Finite Group Databasebeta

← #24917862

Subgroups — #24917862

Order 10624.

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

normal subgroups only

This group has 392 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#19128249C211yesmaximal, characteristic
2#125×D83C211yesmaximal, characteristic, complemented
3#19128071C211yesmaximal, characteristic, complemented
4#19128070C211yesmaximal, characteristic, complemented
5#19128065C211yesmaximal, characteristic, complemented
6#19128054C214yesmaximal, complemented
7#19127699C214yesmaximal, complemented
8#19127653C211yesmaximal, characteristic, complemented
9#490×C83C211yesmaximal, characteristic, complemented
10C2×#13050589C2212yes—
11C2×#13050573C2212yes—
12#14581792C2211yescharacteristic
13#14581784C2211yescharacteristic
14#14581783C2212yes—
15#14581782—22no—
16#14581779C2212yes—
17#14581763—22no—
18D4×Dic83—22no—
19C4⋊C4×D83—22no—
20C4⋊C4×D83C2212yes—
21C42×D83C2211yescharacteristic
22#14581717C2214yes—
23#14581715C2214yes—
24#14581713C2214yes—
25#14581703C2211yescharacteristic
26C8×Dic83C2212yes—
27#127×C83C2211yescharacteristic
28#125×C83C2211yescharacteristic
29#102×C83C2214yes—
30#97×C83C2211yescharacteristic
31#13050589—28no—
32C2×Dic166C2312yes—
33C2×C22⋊D83—22no—
34C22×Dic83—22no—
35C2×C4×D83—22no—
36C2×C4×D83D412yes—
37C2×C4×D83C2311yescharacteristic
38#13050573—24no—
39C2×C83⋊C8C2312yes—
40C4⋊Dic83—22no—
41C4⋊Dic83—22no—
42C4⋊Dic83C2312yes—
43Dic83⋊C4—22no—
44Dic83⋊C4—22no—
45Dic83⋊C4—22no—
46C22⋊Dic83—24no—
47C4×Dic83D412yes—
48C4×Dic83C2311yescharacteristic
49C2×D4×C83—22no—
50C2×D4×C83C2312yes—
51C2×C8×C83C2312yes—
52C4⋊C4×C83—22no—
53C4⋊C4×C83C2312yes—
54C42×C83C2311yescharacteristic
55C23×C83—22no—
56C22×D83C2×D411yescharacteristic
57D4×C83—24no—
58D4×C83—44no—
59C2×C4×C83—22no—
60C2×C4×C83C2×D412yes—
61C2×C4×C83C2411yescharacteristic
62C22⋊D83—44no—
63C2×Dic83—22no—
64C2×Dic83—22no—
65C2×Dic83—44no—
66C2×Dic83C2×D412yes—
67C2×Dic83C2×D411yescharacteristic
68C4×D83—42no—
69C4×D83—44no—
70C4×D83#4114yes—
71Dic166—24no—
72C8×C83—22no—
73C83⋊C8—22no—
74C22×C83—42no—
75C22×C83—44no—
76C22×C83C22×D411yescharacteristic
77C2×D83—22no—
78C2×D83#12412yes—
79C4×C83—22no—
80C4×C83—42no—
81C4×C83C2×#4112yes—
82Dic83—22no—
83Dic83—42no—
84Dic83—42no—
85Dic83#12412yes—
86C2×C83—82no—
87C2×C83C2×#12411yescharacteristic
88C2×C83#57512yes—
89D83—22no—
90#3142—831nomaximal, Sylow
91C83#314211yescharacteristic, complemented, Sylow
92#532—831no—
93C2×#125—831no—
94#491—831no—
95#491—831no—
96#490—831no—
97#490D8311yescharacteristic, complemented
98#488—834no—
99#417—834no—
100C4×M16—831no—
101C2×#41—832no—
102C2×M16—832no—
103C4⋊Q8—831no—
104#127C2×D8311yescharacteristic
105#125—832no—
106#125—831no—
107#125—1662no—
108#125C2×D8311yescharacteristic
109#124—832no—
110#121—1662no—
111C4×D4—1662no—
112C2×C4⋊C4—832no—
113C2×C4⋊C4—1662no—
114C2×C42—831no—
115#103—834no—
116#103—834no—
117#102—834no—
118#102C2×D8314yes—
119#97—831no—
120#97C2×D8311yescharacteristic
121C4×C8—832no—
122#41—1668no—
123C2×Q8—832no—
124C2×D4—22no—
125C2×D4—1662no—
126C2×D4C22×D8312yes—
127C22×C4—832no—
128C22×C4—831no—
129C22×C4—1662no—
130C22×C4—1662no—
131M16—1664no—
132C2×C8—832no—
133C2×C8C22×D8312yes—
134C4⋊C4—22no—
135C4⋊C4—832no—
136C4⋊C4—1662no—
137C4⋊C4—1662no—
138C4⋊C4—1662no—
139C4⋊C4—1662no—
140C4⋊C4—1662no—
141C4⋊C4C22×D8312yes—
142C22⋊C4—1664no—
143C42—832no—
144C42—831no—
145C42C22×D8311yescharacteristic
146C23—22no—
147C23—831no—
148Q8—1664no—
149D4—24no—
150D4—44no—
151D4—3324no—
152C2×C4—22no—
153C2×C4—3322no—
154C2×C4—3324no—
155C2×C4—3324no—
156C2×C4—831no—
157C2×C4—834no—
158C2×C4—832no—
159C2×C4—1662no—
160C2×C4—1662no—
161C2×C4D4×D8312yes—
162C2×C4C23×D8311yescharacteristic
163C8—22no—
164C8—1662no—
165C22—42no—
166C22—44no—
167C22—832no—
168C22—1662no—
169C22C2×D4×D8311yescharacteristic
170C4—22no—
171C4—42no—
172C4—3322no—
173C4—3322no—
174C4—832no—
175C4—1662no—
176C4#41×D8312yes—
177C2—82no—
178C2—1662no—
179C2#124×D8311yescharacteristic
180C2#1912849712yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#490×C83C2
socleC22×C83C22×D4
prime radical#19128249C2
hypercenter#490D83
hypocenterC83#3142
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. #125×D83C2semidirect
3. #19128071C2semidirect
4. #19128070C2semidirect
5. #19128065C2semidirect
6. #19128054C2semidirect
7. #19127699C2semidirect
8. #19127653C2semidirect
9. #490×C83C2semidirect
91. C83#3142semidirect
97. #490D83semidirect

Maximal subgroups

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

subgroupindex
#191282492
#125×D832
#191280712
#191280702
#191280652
#191280542
#191276992
#191276532
#490×C832
#314283