Finite Group Databasebeta

← #24909536

Subgroups — #24909536

Order 10624.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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#19128137C212yesmaximal, complemented
2#19127819C211yesmaximal, characteristic, complemented
3#19127522C211yesmaximal, characteristic, complemented
4#19127514C212yesmaximal, complemented
5#493×C83C211yesmaximal, characteristic, complemented
6#14581814C2211yescharacteristic
7#14581798—21no—
8#14581736—22no—
9#14581733—22no—
10#14581732C2212yes—
11#14581728C2211yescharacteristic
12#14581720—21no—
13C2×D8×C83—21no—
14C2×D8×C83C2212yes—
15#127×C83—21no—
16C4×C8×C83C2211yescharacteristic
17#13050590—21no—
18Dic332D411yescharacteristic, complemented
19#13050579—21no—
20D664D411yescharacteristic, complemented
21#13050574—21no—
22#13050573—41no—
23C8×D83—41no—
24C83⋊C16—22no—
25C2×D4×C83—22no—
26C2×D4×C83—42no—
27D8×C83—22no—
28D8×C83—22no—
29D8×C83—41no—
30D8×C83—41no—
31C2×C8×C83D411yescharacteristic
32C2×C8×C83C2311yescharacteristic
33C42×C83D411yescharacteristic
34C23×C83—42no—
35D4×C83—42no—
36D4×C83—42no—
37D4×C83—84no—
38C2×C4×C83—21no—
39C2×C4×C83C2×D411yescharacteristic
40C22⋊D83—41no—
41D332—21no—
42C4×D83—41no—
43Dic166—21no—
44C8×C83D811yescharacteristic
45C8×C83D811yescharacteristic
46C8×C83C2×D411yescharacteristic
47C8×C83#4111yescharacteristic
48C83⋊C8—41no—
49C22×C83—84no—
50C22×C83—82no—
51C22×C83AGL(1,ℤ/8)11yescharacteristic
52C2×D83—41no—
53C4×C83—22no—
54C4×C83#12111yescharacteristic
55C4×C83C2×D811yescharacteristic
56Dic83—41no—
57C2×C83—21no—
58C2×C83—162no—
59C2×C83#45911yescharacteristic
60D83—81no—
61#2213—831nomaximal, Sylow
62C83#221311yescharacteristic, complemented, Sylow
63#509—832no—
64#493D8311yescharacteristic, complemented
65#443—831no—
66#364—831no—
67#361—832no—
68#135—831no—
69C2×D8—21no—
70C2×D8C2×D8312yes—
71#131—1661no—
72#127—21no—
73SD32—1662no—
74D16—1662no—
75M32—832no—
76#108—831no—
77#104—1661no—
78C4×C8C2×D8311yescharacteristic
79#41—1661no—
80C2×D4—22no—
81C2×D4—42no—
82Q16—831no—
83SD16—1661no—
84D8—22no—
85D8—22no—
86D8—41no—
87D8—41no—
88D8—831no—
89M16—3321no—
90M16—1661no—
91C2×C8—3321no—
92C2×C8C22⋊D8311yescharacteristic
93C2×C8C22×D8311yescharacteristic
94C42C22⋊D8311yescharacteristic
95C16—1662no—
96C23—42no—
97Q8—1661no—
98D4—42no—
99D4—42no—
100D4—84no—
101D4—3321no—
102D4—1661no—
103C2×C4—21no—
104C2×C4—3321no—
105C2×C4C2×C22⋊D8311yescharacteristic
106C8—3321no—
107C8#1305057511yescharacteristic
108C8#1305057511yescharacteristic
109C8D4×D8311yescharacteristic
110C8#1305058911yescharacteristic
111C22—84no—
112C22—82no—
113C22—3321no—
114C22#1458181811yescharacteristic
115C4—22no—
116C4—3321no—
117C4#1458176311yescharacteristic
118C4C2×#1305057511yescharacteristic
119C2—21no—
120C2—162no—
121C2—6641no—
122C2#1912789311yescharacteristic
subgroupG/H
centerC2#19127893
commutator, cosocleC2×C8×C83C23
FrattiniC2×C8C22×D83
Fitting#493×C83C2
socleC2×C83#459
hypercenter#493D83
hypocenterC83#2213
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #19128137C2semidirect
2. #19127819C2semidirect
3. #19127522C2semidirect
4. #19127514C2semidirect
5. #493×C83C2semidirect
18. Dic332D4semidirect
20. D664D4semidirect
62. C83#2213semidirect
64. #493D83semidirect

Maximal subgroups

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

subgroupindex
#191281372
#191278192
#191275222
#191275142
#493×C832
#221383