Finite Group Databasebeta

← #24917394

Subgroups — #24917394

Order 10624.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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#19128212C211yesmaximal, characteristic
2#19128042C214yesmaximal, complemented
3#19128025C214yesmaximal, complemented
4#19127741C212yesmaximal, complemented
5#19127700C212yesmaximal, complemented
6#19127665C211yesmaximal, characteristic, complemented
7C2×#117×C83C211yesmaximal, characteristic, complemented
8C2×#13050590C2211yescharacteristic
9C2×#13050574C2212yes—
10#14581775—21no—
11#14581771C2212yes—
12#14581766C2212yes—
13#14581761—21no—
14C4×C22⋊D83—22no—
15#14581751C2211yescharacteristic
16#14581725C2214yescomplemented
17#14581723C2214yescomplemented
18#14581715C2214yescomplemented
19#14581712C2214yescomplemented
20C22⋊(C83⋊C8)C2214yes—
21C23×C4×C83C2211yescharacteristic
22#117×C83—22no—
23#117×C83C2214yes—
24C2×C4⋊C4×C83C2212yes—
25C2×C22⋊C4×C83—21no—
26C2×C42×C83—21no—
27#13050590—24no—
28C2×Dic166C2311yescharacteristic
29C2×D332C2311yescharacteristic
30C2×C22⋊D83—22no—
31C2×C22⋊D83—21no—
32C2×C4×D83—21no—
33#13050574—24no—
34C2×C83⋊C8C2314yes—
35C4⋊Dic83C2312yes—
36Dic83⋊C4—22no—
37C22⋊Dic83—22no—
38C22⋊Dic83—21no—
39(C2×D83)⋊C4—22no—
40C4×Dic83—21no—
41C24×C83D411yescharacteristic
42C22×C4×C83—24no—
43C22×C4×C83—22no—
44C22×C4×C83D411yescharacteristic
45C22×C4×C83D412yes—
46C22×C4×C83C2312yes—
47C22×C4×C83C2311yescharacteristic
48C4⋊C4×C83—22no—
49C4⋊C4×C83C2314yes—
50C22⋊C4×C83—24no—
51C42×C83—24no—
52C23×C83—22no—
53C23×C83—42no—
54C23×C83C2×D412yes—
55C23×C83C2×D411yescharacteristic
56C22×D83—21no—
57C2×C4×C83—22no—
58C2×C4×C83—24no—
59C2×C4×C83—24no—
60C2×C4×C83—24no—
61C2×C4×C83—44no—
62C2×C4×C83C2×D412yes—
63C2×C4×C83C2×D411yescharacteristic
64C2×C4×C83#4114yes—
65C2×C4×C83C2411yescharacteristic
66C22⋊D83—44no—
67C22⋊D83—42no—
68C2×Dic83—22no—
69C2×Dic83—21no—
70D332—22no—
71C4×D83—42no—
72Dic166—22no—
73C83⋊C8—24no—
74C22×C83—22no—
75C22×C83—22no—
76C22×C83—22no—
77C22×C83—22no—
78C22×C83—44no—
79C22×C83#12312yes—
80C22×C83C22×D411yescharacteristic
81C2×D83—41no—
82C2×D83—42no—
83C4×C83—22no—
84C4×C83—44no—
85C4×C83#12312yes—
86C4×C83C2×#4112yes—
87Dic83—41no—
88Dic83—42no—
89C2×C83—22no—
90C2×C83—42no—
91C2×C83C2×#12311yescharacteristic
92C2×C83#57512yes—
93D83—81no—
94#3093—831nomaximal, Sylow
95C83#309311yescharacteristic, complemented, Sylow
96#525—831no—
97C2×#117D8311yescharacteristic, complemented
98#485—834no—
99#482—834no—
100#428—832no—
101#417—832no—
102#407—831no—
103C2×#41—831no—
104C23×C4C2×D8311yescharacteristic
105C2×M16—832no—
106#123—1661no—
107C22⋊Q8—832no—
108#121—832no—
109#120—1661no—
110C4×D4—1662no—
111#117—22no—
112#117—831no—
113#117C2×D8314yes—
114C2×C4⋊C4C2×D8312yes—
115C2×C22⋊C4—21no—
116C2×C42—21no—
117#107—834no—
118#106—834no—
119#103—834no—
120#102—834no—
121C22⋊C8—834no—
122C24C22⋊D8311yescharacteristic
123#41—1664no—
124C2×Q8—831no—
125C2×D4—831no—
126C2×D4—1662no—
127C2×D4—1661no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—1661no—
131C22×C4C22⋊D8311yescharacteristic
132C22×C4C22⋊D8312yes—
133C22×C4C22×D8312yes—
134C22×C4C22×D8311yescharacteristic
135M16—1664no—
136C2×C8—834no—
137C4⋊C4—22no—
138C4⋊C4—832no—
139C4⋊C4—1662no—
140C4⋊C4C22×D8314yes—
141C22⋊C4—24no—
142C22⋊C4—1661no—
143C22⋊C4—1662no—
144C22⋊C4—1662no—
145C42—24no—
146C42—1661no—
147C23—22no—
148C23—42no—
149C23—1661no—
150C23C2×C22⋊D8312yes—
151C23C2×C22⋊D8311yescharacteristic
152Q8—1662no—
153D4—3324no—
154D4—3322no—
155D4—1662no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—44no—
161C2×C4—3322no—
162C2×C4—1662no—
163C2×C4—1661no—
164C2×C4C2×C22⋊D8312yes—
165C2×C4C2×C22⋊D8311yescharacteristic
166C2×C4#1305059014yes—
167C2×C4C23×D8311yescharacteristic
168C8—1664no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—3321no—
175C22—3322no—
176C22#1458177512yes—
177C22C22×C22⋊D8311yescharacteristic
178C4—22no—
179C4—44no—
180C4—3321no—
181C4—3322no—
182C4#1458177512yes—
183C4C2×#1305059012yes—
184C2—22no—
185C2—42no—
186C2—6641no—
187C2C2×#1458177511yescharacteristic
188C2#1912849912yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#117×C83C2
socleC22×C83C22×D4
prime radical#19128212C2
hypercenterC2×#117D83
hypocenterC83#3093
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #19128042C2semidirect
3. #19128025C2semidirect
4. #19127741C2semidirect
5. #19127700C2semidirect
6. #19127665C2semidirect
7. C2×#117×C83C2semidirect
16. #14581725C22semidirect
17. #14581723C22semidirect
18. #14581715C22semidirect
19. #14581712C22semidirect
95. C83#3093semidirect
97. C2×#117D83semidirect

Maximal subgroups

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

subgroupindex
#191282122
#191280422
#191280252
#191277412
#191277002
#191276652
C2×#117×C832
#309383