Finite Group Databasebeta

← #34259201

Subgroups — #34259201

Order 20224.

602 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773437C211yesmaximal, characteristic
2#23772352C211yesmaximal, characteristic
3#23772342C211yesmaximal, characteristic
4#23762297C211yesmaximal, characteristic, complemented
5#23762295C211yesmaximal, characteristic, complemented
6#23762253C211yesmaximal, characteristic, complemented
7#23762253C211yesmaximal, characteristic, complemented
8#23762252C211yesmaximal, characteristic, complemented
9#23762206C211yesmaximal, characteristic, complemented
10C8×#11929614C211yesmaximal, characteristic, complemented
11#23762173C211yesmaximal, characteristic
12#23761853C211yesmaximal, characteristic
13Q8⋊C8×D79C211yesmaximal, characteristic
14#23761679C211yesmaximal, characteristic
15C79×#1584C211yesmaximal, characteristic, complemented
16#19099117C2211yescharacteristic
17C4×Q8×D79C2211yescharacteristic
18#19098900C411yescharacteristic
19#19098897C411yescharacteristic
20#19098857C411yescharacteristic
21#19098856C411yescharacteristic
22#19098772C411yescharacteristic
23#19098771C411yescharacteristic
24#19098720C411yescharacteristic
25#19098717C411yescharacteristic
26#19098694C2211yescharacteristic
27#19098694C2211yescharacteristic
28#19098693—21no—
29#19098693C2211yescharacteristic
30#19098692—21no—
31#19098692—21no—
32#19098691—21no—
33C8×Dic158C2211yescharacteristic
34#19098687C2211yescharacteristic
35#19098687C2211yescharacteristic
36C4×Dic316C2211yescharacteristic
37C4×#11929614C2211yescharacteristic
38C4×#11929614C2211yescharacteristic
39#19098614C2211yescharacteristic
40#19098588—21no—
41#19098585—21no—
42#19098585C2211yescharacteristic
43C4⋊C8×D79C2211yescharacteristic
44#19098517—21no—
45#19098515—21no—
46C4×#11929607C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098315C2211yescharacteristic
51#19098295C2211yescharacteristic
52#19098295C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54Q8⋊(C79⋊C8)C2211yescharacteristic
55#19098293C2211yescharacteristic
56#19098293C2211yescharacteristic
57#19098285C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C8×Q8×C79C2211yescharacteristic
62C4×Q16×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65Q8⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×Q8×D79C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×#11929614C2×C411yescharacteristic
70C2×#11929614C2×C411yescharacteristic
71C2×#11929607—21no—
72C2×#11929607D411yescharacteristic
73C2×#11929607D411yescharacteristic
74C2×C8×D79—21no—
75C2×C8×D79—21no—
76#14579160C2×C411yescharacteristic
77#14579159C2×C411yescharacteristic
78#14579151—21no—
79#14579141C2×C411yescharacteristic
80#14579139—21no—
81#14579137C2×C411yescharacteristic
82Q8×Dic79C2×C411yescharacteristic
83C4×Dic158—21no—
84C4×Dic158—21no—
85C4×Dic158C2311yescharacteristic
86C4×Dic158C2311yescharacteristic
87#14579126—21no—
88#14579126C2×C411yescharacteristic
89#14579118—21no—
90C4⋊C4×D79—21no—
91C4⋊C4×D79C2×C411yescharacteristic
92C42×D79—21no—
93C42×D79C2311yescharacteristic
94#14579095C2×C411yescharacteristic
95#14579094C2×C411yescharacteristic
96#14579094C2×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⋊C8—21no—
106Dic79⋊C8—21no—
107Dic79⋊C8C2×C411yescharacteristic
108Dic79⋊C8C2×C411yescharacteristic
109#14579086C2×C411yescharacteristic
110#14579086C2×C411yescharacteristic
111#14579085C2×C411yescharacteristic
112#14579085C2×C411yescharacteristic
113#14579084C2×C411yescharacteristic
114#14579084C2×C411yescharacteristic
115(C2×D79)⋊C8C2×C411yescharacteristic
116(C2×D79)⋊C8C2×C411yescharacteristic
117#14579072—21no—
118#14579072—21no—
119#14579071—21no—
120#14579071D411yescharacteristic
121#14579071D411yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C8×Dic79—21no—
126C8×Dic79—21no—
127C8×Dic79—21no—
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
139#97×C79—21no—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158—21no—
146C2×Dic158C22×C411yescharacteristic
147C2×Dic158C22×C411yescharacteristic
148Q8×D79—22no—
149Q8×D79—41no—
150C2×C4×D79—21no—
151C2×C4×D79—21no—
152C2×C4×D79C22×C411yescharacteristic
153C2×C4×D79C2×D411yescharacteristic
154C2×C4×D79#4111yescharacteristic
155Dic316—22no—
156#11929614—22no—
157#11929614—41no—
158#11929607—22no—
159#11929607—42no—
160#11929607Q1614yescomplemented
161C2×C79⋊C8—21no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8—21no—
164C2×C79⋊C8C22×C411yescharacteristic
165C2×C79⋊C8C22×C411yescharacteristic
166C2×C79⋊C8C2×D411yescharacteristic
167C2×C79⋊C8C2×D411yescharacteristic
168C8×D79—42no—
169C8×D79—42no—
170C4⋊Dic79—21no—
171C4⋊Dic79—21no—
172C4⋊Dic79C22×C411yescharacteristic
173C4⋊Dic79C22×C411yescharacteristic
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179Dic79⋊C4—21no—
180Dic79⋊C4—21no—
181Dic79⋊C4—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—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—41no—
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×C79—21no—
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—21no—
228C4×D79—21no—
229C4×D79—41no—
230C4×D79—42no—
231C4×D79—81no—
232C4×D79C2×Q1612yes—
233C4×D79#13512yes—
234Dic158—21no—
235Dic158—21no—
236Dic158—21no—
237Dic158—42no—
238Dic158—81no—
239Dic158#13112yes—
240C8×C79—21no—
241C8×C79—21no—
242C8×C79—21no—
243C8×C79—41no—
244C8×C79—41no—
245C8×C79—41no—
246C8×C79C2×Q1612yes—
247C79⋊C8—22no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—41no—
252C79⋊C8—41no—
253C79⋊C8C2×Q1612yes—
254C22×C79C2×C4×D411yescharacteristic
255C2×D79—21no—
256C2×D79—21no—
257C2×D79C4×Q1612yes—
258C4×C79—21no—
259C4×C79—41no—
260C4×C79—41no—
261C4×C79—41no—
262C4×C79—41no—
263C4×C79C2×#13111yescharacteristic
264C4×C79#56811yescharacteristic
265C4×C79C22×Q1611yescharacteristic
266C4×C79C2×#13511yescharacteristic
267Dic79—22no—
268Dic79—21no—
269Dic79—21no—
270Dic79—41no—
271Dic79—41no—
272Dic79—41no—
273Dic79C4×Q1612yes—
274#17998—791nomaximal, Sylow
275C2×C79#293311yescharacteristic
276C2×C79C2×C4×Q1611yescharacteristic
277C2×C79#295611yescharacteristic
278D79—22no—
279#3104—791no—
280Q8×M16—791no—
281#2962—791no—
282#1591—791no—
283#1591—791no—
284#1584—791no—
285#1584—791no—
286#1584—791no—
287#1584D7911yescharacteristic, complemented
288C8×Q16—791no—
289C8×Q16—791no—
290#1571—791no—
291#1493—791no—
292C2×Q8⋊C8—791no—
293C8×M16—791no—
294C79#1799811yescharacteristic, complemented, Sylow
295#533—791no—
296C2×C4×Q8—791no—
297#484—791no—
298#484—791no—
299#477—791no—
300#477—791no—
301#462—791no—
302#462—791no—
303#451—791no—
304#451—791no—
305#446—791no—
306#446—791no—
307#446—791no—
308#446—1581no—
309#446—1581no—
310#446—1581no—
311#446—1581no—
312#446C2×D7911yescharacteristic
313C8×Q8—791no—
314C8×Q8—791no—
315C8×Q8—791no—
316C8×Q8C2×D7911yescharacteristic
317C4×Q16—791no—
318C4×Q16—791no—
319C4×Q16—791no—
320C4×Q16C2×D7911yescharacteristic
321#431—791no—
322#424—1581no—
323#423—791no—
324#423—1581no—
325C2×C4⋊C8—791no—
326#405—1581no—
327C4×M16—791no—
328C4×M16—1581no—
329C2×C4×C8—791no—
330#335—791no—
331#335—791no—
332#335—791no—
333#335C2×D7911yescharacteristic
334Q8⋊C8—791no—
335Q8⋊C8—791no—
336Q8⋊C8—791no—
337Q8⋊C8—791no—
338Q8⋊C8—791no—
339Q8⋊C8—791no—
340Q8⋊C8C2×D7911yescharacteristic
341Q8⋊C8C2×D7911yescharacteristic
342#322—791no—
343#322C2×D7911yescharacteristic
344C82—791no—
345C82—791no—
346C22×Q8—791no—
347C2×Q16—791no—
348C2×Q16—791no—
349C2×Q16—791no—
350C2×Q16C4×D7911yescharacteristic
351C2×M16—791no—
352C2×M16—791no—
353C2×M16—1581no—
354C22×C8—1581no—
355C22×C8—1581no—
356C4⋊Q8—791no—
357C4⋊Q8—791no—
358#125—1581no—
359C22⋊Q8—791no—
360C22⋊Q8—791no—
361C22⋊Q8—1581no—
362C4×Q8—791no—
363C4×Q8—791no—
364C4×Q8—791no—
365C4×Q8—791no—
366C4×Q8—1581no—
367C4×Q8—1581no—
368C4×Q8—1581no—
369C4×Q8C22×D7911yescharacteristic
370C4×Q8C22×D7911yescharacteristic
371#117—1581no—
372C2×C4⋊C4—791no—
373C2×C4⋊C4—1581no—
374C2×C42—791no—
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⋊C8—1581no—
394C4⋊C8C22×D7911yescharacteristic
395C4⋊C8C22×D7911yescharacteristic
396#103—791no—
397#103—791no—
398#103—791no—
399#103—791no—
400#103—791no—
401#103—791no—
402#103C4×D7911yescharacteristic
403#103C4×D7911yescharacteristic
404C22⋊C8—791no—
405C22⋊C8—791no—
406#97—21no—
407#97—791no—
408#97—791no—
409#97—1581no—
410#97—1581no—
411#97—1581no—
412C4×C8—21no—
413C4×C8—791no—
414C4×C8—791no—
415C4×C8—791no—
416C4×C8—1581no—
417C4×C8—1581no—
418C4×C8—1581no—
419C4×C8—1581no—
420C4×C8C22×D7911yescharacteristic
421C4×C8C22×D7911yescharacteristic
422C4×C8C22×D7911yescharacteristic
423C2×Q8—3161no—
424C2×Q8—791no—
425C2×Q8—791no—
426C2×Q8—1581no—
427C2×Q8—1581no—
428C2×Q8—1582no—
429C2×Q8C2×C4×D7911yescharacteristic
430C2×Q8C2×C4×D7911yescharacteristic
431C22×C4—791no—
432C22×C4—791no—
433C22×C4—791no—
434C22×C4—1581no—
435C22×C4—1581no—
436Q16—41no—
437Q16—3161no—
438Q16—1582no—
439Q16—1582no—
440M16—3162no—
441M16—794no—
442M16—1582no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—3162no—
447C2×C8—3162no—
448C2×C8—791no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—791no—
452C2×C8—1581no—
453C2×C8—1581no—
454C2×C8—1581no—
455C2×C8C2×C4×D7911yescharacteristic
456C2×C8C2×C4×D7911yescharacteristic
457C2×C8D4×D7911yescharacteristic
458C2×C8D4×D7911yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—791no—
462C4⋊C4—791no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4C2×C4×D7911yescharacteristic
474C4⋊C4C2×C4×D7911yescharacteristic
475C22⋊C4—1581no—
476C22⋊C4—1581no—
477C42—21no—
478C42—21no—
479C42—791no—
480C42—791no—
481C42—791no—
482C42—1581no—
483C42—1581no—
484C42—1581no—
485C42—1581no—
486C42C23×D7911yescharacteristic
487C23—791no—
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—3162no—
493Q8—792no—
494Q8—6321no—
495Q8—1581no—
496Q8—1581no—
497Q8—1581no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—3161no—
503C2×C4—3162no—
504C2×C4—791no—
505C2×C4—792no—
506C2×C4—792no—
507C2×C4—791no—
508C2×C4—791no—
509C2×C4—6321no—
510C2×C4—1581no—
511C2×C4—1581no—
512C2×C4—1581no—
513C2×C4—1581no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4C22×C4×D7911yescharacteristic
517C2×C4C2×D4×D7911yescharacteristic
518C2×C4#41×D7911yescharacteristic
519C8—21no—
520C8—21no—
521C8—21no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—3161no—
526C8—3161no—
527C8—792no—
528C8—1582no—
529C8—1581no—
530C8—1581no—
531C8—1581no—
532C8Q16×D7912yes—
533C22—792no—
534C22—1581no—
535C22—1581no—
536C22C4×D4×D7911yescharacteristic
537C4—21no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—3161no—
543C4—3161no—
544C4—3161no—
545C4—792no—
546C4—1581no—
547C4—1582no—
548C4—1581no—
549C4#131×D7911yescharacteristic
550C4#1909932911yescharacteristic
551C4C2×Q16×D7911yescharacteristic
552C4#135×D7911yescharacteristic
553C2—1582no—
554C2#2377212311yescharacteristic
555C2C4×Q16×D7911yescharacteristic
556C2#2377230811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1584C2
socleC22×C79C2×C4×D4
prime radicalC22×D79C4×D4
cosocleC42×C79C24
hypercenter#1584D79
hypocenterC79#17998
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. #23762297C2semidirect
5. #23762295C2semidirect
6. #23762253C2semidirect
7. #23762253C2semidirect
8. #23762252C2semidirect
9. #23762206C2semidirect
10. C8×#11929614C2semidirect
15. C79×#1584C2semidirect
160. #11929607Q16semidirect
287. #1584D79semidirect
294. C79#17998semidirect

Maximal subgroups

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

subgroupindex
#237734372
#237723522
#237723422
#237622972
#237622952
#237622532
#237622522
#237622062
C8×#119296142
#237621732
#237618532
Q8⋊C8×D792
#237616792
C79×#15842
#1799879