Finite Group Databasebeta

← #24918706

Subgroups — #24918706

Order 10624.

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

normal subgroups only

This group has 328 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#19128256C211yesmaximal, characteristic, complemented
2#19128236C211yesmaximal, characteristic
3#19128111C211yesmaximal, characteristic, complemented
4#19128063C211yesmaximal, characteristic, complemented
5#19128061C211yesmaximal, characteristic, complemented
6#19128045C212yesmaximal, complemented
7#19128028C212yesmaximal
8#19128007C212yesmaximal, complemented
9#19128000C212yesmaximal, complemented
10#19127756C211yesmaximal, characteristic, complemented
11#501×C83C211yesmaximal, characteristic, complemented
12#14581792C2211yescharacteristic
13#14581787—21no—
14#14581783C2212yes—
15#14581780C2211yescharacteristic
16#14581777—21no—
17#14581772—21no—
18#14581772C2211yescharacteristic
19#14581768—21no—
20#14581767C2211yescharacteristic
21Q8×Dic83—21no—
22#14581757—21no—
23#14581757C2211yescharacteristic
24#14581754—21no—
25#14581754C2211yescharacteristic
26#14581750—21no—
27#14581750C2212yes—
28C42×D83C2211yescharacteristic
29#14581725C2212yes—
30#14581723C2212yes—
31Dic83⋊C8C2212yes—
32#14581717C2212yes—
33#14581715C2212yes—
34#14581714C2212yes—
35#14581712C2212yes—
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×Dic166C2311yescharacteristic
45C2×D332—21no—
46C2×D332C2311yescharacteristic
47C2×C4×D83—21no—
48C2×C4×D83D411yescharacteristic
49C2×C4×D83D411yescharacteristic
50C2×C4×D83C2311yescharacteristic
51C2×C83⋊C8C2312yes—
52C4⋊Dic83—21no—
53C4⋊Dic83—22no—
54C4⋊Dic83—21no—
55C4⋊Dic83C2312yes—
56Dic83⋊C4—22no—
57Dic83⋊C4—21no—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4—21no—
60(C2×D83)⋊C4—21no—
61(C2×D83)⋊C4—21no—
62(C2×D83)⋊C4—21no—
63(C2×D83)⋊C4—21no—
64C4×Dic83—21no—
65C4×Dic83—22no—
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×C83C2311yescharacteristic
75C4⋊C4×C83C2311yescharacteristic
76C4⋊C4×C83C2312yes—
77C42×C83C2311yescharacteristic
78C22×D83—21no—
79C22×D83C2×D411yescharacteristic
80Q8×C83—42no—
81C2×C4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2411yescharacteristic
87C2×Dic83—21no—
88C2×Dic83—22no—
89C2×Dic83C2×D411yescharacteristic
90C2×Dic83C2×D411yescharacteristic
91C2×Dic83C2×D411yescharacteristic
92D332—22no—
93D332—42no—
94C4×D83—22no—
95C4×D83—41no—
96C4×D83—41no—
97C4×D83—42no—
98Dic166—22no—
99Dic166—42no—
100C8×C83—22no—
101C83⋊C8—22no—
102C22×C83C22×D411yescharacteristic
103C2×D83—21no—
104C2×D83—21no—
105C2×D83—21no—
106C2×D83—41no—
107C2×D83—41no—
108C2×D83—41no—
109C4×C83—21no—
110C4×C83—21no—
111C4×C83—41no—
112C4×C83—41no—
113C4×C83—42no—
114C4×C832−1+412yes—
115Dic83—21no—
116Dic83—21no—
117Dic83—21no—
118Dic83—41no—
119Dic83—42no—
120C2×C83#53611yescharacteristic
121C2×C83#57511yescharacteristic
122C2×C83#57511yescharacteristic
123D83—41no—
124D83—81no—
125#3240—831nomaximal, Sylow
126C83#324011yescharacteristic, complemented, Sylow
127#533—831no—
128#529—831no—
129#501—831no—
130#501D8311yescharacteristic, complemented
131#489—831no—
132#489—831no—
133#485—832no—
134#482—832no—
135#479—832no—
136#478—832no—
137#432—831no—
138C4⋊Q8—831no—
139C4⋊Q8C2×D8311yescharacteristic
140#126—1661no—
141#125—832no—
142#125C2×D8311yescharacteristic
143#124—831no—
144#123—1661no—
145C22⋊Q8—831no—
146C22⋊Q8—1661no—
147C22⋊Q8—1661no—
148#121—831no—
149C4×Q8—831no—
150C4×Q8—1661no—
151C4×Q8—1661no—
152C4×D4—831no—
153C4×D4—1661no—
154#117—832no—
155#117—1661no—
156C2×C42—831no—
157#107—832no—
158#107C2×D8312yes—
159#106—832no—
160#106C2×D8312yes—
161C4⋊C8—832no—
162#103—832no—
163#103—832no—
164#102—832no—
165#102—832no—
166C22⋊C8—832no—
167#97—831no—
168#97C2×D8311yescharacteristic
169C2×Q8—21no—
170C2×Q8—831no—
171C2×Q8—1661no—
172C2×D4—831no—
173C2×D4—1661no—
174C22×C4—831no—
175C22×C4—831no—
176C22×C4—831no—
177C22×C4—1661no—
178C2×C8—832no—
179C2×C8C22×D8312yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—832no—
184C4⋊C4—1661no—
185C4⋊C4—1661no—
186C4⋊C4—1662no—
187C4⋊C4—1662no—
188C4⋊C4—1661no—
189C4⋊C4C22×D8312yes—
190C4⋊C4C22×D8311yescharacteristic
191C4⋊C4C22×D8311yescharacteristic
192C22⋊C4—1661no—
193C22⋊C4—1661no—
194C22⋊C4—1662no—
195C22⋊C4—1661no—
196C22⋊C4—1661no—
197C22⋊C4—1661no—
198C42—831no—
199C42—831no—
200C42—831no—
201C42—1662no—
202C42—1661no—
203C42C22×D8311yescharacteristic
204C23—831no—
205C23—1661no—
206Q8—42no—
207Q8—3322no—
208Q8—1662no—
209D4—3322no—
210D4—1662no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—3321no—
215C2×C4—3321no—
216C2×C4—3322no—
217C2×C4—831no—
218C2×C4—831no—
219C2×C4—831no—
220C2×C4—1661no—
221C2×C4—1662no—
222C2×C4—1662no—
223C2×C4D4×D8311yescharacteristic
224C2×C4D4×D8311yescharacteristic
225C2×C4C23×D8311yescharacteristic
226C8—22no—
227C8—1662no—
228C22—3321no—
229C22—3321no—
230C22—3321no—
231C22—1661no—
232C22—1661no—
233C22—1661no—
234C22C2×D4×D8311yescharacteristic
235C4—21no—
236C4—21no—
237C4—41no—
238C4—42no—
239C4—41no—
240C4—3322no—
241C4—3321no—
242C4—1661no—
243C4—1661no—
244C4—1661no—
245C4#1458184012yes—
246C2—3321no—
247C2—6641no—
248C2#1912827611yescharacteristic
249C2#1912849711yescharacteristic
250C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#501×C83C2
socleC22×C83C22×D4
prime radical#14581767C22
hypercenter#501D83
hypocenterC83#3240
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
1. #19128256C2semidirect
3. #19128111C2semidirect
4. #19128063C2semidirect
5. #19128061C2semidirect
6. #19128045C2semidirect
8. #19128007C2semidirect
9. #19128000C2semidirect
10. #19127756C2semidirect
11. #501×C83C2semidirect
126. C83#3240semidirect
130. #501D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#191282362
#191281112
#191280632
#191280612
#191280452
#191280282
#191280072
#191280002
#191277562
#501×C832
#324083