Finite Group Databasebeta

← #24918642

Subgroups — #24918642

Order 10624.

332 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 121 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#19128235C211yesmaximal, characteristic
2#19128100C212yesmaximal, complemented
3#19128058C212yesmaximal, complemented
4#19128042C212yesmaximal, complemented
5#19128025C212yesmaximal
6#19127993C211yesmaximal, characteristic, complemented
7#19127988C211yesmaximal, characteristic, complemented
8#19127983C211yesmaximal, characteristic, complemented
9#19127978C211yesmaximal, characteristic, complemented
10#19127754C211yesmaximal, characteristic, complemented
11#533×C83C211yesmaximal, characteristic, complemented
12#14581786—21no—
13#14581784C2211yescharacteristic
14#14581780C2211yescharacteristic
15#14581775—21no—
16#14581771C2211yescharacteristic
17#14581766C2211yescharacteristic
18C4×Dic166C2211yescharacteristic
19C4×D332C2211yescharacteristic
20C4×C22⋊D83—21no—
21#14581749—21no—
22#14581725C2212yes—
23#14581725C2212yes—
24#14581723C2212yes—
25#14581723C2212yes—
26C4⋊(C83⋊C8)C2211yescharacteristic
27C4⋊(C83⋊C8)C2211yescharacteristic
28#14581715C2212yes—
29#14581715C2212yes—
30#14581712C2212yes—
31#14581712C2212yes—
32C22⋊(C83⋊C8)C2212yes—
33C4×C83⋊C8C2212yes—
34C4⋊Q8×C83C2211yescharacteristic
35C4⋊Q8×C83C2211yescharacteristic
36#125×C83C2212yes—
37C22⋊Q8×C83—21no—
38C22⋊Q8×C83—21no—
39C4×Q8×C83—21no—
40C4×Q8×C83—21no—
41#117×C83C2212yes—
42C2×C42×C83C2211yescharacteristic
43C2×Dic166C2311yescharacteristic
44C2×D332C2311yescharacteristic
45C2×C22⋊D83—21no—
46C2×C4×D83—21no—
47C2×C83⋊C8C2312yes—
48C2×C83⋊C8C2312yes—
49C4⋊Dic83C2311yescharacteristic
50C4⋊Dic83C2311yescharacteristic
51Dic83⋊C4—21no—
52Dic83⋊C4—21no—
53Dic83⋊C4—21no—
54Dic83⋊C4—21no—
55C22⋊Dic83—21no—
56(C2×D83)⋊C4—21no—
57(C2×D83)⋊C4—21no—
58(C2×D83)⋊C4—21no—
59(C2×D83)⋊C4—21no—
60C4×Dic83—21no—
61C2×Q8×C83—21no—
62C2×Q8×C83—21no—
63C22×C4×C83D411yescharacteristic
64C22×C4×C83D411yescharacteristic
65C22×C4×C83C2311yescharacteristic
66C4⋊C4×C83—22no—
67C4⋊C4×C83—22no—
68C4⋊C4×C83—21no—
69C4⋊C4×C83—21no—
70C4⋊C4×C83C2312yes—
71C4⋊C4×C83C2312yes—
72C22⋊C4×C83—22no—
73C42×C83—22no—
74C42×C83D411yescharacteristic
75C42×C83D411yescharacteristic
76C42×C83C2311yescharacteristic
77C42×C83C2311yescharacteristic
78C23×C83C2×D411yescharacteristic
79C22×D83—21no—
80Q8×C83—42no—
81Q8×C83—42no—
82C2×C4×C83—22no—
83C2×C4×C83—21no—
84C2×C4×C83—21no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—41no—
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83C2×D411yescharacteristic
92C2×C4×C83C2×D411yescharacteristic
93C2×C4×C83C2411yescharacteristic
94C22⋊D83—42no—
95C2×Dic83—21no—
96C2×Dic83—21no—
97C2×Dic83—21no—
98D332—21no—
99D332—21no—
100C4×D83—42no—
101Dic166—21no—
102Dic166—21no—
103C83⋊C8—22no—
104C83⋊C8—22no—
105C22×C83—21no—
106C22×C83—21no—
107C22×C83—21no—
108C22×C83C22×D411yescharacteristic
109C2×D83—41no—
110C2×D83—41no—
111C2×D83—41no—
112C4×C83—21no—
113C4×C83—21no—
114C4×C83—21no—
115C4×C83—21no—
116C4×C83—42no—
117C4×C83—42no—
118C4×C83#13512yes—
119C4×C832−1+411yescharacteristic
120C4×C832−1+411yescharacteristic
121Dic83—41no—
122Dic83—41no—
123Dic83—41no—
124C2×C83—41no—
125C2×C83#53611yescharacteristic
126C2×C83C2×#13511yescharacteristic
127C2×C83#57511yescharacteristic
128D83—81no—
129#3233—831nomaximal, Sylow
130C83#323311yescharacteristic, complemented, Sylow
131#533D8311yescharacteristic, complemented
132#529—831no—
133#499—832no—
134#488—832no—
135#485—832no—
136#482—832no—
137#477—831no—
138#476—831no—
139#475—831no—
140#474—831no—
141#431—831no—
142C4⋊Q8C2×D8311yescharacteristic
143C4⋊Q8C2×D8311yescharacteristic
144#126—1661no—
145#125—831no—
146#125C2×D8312yes—
147#124—831no—
148#123—1661no—
149C22⋊Q8—21no—
150C22⋊Q8—21no—
151C22⋊Q8—831no—
152#121—831no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—831no—
156C4×D4—831no—
157C4×D4—1661no—
158#117—1661no—
159#117C2×D8312yes—
160C2×C42C2×D8311yescharacteristic
161#107—832no—
162#107—832no—
163#106—832no—
164#106—832no—
165C4⋊C8—831no—
166C4⋊C8—831no—
167#103—832no—
168#103—832no—
169#102—832no—
170#102—832no—
171C22⋊C8—832no—
172C4×C8—832no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—831no—
176C2×D4—831no—
177C2×D4—1661no—
178C22×C4—1661no—
179C22×C4C22⋊D8311yescharacteristic
180C22×C4C22⋊D8311yescharacteristic
181C22×C4C22×D8311yescharacteristic
182C2×C8—832no—
183C2×C8—832no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—831no—
189C4⋊C4—831no—
190C4⋊C4—1661no—
191C4⋊C4—1661no—
192C4⋊C4—1661no—
193C4⋊C4—1661no—
194C4⋊C4C22×D8312yes—
195C4⋊C4C22×D8312yes—
196C22⋊C4—22no—
197C22⋊C4—1661no—
198C22⋊C4—1661no—
199C22⋊C4—1661no—
200C22⋊C4—1661no—
201C22⋊C4—1661no—
202C42—22no—
203C42—1661no—
204C42C22⋊D8311yescharacteristic
205C42C22⋊D8311yescharacteristic
206C42C22×D8311yescharacteristic
207C42C22×D8311yescharacteristic
208C23—1661no—
209C23C2×C22⋊D8311yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—1661no—
213Q8—1661no—
214D4—3322no—
215D4—1661no—
216D4—1661no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—3322no—
224C2×C4—1661no—
225C2×C4—1661no—
226C2×C4—1661no—
227C2×C4C2×C22⋊D8311yescharacteristic
228C2×C4C2×C22⋊D8311yescharacteristic
229C2×C4C2×C22⋊D8311yescharacteristic
230C2×C4C2×C22⋊D8311yescharacteristic
231C2×C4C2×C22⋊D8311yescharacteristic
232C2×C4C23×D8311yescharacteristic
233C8—1662no—
234C8—1662no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—3321no—
239C22—3321no—
240C22—3321no—
241C22C22×C22⋊D8311yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—42no—
248C4—3321no—
249C4—3321no—
250C4—3321no—
251C4#1458181212yes—
252C4#1458184011yescharacteristic
253C4#1458184011yescharacteristic
254C2—41no—
255C2—6641no—
256C2#1912827511yescharacteristic
257C2C2×#1458181211yescharacteristic
258C2#1912849911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#533×C83C2
socleC22×C83C22×D4
prime radical#14581766C22
hypercenter#533D83
hypocenterC83#3233
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #19128100C2semidirect
3. #19128058C2semidirect
4. #19128042C2semidirect
6. #19127993C2semidirect
7. #19127988C2semidirect
8. #19127983C2semidirect
9. #19127978C2semidirect
10. #19127754C2semidirect
11. #533×C83C2semidirect
130. C83#3233semidirect
131. #533D83semidirect

Maximal subgroups

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

subgroupindex
#191282352
#191281002
#191280582
#191280422
#191280252
#191279932
#191279882
#191279832
#191279782
#191277542
#533×C832
#323383