Finite Group Databasebeta

← #24917852

Subgroups — #24917852

Order 10624.

400 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#14581780C211yesmaximal, characteristic
2#19128063C214yesmaximal, complemented
3#19128058C214yesmaximal, complemented
4#19127709C212yesmaximal, complemented
5#19127705C212yesmaximal
6C4×#13050574C211yesmaximal, characteristic, complemented
7#533×C83C211yesmaximal, characteristic, complemented
8C22×Dic166C2211yescharacteristic
9C22×D332C2211yescharacteristic
10C2×#13050574C2212yes—
11#14581780—22no—
12#14581780C2212yes—
13#14581780C2212yes—
14C2×(C2×D83)⋊C4—22no—
15#14581715C2218yescomplemented
16#14581712C2218yes—
17#14581703C2212yescomplemented
18C4×C83⋊C8C2212yescomplemented
19C4⋊Q8×C83C2212yescomplemented
20#125×C83C2212yescomplemented
21C22⋊Q8×C83—22no—
22C4×Q8×C83—22no—
23#117×C83C2212yes—
24C2×C42×C83C2211yescharacteristic
25C23×D83—21no—
26C2×Dic166—24no—
27C2×Dic166—21no—
28C2×Dic166C2312yes—
29C2×D332—24no—
30C2×D332—21no—
31C2×D332C2312yes—
32C22×Dic83—21no—
33#13050574—24no—
34C2×C83⋊C8C2314yes—
35(C2×D83)⋊C4—24no—
36(C2×D83)⋊C4—24no—
37C2×Q8×C83—22no—
38C22×C4×C83D412yes—
39C22×C4×C83C2311yescharacteristic
40C4⋊C4×C83—24no—
41C4⋊C4×C83—22no—
42C4⋊C4×C83C2314yes—
43C22⋊C4×C83—22no—
44C42×C83—22no—
45C42×C83D412yes—
46C42×C83C2311yescharacteristic
47C42×C83C2311yescharacteristic
48C23×C83C2×D411yescharacteristic
49C22×D83—22no—
50C22×D83—44no—
51C22×D83—42no—
52Q8×C83—44no—
53C2×C4×C83—24no—
54C2×C4×C83—42no—
55C2×C4×C83C2×D412yes—
56C2×C4×C83C2×D412yes—
57C2×C4×C83C2×D411yescharacteristic
58C2×C4×C83#4114yes—
59C2×C4×C83C2411yescharacteristic
60C2×Dic83—22no—
61C2×Dic83—42no—
62D332—24no—
63D332—42no—
64Dic166—24no—
65Dic166—42no—
66C83⋊C8—24no—
67C22×C83—22no—
68C22×C83#12412yes—
69C22×C83C22×D411yescharacteristic
70C2×D83—42no—
71C2×D83—44no—
72C2×D83—84no—
73C4×C83—22no—
74C4×C83—22no—
75C4×C83—44no—
76C4×C83#12412yes—
77C4×C83C2×#4112yes—
78Dic83—42no—
79C2×C83—22no—
80C2×C83C2×#12411yescharacteristic
81C2×C83#57512yes—
82D83—82no—
83#3140—831nomaximal, Sylow
84C83#314011yescharacteristic, complemented, Sylow
85#533D8311yescharacteristic, complemented
86C2×#124—831no—
87#489—834no—
88#488—834no—
89#419—832no—
90#418—832no—
91C4×M16—831no—
92C22×Q8—831no—
93C22×D4—831no—
94C2×M16—832no—
95C4⋊Q8C2×D8312yescomplemented
96#125C2×D8312yescomplemented
97#124—832no—
98#124—832no—
99#124—1662no—
100C22⋊Q8—22no—
101C4×Q8—22no—
102#117C2×D8312yes—
103C2×C22⋊C4—1662no—
104C2×C42C2×D8311yescharacteristic
105#103—838no—
106#102—838no—
107#97—832no—
108C4×C8—832no—
109C24—1661no—
110C2×Q8—22no—
111C2×Q8—832no—
112C2×Q8—1664no—
113C2×Q8—1661no—
114C2×D4—832no—
115C2×D4—1661no—
116C2×D4—1664no—
117C22×C4—1661no—
118C22×C4C22⋊D8312yes—
119C22×C4C22×D8311yescharacteristic
120M16—1664no—
121C2×C8—834no—
122C4⋊C4—24no—
123C4⋊C4—22no—
124C4⋊C4C22×D8314yes—
125C22⋊C4—22no—
126C22⋊C4—1664no—
127C22⋊C4—1664no—
128C42—22no—
129C42C22⋊D8312yes—
130C42C22×D8311yescharacteristic
131C42C22×D8311yescharacteristic
132C23—3324no—
133C23—3322no—
134C23—1662no—
135C23C2×C22⋊D8311yescharacteristic
136Q8—44no—
137Q8—3322no—
138Q8—1664no—
139D4—3322no—
140D4—1664no—
141C2×C4—24no—
142C2×C4—42no—
143C2×C4—3322no—
144C2×C4—1662no—
145C2×C4C2×C22⋊D8312yes—
146C2×C4C2×C22⋊D8312yes—
147C2×C4C2×C22⋊D8311yescharacteristic
148C2×C4#1305059114yes—
149C2×C4C23×D8311yescharacteristic
150C8—1664no—
151C22—22no—
152C22—3322no—
153C22—3324no—
154C22—6644no—
155C22#1458178112yes—
156C22C22×C22⋊D8311yescharacteristic
157C4—22no—
158C4—22no—
159C4—44no—
160C4—3322no—
161C4#1458178112yes—
162C4C2×#1305059112yes—
163C2—22no—
164C2—6642no—
165C2C2×#1458178111yescharacteristic
166C2#1912849912yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#533×C83C2
socleC22×C83C22×D4
prime radicalC22×D332C22
hypercenter#533D83
hypocenterC83#3140
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #19128063C2semidirect
3. #19128058C2semidirect
4. #19127709C2semidirect
6. C4×#13050574C2semidirect
7. #533×C83C2semidirect
15. #14581715C22semidirect
17. #14581703C22semidirect
18. C4×C83⋊C8C22semidirect
19. C4⋊Q8×C83C22semidirect
20. #125×C83C22semidirect
84. C83#3140semidirect
85. #533D83semidirect
95. C4⋊Q8C2×D83semidirect
96. #125C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817802
#191280632
#191280582
#191277092
#191277052
C4×#130505742
#533×C832
#314083