Finite Group Databasebeta

← #24913333

Subgroups — #24913333

Order 10624.

512 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 129 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×#14581782C211yesmaximal, characteristic, complemented
2C2×#14581754C211yesmaximal, characteristic
3#19127636C212yesmaximal, complemented
4#19127632C211yesmaximal, characteristic, complemented
5#19127630C212yesmaximal, complemented
6#19127619C211yesmaximal, characteristic
7#19127618C211yesmaximal, characteristic
8#19127575C212yesmaximal, complemented
9#19127572C212yesmaximal, complemented
10#19127568C211yesmaximal, characteristic, complemented
11#398×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C4×D83C2212yes—
14#14581782—24no—
15#14581754—24no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2211yescharacteristic
18C2×Dic83⋊C4C2211yescharacteristic
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2212yes—
27C2×C4×Dic83C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2211yescharacteristic
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2212yes—
35#95×C83C2211yescharacteristic
36#95×C83C2212yes—
37C23×D83C2312yes—
38C2×D332—24no—
39C2×D332—22no—
40C22×Dic83C2312yes—
41C22×Dic83C2311yescharacteristic
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2311yescharacteristic
45C2×C4×D83—24no—
46C2×C4×D83—42no—
47C2×C4×D83—42no—
48C4⋊Dic83—22no—
49Dic83⋊C4—22no—
50Dic83⋊C4—22no—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53Dic83⋊C4—42no—
54(C2×D83)⋊C4—24no—
55(C2×D83)⋊C4—24no—
56(C2×D83)⋊C4—24no—
57(C2×D83)⋊C4—42no—
58(C2×D83)⋊C4—42no—
59C4×Dic83D414yes—
60C22×C4×C83C2312yes—
61C22×C4×C83C2312yes—
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2311yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C4⋊C4×C83—22no—
66C4⋊C4×C83—22no—
67C4⋊C4×C83—42no—
68C23×C83C2411yescharacteristic
69C22×D83—22no—
70C22×D83—22no—
71C22×D83—22no—
72C22×D83—22no—
73C22×D83—22no—
74C22×D83—22no—
75C22×D83#4114yes—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—21no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83C2×D412yes—
89C2×Dic83—21no—
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83—21no—
93C2×Dic83—21no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—21no—
97C2×Dic83—22no—
98C2×Dic83—21no—
99C2×Dic83—22no—
100C2×Dic83C2×D412yes—
101C2×Dic83C2×D412yes—
102D332—28no—
103C4×D83—48no—
104C22×C83C22×D411yescharacteristic
105C22×C83C2×#4112yes—
106C22×C83H(5,𝔽2)11yescharacteristic
107C22×C83H(5,𝔽2)11yescharacteristic
108C22×C832−1+412yes—
109C2×D83—24no—
110C2×D83—24no—
111C2×D83—24no—
112C2×D83—42no—
113C2×D83—42no—
114C2×D83—42no—
115C2×D83—42no—
116C4×C83—22no—
117C4×C83—41no—
118C4×C83—42no—
119C4×C83—41no—
120C4×C83—42no—
121Dic83—22no—
122Dic83—22no—
123Dic83—41no—
124Dic83—41no—
125Dic83—42no—
126C2×C83#52511yescharacteristic
127C2×C83#53511yescharacteristic
128C2×C83#53611yescharacteristic
129C2×C83#53912yes—
130C2×C83#53912yes—
131D83—44no—
132#2633—831nomaximal, Sylow
133C83#263311yescharacteristic, complemented, Sylow
134C2×#125—831no—
135C2×C4×D4—831no—
136#399—832no—
137#398—831no—
138#398D8311yescharacteristic, complemented
139#397—832no—
140#394—831no—
141#394—831no—
142#385—832no—
143#385—832no—
144#383—831no—
145C22×D4—831no—
146C23×C4—832no—
147#125—1664no—
148C4×D4—1664no—
149C2×C4⋊C4—832no—
150C2×C4⋊C4—831no—
151C2×C4⋊C4—831no—
152C2×C4⋊C4—831no—
153C2×C4⋊C4—831no—
154C2×C4⋊C4—831no—
155C2×C4⋊C4C2×D8311yescharacteristic
156C2×C4⋊C4C2×D8311yescharacteristic
157C2×C4⋊C4C2×D8312yes—
158C2×C22⋊C4—832no—
159C2×C22⋊C4—832no—
160C2×C22⋊C4—832no—
161C2×C22⋊C4—832no—
162C2×C22⋊C4—832no—
163C2×C42—831no—
164#95—832no—
165#95—831no—
166#95—832no—
167#95—832no—
168#95C2×D8311yescharacteristic
169#95C2×D8312yes—
170C24—832no—
171C2×D4—1664no—
172C2×D4—1662no—
173C22×C4—3322no—
174C22×C4—3322no—
175C22×C4—831no—
176C22×C4—831no—
177C22×C4—832no—
178C22×C4—831no—
179C22×C4—831no—
180C22×C4—1664no—
181C22×C4C22×D8312yes—
182C22×C4C22×D8312yes—
183C22×C4C22×D8311yescharacteristic
184C22×C4C22×D8311yescharacteristic
185C22×C4C22×D8311yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—3322no—
190C4⋊C4—1662no—
191C4⋊C4—1662no—
192C4⋊C4—1662no—
193C4⋊C4—1662no—
194C4⋊C4—1662no—
195C22⋊C4—3322no—
196C22⋊C4—3322no—
197C22⋊C4—1664no—
198C22⋊C4—1664no—
199C22⋊C4—1664no—
200C42—834no—
201C23—834no—
202C23—1662no—
203C23—1662no—
204C23—1662no—
205C23—1662no—
206C23—1662no—
207C23—1662no—
208C23C23×D8311yescharacteristic
209D4—1668no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—3328no—
223C2×C4—832no—
224C2×C4—832no—
225C2×C4—1661no—
226C2×C4—1661no—
227C2×C4—1662no—
228C2×C4—1662no—
229C2×C4—1661no—
230C2×C4—1662no—
231C2×C4—1661no—
232C2×C4—1662no—
233C2×C4—1661no—
234C2×C4—1661no—
235C2×C4—1662no—
236C2×C4D4×D8312yes—
237C22—3322no—
238C22—3322no—
239C22—3322no—
240C22—3322no—
241C22—1664no—
242C22—1664no—
243C22—1664no—
244C22C2×D4×D8311yescharacteristic
245C22#41×D8312yes—
246C22#1458183711yescharacteristic
247C22#1458183811yescharacteristic
248C22#1458184012yes—
249C4—22no—
250C4—41no—
251C4—42no—
252C4—41no—
253C4—42no—
254C4—3321no—
255C4—3321no—
256C4—3322no—
257C4—1662no—
258C4—1662no—
259C2—3324no—
260C2#1912820911yescharacteristic
261C2#1912826711yescharacteristic
262C2#1912827611yescharacteristic
263C2#1912829712yes—
264C2#1912829912yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#398×C83C2
prime radicalC22×D332C22
hypercenter#398D83
hypocenterC83#2633
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#14581782C2semidirect
3. #19127636C2semidirect
4. #19127632C2semidirect
5. #19127630C2semidirect
8. #19127575C2semidirect
9. #19127572C2semidirect
10. #19127568C2semidirect
11. #398×C83C2semidirect
133. C83#2633semidirect
138. #398D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817822
C2×#145817542
#191276362
#191276322
#191276302
#191276192
#191276182
#191275752
#191275722
#191275682
#398×C832
#263383