Finite Group Databasebeta

← #23777057

Subgroups — #23777057

Order 10112.

1536 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14579209C211yesmaximal, characteristic, complemented
2C2×#41×D79C211yesmaximal, characteristic, complemented
3#19099235C211yesmaximal, characteristic, complemented
4#19099234C213yesmaximal, complemented
5#19099232C211yesmaximal, characteristic, complemented
6#19099231C213yesmaximal, complemented
7#19099224C211yesmaximal, characteristic, complemented
8#19099223C213yesmaximal, complemented
9#19099221C213yesmaximal, complemented
10#19099131C213yesmaximal, complemented
11#19099129C213yesmaximal, complemented
12#19099076C213yesmaximal, complemented
13#19099072C213yesmaximal, complemented
14#19099039C211yesmaximal, characteristic, complemented
15#548×C79C211yesmaximal, characteristic, complemented
16#14579209—28no—
17C2×#11929625C2213yes—
18C2×#11929625C2211yescharacteristic
19C2×#11929625C2211yescharacteristic, complemented
20C2×#11929624C2213yes—
21C2×#11929624C2213yes—
22C2×#11929624C2213yescomplemented
23C2×#11929623C2213yes—
24#41×D79—28no—
25C2×Q8×D79C2211yescharacteristic
26C2×Q8×D79C2213yescomplemented
27C2×D4×D79C2213yes—
28C22×C4×D79C2213yes—
29#14579158C2213yescomplemented
30#14579157C2213yescomplemented
31#14579157C2216yescomplemented
32#14579150C2213yescomplemented
33#14579147C2216yescomplemented
34#14579146C2213yes—
35#14579143C2211yescharacteristic
36#14579139C2213yescomplemented
37#14579138C2213yescomplemented
38#14579137C2213yescomplemented
39#14579136C2213yescomplemented
40#14579136C2213yescomplemented
41#14579135C2213yescomplemented
42#14579135C2213yescomplemented
43#14579134C2216yescomplemented
44#14579134C2216yescomplemented
45#14579134C2213yescomplemented
46#14579133C2213yescomplemented
47#14579132C2213yescomplemented
48#14579131C2213yes—
49#14579131C2216yes—
50#14579129C2213yes—
51Q8×Dic79C2211yescharacteristic, complemented
52C4×Dic158C2213yescomplemented
53#14579126C2213yescomplemented
54C4×D316C2213yescomplemented
55#14579123C2216yescomplemented
56C4×C22⋊D79C2213yescomplemented
57C4×C22⋊D79C2213yescomplemented
58(C22⋊D79)⋊C4C2216yescomplemented
59#14579119C2213yes—
60#14579118C2213yes—
61C2×#41×C79C2211yescharacteristic
62C22×Q8×C79C2211yescharacteristic, complemented
63#124×C79C2213yescomplemented
64C22⋊Q8×C79C2213yescomplemented
65#121×C79C2213yescomplemented
66C4×Q8×C79C2211yescharacteristic, complemented
67C4×D4×C79C2213yescomplemented
68C23×D79C2313yes—
69#11929625—212no—
70#11929625—24no—
71#11929625—24no—
72#11929624—212no—
73#11929624—212no—
74#11929624—212no—
75#11929623—212no—
76C2×Dic158C2313yes—
77C2×Dic158C2313yes—
78Q8×D79—212no—
79Q8×D79D418yescomplemented
80C2×D316C2313yes—
81C2×D316C2313yes—
82C2×D316C2313yes—
83C2×D316C2316yes—
84C2×D316C2313yes—
85C2×C22⋊D79C2316yes—
86C2×C22⋊D79C2311yescharacteristic
87C2×C22⋊D79C2313yes—
88C2×C22⋊D79C2316yes—
89C2×C22⋊D79C2313yes—
90C2×C22⋊D79C2313yes—
91D4×D79—212no—
92C22×Dic79C2313yes—
93C2×C4×D79—212no—
94C2×C4×D79C2313yes—
95C2×C4×D79C2316yes—
96C2×C4×D79C2313yes—
97C2×C4×D79C2313yes—
98C2×C4×D79C2311yescharacteristic
99C2×C4×D79C2313yes—
100C4⋊Dic79C2313yes—
101C4⋊Dic79C2311yescharacteristic
102Dic79⋊C4C2313yes—
103Dic79⋊C4C2313yes—
104Dic79⋊C4C2313yes—
105Dic79⋊C4C2311yescharacteristic
106C22⋊Dic79C2311yescharacteristic
107C22⋊Dic79C2313yes—
108(C2×D79)⋊C4C2311yescharacteristic
109(C2×D79)⋊C4C2313yes—
110(C2×D79)⋊C4C2313yes—
111(C2×D79)⋊C4C2313yes—
112(C2×D79)⋊C4C2313yes—
113(C2×D79)⋊C4C2316yes—
114(C2×D79)⋊C4C2313yes—
115C4×Dic79C2313yes—
116C4×Dic79C2316yes—
117#41×C79—24no—
118C2×Q8×C79—24no—
119C2×Q8×C79C2313yes—
120C2×Q8×C79C2311yescharacteristic
121C2×D4×C79C2313yes—
122C2×D4×C79C2313yes—
123C22×C4×C79C2313yes—
124C22×C4×C79C2313yes—
125C4⋊C4×C79C2313yes—
126C4⋊C4×C79C2313yes—
127C22⋊C4×C79C2311yescharacteristic
128C22⋊C4×C79C2313yes—
129C22⋊C4×C79C2316yes—
130C42×C79C2313yes—
131C23×C79C2413yes—
132C23×C79C2411yescharacteristic
133C22×D79—26no—
134C22×D79—43no—
135C22×D79—43no—
136C22×D79C2413yes—
137C22×D79C2413yes—
138C22×D79C2411yescharacteristic
139Q8×C79—26no—
140Q8×C79C2×D414yes—
141D4×C79—26no—
142D4×C79—43no—
143C2×C4×C79—26no—
144C2×C4×C79—26no—
145C2×C4×C79C2413yes—
146C2×C4×C79C2411yescharacteristic
147C2×C4×C79C2413yes—
148C2×C4×C79C2411yescharacteristic
149C2×C4×C79C2413yes—
150C22⋊D79—22no—
151C22⋊D79—212no—
152C22⋊D79—26no—
153C22⋊D79—46no—
154C22⋊D79—43no—
155C22⋊D79—43no—
156C2×Dic79—26no—
157C2×Dic79C2411yescharacteristic
158C2×Dic79C2413yes—
159C2×Dic79C2411yescharacteristic
160C2×Dic79C2411yescharacteristic
161C2×Dic79C2413yes—
162D316—26no—
163D316—26no—
164D316—26no—
165D316—46no—
166D316—43no—
167C4×D79—22no—
168C4×D79—26no—
169C4×D79—26no—
170C4×D79—26no—
171C4×D79—212no—
172C4×D79C2×D4112yes—
173Dic158—26no—
174Dic158C2×D4112yes—
175C22×C79—21no—
176C22×C79—23no—
177C22×C79—21no—
178C22×C79—23no—
179C22×C79—23no—
180C22×C79—21no—
181C22×C79C2511yescharacteristic
182C2×D79—21no—
183C2×D79—21no—
184C2×D79—23no—
185C2×D79—23no—
186C2×D79—23no—
187C2×D79—23no—
188C2×D79—23no—
189C2×D79—23no—
190C2×D79—412no—
191C2×D79C22×D412yes—
192C4×C79—21no—
193C4×C79—23no—
194C4×C79—23no—
195C4×C79—21no—
196C4×C79C22×D416yes—
197Dic79—21no—
198Dic79—21no—
199Dic79—21no—
200Dic79—23no—
201Dic79C22×D416yes—
202C2×C79—43no—
203C2×C79—41no—
204C2×C79C23×D411yescharacteristic
205C2×C79#58511yescharacteristic
206C2×C79#58511yescharacteristic
207D79—22no—
208D79—43no—
209D79—43no—
210#3487—791nomaximal, Sylow
211C79#348711yescharacteristic, complemented, Sylow
212C2×2−1+4—791no—
213C22×#41—791no—
214#550—791no—
215#550—793no—
216#550—791no—
217#550—793no—
218#548—791no—
219#548—793no—
220#548—793no—
221#548D7911yescharacteristic, complemented
222#535—793no—
223#535—793no—
224#525—793no—
225#525—793no—
226#519—791no—
2272−1+4—1588no—
228C2×#41—793no—
229C2×#41—791no—
230C2×#41—791no—
231C2×#41—793no—
232C2×#41—793no—
233C2×#41—793no—
234C2×#41—793no—
235C2×#41—1588no—
236C2×#41C2×D7911yescharacteristic
237C22×Q8—791no—
238C22×Q8—793no—
239C22×Q8C2×D7911yescharacteristic, complemented
240C22×D4—793no—
241C23×C4—793no—
242#127—793no—
243#127—796no—
244#127—793no—
245#124—793no—
246#124—796no—
247#124C2×D7913yescomplemented
248#123—793no—
249#123—791no—
250C22⋊Q8—793no—
251C22⋊Q8—793no—
252C22⋊Q8—793no—
253C22⋊Q8C2×D7913yescomplemented
254#121—793no—
255#121—793no—
256#121—793no—
257#121—793no—
258#121—793no—
259#121—793no—
260#121—793no—
261#121—796no—
262#121—796no—
263#121C2×D7913yescomplemented
264#120—793no—
265#120—796no—
266#120—793no—
267C4×Q8—791no—
268C4×Q8—793no—
269C4×Q8—793no—
270C4×Q8C2×D7911yescharacteristic, complemented
271C4×D4—793no—
272C4×D4—793no—
273C4×D4—796no—
274C4×D4—796no—
275C4×D4—793no—
276C4×D4C2×D7913yescomplemented
277#117—793no—
278#117—793no—
279C24—793no—
280#41—24no—
281#41—15812no—
282#41—15812no—
283#41—15812no—
284#41—15812no—
285#41—15812no—
286#41—1584no—
287#41—1584no—
288C2×Q8—24no—
289C2×Q8—798no—
290C2×Q8—793no—
291C2×Q8—793no—
292C2×Q8—15812no—
293C2×Q8C22×D7913yes—
294C2×Q8C22×D7911yescharacteristic
295C2×D4—793no—
296C2×D4—796no—
297C2×D4—793no—
298C2×D4—791no—
299C2×D4—796no—
300C2×D4—793no—
301C2×D4—796no—
302C2×D4—793no—
303C2×D4—793no—
304C2×D4—793no—
305C2×D4—793no—
306C2×D4—15812no—
307C2×D4C22×D7913yes—
308C2×D4C22×D7913yes—
309C22×C4—793no—
310C22×C4—793no—
311C22×C4—796no—
312C22×C4—793no—
313C22×C4—793no—
314C22×C4—793no—
315C22×C4—791no—
316C22×C4—15812no—
317C22×C4C22×D7913yes—
318C22×C4C22×D7913yes—
319C4⋊C4—793no—
320C4⋊C4—793no—
321C4⋊C4—791no—
322C4⋊C4—793no—
323C4⋊C4—793no—
324C4⋊C4—791no—
325C4⋊C4C22×D7913yes—
326C4⋊C4C22×D7913yes—
327C22⋊C4—791no—
328C22⋊C4—793no—
329C22⋊C4—793no—
330C22⋊C4—791no—
331C22⋊C4—793no—
332C22⋊C4—793no—
333C22⋊C4—793no—
334C22⋊C4—796no—
335C22⋊C4—793no—
336C22⋊C4C22×D7916yes—
337C22⋊C4C22×D7911yescharacteristic
338C22⋊C4C22×D7913yes—
339C42—796no—
340C42—793no—
341C42C22×D7913yes—
342C23—3163no—
343C23—3163no—
344C23—791no—
345C23—793no—
346C23—793no—
347C23—1586no—
348C23C23×D7913yes—
349C23C23×D7911yescharacteristic
350Q8—26no—
351Q8—7912no—
352Q8—1586no—
353Q8D4×D7914yes—
354D4—26no—
355D4—43no—
356D4—3166no—
357D4—3163no—
358D4—3163no—
359D4—3163no—
360D4—3166no—
361D4—1586no—
362D4—15812no—
363D4—1586no—
364D4—1582no—
365D4—1586no—
366D4—1586no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—793no—
370C2×C4—7912no—
371C2×C4—791no—
372C2×C4—793no—
373C2×C4—791no—
374C2×C4—791no—
375C2×C4—1586no—
376C2×C4—15812no—
377C2×C4—1582no—
378C2×C4—1586no—
379C2×C4—1586no—
380C2×C4—1586no—
381C2×C4C23×D7913yes—
382C2×C4C23×D7911yescharacteristic
383C2×C4C23×D7913yes—
384C2×C4C23×D7911yescharacteristic
385C2×C4C23×D7913yes—
386C22—21no—
387C22—23no—
388C22—21no—
389C22—23no—
390C22—21no—
391C22—23no—
392C22—31612no—
393C22—792no—
394C22—1583no—
395C22—1583no—
396C22—1583no—
397C22—1583no—
398C22—1583no—
399C22—1583no—
400C22—1581no—
401C22—1581no—
402C22C24×D7911yescharacteristic
403C4—23no—
404C4—21no—
405C4—21no—
406C4—23no—
407C4—796no—
408C4—1581no—
409C4—1583no—
410C4—1581no—
411C4—1581no—
412C4C2×D4×D7916yes—
413C2—43no—
414C2—41no—
415C2—3163no—
416C2—3163no—
417C2—1582no—
418C2C22×D4×D7911yescharacteristic
419C2#1909939911yescharacteristic
420C2#1909940011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D79
commutator, socle, cosocleC22×C79C25
Fitting#548×C79C2
hypercenter#548D79
hypocenterC79#3487
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#14579209C2semidirect
2. C2×#41×D79C2semidirect
3. #19099235C2semidirect
4. #19099234C2semidirect
5. #19099232C2semidirect
6. #19099231C2semidirect
7. #19099224C2semidirect
8. #19099223C2semidirect
9. #19099221C2semidirect
10. #19099131C2semidirect
11. #19099129C2semidirect
12. #19099076C2semidirect
13. #19099072C2semidirect
14. #19099039C2semidirect
15. #548×C79C2semidirect
19. C2×#11929625C22semidirect
22. C2×#11929624C22semidirect
26. C2×Q8×D79C22semidirect
29. #14579158C22semidirect
30. #14579157C22semidirect
31. #14579157C22semidirect
32. #14579150C22semidirect
33. #14579147C22semidirect
36. #14579139C22semidirect
37. #14579138C22semidirect
38. #14579137C22semidirect
39. #14579136C22semidirect
40. #14579136C22semidirect
41. #14579135C22semidirect
42. #14579135C22semidirect
43. #14579134C22semidirect
44. #14579134C22semidirect
45. #14579134C22semidirect
46. #14579133C22semidirect
47. #14579132C22semidirect
51. Q8×Dic79C22semidirect
52. C4×Dic158C22semidirect
53. #14579126C22semidirect
54. C4×D316C22semidirect
55. #14579123C22semidirect
56. C4×C22⋊D79C22semidirect
57. C4×C22⋊D79C22semidirect
58. (C22⋊D79)⋊C4C22semidirect
62. C22×Q8×C79C22semidirect
63. #124×C79C22semidirect
64. C22⋊Q8×C79C22semidirect
65. #121×C79C22semidirect
66. C4×Q8×C79C22semidirect
67. C4×D4×C79C22semidirect
79. Q8×D79D4semidirect
211. C79#3487semidirect
221. #548D79semidirect
239. C22×Q8C2×D79semidirect
247. #124C2×D79semidirect
253. C22⋊Q8C2×D79semidirect
263. #121C2×D79semidirect
270. C4×Q8C2×D79semidirect
276. C4×D4C2×D79semidirect