Finite Group Databasebeta

← #24912394

Subgroups — #24912394

Order 10624.

900 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#14581775C212yesmaximal, complemented
2C2×#14581766C211yesmaximal, characteristic, complemented
3C2×#14581761C212yesmaximal, complemented
4#19127637C212yesmaximal, complemented
5#19127610C212yesmaximal, complemented
6#19127601C214yesmaximal, complemented
7C4×C22⋊Dic83C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C83C211yesmaximal, characteristic, complemented
9C22×D332C2211yescharacteristic, complemented
10C22×C22⋊D83C2212yes—
11C22×C22⋊D83C2214yes—
12#14581775—28no—
13#14581766—24no—
14#14581761—28no—
15C2×C4⋊Dic83C2211yescharacteristic, complemented
16C2×Dic83⋊C4C2214yescomplemented
17C2×C22⋊Dic83C2214yes—
18C2×C22⋊Dic83C2212yes—
19C2×(C2×D83)⋊C4C2212yescomplemented
20C2×(C2×D83)⋊C4C2214yescomplemented
21C2×C4×Dic83C2212yescomplemented
22C83⋊#95C2212yescomplemented
23C25×C83C2211yescharacteristic
24C23×C4×C83C2211yescharacteristic
25C23×C4×C83C2211yescharacteristic
26C2×C22⋊C4×C83—24no—
27C2×C22⋊C4×C83C2212yes—
28C2×C22⋊C4×C83C2212yes—
29C23×D83C2312yes—
30C2×D332—22no—
31C2×D332—42no—
32C2×C22⋊D83—24no—
33C2×C22⋊D83—24no—
34C2×C22⋊D83—24no—
35C2×C22⋊D83—28no—
36C2×C22⋊D83—28no—
37C2×C22⋊D83—44no—
38C22×Dic83C2312yes—
39C22×Dic83C2314yes—
40C4⋊Dic83—22no—
41Dic83⋊C4—28no—
42C22⋊Dic83—28no—
43C22⋊Dic83D418yescomplemented
44(C2×D83)⋊C4—24no—
45(C2×D83)⋊C4—28no—
46C4×Dic83—42no—
47C24×C83—22no—
48C24×C83—24no—
49C24×C83—24no—
50C24×C83—42no—
51C24×C83C2312yes—
52C24×C83C2311yescharacteristic
53C22×C4×C83—24no—
54C22×C4×C83—22no—
55C22×C4×C83—42no—
56C22×C4×C83D414yescomplemented
57C22×C4×C83C2312yes—
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2311yescharacteristic
60C22⋊C4×C83—24no—
61C22⋊C4×C83—24no—
62C23×C83—22no—
63C23×C83—22no—
64C23×C83—28no—
65C23×C83—22no—
66C23×C83—24no—
67C23×C83—22no—
68C23×C83—24no—
69C23×C83—416no—
70C23×C83—44no—
71C23×C83—44no—
72C23×C83C2×D414yes—
73C23×C83C2×D412yes—
74C23×C83#4114yes—
75C23×C83C2411yescharacteristic
76C22×D83—22no—
77C22×D83—22no—
78C22×D83—22no—
79C22×D83—24no—
80C22×D83—24no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—24no—
85C2×C4×C83—44no—
86C2×C4×C83—44no—
87C2×C4×C83C2×D412yes—
88C2×C4×C83C2×D412yes—
89C22⋊D83—416no—
90C22⋊D83—48no—
91C2×Dic83—24no—
92C2×Dic83—24no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83C2×D418yes—
97D332—44no—
98C22×C83—22no—
99C22×C83—22no—
100C22×C83—28no—
101C22×C83—24no—
102C22×C83—28no—
103C22×C83—416no—
104C22×C83—42no—
105C22×C83—42no—
106C22×C83—44no—
107C22×C83#12714yes—
108C22×C83C22×D412yes—
109C22×C83C22×D411yescharacteristic
110C22×C83C2×#4112yes—
111C22×C83H(5,𝔽2)12yes—
112C2×D83—42no—
113C2×D83—42no—
114C2×D83—42no—
115C2×D83—44no—
116C2×D83—44no—
117C4×C83—22no—
118C4×C83—22no—
119C4×C83—42no—
120Dic83—28no—
121Dic83—42no—
122C2×C83—24no—
123C2×C83—44no—
124C2×C83#52511yescharacteristic
125C2×C83C2×#12711yescharacteristic
126C2×C83#53411yescharacteristic
127C2×C83#54614yes—
128D83—82no—
129#2539—831nomaximal, Sylow
130C83#253911yescharacteristic, complemented, Sylow
131C2×#123—832no—
132C2×#121—831no—
133C2×#120—832no—
134C22×C22⋊C4D8311yescharacteristic, complemented
135#399—832no—
136#392—832no—
137#390—834no—
138C4×C22⋊C4—831no—
139C25C2×D8311yescharacteristic
140C22×D4—834no—
141C22×D4—832no—
142C22×D4—831no—
143C23×C4C2×D8311yescharacteristic
144C23×C4C2×D8311yescharacteristic
145#123—1668no—
146#121—1664no—
147#120—1668no—
148C2×C4⋊C4—831no—
149C2×C4⋊C4—834no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—832no—
152C2×C22⋊C4—834no—
153C2×C22⋊C4—832no—
154C2×C22⋊C4—834no—
155C2×C22⋊C4C2×D8312yes—
156C2×C22⋊C4C2×D8312yes—
157C2×C42—832no—
158#95—832no—
159C24—22no—
160C24—24no—
161C24—24no—
162C24—42no—
163C24—832no—
164C24C22×D8312yes—
165C24C22×D8311yescharacteristic
166C2×D4—3322no—
167C2×D4—3324no—
168C2×D4—1668no—
169C2×D4—1664no—
170C2×D4—1664no—
171C2×D4—1664no—
172C2×D4—1668no—
173C2×D4—1662no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—42no—
177C22×C4—834no—
178C22×C4—832no—
179C22×C4C22⋊D8314yescomplemented
180C22×C4C22×D8312yes—
181C22×C4C22×D8311yescharacteristic
182C22×C4C22×D8311yescharacteristic
183C4⋊C4—1668no—
184C4⋊C4—1662no—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—838no—
188C22⋊C4—1668no—
189C22⋊C4—1668no—
190C22⋊C4—1664no—
191C42—3322no—
192C23—22no—
193C23—22no—
194C23—28no—
195C23—24no—
196C23—22no—
197C23—22no—
198C23—24no—
199C23—416no—
200C23—44no—
201C23—44no—
202C23—1662no—
203C23—1662no—
204C23—1662no—
205C23—1664no—
206C23—1664no—
207C23D4×D8314yes—
208C23C2×C22⋊D8312yes—
209C23#1305059014yes—
210C23C23×D8311yescharacteristic
211D4—3324no—
212D4—3328no—
213D4—33216no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—838no—
221C2×C4—1662no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4—1664no—
225C2×C4—1664no—
226C2×C4C2×C22⋊D8312yes—
227C2×C4C2×C22⋊D8312yes—
228C22—28no—
229C22—22no—
230C22—22no—
231C22—28no—
232C22—24no—
233C22—416no—
234C22—44no—
235C22—42no—
236C22—42no—
237C22—3324no—
238C22—3324no—
239C22—3322no—
240C22—3322no—
241C22—3322no—
242C22#1458178814yes—
243C22C2×D4×D8312yes—
244C22C22×C22⋊D8311yescharacteristic
245C22C2×#1305059012yes—
246C22#1458183712yes—
247C4—22no—
248C4—22no—
249C4—42no—
250C4—3322no—
251C4—1668no—
252C2—24no—
253C2—44no—
254C2—6642no—
255C2#1912821211yescharacteristic
256C2C2×#1458178811yescharacteristic
257C2#1912826011yescharacteristic
258C2#1912834114yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC22×C22⋊C4×C83C2
hypercenterC22×C22⋊C4D83
hypocenterC83#2539
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#14581775C2semidirect
2. C2×#14581766C2semidirect
3. C2×#14581761C2semidirect
4. #19127637C2semidirect
5. #19127610C2semidirect
6. #19127601C2semidirect
7. C4×C22⋊Dic83C2semidirect
8. C22×C22⋊C4×C83C2semidirect
9. C22×D332C22semidirect
15. C2×C4⋊Dic83C22semidirect
16. C2×Dic83⋊C4C22semidirect
19. C2×(C2×D83)⋊C4C22semidirect
20. C2×(C2×D83)⋊C4C22semidirect
21. C2×C4×Dic83C22semidirect
22. C83⋊#95C22semidirect
43. C22⋊Dic83D4semidirect
56. C22×C4×C83D4semidirect
130. C83#2539semidirect
134. C22×C22⋊C4D83semidirect
179. C22×C4C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817752
C2×#145817662
C2×#145817612
#191276372
#191276102
#191276012
C4×C22⋊Dic832
C22×C22⋊C4×C832
#253983