yes—
95#14579120C2312yes—
96#14579119C2×C414yes—
97#14579118C2314yes—
98#14579117C2×C414yes—
99C4⋊C4×D79C2×C414yes—
100C4⋊C4×D79C2×C418yes—
101C4⋊C4×D79C2×C414yes—
102C22⋊C4×D79C2×C414yes—
103C2×C4×Dic79C2×C412yes—
104C2×C4×Dic79C2311yescharacteristic
105C42×D79C2×C411yescharacteristic
106C42×D79C2×C411yescharacteristic
107C42×D79C2×C411yescharacteristic
108C42×D79C2×C411yescharacteristic
109C42×D79C2312yes—
110C42×D79C2312yes—
111C42×D79C2314yes—
112C4⋊(C79⋊C8)C2318yes—
113C4⋊(C79⋊C8)C2314yes—
114Dic79⋊C8C2318yes—
115Dic79⋊C8C2318yes—
116Dic79⋊C8C2318yes—
117#14579072C2314yes—
118#14579072C2314yes—
119#14579071C2314yes—
120#14579071C2314yes—
121C4×C79⋊C8C2314yes—
122C8×Dic79C2314yes—
123C8×Dic79C2314yes—
124C8×Dic79C2314yes—
125C8×Dic79C2314yes—
126C2×M16×C79C2312yes—
127C22×C8×C79C2311yescharacteristic
128C22×C8×C79C2311yescharacteristic
129C4⋊Q8×C79C2×C412yes—
130#125×C79C2×C412yes—
131C22⋊Q8×C79C2×C414yes—
132C4×Q8×C79C2314yes—
133#117×C79C2312yes—
134C2×C42×C79C2311yescharacteristic
135C4⋊C8×C79C2314yes—
136C4⋊C8×C79C2318yes—
137#97×C79C2314yes—
138#97×C79C2314yes—
139C4×C8×C79C2314yes—
140C23×D79C22×C411yescharacteristic
141C2×Dic158C22×C418yes—
142C2×Dic158C22×C414yes—
143Q8×D79—48no—
144C22×Dic79C22×C412yes—
145C22×Dic79C22×C411yescharacteristic
146C2×C4×D79—44no—
147C2×C4×D79C22×C412yes—
148C2×C4×D79C22×C412yes—
149C2×C4×D79C22×C418yes—
150C2×C4×D79C22×C412yes—
151C2×C4×D79C22×C412yes—
152C2×C4×D79C22×C411yescharacteristic
153C2×C4×D79C22×C411yescharacteristic
154C2×C4×D79C2×Q814yes—
155C2×C4×D79#4114yes—
156C2×C4×D79C2411yescharacteristic
157C2×C4×D79C2411yescharacteristic
158#11929608—24no—
159#11929607—28no—
160#11929607—24no—
161#11929607—24no—
162#11929607C2×Q818yes—
163#11929607C2×Q818yescomplemented
164C2×C79⋊C8—22no—
165C2×C79⋊C8C2×Q814yescomplemented
166C2×C79⋊C8C2412yes—
167C2×C79⋊C8C2414yes—
168C2×C79⋊C8C2412yes—
169C8×D79—28no—
170C4⋊Dic79C22×C414yes—
171C4⋊Dic79C22×C414yes—
172C4⋊Dic79C22×C418yes—
173Dic79⋊C4C22×C418yes—
174Dic79⋊C4C22×C418yes—
175Dic79⋊C4C22×C418yes—
176Dic79⋊C4C22×C418yes—
177C22⋊Dic79C22×C414yes—
178(C2×D79)⋊C4C22×C418yes—
179C4×Dic79C22×C411yescharacteristic
180C4×Dic79C22×C411yescharacteristic
181C4×Dic79C22×C412yes—
182C4×Dic79C22×C412yes—
183C4×Dic79C22×C412yes—
184C4×Dic79C22×C412yes—
185C4×Dic79C22×C418yes—
186C4×Dic79C2412yes—
187C4×Dic79C2414yes—
188C2×Q8×C79C22×C414yes—
189C22×C4×C79C22×C412yes—
190C22×C4×C79C2411yescharacteristic
191M16×C79—24no—
192C2×C8×C79—22no—
193C2×C8×C79C2×Q814yescomplemented
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—42no—
207C22×D79C4×Q814yes—
208C22×D79C23×C411yescharacteristic
209C22×D79C23×C411yescharacteristic
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×C79C22×Q812yes—
217C2×C4×C79C2×#4112yes—
218C2×C4×C79C2511yescharacteristic
219C2×Dic79—24no—
220C2×Dic79C4×Q814yes—
221C2×Dic79C23×C412yes—
222C2×Dic79C23×C418yes—
223C2×Dic79C23×C411yescharacteristic
224C2×Dic79C23×C411yescharacteristic
225C2×Dic79C23×C412yes—
226C4×D79—24no—
227C4×D79—24no—
228C4×D79—42no—
229C4×D79—42no—
230C4×D79—48no—
231C4×D79C4×Q814yes—
232C4×D79C4×Q814yes—
233C4×D79C22×Q812yes—
234C4×D79C22×Q812yes—
235C4×D79C2×#4112yes—
236C4×D79C2×#4112yes—
237Dic158—216no—
238Dic158—44no—
239C8×C79—24no—
240C8×C79—22no—
241C8×C79C22×Q814yes—
242C79⋊C8—24no—
243C79⋊C8—22no—
244C79⋊C8C22×Q814yes—
245C22×C79—22no—
246C22×C79C2×C4×Q812yes—
247C22×C79C24×C411yescharacteristic
248C2×D79—22no—
249C2×D79—22no—
250C2×D79—44no—
251C2×D79C2×C4×Q812yes—
252C2×D79C2×C4×Q812yes—
253C4×C79—22no—
254C4×C79—22no—
255C4×C79—28no—
256C4×C79C2×C4×Q812yes—
257C4×C79C23×Q811yescharacteristic
258C4×C79C22×#4111yescharacteristic
259Dic79—28no—
260Dic79—22no—
261Dic79C2×C4×Q812yes—
262Dic79C2×C4×Q812yes—
263Dic79#56814yes—
264#60192—791nomaximal, Sylow
265C2×C79—22no—
266C2×C79C22×C4×Q811yescharacteristic
267C2×C79C2×#56812yes—
268D79—22no—
269D79—22no—
270C2×#533—791no—
271Q8×M16—798no—
272#2962—798no—
273#2961—791no—
274#2961—791no—
275#2961—791no—
276#2961D7911yescharacteristic, complemented
277#2960—792no—
278#2960—792no—
279#2905—792no—
280C2×#423—791no—
281#2873—792no—
282C2×C4×M16—791no—
283C79#6019211yescharacteristic, complemented, Sylow
284C22×M16—792no—
285C22×M16—791no—
286C22×M16—791no—
287#533—792no—
288#533—792no—
289#533—794no—
290#533—794no—
291#533—792no—
292#533—791no—
293#533C2×D7911yescharacteristic
294C2×C4⋊Q8—792no—
295C2×#125—792no—
296C2×C22⋊Q8—794no—
297C2×C4×Q8—794no—
298C2×#117—792no—
299C22×C42—791no—
300#446—798no—
301#446—798no—
302#446—798no—
303#446—798no—
304#446—798no—
305#446C2×D7918yescomplemented
306C8×Q8—798no—
307C8×Q8—798no—
308#424—792no—
309#424—792no—
310#424—792no—
311#424—798no—
312#424C2×D7912yes—
313#423—792no—
314#423—791no—
315#423—794no—
316#423—798no—
317#423C2×D7911yescharacteristic
318C2×C4⋊C8—794no—
319#405—792no—
320#405—792no—
321#405—792no—
322#405—794no—
323#405—794no—
324#405C2×D7912yes—
325C4×M16—794no—
326C4×M16—794no—
327C4×M16—794no—
328C4×M16—794no—
329C2×#97—791no—
330C2×#97C2×D7911yescharacteristic
331C2×C4×C8—791no—
332C2×C4×C8—791no—
333C22×Q8—794no—
334C23×C4—792no—
335C23×C4—791no—
336C2×M16—792no—
337C2×M16—798no—
338C2×M16—792no—
339C2×M16—792no—
340C2×M16—792no—
341C2×M16—794no—
342C2×M16—792no—
343C2×M16—1584no—
344C2×M16—1588no—
345C2×M16C22×D7912yes—
346C22×C8—791no—
347C22×C8—791no—
348C22×C8—794no—
349C22×C8C22×D7911yescharacteristic
350C22×C8C22×D7911yescharacteristic
351C4⋊Q8—794no—
352C4⋊Q8—794no—
353C4⋊Q8—794no—
354C4⋊Q8—792no—
355C4⋊Q8C4×D7912yes—
356#125—798no—
357#125—794no—
358#125—792no—
359#125C4×D7912yes—
360C22⋊Q8—794no—
361C22⋊Q8—794no—
362C22⋊Q8—798no—
363C22⋊Q8—794no—
364C22⋊Q8—798no—
365C22⋊Q8C4×D7914yes—
366C4×Q8—794no—
367C4×Q8—794no—
368C4×Q8—798no—
369C4×Q8—798no—
370C4×Q8—794no—
371C4×Q8C22×D7914yes—
372#117—792no—
373#117—794no—
374#117—794no—
375#117—794no—
376#117C22×D7912yes—
377C2×C4⋊C4—794no—
378C2×C4⋊C4—798no—
379C2×C4⋊C4—794no—
380C2×C22⋊C4—794no—
381C2×C42—792no—
382C2×C42—791no—
383C2×C42—791no—
384C2×C42—791no—
385C2×C42—791no—
386C2×C42—791no—
387C2×C42—794no—
388C2×C42—792no—
389C2×C42—792no—
390C2×C42C22×D7911yescharacteristic
391C4⋊C8—798no—
392C4⋊C8—798no—
393C4⋊C8—798no—
394C4⋊C8—794no—
395C4⋊C8—798no—
396C4⋊C8C22×D7914yes—
397C4⋊C8C22×D7918yes—
398#97—794no—
399#97—794no—
400#97—794no—
401#97—794no—
402#97C22×D7914yes—
403#97C22×D7914yes—
404C4×C8—794no—
405C4×C8—794no—
406C4×C8—794no—
407C4×C8—794no—
408C4×C8—794no—
409C4×C8C22×D7914yes—
410C24—791no—
411C2×Q8—3168no—
412C2×Q8—794no—
413C2×Q8—798no—
414C2×Q8C2×C4×D7914yes—
415C22×C4—3164no—
416C22×C4—792no—
417C22×C4—792no—
418C22×C4—798no—
419C22×C4—791no—
420C22×C4—794no—
421C22×C4—794no—
422C22×C4—791no—
423C22×C4—791no—
424C22×C4—791no—
425C22×C4—791no—
426C22×C4—792no—
427C22×C4—792no—
428C22×C4—792no—
429C22×C4C2×C4×D7912yes—
430C22×C4C23×D7911yescharacteristic
431M16—24no—
432M16—798no—
433M16—798no—
434M16—1584no—
435M16—1584no—
436M16—1588no—
437M16—1584no—
438C2×C8—22no—
439C2×C8—794no—
440C2×C8—792no—
441C2×C8—792no—
442C2×C8—794no—
443C2×C8—1588no—
444C2×C8—1582no—
445C2×C8Q8×D7914yescomplemented
446C2×C8C23×D7912yes—
447C2×C8C23×D7914yes—
448C2×C8C23×D7912yes—
449C4⋊C4—798no—
450C4⋊C4—798no—
451C4⋊C4—798no—
452C4⋊C4—798no—
453C4⋊C4—794no—
454C4⋊C4—798no—
455C4⋊C4—794no—
456C4⋊C4C2×C4×D7914yes—
457C4⋊C4C2×C4×D7918yes—
458C4⋊C4C2×C4×D7914yes—
459C22⋊C4—794no—
460C22⋊C4—798no—
461C22⋊C4C2×C4×D7914yes—
462C42—792no—
463C42—794no—
464C42—792no—
465C42—792no—
466C42—792no—
467C42—791no—
468C42—792no—
469C42—791no—
470C42—798no—
471C42C2×C4×D7911yescharacteristic
472C42C2×C4×D7911yescharacteristic
473C42C23×D7914yes—
474C42C23×D7912yes—
475C23—3162no—
476C23—794no—
477C23—791no—
478C23—791no—
479C23C22×C4×D7911yescharacteristic
480Q8—44no—
481Q8—3164no—
482Q8—15816no—
483C2×C4—42no—
484C2×C4—3162no—
485C2×C4—3162no—
486C2×C4—3168no—
487C2×C4—794no—
488C2×C4—791no—
489C2×C4—791no—
490C2×C4—792no—
491C2×C4—794no—
492C2×C4—794no—
493C2×C4—792no—
494C2×C4—792no—
495C2×C4—792no—
496C2×C4—792no—
497C2×C4—792no—
498C2×C4—798no—
499C2×C4—1584no—
500C2×C4—1584no—
501C2×C4—1584no—
502C2×C4C22×C4×D7912yes—
503C2×C4C22×C4×D7912yes—
504C2×C4C22×C4×D7918yes—
505C2×C4C22×C4×D7911yescharacteristic
506C2×C4C2×Q8×D7912yes—
507C2×C4#41×D7912yes—
508C2×C4C24×D7911yescharacteristic
509C8—22no—
510C8—24no—
511C8—794no—
512C8—1584no—
513C8—1582no—
514C8C2×Q8×D7914yes—
515C22—22no—
516C22—3164no—
517C22—792no—
518C22—792no—
519C22—1582no—
520C22—1582no—
521C22C4×Q8×D7912yes—
522C22C23×C4×D7911yescharacteristic
523C4—22no—
524C4—22no—
525C4—28no—
526C4—794no—
527C4—792no—
528C4—792no—
529C4—1582no—
530C4—1588no—
531C4C4×Q8×D7912yes—
532C4C22×Q8×D7911yescharacteristic
533C4C2×#41×D7911yescharacteristic
534C2—22no—
535C2—1582no—
536C2—1582no—
537C2C2×C4×Q8×D7911yescharacteristic
538C2#568×D7912yes—
subgroupG/H
center, FrattiniC2×C4C24×D79
commutator, socleC22×C79C24×C4
FittingC79×#2961C2
prime radicalC23×D79C22×C4
cosocleC2×C4×C79C25
hypercenter#2961D79
hypocenterC79#60192
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #2961

Complements

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

NH ≅ G/Ntype
2. #23772352C2semidirect
3. #23772345C2semidirect
4. #23772341C2semidirect
5. #23772340C2semidirect
6. #23772337C2semidirect
7. #23772333C2semidirect
8. #23772332C2semidirect
13. C79×#2961C2semidirect
29. #19098694C22semidirect
30. #19098692C22semidirect
31. #19098692C22semidirect
32. #19098691C22semidirect
33. #19098691C22semidirect
34. C8×Dic158C22semidirect
35. #19098687C22semidirect
58. #446×C79C22semidirect
68. C2×#11929607Q8semidirect
163. #11929607C2×Q8semidirect
165. C2×C79⋊C8C2×Q8semidirect
193. C2×C8×C79C2×Q8semidirect
276. #2961D79semidirect
283. C79#60192semidirect
305. #446C2×D79semidirect
445. C2×C8Q8×D79semidirect

Maximal subgroups

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

subgroupindex
#533×D792
#237723522
#237723452
#237723412
#237723402
#237723372
#237723332
#237723322
#237719612
#423×D792
#237718162
C2×#190985152
C79×#29612
#6019279