Finite Group Databasebeta

← #34259360

Subgroups — #34259360

Order 20224.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773484C211yesmaximal, characteristic
2#23772403C211yesmaximal, characteristic
3Q8×#11929607C211yesmaximal, characteristic, complemented
4#23762304C211yesmaximal, characteristic, complemented
5#23762294C211yesmaximal, characteristic, complemented
6#23762278C211yesmaximal, characteristic, complemented
7#23762271C211yesmaximal, characteristic, complemented
8#23762251C211yesmaximal, characteristic, complemented
9#23762235C211yesmaximal, characteristic, complemented
10#23762197C211yesmaximal, characteristic, complemented
11#23762187C211yesmaximal, characteristic, complemented
12#23761864C211yesmaximal, characteristic
13#23761793C211yesmaximal, characteristic, complemented
14#23761696C211yesmaximal, characteristic, complemented
15C79×#1579C211yesmaximal, characteristic, complemented
16#19099100C2211yescharacteristic
17C4×Q8×D79C2211yescharacteristic
18#19098892C411yescharacteristic
19#19098878C411yescharacteristic
20#19098869C411yescharacteristic
21#19098868C411yescharacteristic
22#19098788C411yescharacteristic
23#19098787C411yescharacteristic
24#19098717C411yescharacteristic
25#19098703C411yescharacteristic
26#19098694C2211yescharacteristic
27#19098693—21no—
28#19098693C2211yescharacteristic
29#19098690—21no—
30#19098690C2211yescharacteristic
31Q8×C79⋊C8C2211yescharacteristic
32C4×#11929614C2211yescharacteristic
33C4×#11929613C2211yescharacteristic
34C4×#11929611C2211yescharacteristic
35#19098642—21no—
36#19098642C2211yescharacteristic
37#19098633—21no—
38#19098628C2211yescharacteristic
39#19098623—21no—
40#19098615C2211yescharacteristic
41#19098614C2211yescharacteristic
42#19098614C2211yescharacteristic
43#19098585—21no—
44#19098585C2211yescharacteristic
45#19098517—21no—
46C4×#11929607—21no—
47C4×#11929607C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54#19098293C2211yescharacteristic
55#19098292C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C8×Q8×C79C2211yescharacteristic
62C4×Q16×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65Q8⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×Q8×D79C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71C2×#11929607—21no—
72C2×#11929607—21no—
73C2×#11929607D411yescharacteristic
74C2×#11929607D411yescharacteristic
75C2×C8×D79—21no—
76#14579156—21no—
77#14579152C2×C411yescharacteristic
78#14579150C2×C411yescharacteristic
79#14579146—21no—
80#14579137C2×C411yescharacteristic
81#14579136C2×C411yescharacteristic
82Q8×Dic79C2×C411yescharacteristic
83C4×Dic158—21no—
84C4×Dic158C2311yescharacteristic
85#14579126—21no—
86#14579126C2×C411yescharacteristic
87C4×D316—21no—
88C4×D316C2311yescharacteristic
89#14579118—21no—
90C4⋊C4×D79—21no—
91C4⋊C4×D79C2×C411yescharacteristic
92C42×D79—21no—
93C42×D79C2311yescharacteristic
94#14579094C2×C411yescharacteristic
95#14579093C2×C411yescharacteristic
96#14579092C2×C411yescharacteristic
97C4⋊(C79⋊C8)—21no—
98C4⋊(C79⋊C8)—21no—
99C4⋊(C79⋊C8)C2311yescharacteristic
100C4⋊(C79⋊C8)C2311yescharacteristic
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8C2×C411yescharacteristic
106Dic79⋊C8C2×C411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579085C2×C411yescharacteristic
109#14579085C2×C411yescharacteristic
110#14579084C2×C411yescharacteristic
111#14579083C2×C411yescharacteristic
112#14579081C2×C411yescharacteristic
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8C2×C411yescharacteristic
118(C2×D79)⋊C8C2×C411yescharacteristic
119#14579072—21no—
120#14579071—21no—
121#14579071—21no—
122#14579071D411yescharacteristic
123#14579071D411yescharacteristic
124C4×C79⋊C8—21no—
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C4×C79⋊C8C2311yescharacteristic
128C8×Dic79—21no—
129C2×Q16×C79C2×C411yescharacteristic
130C4×Q8×C79C2311yescharacteristic
131C4×Q8×C79C2311yescharacteristic
132#107×C79C2×C411yescharacteristic
133C4⋊C8×C79—21no—
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#103×C79C2×C411yescharacteristic
139C4×C8×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158C22×C411yescharacteristic
146Q8×D79—22no—
147Q8×D79—41no—
148C2×D316—21no—
149C2×D316C22×C411yescharacteristic
150C2×C4×D79—21no—
151C2×C4×D79—21no—
152C2×C4×D79—21no—
153C2×C4×D79C22×C411yescharacteristic
154C2×C4×D79C2×D411yescharacteristic
155C2×C4×D79#4111yescharacteristic
156#11929614—41no—
157#11929613—41no—
158#11929611—22no—
159#11929607—22no—
160#11929607—41no—
161#11929607—42no—
162#11929607—42no—
163C2×C79⋊C8—21no—
164C2×C79⋊C8—21no—
165C2×C79⋊C8—21no—
166C2×C79⋊C8C22×C411yescharacteristic
167C2×C79⋊C8C22×C411yescharacteristic
168C2×C79⋊C8C2×D411yescharacteristic
169C2×C79⋊C8C2×D411yescharacteristic
170C8×D79—42no—
171C4⋊Dic79—21no—
172C4⋊Dic79—21no—
173C4⋊Dic79C22×C411yescharacteristic
174C4⋊Dic79C22×C411yescharacteristic
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179Dic79⋊C4—21no—
180(C2×D79)⋊C4—21no—
181(C2×D79)⋊C4—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185C4×Dic79—21no—
186C4×Dic79—21no—
187C4×Dic79—21no—
188C4×Dic79C22×C411yescharacteristic
189C4×Dic79C2×D411yescharacteristic
190C4×Dic79#4111yescharacteristic
191C2×Q8×C79C22×C411yescharacteristic
192C2×Q8×C79C22×C411yescharacteristic
193Q16×C79—22no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79C22×C411yescharacteristic
198C2×C8×C79C22×C411yescharacteristic
199C2×C8×C79C2×D411yescharacteristic
200C2×C8×C79C2×D411yescharacteristic
201C4⋊C4×C79—21no—
202C4⋊C4×C79—21no—
203C4⋊C4×C79C22×C411yescharacteristic
204C4⋊C4×C79C22×C411yescharacteristic
205C42×C79—21no—
206C42×C79—21no—
207C42×C79C2411yescharacteristic
208C22×D79—21no—
209C22×D79C4×D411yescharacteristic
210Q8×C79—21no—
211Q8×C79—21no—
212Q8×C79—21no—
213Q8×C79#13112yes—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79C23×C411yescharacteristic
219C2×C4×C79C22×D411yescharacteristic
220C2×C4×C79C2×#4111yescharacteristic
221C2×Dic79—21no—
222C2×Dic79—21no—
223C2×Dic79—21no—
224C2×Dic79C4×D411yescharacteristic
225C2×Dic79C4×D411yescharacteristic
226C2×Dic79C4×D411yescharacteristic
227D316—21no—
228D316—21no—
229D316—81no—
230C4×D79—21no—
231C4×D79—21no—
232C4×D79—21no—
233C4×D79—21no—
234C4×D79—41no—
235C4×D79—42no—
236C4×D79—41no—
237C4×D79—41no—
238C4×D79—81no—
239Dic158—21no—
240Dic158—21no—
241Dic158—42no—
242C8×C79—21no—
243C8×C79—21no—
244C8×C79—21no—
245C8×C79—21no—
246C8×C79—41no—
247C8×C79—41no—
248C8×C79—41no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C79⋊C8#13712yes—
256C22×C79C2×C4×D411yescharacteristic
257C2×D79—21no—
258C2×D79—21no—
259C2×D79—21no—
260C2×D79—41no—
261C2×D79—41no—
262C2×D79—41no—
263C4×C79—22no—
264C4×C79—21no—
265C4×C79—41no—
266C4×C79—41no—
267C4×C79—41no—
268C4×C79C2×#13111yescharacteristic
269C4×C79#56811yescharacteristic
270C4×C79C2×#13711yescharacteristic
271C4×C79#57511yescharacteristic
272Dic79—21no—
273Dic79—21no—
274Dic79—21no—
275Dic79—41no—
276Dic79—41no—
277Dic79—41no—
278#18010—791nomaximal, Sylow
279C2×C79#293311yescharacteristic
280C2×C79C4×#13711yescharacteristic
281C2×C79#295711yescharacteristic
282D79—41no—
283D79—81no—
284#3108—791no—
285#2969—791no—
286Q8×M16—791no—
287#1592—791no—
288#1591—791no—
289#1588—791no—
290#1587—791no—
291#1584—791no—
292#1582—791no—
293#1579D7911yescharacteristic, complemented
294C8×SD16—791no—
295#1574—791no—
296#1495—791no—
297#1479—791no—
298#1453—791no—
299C79#1801011yescharacteristic, complemented, Sylow
300#529—791no—
301C2×C4×Q8—791no—
302#483—791no—
303#480—791no—
304#479—791no—
305#479—791no—
306#464—791no—
307#464—791no—
308#451—791no—
309#448—791no—
310#446—791no—
311#446—791no—
312#446—791no—
313#446—1581no—
314#446—1581no—
315C8×Q8—791no—
316C8×Q8C2×D7911yescharacteristic
317C8×Q8C2×D7911yescharacteristic
318C4×Q16—791no—
319C4×Q16C2×D7911yescharacteristic
320C4×SD16—791no—
321C4×SD16—791no—
322#436—791no—
323#436—1581no—
324#435—1581no—
325C8×D4—791no—
326#433—1581no—
327#431—791no—
328#431—791no—
329#431—791no—
330#423—791no—
331#423—1581no—
332#405—1581no—
333C4×M16—791no—
334C4×M16—1581no—
335#335—791no—
336#335—791no—
337#334—791no—
338#334C2×D7911yescharacteristic
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8—791no—
342Q8⋊C8—791no—
343Q8⋊C8C2×D7911yescharacteristic
344Q8⋊C8C2×D7911yescharacteristic
345D4⋊C8—791no—
346D4⋊C8—791no—
347#322—791no—
348#322—791no—
349#322C2×D7911yescharacteristic
350C82—791no—
351C22×Q8—791no—
352C2×Q16—791no—
353C2×Q16C4×D7911yescharacteristic
354C2×SD16—791no—
355C2×SD16—791no—
356C2×M16—791no—
357C2×M16—791no—
358C2×M16—1581no—
359C2×M16—1581no—
360C22×C8—1581no—
361#126—1581no—
362#125—791no—
363#124—791no—
364#123—1581no—
365C22⋊Q8—791no—
366#121—791no—
367C4×Q8—791no—
368C4×Q8—791no—
369C4×Q8—791no—
370C4×Q8—1581no—
371C4×Q8—1581no—
372C4×Q8C22×D7911yescharacteristic
373C4×Q8C22×D7911yescharacteristic
374C4×D4—791no—
375C4×D4—1581no—
376#117—1581no—
377C2×C4⋊C4—791no—
378C2×C4⋊C4—1581no—
379C2×C42—791no—
380C2×C42—1581no—
381#107—791no—
382#107C4×D7911yescharacteristic
383#106—791no—
384#106—791no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—791no—
388C4⋊C8—791no—
389C4⋊C8—791no—
390C4⋊C8—791no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8—1581no—
395C4⋊C8—1581no—
396C4⋊C8—1581no—
397C4⋊C8C22×D7911yescharacteristic
398C4⋊C8C22×D7911yescharacteristic
399#103—791no—
400#103—791no—
401#103—791no—
402#103—791no—
403#103C4×D7911yescharacteristic
404#103C4×D7911yescharacteristic
405#102—791no—
406#102—791no—
407C22⋊C8—791no—
408C22⋊C8—791no—
409C22⋊C8—1581no—
410C22⋊C8—1581no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413#97—791no—
414#97—791no—
415#97—1581no—
416#97—1581no—
417#97—1581no—
418C4×C8—21no—
419C4×C8—21no—
420C4×C8—791no—
421C4×C8—791no—
422C4×C8—791no—
423C4×C8—1581no—
424C4×C8—1581no—
425C4×C8C22×D7911yescharacteristic
426C4×C8C22×D7911yescharacteristic
427C4×C8C22×D7911yescharacteristic
428C2×Q8—3161no—
429C2×Q8—791no—
430C2×Q8—1581no—
431C2×Q8—1582no—
432C2×Q8C2×C4×D7911yescharacteristic
433C2×Q8C2×C4×D7911yescharacteristic
434C2×D4—791no—
435C2×D4—1581no—
436C22×C4—791no—
437C22×C4—791no—
438C22×C4—791no—
439C22×C4—1581no—
440C22×C4—1581no—
441C22×C4—1581no—
442Q16—22no—
443Q16—3161no—
444SD16—3161no—
445SD16—1582no—
446M16—3162no—
447M16—3161no—
448M16—3162no—
449M16—1582no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—3162no—
454C2×C8—791no—
455C2×C8—791no—
456C2×C8—791no—
457C2×C8—791no—
458C2×C8—1581no—
459C2×C8—1581no—
460C2×C8—1581no—
461C2×C8C2×C4×D7911yescharacteristic
462C2×C8C2×C4×D7911yescharacteristic
463C2×C8D4×D7911yescharacteristic
464C2×C8D4×D7911yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—791no—
468C4⋊C4—791no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4C2×C4×D7911yescharacteristic
477C4⋊C4C2×C4×D7911yescharacteristic
478C22⋊C4—1581no—
479C22⋊C4—1581no—
480C22⋊C4—1581no—
481C22⋊C4—1581no—
482C22⋊C4—1581no—
483C42—21no—
484C42—21no—
485C42—791no—
486C42—791no—
487C42—791no—
488C42—1581no—
489C42—1581no—
490C42—1581no—
491C42C23×D7911yescharacteristic
492C23—791no—
493C23—1581no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—3162no—
498Q8—1581no—
499Q8—1581no—
500Q8#1457916712yes—
501D4—6321no—
502D4—1581no—
503D4—1581no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—3162no—
509C2×C4—3161no—
510C2×C4—3161no—
511C2×C4—3161no—
512C2×C4—791no—
513C2×C4—791no—
514C2×C4—791no—
515C2×C4—6321no—
516C2×C4—1581no—
517C2×C4—1581no—
518C2×C4—1581no—
519C2×C4—1581no—
520C2×C4—1581no—
521C2×C4—1581no—
522C2×C4—1581no—
523C2×C4C22×C4×D7911yescharacteristic
524C2×C4C2×D4×D7911yescharacteristic
525C2×C4#41×D7911yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—3161no—
534C8—3161no—
535C8—3161no—
536C8—792no—
537C8—1581no—
538C8—1581no—
539C8—1581no—
540C22—3161no—
541C22—3161no—
542C22—3161no—
543C22—1581no—
544C22—1581no—
545C22—1581no—
546C22C4×D4×D7911yescharacteristic
547C4—21no—
548C4—22no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—3161no—
553C4—3161no—
554C4—3161no—
555C4—1581no—
556C4—1581no—
557C4—1581no—
558C4C2×#1457916711yescharacteristic
559C4#1909932811yescharacteristic
560C4#137×D7911yescharacteristic
561C4#1909936211yescharacteristic
562C2—3161no—
563C2—6321no—
564C2#2377212011yescharacteristic
565C2#2377221611yescharacteristic
566C2#2377231411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1579C2
socleC22×C79C2×C4×D4
prime radical#14579136C2×C4
cosocleC42×C79C24
hypercenter#1579D79
hypocenterC79#18010
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1579

Complements

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

NH ≅ G/Ntype
3. Q8×#11929607C2semidirect
4. #23762304C2semidirect
5. #23762294C2semidirect
6. #23762278C2semidirect
7. #23762271C2semidirect
8. #23762251C2semidirect
9. #23762235C2semidirect
10. #23762197C2semidirect
11. #23762187C2semidirect
13. #23761793C2semidirect
14. #23761696C2semidirect
15. C79×#1579C2semidirect
293. #1579D79semidirect
299. C79#18010semidirect

Maximal subgroups

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

subgroupindex
#237734842
#237724032
Q8×#119296072
#237623042
#237622942
#237622782
#237622712
#237622512
#237622352
#237621972
#237621872
#237618642
#237617932
#237616962
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