Finite Group Databasebeta

← #24909129

Subgroups — #24909129

Order 10624.

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

normal subgroups only

This group has 286 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#99×D83C212yesmaximal, complemented
3#19127495C212yesmaximal, complemented
4#19127494C211yesmaximal, characteristic, complemented
5#353×C83C211yesmaximal, characteristic, complemented
6C2×#13050589C411yescharacteristic, complemented
7C2×D4×D83C2211yescharacteristic
8#14581788C411yescharacteristic, complemented
9#14581780C411yescharacteristic, complemented
10#14581779C2211yescharacteristic
11#14581764—22no—
12#14581760—22no—
13#14581749C411yescharacteristic, complemented
14C22⋊C4×D83—22no—
15#14581707C2212yes—
16#14581706—22no—
17#127×C83C2211yescharacteristic
18#99×C83C2212yes—
19C23×D83D412yes—
20#13050589—42no—
21C2×Dic166C2×C411yescharacteristic, complemented
22C2×D332C2×C411yescharacteristic
23C2×C22⋊D83—21no—
24C2×C22⋊D83—22no—
25C2×C22⋊D83D412yes—
26D4×D83—42no—
27C22×Dic83—21no—
28C2×C4×D83—42no—
29C2×C4×D83C2×C411yescharacteristic
30Dic83⋊C4—21no—
31C22⋊Dic83—22no—
32C22⋊Dic83—22no—
33(C2×D83)⋊C4—21no—
34(C2×D83)⋊C4—22no—
35(C2×D83)⋊C4—42no—
36C4×Dic83C2×C411yescharacteristic, complemented
37C2×D4×C83—22no—
38C2×D4×C83C2×C411yescharacteristic, complemented
39C2×D4×C83C2311yescharacteristic
40C22⋊C4×C83—22no—
41C42×C83C2×C411yescharacteristic, complemented
42C23×C83—21no—
43C23×C83C2×D412yes—
44C22×D83—22no—
45C22×D83—22no—
46C22×D83—82no—
47C22×D83C22⋊C412yes—
48C22×D83C22⋊C411yescharacteristic
49D4×C83—41no—
50D4×C83—41no—
51D4×C83—82no—
52C2×C4×C83—21no—
53C2×C4×C83—42no—
54C2×C4×C83C22×C411yescharacteristic
55C22⋊D83—42no—
56C22⋊D83—42no—
57C22⋊D83—82no—
58C2×Dic83—21no—
59C2×Dic83—21no—
60C2×Dic83—42no—
61C2×Dic83—42no—
62C2×Dic83C22⋊C411yescharacteristic
63D332—41no—
64C4×D83—22no—
65C4×D83—84no—
66Dic166—41no—
67C22×C83—22no—
68C22×C83—42no—
69C22×C83—42no—
70C22×C83—41no—
71C22×C83C2×C22⋊C411yescharacteristic
72C2×D83—22no—
73C2×D83—41no—
74C2×D83—44no—
75C2×D83—42no—
76C2×D83—82no—
77C2×D83#9912yes—
78C4×C83—21no—
79C4×C83—41no—
80C4×C83—82no—
81Dic83—21no—
82Dic83—41no—
83Dic83—41no—
84Dic83—82no—
85C2×C83—21no—
86C2×C83—42no—
87C2×C83—81no—
88C2×C83C2×#9911yescharacteristic
89D83—22no—
90D83—42no—
91#2127—831nomaximal, Sylow
92C83#212711yescharacteristic, complemented, Sylow
93#535—831no—
94C2×#99—832no—
95#354—832no—
96#353—831no—
97#353D8311yescharacteristic, complemented
98C2×#41—831no—
99C22×D4—831no—
100#127—831no—
101#127C2×D8311yescharacteristic
102#124—831no—
103#124—831no—
104#121—1662no—
105#120—1662no—
106#117—831no—
107C2×C22⋊C4—1662no—
108#99—832no—
109#99—1662no—
110#99C2×D8312yes—
111C24—832no—
112#41—3322no—
113C2×Q8—831no—
114C2×D4—22no—
115C2×D4—3322no—
116C2×D4—832no—
117C2×D4—831no—
118C2×D4—1661no—
119C2×D4—1662no—
120C2×D4C4×D8311yescharacteristic, complemented
121C2×D4C22×D8311yescharacteristic
122C22×C4—3322no—
123C22×C4—831no—
124C22×C4—1661no—
125C4⋊C4—1661no—
126C22⋊C4—22no—
127C22⋊C4—3322no—
128C22⋊C4—1661no—
129C22⋊C4—1662no—
130C22⋊C4—1662no—
131C22⋊C4—1662no—
132C42—831no—
133C42C4×D8311yescharacteristic, complemented
134C23—21no—
135C23—831no—
136C23—832no—
137C23—6642no—
138C23—1662no—
139C23—1662no—
140C23D4×D8312yes—
141Q8—3321no—
142D4—41no—
143D4—41no—
144D4—82no—
145D4—3321no—
146D4—3322no—
147D4—3322no—
148D4—6642no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—3322no—
152C2×C4—3322no—
153C2×C4—831no—
154C2×C4—6644no—
155C2×C4—1662no—
156C2×C4—1661no—
157C2×C4—1661no—
158C2×C4C2×C4×D8311yescharacteristic
159C22—22no—
160C22—42no—
161C22—42no—
162C22—41no—
163C22—3322no—
164C22—3321no—
165C22—3324no—
166C22—832no—
167C22—6642no—
168C22—1662no—
169C22C22⋊C4×D8311yescharacteristic
170C4—21no—
171C4—41no—
172C4—82no—
173C4—3321no—
174C4—3321no—
175C4—6642no—
176C4—1661no—
177C2—21no—
178C2—42no—
179C2—81no—
180C2—3322no—
181C2—1662no—
182C2#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#2127
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. #99×D83C2semidirect
3. #19127495C2semidirect
4. #19127494C2semidirect
5. #353×C83C2semidirect
6. C2×#13050589C4semidirect
8. #14581788C4semidirect
9. #14581780C4semidirect
13. #14581749C4semidirect
21. C2×Dic166C2×C4semidirect
36. C4×Dic83C2×C4semidirect
38. C2×D4×C83C2×C4semidirect
41. C42×C83C2×C4semidirect
92. C83#2127semidirect
97. #353D83semidirect
120. C2×D4C4×D83semidirect
133. C42C4×D83semidirect

Maximal subgroups

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

subgroupindex
#191282642
#99×D832
#191274952
#191274942
#353×C832
#212783