Finite Group Databasebeta

← #24910752

Subgroups — #24910752

Order 10624.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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×#14581754C211yesmaximal, characteristic
3#19127639C211yesmaximal, characteristic, complemented
4#19127630C212yesmaximal, complemented
5#19127619C211yesmaximal, characteristic
6#19127618C211yesmaximal, characteristic
7#19127614C212yesmaximal, complemented
8#19127548C212yesmaximal, complemented
9#19127548C212yesmaximal, complemented
10#19127546C211yesmaximal, characteristic, complemented
11#400×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×D332C2211yescharacteristic
14C22×C4×D83C2212yes—
15#14581757—24no—
16#14581754—24no—
17C2×Dic83⋊C4C2211yescharacteristic
18C2×Dic83⋊C4C2211yescharacteristic
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2212yes—
24C2×C4×Dic83C2211yescharacteristic
25C2×C4×Dic83C2211yescharacteristic
26C2×C4×Dic83C2212yes—
27C2×C4×Dic83C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2211yescharacteristic
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34#95×C83C2212yes—
35#95×C83C2211yescharacteristic
36#95×C83C2212yes—
37C23×D83C2312yes—
38C2×Dic166—22no—
39C2×Dic166—22no—
40C2×Dic166—22no—
41C2×D332—24no—
42C2×D332—22no—
43C22×Dic83C2312yes—
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2311yescharacteristic
48C2×C4×D83—24no—
49C2×C4×D83—24no—
50C2×C4×D83—24no—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53(C2×D83)⋊C4—24no—
54(C2×D83)⋊C4—24no—
55(C2×D83)⋊C4—24no—
56(C2×D83)⋊C4—24no—
57(C2×D83)⋊C4—24no—
58C4×Dic83—22no—
59C4×Dic83—22no—
60C4×Dic83—24no—
61C4×Dic83D414yes—
62C22×C4×C83C2312yes—
63C22×C4×C83C2312yes—
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2311yescharacteristic
66C22×C4×C83C2311yescharacteristic
67C4⋊C4×C83—22no—
68C4⋊C4×C83—22no—
69C23×C83C2411yescharacteristic
70C22×D83—22no—
71C22×D83—22no—
72C22×D83—22no—
73C22×D83—22no—
74C22×D83—22no—
75C22×D83—22no—
76C22×D83#4114yes—
77C2×C4×C83—21no—
78C2×C4×C83—22no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83C2×D412yes—
89C2×C4×C83#4112yes—
90C2×Dic83—21no—
91C2×Dic83—21no—
92C2×Dic83—21no—
93C2×Dic83—21no—
94C2×Dic83—21no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83C2×D412yes—
100C2×Dic83C2×D412yes—
101C2×Dic83#4112yes—
102C2×Dic83#4114yes—
103D332—28no—
104C4×D83—48no—
105Dic166—28no—
106C22×C83C22×D411yescharacteristic
107C22×C83C2×#4112yes—
108C22×C83C2×#4111yescharacteristic
109C22×C83C2×#4112yes—
110C22×C83C2×#4111yescharacteristic
111C2×D83—24no—
112C2×D83—24no—
113C2×D83—24no—
114C2×D83—42no—
115C2×D83—42no—
116C2×D83—42no—
117C2×D83—42no—
118C4×C83—22no—
119C4×C83—22no—
120C4×C83—41no—
121C4×C83—42no—
122C4×C83—42no—
123Dic83—22no—
124Dic83—22no—
125Dic83—41no—
126Dic83—42no—
127Dic83#12414yes—
128C2×C83#52511yescharacteristic
129C2×C83#52511yescharacteristic
130C2×C83C2×#12411yescharacteristic
131C2×C83#52912yes—
132C2×C83#52912yes—
133D83—44no—
134#2398—831nomaximal, Sylow
135C83#239811yescharacteristic, complemented, Sylow
136C2×C4×Q8—831no—
137C2×C4×D4—831no—
138#400—831no—
139#400D8311yescharacteristic, complemented
140#397—832no—
141#394—831no—
142#394—831no—
143#393—832no—
144C4×C22⋊C4—832no—
145C4×C22⋊C4—832no—
146#376—831no—
147C22×Q8—831no—
148C22×D4—831no—
149C23×C4—832no—
150C4×Q8—1664no—
151C4×D4—1664no—
152C2×C4⋊C4—831no—
153C2×C4⋊C4—831no—
154C2×C4⋊C4C2×D8311yescharacteristic
155C2×C4⋊C4C2×D8311yescharacteristic
156C2×C22⋊C4—832no—
157C2×C22⋊C4—832no—
158C2×C22⋊C4—832no—
159C2×C22⋊C4—832no—
160C2×C22⋊C4—832no—
161C2×C42—831no—
162C2×C42—831no—
163C2×C42—831no—
164C2×C42—832no—
165#95—832no—
166#95—831no—
167#95—832no—
168#95—832no—
169#95C2×D8311yescharacteristic
170#95C2×D8312yes—
171#95C2×D8312yes—
172C24—832no—
173C2×Q8—1662no—
174C2×Q8—1662no—
175C2×Q8—1662no—
176C2×D4—1664no—
177C2×D4—1662no—
178C22×C4—831no—
179C22×C4—831no—
180C22×C4—831no—
181C22×C4—832no—
182C22×C4—831no—
183C22×C4—1664no—
184C22×C4—1664no—
185C22×C4—1664no—
186C22×C4C22×D8312yes—
187C22×C4C22×D8312yes—
188C22×C4C22×D8311yescharacteristic
189C22×C4C22×D8311yescharacteristic
190C22×C4C22×D8311yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—1662no—
194C4⋊C4—1662no—
195C22⋊C4—1664no—
196C22⋊C4—1664no—
197C22⋊C4—1664no—
198C22⋊C4—1664no—
199C22⋊C4—1664no—
200C42—834no—
201C42—1662no—
202C42—1662no—
203C42—1664no—
204C23—834no—
205C23—1662no—
206C23—1662no—
207C23—1662no—
208C23—1662no—
209C23—1662no—
210C23—1662no—
211C23C23×D8311yescharacteristic
212Q8—1668no—
213D4—1668no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—3328no—
226C2×C4—832no—
227C2×C4—834no—
228C2×C4—832no—
229C2×C4—832no—
230C2×C4—1661no—
231C2×C4—1662no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1661no—
236C2×C4—1661no—
237C2×C4—1661no—
238C2×C4—1661no—
239C2×C4D4×D8312yes—
240C2×C4#1305059112yes—
241C22—3322no—
242C22—3322no—
243C22—3322no—
244C22—3322no—
245C22—1664no—
246C22—1664no—
247C22—1664no—
248C22C2×D4×D8311yescharacteristic
249C22#41×D8312yes—
250C22#41×D8311yescharacteristic
251C22#41×D8312yes—
252C22C2×#1305059111yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—42no—
258C4—3321no—
259C4—3322no—
260C4—834no—
261C4—1662no—
262C4—1662no—
263C2—3324no—
264C2#1912820911yescharacteristic
265C2#1912821411yescharacteristic
266C2#124×D8311yescharacteristic
267C2#1912823412yes—
268C2#1912823612yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#400×C83C2
prime radicalC22×D332C22
hypercenter#400D83
hypocenterC83#2398
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#14581757C2semidirect
3. #19127639C2semidirect
4. #19127630C2semidirect
7. #19127614C2semidirect
8. #19127548C2semidirect
9. #19127548C2semidirect
10. #19127546C2semidirect
11. #400×C83C2semidirect
135. C83#2398semidirect
139. #400D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817572
C2×#145817542
#191276392
#191276302
#191276192
#191276182
#191276142
#191275482
#191275462
#400×C832
#239883