Finite Group Databasebeta

← #24918550

Subgroups — #24918550

Order 10624.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128320C211yesmaximal, characteristic, complemented
2#19128269C211yesmaximal, characteristic, complemented
3#19128091C211yesmaximal, characteristic, complemented
4#19128060C211yesmaximal, characteristic, complemented
5#19128048C211yesmaximal, characteristic, complemented
6#19127925C211yesmaximal, characteristic, complemented
7#19127918C211yesmaximal, characteristic, complemented
8#19127899C211yesmaximal, characteristic, complemented
9#19127894C211yesmaximal, characteristic, complemented
10#19127862C211yesmaximal, characteristic, complemented
11#19127850C211yesmaximal, characteristic, complemented
12#19127845C211yesmaximal, characteristic, complemented
13#19127833C211yesmaximal, characteristic, complemented
14#19127760C211yesmaximal, characteristic, complemented
15#495×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic
17C2×D4×D83C2211yescharacteristic
18C2×#13050577C2211yescharacteristic
19C2×#13050577C2211yescharacteristic, complemented
20C2×#13050575C2211yescharacteristic
21C2×#13050575C2211yescharacteristic, complemented
22#14581792C2211yescharacteristic, complemented
23#14581789C2211yescharacteristic
24#14581788C2211yescharacteristic
25#14581785—21no—
26#14581781C2211yescharacteristic
27#14581778—21no—
28#14581776—21no—
29#14581769—21no—
30#14581768C2211yescharacteristic
31#14581765—21no—
32#14581765—21no—
33#14581763C2211yescharacteristic
34#14581762—21no—
35#14581762—21no—
36Q8×Dic83C2211yescharacteristic, complemented
37D4×Dic83C2211yescharacteristic, complemented
38#14581749C2211yescharacteristic
39Dic83⋊C8C2211yescharacteristic, complemented
40Dic83⋊C8C2211yescharacteristic, complemented
41#14581716C2211yescharacteristic, complemented
42#14581716C2211yescharacteristic, complemented
43#14581714C2211yescharacteristic
44#14581714C2211yescharacteristic
45#14581714C2211yescharacteristic, complemented
46#14581714C2211yescharacteristic, complemented
47#14581713C2211yescharacteristic
48#14581713C2211yescharacteristic
49(C2×D83)⋊C8C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51#14581703C2211yescharacteristic, complemented
52C2×Q16×C83C2211yescharacteristic, complemented
53C2×SD16×C83C2211yescharacteristic
54C2×SD16×C83C2211yescharacteristic, complemented
55C2×D8×C83C2211yescharacteristic
56#124×C83C2211yescharacteristic
57#124×C83C2211yescharacteristic, complemented
58C4×C8×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60#13050591—41no—
61#13050591—41no—
62C2×Dic166—21no—
63C2×D332—21no—
64C2×D332—21no—
65C2×D332—21no—
66C2×D332C2311yescharacteristic
67C2×D332C2311yescharacteristic
68C2×C22⋊D83—21no—
69C2×C22⋊D83—21no—
70C2×C22⋊D83—21no—
71C2×C22⋊D83—21no—
72D4×D83—41no—
73D4×D83—41no—
74C22×Dic83—21no—
75C2×C4×D83—21no—
76C2×C4×D83C2311yescharacteristic
77#13050577—41no—
78#13050577—41no—
79#13050575—41no—
80#13050575—41no—
81C2×C83⋊C8C2311yescharacteristic
82C2×C83⋊C8C2311yescharacteristic
83C4⋊Dic83—21no—
84C4⋊Dic83—21no—
85C4⋊Dic83C2311yescharacteristic
86C4⋊Dic83C2311yescharacteristic
87Dic83⋊C4—21no—
88Dic83⋊C4D411yescharacteristic
89Dic83⋊C4D411yescharacteristic
90C22⋊Dic83—21no—
91C22⋊Dic83—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4D411yescharacteristic
96(C2×D83)⋊C4D411yescharacteristic
97C4×Dic83—21no—
98C4×Dic83C2311yescharacteristic
99C2×Q8×C83C2311yescharacteristic
100C2×Q8×C83C2311yescharacteristic
101C2×D4×C83C2311yescharacteristic
102C2×D4×C83C2311yescharacteristic
103Q16×C83—41no—
104SD16×C83—41no—
105SD16×C83—41no—
106D8×C83—41no—
107C2×C8×C83C2311yescharacteristic
108C2×C8×C83C2311yescharacteristic
109C22⋊C4×C83—21no—
110C22⋊C4×C83—21no—
111C22⋊C4×C83—21no—
112C22⋊C4×C83—21no—
113C42×C83C2311yescharacteristic
114C23×C83—21no—
115C23×C83—21no—
116C22×D83—21no—
117C22×D83—21no—
118C22×D83—42no—
119C22×D83—81no—
120C22×D83—81no—
121C22×D83C2×D411yescharacteristic
122Q8×C83—21no—
123Q8×C83—21no—
124Q8×C83—21no—
125Q8×C83—21no—
126D4×C83—21no—
127D4×C83—21no—
128D4×C83—21no—
129D4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83C2×D411yescharacteristic
133C2×C4×C83C2×D411yescharacteristic
134C2×C4×C83C2411yescharacteristic
135C22⋊D83—42no—
136C22⋊D83—81no—
137C22⋊D83—81no—
138C22⋊D83—81no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83—81no—
142C2×Dic83C2×D411yescharacteristic
143C2×Dic83C2×D411yescharacteristic
144C2×Dic83C2×D411yescharacteristic
145D332—21no—
146D332—21no—
147D332—21no—
148D332—21no—
149D332—42no—
150D332—81no—
151D332—81no—
152C4×D83—21no—
153C4×D83—21no—
154C4×D83—42no—
155Dic166—81no—
156C8×C83—21no—
157C8×C83—21no—
158C83⋊C8—21no—
159C83⋊C8—21no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83C22×D411yescharacteristic
167C2×D83—21no—
168C2×D83—21no—
169C2×D83—21no—
170C2×D83—41no—
171C2×D83—41no—
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—41no—
175C2×D83—41no—
176C2×D83—84no—
177C4×C83—21no—
178C4×C83—21no—
179C4×C83—41no—
180C4×C83—41no—
181C4×C83H(5,𝔽2)11yescharacteristic
182C4×C83H(5,𝔽2)11yescharacteristic
183Dic83—21no—
184Dic83—21no—
185Dic83—21no—
186Dic83—41no—
187Dic83—41no—
188C2×C83—81no—
189C2×C83—81no—
190C2×C83#53511yescharacteristic
191C2×C83#57611yescharacteristic
192C2×C83#57711yescharacteristic
193D83—41no—
194D83—81no—
195D83—81no—
196#3224—831nomaximal, Sylow
197C83#322411yescharacteristic, complemented, Sylow
198#542—831no—
199#535—831no—
200#496—831no—
201#495D8311yescharacteristic, complemented
202#489—831no—
203#486—831no—
204#464—831no—
205#463—831no—
206#460—831no—
207#459—831no—
208#452—831no—
209#450—831no—
210#449—831no—
211#447—831no—
212#433—831no—
213C2×#41—831no—
214C22×D4—831no—
215C2×Q16C2×D8311yescharacteristic, complemented
216C2×SD16—831no—
217C2×SD16—831no—
218C2×SD16C2×D8311yescharacteristic
219C2×SD16C2×D8311yescharacteristic, complemented
220C2×D8—831no—
221C2×D8—831no—
222C2×D8C2×D8311yescharacteristic
223C4⋊Q8—831no—
224#127—831no—
225#127—831no—
226#126—1661no—
227#124—831no—
228#124—1661no—
229#124C2×D8311yescharacteristic
230#124C2×D8311yescharacteristic, complemented
231#123—1661no—
232C22⋊Q8—831no—
233C22⋊Q8—1661no—
234#121—831no—
235#121—1661no—
236#121—1661no—
237#120—1661no—
238#120—1661no—
239C4×Q8—831no—
240C4×D4—831no—
241#117—831no—
242C4⋊C8—831no—
243C4⋊C8—831no—
244#103—831no—
245#103—831no—
246#102—831no—
247#102—831no—
248#102—831no—
249#102—831no—
250#102—831no—
251#102—831no—
252C22⋊C8—831no—
253C22⋊C8—831no—
254#97—831no—
255C4×C8C2×D8311yescharacteristic, complemented
256C24—1661no—
257#41—3321no—
258#41—3321no—
259C2×Q8—1661no—
260C2×Q8C22×D8311yescharacteristic
261C2×Q8C22×D8311yescharacteristic
262C2×D4—3321no—
263C2×D4—3321no—
264C2×D4—831no—
265C2×D4—831no—
266C2×D4—1661no—
267C2×D4—1661no—
268C2×D4—1661no—
269C2×D4—1661no—
270C2×D4—1661no—
271C2×D4—1661no—
272C2×D4—1661no—
273C2×D4C22×D8311yescharacteristic
274C2×D4C22×D8311yescharacteristic
275C22×C4—831no—
276C22×C4—1661no—
277C22×C4—1661no—
278Q16—41no—
279SD16—41no—
280SD16—41no—
281SD16—3321no—
282SD16—3321no—
283D8—41no—
284D8—3321no—
285D8—3321no—
286C2×C8—831no—
287C2×C8—831no—
288C2×C8C22×D8311yescharacteristic
289C2×C8C22×D8311yescharacteristic
290C4⋊C4—831no—
291C4⋊C4—831no—
292C4⋊C4—831no—
293C4⋊C4—831no—
294C4⋊C4—1661no—
295C4⋊C4—1661no—
296C4⋊C4—1661no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—831no—
302C22⋊C4—831no—
303C22⋊C4—1661no—
304C22⋊C4—1661no—
305C22⋊C4—1661no—
306C22⋊C4—1661no—
307C22⋊C4—1661no—
308C42—831no—
309C42—1661no—
310C42C22×D8311yescharacteristic
311C23—21no—
312C23—21no—
313C23—3322no—
314C23—831no—
315C23—6641no—
316C23—6641no—
317C23—1661no—
318C23—1661no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—6641no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—3322no—
329D4—3322no—
330D4—6641no—
331D4—6641no—
332D4—6641no—
333D4—6641no—
334D4—6641no—
335D4—1661no—
336D4—1661no—
337D4—1661no—
338D4—1661no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—3322no—
342C2×C4—831no—
343C2×C4—831no—
344C2×C4—831no—
345C2×C4—6641no—
346C2×C4—1661no—
347C2×C4—1661no—
348C2×C4—1661no—
349C2×C4—1661no—
350C2×C4D4×D8311yescharacteristic
351C2×C4D4×D8311yescharacteristic
352C2×C4C23×D8311yescharacteristic
353C8—21no—
354C8—21no—
355C8—1661no—
356C8—1661no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—3321no—
364C22—3321no—
365C22—3321no—
366C22—3321no—
367C22—3321no—
368C22—3321no—
369C22—6644no—
370C22—1661no—
371C22—1661no—
372C22—1661no—
373C22C2×D4×D8311yescharacteristic
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—3321no—
379C4—3321no—
380C4—1661no—
381C4—1661no—
382C4—1661no—
383C4#1458183711yescharacteristic
384C4#1458183711yescharacteristic
385C2—81no—
386C2—81no—
387C2—3321no—
388C2—6641no—
389C2—6641no—
390C2#1912826411yescharacteristic
391C2#1912850311yescharacteristic
392C2#1912850811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#495×C83C2
socleC22×C83C22×D4
hypercenter#495D83
hypocenterC83#3224
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #19128320C2semidirect
2. #19128269C2semidirect
3. #19128091C2semidirect
4. #19128060C2semidirect
5. #19128048C2semidirect
6. #19127925C2semidirect
7. #19127918C2semidirect
8. #19127899C2semidirect
9. #19127894C2semidirect
10. #19127862C2semidirect
11. #19127850C2semidirect
12. #19127845C2semidirect
13. #19127833C2semidirect
14. #19127760C2semidirect
15. #495×C83C2semidirect
19. C2×#13050577C22semidirect
21. C2×#13050575C22semidirect
22. #14581792C22semidirect
36. Q8×Dic83C22semidirect
37. D4×Dic83C22semidirect
39. Dic83⋊C8C22semidirect
40. Dic83⋊C8C22semidirect
41. #14581716C22semidirect
42. #14581716C22semidirect
45. #14581714C22semidirect
46. #14581714C22semidirect
51. #14581703C22semidirect
52. C2×Q16×C83C22semidirect
54. C2×SD16×C83C22semidirect
57. #124×C83C22semidirect
58. C4×C8×C83C22semidirect
197. C83#3224semidirect
201. #495D83semidirect
215. C2×Q16C2×D83semidirect
219. C2×SD16C2×D83semidirect
230. #124C2×D83semidirect
255. C4×C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283202
#191282692
#191280912
#191280602
#191280482
#191279252
#191279182
#191278992
#191278942
#191278622
#191278502
#191278452
#191278332
#191277602
#495×C832
#322483