Finite Group Databasebeta

← #24909538

Subgroups — #24909538

Order 10624.

170 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 43 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#19128139C211yesmaximal, characteristic, complemented
2#19128137C211yesmaximal, characteristic, complemented
3#19128076C211yesmaximal, characteristic, complemented
4#19127817C211yesmaximal, characteristic, complemented
5#19127521C211yesmaximal, characteristic, complemented
6#19127513C211yesmaximal, characteristic, complemented
7#361×C83C211yesmaximal, characteristic, complemented
8#14581814C2211yescharacteristic
9C2×D664C2211yescharacteristic
10C2×#13050575—21no—
11#14581800—21no—
12#14581788—21no—
13#14581737—21no—
14#14581736—21no—
15D1328—21no—
16#14581733—21no—
17#14581732C2211yescharacteristic
18#14581718—21no—
19C8×Dic83C2211yescharacteristic
20C2×D8×C83C2211yescharacteristic
21M32×C83C2211yescharacteristic
22#108×C83C2211yescharacteristic
23#13050590—21no—
24C2×D332—21no—
25C2×C22⋊D83—42no—
26Dic332D411yescharacteristic, complemented
27#13050579—21no—
28D664—21no—
29D664—21no—
30D664D411yescharacteristic, complemented
31#13050575—41no—
32#13050575—41no—
33#13050573—41no—
34C2×C83⋊C8D411yescharacteristic
35C8×D83—41no—
36C4×Dic83D411yescharacteristic
37C83⋊C16—21no—
38C2×D4×C83—21no—
39D8×C83—21no—
40D8×C83—21no—
41M16×C83—21no—
42C2×C8×C83C2311yescharacteristic
43C16×C83—21no—
44C23×C83—41no—
45C22×D83—41no—
46D4×C83—41no—
47D4×C83—41no—
48C2×C4×C83C2×D411yescharacteristic
49C22⋊D83—41no—
50C22⋊D83—84no—
51C2×Dic83—21no—
52D332—21no—
53D332—41no—
54D332—41no—
55C4×D83—41no—
56Dic166—21no—
57C8×C83—41no—
58C8×C83C2×D411yescharacteristic
59C8×C83#4111yescharacteristic
60C83⋊C8D811yescharacteristic
61C83⋊C8D811yescharacteristic
62C22×C83—82no—
63C22×C83—81no—
64C22×C83AGL(1,ℤ/8)11yescharacteristic
65C2×D83—41no—
66C2×D83—82no—
67C2×D83—81no—
68C4×C83#12111yescharacteristic
69C4×C83C2×D811yescharacteristic
70Dic83—22no—
71Dic83—41no—
72C2×C83—21no—
73C2×C83—161no—
74C2×C83#45911yescharacteristic
75D83—81no—
76D83—161no—
77#2213—831nomaximal, Sylow
78C83#221311yescharacteristic, complemented, Sylow
79#509—831no—
80#509—831no—
81#493—831no—
82#443—831no—
83#364—831no—
84#361—831no—
85#361D8311yescharacteristic, complemented
86#135—831no—
87C2×D8—831no—
88C2×D8—1661no—
89C2×D8C2×D8311yescharacteristic
90#131—1661no—
91#127—1661no—
92SD32—1661no—
93SD32—1661no—
94D16—1661no—
95D16—1661no—
96M32—831no—
97M32C2×D8311yescharacteristic
98#108C2×D8311yescharacteristic
99#104—1661no—
100C4×C8—831no—
101#41—1661no—
102C2×D4—21no—
103C2×D4—3322no—
104C2×D4—1661no—
105Q16—831no—
106SD16—1661no—
107D8—21no—
108D8—21no—
109D8—3321no—
110D8—3321no—
111D8—831no—
112D8—1661no—
113D8—1661no—
114M16—21no—
115M16—3321no—
116C2×C8—3321no—
117C2×C8—831no—
118C2×C8C22×D8311yescharacteristic
119C42—831no—
120C16—21no—
121C16—1661no—
122C23—41no—
123C23—3321no—
124Q8—1661no—
125D4—41no—
126D4—41no—
127D4—3321no—
128D4—3321no—
129D4—3321no—
130D4—6644no—
131D4—1661no—
132C2×C4—3321no—
133C2×C4—1661no—
134C2×C4D4×D8311yescharacteristic
135C8—41no—
136C8—831no—
137C8—831no—
138C8D4×D8311yescharacteristic
139C8#1305059011yescharacteristic
140C22—82no—
141C22—81no—
142C22—3321no—
143C22—6642no—
144C22—6641no—
145C22#1458181911yescharacteristic
146C4—3321no—
147C4—1662no—
148C4#1458176511yescharacteristic
149C4D8×D8311yescharacteristic
150C2—21no—
151C2—161no—
152C2—13281no—
153C2—6641no—
154C2#1912789511yescharacteristic
subgroupG/H
centerC2#19127895
commutator, cosocleC2×C8×C83C23
FrattiniC2×C8C22×D83
Fitting#361×C83C2
socleC2×C83#459
hypercenter#361D83
hypocenterC83#2213
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
1. #19128139C2semidirect
2. #19128137C2semidirect
3. #19128076C2semidirect
4. #19127817C2semidirect
5. #19127521C2semidirect
6. #19127513C2semidirect
7. #361×C83C2semidirect
26. Dic332D4semidirect
30. D664D4semidirect
78. C83#2213semidirect
85. #361D83semidirect

Maximal subgroups

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

subgroupindex
#191281392
#191281372
#191280762
#191278172
#191275212
#191275132
#361×C832
#221383