Finite Group Databasebeta

← #24911081

Subgroups — #24911081

Order 10624.

952 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C22×#13050589C211yesmaximal, characteristic, complemented
2C2×#14581788C211yesmaximal, characteristic, complemented
3C2×#14581780C211yesmaximal, characteristic, complemented
4C2×#14581767C211yesmaximal, characteristic, complemented
5#19127610C214yesmaximal, complemented
6#19127606C211yesmaximal, characteristic, complemented
7#19127578C214yesmaximal, complemented
8C4×(C2×D83)⋊C4C211yesmaximal, characteristic, complemented
9#390×C83C211yesmaximal, characteristic, complemented
10C2×#13050589—24no—
11C2×#13050589—24no—
12C2×#13050589—24no—
13C22×Dic166C2211yescharacteristic
14C22×D332C2211yescharacteristic
15C22×D332C2212yes—
16C22×C22⋊D83C2212yes—
17C22×C22⋊D83C2214yes—
18C23×Dic83C2212yes—
19C22×C4×D83C2211yescharacteristic
20#14581788—24no—
21#14581780—24no—
22#14581767—24no—
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2214yes—
25C2×(C2×D83)⋊C4C2212yes—
26C2×C4×Dic83C2211yescharacteristic
27C83⋊#95C2212yes—
28C83⋊#95C2214yes—
29C22×D4×C83C2211yescharacteristic
30C2×C4⋊C4×C83C2211yescharacteristic
31C2×C22⋊C4×C83C2214yes—
32C2×C42×C83C2211yescharacteristic
33C23×D83C2312yes—
34C23×D83C2311yescharacteristic
35#13050589—232no—
36C2×Dic166—22no—
37C2×Dic166—22no—
38C2×Dic166D414yescomplemented
39C2×D332—24no—
40C2×D332—22no—
41C2×D332—22no—
42C2×D332—22no—
43C2×D332—44no—
44C2×C22⋊D83—24no—
45C2×C22⋊D83—24no—
46C2×C22⋊D83—28no—
47C2×C22⋊D83—24no—
48C2×C22⋊D83—28no—
49C2×C22⋊D83—44no—
50C22×Dic83—24no—
51C22×Dic83—24no—
52C22×Dic83—24no—
53C22×Dic83C2311yescharacteristic
54C22×Dic83C2312yes—
55C22×Dic83C2312yes—
56C2×C4×D83—22no—
57C2×C4×D83—22no—
58C2×C4×D83D414yescomplemented
59(C2×D83)⋊C4—24no—
60(C2×D83)⋊C4—28no—
61(C2×D83)⋊C4—24no—
62C4×Dic83—22no—
63C24×C83C2312yes—
64C2×D4×C83—22no—
65C2×D4×C83—22no—
66C2×D4×C83D414yescomplemented
67C22×C4×C83C2312yes—
68C22×C4×C83C2312yes—
69C22×C4×C83C2311yescharacteristic
70C4⋊C4×C83—22no—
71C22⋊C4×C83—44no—
72C42×C83—22no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83—24no—
76C23×C83—24no—
77C23×C83C2×D414yes—
78C23×C83C2411yescharacteristic
79C22×D83—21no—
80C22×D83—21no—
81C22×D83—22no—
82C22×D83—22no—
83C22×D83—22no—
84C22×D83—22no—
85C22×D83—22no—
86C22×D83—24no—
87C22×D83—24no—
88C22×D83C2×D412yes—
89D4×C83—28no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—24no—
95C2×C4×C83C2×D412yes—
96C2×C4×C83C2×D412yes—
97C2×C4×C83C2×D412yes—
98C22⋊D83—216no—
99C22⋊D83—416no—
100C2×Dic83—216no—
101C2×Dic83—22no—
102C2×Dic83—21no—
103C2×Dic83—21no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—22no—
107C2×Dic83C2×D412yes—
108C2×Dic83C2×D414yes—
109C2×Dic83#4114yescomplemented
110D332—48no—
111D332—44no—
112C4×D83—28no—
113Dic166—28no—
114C22×C83—24no—
115C22×C83—28no—
116C22×C83—42no—
117C22×C83—42no—
118C22×C83—44no—
119C22×C83C22×D411yescharacteristic
120C22×C83C22×D411yescharacteristic
121C22×C83C22×D411yescharacteristic
122C22×C83C2×#4112yes—
123C22×C83H(5,𝔽2)12yes—
124C2×D83—22no—
125C2×D83—22no—
126C2×D83—22no—
127C2×D83—42no—
128C2×D83—42no—
129C2×D83—42no—
130C2×D83—42no—
131C2×D83—42no—
132C2×D83—44no—
133C2×D83—44no—
134C4×C83—42no—
135C4×C83—42no—
136C4×C83#12014yes—
137Dic83—24no—
138Dic83—22no—
139Dic83—24no—
140C2×C83—44no—
141C2×C83C2×#12011yescharacteristic
142C2×C83#53211yescharacteristic
143C2×C83#53511yescharacteristic
144C2×C83#54612yes—
145C2×C83#55012yes—
146D83—42no—
147D83—82no—
148#2431—831nomaximal, Sylow
149C83#243111yescharacteristic, complemented, Sylow
150C22×#41—831no—
151C2×#127—831no—
152C2×#124—831no—
153C2×#121—831no—
154#392—834no—
155#391—831no—
156#390D8311yescharacteristic, complemented
157#386—834no—
158C4×C22⋊C4—831no—
159C2×#41—1664no—
160C2×#41—1664no—
161C2×#41—1664no—
162C22×Q8—831no—
163C22×D4—834no—
164C22×D4—832no—
165C22×D4—832no—
166C22×D4—831no—
167C22×D4C2×D8311yescharacteristic
168C23×C4—831no—
169C23×C4—832no—
170#127—1664no—
171#124—1664no—
172#121—1664no—
173C2×C4⋊C4C2×D8311yescharacteristic
174C2×C22⋊C4—832no—
175C2×C22⋊C4—834no—
176C2×C22⋊C4—832no—
177C2×C22⋊C4C2×D8314yes—
178C2×C42—831no—
179C2×C42C2×D8311yescharacteristic
180#95—832no—
181#95—834no—
182C24—831no—
183C24—832no—
184C24C22×D8312yes—
185#41—16632no—
186C2×Q8—834no—
187C2×Q8—1662no—
188C2×Q8—1662no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—3324no—
192C2×D4—3324no—
193C2×D4—1664no—
194C2×D4—1668no—
195C2×D4—1664no—
196C2×D4—1668no—
197C2×D4—1664no—
198C2×D4—1664no—
199C2×D4—1662no—
200C2×D4—1662no—
201C2×D4—1662no—
202C2×D4D33214yescomplemented
203C22×C4—832no—
204C22×C4—831no—
205C22×C4—832no—
206C22×C4—834no—
207C22×C4—1664no—
208C22×C4—1664no—
209C22×C4—1664no—
210C22×C4—1662no—
211C22×C4—1662no—
212C22×C4C22×D8312yes—
213C22×C4C22×D8312yes—
214C22×C4C22×D8311yescharacteristic
215C4⋊C4—22no—
216C22⋊C4—44no—
217C22⋊C4—1664no—
218C22⋊C4—1668no—
219C22⋊C4—1664no—
220C42—22no—
221C42—1662no—
222C23—22no—
223C23—22no—
224C23—24no—
225C23—24no—
226C23—832no—
227C23—1662no—
228C23—1662no—
229C23—1662no—
230C23—1664no—
231C23—1664no—
232C23—1661no—
233C23—1661no—
234C23—1662no—
235C23—1662no—
236C23C2×D33214yes—
237C23C23×D8311yescharacteristic
238Q8—1668no—
239D4—28no—
240D4—3328no—
241D4—33216no—
242D4—3324no—
243D4—16616no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—832no—
250C2×C4—834no—
251C2×C4—834no—
252C2×C4—16616no—
253C2×C4—1661no—
254C2×C4—1662no—
255C2×C4—1662no—
256C2×C4—1661no—
257C2×C4—1662no—
258C2×C4—1662no—
259C2×C4—1668no—
260C2×C4D4×D8312yes—
261C2×C4D4×D8312yes—
262C2×C4C2×D33212yes—
263C22—28no—
264C22—24no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—3322no—
269C22—3322no—
270C22—3324no—
271C22—3324no—
272C22—3322no—
273C22—3322no—
274C22—3322no—
275C22—1662no—
276C22—1662no—
277C22—1662no—
278C22C2×D4×D8311yescharacteristic
279C22C2×D4×D8311yescharacteristic
280C22C22×D33211yescharacteristic
281C22#41×D8312yes—
282C22#1458183712yes—
283C4—42no—
284C4—42no—
285C4—1664no—
286C4—1662no—
287C4—1664no—
288C4#1458176214yes—
289C2—44no—
290C2—3322no—
291C2—6642no—
292C2C2×#1458176211yescharacteristic
293C2#1912825011yescharacteristic
294C2#1912826411yescharacteristic
295C2#1912834612yes—
296C2#1912836912yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#390×C83C2
hypercenter#390D83
hypocenterC83#2431
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C22×#13050589C2semidirect
2. C2×#14581788C2semidirect
3. C2×#14581780C2semidirect
4. C2×#14581767C2semidirect
5. #19127610C2semidirect
6. #19127606C2semidirect
7. #19127578C2semidirect
8. C4×(C2×D83)⋊C4C2semidirect
9. #390×C83C2semidirect
38. C2×Dic166D4semidirect
58. C2×C4×D83D4semidirect
66. C2×D4×C83D4semidirect
109. C2×Dic83#41semidirect
149. C83#2431semidirect
156. #390D83semidirect
202. C2×D4D332semidirect

Maximal subgroups

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

subgroupindex
C22×#130505892
C2×#145817882
C2×#145817802
C2×#145817672
#191276102
#191276062
#191275782
C4×(C2×D83)⋊C42
#390×C832
#243183