Finite Group Databasebeta

← #24918306

Subgroups — #24918306

Order 10624.

418 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 109 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#19128214C212yesmaximal, complemented
2#19128040C212yesmaximal, complemented
3#19128027C212yesmaximal, complemented
4#19127954C211yesmaximal, characteristic, complemented
5#19127949C212yesmaximal, complemented
6#19127862C212yesmaximal, complemented
7#19127849C212yesmaximal, complemented
8#19127664C211yesmaximal, characteristic, complemented
9#470×C83C211yesmaximal, characteristic, complemented
10C2×#13050591C2212yes—
11C22×C4×D83C2211yescharacteristic
12C2×#13050577C2212yescomplemented
13C2×D664C2211yescharacteristic, complemented
14C2×#13050574C2211yescharacteristic
15#14581777—22no—
16#14581768C2212yes—
17#14581767C2212yes—
18#14581766C2212yes—
19#14581762—22no—
20#14581754—22no—
21(C22⋊D83)⋊C4—22no—
22#14581750C2212yes—
23#14581725C2212yescomplemented
24#14581724C2211yescharacteristic, complemented
25#14581716C2212yescomplemented
26#14581714C2212yescomplemented
27#14581712C2212yescomplemented
28(C2×D83)⋊C8C2212yes—
29(C2×D83)⋊C8C2212yes—
30C2×Q16×C83C2211yescharacteristic, complemented
31C2×M16×C83C2211yescharacteristic
32C22⋊Q8×C83C2212yes—
33#106×C83C2211yescharacteristic, complemented
34#103×C83C2212yescomplemented
35C23×D83D411yescharacteristic
36#13050591—42no—
37#13050591—42no—
38C2×D332—22no—
39C2×D332—22no—
40C2×D332C2312yes—
41C2×C22⋊D83—22no—
42C22×Dic83D411yescharacteristic
43C2×C4×D83—22no—
44C2×C4×D83—41no—
45C2×C4×D83—41no—
46C2×C4×D83D412yes—
47C2×C4×D83C2312yes—
48#13050577—42no—
49D664—41no—
50#13050574—41no—
51C2×C83⋊C8C2312yes—
52C4⋊Dic83C2312yes—
53Dic83⋊C4—22no—
54(C2×D83)⋊C4—22no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—22no—
58C4×Dic83—22no—
59C2×Q8×C83C2312yes—
60C22×C4×C83C2311yescharacteristic
61Q16×C83—41no—
62M16×C83—41no—
63C2×C8×C83C2311yescharacteristic
64C2×C8×C83C2311yescharacteristic
65C4⋊C4×C83—22no—
66C4⋊C4×C83C2312yes—
67C22⋊C4×C83—22no—
68C23×C83C2×D411yescharacteristic
69C22×D83—22no—
70C22×D83—41no—
71C22×D83—42no—
72C22×D83C2×D412yes—
73Q8×C83—22no—
74Q8×C83—22no—
75C2×C4×C83—21no—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2411yescharacteristic
81C22⋊D83—44no—
82C2×Dic83—22no—
83C2×Dic83—41no—
84C2×Dic83C2×D412yes—
85D332—22no—
86D332—22no—
87D332—44no—
88D332—44no—
89C4×D83—22no—
90C4×D83—22no—
91C4×D83—42no—
92C4×D83—44no—
93C8×C83—21no—
94C8×C83—21no—
95C83⋊C8—22no—
96C22×C83—21no—
97C22×C83—22no—
98C22×C83C22×D411yescharacteristic
99C2×D83—22no—
100C2×D83—22no—
101C2×D83—22no—
102C2×D83—44no—
103C2×D83—42no—
104C2×D83—42no—
105C2×D83—42no—
106C4×C83—21no—
107C4×C83—42no—
108C4×C83—42no—
109C4×C83H(5,𝔽2)11yescharacteristic
110C4×C83H(5,𝔽2)11yescharacteristic
111Dic83—22no—
112Dic83—42no—
113C2×C83—41no—
114C2×C83#53411yescharacteristic
115C2×C83#57512yes—
116D83—42no—
117D83—82no—
118#3197—831nomaximal, Sylow
119C83#319711yescharacteristic, complemented, Sylow
120#525—832no—
121#485—832no—
122#482—832no—
123#470D8311yescharacteristic, complemented
124#469—831no—
125#468—832no—
126#452—832no—
127#449—832no—
128#407—831no—
129C2×#41—832no—
130C23×C4—831no—
131C2×Q16C2×D8311yescharacteristic, complemented
132C2×SD16—832no—
133C2×D8—831no—
134C2×M16—831no—
135C2×M16C2×D8311yescharacteristic
136#123—1662no—
137C22⋊Q8—832no—
138C22⋊Q8C2×D8312yes—
139#121—832no—
140#121—832no—
141#120—1662no—
142C4×D4—1662no—
143C4×D4—1662no—
144#117—832no—
145#107—832no—
146#106—831no—
147#106C2×D8311yescharacteristic, complemented
148#103—832no—
149#103C2×D8312yescomplemented
150#102—832no—
151#102—832no—
152C22⋊C8—832no—
153C22⋊C8—832no—
154C24—831no—
155#41—3322no—
156#41—3322no—
157C2×Q8C22×D8312yes—
158C2×D4—832no—
159C2×D4—1662no—
160C2×D4—1662no—
161C2×D4—1662no—
162C22×C4—3321no—
163C22×C4—3321no—
164C22×C4—831no—
165C22×C4—832no—
166C22×C4—832no—
167C22×C4—1662no—
168C22×C4C22×D8311yescharacteristic
169Q16—41no—
170SD16—3322no—
171D8—3321no—
172M16—41no—
173M16—3321no—
174C2×C8—832no—
175C2×C8C22×D8311yescharacteristic
176C2×C8C22×D8311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—832no—
179C4⋊C4—1662no—
180C4⋊C4C22×D8312yes—
181C22⋊C4—22no—
182C22⋊C4—1662no—
183C22⋊C4—1662no—
184C22⋊C4—1662no—
185C22⋊C4—1662no—
186C42—1662no—
187C23—3321no—
188C23—3322no—
189C23—832no—
190C23—1662no—
191C23D4×D8311yescharacteristic
192Q8—22no—
193Q8—22no—
194D4—3324no—
195D4—3324no—
196D4—3324no—
197D4—1662no—
198D4—1662no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—3322no—
204C2×C4—3324no—
205C2×C4—3321no—
206C2×C4—832no—
207C2×C4—1662no—
208C2×C4—1662no—
209C2×C4—1662no—
210C2×C4D4×D8311yescharacteristic
211C2×C4C23×D8311yescharacteristic
212C8—21no—
213C8—21no—
214C8—1662no—
215C22—22no—
216C22—21no—
217C22—3324no—
218C22—3322no—
219C22—3322no—
220C22—3322no—
221C22—1662no—
222C22—1662no—
223C22—1662no—
224C22C2×D4×D8311yescharacteristic
225C4—21no—
226C4—42no—
227C4—42no—
228C4—3322no—
229C4—1662no—
230C4#1458183711yescharacteristic
231C4#1458183811yescharacteristic
232C2—41no—
233C2—3322no—
234C2—6642no—
235C2#1912825911yescharacteristic
236C2#1912850012yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#470×C83C2
socleC22×C83C22×D4
hypercenter#470D83
hypocenterC83#3197
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #19128214C2semidirect
2. #19128040C2semidirect
3. #19128027C2semidirect
4. #19127954C2semidirect
5. #19127949C2semidirect
6. #19127862C2semidirect
7. #19127849C2semidirect
8. #19127664C2semidirect
9. #470×C83C2semidirect
12. C2×#13050577C22semidirect
13. C2×D664C22semidirect
23. #14581725C22semidirect
24. #14581724C22semidirect
25. #14581716C22semidirect
26. #14581714C22semidirect
27. #14581712C22semidirect
30. C2×Q16×C83C22semidirect
33. #106×C83C22semidirect
34. #103×C83C22semidirect
119. C83#3197semidirect
123. #470D83semidirect
131. C2×Q16C2×D83semidirect
147. #106C2×D83semidirect
149. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282142
#191280402
#191280272
#191279542
#191279492
#191278622
#191278492
#191276642
#470×C832
#319783