Finite Group Databasebeta

← #24917392

Subgroups — #24917392

Order 10624.

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

normal subgroups only

This group has 414 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×#14581751C211yesmaximal, characteristic, complemented
2#19128039C214yesmaximal, complemented
3#19128022C214yesmaximal, complemented
4#19127744C212yesmaximal, complemented
5#19127702C212yesmaximal, complemented
6#19127665C211yesmaximal, characteristic, complemented
7#525×C83C211yesmaximal, characteristic
8C2×#13050574C2212yes—
9#14581751—22no—
10#14581751C2214yes—
11C2×C4⋊Dic83C2212yes—
12C2×C22⋊Dic83—21no—
13C2×C4×Dic83—21no—
14#14581725C2214yescomplemented
15#14581723C2214yescomplemented
16#14581716C2214yescomplemented
17#14581713C2214yescomplemented
18C22⋊(C83⋊C8)C2214yes—
19C2×#41×C83C2211yescharacteristic
20C23×C4×C83C2211yescharacteristic
21#123×C83—21no—
22C22⋊Q8×C83C2212yes—
23#121×C83C2212yes—
24#120×C83—21no—
25C4×D4×C83—22no—
26#117×C83C2211yescharacteristic
27C22×Dic83—22no—
28#13050574—24no—
29C2×C83⋊C8C2314yes—
30C4⋊Dic83—22no—
31C4⋊Dic83C2314yes—
32C22⋊Dic83—24no—
33C4×Dic83—24no—
34C24×C83D411yescharacteristic
35#41×C83—24no—
36C2×Q8×C83C2311yescharacteristic
37C2×D4×C83—22no—
38C2×D4×C83—21no—
39C2×D4×C83C2311yescharacteristic
40C22×C4×C83—24no—
41C22×C4×C83—21no—
42C22×C4×C83D412yes—
43C22×C4×C83D411yescharacteristic
44C22×C4×C83C2312yes—
45C22×C4×C83C2311yescharacteristic
46C4⋊C4×C83—22no—
47C4⋊C4×C83C2312yes—
48C22⋊C4×C83—21no—
49C22⋊C4×C83—22no—
50C22⋊C4×C83—22no—
51C42×C83—21no—
52C23×C83—22no—
53C23×C83—21no—
54C23×C83—42no—
55C23×C83C2×D412yes—
56C23×C83C2×D411yescharacteristic
57Q8×C83—22no—
58D4×C83—22no—
59D4×C83—44no—
60D4×C83—42no—
61C2×C4×C83—22no—
62C2×C4×C83—24no—
63C2×C4×C83—24no—
64C2×C4×C83—22no—
65C2×C4×C83—21no—
66C2×C4×C83—42no—
67C2×C4×C83C2×D412yes—
68C2×C4×C83C2×D411yescharacteristic
69C2×C4×C83#4114yes—
70C2×C4×C83C2411yescharacteristic
71C2×Dic83—24no—
72C2×Dic83—44no—
73C83⋊C8—24no—
74C22×C83—22no—
75C22×C83—22no—
76C22×C83—22no—
77C22×C83—22no—
78C22×C83—44no—
79C22×C83—41no—
80C22×C83—42no—
81C22×C83#12312yes—
82C22×C83C22×D411yescharacteristic
83C4×C83—22no—
84C4×C83—41no—
85C4×C83—42no—
86C4×C83#12312yes—
87C4×C83C2×#4112yes—
88Dic83—44no—
89C2×C83—22no—
90C2×C83—42no—
91C2×C83—81no—
92C2×C83C2×#12311yescharacteristic
93C2×C83#57512yes—
94#3093—831nomaximal, Sylow
95C83#309311yescharacteristic, complemented, Sylow
96#525D8311yescharacteristic
97C2×#117—831no—
98#485—834no—
99#482—834no—
100#428—832no—
101#417—832no—
102#407—831no—
103C2×#41C2×D8311yescharacteristic
104C23×C4C2×D8311yescharacteristic
105C2×M16—832no—
106#123—21no—
107C22⋊Q8C2×D8312yes—
108#121C2×D8312yes—
109#120—21no—
110C4×D4—22no—
111#117—834no—
112#117—1662no—
113#117C2×D8311yescharacteristic
114C2×C4⋊C4—832no—
115C2×C22⋊C4—1661no—
116C2×C42—1661no—
117#107—834no—
118#106—834no—
119#103—834no—
120#102—834no—
121C22⋊C8—834no—
122C24C22⋊D8311yescharacteristic
123#41—24no—
124C2×Q8C22×D8311yescharacteristic
125C2×D4—22no—
126C2×D4—21no—
127C2×D4C22×D8311yescharacteristic
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—1662no—
131C22×C4C22⋊D8311yescharacteristic
132C22×C4C22⋊D8312yes—
133C22×C4C22×D8312yes—
134C22×C4C22×D8311yescharacteristic
135M16—1664no—
136C2×C8—834no—
137C4⋊C4—22no—
138C4⋊C4—834no—
139C4⋊C4—1662no—
140C4⋊C4C22×D8312yes—
141C22⋊C4—21no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—1664no—
145C42—21no—
146C42—1664no—
147C23—22no—
148C23—21no—
149C23—42no—
150C23C2×C22⋊D8312yes—
151C23C2×C22⋊D8311yescharacteristic
152Q8—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—3324no—
163C2×C4—1664no—
164C2×C4C2×C22⋊D8312yes—
165C2×C4C2×C22⋊D8311yescharacteristic
166C2×C4#1305058914yes—
167C2×C4C23×D8311yescharacteristic
168C8—1664no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—41no—
175C22—42no—
176C22#1458177312yes—
177C22C22×C22⋊D8311yescharacteristic
178C4—22no—
179C4—41no—
180C4—42no—
181C4—3324no—
182C4#1458177312yes—
183C4C2×#1305058912yes—
184C2—22no—
185C2—42no—
186C2—81no—
187C2C2×#1458177311yescharacteristic
188C2#1912849912yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radical#525×C83C2
socleC22×C83C22×D4
hypercenter#525D83
hypocenterC83#3093
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#14581751C2semidirect
2. #19128039C2semidirect
3. #19128022C2semidirect
4. #19127744C2semidirect
5. #19127702C2semidirect
6. #19127665C2semidirect
14. #14581725C22semidirect
15. #14581723C22semidirect
16. #14581716C22semidirect
17. #14581713C22semidirect
95. C83#3093semidirect

Maximal subgroups

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

subgroupindex
C2×#145817512
#191280392
#191280222
#191277442
#191277022
#191276652
#525×C832
#309383