Finite Group Databasebeta

← #24918640

Subgroups — #24918640

Order 10624.

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

normal subgroups only

This group has 332 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#19128256C211yesmaximal, characteristic, complemented
2#19128233C211yesmaximal, characteristic
3#19128102C212yesmaximal, complemented
4#19128054C212yesmaximal, complemented
5#19128044C212yesmaximal, complemented
6#19128023C212yesmaximal
7#19127997C211yesmaximal, characteristic, complemented
8#19127991C211yesmaximal, characteristic, complemented
9#19127983C211yesmaximal, characteristic, complemented
10#19127753C211yesmaximal, characteristic, complemented
11#474×C83C211yesmaximal, characteristic, complemented
12#14581792C2211yescharacteristic
13#14581785—21no—
14#14581783C2211yescharacteristic
15#14581783C2212yes—
16#14581779C2211yescharacteristic
17#14581774—21no—
18#14581772—21no—
19#14581772C2211yescharacteristic
20#14581768—21no—
21#14581763C2211yescharacteristic
22Q8×Dic83—21no—
23C4×Dic166C2211yescharacteristic
24#14581757—21no—
25C4×C22⋊D83—21no—
26#14581751—21no—
27#14581750C2212yes—
28C42×D83C2211yescharacteristic
29#14581726C2212yes—
30#14581723C2212yes—
31#14581723C2212yes—
32C4⋊(C83⋊C8)C2211yescharacteristic
33#14581717C2212yes—
34#14581715C2212yes—
35#14581713C2212yes—
36(C2×D83)⋊C8C2212yes—
37C8×Dic83C2212yes—
38C4⋊Q8×C83C2211yescharacteristic
39C4×D4×C83C2211yescharacteristic
40#107×C83C2212yes—
41C4⋊C8×C83C2211yescharacteristic
42#102×C83C2212yes—
43C2×Dic166—21no—
44C2×Dic166C2311yescharacteristic
45C2×C22⋊D83—21no—
46C2×C4×D83D411yescharacteristic
47C2×C4×D83D411yescharacteristic
48C2×C4×D83C2311yescharacteristic
49C2×C83⋊C8C2312yes—
50C4⋊Dic83—21no—
51C4⋊Dic83—22no—
52C4⋊Dic83—21no—
53C4⋊Dic83—21no—
54C4⋊Dic83C2311yescharacteristic
55C4⋊Dic83C2312yes—
56Dic83⋊C4—21no—
57Dic83⋊C4—22no—
58Dic83⋊C4—21no—
59C22⋊Dic83—21no—
60C22⋊Dic83—21no—
61C22⋊Dic83—21no—
62(C2×D83)⋊C4—22no—
63(C2×D83)⋊C4—21no—
64C4×Dic83—21no—
65C4×Dic83—22no—
66C4×Dic83D411yescharacteristic
67C4×Dic83D411yescharacteristic
68C4×Dic83C2311yescharacteristic
69C2×Q8×C83—21no—
70C2×D4×C83C2311yescharacteristic
71C22×C4×C83—21no—
72C2×C8×C83C2312yes—
73C4⋊C4×C83—21no—
74C4⋊C4×C83C2311yescharacteristic
75C4⋊C4×C83C2312yes—
76C22⋊C4×C83—21no—
77C42×C83C2311yescharacteristic
78C23×C83—21no—
79C22×D83C2×D411yescharacteristic
80Q8×C83—42no—
81D4×C83—21no—
82D4×C83—21no—
83C2×C4×C83—21no—
84C2×C4×C83—22no—
85C2×C4×C83—42no—
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2411yescharacteristic
89C22⋊D83—42no—
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83—21no—
93C2×Dic83C2×D411yescharacteristic
94C2×Dic83C2×D411yescharacteristic
95C2×Dic83C2×D411yescharacteristic
96C4×D83—22no—
97C4×D83—21no—
98C4×D83—21no—
99C4×D83—41no—
100Dic166—21no—
101Dic166—21no—
102Dic166—42no—
103C8×C83—22no—
104C83⋊C8—22no—
105C22×C83—41no—
106C22×C83—41no—
107C22×C83—41no—
108C22×C83C22×D411yescharacteristic
109C2×D83—21no—
110C2×D83—21no—
111C2×D83—21no—
112C4×C83—21no—
113C4×C83—41no—
114C4×C83—42no—
115C4×C83#13512yes—
116C4×C832−1+411yescharacteristic
117C4×C832−1+411yescharacteristic
118Dic83—21no—
119Dic83—21no—
120Dic83—21no—
121Dic83—41no—
122Dic83—42no—
123Dic83—41no—
124C2×C83—81no—
125C2×C83#53611yescharacteristic
126C2×C83C2×#13511yescharacteristic
127C2×C83#57511yescharacteristic
128D83—41no—
129#3233—831nomaximal, Sylow
130C83#323311yescharacteristic, complemented, Sylow
131#533—831no—
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#474D8311yescharacteristic, complemented
141#431—831no—
142C4⋊Q8—831no—
143C4⋊Q8C2×D8311yescharacteristic
144#126—1661no—
145#125—831no—
146#125—832no—
147#124—831no—
148#123—1661no—
149C22⋊Q8—831no—
150C22⋊Q8—1661no—
151C22⋊Q8—1661no—
152#121—831no—
153C4×Q8—831no—
154C4×Q8—1661no—
155C4×Q8—1661no—
156C4×D4—1661no—
157C4×D4C2×D8311yescharacteristic
158#117—832no—
159#117—1661no—
160C2×C42—831no—
161#107—832no—
162#107C2×D8312yes—
163#106—832no—
164#106—832no—
165C4⋊C8—831no—
166C4⋊C8C2×D8311yescharacteristic
167#103—832no—
168#103—832no—
169#102—832no—
170#102C2×D8312yes—
171C22⋊C8—832no—
172C4×C8—832no—
173C2×Q8—21no—
174C2×Q8—831no—
175C2×Q8—1661no—
176C2×D4—1661no—
177C2×D4C22×D8311yescharacteristic
178C22×C4—21no—
179C22×C4—831no—
180C22×C4—831no—
181C22×C4—831no—
182C2×C8—832no—
183C2×C8C22×D8312yes—
184C4⋊C4—21no—
185C4⋊C4—832no—
186C4⋊C4—831no—
187C4⋊C4—1661no—
188C4⋊C4—1661no—
189C4⋊C4—1661no—
190C4⋊C4—1662no—
191C4⋊C4—1662no—
192C4⋊C4—1661no—
193C4⋊C4—1661no—
194C4⋊C4C22×D8312yes—
195C4⋊C4C22×D8311yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—1662no—
198C22⋊C4—1661no—
199C22⋊C4—1661no—
200C22⋊C4—1661no—
201C22⋊C4—1661no—
202C42—831no—
203C42—831no—
204C42—831no—
205C42—1662no—
206C42—1661no—
207C42C22×D8311yescharacteristic
208C23—21no—
209C23—831no—
210Q8—42no—
211Q8—3322no—
212Q8—1661no—
213Q8—1661no—
214D4—21no—
215D4—21no—
216D4—3322no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—3321no—
221C2×C4—831no—
222C2×C4—831no—
223C2×C4—831no—
224C2×C4—1661no—
225C2×C4—1661no—
226C2×C4—1662no—
227C2×C4—1662no—
228C2×C4—1661no—
229C2×C4—1661no—
230C2×C4D4×D8311yescharacteristic
231C2×C4D4×D8311yescharacteristic
232C2×C4C23×D8311yescharacteristic
233C8—22no—
234C8—1662no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—1661no—
239C22—1661no—
240C22—1661no—
241C22C2×D4×D8311yescharacteristic
242C4—21no—
243C4—41no—
244C4—42no—
245C4—3321no—
246C4—3322no—
247C4—3321no—
248C4—1661no—
249C4—1661no—
250C4—1661no—
251C4#1458181112yes—
252C4#1458183911yescharacteristic
253C4#1458184011yescharacteristic
254C2—81no—
255C2—3321no—
256C2#1912827411yescharacteristic
257C2C2×#1458181111yescharacteristic
258C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#474×C83C2
socleC22×C83C22×D4
prime radical#14581763C22
hypercenter#474D83
hypocenterC83#3233
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #19128256C2semidirect
3. #19128102C2semidirect
4. #19128054C2semidirect
5. #19128044C2semidirect
7. #19127997C2semidirect
8. #19127991C2semidirect
9. #19127983C2semidirect
10. #19127753C2semidirect
11. #474×C83C2semidirect
130. C83#3233semidirect
140. #474D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#191282332
#191281022
#191280542
#191280442
#191280232
#191279972
#191279912
#191279832
#191277532
#474×C832
#323383