Finite Group Databasebeta

← #24913043

Subgroups — #24913043

Order 10624.

492 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 135 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×#14581757C211yesmaximal, characteristic, complemented
2C2×#14581750C211yesmaximal, characteristic
3#19127627C211yesmaximal, characteristic
4#19127621C211yesmaximal, characteristic, complemented
5#19127614C211yesmaximal, characteristic
6#19127590C212yesmaximal
7#19127587C212yesmaximal, complemented
8#19127583C212yesmaximal, complemented
9#19127572C212yesmaximal
10#19127568C211yesmaximal, characteristic, complemented
11#398×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14#14581757—24no—
15#14581750—24no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2211yescharacteristic
22C2×(C2×D83)⋊C4C2211yescharacteristic
23C2×C4×Dic83C2211yescharacteristic
24C2×C4×Dic83C2212yes—
25C2×C4×Dic83C2211yescharacteristic
26C83⋊#95C2212yes—
27C83⋊#95C2212yes—
28C83⋊#95C2211yescharacteristic
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2212yes—
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2211yescharacteristic
35#95×C83C2211yescharacteristic
36#95×C83C2212yes—
37C23×D83C2311yescharacteristic
38C2×Dic166—22no—
39C2×Dic166—24no—
40C22×Dic83C2312yes—
41C22×Dic83C2311yescharacteristic
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2312yes—
45C2×C4×D83—22no—
46C2×C4×D83—22no—
47C2×C4×D83—22no—
48C4⋊Dic83—22no—
49Dic83⋊C4—24no—
50Dic83⋊C4—24no—
51(C2×D83)⋊C4—24no—
52(C2×D83)⋊C4—22no—
53(C2×D83)⋊C4—22no—
54(C2×D83)⋊C4—24no—
55C4×Dic83—22no—
56C4×Dic83—24no—
57C4×Dic83D414yes—
58C22×C4×C83C2312yes—
59C22×C4×C83C2311yescharacteristic
60C22×C4×C83C2312yes—
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2311yescharacteristic
63C4⋊C4×C83—22no—
64C4⋊C4×C83—22no—
65C4⋊C4×C83—24no—
66C23×C83C2411yescharacteristic
67C22×D83—21no—
68C22×D83—22no—
69C22×D83—21no—
70C22×D83—22no—
71C22×D83#4112yes—
72C2×C4×C83—21no—
73C2×C4×C83—21no—
74C2×C4×C83—22no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83C2×D412yes—
84C2×C4×C83#4112yes—
85C2×Dic83—22no—
86C2×Dic83—21no—
87C2×Dic83—21no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83C2×D412yes—
96C2×Dic83C2×D412yes—
97C2×Dic83#4114yes—
98C4×D83—28no—
99Dic166—28no—
100C22×C83C22×D411yescharacteristic
101C22×C83C2×#4111yescharacteristic
102C22×C83C2×#4112yes—
103C22×C83C2×#4111yescharacteristic
104C22×C832−1+412yes—
105C2×D83—22no—
106C2×D83—24no—
107C2×D83—41no—
108C2×D83—41no—
109C2×D83—42no—
110C4×C83—22no—
111C4×C83—22no—
112C4×C83—41no—
113C4×C83—42no—
114C4×C83—42no—
115Dic83—24no—
116Dic83—22no—
117Dic83—22no—
118Dic83—41no—
119Dic83—42no—
120C2×C83#52511yescharacteristic
121C2×C83#52511yescharacteristic
122C2×C83#53611yescharacteristic
123C2×C83#55212yes—
124C2×C83#55612yes—
125D83—42no—
126#2599—831nomaximal, Sylow
127C83#259911yescharacteristic, complemented, Sylow
128C2×C4×Q8—831no—
129C2×#117—831no—
130#398D8311yescharacteristic, complemented
131#397—831no—
132#395—831no—
133#393—831no—
134#388—832no—
135#387—832no—
136#387—832no—
137#385—832no—
138#383—831no—
139C22×Q8—831no—
140C23×C4—831no—
141C4×Q8—1664no—
142#117—1664no—
143C2×C4⋊C4—831no—
144C2×C4⋊C4—832no—
145C2×C4⋊C4—832no—
146C2×C4⋊C4C2×D8311yescharacteristic
147C2×C4⋊C4C2×D8311yescharacteristic
148C2×C4⋊C4C2×D8312yes—
149C2×C22⋊C4—831no—
150C2×C22⋊C4—831no—
151C2×C22⋊C4—832no—
152C2×C22⋊C4—832no—
153C2×C42—831no—
154C2×C42—831no—
155C2×C42—832no—
156#95—832no—
157#95—832no—
158#95—831no—
159#95—832no—
160#95—832no—
161#95—832no—
162#95C2×D8311yescharacteristic
163#95C2×D8312yes—
164C24—831no—
165C2×Q8—1664no—
166C2×Q8—1662no—
167C22×C4—831no—
168C22×C4—831no—
169C22×C4—832no—
170C22×C4—832no—
171C22×C4—831no—
172C22×C4—1662no—
173C22×C4—1662no—
174C22×C4—1662no—
175C22×C4C22×D8312yes—
176C22×C4C22×D8312yes—
177C22×C4C22×D8311yescharacteristic
178C22×C4C22×D8311yescharacteristic
179C22×C4C22×D8311yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—24no—
183C4⋊C4—1662no—
184C4⋊C4—1664no—
185C4⋊C4—1664no—
186C22⋊C4—1662no—
187C22⋊C4—1664no—
188C22⋊C4—1664no—
189C22⋊C4—1662no—
190C42—834no—
191C42—1662no—
192C42—1664no—
193C23—832no—
194C23—1661no—
195C23—1662no—
196C23—1661no—
197C23—1662no—
198C23C23×D8311yescharacteristic
199Q8—1668no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—832no—
212C2×C4—832no—
213C2×C4—834no—
214C2×C4—1668no—
215C2×C4—1661no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4—1661no—
222C2×C4—1662no—
223C2×C4—1661no—
224C2×C4—1662no—
225C2×C4D4×D8312yes—
226C2×C4#1305058912yes—
227C22—3322no—
228C22—3321no—
229C22—3321no—
230C22—1664no—
231C22—1662no—
232C22C2×D4×D8311yescharacteristic
233C22#41×D8311yescharacteristic
234C22#41×D8312yes—
235C22C2×#1305058911yescharacteristic
236C22#1458184012yes—
237C4—22no—
238C4—22no—
239C4—42no—
240C4—41no—
241C4—42no—
242C4—3322no—
243C4—3321no—
244C4—1664no—
245C4—1662no—
246C4—1662no—
247C2—3322no—
248C2#1912820911yescharacteristic
249C2#1912821111yescharacteristic
250C2#1912827611yescharacteristic
251C2#1912838312yes—
252C2#1912842212yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#398×C83C2
prime radicalC23×D83C23
hypercenter#398D83
hypocenterC83#2599
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#14581757C2semidirect
4. #19127621C2semidirect
7. #19127587C2semidirect
8. #19127583C2semidirect
10. #19127568C2semidirect
11. #398×C83C2semidirect
127. C83#2599semidirect
130. #398D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817572
C2×#145817502
#191276272
#191276212
#191276142
#191275902
#191275872
#191275832
#191275722
#191275682
#398×C832
#259983