Finite Group Databasebeta

← #34443066

Subgroups — #34443066

Order 20224.

590 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772977C211yesmaximal, characteristic
2#427×D79C211yesmaximal, characteristic, complemented
3#23765607C211yesmaximal, characteristic, complemented
4#23765606C211yesmaximal, characteristic, complemented
5#23765601C211yesmaximal, characteristic, complemented
6#23765596C211yesmaximal, characteristic, complemented
7#23765590C211yesmaximal, characteristic, complemented
8#23765589C211yesmaximal, characteristic, complemented
9#23765584C211yesmaximal, characteristic, complemented
10#23765579C211yesmaximal, characteristic, complemented
11#23765184C211yesmaximal, characteristic, complemented
12#23765182C211yesmaximal, characteristic, complemented
13#23765107C211yesmaximal, characteristic, complemented
14C2×(C2×D79)⋊C16C211yesmaximal, characteristic, complemented
15C79×#2138C211yesmaximal, characteristic, complemented
16C22×C8×D79C2211yescharacteristic
17#19099070—21no—
18#117×D79—21no—
19C22×C79⋊C16C2211yescharacteristic
20#19098801C2211yescharacteristic
21#19098801C2211yescharacteristic
22#19098796C2211yescharacteristic
23#19098796C2211yescharacteristic
24#19098789—21no—
25#19098789—21no—
26#19098601C2211yescharacteristic
27#19098601C2211yescharacteristic
28#19098600C2211yescharacteristic
29#19098600C2211yescharacteristic
30#19098599—21no—
31#19098598—21no—
32#107×D79C2211yescharacteristic
33#107×D79C2211yescharacteristic
34#106×D79—21no—
35#19098559—21no—
36#19098391C2211yescharacteristic, complemented
37#19098391C2211yescharacteristic, complemented
38#19098390C2211yescharacteristic, complemented
39#19098390C2211yescharacteristic, complemented
40#19098389C2211yescharacteristic, complemented
41#19098389C2211yescharacteristic, complemented
42#19098388C2211yescharacteristic, complemented
43#19098388C2211yescharacteristic, complemented
44#19098367C2211yescharacteristic, complemented
45#19098367C2211yescharacteristic, complemented
46#19098364C2211yescharacteristic, complemented
47#19098364C2211yescharacteristic, complemented
48(C2×D79)⋊C16C2211yescharacteristic
49(C2×D79)⋊C16C2211yescharacteristic
50(C2×D79)⋊C16C2211yescharacteristic
51(C2×D79)⋊C16C2211yescharacteristic
52C2×#135×C79C2211yescharacteristic
53C22×C16×C79C2211yescharacteristic
54#427×C79C2211yescharacteristic
55#358×C79C2211yescharacteristic, complemented
56#358×C79C2211yescharacteristic, complemented
57#357×C79C2211yescharacteristic, complemented
58#357×C79C2211yescharacteristic, complemented
59C22×C4×D79D411yescharacteristic
60C22×C79⋊C8D411yescharacteristic
61C2×C8×D79—22no—
62C2×C8×D79—22no—
63C2×C8×D79D411yescharacteristic
64C2×C8×D79D411yescharacteristic
65C2×C8×D79C2311yescharacteristic
66C2×C8×D79C2311yescharacteristic
67#14579142—41no—
68#14579137—21no—
69#14579137—21no—
70#14579132—21no—
71#14579132—21no—
72#14579129—41no—
73C4×C22⋊D79—41no—
74C4×C22⋊D79—41no—
75#14579120—21no—
76#14579120—21no—
77#14579119—21no—
78#14579119—21no—
79#14579118—41no—
80#14579117—41no—
81C4⋊C4×D79—21no—
82C4⋊C4×D79—21no—
83C22⋊C4×D79—41no—
84C42×D79—41no—
85C2×C79⋊C16—22no—
86C2×C79⋊C16C2311yescharacteristic
87C2×C79⋊C16C2311yescharacteristic
88#14579095C2311yescharacteristic
89#14579095C2311yescharacteristic
90#14579095C2311yescharacteristic
91#14579095C2311yescharacteristic
92#14579094—21no—
93#14579094—21no—
94#14579093—21no—
95#14579093—21no—
96#14579092—21no—
97#14579092—21no—
98#14579085—21no—
99#14579085—21no—
100#14579082—21no—
101#14579082—21no—
102C2×#41×C79—21no—
103#135×C79—22no—
104#135×C79—22no—
105C2×Q16×C79C2311yescharacteristic
106C2×SD16×C79—21no—
107C2×D8×C79C2311yescharacteristic
108C22×C8×C79C2311yescharacteristic
109#117×C79—21no—
110C2×C16×C79—22no—
111C2×C16×C79C2311yescharacteristic
112C2×C16×C79C2311yescharacteristic
113#107×C79C2311yescharacteristic
114#107×C79C2311yescharacteristic
115#106×C79—21no—
116C23×D79D811yescharacteristic
117C2×C22⋊D79—41no—
118C2×C22⋊D79—41no—
119C22×Dic79D811yescharacteristic
120C2×C4×D79—22no—
121C2×C4×D79—22no—
122C2×C4×D79—41no—
123C2×C4×D79—41no—
124C2×C4×D79D811yescharacteristic
125C2×C4×D79D811yescharacteristic
126C2×C4×D79C2×D411yescharacteristic
127C2×C4×D79C2×D411yescharacteristic
128C2×C79⋊C8—22no—
129C2×C79⋊C8C2×D411yescharacteristic
130C2×C79⋊C8C2×D411yescharacteristic
131C8×D79—22no—
132C8×D79—22no—
133C8×D79—21no—
134C8×D79—21no—
135C8×D79—21no—
136C8×D79—21no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141Dic79⋊C4—41no—
142Dic79⋊C4—41no—
143Dic79⋊C4—41no—
144Dic79⋊C4—41no—
145C22⋊Dic79—41no—
146C22⋊Dic79—41no—
147C22⋊Dic79—41no—
148C22⋊Dic79—41no—
149(C2×D79)⋊C4—41no—
150(C2×D79)⋊C4—41no—
151(C2×D79)⋊C4—41no—
152(C2×D79)⋊C4—41no—
153C4×Dic79—41no—
154C4×Dic79—41no—
155C4×Dic79—41no—
156C4×Dic79—41no—
157C79⋊C16—21no—
158C79⋊C16—21no—
159#41×C79—42no—
160#41×C79—42no—
161C2×Q8×C79—21no—
162C2×D4×C79—21no—
163C2×D4×C79—41no—
164C22×C4×C79—41no—
165C22×C4×C79C2×D411yescharacteristic
166Q16×C79—21no—
167Q16×C79—21no—
168SD16×C79—42no—
169D8×C79—21no—
170D8×C79—21no—
171C2×C8×C79C2×D411yescharacteristic
172C2×C8×C79#4112yes—
173C2×C8×C79#4112yes—
174C2×C8×C79C2411yescharacteristic
175C4⋊C4×C79—21no—
176C4⋊C4×C79—21no—
177C22⋊C4×C79—41no—
178C42×C79—41no—
179C16×C79—21no—
180C16×C79—21no—
181C23×C79—41no—
182C23×C79C2×D811yescharacteristic
183C22×D79—22no—
184C22×D79—22no—
185C22×D79—41no—
186C22×D79C2×D811yescharacteristic
187C22×D79C2×D811yescharacteristic
188Q8×C79—41no—
189Q8×C79—41no—
190D4×C79—41no—
191D4×C79—41no—
192D4×C79—82no—
193C2×C4×C79—41no—
194C2×C4×C79—41no—
195C2×C4×C79—41no—
196C2×C4×C79—82no—
197C2×C4×C79#12312yes—
198C2×C4×C79C2×D811yescharacteristic
199C2×C4×C79#13712yes—
200C2×C4×C79C22×D411yescharacteristic
201C22⋊D79—82no—
202C22⋊D79—82no—
203C2×Dic79—22no—
204C2×Dic79—41no—
205C2×Dic79—41no—
206C2×Dic79—41no—
207C2×Dic79—41no—
208C2×Dic79C2×D811yescharacteristic
209C2×Dic79C2×D811yescharacteristic
210C4×D79—21no—
211C4×D79—21no—
212C4×D79—21no—
213C4×D79—21no—
214C4×D79—22no—
215C4×D79—22no—
216C4×D79—82no—
217C4×D79—82no—
218C8×C79#12312yes—
219C8×C79C2×#4111yescharacteristic
220C8×C79C2×#4111yescharacteristic
221C79⋊C8—21no—
222C79⋊C8—21no—
223C22×C79—21no—
224C22×C79—21no—
225C22×C79—81no—
226C22×C79—81no—
227C22×C79—81no—
228C22×C79#48012yes—
229C22×C79C22×D811yescharacteristic
230C2×D79—21no—
231C2×D79—21no—
232C2×D79—21no—
233C2×D79—21no—
234C2×D79—21no—
235C2×D79—44no—
236C2×D79#50812yes—
237C4×C79—81no—
238C4×C79—81no—
239C4×C79—81no—
240C4×C79#48012yes—
241C4×C79C2×#12311yescharacteristic
242C4×C79C2×#13711yescharacteristic
243Dic79—21no—
244Dic79—21no—
245Dic79—81no—
246Dic79—81no—
247Dic79—81no—
248Dic79—81no—
249#33645—791nomaximal, Sylow
250C2×C79—22no—
251C2×C79—161no—
252C2×C79C2×#48011yescharacteristic
253C2×C79C2×#50811yescharacteristic
254C2×C79#341411yescharacteristic
255D79—22no—
256D79—41no—
257#3050—791no—
258C2×#427—791no—
259#2237—791no—
260#2237—791no—
261#2236—791no—
262#2235—791no—
263#2234—791no—
264#2234—791no—
265#2233—791no—
266#2232—791no—
267#2158—791no—
268#2158—791no—
269#2138—791no—
270#2138D7911yescharacteristic, complemented
271C2×C22⋊C16—791no—
272C79#3364511yescharacteristic, complemented, Sylow
273C2×#135C2×D7911yescharacteristic
274C23×C8—791no—
275#525—1581no—
276C2×#117—1581no—
277C22×C16—791no—
278C22×C16C2×D7911yescharacteristic
279#467—791no—
280#467—791no—
281#466—791no—
282#466—791no—
283#465—1581no—
284#465—1581no—
285#427—791no—
286#427—791no—
287#427—791no—
288#427—791no—
289#427—1581no—
290#427—1581no—
291#427C2×D7911yescharacteristic
292C2×#107—791no—
293C2×#107—791no—
294C2×#106—1581no—
295#416—1581no—
296#367—791no—
297#367—791no—
298#367—791no—
299#367—791no—
300#366—791no—
301#366—791no—
302#366—791no—
303#366—791no—
304#358—791no—
305#358—791no—
306#358C2×D7911yescharacteristic, complemented
307#358C2×D7911yescharacteristic, complemented
308#357—791no—
309#357—791no—
310#357C2×D7911yescharacteristic, complemented
311#357C2×D7911yescharacteristic, complemented
312C22⋊C16—791no—
313C22⋊C16—791no—
314C22⋊C16—791no—
315C22⋊C16—791no—
316C2×#41—21no—
317C23×C4—791no—
318#135—22no—
319#135—22no—
320C2×Q16C22×D7911yescharacteristic
321C2×SD16—21no—
322C2×D8C22×D7911yescharacteristic
323C22×C8—791no—
324C22×C8—791no—
325C22×C8—791no—
326C22×C8—791no—
327C22×C8—791no—
328C22×C8—1582no—
329C22×C8—1582no—
330C22×C8C22×D7911yescharacteristic
331#123—3161no—
332C22⋊Q8—1581no—
333C22⋊Q8—1581no—
334#121—1581no—
335#121—1581no—
336#120—3161no—
337C4×D4—3161no—
338C4×D4—3161no—
339#117—21no—
340#117—3161no—
341#117—3161no—
342#117—1581no—
343#117—1581no—
344#117—1581no—
345#117—1581no—
346C2×C4⋊C4—1581no—
347C2×C4⋊C4—1581no—
348C2×C22⋊C4—3161no—
349C2×C42—3161no—
350C2×C16—22no—
351C2×C16—791no—
352C2×C16—791no—
353C2×C16—1582no—
354C2×C16C22×D7911yescharacteristic
355C2×C16C22×D7911yescharacteristic
356#107—791no—
357#107—791no—
358#107—791no—
359#107—791no—
360#107—1581no—
361#107—1581no—
362#107C22×D7911yescharacteristic
363#107C22×D7911yescharacteristic
364#106—21no—
365#106—1581no—
366#106—1581no—
367#106—1581no—
368#106—1581no—
369#103—1581no—
370#103—1581no—
371#102—1581no—
372#102—1581no—
373C24—791no—
374#41—42no—
375#41—42no—
376C2×Q8—21no—
377C2×D4—21no—
378C2×D4—41no—
379C2×D4—3161no—
380C2×D4—3161no—
381C22×C4—41no—
382C22×C4—3161no—
383C22×C4—3161no—
384C22×C4—791no—
385C22×C4—791no—
386C22×C4—791no—
387C22×C4—791no—
388C22×C4—791no—
389C22×C4—1582no—
390C22×C4—1582no—
391C22×C4D4×D7911yescharacteristic
392Q16—21no—
393Q16—21no—
394SD16—42no—
395D8—21no—
396D8—21no—
397C2×C8—791no—
398C2×C8—791no—
399C2×C8—1582no—
400C2×C8—1581no—
401C2×C8—1581no—
402C2×C8—1581no—
403C2×C8—1581no—
404C2×C8—1582no—
405C2×C8—1582no—
406C2×C8D4×D7911yescharacteristic
407C2×C8#1192962312yes—
408C2×C8#1192962412yes—
409C2×C8C23×D7911yescharacteristic
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—3161no—
413C4⋊C4—3161no—
414C4⋊C4—3161no—
415C4⋊C4—3161no—
416C4⋊C4—1581no—
417C4⋊C4—1581no—
418C4⋊C4—1581no—
419C4⋊C4—1581no—
420C22⋊C4—41no—
421C22⋊C4—3161no—
422C22⋊C4—3161no—
423C22⋊C4—3161no—
424C22⋊C4—3161no—
425C22⋊C4—3161no—
426C22⋊C4—3161no—
427C22⋊C4—3161no—
428C22⋊C4—3161no—
429C42—41no—
430C42—3161no—
431C42—3161no—
432C42—3161no—
433C42—3161no—
434C16—21no—
435C16—21no—
436C16—1581no—
437C16—1581no—
438C23—41no—
439C23—3161no—
440C23—791no—
441C23—791no—
442C23—1582no—
443C23—1582no—
444C23D8×D7911yescharacteristic
445Q8—41no—
446Q8—41no—
447D4—41no—
448D4—41no—
449D4—82no—
450D4—6322no—
451D4—6322no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—82no—
456C2×C4—3161no—
457C2×C4—3161no—
458C2×C4—3161no—
459C2×C4—3161no—
460C2×C4—791no—
461C2×C4—791no—
462C2×C4—6322no—
463C2×C4—6322no—
464C2×C4—1582no—
465C2×C4—1581no—
466C2×C4—1581no—
467C2×C4—1581no—
468C2×C4—1581no—
469C2×C4—1582no—
470C2×C4—1582no—
471C2×C4#1457914312yes—
472C2×C4D8×D7911yescharacteristic
473C2×C4#1457919012yes—
474C2×C4C2×D4×D7911yescharacteristic
475C8—1581no—
476C8—1581no—
477C8#1457914312yes—
478C8C2×#1192962311yescharacteristic
479C8C2×#1192962411yescharacteristic
480C22—21no—
481C22—21no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—3164no—
486C22—792no—
487C22—1581no—
488C22—1581no—
489C22—1581no—
490C22—1581no—
491C22—1581no—
492C22#1909887512yes—
493C22C2×D8×D7911yescharacteristic
494C4—81no—
495C4—81no—
496C4—81no—
497C4—6321no—
498C4—6321no—
499C4—6321no—
500C4—6321no—
501C4—1581no—
502C4—1581no—
503C4#1909887512yes—
504C4C2×#1457914311yescharacteristic
505C4C2×#1457919011yescharacteristic
506C2—22no—
507C2—161no—
508C2—3161no—
509C2—1582no—
510C2C2×#1909887511yescharacteristic
511C2#508×D7911yescharacteristic
512C2#2377653311yescharacteristic
subgroupG/H
centerC2×C4D8×D79
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#2138C2
socleC22×C79C22×D8
prime radical#23772977C2
hypercenter#2138D79
hypocenterC79#33645
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
2. #427×D79C2semidirect
3. #23765607C2semidirect
4. #23765606C2semidirect
5. #23765601C2semidirect
6. #23765596C2semidirect
7. #23765590C2semidirect
8. #23765589C2semidirect
9. #23765584C2semidirect
10. #23765579C2semidirect
11. #23765184C2semidirect
12. #23765182C2semidirect
13. #23765107C2semidirect
14. C2×(C2×D79)⋊C16C2semidirect
15. C79×#2138C2semidirect
36. #19098391C22semidirect
37. #19098391C22semidirect
38. #19098390C22semidirect
39. #19098390C22semidirect
40. #19098389C22semidirect
41. #19098389C22semidirect
42. #19098388C22semidirect
43. #19098388C22semidirect
44. #19098367C22semidirect
45. #19098367C22semidirect
46. #19098364C22semidirect
47. #19098364C22semidirect
55. #358×C79C22semidirect
56. #358×C79C22semidirect
57. #357×C79C22semidirect
58. #357×C79C22semidirect
270. #2138D79semidirect
272. C79#33645semidirect
306. #358C2×D79semidirect
307. #358C2×D79semidirect
310. #357C2×D79semidirect
311. #357C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237729772
#427×D792
#237656072
#237656062
#237656012
#237655962
#237655902
#237655892
#237655842
#237655792
#237651842
#237651822
#237651072
C2×(C2×D79)⋊C162
C79×#21382
#3364579