Finite Group Databasebeta

← #24906786

Subgroups — #24906786

Order 10624.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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#19128071C211yesmaximal, characteristic, complemented
2#19128060C212yesmaximal, complemented
3#19127443C211yesmaximal, characteristic, complemented
4#19127441C212yesmaximal, complemented
5#562×C83C211yesmaximal, characteristic
6#14581792C2211yescharacteristic
7#14581716—22no—
8#14581715—22no—
9#14581713—22no—
10#14581703C2211yescharacteristic
11#14581703C2212yes—
12#125×C83C2211yescharacteristic
13#124×C83C2212yes—
14#123×C83—22no—
15C22⋊Q8×C83—22no—
16#121×C83—22no—
17C2×Dic166—21no—
18C2×C83⋊C8—22no—
19C2×C83⋊C8—21no—
20C4⋊Dic83—22no—
21C2×Q8×C83D412yes—
22C2×D4×C83—22no—
23C2×D4×C83D412yes—
24C22×C4×C83—22no—
25C4⋊C4×C83—22no—
26C4⋊C4×C83—22no—
27C4⋊C4×C83D412yes—
28C22⋊C4×C83—22no—
29C22⋊C4×C83—22no—
30C22⋊C4×C83—22no—
31C42×C83C2311yescharacteristic
32C23×C83—22no—
33Q8×C83—42no—
34D4×C83—42no—
35D4×C83—82no—
36C2×C4×C83—22no—
37C2×C4×C83—22no—
38C2×C4×C83—82no—
39C2×C4×C83C2×D411yescharacteristic
40C2×C4×C83C2×D412yes—
41C2×Dic83—41no—
42Dic166—42no—
43C83⋊C8—41no—
44C83⋊C8—42no—
45C22×C83—42no—
46C22×C83—42no—
47C22×C83—42no—
48C22×C83#12011yescharacteristic
49C4×C83—21no—
50C4×C83—22no—
51C4×C83—42no—
52C4×C83—42no—
53Dic83—81no—
54C2×C83—82no—
55C2×C83#45411yescharacteristic
56C2×C83#45512yes—
57#1662—831nomaximal, Sylow
58C83#166211yescharacteristic, complemented, Sylow
59#562D8311yescharacteristic
60#491—831no—
61#489—832no—
62#330—831no—
63#329—832no—
64C4⋊Q8—831no—
65#125C2×D8311yescharacteristic
66#124C2×D8312yes—
67#123—22no—
68C22⋊Q8—22no—
69#121—22no—
70#103—1662no—
71#103—1662no—
72#102—1662no—
73#97—831no—
74#97—832no—
75C2×Q8—1661no—
76C2×Q8C22⋊D8312yes—
77C2×D4—22no—
78C2×D4C22⋊D8312yes—
79C22×C4—22no—
80C2×C8—1662no—
81C2×C8—1661no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—1662no—
85C4⋊C4C22⋊D8312yes—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C22⋊C4—22no—
89C42C22×D8311yescharacteristic
90C23—22no—
91Q8—42no—
92Q8—3322no—
93D4—42no—
94D4—82no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—82no—
98C2×C4—3321no—
99C2×C4C2×C22⋊D8311yescharacteristic
100C2×C4C2×C22⋊D8312yes—
101C8—3321no—
102C8—3322no—
103C22—42no—
104C22—42no—
105C22—42no—
106C22#1458176111yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—42no—
111C4—6641no—
112C2—82no—
113C2#1912787411yescharacteristic
114C2#1912787912yes—
subgroupG/H
centerC22#14581761
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting, prime radical#562×C83C2
socleC22×C83#120
hypercenter#562D83
hypocenterC83#1662
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #19128071C2semidirect
2. #19128060C2semidirect
3. #19127443C2semidirect
4. #19127441C2semidirect
58. C83#1662semidirect

Maximal subgroups

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

subgroupindex
#191280712
#191280602
#191274432
#191274412
#562×C832
#166283