Finite Group Databasebeta

← #24906753

Subgroups — #24906753

Order 10624.

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

normal subgroups only

This group has 262 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#19128443C211yesmaximal, characteristic, complemented
2#19128091C211yesmaximal, characteristic, complemented
3#19128090C212yesmaximal, complemented
4#19127440C212yesmaximal, complemented
5#329×C83C211yesmaximal, characteristic, complemented
6C2×#13050579—22no—
7C2×#13050577—21no—
8C2×D664—22no—
9C2×#13050575—21no—
10#14581789C2211yescharacteristic, complemented
11#14581781—21no—
12#14581780C2212yes—
13#14581778—22no—
14#14581762—22no—
15#14581760—21no—
16#14581703C2211yescharacteristic, complemented
17#124×C83C2211yescharacteristic
18#97×C83C2212yescomplemented
19C23×D83—21no—
20C2×Dic166D412yes—
21C2×D332—21no—
22C2×D332—22no—
23C2×D332—21no—
24C2×D332D412yes—
25C2×C22⋊D83—22no—
26#13050579—44no—
27#13050577—42no—
28D664—44no—
29#13050575—42no—
30C2×C83⋊C8—21no—
31C22⋊Dic83—21no—
32(C2×D83)⋊C4—22no—
33(C2×D83)⋊C4—22no—
34(C2×D83)⋊C4—22no—
35C4×Dic83—21no—
36C2×Q8×C83D411yescharacteristic
37C2×D4×C83D411yescharacteristic
38C2×C8×C83—22no—
39C22⋊C4×C83—22no—
40C42×C83C2311yescharacteristic
41C23×C83—21no—
42C22×D83—22no—
43C22×D83—41no—
44C22×D83—82no—
45C22×D83—81no—
46Q8×C83—41no—
47D4×C83—41no—
48C2×C4×C83—21no—
49C2×C4×C83C2×D412yes—
50C2×C4×C83C2×D411yescharacteristic
51C22⋊D83—82no—
52C2×Dic83—22no—
53D332—42no—
54D332—41no—
55D332—42no—
56D332—41no—
57D332—42no—
58D332—81no—
59Dic166—42no—
60C8×C83—24no—
61C83⋊C8—22no—
62C22×C83—42no—
63C22×C83—41no—
64C22×C83#12011yescharacteristic
65C2×D83—42no—
66C2×D83—42no—
67C2×D83—42no—
68C2×D83—84no—
69C2×D83—82no—
70C2×D83—81no—
71C4×C83—21no—
72C4×C83—22no—
73C4×C83—41no—
74Dic83—42no—
75C2×C83—81no—
76C2×C83#45312yes—
77C2×C83#45311yescharacteristic
78D83—82no—
79D83—161no—
80#1655—831nomaximal, Sylow
81C83#165511yescharacteristic, complemented, Sylow
82H(3,𝔽4)—831no—
83#496—831no—
84#496—832no—
85#329—832no—
86#329D8311yescharacteristic, complemented
87C2×SD16—1662no—
88C2×SD16—1661no—
89C2×D8—1662no—
90C2×D8—1661no—
91#127—831no—
92#124—832no—
93#124—1662no—
94#124—1661no—
95#124C2×D8311yescharacteristic
96#120—1662no—
97#120—1661no—
98#97—831no—
99#97C2×D8312yescomplemented
100C24—1661no—
101C2×Q8—832no—
102C2×Q8D33211yescharacteristic
103C2×D4—832no—
104C2×D4—1661no—
105C2×D4—1662no—
106C2×D4—1662no—
107C2×D4—1661no—
108C2×D4D33211yescharacteristic
109SD16—3324no—
110SD16—3322no—
111D8—3324no—
112D8—3322no—
113C2×C8—22no—
114C2×C8—1661no—
115C22⋊C4—22no—
116C22⋊C4—1662no—
117C22⋊C4—1661no—
118C22⋊C4—1662no—
119C22⋊C4—1662no—
120C42—1661no—
121C42C22×D8311yescharacteristic
122C23—21no—
123C23—3321no—
124C23—6642no—
125C23—6641no—
126C23—1662no—
127Q8—41no—
128Q8—3322no—
129D4—41no—
130D4—3322no—
131D4—3322no—
132D4—3321no—
133D4—3322no—
134D4—3321no—
135D4—6641no—
136D4—6642no—
137C2×C4—21no—
138C2×C4—1662no—
139C2×C4D4×D8312yes—
140C2×C4C2×D33211yescharacteristic
141C8—24no—
142C8—1662no—
143C22—42no—
144C22—41no—
145C22—3322no—
146C22—3322no—
147C22—3322no—
148C22—6644no—
149C22—6642no—
150C22—6641no—
151C22#1458176211yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—3322no—
156C2—81no—
157C2—13281no—
158C2—6642no—
159C2#1912787012yes—
160C2#1912787111yescharacteristic
subgroupG/H
centerC22#14581762
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#329×C83C2
socleC22×C83#120
hypercenter#329D83
hypocenterC83#1655
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #19128443C2semidirect
2. #19128091C2semidirect
3. #19128090C2semidirect
4. #19127440C2semidirect
5. #329×C83C2semidirect
10. #14581789C22semidirect
16. #14581703C22semidirect
18. #97×C83C22semidirect
81. C83#1655semidirect
86. #329D83semidirect
99. #97C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284432
#191280912
#191280902
#191274402
#329×C832
#165583