Finite Group Databasebeta

← #24915785

Subgroups — #24915785

Order 10624.

582 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14581775C211yesmaximal, characteristic
2#19127592C214yesmaximal, complemented
3#19127585C214yesmaximal, complemented
4#19127580C212yesmaximal
5#19127574C212yesmaximal, complemented
6#19127564C211yesmaximal, characteristic, complemented
7C22×C42×C83C211yesmaximal, characteristic, complemented
8C22×C22⋊D83C2211yescharacteristic
9#14581775—24no—
10C2×Dic83⋊C4C2214yes—
11C2×Dic83⋊C4C2212yes—
12C2×C22⋊Dic83C2212yes—
13C2×C22⋊Dic83C2211yescharacteristic
14C2×(C2×D83)⋊C4C2214yes—
15C2×(C2×D83)⋊C4C2212yes—
16C83⋊#95C2214yes—
17C83⋊#95C2214yes—
18C83⋊#95C2214yes—
19C23×C4×C83C2212yes—
20C23×C4×C83C2211yescharacteristic
21C2×C42×C83—24no—
22C2×C42×C83C2214yes—
23C23×D83C2311yescharacteristic
24C2×C22⋊D83—22no—
25C2×C22⋊D83—24no—
26C22×Dic83C2311yescharacteristic
27C22×Dic83C2314yes—
28C22×Dic83C2312yes—
29Dic83⋊C4—24no—
30Dic83⋊C4—28no—
31C22⋊Dic83—22no—
32C22⋊Dic83—24no—
33(C2×D83)⋊C4—24no—
34(C2×D83)⋊C4—28no—
35C24×C83C2311yescharacteristic
36C22×C4×C83—24no—
37C22×C4×C83—24no—
38C22×C4×C83—28no—
39C22×C4×C83D414yes—
40C22×C4×C83C2312yes—
41C22×C4×C83C2314yes—
42C42×C83—28no—
43C23×C83—24no—
44C23×C83C2×D412yes—
45C23×C83#4114yes—
46C23×C83C2411yescharacteristic
47C22×D83—21no—
48C22×D83—24no—
49C22×D83—22no—
50C2×C4×C83—216no—
51C2×C4×C83—28no—
52C2×C4×C83—24no—
53C2×C4×C83—24no—
54C2×C4×C83—24no—
55C2×C4×C83C2×D414yes—
56C2×C4×C83#4118yes—
57C22⋊D83—44no—
58C2×Dic83—22no—
59C2×Dic83—21no—
60C2×Dic83—24no—
61C2×Dic83—24no—
62C2×Dic83—24no—
63C2×Dic83—24no—
64C2×Dic83—22no—
65C22×C83—24no—
66C22×C83—28no—
67C22×C83#12314yes—
68C22×C83C22×D411yescharacteristic
69C22×C83C2×#4112yes—
70C22×C83C2×#4114yes—
71C2×D83—41no—
72C2×D83—42no—
73C2×D83—44no—
74C4×C83—24no—
75C4×C83—28no—
76Dic83—41no—
77Dic83—42no—
78Dic83—44no—
79C2×C83—24no—
80C2×C83C2×#12311yescharacteristic
81C2×C83#52512yes—
82C2×C83#52914yes—
83D83—81no—
84#2853—831nomaximal, Sylow
85C83#285311yescharacteristic, complemented, Sylow
86C2×#123—831no—
87C22×C42D8311yescharacteristic, complemented
88#388—834no—
89#387—834no—
90#386—832no—
91#385—832no—
92#381—831no—
93C22×D4—831no—
94C23×C4C2×D8312yes—
95C23×C4C2×D8311yescharacteristic
96#123—1664no—
97C2×C4⋊C4—832no—
98C2×C4⋊C4—834no—
99C2×C22⋊C4—831no—
100C2×C22⋊C4—832no—
101C2×C22⋊C4—832no—
102C2×C22⋊C4—834no—
103C2×C42—24no—
104C2×C42C2×D8314yes—
105#95—834no—
106#95—834no—
107#95—834no—
108C24—831no—
109C24C22×D8311yescharacteristic
110C2×D4—1662no—
111C2×D4—1664no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—28no—
115C22×C4—834no—
116C22×C4—831no—
117C22×C4—832no—
118C22×C4C22⋊D8314yes—
119C22×C4C22×D8312yes—
120C22×C4C22×D8314yes—
121C4⋊C4—1664no—
122C4⋊C4—1668no—
123C22⋊C4—1662no—
124C22⋊C4—1664no—
125C22⋊C4—1664no—
126C22⋊C4—1668no—
127C42—28no—
128C23—24no—
129C23—1661no—
130C23—1664no—
131C23—1662no—
132C23C2×C22⋊D8312yes—
133C23#1305059014yes—
134C23C23×D8311yescharacteristic
135D4—3324no—
136C2×C4—216no—
137C2×C4—28no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—1662no—
142C2×C4—1661no—
143C2×C4—1664no—
144C2×C4—1664no—
145C2×C4—1664no—
146C2×C4—1664no—
147C2×C4—1662no—
148C2×C4C2×C22⋊D8314yes—
149C2×C4#1305059018yes—
150C22—24no—
151C22—28no—
152C22—3321no—
153C22—3322no—
154C22—3324no—
155C22#1458177514yes—
156C22C22×C22⋊D8311yescharacteristic
157C22C2×#1305059012yes—
158C22C2×#1305059014yes—
159C4—24no—
160C4—28no—
161C4—3321no—
162C4—3322no—
163C4—3324no—
164C2—24no—
165C2—6641no—
166C2C2×#1458177511yescharacteristic
167C2#1912821212yes—
168C2#1912823514yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC22×C42×C83C2
prime radicalC22×C22⋊D83C22
hypercenterC22×C42D83
hypocenterC83#2853
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #19127592C2semidirect
3. #19127585C2semidirect
5. #19127574C2semidirect
6. #19127564C2semidirect
7. C22×C42×C83C2semidirect
85. C83#2853semidirect
87. C22×C42D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817752
#191275922
#191275852
#191275802
#191275742
#191275642
C22×C42×C832
#285383