Finite Group Databasebeta

← #24906756

Subgroups — #24906756

Order 10624.

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

normal subgroups only

This group has 230 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#19128091C212yesmaximal, complemented
2#19128067C211yesmaximal, characteristic, complemented
3#19127443C211yesmaximal, characteristic, complemented
4#19127441C212yesmaximal, complemented
5#562×C83C211yesmaximal, characteristic, complemented
6C2×#13050577—22no—
7C2×#13050575—22no—
8#14581789C2211yescharacteristic
9#14581712—22no—
10#14581703C2211yescharacteristic
11#14581703C2212yes—
12#125×C83C2211yescharacteristic
13#124×C83C2212yes—
14#123×C83—22no—
15C22⋊Q8×C83—22no—
16#121×C83—22no—
17C2×D332—22no—
18C2×D332—21no—
19#13050577—44no—
20#13050575—44no—
21C2×C83⋊C8—22no—
22C2×C83⋊C8—21no—
23C2×Q8×C83D412yes—
24C2×D4×C83—22no—
25C2×D4×C83D412yes—
26C22×C4×C83—22no—
27C4⋊C4×C83—22no—
28C4⋊C4×C83—22no—
29C4⋊C4×C83D412yes—
30C22⋊C4×C83—22no—
31C22⋊C4×C83—22no—
32C22⋊C4×C83—22no—
33C42×C83C2311yescharacteristic
34C23×C83—22no—
35C22×D83—41no—
36Q8×C83—42no—
37D4×C83—42no—
38D4×C83—82no—
39C2×C4×C83—22no—
40C2×C4×C83—22no—
41C2×C4×C83—82no—
42C2×C4×C83C2×D411yescharacteristic
43C2×C4×C83C2×D412yes—
44D332—42no—
45D332—42no—
46D332—42no—
47C83⋊C8—24no—
48C83⋊C8—41no—
49C22×C83—42no—
50C22×C83—42no—
51C22×C83—42no—
52C22×C83#12011yescharacteristic
53C2×D83—82no—
54C2×D83—81no—
55C4×C83—22no—
56C4×C83—21no—
57C4×C83—42no—
58C4×C83—42no—
59C2×C83—82no—
60C2×C83#45312yes—
61C2×C83#45411yescharacteristic
62D83—161no—
63#1656—831nomaximal, Sylow
64C83#165611yescharacteristic, complemented, Sylow
65#562D8311yescharacteristic, complemented
66#496—832no—
67#490—831no—
68#330—831no—
69#329—832no—
70C2×SD16—1662no—
71C2×D8—1662no—
72#127—831no—
73#125C2×D8311yescharacteristic
74#124C2×D8312yes—
75#123—22no—
76C22⋊Q8—22no—
77#121—22no—
78#102—1662no—
79#97—831no—
80#97—832no—
81C2×Q8C22⋊D8312yes—
82C2×D4—22no—
83C2×D4—1661no—
84C2×D4—1662no—
85C2×D4C22⋊D8312yes—
86C22×C4—22no—
87SD16—3324no—
88D8—3324no—
89C2×C8—1661no—
90C2×C8—1662no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D8312yes—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C42C22×D8311yescharacteristic
98C23—22no—
99C23—3321no—
100Q8—42no—
101D4—42no—
102D4—82no—
103D4—3322no—
104D4—3322no—
105D4—3322no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—82no—
109C2×C4C2×C22⋊D8311yescharacteristic
110C2×C4C2×C22⋊D8312yes—
111C8—3321no—
112C8—1664no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—6642no—
117C22—6641no—
118C22#1458176111yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C2—82no—
124C2—13281no—
125C2#1912786912yes—
126C2#1912787411yescharacteristic
subgroupG/H
centerC22#14581761
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#562×C83C2
socleC22×C83#120
hypercenter#562D83
hypocenterC83#1656
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. #19128091C2semidirect
2. #19128067C2semidirect
3. #19127443C2semidirect
4. #19127441C2semidirect
5. #562×C83C2semidirect
64. C83#1656semidirect
65. #562D83semidirect

Maximal subgroups

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

subgroupindex
#191280912
#191280672
#191274432
#191274412
#562×C832
#165683