Finite Group Databasebeta

← #24915753

Subgroups — #24915753

Order 10624.

1102 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C24×Dic83C211yesmaximal, characteristic, complemented
2C2×#14581773C212yesmaximal, complemented
3#19127581C214yesmaximal, complemented
4#19127573C214yesmaximal, complemented
5#19127562C212yesmaximal, complemented
6#19127557C211yesmaximal, characteristic, complemented
7C2×#120×C83C211yesmaximal, characteristic
8C23×Dic83—28no—
9C23×Dic83—24no—
10C23×Dic83C2214yes—
11C23×Dic83C2212yes—
12#14581773—28no—
13C2×Dic83⋊C4C2214yescomplemented
14C2×C22⋊Dic83C2212yes—
15C2×C22⋊Dic83C2214yes—
16C2×C22⋊Dic83C2214yescomplemented
17C83⋊#95C2218yescomplemented
18C25×C83C2211yescharacteristic
19C22×D4×C83C2212yes—
20C22×D4×C83C2211yescharacteristic
21#120×C83—24no—
22C2×C22⋊C4×C83C2211yescharacteristic
23C2×C22⋊C4×C83C2212yes—
24C22×Dic83—232no—
25C22×Dic83—24no—
26C22×Dic83—28no—
27C22×Dic83—24no—
28C22×Dic83—216no—
29C22×Dic83D418yescomplemented
30C22×Dic83C2314yes—
31C22×Dic83C2314yes—
32Dic83⋊C4—28no—
33C22⋊Dic83—24no—
34C22⋊Dic83—28no—
35C22⋊Dic83—28no—
36C24×C83—24no—
37C24×C83—28no—
38C24×C83D414yes—
39C24×C83C2311yescharacteristic
40C24×C83C2311yescharacteristic
41C24×C83C2312yes—
42C2×D4×C83—24no—
43C2×D4×C83—22no—
44C2×D4×C83—24no—
45C2×D4×C83—28no—
46C22×C4×C83C2312yes—
47C22×C4×C83C2311yescharacteristic
48C22⋊C4×C83—22no—
49C22⋊C4×C83—24no—
50C23×C83—232no—
51C23×C83—28no—
52C23×C83—216no—
53C23×C83—24no—
54C23×C83—24no—
55C23×C83—24no—
56C23×C83—21no—
57C23×C83—22no—
58C23×C83—24no—
59C23×C83C2×D414yes—
60C23×C83C2×D412yes—
61C23×C83C2×D414yes—
62C23×C83#4118yes—
63C23×C83C2411yescharacteristic
64D4×C83—44no—
65D4×C83—48no—
66C2×C4×C83—21no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—24no—
70C2×Dic83—232no—
71C2×Dic83—216no—
72C2×Dic83—24no—
73C2×Dic83—24no—
74C2×Dic83—24no—
75C2×Dic83—28no—
76C2×Dic83C2×D418yes—
77C22×C83—232no—
78C22×C83—24no—
79C22×C83—24no—
80C22×C83—24no—
81C22×C83—28no—
82C22×C83—216no—
83C22×C83—41no—
84C22×C83—42no—
85C22×C83—44no—
86C22×C83#12014yes—
87C22×C83#12318yescomplemented
88C22×C83C22×D412yes—
89C22×C83C22×D411yescharacteristic
90C22×C83C2×#4114yes—
91C4×C83—41no—
92C4×C83—42no—
93Dic83—28no—
94Dic83—44no—
95C2×C83—24no—
96C2×C83—28no—
97C2×C83—81no—
98C2×C83C2×#12011yescharacteristic
99C2×C83C2×#12312yes—
100C2×C83#52514yes—
101#2849—831nomaximal, Sylow
102C83#284911yescharacteristic, complemented, Sylow
103C24×C4—831no—
104C2×#123—832no—
105C2×#120D8311yescharacteristic
106#386—834no—
107#385—834no—
108#381—832no—
109#379—831no—
110C25C2×D8311yescharacteristic
111C22×D4C2×D8311yescharacteristic
112C22×D4C2×D8312yes—
113C23×C4—832no—
114C23×C4—834no—
115C23×C4—1668no—
116C23×C4—1664no—
117#123—1668no—
118#120—24no—
119C2×C4⋊C4—834no—
120C2×C22⋊C4—832no—
121C2×C22⋊C4—834no—
122C2×C22⋊C4—834no—
123C2×C22⋊C4C2×D8311yescharacteristic
124C2×C22⋊C4C2×D8312yes—
125#95—838no—
126C24—24no—
127C24—28no—
128C24C22⋊D8314yes—
129C24C22×D8311yescharacteristic
130C24C22×D8311yescharacteristic
131C24C22×D8312yes—
132C2×D4—24no—
133C2×D4—22no—
134C2×D4—24no—
135C2×D4—28no—
136C22×C4—838no—
137C22×C4—834no—
138C22×C4—834no—
139C22×C4—16616no—
140C22×C4—16632no—
141C22×C4—1664no—
142C22×C4—1668no—
143C22×C4—1664no—
144C22×C4C22×D8312yes—
145C22×C4C22×D8311yescharacteristic
146C4⋊C4—1668no—
147C22⋊C4—22no—
148C22⋊C4—24no—
149C22⋊C4—1668no—
150C22⋊C4—1664no—
151C22⋊C4—1668no—
152C23—232no—
153C23—28no—
154C23—216no—
155C23—24no—
156C23—24no—
157C23—24no—
158C23—22no—
159C23—21no—
160C23—24no—
161C23D4×D8314yes—
162C23C2×C22⋊D8312yes—
163C23C2×C22⋊D8314yes—
164C23#1305058918yes—
165C23C23×D8311yescharacteristic
166D4—44no—
167D4—48no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—838no—
173C2×C4—1664no—
174C2×C4—1668no—
175C2×C4—16632no—
176C2×C4—16616no—
177C2×C4—1664no—
178C2×C4—1664no—
179C22—232no—
180C22—24no—
181C22—24no—
182C22—24no—
183C22—28no—
184C22—216no—
185C22—41no—
186C22—42no—
187C22—44no—
188C22#1458176014yes—
189C22#1458177318yescomplemented
190C22C2×D4×D8312yes—
191C22C22×C22⋊D8311yescharacteristic
192C22C2×#1305058914yes—
193C4—41no—
194C4—42no—
195C4—3324no—
196C4—1668no—
197C2—24no—
198C2—28no—
199C2—81no—
200C2C2×#1458176011yescharacteristic
201C2C2×#1458177312yes—
202C2#1912820714yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting, prime radicalC2×#120×C83C2
hypercenterC2×#120D83
hypocenterC83#2849
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C24×Dic83C2semidirect
2. C2×#14581773C2semidirect
3. #19127581C2semidirect
4. #19127573C2semidirect
5. #19127562C2semidirect
6. #19127557C2semidirect
13. C2×Dic83⋊C4C22semidirect
16. C2×C22⋊Dic83C22semidirect
17. C83⋊#95C22semidirect
29. C22×Dic83D4semidirect
87. C22×C83#123semidirect
102. C83#2849semidirect
189. C22#14581773semidirect

Maximal subgroups

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

subgroupindex
C24×Dic832
C2×#145817732
#191275812
#191275732
#191275622
#191275572
C2×#120×C832
#284983