Finite Group Databasebeta

← #24916953

Subgroups — #24916953

Order 10624.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14581823C211yesmaximal, characteristic, complemented
2C2×#14581818C211yesmaximal, characteristic
3C4×#13050590C211yesmaximal, characteristic, complemented
4#19127920C214yesmaximal, complemented
5#19127913C214yesmaximal, complemented
6#19127725C211yesmaximal, characteristic, complemented
7#19127710C211yesmaximal, characteristic, complemented
8#19127706C211yesmaximal, characteristic, complemented
9C2×#124×C83C211yesmaximal, characteristic, complemented
10C2×#13050590C2211yescharacteristic
11#14581823—24no—
12#14581818—24no—
13C2×#13050580C2212yescomplemented
14C2×#13050578C2212yes—
15C2×#13050577C2212yescomplemented
16C2×#13050575C2212yes—
17C2×#13050574C2211yescharacteristic
18C2×#13050574C2211yescharacteristic
19C4×Dic166C2212yescomplemented
20C4×D332C2212yescomplemented
21C4×C22⋊D83—22no—
22#14581751C2211yescharacteristic
23#14581749—21no—
24C42×D83—21no—
25C4⋊(C83⋊C8)C2214yescomplemented
26#14581716C2214yescomplemented
27#14581713C2214yes—
28C22×Q8×C83C2211yescharacteristic
29C22×D4×C83C2211yescharacteristic
30#124×C83—21no—
31#124×C83—21no—
32#124×C83C2214yes—
33C2×C22⋊C4×C83—22no—
34C2×C42×C83C2211yescharacteristic
35#13050590—22no—
36#13050590D414yes—
37C2×Dic166C2311yescharacteristic
38C2×D332C2311yescharacteristic
39C2×C22⋊D83—21no—
40C2×C4×D83—21no—
41C2×C4×D83—22no—
42#13050580—24no—
43#13050578—24no—
44#13050577—24no—
45#13050575—24no—
46#13050574—22no—
47#13050574—22no—
48C2×C83⋊C8C2312yes—
49C2×C83⋊C8C2312yes—
50C4⋊Dic83C2312yes—
51Dic83⋊C4—22no—
52C22⋊Dic83—21no—
53(C2×D83)⋊C4—22no—
54C4×Dic83—21no—
55C4×Dic83—22no—
56C24×C83—21no—
57C2×Q8×C83—21no—
58C2×Q8×C83—22no—
59C2×Q8×C83—22no—
60C2×Q8×C83C2312yes—
61C2×D4×C83—22no—
62C2×D4×C83—21no—
63C2×D4×C83—22no—
64C2×D4×C83C2312yes—
65C22×C4×C83—21no—
66C22×C4×C83D412yes—
67C22×C4×C83C2311yescharacteristic
68C22⋊C4×C83—24no—
69C22⋊C4×C83—24no—
70C42×C83D411yescharacteristic
71C42×C83D411yescharacteristic
72C42×C83C2312yes—
73C23×C83—22no—
74C23×C83—44no—
75C23×C83—42no—
76C23×C83C2×D411yescharacteristic
77C22×D83—21no—
78Q8×C83—22no—
79Q8×C83—22no—
80Q8×C83—42no—
81D4×C83—22no—
82D4×C83—22no—
83D4×C83—42no—
84C2×C4×C83—22no—
85C2×C4×C83—42no—
86C2×C4×C83—42no—
87C2×C4×C83C2×D412yes—
88C2×C4×C83C2×D412yes—
89C2×C4×C83C2×D412yes—
90C2×C4×C83C2×D411yescharacteristic
91C2×C4×C83#4112yes—
92C2×C4×C83C2411yescharacteristic
93C22⋊D83—24no—
94C2×Dic83—21no—
95C2×Dic83—22no—
96D332—21no—
97D332C2×D412yes—
98C4×D83—24no—
99C4×D83—44no—
100Dic166—21no—
101Dic166C2×D412yes—
102C83⋊C8—22no—
103C83⋊C8—22no—
104C22×C83—22no—
105C22×C83—42no—
106C22×C83—44no—
107C22×C83—84no—
108C22×C83#12112yes—
109C22×C83C22×D411yescharacteristic
110C2×D83—22no—
111C2×D83—42no—
112C4×C83—22no—
113C4×C83—22no—
114C4×C83—42no—
115C4×C83#12112yes—
116C4×C83C22×D411yescharacteristic
117C4×C83C2×#4111yescharacteristic
118Dic83—22no—
119Dic83—42no—
120C2×C83—22no—
121C2×C83—82no—
122C2×C83C2×#12111yescharacteristic
123C2×C83#57512yes—
124D83—42no—
125#3040—831nomaximal, Sylow
126C83#304011yescharacteristic, complemented, Sylow
127C2×#137—831no—
128C2×AGL(1,ℤ/8)—831no—
129C2×#124D8311yescharacteristic, complemented
130C4×#41—831no—
131#464—834no—
132#463—834no—
133#423—831no—
134#419—831no—
135#418—831no—
136C2×#41—831no—
137C22×Q8C2×D8311yescharacteristic
138C22×D4C2×D8311yescharacteristic
139#137—1664no—
140AGL(1,ℤ/8)—1664no—
141C2×Q16—832no—
142C2×SD16—832no—
143C2×SD16—832no—
144C2×D8—832no—
145C2×M16—831no—
146C2×M16—831no—
147#124—21no—
148#124—21no—
149#124C2×D8314yes—
150C4×Q8—832no—
151C4×D4—832no—
152C4×D4—1662no—
153#117—831no—
154#117—1661no—
155C2×C22⋊C4—22no—
156C2×C42—1661no—
157C2×C42C2×D8311yescharacteristic
158C4⋊C8—834no—
159#103—834no—
160#102—834no—
161C24—21no—
162#41—834no—
163#41—1662no—
164C2×Q8—21no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—831no—
168C2×Q8C22×D8312yes—
169C2×D4—22no—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—831no—
173C2×D4—1661no—
174C2×D4C22×D8312yes—
175C22×C4—21no—
176C22×C4—1662no—
177C22×C4—1661no—
178C22×C4C22⋊D8312yes—
179C22×C4C22×D8311yescharacteristic
180Q16—1664no—
181SD16—1664no—
182SD16—1664no—
183D8—1664no—
184M16—1662no—
185M16—1662no—
186C2×C8—832no—
187C2×C8—832no—
188C4⋊C4—832no—
189C4⋊C4—1662no—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—1661no—
193C22⋊C4—1662no—
194C42—1661no—
195C42—1662no—
196C42C22⋊D8311yescharacteristic
197C42C22⋊D8311yescharacteristic
198C42C22×D8312yes—
199C23—22no—
200C23—44no—
201C23—42no—
202C23—1661no—
203C23C2×C22⋊D8311yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—42no—
207Q8—832no—
208Q8—1661no—
209D4—22no—
210D4—22no—
211D4—42no—
212D4—832no—
213D4—1664no—
214D4—1661no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—3324no—
219C2×C4—1662no—
220C2×C4—1661no—
221C2×C4—1664no—
222C2×C4D4×D8312yes—
223C2×C4C2×C22⋊D8312yes—
224C2×C4C2×C22⋊D8312yes—
225C2×C4C2×C22⋊D8311yescharacteristic
226C2×C4#1305058912yes—
227C2×C4C23×D8311yescharacteristic
228C8—1662no—
229C8—1662no—
230C22—22no—
231C22—42no—
232C22—44no—
233C22—84no—
234C22—3322no—
235C22—1662no—
236C22#1458176312yes—
237C22C22×C22⋊D8311yescharacteristic
238C4—22no—
239C4—22no—
240C4—42no—
241C4—3322no—
242C4—1662no—
243C4#1458176312yes—
244C4C2×D4×D8311yescharacteristic
245C4C2×#1305058911yescharacteristic
246C2—22no—
247C2—82no—
248C2—3322no—
249C2C2×#1458176311yescharacteristic
250C2#1912849912yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#124×C83C2
socleC22×C83C22×D4
prime radicalC2×#14581818C2
hypercenterC2×#124D83
hypocenterC83#3040
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#14581823C2semidirect
3. C4×#13050590C2semidirect
4. #19127920C2semidirect
5. #19127913C2semidirect
6. #19127725C2semidirect
7. #19127710C2semidirect
8. #19127706C2semidirect
9. C2×#124×C83C2semidirect
13. C2×#13050580C22semidirect
15. C2×#13050577C22semidirect
19. C4×Dic166C22semidirect
20. C4×D332C22semidirect
25. C4⋊(C83⋊C8)C22semidirect
26. #14581716C22semidirect
126. C83#3040semidirect
129. C2×#124D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818232
C2×#145818182
C4×#130505902
#191279202
#191279132
#191277252
#191277102
#191277062
C2×#124×C832
#304083