Finite Group Databasebeta

← #24918465

Subgroups — #24918465

Order 10624.

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

normal subgroups only

This group has 400 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#19128253C211yesmaximal, characteristic, complemented
2C2×#14581781C211yesmaximal, characteristic
3#19128094C212yesmaximal, complemented
4#19128062C212yesmaximal, complemented
5#19127926C212yesmaximal, complemented
6#19127925C212yesmaximal, complemented
7#19127857C211yesmaximal, characteristic, complemented
8#19127854C211yesmaximal, characteristic
9#19127851C211yesmaximal, characteristic
10#19127755C211yesmaximal, characteristic, complemented
11#451×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C2×#13050580C2212yescomplemented
14C2×#13050579C2212yes—
15C2×#13050577C2212yes—
16#14581791C2211yescharacteristic, complemented
17#14581790C2211yescharacteristic, complemented
18#14581782—21no—
19#14581781—21no—
20#14581781—21no—
21#14581781C2212yes—
22#14581781C2212yes—
23#14581771C2211yescharacteristic
24#14581769—21no—
25Q8×Dic83C2211yescharacteristic, complemented
26#14581757—21no—
27#14581757C2211yescharacteristic, complemented
28#14581748—21no—
29C2×(C2×D83)⋊C4—21no—
30C2×(C2×D83)⋊C4—21no—
31C2×C4×Dic83C2211yescharacteristic
32Dic83⋊C8C2212yescomplemented
33#14581716C2212yescomplemented
34#14581714C2212yes—
35#14581712C2212yes—
36C22⋊(C83⋊C8)C2211yescharacteristic
37#14581702C2212yescomplemented
38C22×Q8×C83C2211yescharacteristic
39C2×Q16×C83C2212yescomplemented
40C22⋊Q8×C83C2211yescharacteristic
41#103×C83C2212yescomplemented
42C22⋊C8×C83C2211yescharacteristic
43C23×D83—21no—
44C2×Dic166—21no—
45C2×Dic166C2311yescharacteristic
46C2×D332—21no—
47C2×D332—42no—
48C2×D332C2312yes—
49C22×Dic83D411yescharacteristic
50C22×Dic83D411yescharacteristic
51#13050580—42no—
52#13050579—42no—
53#13050577—42no—
54C2×C83⋊C8C2312yes—
55C4⋊Dic83—21no—
56C4⋊Dic83C2311yescharacteristic
57Dic83⋊C4—21no—
58Dic83⋊C4—21no—
59Dic83⋊C4—21no—
60Dic83⋊C4—21no—
61Dic83⋊C4—21no—
62C22⋊Dic83—21no—
63(C2×D83)⋊C4—22no—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4—22no—
66(C2×D83)⋊C4—22no—
67C4×Dic83—21no—
68C4×Dic83—21no—
69C4×Dic83D411yescharacteristic
70C4×Dic83D411yescharacteristic
71C4×Dic83C2311yescharacteristic
72C4×Dic83C2311yescharacteristic
73C2×Q8×C83—21no—
74C2×Q8×C83—42no—
75C2×Q8×C83C2312yes—
76C2×Q8×C83C2311yescharacteristic
77C22×C4×C83—21no—
78C22×C4×C83C2311yescharacteristic
79Q16×C83—42no—
80C2×C8×C83C2312yes—
81C4⋊C4×C83—21no—
82C4⋊C4×C83C2311yescharacteristic
83C22⋊C4×C83—21no—
84C23×C83C2×D411yescharacteristic
85C22×D83—22no—
86C22×D83—42no—
87C22×D83—42no—
88C22×D83—42no—
89Q8×C83—22no—
90Q8×C83—22no—
91Q8×C83—22no—
92Q8×C83—42no—
93C2×C4×C83—22no—
94C2×C4×C83—21no—
95C2×C4×C83—21no—
96C2×C4×C83—22no—
97C2×C4×C83—42no—
98C2×C4×C83C2×D411yescharacteristic
99C2×C4×C83C2411yescharacteristic
100C2×Dic83—22no—
101C2×Dic83—21no—
102C2×Dic83—21no—
103C2×Dic83—41no—
104C2×Dic83C2×D411yescharacteristic
105C2×Dic83C2×D411yescharacteristic
106C2×Dic83C2×D411yescharacteristic
107C2×Dic83C2×D411yescharacteristic
108D332—22no—
109D332—22no—
110D332—42no—
111Dic166—22no—
112Dic166—42no—
113C8×C83—22no—
114C83⋊C8—22no—
115C22×C83—21no—
116C22×C83—21no—
117C22×C83#13712yes—
118C22×C83C22×D411yescharacteristic
119C2×D83—42no—
120C2×D83—42no—
121C2×D83—42no—
122C2×D83—84no—
123C4×C83—21no—
124C4×C83—42no—
125C4×C83—41no—
126C4×C83—41no—
127C4×C83H(5,𝔽2)12yes—
128Dic83—21no—
129Dic83—21no—
130Dic83—21no—
131Dic83—21no—
132Dic83—41no—
133Dic83—41no—
134C2×C83—22no—
135C2×C83#53511yescharacteristic
136C2×C83C2×#13711yescharacteristic
137C2×C83#57511yescharacteristic
138D83—82no—
139#3215—831nomaximal, Sylow
140C83#321511yescharacteristic, complemented, Sylow
141#533—831no—
142C2×#124—831no—
143#497—832no—
144#489—832no—
145#464—832no—
146#464—832no—
147#451—831no—
148#451D8311yescharacteristic, complemented
149#450—831no—
150#450—831no—
151#432—831no—
152C22×Q8C2×D8311yescharacteristic
153C22×D4—831no—
154C2×Q16—832no—
155C2×Q16C2×D8312yescomplemented
156C2×SD16—832no—
157C2×SD16—832no—
158C4⋊Q8—831no—
159C4⋊Q8—831no—
160#125—1661no—
161#124—832no—
162#124—832no—
163#124—1661no—
164#124—1661no—
165C22⋊Q8—831no—
166C22⋊Q8—1661no—
167C22⋊Q8C2×D8311yescharacteristic
168C4×Q8—831no—
169C4×Q8—831no—
170C4×Q8—1661no—
171#117—1661no—
172C2×C22⋊C4—1661no—
173C2×C22⋊C4—1661no—
174C2×C42—831no—
175C4⋊C8—832no—
176#103—832no—
177#103C2×D8312yescomplemented
178#102—832no—
179#102—832no—
180C22⋊C8—831no—
181C22⋊C8C2×D8311yescharacteristic
182#97—832no—
183C24—1661no—
184C2×Q8—21no—
185C2×Q8—42no—
186C2×Q8—831no—
187C2×Q8—1661no—
188C2×Q8C22×D8312yes—
189C2×Q8C22×D8311yescharacteristic
190C2×D4—3322no—
191C2×D4—832no—
192C2×D4—1661no—
193C22×C4—21no—
194C22×C4—831no—
195C22×C4—831no—
196C22×C4C22×D8311yescharacteristic
197Q16—42no—
198Q16—3322no—
199SD16—3322no—
200SD16—3322no—
201C2×C8—832no—
202C2×C8C22×D8312yes—
203C4⋊C4—21no—
204C4⋊C4—831no—
205C4⋊C4—1661no—
206C4⋊C4—1661no—
207C4⋊C4—1661no—
208C4⋊C4—1661no—
209C4⋊C4—1661no—
210C4⋊C4—1661no—
211C4⋊C4C22×D8311yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—1662no—
214C22⋊C4—1662no—
215C22⋊C4—1662no—
216C22⋊C4—1662no—
217C22⋊C4—1661no—
218C42—831no—
219C42—831no—
220C42—831no—
221C42—831no—
222C42—1661no—
223C42—1661no—
224C23—3322no—
225C23—3322no—
226C23—3322no—
227C23—1662no—
228C23D4×D8311yescharacteristic
229Q8—22no—
230Q8—22no—
231Q8—22no—
232Q8—42no—
233Q8—3322no—
234Q8—1662no—
235D4—3322no—
236D4—1662no—
237D4—1662no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—3321no—
244C2×C4—831no—
245C2×C4—831no—
246C2×C4—831no—
247C2×C4—831no—
248C2×C4—1661no—
249C2×C4—1661no—
250C2×C4—1662no—
251C2×C4D4×D8311yescharacteristic
252C2×C4C23×D8311yescharacteristic
253C8—22no—
254C8—1662no—
255C22—21no—
256C22—21no—
257C22—3322no—
258C22—3322no—
259C22—3322no—
260C22—6644no—
261C22#1458182412yes—
262C22C2×D4×D8311yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—3321no—
268C4—3321no—
269C4—1661no—
270C4—1661no—
271C4—1661no—
272C4—1661no—
273C4#1458183712yes—
274C2—22no—
275C2—6642no—
276C2#1912826611yescharacteristic
277C2C2×#1458182411yescharacteristic
278C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#451×C83C2
socleC22×C83C22×D4
prime radicalC22×D332C22
hypercenter#451D83
hypocenterC83#3215
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #19128253C2semidirect
3. #19128094C2semidirect
4. #19128062C2semidirect
5. #19127926C2semidirect
6. #19127925C2semidirect
7. #19127857C2semidirect
10. #19127755C2semidirect
11. #451×C83C2semidirect
13. C2×#13050580C22semidirect
16. #14581791C22semidirect
17. #14581790C22semidirect
25. Q8×Dic83C22semidirect
27. #14581757C22semidirect
32. Dic83⋊C8C22semidirect
33. #14581716C22semidirect
37. #14581702C22semidirect
39. C2×Q16×C83C22semidirect
41. #103×C83C22semidirect
140. C83#3215semidirect
148. #451D83semidirect
155. C2×Q16C2×D83semidirect
177. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282532
C2×#145817812
#191280942
#191280622
#191279262
#191279252
#191278572
#191278542
#191278512
#191277552
#451×C832
#321583