Finite Group Databasebeta

← #24917706

Subgroups — #24917706

Order 10624.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128306C211yesmaximal, characteristic, complemented
2#19128025C212yesmaximal, complemented
3#19128001C212yesmaximal, complemented
4#19127896C212yesmaximal, complemented
5#19127835C212yesmaximal, complemented
6#19127811C212yesmaximal, complemented
7C4×#13050575C212yesmaximal, complemented
8#19127762C211yesmaximal, characteristic, complemented
9#518×C83C211yesmaximal, characteristic, complemented
10C2×#13050575C2212yes—
11C2×#13050575C2212yes—
12#14581789C2211yescharacteristic, complemented
13#14581784C2211yescharacteristic, complemented
14#14581777—22no—
15#14581766C2212yes—
16#14581765—22no—
17C4×D332C2212yescomplemented
18#14581725C2212yescomplemented
19#14581723C2212yescomplemented
20C4⋊(C83⋊C8)C2212yescomplemented
21#14581713C2212yes—
22#14581713C2212yes—
23#14581712C2212yescomplemented
24#14581712C2212yescomplemented
25C22⋊(C83⋊C8)C2212yes—
26#14581703C2211yescharacteristic, complemented
27C4×C83⋊C8C2211yescharacteristic, complemented
28C22×D4×C83C2211yescharacteristic
29C4×D4×C83—22no—
30C4×D4×C83C2212yes—
31C4×D4×C83C2212yes—
32#117×C83C2211yescharacteristic
33#117×C83C2211yescharacteristic
34C2×C22⋊C4×C83—22no—
35C2×D332—22no—
36C2×D332C2312yes—
37C2×C22⋊D83—22no—
38C2×C4×D83—22no—
39#13050575—24no—
40#13050575—24no—
41C2×C83⋊C8C2312yes—
42C2×C83⋊C8C2312yes—
43C4⋊Dic83C2312yes—
44Dic83⋊C4—22no—
45(C2×D83)⋊C4—22no—
46(C2×D83)⋊C4—22no—
47C24×C83—21no—
48C2×D4×C83—24no—
49C2×D4×C83—21no—
50C2×D4×C83C2312yes—
51C22×C4×C83—22no—
52C22×C4×C83—22no—
53C22×C4×C83C2311yescharacteristic
54C4⋊C4×C83D412yes—
55C4⋊C4×C83C2312yes—
56C22⋊C4×C83—21no—
57C22⋊C4×C83—22no—
58C22⋊C4×C83—22no—
59C22⋊C4×C83D412yes—
60C42×C83—21no—
61C42×C83C2311yescharacteristic
62C42×C83C2311yescharacteristic
63C23×C83—24no—
64C23×C83—22no—
65C23×C83—82no—
66C23×C83C2×D411yescharacteristic
67C22×D83—22no—
68D4×C83—24no—
69D4×C83—22no—
70D4×C83#4114yes—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—82no—
74C2×C4×C83—82no—
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83C2×D412yes—
77C2×C4×C83C2×D412yes—
78C2×C4×C83C2411yescharacteristic
79C22⋊D83—82no—
80C2×Dic83—22no—
81D332—22no—
82D332—22no—
83D332—82no—
84C4×D83—82no—
85C83⋊C8—22no—
86C83⋊C8—22no—
87C22×C83—21no—
88C22×C83—24no—
89C22×C83—22no—
90C22×C83—42no—
91C22×C83—42no—
92C22×C83—84no—
93C22×C83C22×D411yescharacteristic
94C2×D83—42no—
95C2×D83—42no—
96C2×D83—42no—
97C4×C83—21no—
98C4×C83—22no—
99C4×C83—22no—
100C4×C83—42no—
101C4×C83C2×#4112yes—
102Dic83—42no—
103C2×C83—41no—
104C2×C83—44no—
105C2×C83#52511yescharacteristic
106C2×C83#57612yes—
107D83—82no—
108#3122—831nomaximal, Sylow
109C83#312211yescharacteristic, complemented, Sylow
110#540—831no—
111#518D8311yescharacteristic, complemented
112#482—832no—
113#478—832no—
114#459—832no—
115#447—832no—
116#442—832no—
117C4×D8—832no—
118#433—831no—
119C22×D4C2×D8311yescharacteristic
120C2×D8—832no—
121C2×D8—832no—
122#127—831no—
123#125—831no—
124#123—1662no—
125#121—832no—
126#121—1662no—
127C4×D4—22no—
128C4×D4—832no—
129C4×D4C2×D8312yes—
130C4×D4C2×D8312yes—
131#117C2×D8311yescharacteristic
132#117C2×D8311yescharacteristic
133C2×C22⋊C4—22no—
134#107—832no—
135#106—832no—
136C4⋊C8—832no—
137#102—832no—
138#102—832no—
139#102—832no—
140#102—832no—
141C22⋊C8—832no—
142#97—831no—
143C4×C8—831no—
144C24—21no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—832no—
148C2×D4—1662no—
149C2×D4—1662no—
150C2×D4C22×D8312yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—1662no—
154C22×C4C22×D8311yescharacteristic
155D8—1664no—
156D8—1664no—
157C2×C8—832no—
158C2×C8—832no—
159C4⋊C4—832no—
160C4⋊C4—1662no—
161C4⋊C4C22⋊D8312yes—
162C4⋊C4C22×D8312yes—
163C22⋊C4—21no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—1662no—
167C22⋊C4—1662no—
168C22⋊C4C22⋊D8312yes—
169C42—21no—
170C42C22×D8311yescharacteristic
171C42C22×D8311yescharacteristic
172C23—24no—
173C23—22no—
174C23—82no—
175C23—1662no—
176C23C2×C22⋊D8311yescharacteristic
177D4—24no—
178D4—22no—
179D4—6642no—
180D4—6642no—
181D4—1662no—
182D4—1662no—
183D4#1305059014yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—6642no—
189C2×C4—1662no—
190C2×C4C2×C22⋊D8311yescharacteristic
191C2×C4C2×C22⋊D8312yes—
192C2×C4C2×C22⋊D8312yes—
193C2×C4C23×D8311yescharacteristic
194C8—1662no—
195C8—1662no—
196C22—21no—
197C22—24no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—3322no—
203C22—3322no—
204C22—3322no—
205C22C22×C22⋊D8311yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—3322no—
211C4C2×#1305059012yes—
212C2—41no—
213C2—44no—
214C2—6642no—
215C2#1912821211yescharacteristic
216C2#1912850412yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#518×C83C2
socleC22×C83C22×D4
hypercenter#518D83
hypocenterC83#3122
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #19128306C2semidirect
2. #19128025C2semidirect
3. #19128001C2semidirect
4. #19127896C2semidirect
5. #19127835C2semidirect
6. #19127811C2semidirect
7. C4×#13050575C2semidirect
8. #19127762C2semidirect
9. #518×C83C2semidirect
12. #14581789C22semidirect
13. #14581784C22semidirect
17. C4×D332C22semidirect
18. #14581725C22semidirect
19. #14581723C22semidirect
20. C4⋊(C83⋊C8)C22semidirect
23. #14581712C22semidirect
24. #14581712C22semidirect
26. #14581703C22semidirect
27. C4×C83⋊C8C22semidirect
109. C83#3122semidirect
111. #518D83semidirect

Maximal subgroups

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

subgroupindex
#191283062
#191280252
#191280012
#191278962
#191278352
#191278112
C4×#130505752
#191277622
#518×C832
#312283