Finite Group Databasebeta

← #24917631

Subgroups — #24917631

Order 10624.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#19128232C211yesmaximal, characteristic
2C2×#14581757C211yesmaximal, characteristic, complemented
3#19128036C211yesmaximal, characteristic, complemented
4#19128011C212yesmaximal, complemented
5#19127922C212yesmaximal, complemented
6#19127859C211yesmaximal, characteristic, complemented
7#19127839C211yesmaximal, characteristic
8#19127807C212yesmaximal, complemented
9#19127798C212yesmaximal, complemented
10#19127755C211yesmaximal, characteristic, complemented
11#481×C83C211yesmaximal, characteristic
12C22×Dic166C2211yescharacteristic
13C2×Dic332C2212yes—
14C2×#13050578C2212yes—
15#14581785—21no—
16#14581783C2211yescharacteristic
17#14581779C2211yescharacteristic
18#14581773—21no—
19#14581771C2211yescharacteristic
20#14581757—21no—
21#14581757—21no—
22#14581757C2212yes—
23#14581757C2212yes—
24#14581757C2211yescharacteristic
25D4×Dic83—21no—
26D4×Dic83C2211yescharacteristic
27#14581748—21no—
28C2×Dic83⋊C4—21no—
29C2×C4×Dic83—21no—
30C2×C4×Dic83C2211yescharacteristic
31#14581725C2212yes—
32Dic83⋊C8C2212yes—
33#14581717C2212yes—
34#14581715C2212yes—
35#14581715C2212yes—
36C22⋊(C83⋊C8)C2211yescharacteristic
37#14581702C2212yes—
38#121×C83C2211yescharacteristic
39C2×C4⋊C4×C83C2211yescharacteristic
40#106×C83C2212yes—
41#102×C83C2212yes—
42C22⋊C8×C83C2211yescharacteristic
43C2×Dic166—21no—
44C2×Dic166—24no—
45C2×Dic166C2311yescharacteristic
46C2×Dic166C2312yes—
47C22×Dic83—21no—
48C22×Dic83—21no—
49C22×Dic83D411yescharacteristic
50C22×Dic83D411yescharacteristic
51Dic332—24no—
52#13050578—24no—
53C2×C83⋊C8C2312yes—
54C4⋊Dic83—21no—
55C4⋊Dic83C2311yescharacteristic
56Dic83⋊C4—21no—
57Dic83⋊C4—21no—
58Dic83⋊C4—21no—
59Dic83⋊C4—22no—
60Dic83⋊C4—22no—
61C22⋊Dic83—21no—
62C22⋊Dic83—21no—
63C22⋊Dic83—21no—
64C22⋊Dic83—21no—
65C4×Dic83—22no—
66C4×Dic83—22no—
67C4×Dic83—21no—
68C4×Dic83D411yescharacteristic
69C4×Dic83D411yescharacteristic
70C4×Dic83C2311yescharacteristic
71C4×Dic83C2311yescharacteristic
72C2×D4×C83—21no—
73C2×D4×C83C2311yescharacteristic
74C22×C4×C83—21no—
75C22×C4×C83C2311yescharacteristic
76C2×C8×C83C2312yes—
77C4⋊C4×C83—21no—
78C4⋊C4×C83C2311yescharacteristic
79C4⋊C4×C83C2312yes—
80C22⋊C4×C83—21no—
81C23×C83—21no—
82C23×C83C2×D411yescharacteristic
83D4×C83—22no—
84D4×C83—42no—
85C2×C4×C83—22no—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83—42no—
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2411yescharacteristic
91C2×Dic83—24no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—21no—
95C2×Dic83—21no—
96C2×Dic83—41no—
97C2×Dic83—42no—
98C2×Dic83C2×D411yescharacteristic
99C2×Dic83C2×D411yescharacteristic
100C2×Dic83C2×D411yescharacteristic
101C2×Dic83C2×D411yescharacteristic
102Dic166—24no—
103Dic166—22no—
104Dic166—22no—
105Dic166#4114yes—
106C8×C83—22no—
107C83⋊C8—22no—
108C22×C83—21no—
109C22×C83—21no—
110C22×C83—41no—
111C22×C83—41no—
112C22×C83—41no—
113C22×C83#13712yes—
114C22×C83C22×D411yescharacteristic
115C4×C83—21no—
116C4×C83—41no—
117C4×C83—42no—
118C4×C83C2×#4112yes—
119Dic83—24no—
120Dic83—21no—
121Dic83—21no—
122Dic83—21no—
123Dic83—21no—
124Dic83—41no—
125Dic83—41no—
126C2×C83—22no—
127C2×C83—81no—
128C2×C83#52511yescharacteristic
129C2×C83C2×#13711yescharacteristic
130C2×C83#57511yescharacteristic
131#3117—831nomaximal, Sylow
132C83#311711yescharacteristic, complemented, Sylow
133#529—831no—
134C2×C4×Q8—831no—
135#484—831no—
136#481D8311yescharacteristic
137#479—832no—
138#464—832no—
139#451—831no—
140#448—831no—
141#441—832no—
142#440—832no—
143#432—831no—
144C22×Q8—831no—
145C2×Q16—832no—
146C2×SD16—832no—
147#126—1661no—
148#125—831no—
149#124—831no—
150#123—1661no—
151C22⋊Q8—831no—
152#121C2×D8311yescharacteristic
153C4×Q8—832no—
154C4×Q8—832no—
155C4×Q8—831no—
156C4×Q8—1661no—
157C4×Q8—1661no—
158C4×D4—831no—
159C4×D4—1661no—
160#117—1661no—
161C2×C4⋊C4—1661no—
162C2×C4⋊C4C2×D8311yescharacteristic
163C2×C42—831no—
164C2×C42—1661no—
165#107—832no—
166#106C2×D8312yes—
167C4⋊C8—832no—
168#103—832no—
169#103—832no—
170#103—832no—
171#102C2×D8312yes—
172C22⋊C8—831no—
173C22⋊C8C2×D8311yescharacteristic
174#97—832no—
175C2×Q8—832no—
176C2×Q8—831no—
177C2×Q8—1664no—
178C2×Q8—1661no—
179C2×D4—21no—
180C2×D4C22×D8311yescharacteristic
181C22×C4—21no—
182C22×C4—831no—
183C22×C4—831no—
184C22×C4—1661no—
185C22×C4—1661no—
186C22×C4C22×D8311yescharacteristic
187Q16—1664no—
188SD16—1664no—
189C2×C8—832no—
190C2×C8C22×D8312yes—
191C4⋊C4—21no—
192C4⋊C4—831no—
193C4⋊C4—1662no—
194C4⋊C4—1662no—
195C4⋊C4—1661no—
196C4⋊C4—1661no—
197C4⋊C4—1661no—
198C4⋊C4—1661no—
199C4⋊C4C22×D8312yes—
200C4⋊C4C22×D8311yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—1661no—
203C22⋊C4—1661no—
204C22⋊C4—1661no—
205C22⋊C4—1661no—
206C42—831no—
207C42—831no—
208C42—831no—
209C42—831no—
210C42—1662no—
211C42—1662no—
212C42—1661no—
213C23—21no—
214C23D4×D8311yescharacteristic
215Q8—834no—
216Q8—1664no—
217Q8—1662no—
218Q8—1662no—
219D4—22no—
220D4—42no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—3322no—
226C2×C4—3321no—
227C2×C4—831no—
228C2×C4—831no—
229C2×C4—831no—
230C2×C4—831no—
231C2×C4—1661no—
232C2×C4—1661no—
233C2×C4—1664no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4D4×D8311yescharacteristic
237C2×C4C23×D8311yescharacteristic
238C8—22no—
239C8—1662no—
240C22—21no—
241C22—21no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22#1458182112yes—
246C22C2×D4×D8311yescharacteristic
247C4—21no—
248C4—41no—
249C4—42no—
250C4—3321no—
251C4—3321no—
252C4—1664no—
253C4—1661no—
254C4—1661no—
255C4—1661no—
256C4—1661no—
257C4#41×D8312yes—
258C2—22no—
259C2—81no—
260C2#1912820911yescharacteristic
261C2C2×#1458182111yescharacteristic
262C2#1912849711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#481×C83C2
socleC22×C83C22×D4
prime radical#121×C83C22
hypercenter#481D83
hypocenterC83#3117
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#14581757C2semidirect
3. #19128036C2semidirect
4. #19128011C2semidirect
5. #19127922C2semidirect
6. #19127859C2semidirect
8. #19127807C2semidirect
9. #19127798C2semidirect
10. #19127755C2semidirect
132. C83#3117semidirect

Maximal subgroups

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

subgroupindex
#191282322
C2×#145817572
#191280362
#191280112
#191279222
#191278592
#191278392
#191278072
#191277982
#191277552
#481×C832
#311783