Finite Group Databasebeta

← #24909133

Subgroups — #24909133

Order 10624.

270 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 55 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#19128264C211yesmaximal, characteristic
2#19127673C212yesmaximal, complemented
3#19127494C211yesmaximal, characteristic, complemented
4#19127493C212yesmaximal, complemented
5#353×C83C211yesmaximal, characteristic, complemented
6C2×#13050589C2211yescharacteristic
7C2×D4×D83C411yescharacteristic, complemented
8#14581788C2211yescharacteristic
9#14581780C411yescharacteristic, complemented
10#14581779C411yescharacteristic, complemented
11#14581764—22no—
12#14581760—22no—
13#14581749C411yescharacteristic, complemented
14#14581748—22no—
15#14581707C2212yescomplemented
16#14581706—22no—
17#127×C83C2211yescharacteristic
18#99×C83C2212yescomplemented
19C23×D83—21no—
20#13050589—42no—
21C2×Dic166C2×C411yescharacteristic
22C2×D332C2×C411yescharacteristic
23C2×C22⋊D83—22no—
24C2×C22⋊D83—21no—
25C2×C22⋊D83D412yes—
26D4×D83—42no—
27C22×Dic83D412yes—
28C2×C4×D83C2×C411yescharacteristic
29Dic83⋊C4—21no—
30Dic83⋊C4—42no—
31C22⋊Dic83—22no—
32C22⋊Dic83—22no—
33(C2×D83)⋊C4—21no—
34(C2×D83)⋊C4—22no—
35C4×Dic83—42no—
36C4×Dic83C2×C411yescharacteristic
37C2×D4×C83—22no—
38C2×D4×C83C2×C411yescharacteristic
39C2×D4×C83C2311yescharacteristic
40C22⋊C4×C83—22no—
41C42×C83C2×C411yescharacteristic
42C23×C83—21no—
43C23×C83C2×D412yes—
44C22×D83—21no—
45C22×D83—42no—
46C22×D83—82no—
47C22×D83C22⋊C411yescharacteristic
48D4×C83—41no—
49D4×C83—41no—
50D4×C83—82no—
51C2×C4×C83—21no—
52C2×C4×C83—42no—
53C2×C4×C83C22×C411yescharacteristic
54C22⋊D83—42no—
55C22⋊D83—42no—
56C22⋊D83—82no—
57C2×Dic83—22no—
58C2×Dic83—22no—
59C2×Dic83—21no—
60C2×Dic83—42no—
61C2×Dic83C22⋊C412yes—
62C2×Dic83C22⋊C411yescharacteristic
63D332—41no—
64C4×D83—22no—
65Dic166—41no—
66C22×C83—22no—
67C22×C83—42no—
68C22×C83—42no—
69C22×C83—41no—
70C22×C83C2×C22⋊C411yescharacteristic
71C2×D83—21no—
72C2×D83—41no—
73C2×D83—42no—
74C2×D83—41no—
75C2×D83—84no—
76C4×C83—21no—
77C4×C83—41no—
78C4×C83—82no—
79Dic83—22no—
80Dic83—41no—
81Dic83—82no—
82Dic83#9912yes—
83C2×C83—21no—
84C2×C83—42no—
85C2×C83—81no—
86C2×C83C2×#9911yescharacteristic
87D83—41no—
88D83—81no—
89#2128—831nomaximal, Sylow
90C83#212811yescharacteristic, complemented, Sylow
91#535—831no—
92#410—832no—
93#353—831no—
94#353—832no—
95#353D8311yescharacteristic, complemented
96C2×#41—831no—
97C22×D4—831no—
98#127—831no—
99#127C2×D8311yescharacteristic
100#124—831no—
101#124—831no—
102#121—1662no—
103#120—1662no—
104#117—831no—
105#117—1662no—
106#99—832no—
107#99—1662no—
108#99C2×D8312yescomplemented
109C24—1661no—
110#41—3322no—
111C2×Q8—831no—
112C2×D4—22no—
113C2×D4—3322no—
114C2×D4—832no—
115C2×D4—831no—
116C2×D4—1661no—
117C2×D4—1662no—
118C2×D4C4×D8311yescharacteristic
119C2×D4C22×D8311yescharacteristic
120C22×C4—832no—
121C22×C4—831no—
122C4⋊C4—3322no—
123C4⋊C4—1661no—
124C22⋊C4—22no—
125C22⋊C4—1662no—
126C22⋊C4—1661no—
127C22⋊C4—1662no—
128C22⋊C4—1662no—
129C42—3322no—
130C42—831no—
131C42C4×D8311yescharacteristic
132C23—21no—
133C23—3322no—
134C23—831no—
135C23—6642no—
136C23—1661no—
137C23D4×D8312yes—
138Q8—3321no—
139D4—41no—
140D4—41no—
141D4—82no—
142D4—3321no—
143D4—3322no—
144D4—3322no—
145D4—6642no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—3322no—
149C2×C4—831no—
150C2×C4—832no—
151C2×C4—1662no—
152C2×C4—1661no—
153C2×C4—1662no—
154C2×C4—1662no—
155C2×C4C2×C4×D8311yescharacteristic
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—3321no—
161C22—3322no—
162C22—3321no—
163C22—6644no—
164C22—1661no—
165C22C22⋊C4×D8311yescharacteristic
166C4—21no—
167C4—41no—
168C4—82no—
169C4—3321no—
170C4—832no—
171C4—6642no—
172C4—1662no—
173C2—21no—
174C2—42no—
175C2—81no—
176C2—3321no—
177C2—6641no—
178C2#99×D8311yescharacteristic
subgroupG/H
centerC2#99×D83
commutatorC2×C4×C83C22×C4
FrattiniC2×D4C22×D83
Fitting#353×C83C2
socleC2×C83C2×#99
prime radical#19128264C2
cosocleC2×D4×C83C23
hypercenter#353D83
hypocenterC83#2128
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #19127673C2semidirect
3. #19127494C2semidirect
4. #19127493C2semidirect
5. #353×C83C2semidirect
7. C2×D4×D83C4semidirect
9. #14581780C4semidirect
10. #14581779C4semidirect
13. #14581749C4semidirect
15. #14581707C22semidirect
18. #99×C83C22semidirect
90. C83#2128semidirect
95. #353D83semidirect
108. #99C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282642
#191276732
#191274942
#191274932
#353×C832
#212883