Finite Group Databasebeta

← #23777068

Subgroups — #23777068

Order 10112.

1658 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14579207C211yesmaximal, characteristic, complemented
2C2×#41×D79C211yesmaximal, characteristic, complemented
3#19099232C211yesmaximal, characteristic, complemented
4#19099225C211yesmaximal, characteristic, complemented
5#19099220C211yesmaximal, characteristic, complemented
6#19099210C213yesmaximal, complemented
7#19099209C213yesmaximal, complemented
8#19099198C213yesmaximal, complemented
9#19099197C213yesmaximal, complemented
10#19099142C213yesmaximal, complemented
11#19099126C213yesmaximal, complemented
12#19099072C213yesmaximal, complemented
13#121×D79C213yesmaximal, complemented
14#19099045C211yesmaximal, characteristic, complemented
15#550×C79C211yesmaximal, characteristic, complemented
16#14579207—28no—
17C2×#11929625C2211yescharacteristic
18C2×#11929625C2211yescharacteristic, complemented
19C2×#11929624C2213yescomplemented
20C2×#11929624C2213yescomplemented
21C2×#11929623C2213yescomplemented
22#41×D79—28no—
23C2×Q8×D79C2211yescharacteristic, complemented
24C22×D316C2213yescomplemented
25C2×D4×D79C2213yescomplemented
26C2×D4×D79C2213yescomplemented
27C2×D4×D79C2213yescomplemented
28C2×D4×D79C2216yescomplemented
29C22×C4×D79C2213yescomplemented
30#14579157C2213yescomplemented
31#14579149C2213yescomplemented
32#14579148C2213yescomplemented
33#14579146C2213yescomplemented
34#14579143C2213yescomplemented
35#14579141C2213yescomplemented
36#14579140C2213yescomplemented
37#14579139C2211yescharacteristic, complemented
38#14579138C2213yescomplemented
39#14579136C2213yescomplemented
40#14579135C2213yescomplemented
41#14579134C2216yescomplemented
42#14579134C2213yescomplemented
43#14579133C2216yescomplemented
44#14579133C2213yescomplemented
45#14579132C2216yescomplemented
46#14579132C2213yescomplemented
47#14579132C2213yescomplemented
48#14579131C2213yescomplemented
49#14579129C2216yescomplemented
50#14579129C2213yescomplemented
51#14579129C2216yescomplemented
52Q8×Dic79C2211yescharacteristic, complemented
53C4×D316C2213yescomplemented
54C4×D316C2213yescomplemented
55#14579123C2213yescomplemented
56C4×C22⋊D79C2213yescomplemented
57C4×C22⋊D79C2213yescomplemented
58#14579118C2213yescomplemented
59C4⋊C4×D79C2213yescomplemented
60C22⋊C4×D79C2216yescomplemented
61C2×#41×C79C2211yescharacteristic, complemented
62C2×#41×C79C2211yescharacteristic, complemented
63#127×C79C2213yescomplemented
64#121×C79C2213yescomplemented
65#121×C79C2213yescomplemented
66C4×Q8×C79C2211yescharacteristic, complemented
67C4×D4×C79C2213yescomplemented
68C23×D79C2313yes—
69C23×D79C2316yes—
70#11929625—24no—
71#11929625D418yescomplemented
72#11929624—212no—
73#11929624—212no—
74#11929623—212no—
75C2×Dic158C2313yes—
76C2×Dic158C2311yescharacteristic
77Q8×D79—24no—
78C2×D316—212no—
79C2×D316C2313yes—
80C2×D316C2313yes—
81C2×D316C2313yes—
82C2×D316C2313yes—
83C2×C22⋊D79C2316yes—
84C2×C22⋊D79C2313yes—
85C2×C22⋊D79C2313yes—
86C2×C22⋊D79C2313yes—
87C2×C22⋊D79C2313yes—
88C2×C22⋊D79C2316yes—
89D4×D79—212no—
90D4×D79—212no—
91D4×D79—212no—
92D4×D79—412no—
93C22×Dic79C2313yes—
94C2×C4×D79—212no—
95C2×C4×D79C2313yes—
96C2×C4×D79C2313yes—
97C2×C4×D79C2313yes—
98C2×C4×D79C2311yescharacteristic
99C2×C4×D79C2316yes—
100C2×C4×D79C2313yes—
101C4⋊Dic79C2313yes—
102C4⋊Dic79C2313yes—
103Dic79⋊C4C2313yes—
104Dic79⋊C4C2311yescharacteristic
105Dic79⋊C4C2311yescharacteristic
106Dic79⋊C4C2313yes—
107C22⋊Dic79C2316yes—
108C22⋊Dic79C2313yes—
109C22⋊Dic79C2313yes—
110(C2×D79)⋊C4C2313yes—
111(C2×D79)⋊C4C2316yes—
112(C2×D79)⋊C4C2313yes—
113(C2×D79)⋊C4C2313yes—
114(C2×D79)⋊C4C2311yescharacteristic
115(C2×D79)⋊C4C2311yescharacteristic
116(C2×D79)⋊C4C2313yes—
117C4×Dic79C2313yes—
118#41×C79—24no—
119#41×C79—24no—
120C2×Q8×C79C2311yescharacteristic
121C2×D4×C79C2313yes—
122C2×D4×C79C2313yes—
123C2×D4×C79C2313yes—
124C2×D4×C79C2316yes—
125C22×C4×C79C2313yes—
126C22×C4×C79C2313yes—
127C4⋊C4×C79C2311yescharacteristic
128C4⋊C4×C79C2313yes—
129C22⋊C4×C79C2313yes—
130C22⋊C4×C79C2313yes—
131C42×C79C2313yes—
132C23×C79C2413yes—
133C23×C79C2413yes—
134C22×D79—26no—
135C22×D79—26no—
136C22×D79—212no—
137C22×D79—43no—
138C22×D79—46no—
139C22×D79—46no—
140C22×D79C2413yes—
141C22×D79C2413yes—
142C22×D79C2411yescharacteristic
143Q8×C79C2×D414yescomplemented
144D4×C79—26no—
145D4×C79—26no—
146D4×C79—46no—
147D4×C79—43no—
148C2×C4×C79—26no—
149C2×C4×C79—26no—
150C2×C4×C79C2411yescharacteristic
151C2×C4×C79C2413yes—
152C2×C4×C79C2411yescharacteristic
153C2×C4×C79C2411yescharacteristic
154C2×C4×C79C2413yes—
155C22⋊D79—26no—
156C22⋊D79—212no—
157C22⋊D79—26no—
158C22⋊D79—26no—
159C22⋊D79—46no—
160C22⋊D79—43no—
161C2×Dic79—26no—
162C2×Dic79C2411yescharacteristic
163C2×Dic79C2413yes—
164C2×Dic79C2413yes—
165C2×Dic79C2411yescharacteristic
166C2×Dic79C2411yescharacteristic
167D316—26no—
168D316—26no—
169D316—26no—
170D316C2×D4112yescomplemented
171C4×D79—26no—
172C4×D79—26no—
173C4×D79—26no—
174C4×D79—22no—
175C4×D79—46no—
176C4×D79C2×D4112yescomplemented
177Dic158—26no—
178Dic158—22no—
179C22×C79—23no—
180C22×C79—23no—
181C22×C79—23no—
182C22×C79—23no—
183C22×C79—23no—
184C22×C79—23no—
185C22×C79C2511yescharacteristic
186C2×D79—21no—
187C2×D79—23no—
188C2×D79—21no—
189C2×D79—23no—
190C2×D79—23no—
191C2×D79—23no—
192C2×D79—23no—
193C2×D79—412no—
194C2×D79—812no—
195C2×D79C22×D416yes—
196C2×D79H(5,𝔽2)12yescomplemented
197C4×C79—21no—
198C4×C79—21no—
199C4×C79—23no—
200C4×C79—21no—
201C4×C79C22×D416yes—
202Dic79—21no—
203Dic79—23no—
204Dic79—23no—
205Dic79—21no—
206Dic79C22×D412yes—
207C2×C79—43no—
208C2×C79—43no—
209C2×C79C23×D411yescharacteristic
210C2×C79C2×H(5,𝔽2)11yescharacteristic
211C2×C79#58511yescharacteristic
212D79—22no—
213D79—26no—
214D79—43no—
215#3488—791nomaximal, Sylow
216C79#348811yescharacteristic, complemented, Sylow
217C2×H(5,𝔽2)—791no—
218C22×#41—791no—
219#550—791no—
220#550D7911yescharacteristic, complemented
221#548—791no—
222#548—791no—
223#546—793no—
224#546—793no—
225D42—793no—
226D42—793no—
227#537—793no—
228#535—793no—
229#525—793no—
230C2×#121—793no—
231#520—791no—
232H(5,𝔽2)—1588no—
233C2×#41—791no—
234C2×#41—791no—
235C2×#41—793no—
236C2×#41—793no—
237C2×#41—793no—
238C2×#41—1588no—
239C2×#41C2×D7911yescharacteristic, complemented
240C2×#41C2×D7911yescharacteristic, complemented
241C22×Q8—791no—
242C22×D4—796no—
243C22×D4—793no—
244C22×D4—793no—
245C22×D4—793no—
246C22×D4—793no—
247C23×C4—793no—
248#127—793no—
249#127C2×D7913yescomplemented
250#124—793no—
251#124—793no—
252#123—793no—
253#123—793no—
254C22⋊Q8—793no—
255C22⋊Q8—791no—
256C22⋊Q8—793no—
257C22⋊Q8—793no—
258#121—796no—
259#121—793no—
260#121—793no—
261#121—796no—
262#121—793no—
263#121—793no—
264#121—793no—
265#121—793no—
266#121—796no—
267#121C2×D7913yescomplemented
268#121C2×D7913yescomplemented
269#120—793no—
270#120—796no—
271#120—796no—
272#120—793no—
273C4×Q8—791no—
274C4×Q8C2×D7911yescharacteristic, complemented
275C4×D4—793no—
276C4×D4—793no—
277C4×D4—793no—
278C4×D4—793no—
279C4×D4—793no—
280C4×D4C2×D7913yescomplemented
281#117—793no—
282C2×C4⋊C4—793no—
283C2×C22⋊C4—796no—
284C24—796no—
285C24—793no—
286#41—24no—
287#41—24no—
288#41—798no—
289#41—15812no—
290#41—15812no—
291#41—15812no—
292#41—1584no—
293C2×Q8—791no—
294C2×Q8—793no—
295C2×Q8—1584no—
296C2×Q8C22×D7911yescharacteristic
297C2×D4—31612no—
298C2×D4—793no—
299C2×D4—796no—
300C2×D4—793no—
301C2×D4—796no—
302C2×D4—793no—
303C2×D4—793no—
304C2×D4—793no—
305C2×D4—793no—
306C2×D4—793no—
307C2×D4—793no—
308C2×D4—15812no—
309C2×D4—15812no—
310C2×D4—15812no—
311C2×D4—15812no—
312C2×D4C22×D7913yes—
313C2×D4C22×D7913yes—
314C2×D4C22×D7913yes—
315C2×D4C22×D7916yes—
316C22×C4—796no—
317C22×C4—793no—
318C22×C4—793no—
319C22×C4—793no—
320C22×C4—793no—
321C22×C4—793no—
322C22×C4—791no—
323C22×C4—15812no—
324C22×C4C22×D7913yes—
325C22×C4C22×D7913yes—
326C4⋊C4—793no—
327C4⋊C4—791no—
328C4⋊C4—791no—
329C4⋊C4—793no—
330C4⋊C4—793no—
331C4⋊C4—793no—
332C4⋊C4C22×D7913yes—
333C4⋊C4C22×D7911yescharacteristic
334C22⋊C4—796no—
335C22⋊C4—793no—
336C22⋊C4—793no—
337C22⋊C4—793no—
338C22⋊C4—796no—
339C22⋊C4—793no—
340C22⋊C4—793no—
341C22⋊C4—791no—
342C22⋊C4—791no—
343C22⋊C4—793no—
344C22⋊C4C22×D7913yes—
345C22⋊C4C22×D7913yes—
346C42—793no—
347C42C22×D7913yes—
348C23—3163no—
349C23—3166no—
350C23—3166no—
351C23—791no—
352C23—793no—
353C23—793no—
354C23—1586no—
355C23—1586no—
356C23—15812no—
357C23C23×D7913yes—
358C23C23×D7913yes—
359Q8—1582no—
360Q8—1586no—
361Q8D4×D7914yescomplemented
362D4—26no—
363D4—26no—
364D4—46no—
365D4—43no—
366D4—3166no—
367D4—3163no—
368D4—7912no—
369D4—1586no—
370D4—1586no—
371D4—1586no—
372D4—1586no—
373D4—1586no—
374D4—15812no—
375D4—1586no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—3166no—
379C2×C4—791no—
380C2×C4—7912no—
381C2×C4—791no—
382C2×C4—793no—
383C2×C4—793no—
384C2×C4—791no—
385C2×C4—1582no—
386C2×C4—1586no—
387C2×C4—1586no—
388C2×C4—1586no—
389C2×C4—1586no—
390C2×C4C23×D7911yescharacteristic
391C2×C4C23×D7913yes—
392C2×C4C23×D7911yescharacteristic
393C2×C4C23×D7911yescharacteristic
394C2×C4C23×D7913yes—
395C22—23no—
396C22—23no—
397C22—23no—
398C22—23no—
399C22—23no—
400C22—23no—
401C22—31612no—
402C22—792no—
403C22—796no—
404C22—63212no—
405C22—1583no—
406C22—1583no—
407C22—1583no—
408C22—1583no—
409C22—1581no—
410C22—1583no—
411C22—1581no—
412C22C24×D7911yescharacteristic
413C4—23no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—792no—
418C4—1583no—
419C4—1581no—
420C4—1581no—
421C4—1583no—
422C4C2×D4×D7916yes—
423C2—43no—
424C2—43no—
425C2—3163no—
426C2—1582no—
427C2—1586no—
428C2C22×D4×D7911yescharacteristic
429C2H(5,𝔽2)×D7911yescharacteristic
430C2#1909939911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D79
commutator, socle, cosocleC22×C79C25
Fitting#550×C79C2
hypercenter#550D79
hypocenterC79#3488
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#14579207C2semidirect
2. C2×#41×D79C2semidirect
3. #19099232C2semidirect
4. #19099225C2semidirect
5. #19099220C2semidirect
6. #19099210C2semidirect
7. #19099209C2semidirect
8. #19099198C2semidirect
9. #19099197C2semidirect
10. #19099142C2semidirect
11. #19099126C2semidirect
12. #19099072C2semidirect
13. #121×D79C2semidirect
14. #19099045C2semidirect
15. #550×C79C2semidirect
18. C2×#11929625C22semidirect
19. C2×#11929624C22semidirect
20. C2×#11929624C22semidirect
21. C2×#11929623C22semidirect
23. C2×Q8×D79C22semidirect
24. C22×D316C22semidirect
25. C2×D4×D79C22semidirect
26. C2×D4×D79C22semidirect
27. C2×D4×D79C22semidirect
28. C2×D4×D79C22semidirect
29. C22×C4×D79C22semidirect
30. #14579157C22semidirect
31. #14579149C22semidirect
32. #14579148C22semidirect
33. #14579146C22semidirect
34. #14579143C22semidirect
35. #14579141C22semidirect
36. #14579140C22semidirect
37. #14579139C22semidirect
38. #14579138C22semidirect
39. #14579136C22semidirect
40. #14579135C22semidirect
41. #14579134C22semidirect
42. #14579134C22semidirect
43. #14579133C22semidirect
44. #14579133C22semidirect
45. #14579132C22semidirect
46. #14579132C22semidirect
47. #14579132C22semidirect
48. #14579131C22semidirect
49. #14579129C22semidirect
50. #14579129C22semidirect
51. #14579129C22semidirect
52. Q8×Dic79C22semidirect
53. C4×D316C22semidirect
54. C4×D316C22semidirect
55. #14579123C22semidirect
56. C4×C22⋊D79C22semidirect
57. C4×C22⋊D79C22semidirect
58. #14579118C22semidirect
59. C4⋊C4×D79C22semidirect
60. C22⋊C4×D79C22semidirect
61. C2×#41×C79C22semidirect
62. C2×#41×C79C22semidirect
63. #127×C79C22semidirect
64. #121×C79C22semidirect
65. #121×C79C22semidirect
66. C4×Q8×C79C22semidirect
67. C4×D4×C79C22semidirect
71. #11929625D4semidirect
143. Q8×C79C2×D4semidirect
170. D316C2×D4semidirect
176. C4×D79C2×D4semidirect
196. C2×D79H(5,𝔽2)semidirect
216. C79#3488semidirect
220. #550D79semidirect
239. C2×#41C2×D79semidirect
240. C2×#41C2×D79semidirect
249. #127C2×D79semidirect
267. #121C2×D79semidirect
268. #121C2×D79semidirect
274. C4×Q8C2×D79semidirect
280. C4×D4C2×D79semidirect
361. Q8D4×D79semidirect