Finite Group Databasebeta

← #24918678

Subgroups — #24918678

Order 10624.

322 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 111 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#19128257C211yesmaximal, characteristic
2#19128256C211yesmaximal, characteristic
3#19128107C211yesmaximal, characteristic, complemented
4#19128071C211yesmaximal, characteristic, complemented
5#19128070C211yesmaximal, characteristic, complemented
6#19128045C212yesmaximal
7#19128044C212yesmaximal
8#19127995C212yesmaximal, complemented
9#19127985C212yesmaximal, complemented
10#19127756C211yesmaximal, characteristic
11#501×C83C211yesmaximal, characteristic, complemented
12#14581792C2211yescharacteristic
13#14581791C2211yescharacteristic
14#14581791C2211yescharacteristic
15#14581784C2211yescharacteristic
16#14581783—21no—
17#14581772C2212yes—
18#14581770—21no—
19#14581770—21no—
20#14581768—21no—
21Q8×Dic83—21no—
22#14581757—21no—
23#14581757—21no—
24#14581757C2212yes—
25#14581750—21no—
26#14581750C2211yescharacteristic
27#14581750C2211yescharacteristic
28C42×D83C2211yescharacteristic
29#14581725C2212yes—
30#14581723C2212yes—
31Dic83⋊C8C2212yes—
32#14581717C2212yes—
33#14581717C2212yes—
34#14581715C2212yes—
35#14581715C2212yes—
36(C2×D83)⋊C8C2212yes—
37#14581703C2211yescharacteristic
38C4⋊Q8×C83C2211yescharacteristic
39#125×C83C2211yescharacteristic
40#107×C83C2212yes—
41#106×C83C2212yes—
42#97×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×Dic166—21no—
45C2×Dic166C2312yes—
46C2×C4×D83D411yescharacteristic
47C2×C4×D83D411yescharacteristic
48C2×C4×D83C2311yescharacteristic
49C2×C83⋊C8C2312yes—
50C4⋊Dic83—21no—
51C4⋊Dic83—21no—
52C4⋊Dic83—22no—
53C4⋊Dic83C2311yescharacteristic
54C4⋊Dic83C2311yescharacteristic
55Dic83⋊C4—21no—
56Dic83⋊C4—21no—
57Dic83⋊C4—22no—
58Dic83⋊C4—21no—
59Dic83⋊C4—21no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—21no—
62(C2×D83)⋊C4—21no—
63C4×Dic83—22no—
64C4×Dic83—21no—
65C4×Dic83—21no—
66C4×Dic83D411yescharacteristic
67C4×Dic83D411yescharacteristic
68C4×Dic83C2311yescharacteristic
69C2×Q8×C83—21no—
70C2×C8×C83C2312yes—
71C4⋊C4×C83—21no—
72C4⋊C4×C83—21no—
73C4⋊C4×C83—21no—
74C4⋊C4×C83C2312yes—
75C4⋊C4×C83C2311yescharacteristic
76C4⋊C4×C83C2311yescharacteristic
77C42×C83C2311yescharacteristic
78C22×D83C2×D411yescharacteristic
79Q8×C83—42no—
80C2×C4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—22no—
83C2×C4×C83C2×D411yescharacteristic
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2411yescharacteristic
86C2×Dic83—21no—
87C2×Dic83—21no—
88C2×Dic83—22no—
89C2×Dic83C2×D411yescharacteristic
90C2×Dic83C2×D411yescharacteristic
91C2×Dic83C2×D411yescharacteristic
92C4×D83—22no—
93C4×D83—22no—
94C4×D83—41no—
95Dic166—24no—
96Dic166—42no—
97Dic166—42no—
98C8×C83—22no—
99C83⋊C8—22no—
100C22×C83C22×D411yescharacteristic
101C2×D83—21no—
102C2×D83—21no—
103C2×D83—21no—
104C4×C83—21no—
105C4×C83—21no—
106C4×C83—41no—
107C4×C83—41no—
108C4×C83—42no—
109C4×C832−1+412yes—
110Dic83—21no—
111Dic83—21no—
112Dic83—42no—
113Dic83—41no—
114Dic83—41no—
115Dic83#13712yes—
116C2×C83#53611yescharacteristic
117C2×C83C2×#13711yescharacteristic
118C2×C83#57511yescharacteristic
119D83—41no—
120#3237—831nomaximal, Sylow
121C83#323711yescharacteristic, complemented, Sylow
122#533—831no—
123#533—831no—
124#501—831no—
125#501D8311yescharacteristic, complemented
126#491—831no—
127#491—831no—
128#485—832no—
129#485—832no—
130#477—832no—
131#475—832no—
132#432—831no—
133C4⋊Q8—831no—
134C4⋊Q8—831no—
135C4⋊Q8—831no—
136C4⋊Q8C2×D8311yescharacteristic
137#125—831no—
138#125—1661no—
139#125C2×D8311yescharacteristic
140C22⋊Q8—832no—
141C22⋊Q8—1661no—
142C22⋊Q8—1661no—
143C22⋊Q8—1661no—
144C4×Q8—832no—
145C4×Q8—1661no—
146C4×Q8—1661no—
147C4×Q8—1661no—
148#117—831no—
149#117—831no—
150#117—1661no—
151C2×C42—831no—
152#107—832no—
153#107C2×D8312yes—
154#106—832no—
155#106C2×D8312yes—
156C4⋊C8—832no—
157#103—832no—
158#103—832no—
159#103—832no—
160#103—832no—
161C22⋊C8—832no—
162#97—831no—
163#97C2×D8311yescharacteristic
164C2×Q8—21no—
165C2×Q8—832no—
166C2×Q8—1661no—
167C2×Q8—1661no—
168C22×C4—831no—
169C22×C4—831no—
170C22×C4—831no—
171C2×C8—832no—
172C2×C8C22×D8312yes—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—831no—
177C4⋊C4—831no—
178C4⋊C4—1662no—
179C4⋊C4—1662no—
180C4⋊C4—1661no—
181C4⋊C4—1661no—
182C4⋊C4—1661no—
183C4⋊C4—1661no—
184C4⋊C4—1661no—
185C4⋊C4—1661no—
186C4⋊C4C22×D8311yescharacteristic
187C4⋊C4C22×D8311yescharacteristic
188C4⋊C4C22×D8312yes—
189C22⋊C4—1661no—
190C22⋊C4—1661no—
191C22⋊C4—1662no—
192C42—831no—
193C42—831no—
194C42—831no—
195C42—1661no—
196C42—1661no—
197C42—1662no—
198C42C22×D8311yescharacteristic
199C23—831no—
200Q8—42no—
201Q8—3322no—
202Q8—3322no—
203Q8—1664no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—3321no—
208C2×C4—831no—
209C2×C4—831no—
210C2×C4—831no—
211C2×C4—1662no—
212C2×C4—1661no—
213C2×C4—1661no—
214C2×C4—1662no—
215C2×C4—1662no—
216C2×C4D4×D8311yescharacteristic
217C2×C4D4×D8311yescharacteristic
218C2×C4C23×D8311yescharacteristic
219C8—22no—
220C8—1662no—
221C22—1661no—
222C22—1661no—
223C22—1661no—
224C22C2×D4×D8311yescharacteristic
225C4—21no—
226C4—21no—
227C4—41no—
228C4—42no—
229C4—41no—
230C4—3321no—
231C4—3321no—
232C4—3322no—
233C4—832no—
234C4—1661no—
235C4—1661no—
236C4#1458184012yes—
237C2—3321no—
238C2#1912827611yescharacteristic
239C2#137×D8311yescharacteristic
240C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#501×C83C2
socleC22×C83C22×D4
prime radicalC22×D83C2×D4
hypercenter#501D83
hypocenterC83#3237
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #19128107C2semidirect
4. #19128071C2semidirect
5. #19128070C2semidirect
8. #19127995C2semidirect
9. #19127985C2semidirect
11. #501×C83C2semidirect
121. C83#3237semidirect
125. #501D83semidirect

Maximal subgroups

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

subgroupindex
#191282572
#191282562
#191281072
#191280712
#191280702
#191280452
#191280442
#191279952
#191279852
#191277562
#501×C832
#323783