Finite Group Databasebeta

← #24913161

Subgroups — #24913161

Order 10624.

476 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 145 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×#14581784C211yesmaximal, characteristic, complemented
2#19127640C211yesmaximal, characteristic, complemented
3#19127637C211yesmaximal, characteristic
4#19127631C211yesmaximal, characteristic, complemented
5#19127626C211yesmaximal, characteristic
6#19127596C212yesmaximal, complemented
7#19127595C212yesmaximal
8#19127585C212yesmaximal, complemented
9#19127574C212yesmaximal
10#19127547C211yesmaximal, characteristic, complemented
11C2×#117×C83C211yesmaximal, characteristic
12#14581784—24no—
13C2×C4⋊Dic83C2211yescharacteristic
14C2×C4⋊Dic83C2211yescharacteristic
15C2×Dic83⋊C4C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2212yes—
21C2×C4×Dic83C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2211yescharacteristic
28C83⋊#95C2211yescharacteristic
29C23×C4×C83C2211yescharacteristic
30#117×C83—24no—
31C2×C4⋊C4×C83C2211yescharacteristic
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C22⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C42×C83C2211yescharacteristic
36C2×C42×C83C2211yescharacteristic
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C4⋊Dic83—22no—
42C4⋊Dic83—22no—
43Dic83⋊C4—24no—
44Dic83⋊C4—24no—
45Dic83⋊C4—24no—
46Dic83⋊C4—24no—
47C22⋊Dic83—24no—
48C22⋊Dic83—24no—
49C4×Dic83—24no—
50C24×C83C2311yescharacteristic
51C22×C4×C83—22no—
52C22×C4×C83—22no—
53C22×C4×C83—22no—
54C22×C4×C83C2311yescharacteristic
55C22×C4×C83C2311yescharacteristic
56C22×C4×C83C2311yescharacteristic
57C22×C4×C83C2311yescharacteristic
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2311yescharacteristic
60C4⋊C4×C83—22no—
61C4⋊C4×C83—22no—
62C22⋊C4×C83—22no—
63C22⋊C4×C83—22no—
64C42×C83—22no—
65C42×C83D414yes—
66C23×C83—21no—
67C23×C83—21no—
68C23×C83—22no—
69C23×C83—22no—
70C23×C83#4112yes—
71C23×C83C2411yescharacteristic
72C2×C4×C83—28no—
73C2×C4×C83—21no—
74C2×C4×C83—21no—
75C2×C4×C83—21no—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83C2×D412yes—
84C2×C4×C83C2×D412yes—
85C2×C4×C83C2×D412yes—
86C2×C4×C83#4112yes—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83#4114yes—
99C22×C83—22no—
100C22×C83—24no—
101C22×C83—41no—
102C22×C83—41no—
103C22×C83—42no—
104C22×C83C22×D411yescharacteristic
105C22×C83C2×#4112yes—
106C22×C83C2×#4111yescharacteristic
107C22×C83C2×#4111yescharacteristic
108C22×C832−1+412yes—
109C4×C83—22no—
110C4×C83—22no—
111C4×C83—22no—
112C4×C83—22no—
113C4×C83—41no—
114C4×C83—41no—
115Dic83—24no—
116Dic83—42no—
117Dic83—42no—
118Dic83—42no—
119C2×C83—42no—
120C2×C83#52511yescharacteristic
121C2×C83#53211yescharacteristic
122C2×C83#53611yescharacteristic
123C2×C83#55212yes—
124C2×C83#55212yes—
125#2611—831nomaximal, Sylow
126C83#261111yescharacteristic, complemented, Sylow
127C2×#125—831no—
128C2×#117D8311yescharacteristic
129#400—831no—
130#399—831no—
131#398—831no—
132#397—831no—
133#389—832no—
134#388—832no—
135#387—832no—
136#385—832no—
137#376—831no—
138C23×C4C2×D8311yescharacteristic
139#125—1664no—
140#117—24no—
141C2×C4⋊C4—832no—
142C2×C4⋊C4—832no—
143C2×C4⋊C4—831no—
144C2×C4⋊C4—831no—
145C2×C4⋊C4—832no—
146C2×C4⋊C4—832no—
147C2×C4⋊C4C2×D8311yescharacteristic
148C2×C4⋊C4C2×D8311yescharacteristic
149C2×C22⋊C4—832no—
150C2×C22⋊C4—832no—
151C2×C22⋊C4C2×D8311yescharacteristic
152C2×C22⋊C4C2×D8311yescharacteristic
153C2×C42—832no—
154C2×C42C2×D8311yescharacteristic
155C2×C42C2×D8311yescharacteristic
156#95—832no—
157#95—831no—
158#95—831no—
159#95—832no—
160#95—832no—
161#95—832no—
162#95—832no—
163C24C22×D8311yescharacteristic
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—832no—
168C22×C4—832no—
169C22×C4—832no—
170C22×C4—832no—
171C22×C4C22×D8311yescharacteristic
172C22×C4C22×D8311yescharacteristic
173C22×C4C22×D8311yescharacteristic
174C22×C4C22×D8311yescharacteristic
175C22×C4C22×D8311yescharacteristic
176C22×C4C22×D8311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—1664no—
180C4⋊C4—1664no—
181C4⋊C4—1664no—
182C4⋊C4—1662no—
183C4⋊C4—1662no—
184C4⋊C4—1664no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—1664no—
188C22⋊C4—1664no—
189C42—22no—
190C42—1664no—
191C42C22⋊D8314yes—
192C23—21no—
193C23—22no—
194C23—21no—
195C23—22no—
196C23#1305059012yes—
197C23C23×D8311yescharacteristic
198C2×C4—28no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—834no—
210C2×C4—1662no—
211C2×C4—1662no—
212C2×C4—1662no—
213C2×C4—1662no—
214C2×C4—1662no—
215C2×C4—1662no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4C2×C22⋊D8312yes—
222C2×C4C2×C22⋊D8312yes—
223C2×C4C2×C22⋊D8312yes—
224C2×C4#1305059012yes—
225C22—24no—
226C22—22no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22C22×C22⋊D8311yescharacteristic
231C22#41×D8312yes—
232C22C2×#1305059011yescharacteristic
233C22C2×#1305059011yescharacteristic
234C22#1458183912yes—
235C4—22no—
236C4—22no—
237C4—22no—
238C4—22no—
239C4—41no—
240C4—41no—
241C4—3322no—
242C4—3322no—
243C4—3322no—
244C4—1664no—
245C2—42no—
246C2#1912821211yescharacteristic
247C2#1912824711yescharacteristic
248C2#1912827211yescharacteristic
249C2#1912838412yes—
250C2#1912839012yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#117×C83C2
prime radicalC24×C83C23
hypercenterC2×#117D83
hypocenterC83#2611
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#14581784C2semidirect
2. #19127640C2semidirect
4. #19127631C2semidirect
6. #19127596C2semidirect
8. #19127585C2semidirect
10. #19127547C2semidirect
126. C83#2611semidirect

Maximal subgroups

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

subgroupindex
C2×#145817842
#191276402
#191276372
#191276312
#191276262
#191275962
#191275952
#191275852
#191275742
#191275472
C2×#117×C832
#261183