Finite Group Databasebeta

← #24918638

Subgroups — #24918638

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#19128258C211yesmaximal, characteristic, complemented
2#19128100C212yesmaximal, complemented
3#19128055C212yesmaximal, complemented
4#19128039C212yesmaximal, complemented
5#19128022C212yesmaximal
6#19127996C211yesmaximal, characteristic, complemented
7#19127991C211yesmaximal, characteristic, complemented
8#19127986C211yesmaximal, characteristic, complemented
9#19127981C211yesmaximal, characteristic, complemented
10#19127754C211yesmaximal, characteristic, complemented
11#529×C83C211yesmaximal, characteristic
12#14581792C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581784C2212yes—
15#14581771—21no—
16#14581771—21no—
17C4×Dic166—21no—
18C4×Dic166—21no—
19#14581751C2212yes—
20#14581725C2212yes—
21#14581725C2212yes—
22#14581723C2212yes—
23#14581723C2212yes—
24C4⋊(C83⋊C8)C2211yescharacteristic
25C4⋊(C83⋊C8)C2211yescharacteristic
26#14581716C2212yes—
27#14581716C2212yes—
28#14581713C2212yes—
29#14581713C2212yes—
30C22⋊(C83⋊C8)C2212yes—
31C4×C83⋊C8C2212yes—
32#126×C83—21no—
33#125×C83C2211yescharacteristic
34#124×C83C2211yescharacteristic
35#123×C83—21no—
36C22⋊Q8×C83C2211yescharacteristic
37#121×C83C2211yescharacteristic
38C4×Q8×C83C2211yescharacteristic
39C4×D4×C83—21no—
40C4×D4×C83C2211yescharacteristic
41#117×C83—21no—
42C2×C42×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×Dic166—21no—
45C2×C83⋊C8C2312yes—
46C2×C83⋊C8C2312yes—
47C4⋊Dic83—21no—
48C4⋊Dic83—21no—
49C4⋊Dic83C2312yes—
50C4⋊Dic83C2312yes—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53C22⋊Dic83—22no—
54C4×Dic83—22no—
55C2×Q8×C83C2311yescharacteristic
56C2×D4×C83—21no—
57C2×D4×C83C2311yescharacteristic
58C22×C4×C83—21no—
59C22×C4×C83D411yescharacteristic
60C22×C4×C83D411yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C4⋊C4×C83—21no—
63C4⋊C4×C83—21no—
64C4⋊C4×C83—21no—
65C4⋊C4×C83—21no—
66C4⋊C4×C83C2311yescharacteristic
67C4⋊C4×C83C2311yescharacteristic
68C22⋊C4×C83—21no—
69C22⋊C4×C83—21no—
70C22⋊C4×C83—21no—
71C22⋊C4×C83—21no—
72C22⋊C4×C83—21no—
73C42×C83—21no—
74C42×C83D411yescharacteristic
75C42×C83D411yescharacteristic
76C42×C83C2311yescharacteristic
77C42×C83C2311yescharacteristic
78C23×C83—21no—
79C23×C83C2×D411yescharacteristic
80Q8×C83—21no—
81Q8×C83—21no—
82D4×C83—21no—
83D4×C83—21no—
84D4×C83—42no—
85C2×C4×C83—22no—
86C2×C4×C83—21no—
87C2×C4×C83—21no—
88C2×C4×C83—21no—
89C2×C4×C83—21no—
90C2×C4×C83—21no—
91C2×C4×C83—41no—
92C2×C4×C83—42no—
93C2×C4×C83C2×D411yescharacteristic
94C2×C4×C83C2×D411yescharacteristic
95C2×C4×C83C2×D411yescharacteristic
96C2×C4×C83C2×D411yescharacteristic
97C2×C4×C83C2×D411yescharacteristic
98C2×C4×C83C2411yescharacteristic
99C2×Dic83—22no—
100C2×Dic83—22no—
101Dic166—42no—
102Dic166—42no—
103C83⋊C8—22no—
104C83⋊C8—22no—
105C22×C83—21no—
106C22×C83—21no—
107C22×C83—21no—
108C22×C83—41no—
109C22×C83—41no—
110C22×C83—41no—
111C22×C83C22×D411yescharacteristic
112C4×C83—21no—
113C4×C83—21no—
114C4×C83—21no—
115C4×C83—21no—
116C4×C83—41no—
117C4×C83—41no—
118C4×C83—41no—
119C4×C83#13512yes—
120C4×C832−1+411yescharacteristic
121C4×C832−1+411yescharacteristic
122Dic83—42no—
123Dic83—42no—
124C2×C83—41no—
125C2×C83—81no—
126C2×C83#53611yescharacteristic
127C2×C83C2×#13511yescharacteristic
128C2×C83#57511yescharacteristic
129#3233—831nomaximal, Sylow
130C83#323311yescharacteristic, complemented, Sylow
131#533—831no—
132#529D8311yescharacteristic
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⋊Q8—831no—
143C4⋊Q8—831no—
144#126—21no—
145#125—832no—
146#125C2×D8311yescharacteristic
147#124C2×D8311yescharacteristic
148#123—21no—
149C22⋊Q8—1661no—
150C22⋊Q8—1661no—
151C22⋊Q8C2×D8311yescharacteristic
152#121C2×D8311yescharacteristic
153C4×Q8—1661no—
154C4×Q8—1661no—
155C4×Q8C2×D8311yescharacteristic
156C4×D4—21no—
157C4×D4C2×D8311yescharacteristic
158#117—21no—
159#117—832no—
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—1661no—
174C2×Q8—1661no—
175C2×Q8C22×D8311yescharacteristic
176C2×D4—21no—
177C2×D4C22×D8311yescharacteristic
178C22×C4—21no—
179C22×C4C22⋊D8311yescharacteristic
180C22×C4C22⋊D8311yescharacteristic
181C22×C4C22×D8311yescharacteristic
182C2×C8—832no—
183C2×C8—832no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—832no—
189C4⋊C4—832no—
190C4⋊C4—1662no—
191C4⋊C4—1662no—
192C4⋊C4—1661no—
193C4⋊C4—1661no—
194C4⋊C4C22×D8311yescharacteristic
195C4⋊C4C22×D8311yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—1662no—
202C42—21no—
203C42—1662no—
204C42C22⋊D8311yescharacteristic
205C42C22⋊D8311yescharacteristic
206C42C22×D8311yescharacteristic
207C42C22×D8311yescharacteristic
208C23—21no—
209C23C2×C22⋊D8311yescharacteristic
210Q8—21no—
211Q8—21no—
212Q8—3322no—
213Q8—3322no—
214D4—21no—
215D4—21no—
216D4—42no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—1662no—
226C2×C4—1662no—
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—41no—
239C22—41no—
240C22—41no—
241C22C22×C22⋊D8311yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—3322no—
250C4—3322no—
251C4#1458181212yes—
252C4#1458183911yescharacteristic
253C4#1458183911yescharacteristic
254C2—41no—
255C2—81no—
256C2#1912827211yescharacteristic
257C2C2×#1458181211yescharacteristic
258C2#1912849911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#529×C83C2
socleC22×C83C22×D4
prime radical#121×C83C22
hypercenter#529D83
hypocenterC83#3233
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #19128258C2semidirect
2. #19128100C2semidirect
3. #19128055C2semidirect
4. #19128039C2semidirect
6. #19127996C2semidirect
7. #19127991C2semidirect
8. #19127986C2semidirect
9. #19127981C2semidirect
10. #19127754C2semidirect
130. C83#3233semidirect

Maximal subgroups

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

subgroupindex
#191282582
#191281002
#191280552
#191280392
#191280222
#191279962
#191279912
#191279862
#191279812
#191277542
#529×C832
#323383