Finite Group Databasebeta

← #24906759

Subgroups — #24906759

Order 10624.

206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 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#19128451C211yesmaximal, characteristic, complemented
2#19128065C212yesmaximal, complemented
3#19127442C212yesmaximal, complemented
4#19127441C211yesmaximal, characteristic, complemented
5#496×C83C211yesmaximal, characteristic
6#14581790—21no—
7#14581785—22no—
8#14581784C2212yes—
9#14581773—21no—
10#14581769—22no—
11#14581713—22no—
12#14581713—22no—
13#14581703C2212yes—
14C2×SD16×C83—21no—
15C2×D8×C83—21no—
16#127×C83C2211yescharacteristic
17#124×C83C2211yescharacteristic
18#97×C83C2211yescharacteristic
19C2×Dic166—21no—
20C22×Dic83—21no—
21C2×C83⋊C8—22no—
22C4⋊Dic83D412yes—
23C4⋊Dic83D412yes—
24Dic83⋊C4—22no—
25Dic83⋊C4—22no—
26Dic83⋊C4—22no—
27C22⋊Dic83—22no—
28C22⋊Dic83—21no—
29C4×Dic83—21no—
30C2×Q8×C83D411yescharacteristic
31C2×D4×C83—21no—
32C2×D4×C83—22no—
33C2×D4×C83D411yescharacteristic
34SD16×C83—42no—
35D8×C83—42no—
36C2×C8×C83—21no—
37C22⋊C4×C83—22no—
38C42×C83C2311yescharacteristic
39C23×C83—21no—
40C23×C83—41no—
41Q8×C83—41no—
42D4×C83—41no—
43D4×C83—41no—
44D4×C83—41no—
45D4×C83—44no—
46C2×C4×C83—21no—
47C2×C4×C83C2×D412yes—
48C2×C4×C83C2×D411yescharacteristic
49C2×Dic83—22no—
50C2×Dic83—22no—
51C2×Dic83—81no—
52Dic166—81no—
53C8×C83—22no—
54C83⋊C8—42no—
55C22×C83—41no—
56C22×C83—42no—
57C22×C83—81no—
58C22×C83—82no—
59C22×C83#12011yescharacteristic
60C4×C83—21no—
61C4×C83—22no—
62C4×C83—41no—
63Dic83—42no—
64Dic83—42no—
65C2×C83—81no—
66C2×C83—161no—
67C2×C83#45311yescharacteristic
68C2×C83#45412yes—
69#1657—831nomaximal, Sylow
70C83#165711yescharacteristic, complemented, Sylow
71#563—831no—
72#496D8311yescharacteristic
73#490—832no—
74#330—832no—
75#329—831no—
76C2×SD16—21no—
77C2×D8—21no—
78C4⋊Q8—1661no—
79#127C2×D8311yescharacteristic
80#126—1662no—
81#125—832no—
82#124C2×D8311yescharacteristic
83#123—1661no—
84C22⋊Q8—1662no—
85#102—1662no—
86#102—1662no—
87#97—832no—
88#97C2×D8311yescharacteristic
89C2×Q8—1661no—
90C2×Q8C22⋊D8311yescharacteristic
91C2×D4—22no—
92C2×D4—21no—
93C2×D4C22⋊D8311yescharacteristic
94C22×C4—1661no—
95SD16—42no—
96D8—42no—
97C2×C8—21no—
98C2×C8—1662no—
99C4⋊C4—832no—
100C4⋊C4—832no—
101C4⋊C4—1662no—
102C4⋊C4—1662no—
103C4⋊C4—1662no—
104C22⋊C4—22no—
105C22⋊C4—1662no—
106C22⋊C4—1661no—
107C42—1661no—
108C42C22×D8311yescharacteristic
109C23—21no—
110C23—41no—
111Q8—41no—
112Q8—6641no—
113D4—41no—
114D4—41no—
115D4—41no—
116D4—44no—
117C2×C4—21no—
118C2×C4—6641no—
119C2×C4—1662no—
120C2×C4—1662no—
121C2×C4D4×D8312yes—
122C2×C4C2×C22⋊D8311yescharacteristic
123C8—22no—
124C8—3322no—
125C22—41no—
126C22—42no—
127C22—81no—
128C22—82no—
129C22#1458176011yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—3322no—
134C4—3322no—
135C2—81no—
136C2—161no—
137C2#1912786811yescharacteristic
138C2#1912787212yes—
subgroupG/H
centerC22#14581760
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting, prime radical#496×C83C2
socleC22×C83#120
hypercenter#496D83
hypocenterC83#1657
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #19128451C2semidirect
2. #19128065C2semidirect
3. #19127442C2semidirect
4. #19127441C2semidirect
70. C83#1657semidirect

Maximal subgroups

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

subgroupindex
#191284512
#191280652
#191274422
#191274412
#496×C832
#165783