Finite Group Databasebeta

← #24913175

Subgroups — #24913175

Order 10624.

1134 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14581775C211yesmaximal, characteristic, complemented
2C2×#14581761C212yesmaximal, complemented
3#19127615C214yesmaximal, complemented
4#19127609C214yesmaximal, complemented
5#19127564C211yesmaximal, characteristic, complemented
6#19127557C212yesmaximal, complemented
7C23×D4×C83C211yesmaximal, characteristic, complemented
8C22×C22⋊D83C2211yescharacteristic
9C22×C22⋊D83C2214yescomplemented
10#14581775—24no—
11#14581761—28no—
12C2×Dic83⋊C4C2212yescomplemented
13C2×C22⋊Dic83C2211yescharacteristic
14C2×C22⋊Dic83C2212yes—
15C2×C22⋊Dic83C2214yescomplemented
16C2×C22⋊Dic83C2218yescomplemented
17C2×(C2×D83)⋊C4C2212yescomplemented
18C83⋊#95C2214yescomplemented
19C25×C83C2212yes—
20C22×D4×C83—28no—
21C22×D4×C83—24no—
22C22×D4×C83C2214yes—
23C23×C4×C83C2211yescharacteristic
24C23×D83C2311yescharacteristic
25C2×C22⋊D83—24no—
26C2×C22⋊D83—28no—
27C2×C22⋊D83—22no—
28C2×C22⋊D83—48no—
29C22×Dic83C2311yescharacteristic
30C22×Dic83C2314yes—
31C22×Dic83C2312yes—
32Dic83⋊C4—24no—
33C22⋊Dic83—24no—
34C22⋊Dic83—28no—
35C22⋊Dic83—22no—
36C22⋊Dic83—48no—
37(C2×D83)⋊C4—24no—
38C24×C83—24no—
39C24×C83—24no—
40C24×C83—48no—
41C24×C83D418yescomplemented
42C24×C83C2314yes—
43C24×C83C2311yescharacteristic
44C2×D4×C83—232no—
45C2×D4×C83—216no—
46C2×D4×C83—28no—
47C22×C4×C83—24no—
48C22×C4×C83D414yescomplemented
49C22×C4×C83C2312yes—
50C23×C83—24no—
51C23×C83—216no—
52C23×C83—24no—
53C23×C83—24no—
54C23×C83—28no—
55C23×C83—28no—
56C23×C83—416no—
57C23×C83—416no—
58C23×C83—416no—
59C23×C83C2×D418yes—
60C23×C83C2×D414yes—
61C23×C83C2×D412yes—
62C23×C83C2411yescharacteristic
63C22×D83—21no—
64C22×D83—22no—
65C22×D83—24no—
66D4×C83—232no—
67C2×C4×C83—28no—
68C2×C4×C83—24no—
69C2×C4×C83C2×D414yes—
70C22⋊D83—416no—
71C22⋊D83—44no—
72C2×Dic83—22no—
73C2×Dic83—22no—
74C2×Dic83—24no—
75C2×Dic83—21no—
76C2×Dic83—24no—
77C2×Dic83—28no—
78C22×C83—28no—
79C22×C83—24no—
80C22×C83—28no—
81C22×C83—28no—
82C22×C83—28no—
83C22×C83—432no—
84C22×C83—48no—
85C22×C83—48no—
86C22×C83#12014yes—
87C22×C83C22×D412yes—
88C22×C83C22×D411yescharacteristic
89C22×C83H(5,𝔽2)14yes—
90C2×D83—41no—
91C2×D83—44no—
92C2×D83—42no—
93C4×C83—24no—
94Dic83—41no—
95Dic83—44no—
96Dic83—42no—
97C2×C83—24no—
98C2×C83—48no—
99C2×C83C2×#12011yescharacteristic
100C2×C83#53414yes—
101C2×C83#53512yes—
102D83—81no—
103#2613—831nomaximal, Sylow
104C83#261311yescharacteristic, complemented, Sylow
105C23×D4D8311yescharacteristic, complemented
106C2×#123—831no—
107C2×#120—832no—
108#394—834no—
109#392—834no—
110#381—831no—
111#379—832no—
112C25C2×D8312yes—
113C22×D4—28no—
114C22×D4—24no—
115C22×D4—834no—
116C22×D4—831no—
117C22×D4C2×D8314yes—
118C23×C4C2×D8311yescharacteristic
119#123—1664no—
120#120—1668no—
121C2×C4⋊C4—832no—
122C2×C22⋊C4—838no—
123C2×C22⋊C4—832no—
124C2×C22⋊C4—834no—
125C2×C22⋊C4—831no—
126C2×C22⋊C4—832no—
127#95—834no—
128C24—24no—
129C24—24no—
130C24—48no—
131C24—831no—
132C24C22⋊D8318yescomplemented
133C24C22×D8314yes—
134C24C22×D8311yescharacteristic
135C2×D4—232no—
136C2×D4—216no—
137C2×D4—28no—
138C2×D4—3328no—
139C2×D4—1668no—
140C2×D4—1664no—
141C2×D4—1662no—
142C22×C4—24no—
143C22×C4—834no—
144C22×C4—832no—
145C22×C4—831no—
146C22×C4C22⋊D8314yescomplemented
147C22×C4C22×D8312yes—
148C4⋊C4—1664no—
149C22⋊C4—3328no—
150C22⋊C4—1668no—
151C22⋊C4—1664no—
152C22⋊C4—1662no—
153C22⋊C4—1664no—
154C23—24no—
155C23—216no—
156C23—24no—
157C23—24no—
158C23—28no—
159C23—28no—
160C23—416no—
161C23—416no—
162C23—416no—
163C23—1661no—
164C23—1662no—
165C23—1664no—
166C23C2×C22⋊D8318yes—
167C23C2×C22⋊D8314yes—
168C23C2×C22⋊D8312yes—
169C23C23×D8311yescharacteristic
170D4—232no—
171D4—33216no—
172D4—3324no—
173C2×C4—28no—
174C2×C4—24no—
175C2×C4—1662no—
176C2×C4—1662no—
177C2×C4—1664no—
178C2×C4—1661no—
179C2×C4—1664no—
180C2×C4—1668no—
181C2×C4C2×C22⋊D8314yes—
182C22—28no—
183C22—24no—
184C22—28no—
185C22—28no—
186C22—28no—
187C22—432no—
188C22—48no—
189C22—48no—
190C22—3321no—
191C22—3324no—
192C22—3322no—
193C22#1458176114yes—
194C22C22×C22⋊D8312yes—
195C22C22×C22⋊D8311yescharacteristic
196C22#1458183714yes—
197C4—24no—
198C4—3321no—
199C4—3324no—
200C4—3322no—
201C2—24no—
202C2—48no—
203C2—6641no—
204C2C2×#1458176111yescharacteristic
205C2#1912826014yes—
206C2#1912826812yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC23×D4×C83C2
hypercenterC23×D4D83
hypocenterC83#2613
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#14581775C2semidirect
2. C2×#14581761C2semidirect
3. #19127615C2semidirect
4. #19127609C2semidirect
5. #19127564C2semidirect
6. #19127557C2semidirect
7. C23×D4×C83C2semidirect
9. C22×C22⋊D83C22semidirect
12. C2×Dic83⋊C4C22semidirect
15. C2×C22⋊Dic83C22semidirect
16. C2×C22⋊Dic83C22semidirect
17. C2×(C2×D83)⋊C4C22semidirect
18. C83⋊#95C22semidirect
41. C24×C83D4semidirect
48. C22×C4×C83D4semidirect
104. C83#2613semidirect
105. C23×D4D83semidirect
132. C24C22⋊D83semidirect
146. C22×C4C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817752
C2×#145817612
#191276152
#191276092
#191275642
#191275572
C23×D4×C832
#261383