Finite Group Databasebeta

← #23775163

Subgroups — #23775163

Order 10112.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19099375C211yesmaximal, characteristic, complemented
2#19099373C211yesmaximal, characteristic, complemented
3#19099367C211yesmaximal, characteristic, complemented
4#19099364C211yesmaximal, characteristic, complemented
5#19098951C211yesmaximal, characteristic, complemented
6#19098832C211yesmaximal, characteristic, complemented
7#19098827C211yesmaximal, characteristic, complemented
8#19098825C211yesmaximal, characteristic, complemented
9#19098743C211yesmaximal, characteristic, complemented
10#19098734C211yesmaximal, characteristic, complemented
11#19098733C211yesmaximal, characteristic, complemented
12#19098729C211yesmaximal, characteristic, complemented
13#19098684C211yesmaximal, characteristic, complemented
14#19098683C211yesmaximal, characteristic, complemented
15#472×C79C211yesmaximal, characteristic, complemented
16#14579208C2211yescharacteristic, complemented
17#14579207C2211yescharacteristic, complemented
18#14579206C2211yescharacteristic
19#14579206C2211yescharacteristic, complemented
20#14579194—21no—
21#14579194C2211yescharacteristic, complemented
22#14579194C2211yescharacteristic, complemented
23#14579190—21no—
24#14579189—21no—
25#14579189C2211yescharacteristic
26#14579189C2211yescharacteristic, complemented
27#14579188—21no—
28#14579187C2211yescharacteristic
29#14579187C2211yescharacteristic, complemented
30#14579185—21no—
31#14579185—21no—
32#14579185—21no—
33#14579185—21no—
34#14579183—21no—
35#14579183C2211yescharacteristic
36#14579182—21no—
37#14579181C2211yescharacteristic
38#14579180—21no—
39#14579180—21no—
40C2×#11929613—21no—
41C2×#11929613C2211yescharacteristic, complemented
42C2×#11929612C2211yescharacteristic, complemented
43SD16×D79—21no—
44C2×#11929609C2211yescharacteristic, complemented
45D8×D79—21no—
46D8×D79—21no—
47#14579169C2211yescharacteristic, complemented
48#14579169C2211yescharacteristic, complemented
49#14579167C2211yescharacteristic, complemented
50#14579157C2211yescharacteristic, complemented
51#14579148C2211yescharacteristic, complemented
52#14579096C2211yescharacteristic, complemented
53#14579088C2211yescharacteristic, complemented
54#14579087C2211yescharacteristic, complemented
55#14579087C2211yescharacteristic, complemented
56#14579087C2211yescharacteristic, complemented
57#14579079C2211yescharacteristic, complemented
58#14579077C2211yescharacteristic, complemented
59#14579071C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C79C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C79C2211yescharacteristic, complemented
62C2×D8×C79C2211yescharacteristic, complemented
63#131×C79C2211yescharacteristic, complemented
64#108×C79C2211yescharacteristic, complemented
65#100×C79C2211yescharacteristic, complemented
66#100×C79C2211yescharacteristic, complemented
67#11929625—21no—
68#11929624—21no—
69#11929624—21no—
70#11929624C2311yescharacteristic
71#11929624C2311yescharacteristic
72#11929624C2311yescharacteristic
73#11929623—21no—
74#11929623—21no—
75#11929623—21no—
76#11929623—21no—
77#11929623—21no—
78#11929623—41no—
79C2×Dic158—21no—
80C2×Dic158C2311yescharacteristic
81Q8×D79—21no—
82C2×D316—21no—
83C2×D316C2311yescharacteristic
84C2×C22⋊D79—41no—
85C2×C22⋊D79—41no—
86C2×C22⋊D79—41no—
87D4×D79—21no—
88D4×D79—21no—
89D4×D79—21no—
90D4×D79—21no—
91D4×D79—21no—
92D4×D79—41no—
93Dic316—21no—
94Dic316—21no—
95Dic316D411yescharacteristic, complemented
96#11929614D411yescharacteristic, complemented
97#11929613—21no—
98#11929613—21no—
99#11929613—21no—
100#11929613—21no—
101#11929613—21no—
102#11929613—21no—
103#11929613D411yescharacteristic, complemented
104#11929613D411yescharacteristic, complemented
105#11929612—21no—
106#11929612—21no—
107#11929612—21no—
108#11929612—21no—
109#11929612D411yescharacteristic, complemented
110#11929611D411yescharacteristic, complemented
111D632—21no—
112D632—21no—
113D632D411yescharacteristic, complemented
114#11929609—21no—
115#11929609—21no—
116#11929609—21no—
117#11929609—21no—
118#11929609D411yescharacteristic, complemented
119#11929608C2311yescharacteristic
120#11929607—21no—
121#11929607—21no—
122#11929607—21no—
123C2×C79⋊C8C2311yescharacteristic
124C8×D79—21no—
125C8×D79—21no—
126C8×D79—21no—
127C22⋊Dic79—41no—
128C4×Dic79C2311yescharacteristic
129#41×C79C2311yescharacteristic
130C2×D4×C79C2311yescharacteristic
131C2×D4×C79C2311yescharacteristic
132SD16×C79—21no—
133SD16×C79—21no—
134D8×C79—21no—
135D8×C79—21no—
136D8×C79—21no—
137D8×C79—21no—
138M16×C79—21no—
139M16×C79C2311yescharacteristic
140M16×C79C2311yescharacteristic
141M16×C79C2311yescharacteristic
142C2×C8×C79—21no—
143C2×C8×C79C2311yescharacteristic
144C23×C79—21no—
145C23×C79—21no—
146C22×D79—21no—
147C22×D79—41no—
148C22×D79—41no—
149C22×D79—41no—
150Q8×C79C2×D411yescharacteristic
151D4×C79—21no—
152D4×C79—21no—
153D4×C79—21no—
154D4×C79—21no—
155D4×C79—21no—
156D4×C79C2×D411yescharacteristic
157C2×C4×C79—21no—
158C2×C4×C79C2411yescharacteristic
159C22⋊D79—21no—
160C22⋊D79—21no—
161C22⋊D79—21no—
162C22⋊D79—41no—
163C22⋊D79—41no—
164C22⋊D79—41no—
165C22⋊D79—41no—
166C22⋊D79—41no—
167C22⋊D79—41no—
168C22⋊D79—82no—
169C2×Dic79—21no—
170C2×Dic79—21no—
171C2×Dic79—41no—
172C2×Dic79—41no—
173C2×Dic79—41no—
174D316—21no—
175D316—21no—
176D316—41no—
177D316C2×D411yescharacteristic
178D316C2×D411yescharacteristic
179D316C2×D411yescharacteristic
180C4×D79—21no—
181C4×D79—21no—
182C4×D79—21no—
183C4×D79—41no—
184C4×D79—41no—
185Dic158—21no—
186Dic158—21no—
187Dic158—41no—
188Dic158C2×D411yescharacteristic
189Dic158C2×D411yescharacteristic
190Dic158C2×D411yescharacteristic
191C8×C79—21no—
192C8×C79—21no—
193C8×C79—21no—
194C8×C79C2×D411yescharacteristic
195C8×C79C2×D411yescharacteristic
196C79⋊C8—21no—
197C79⋊C8C2×D411yescharacteristic
198C79⋊C8C2×D411yescharacteristic
199C22×C79—21no—
200C22×C79—41no—
201C22×C79—41no—
202C22×C79—81no—
203C22×C79—81no—
204C22×C79H(5,𝔽2)11yescharacteristic
205C2×D79—21no—
206C2×D79—21no—
207C2×D79—21no—
208C2×D79—41no—
209C2×D79—81no—
210C2×D79—161no—
211C2×D79—161no—
212C2×D79—161no—
213C4×C79—21no—
214C4×C79C22×D411yescharacteristic
215C4×C79C22×D411yescharacteristic
216Dic79—21no—
217Dic79—21no—
218Dic79—21no—
219Dic79—41no—
220Dic79—41no—
221C2×C79—21no—
222C2×C79—41no—
223C2×C79—81no—
224C2×C79—81no—
225C2×C79D4211yescharacteristic
226D79—41no—
227D79—41no—
228D79—41no—
229D79—81no—
230#3288—791nomaximal, Sylow
231C79#328811yescharacteristic, complemented, Sylow
232#577—791no—
233#577—791no—
234#576—791no—
235#576—791no—
236#496—791no—
237#472—791no—
238#472D7911yescharacteristic, complemented
239#471—791no—
240#471—791no—
241#455—791no—
242#454—791no—
243#453—791no—
244#453—791no—
245#444—791no—
246#444—791no—
2472−1+4—791no—
248H(5,𝔽2)—791no—
249H(5,𝔽2)—791no—
250H(5,𝔽2)—791no—
251#137—791no—
252#137—791no—
253#137—1581no—
254#137—1581no—
255AGL(1,ℤ/8)—791no—
256AGL(1,ℤ/8)—791no—
257AGL(1,ℤ/8)—791no—
258AGL(1,ℤ/8)—791no—
259AGL(1,ℤ/8)—1581no—
260AGL(1,ℤ/8)—1581no—
261AGL(1,ℤ/8)—1581no—
262AGL(1,ℤ/8)—1581no—
263AGL(1,ℤ/8)—1581no—
264AGL(1,ℤ/8)—1581no—
265AGL(1,ℤ/8)C2×D7911yescharacteristic, complemented
266AGL(1,ℤ/8)C2×D7911yescharacteristic, complemented
267#135—791no—
268#135—791no—
269#135—1581no—
270#135—1581no—
271#135—1581no—
272#135—1581no—
273C2×SD16—791no—
274C2×SD16—791no—
275C2×SD16—1581no—
276C2×SD16—1581no—
277C2×D8—791no—
278C2×D8—1581no—
279C2×D8—1581no—
280C2×D8C2×D7911yescharacteristic, complemented
281#131—791no—
282#131—791no—
283#131—791no—
284#131C2×D7911yescharacteristic, complemented
285#127—791no—
286#124—791no—
287#108—791no—
288#108C2×D7911yescharacteristic, complemented
289#104—791no—
290#104—791no—
291#104—791no—
292#104—791no—
293#101—791no—
294#100—791no—
295#100C2×D7911yescharacteristic, complemented
296#100C2×D7911yescharacteristic, complemented
297#97—791no—
298#41—3161no—
299#41—791no—
300#41—791no—
301#41—791no—
302#41—1581no—
303#41—1581no—
304#41—1581no—
305#41—1581no—
306#41—1581no—
307#41—1581no—
308#41—1581no—
309#41—1581no—
310#41C22×D7911yescharacteristic
311C2×Q8—791no—
312C2×Q8—1581no—
313C2×Q8—1581no—
314C2×D4—3161no—
315C2×D4—3161no—
316C2×D4—3161no—
317C2×D4—3161no—
318C2×D4—791no—
319C2×D4—1581no—
320C2×D4—1581no—
321C2×D4—1581no—
322C2×D4—1581no—
323C2×D4—1581no—
324C2×D4—1581no—
325C2×D4C22×D7911yescharacteristic
326C2×D4C22×D7911yescharacteristic
327Q16—791no—
328Q16—791no—
329Q16—1581no—
330Q16—1581no—
331SD16—21no—
332SD16—21no—
333SD16—791no—
334SD16—791no—
335SD16—791no—
336SD16—791no—
337SD16—1581no—
338SD16—1581no—
339SD16—1581no—
340SD16—1581no—
341SD16—1581no—
342SD16—1581no—
343SD16—1581no—
344SD16—1581no—
345SD16—1581no—
346SD16—1581no—
347D8—21no—
348D8—21no—
349D8—21no—
350D8—21no—
351D8—791no—
352D8—791no—
353D8—1581no—
354D8—1581no—
355D8—1581no—
356D8—1581no—
357D8—1581no—
358D8—1581no—
359M16—21no—
360M16—791no—
361M16—1581no—
362M16—1581no—
363M16—1581no—
364M16C22×D7911yescharacteristic
365M16C22×D7911yescharacteristic
366M16C22×D7911yescharacteristic
367C2×C8—21no—
368C2×C8—791no—
369C2×C8—1581no—
370C2×C8—1581no—
371C2×C8—1581no—
372C2×C8C22×D7911yescharacteristic
373C22⋊C4—3161no—
374C42—791no—
375C23—21no—
376C23—21no—
377C23—3161no—
378C23—3161no—
379C23—3161no—
380C23—1581no—
381Q8—3161no—
382Q8—791no—
383Q8—791no—
384Q8—791no—
385Q8—1581no—
386Q8—1581no—
387Q8D4×D7911yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—3161no—
394D4—3161no—
395D4—3161no—
396D4—3161no—
397D4—3161no—
398D4—3161no—
399D4—3161no—
400D4—791no—
401D4—791no—
402D4—791no—
403D4—6322no—
404D4—1581no—
405D4—1581no—
406D4—1581no—
407D4—1581no—
408D4—1581no—
409D4D4×D7911yescharacteristic
410C2×C4—21no—
411C2×C4—3161no—
412C2×C4—3161no—
413C2×C4—3161no—
414C2×C4—3161no—
415C2×C4—3161no—
416C2×C4—1581no—
417C2×C4—1581no—
418C2×C4—1581no—
419C2×C4—1581no—
420C2×C4—1581no—
421C2×C4C23×D7911yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—791no—
426C8—791no—
427C8—1581no—
428C8D4×D7911yescharacteristic
429C8D4×D7911yescharacteristic
430C22—21no—
431C22—41no—
432C22—41no—
433C22—81no—
434C22—81no—
435C22—3161no—
436C22—6321no—
437C22—1581no—
438C22—1581no—
439C22—1581no—
440C22—12641no—
441C22—12641no—
442C22—12641no—
443C22#1457920711yescharacteristic
444C4—21no—
445C4—3161no—
446C4—3161no—
447C4—1581no—
448C4—1581no—
449C4—1581no—
450C4C2×D4×D7911yescharacteristic
451C4C2×D4×D7911yescharacteristic
452C2—21no—
453C2—41no—
454C2—81no—
455C2—81no—
456C2—3161no—
457C2—3161no—
458C2—3161no—
459C2—6321no—
460C2#1909919711yescharacteristic
subgroupG/H
centerC2#19099197
commutator, cosocleC2×C4×C79C24
FrattiniC2×C4C23×D79
Fitting#472×C79C2
socleC2×C79D42
hypercenter#472D79
hypocenterC79#3288
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #19099375C2semidirect
2. #19099373C2semidirect
3. #19099367C2semidirect
4. #19099364C2semidirect
5. #19098951C2semidirect
6. #19098832C2semidirect
7. #19098827C2semidirect
8. #19098825C2semidirect
9. #19098743C2semidirect
10. #19098734C2semidirect
11. #19098733C2semidirect
12. #19098729C2semidirect
13. #19098684C2semidirect
14. #19098683C2semidirect
15. #472×C79C2semidirect
16. #14579208C22semidirect
17. #14579207C22semidirect
19. #14579206C22semidirect
21. #14579194C22semidirect
22. #14579194C22semidirect
26. #14579189C22semidirect
29. #14579187C22semidirect
41. C2×#11929613C22semidirect
42. C2×#11929612C22semidirect
44. C2×#11929609C22semidirect
47. #14579169C22semidirect
48. #14579169C22semidirect
49. #14579167C22semidirect
50. #14579157C22semidirect
51. #14579148C22semidirect
52. #14579096C22semidirect
53. #14579088C22semidirect
54. #14579087C22semidirect
55. #14579087C22semidirect
56. #14579087C22semidirect
57. #14579079C22semidirect
58. #14579077C22semidirect
59. #14579071C22semidirect
60. AGL(1,ℤ/8)×C79C22semidirect
61. AGL(1,ℤ/8)×C79C22semidirect
62. C2×D8×C79C22semidirect
63. #131×C79C22semidirect
64. #108×C79C22semidirect
65. #100×C79C22semidirect
66. #100×C79C22semidirect
95. Dic316D4semidirect
96. #11929614D4semidirect
103. #11929613D4semidirect
104. #11929613D4semidirect
109. #11929612D4semidirect
110. #11929611D4semidirect
113. D632D4semidirect
118. #11929609D4semidirect
231. C79#3288semidirect
238. #472D79semidirect
265. AGL(1,ℤ/8)C2×D79semidirect
266. AGL(1,ℤ/8)C2×D79semidirect
280. C2×D8C2×D79semidirect
284. #131C2×D79semidirect
288. #108C2×D79semidirect
295. #100C2×D79semidirect
296. #100C2×D79semidirect