Finite Group Databasebeta

← #24908810

Subgroups — #24908810

Order 10624.

230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 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#19127744C212yesmaximal
2#19127566C211yesmaximal, characteristic
3#19127470C211yesmaximal, characteristic
4#19127460C212yesmaximal
5#544×C83C211yesmaximal, characteristic
6C2×#13050574C2212yes—
7C2×C4⋊Dic83C2211yescharacteristic
8C2×C4×Dic83C2211yescharacteristic
9#14581725—22no—
10#14581723—22no—
11C83⋊#95—22no—
12C4⋊Q8×C83—22no—
13#125×C83—22no—
14C22⋊Q8×C83—22no—
15#117×C83C2212yes—
16C2×C4⋊C4×C83C2211yescharacteristic
17C22×Dic83—21no—
18C22×Dic83—21no—
19#13050574—22no—
20#13050574Q814yescomplemented
21C2×C83⋊C8—22no—
22C4⋊Dic83—22no—
23C4×Dic83—21no—
24C4×Dic83Q812yescomplemented
25C2×Q8×C83—22no—
26C22×C4×C83D412yes—
27C22×C4×C83C2311yescharacteristic
28C4⋊C4×C83—22no—
29C4⋊C4×C83—24no—
30C4⋊C4×C83—22no—
31C4⋊C4×C83D412yes—
32C4⋊C4×C83D412yes—
33C22⋊C4×C83—22no—
34C42×C83—22no—
35C23×C83C2×D411yescharacteristic
36Q8×C83—82no—
37C2×C4×C83—22no—
38C2×C4×C83—22no—
39C2×C4×C83—22no—
40C2×C4×C83—42no—
41C2×C4×C83C2×D412yes—
42C2×C4×C83C2×Q812yes—
43C2×C4×C83C2×Q811yescharacteristic
44C2×C4×C83#4111yescharacteristic
45C2×Dic83—22no—
46C2×Dic83—41no—
47C2×Dic83—41no—
48C2×Dic83—42no—
49C2×Dic83—41no—
50C83⋊C8—24no—
51C22×C83—21no—
52C22×C83—21no—
53C22×C83#12011yescharacteristic
54C22×C83C22⋊Q811yescharacteristic
55C22×C83C4⋊Q811yescharacteristic
56C4×C83—42no—
57C4×C83—42no—
58C4×C83—42no—
59C4×C83C22⋊Q812yes—
60C4×C83C4⋊Q812yes—
61Dic83—42no—
62Dic83—81no—
63C2×C83—21no—
64C2×C83—21no—
65C2×C83#39511yescharacteristic
66C2×C83#45411yescharacteristic
67C2×C83#45511yescharacteristic
68#2061—831nomaximal, Sylow
69C83#206111yescharacteristic, complemented, Sylow
70#544D8311yescharacteristic
71#428—832no—
72#382—831no—
73#343—831no—
74#339—832no—
75C2×M16—832no—
76C4⋊Q8—22no—
77#125—22no—
78C22⋊Q8—22no—
79#117C2×D8312yes—
80C2×C4⋊C4—831no—
81C2×C4⋊C4C2×D8311yescharacteristic
82C2×C42—831no—
83#107—1662no—
84#106—1662no—
85#95—1662no—
86C2×Q8—22no—
87C22×C4—1661no—
88C22×C4—1661no—
89C22×C4C22⋊D8312yes—
90C22×C4C22×D8311yescharacteristic
91M16—834no—
92M16—1662no—
93C2×C8—1662no—
94C4⋊C4—22no—
95C4⋊C4—24no—
96C4⋊C4—22no—
97C4⋊C4—1662no—
98C4⋊C4C22⋊D8312yes—
99C4⋊C4C22⋊D8312yes—
100C22⋊C4—22no—
101C42—22no—
102C42—832no—
103C42—1661no—
104C23C2×C22⋊D8311yescharacteristic
105Q8—82no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—3321no—
111C2×C4—3321no—
112C2×C4—3322no—
113C2×C4—3321no—
114C2×C4—1662no—
115C2×C4C2×C22⋊D8312yes—
116C2×C4Q8×D8312yes—
117C2×C4Q8×D8311yescharacteristic
118C2×C4#1305059111yescharacteristic
119C8—1664no—
120C22—21no—
121C22—21no—
122C22#1458176111yescharacteristic
123C22#1458176811yescharacteristic
124C22#1458179011yescharacteristic
125C4—42no—
126C4—42no—
127C4—42no—
128C4—3322no—
129C4—6641no—
130C4#1458176812yes—
131C4#1458179012yes—
132C2—21no—
133C2—21no—
134C2#1912762011yescharacteristic
135C2#1912787411yescharacteristic
136C2#1912787911yescharacteristic
subgroupG/H
centerC22#14581761
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#544×C83C2
socleC22×C83#120
prime radicalC23×C83C2×D4
hypercenter#544D83
hypocenterC83#2061
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
20. #13050574Q8semidirect
24. C4×Dic83Q8semidirect
69. C83#2061semidirect

Maximal subgroups

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

subgroupindex
#191277442
#191275662
#191274702
#191274602
#544×C832
#206183