Finite Group Databasebeta

← #34257964

Subgroups — #34257964

Order 20224.

626 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772482C211yesmaximal, characteristic
2#23772479C211yesmaximal, characteristic
3C4×Q16×D79C211yesmaximal, characteristic
4#23762273C211yesmaximal, characteristic, complemented
5#23762272C211yesmaximal, characteristic, complemented
6#23762270C211yesmaximal, characteristic, complemented
7#23762253C211yesmaximal, characteristic, complemented
8#23762250C211yesmaximal, characteristic, complemented
9#23762248C211yesmaximal, characteristic, complemented
10#23762247C211yesmaximal, characteristic, complemented
11#23762145C211yesmaximal, characteristic
12#23761793C211yesmaximal, characteristic
13#23761793C211yesmaximal, characteristic
14#23761691C211yesmaximal, characteristic
15C79×#1587C211yesmaximal, characteristic, complemented
16C2×Q16×D79C411yescharacteristic
17C4×Q8×D79C2211yescharacteristic
18C4×Q8×D79C2211yescharacteristic
19#19098694C2211yescharacteristic
20#19098693C2211yescharacteristic
21#19098692—21no—
22#19098691—21no—
23#19098690C2211yescharacteristic
24Q8×C79⋊C8C2211yescharacteristic
25C8×Dic158—21no—
26C8×Dic158C2211yescharacteristic
27#19098687—21no—
28#19098687C2211yescharacteristic
29C4×Dic316C2211yescharacteristic
30Q16×Dic79C411yescharacteristic
31#19098657C411yescharacteristic
32C4×#11929614C2211yescharacteristic
33C4×#11929614C2211yescharacteristic
34#19098655C411yescharacteristic
35#19098655C411yescharacteristic
36#19098619—21no—
37#19098618—21no—
38#19098615—21no—
39#19098615C2211yescharacteristic
40#19098615C2211yescharacteristic
41#19098614—21no—
42#19098614C2211yescharacteristic
43#19098614C2211yescharacteristic
44#107×D79C411yescharacteristic
45#103×D79C411yescharacteristic
46#103×D79C411yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098315C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098314C2211yescharacteristic
51#19098295C2211yescharacteristic
52#19098295C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54Q8⋊(C79⋊C8)C2211yescharacteristic
55#19098293C2211yescharacteristic
56#19098293C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59#19098284C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C8×Q8×C79C2211yescharacteristic
62C4×Q16×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65Q8⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×Q8×D79C2×C411yescharacteristic
68C2×Q8×D79C2×C411yescharacteristic
69C2×Dic316C2×C411yescharacteristic
70C2×#11929614C2×C411yescharacteristic
71C2×#11929614C2×C411yescharacteristic
72Q16×D79—24no—
73C2×C8×D79C2×C411yescharacteristic
74C2×C8×D79C2×C411yescharacteristic
75Q8×Dic79C2×C411yescharacteristic
76Q8×Dic79C2×C411yescharacteristic
77C4×Dic158—21no—
78C4×Dic158—21no—
79C4×Dic158C2311yescharacteristic
80C4×Dic158C2311yescharacteristic
81#14579126—21no—
82#14579126—21no—
83#14579126C2×C411yescharacteristic
84#14579126C2×C411yescharacteristic
85C4⋊C4×D79—21no—
86C4⋊C4×D79—21no—
87C4⋊C4×D79C2×C411yescharacteristic
88C4⋊C4×D79C2×C411yescharacteristic
89C42×D79—21no—
90C42×D79—21no—
91C42×D79C2311yescharacteristic
92#14579095C2×C411yescharacteristic
93#14579094C2×C411yescharacteristic
94#14579094C2×C411yescharacteristic
95C4⋊(C79⋊C8)—21no—
96C4⋊(C79⋊C8)—21no—
97C4⋊(C79⋊C8)C2311yescharacteristic
98C4⋊(C79⋊C8)C2311yescharacteristic
99Dic79⋊C8—21no—
100Dic79⋊C8—21no—
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8D411yescharacteristic
105Dic79⋊C8D411yescharacteristic
106#14579086C2×C411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579085C2×C411yescharacteristic
109#14579085C2×C411yescharacteristic
110#14579084C2×C411yescharacteristic
111#14579084C2×C411yescharacteristic
112(C2×D79)⋊C8—21no—
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8D411yescharacteristic
116(C2×D79)⋊C8D411yescharacteristic
117#14579072—21no—
118#14579071—21no—
119#14579071—21no—
120C4×C79⋊C8—21no—
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C8×Dic79—21no—
125C8×Dic79—21no—
126C8×Dic79—21no—
127C8×Dic79—21no—
128C8×Dic79C2×C411yescharacteristic
129C8×Dic79C2×C411yescharacteristic
130C2×Q16×C79C2×C411yescharacteristic
131C4×Q8×C79C2311yescharacteristic
132C4×Q8×C79C2311yescharacteristic
133#107×C79C2×C411yescharacteristic
134C4⋊C8×C79—21no—
135C4⋊C8×C79—21no—
136C4⋊C8×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138#103×C79C2×C411yescharacteristic
139#103×C79C2×C411yescharacteristic
140#97×C79—21no—
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C4×C8×C79C2311yescharacteristic
145C2×Dic158—21no—
146C2×Dic158—21no—
147C2×Dic158C22×C411yescharacteristic
148C2×Dic158C22×C411yescharacteristic
149Q8×D79—22no—
150Q8×D79—22no—
151Q8×D79—22no—
152Q8×D79M1614yes—
153C2×C4×D79—21no—
154C2×C4×D79—21no—
155C2×C4×D79—21no—
156C2×C4×D79—21no—
157C2×C4×D79C22×C411yescharacteristic
158C2×C4×D79C2×D411yescharacteristic
159C2×C4×D79#4111yescharacteristic
160Dic316—22no—
161#11929614—22no—
162#11929614—22no—
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—22no—
171C8×D79—22no—
172C4⋊Dic79—21no—
173C4⋊Dic79—21no—
174C4⋊Dic79C22×C411yescharacteristic
175C4⋊Dic79C22×C411yescharacteristic
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179Dic79⋊C4—21no—
180Dic79⋊C4—21no—
181Dic79⋊C4—21no—
182C4×Dic79—21no—
183C4×Dic79—21no—
184C4×Dic79—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×D79C4×D411yescharacteristic
209Q8×C79—21no—
210Q8×C79—21no—
211Q8×C79—21no—
212Q8×C79C2×M1612yes—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79C23×C411yescharacteristic
218C2×C4×C79C22×D411yescharacteristic
219C2×C4×C79C2×#4111yescharacteristic
220C2×Dic79—21no—
221C2×Dic79—21no—
222C2×Dic79—21no—
223C2×Dic79—21no—
224C2×Dic79C4×D411yescharacteristic
225C2×Dic79C4×D411yescharacteristic
226C2×Dic79C4×D411yescharacteristic
227C4×D79—24no—
228C4×D79—21no—
229C4×D79—21no—
230C4×D79—41no—
231C4×D79—42no—
232C4×D79—81no—
233C4×D79—81no—
234C4×D79C2×M1612yes—
235C4×D79#13112yes—
236Dic158—21no—
237Dic158—24no—
238Dic158—21no—
239Dic158—21no—
240Dic158—42no—
241Dic158C2×M1612yes—
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—21no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C79⋊C8—41no—
256C22×C79C2×C4×D411yescharacteristic
257C2×D79—21no—
258C2×D79—21no—
259C2×D79#43512yes—
260C4×C79—22no—
261C4×C79—21no—
262C4×C79—41no—
263C4×C79—41no—
264C4×C79—41no—
265C4×C79C22×M1611yescharacteristic
266C4×C79C2×#13111yescharacteristic
267C4×C79#57811yescharacteristic
268C4×C79#57811yescharacteristic
269Dic79—22no—
270Dic79—21no—
271Dic79—21no—
272Dic79—41no—
273Dic79—41no—
274Dic79—41no—
275Dic79#43512yes—
276#17880—791nomaximal, Sylow
277C2×C79C2×#43511yescharacteristic
278C2×C79#295011yescharacteristic
279C2×C79#295711yescharacteristic
280D79—22no—
281#2983—791no—
282#2982—791no—
283C2×C4×Q16—791no—
284#1587—791no—
285#1587—791no—
286#1587—791no—
287#1587D7911yescharacteristic, complemented
288#1584—791no—
289#1584—791no—
290#1584—791no—
291#1584—791no—
292#1564—791no—
293#1479—791no—
294#1479—791no—
295#1452—791no—
296C79#1788011yescharacteristic, complemented, Sylow
297C22×Q16—791no—
298C2×C4×Q8—791no—
299C2×C4×Q8—791no—
300#446—791no—
301#446—791no—
302#446—791no—
303#446—1581no—
304#446—1581no—
305#446C2×D7911yescharacteristic
306C8×Q8—791no—
307C8×Q8—791no—
308C8×Q8—791no—
309C8×Q8—1581no—
310C8×Q8—1581no—
311C8×Q8C2×D7911yescharacteristic
312C4×Q16—791no—
313C4×Q16—791no—
314C4×Q16—791no—
315C4×Q16—791no—
316C4×Q16—791no—
317C4×Q16—791no—
318C4×Q16—791no—
319C4×Q16C2×D7911yescharacteristic
320#432—1581no—
321#432—1581no—
322#431—791no—
323#431—791no—
324#431—791no—
325#431—791no—
326#431—1581no—
327#431—1581no—
328C2×#107—791no—
329C2×#103—791no—
330C2×#103—791no—
331C2×C4×C8—791no—
332#335—791no—
333#335—791no—
334#334—791no—
335#334C2×D7911yescharacteristic
336Q8⋊C8—791no—
337Q8⋊C8—791no—
338Q8⋊C8—791no—
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8—791no—
342Q8⋊C8C2×D7911yescharacteristic
343Q8⋊C8C2×D7911yescharacteristic
344#322—791no—
345#322—791no—
346#322—791no—
347#322C2×D7911yescharacteristic
348C22×Q8—791no—
349C22×Q8—791no—
350C2×Q16—791no—
351C2×Q16—791no—
352C2×Q16—791no—
353C2×Q16—1584no—
354C2×Q16C4×D7911yescharacteristic
355C22×C8—791no—
356C22×C8—791no—
357C4×Q8—791no—
358C4×Q8—791no—
359C4×Q8—791no—
360C4×Q8—791no—
361C4×Q8—791no—
362C4×Q8—791no—
363C4×Q8—1581no—
364C4×Q8—1581no—
365C4×Q8—1581no—
366C4×Q8—1581no—
367C4×Q8C22×D7911yescharacteristic
368C4×Q8C22×D7911yescharacteristic
369C2×C4⋊C4—791no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—1581no—
372C2×C4⋊C4—1581no—
373C2×C42—791no—
374C2×C42—1581no—
375C2×C42—1581no—
376#107—791no—
377#107—791no—
378#107—791no—
379#107C4×D7911yescharacteristic
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8—791no—
383C4⋊C8—791no—
384C4⋊C8—791no—
385C4⋊C8—791no—
386C4⋊C8—1581no—
387C4⋊C8—1581no—
388C4⋊C8—1581no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8C22×D7911yescharacteristic
394C4⋊C8C22×D7911yescharacteristic
395#103—791no—
396#103—791no—
397#103—791no—
398#103—791no—
399#103—791no—
400#103—791no—
401#103C4×D7911yescharacteristic
402#103C4×D7911yescharacteristic
403C22⋊C8—791no—
404C22⋊C8—791no—
405C22⋊C8—1581no—
406C22⋊C8—1581no—
407C22⋊C8—1581no—
408#97—21no—
409#97—1581no—
410#97—1581no—
411#97—1581no—
412C4×C8—21no—
413C4×C8—791no—
414C4×C8—791no—
415C4×C8—791no—
416C4×C8—791no—
417C4×C8—791no—
418C4×C8—1581no—
419C4×C8—1581no—
420C4×C8—1581no—
421C4×C8—1581no—
422C4×C8—1581no—
423C4×C8C22×D7911yescharacteristic
424C4×C8C22×D7911yescharacteristic
425C4×C8C22×D7911yescharacteristic
426C2×Q8—794no—
427C2×Q8—791no—
428C2×Q8—791no—
429C2×Q8—1581no—
430C2×Q8—1581no—
431C2×Q8—1582no—
432C2×Q8—1582no—
433C2×Q8—1582no—
434C2×Q8C2×C4×D7911yescharacteristic
435C2×Q8C2×C4×D7911yescharacteristic
436C22×C4—791no—
437C22×C4—791no—
438C22×C4—791no—
439C22×C4—1581no—
440C22×C4—1581no—
441C22×C4—1581no—
442C22×C4—1581no—
443Q16—22no—
444Q16—1582no—
445Q16—1582no—
446Q16—1582no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—791no—
451C2×C8—791no—
452C2×C8—791no—
453C2×C8—791no—
454C2×C8—1582no—
455C2×C8—1582no—
456C2×C8—1581no—
457C2×C8—1581no—
458C2×C8—1581no—
459C2×C8C2×C4×D7911yescharacteristic
460C2×C8C2×C4×D7911yescharacteristic
461C2×C8D4×D7911yescharacteristic
462C2×C8D4×D7911yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—791no—
466C4⋊C4—791no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4C2×C4×D7911yescharacteristic
476C4⋊C4C2×C4×D7911yescharacteristic
477C42—21no—
478C42—21no—
479C42—791no—
480C42—791no—
481C42—791no—
482C42—1581no—
483C42—1581no—
484C42—1581no—
485C42—1581no—
486C42—1581no—
487C42—1581no—
488C42C23×D7911yescharacteristic
489C23—791no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—3162no—
494Q8—792no—
495Q8—1584no—
496Q8—1581no—
497Q8—1581no—
498Q8—1581no—
499Q8M16×D7912yes—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—3161no—
505C2×C4—3162no—
506C2×C4—791no—
507C2×C4—792no—
508C2×C4—792no—
509C2×C4—791no—
510C2×C4—791no—
511C2×C4—6321no—
512C2×C4—6321no—
513C2×C4—1584no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4—1581no—
517C2×C4—1581no—
518C2×C4—1581no—
519C2×C4—1581no—
520C2×C4C22×C4×D7911yescharacteristic
521C2×C4C2×D4×D7911yescharacteristic
522C2×C4#41×D7911yescharacteristic
523C8—21no—
524C8—21no—
525C8—21no—
526C8—21no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—3161no—
531C8—3161no—
532C8—3161no—
533C8—1581no—
534C8—1581no—
535C8—1581no—
536C8—1581no—
537C22—792no—
538C22—1581no—
539C22—1581no—
540C22C4×D4×D7911yescharacteristic
541C4—21no—
542C4—22no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—3161no—
547C4—3161no—
548C4—3161no—
549C4—792no—
550C4—1581no—
551C4—1581no—
552C4—1582no—
553C4C2×M16×D7911yescharacteristic
554C4#131×D7911yescharacteristic
555C4#1909937911yescharacteristic
556C4#1909937911yescharacteristic
557C2—1582no—
558C2#435×D7911yescharacteristic
559C2#2377226511yescharacteristic
560C2#2377231211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1587C2
socleC22×C79C2×C4×D4
prime radicalC22×D79C4×D4
cosocleC42×C79C24
hypercenter#1587D79
hypocenterC79#17880
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1587

Complements

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

NH ≅ G/Ntype
4. #23762273C2semidirect
5. #23762272C2semidirect
6. #23762270C2semidirect
7. #23762253C2semidirect
8. #23762250C2semidirect
9. #23762248C2semidirect
10. #23762247C2semidirect
15. C79×#1587C2semidirect
287. #1587D79semidirect
296. C79#17880semidirect

Maximal subgroups

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

subgroupindex
#237724822
#237724792
C4×Q16×D792
#237622732
#237622722
#237622702
#237622532
#237622502
#237622482
#237622472
#237621452
#237617932
#237616912
C79×#15872
#1788079