Finite Group Databasebeta

← #24915535

Subgroups — #24915535

Order 10624.

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

normal subgroups only

This group has 456 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
1C2×#14581786C211yesmaximal, characteristic, complemented
2#19127634C213yesmaximal, complemented
3#19127624C213yesmaximal, complemented
4#19127623C213yesmaximal, complemented
5#19127617C213yesmaximal
6#19127571C211yesmaximal, characteristic, complemented
7C2×#126×C83C211yesmaximal, characteristic, complemented
8C22×C22⋊D83C2211yescharacteristic
9#14581786—24no—
10C2×C4⋊Dic83C2213yes—
11C2×C4⋊Dic83C2213yes—
12C2×Dic83⋊C4C2213yes—
13C2×Dic83⋊C4C2213yes—
14C2×C22⋊Dic83C2213yes—
15C2×(C2×D83)⋊C4C2213yes—
16C2×(C2×D83)⋊C4C2213yes—
17C83⋊#95C2213yes—
18C83⋊#95C2213yes—
19#126×C83—24no—
20C2×C4⋊C4×C83C2213yes—
21C2×C22⋊C4×C83C2213yes—
22C2×C42×C83C2211yescharacteristic
23C23×D83C2311yescharacteristic
24C2×C22⋊D83—43no—
25C22×Dic83C2311yescharacteristic
26C22×Dic83C2313yes—
27C22×Dic83C2313yes—
28C4⋊Dic83—43no—
29C4⋊Dic83—43no—
30Dic83⋊C4—26no—
31Dic83⋊C4—43no—
32C22⋊Dic83—43no—
33(C2×D83)⋊C4—26no—
34(C2×D83)⋊C4—43no—
35C24×C83C2311yescharacteristic
36C22×C4×C83C2313yes—
37C22×C4×C83C2313yes—
38C4⋊C4×C83—26no—
39C22⋊C4×C83—26no—
40C42×C83D414yescomplemented
41C23×C83—21no—
42C23×C83—23no—
43C23×C83—23no—
44C23×C83C2411yescharacteristic
45C22×D83—21no—
46C22×D83—23no—
47C22×D83—23no—
48C2×C4×C83—23no—
49C2×C4×C83—23no—
50C2×C4×C83—23no—
51C2×C4×C83—23no—
52C2×C4×C83—23no—
53C2×C4×C83C2×D416yes—
54C22⋊D83—44no—
55C2×Dic83—23no—
56C2×Dic83—23no—
57C2×Dic83—23no—
58C2×Dic83—23no—
59C2×Dic83—23no—
60C2×Dic83—23no—
61C2×Dic83—23no—
62C22×C83—43no—
63C22×C83—41no—
64C22×C83—43no—
65C22×C83C22×D411yescharacteristic
66C22×C83H(5,𝔽2)13yes—
67C22×C832−1+413yes—
68C2×D83—43no—
69C2×D83—41no—
70C2×D83—43no—
71C4×C83—26no—
72C4×C83—43no—
73Dic83—43no—
74Dic83—43no—
75Dic83—41no—
76C2×C83—81no—
77C2×C83#53713yes—
78C2×C83#56011yescharacteristic
79C2×C83#56313yes—
80D83—81no—
81#2823—831nomaximal, Sylow
82C83#282311yescharacteristic, complemented, Sylow
83C2×#126—831no—
84C2×#126D8311yescharacteristic, complemented
85#398—833no—
86#396—833no—
87#396—833no—
88#394—833no—
89#384—831no—
90C22×D4—831no—
91#126—24no—
92#126—1664no—
93C2×C4⋊C4—833no—
94C2×C4⋊C4—833no—
95C2×C4⋊C4—833no—
96C2×C4⋊C4—833no—
97C2×C4⋊C4C2×D8313yes—
98C2×C22⋊C4—833no—
99C2×C22⋊C4—833no—
100C2×C22⋊C4—833no—
101C2×C22⋊C4C2×D8313yes—
102C2×C42C2×D8311yescharacteristic
103#95—833no—
104#95—833no—
105C24—831no—
106C24C22×D8311yescharacteristic
107C2×D4—3323no—
108C22×C4—831no—
109C22×C4—833no—
110C22×C4—833no—
111C22×C4C22×D8313yes—
112C22×C4C22×D8313yes—
113C4⋊C4—26no—
114C4⋊C4—3323no—
115C4⋊C4—3323no—
116C4⋊C4—3323no—
117C4⋊C4—1666no—
118C22⋊C4—26no—
119C22⋊C4—3323no—
120C22⋊C4—3323no—
121C22⋊C4—1666no—
122C42C22⋊D8314yescomplemented
123C23—21no—
124C23—23no—
125C23—23no—
126C23—1661no—
127C23—1663no—
128C23—1663no—
129C23C23×D8311yescharacteristic
130D4—3324no—
131C2×C4—23no—
132C2×C4—23no—
133C2×C4—23no—
134C2×C4—23no—
135C2×C4—23no—
136C2×C4—1663no—
137C2×C4—1663no—
138C2×C4—1663no—
139C2×C4—1663no—
140C2×C4—1663no—
141C2×C4—1663no—
142C2×C4—1663no—
143C2×C4C2×C22⋊D8316yes—
144C22—41no—
145C22—43no—
146C22—43no—
147C22—3321no—
148C22—3323no—
149C22—3323no—
150C22C22×C22⋊D8311yescharacteristic
151C22#1458183813yes—
152C22#1458183913yes—
153C4—26no—
154C4—43no—
155C4—3321no—
156C4—3323no—
157C4—3323no—
158C2—81no—
159C2—6641no—
160C2#1912827913yes—
161C2#1912844211yescharacteristic
162C2#1912845313yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#126×C83C2
prime radicalC22×C22⋊D83C22
hypercenterC2×#126D83
hypocenterC83#2823
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#14581786C2semidirect
2. #19127634C2semidirect
3. #19127624C2semidirect
4. #19127623C2semidirect
6. #19127571C2semidirect
7. C2×#126×C83C2semidirect
40. C42×C83D4semidirect
82. C83#2823semidirect
84. C2×#126D83semidirect
122. C42C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817862
#191276342
#191276242
#191276232
#191276172
#191275712
C2×#126×C832
#282383