Finite Group Databasebeta

← #24918431

Subgroups — #24918431

Order 10624.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128237C211yesmaximal, characteristic, complemented
2C2×#14581779C211yesmaximal, characteristic, complemented
3#19128082C212yesmaximal, complemented
4#19128054C212yesmaximal, complemented
5#19127921C212yesmaximal, complemented
6#19127915C212yesmaximal, complemented
7#19127856C211yesmaximal, characteristic, complemented
8#19127842C211yesmaximal, characteristic, complemented
9#19127835C211yesmaximal, characteristic
10#19127751C211yesmaximal, characteristic, complemented
11#450×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C2×#13050580C2212yescomplemented
14C2×#13050579C2212yescomplemented
15C2×#13050575C2212yes—
16#14581787—21no—
17#14581783C2211yescharacteristic, complemented
18#14581781C2211yescharacteristic, complemented
19#14581779—21no—
20#14581779—21no—
21#14581779C2212yes—
22#14581779C2212yes—
23#14581776—21no—
24#14581766C2211yescharacteristic
25Q8×Dic83C2211yescharacteristic, complemented
26#14581754C2211yescharacteristic, complemented
27(C22⋊D83)⋊C4—21no—
28#14581750—21no—
29C2×C22⋊Dic83—21no—
30C2×C22⋊Dic83—21no—
31C2×C4×Dic83C2211yescharacteristic
32Dic83⋊C8C2212yescomplemented
33#14581717C2212yescomplemented
34#14581715C2212yescomplemented
35#14581713C2212yes—
36C22⋊(C83⋊C8)C2211yescharacteristic
37C8×Dic83C2212yescomplemented
38C22×D4×C83C2211yescharacteristic
39C2×SD16×C83C2212yes—
40C22⋊Q8×C83C2211yescharacteristic
41#102×C83C2212yes—
42C22⋊C8×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×Dic166—42no—
45C2×Dic166C2312yes—
46C2×D332C2311yescharacteristic
47C2×C22⋊D83—21no—
48C22×Dic83—21no—
49C22×Dic83D411yescharacteristic
50C22×Dic83D411yescharacteristic
51C2×C4×D83—21no—
52#13050580—42no—
53#13050579—42no—
54#13050575—42no—
55C2×C83⋊C8C2312yes—
56C4⋊Dic83—21no—
57C4⋊Dic83—21no—
58C4⋊Dic83C2311yescharacteristic
59Dic83⋊C4—21no—
60C22⋊Dic83—22no—
61C22⋊Dic83—22no—
62C22⋊Dic83—22no—
63C22⋊Dic83—22no—
64(C2×D83)⋊C4—21no—
65(C2×D83)⋊C4—21no—
66(C2×D83)⋊C4—21no—
67(C2×D83)⋊C4—21no—
68C4×Dic83—21no—
69C4×Dic83D411yescharacteristic
70C4×Dic83D411yescharacteristic
71C4×Dic83C2311yescharacteristic
72C4×Dic83C2311yescharacteristic
73C24×C83—21no—
74C2×Q8×C83C2311yescharacteristic
75C2×D4×C83—21no—
76C2×D4×C83—42no—
77C2×D4×C83C2312yes—
78C22×C4×C83C2311yescharacteristic
79SD16×C83—42no—
80C2×C8×C83C2312yes—
81C4⋊C4×C83—21no—
82C4⋊C4×C83C2311yescharacteristic
83C22⋊C4×C83—21no—
84C23×C83—22no—
85C23×C83—42no—
86C23×C83—42no—
87C23×C83—42no—
88C23×C83C2×D411yescharacteristic
89C22×D83—21no—
90Q8×C83—22no—
91D4×C83—22no—
92D4×C83—22no—
93D4×C83—42no—
94C2×C4×C83—22no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2411yescharacteristic
99C22⋊D83—42no—
100C2×Dic83—22no—
101C2×Dic83—21no—
102C2×Dic83—22no—
103C2×Dic83—41no—
104C2×Dic83—42no—
105C2×Dic83C2×D411yescharacteristic
106C2×Dic83C2×D411yescharacteristic
107C2×Dic83C2×D411yescharacteristic
108C2×Dic83C2×D411yescharacteristic
109D332—22no—
110C4×D83—42no—
111Dic166—22no—
112Dic166—22no—
113Dic166—42no—
114C8×C83—22no—
115C83⋊C8—22no—
116C22×C83—21no—
117C22×C83—21no—
118C22×C83—42no—
119C22×C83—42no—
120C22×C83—42no—
121C22×C83—84no—
122C22×C83#13512yes—
123C22×C83C22×D411yescharacteristic
124C2×D83—41no—
125C2×D83—41no—
126C2×D83—41no—
127C4×C83—21no—
128C4×C83—41no—
129C4×C83—41no—
130C4×C83H(5,𝔽2)12yes—
131Dic83—21no—
132Dic83—21no—
133Dic83—21no—
134Dic83—21no—
135Dic83—42no—
136Dic83—41no—
137C2×C83—22no—
138C2×C83—82no—
139C2×C83#53511yescharacteristic
140C2×C83C2×#13511yescharacteristic
141C2×C83#57511yescharacteristic
142D83—81no—
143#3211—831nomaximal, Sylow
144C83#321111yescharacteristic, complemented, Sylow
145#529—831no—
146C2×#124—831no—
147#495—832no—
148#488—832no—
149#464—832no—
150#463—832no—
151#451—831no—
152#450D8311yescharacteristic, complemented
153#448—831no—
154#447—831no—
155#431—831no—
156C22×Q8—831no—
157C22×D4C2×D8311yescharacteristic
158C2×Q16—832no—
159C2×SD16—832no—
160C2×SD16C2×D8312yes—
161C2×D8—832no—
162#126—1661no—
163#125—831no—
164#124—831no—
165#124—832no—
166#124—832no—
167#124—1661no—
168#124—1661no—
169#123—1661no—
170C22⋊Q8C2×D8311yescharacteristic
171#121—831no—
172C4×Q8—831no—
173C4×D4—831no—
174C4×D4—1661no—
175#117—1661no—
176C2×C22⋊C4—1661no—
177C2×C22⋊C4—1661no—
178C2×C42—831no—
179C4⋊C8—832no—
180#103—832no—
181#103—832no—
182#102—832no—
183#102C2×D8312yes—
184C22⋊C8—831no—
185C22⋊C8C2×D8311yescharacteristic
186C4×C8—832no—
187C24—21no—
188C2×Q8—3322no—
189C2×Q8—832no—
190C2×Q8—1661no—
191C2×Q8C22×D8311yescharacteristic
192C2×D4—21no—
193C2×D4—42no—
194C2×D4—831no—
195C2×D4—1661no—
196C2×D4C22×D8312yes—
197C22×C4—831no—
198C22×C4—831no—
199C22×C4—1661no—
200C22×C4—1661no—
201C22×C4C22×D8311yescharacteristic
202Q16—3322no—
203SD16—42no—
204SD16—3322no—
205D8—3322no—
206C2×C8—832no—
207C2×C8C22×D8312yes—
208C4⋊C4—21no—
209C4⋊C4—831no—
210C4⋊C4—1661no—
211C4⋊C4—1661no—
212C4⋊C4—1661no—
213C4⋊C4C22×D8311yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—1661no—
216C22⋊C4—1661no—
217C22⋊C4—1661no—
218C22⋊C4—1662no—
219C22⋊C4—1662no—
220C22⋊C4—1662no—
221C22⋊C4—1662no—
222C22⋊C4—1661no—
223C42—831no—
224C42—831no—
225C42—831no—
226C42—831no—
227C42—1661no—
228C23—22no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—1661no—
233C23D4×D8311yescharacteristic
234Q8—22no—
235Q8—3322no—
236Q8—1662no—
237Q8—1662no—
238D4—22no—
239D4—22no—
240D4—42no—
241D4—3322no—
242D4—1662no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—3322no—
247C2×C4—3322no—
248C2×C4—3321no—
249C2×C4—831no—
250C2×C4—831no—
251C2×C4—831no—
252C2×C4—831no—
253C2×C4—1661no—
254C2×C4—1662no—
255C2×C4—1662no—
256C2×C4D4×D8311yescharacteristic
257C2×C4C23×D8311yescharacteristic
258C8—22no—
259C8—1662no—
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—3321no—
267C22—3321no—
268C22—3321no—
269C22#1458181312yes—
270C22C2×D4×D8311yescharacteristic
271C4—21no—
272C4—41no—
273C4—41no—
274C4—3321no—
275C4—3322no—
276C4—1661no—
277C4—1661no—
278C4—1661no—
279C4—1661no—
280C4#1458183712yes—
281C2—22no—
282C2—82no—
283C2—6641no—
284C2#1912826611yescharacteristic
285C2C2×#1458181311yescharacteristic
286C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#450×C83C2
socleC22×C83C22×D4
prime radical#19127835C2
hypercenter#450D83
hypocenterC83#3211
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #19128237C2semidirect
2. C2×#14581779C2semidirect
3. #19128082C2semidirect
4. #19128054C2semidirect
5. #19127921C2semidirect
6. #19127915C2semidirect
7. #19127856C2semidirect
8. #19127842C2semidirect
10. #19127751C2semidirect
11. #450×C83C2semidirect
13. C2×#13050580C22semidirect
14. C2×#13050579C22semidirect
17. #14581783C22semidirect
18. #14581781C22semidirect
25. Q8×Dic83C22semidirect
26. #14581754C22semidirect
32. Dic83⋊C8C22semidirect
33. #14581717C22semidirect
34. #14581715C22semidirect
37. C8×Dic83C22semidirect
144. C83#3211semidirect
152. #450D83semidirect

Maximal subgroups

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

subgroupindex
#191282372
C2×#145817792
#191280822
#191280542
#191279212
#191279152
#191278562
#191278422
#191278352
#191277512
#450×C832
#321183