Finite Group Databasebeta

← #24918328

Subgroups — #24918328

Order 10624.

442 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128265C212yesmaximal, complemented
2#19128026C214yesmaximal, complemented
3#19127954C212yesmaximal, complemented
4#19127937C212yesmaximal, complemented
5#19127837C214yesmaximal, complemented
6#408×C83C211yesmaximal, characteristic, complemented
7C22×D332C2212yescomplemented
8C2×D664C2212yes—
9C2×D664C2212yes—
10#14581788—22no—
11#14581781—22no—
12#14581766C2214yes—
13#14581766C2214yescomplemented
14#14581760—22no—
15#14581760—22no—
16#14581751C2212yescomplemented
17#14581726C2212yescomplemented
18#14581724C2212yescomplemented
19#14581714C2214yescomplemented
20#14581714C2214yescomplemented
21C2×#41×C83C2211yescharacteristic
22C2×M16×C83C2211yescharacteristic, complemented
23C22×C8×C83C2211yescharacteristic, complemented
24C22⋊C8×C83C2214yescomplemented
25C23×D83—22no—
26C2×D332—22no—
27C2×D332—42no—
28C2×D332—42no—
29C2×D332C2314yes—
30C2×C22⋊D83—24no—
31C2×C22⋊D83—24no—
32D664—42no—
33D664—42no—
34C4⋊Dic83C2314yes—
35C22⋊Dic83—22no—
36(C2×D83)⋊C4—24no—
37(C2×D83)⋊C4—24no—
38C4×Dic83—22no—
39#41×C83—41no—
40#41×C83—41no—
41C2×Q8×C83D411yescharacteristic
42C2×D4×C83D411yescharacteristic
43C2×D4×C83D412yes—
44C22×C4×C83C2312yes—
45C22×C4×C83C2311yescharacteristic
46M16×C83—41no—
47C2×C8×C83—41no—
48C2×C8×C83C2312yes—
49C2×C8×C83C2312yes—
50C23×C83C2×D412yes—
51C23×C83C2×D411yescharacteristic
52C22×D83—24no—
53C22×D83—44no—
54C22×D83—84no—
55Q8×C83—41no—
56D4×C83—42no—
57D4×C83—41no—
58C2×C4×C83—21no—
59C2×C4×C83—21no—
60C2×C4×C83—22no—
61C2×C4×C83—22no—
62C2×C4×C83C2×D412yes—
63C2×C4×C83C2×D411yescharacteristic
64C2×C4×C83C2411yescharacteristic
65C22⋊D83—84no—
66C22⋊D83—84no—
67C2×Dic83—24no—
68D332—24no—
69D332—24no—
70D332—44no—
71C8×C83—22no—
72C8×C83—22no—
73C22×C83—22no—
74C22×C83—21no—
75C22×C83—22no—
76C22×C83—24no—
77C22×C83C22×D411yescharacteristic
78C2×D83—44no—
79C2×D83—44no—
80C2×D83—44no—
81C2×D83—88no—
82C4×C83—22no—
83C4×C83—21no—
84C4×C83H(5,𝔽2)11yescharacteristic
85C4×C83H(5,𝔽2)11yescharacteristic
86Dic83—44no—
87C2×C83—42no—
88C2×C83—41no—
89C2×C83#53411yescharacteristic
90C2×C83#57612yes—
91D83—84no—
92#3199—831nomaximal, Sylow
93C83#319911yescharacteristic, complemented, Sylow
94#535—832no—
95#482—834no—
96#469—832no—
97#466—832no—
98#447—834no—
99#408D8311yescharacteristic, complemented
100C2×#41C2×D8311yescharacteristic
101C22×D4—832no—
102C2×D8—832no—
103C2×D8—832no—
104C2×M16C2×D8311yescharacteristic, complemented
105C22×C8C2×D8311yescharacteristic, complemented
106#127—1662no—
107#124—1662no—
108#121—834no—
109#121—834no—
110#120—1662no—
111#120—1662no—
112#117—832no—
113#107—832no—
114#106—832no—
115#102—834no—
116#102—834no—
117C22⋊C8C2×D8314yescomplemented
118C24—1662no—
119#41—41no—
120#41—41no—
121C2×Q8D33211yescharacteristic
122C2×D4—3322no—
123C2×D4—3322no—
124C2×D4—834no—
125C2×D4—1664no—
126C2×D4—1664no—
127C2×D4—1662no—
128C2×D4D33211yescharacteristic
129C2×D4D33212yes—
130C22×C4C22×D8312yes—
131C22×C4C22×D8311yescharacteristic
132D8—3322no—
133D8—3322no—
134M16—41no—
135C2×C8—41no—
136C2×C8C22×D8312yes—
137C2×C8C22×D8312yes—
138C4⋊C4—834no—
139C22⋊C4—1662no—
140C22⋊C4—1664no—
141C22⋊C4—1664no—
142C42—1662no—
143C23—3324no—
144C23—6644no—
145C23—1664no—
146C23C2×D33212yes—
147C23C2×D33211yescharacteristic
148Q8—41no—
149D4—42no—
150D4—41no—
151D4—3324no—
152D4—6644no—
153D4—6644no—
154D4—1664no—
155D4—1664no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—1664no—
161C2×C4C2×D33212yes—
162C2×C4C2×D33211yescharacteristic
163C2×C4C23×D8311yescharacteristic
164C8—22no—
165C8—22no—
166C22—22no—
167C22—24no—
168C22—22no—
169C22—21no—
170C22—3324no—
171C22—3324no—
172C22—3324no—
173C22—6648no—
174C22C22×D33211yescharacteristic
175C4—22no—
176C4—21no—
177C4—3324no—
178C4#1458183711yescharacteristic
179C4#1458183711yescharacteristic
180C2—42no—
181C2—41no—
182C2—6644no—
183C2#1912826311yescharacteristic
184C2#1912850612yes—
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#408×C83C2
socleC22×C83C22×D4
hypercenter#408D83
hypocenterC83#3199
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #19128265C2semidirect
2. #19128026C2semidirect
3. #19127954C2semidirect
4. #19127937C2semidirect
5. #19127837C2semidirect
6. #408×C83C2semidirect
7. C22×D332C22semidirect
13. #14581766C22semidirect
16. #14581751C22semidirect
17. #14581726C22semidirect
18. #14581724C22semidirect
19. #14581714C22semidirect
20. #14581714C22semidirect
22. C2×M16×C83C22semidirect
23. C22×C8×C83C22semidirect
24. C22⋊C8×C83C22semidirect
93. C83#3199semidirect
99. #408D83semidirect
104. C2×M16C2×D83semidirect
105. C22×C8C2×D83semidirect
117. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282652
#191280262
#191279542
#191279372
#191278372
#408×C832
#319983