Finite Group Databasebeta

← #24912621

Subgroups — #24912621

Order 10624.

672 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14581771C211yesmaximal, characteristic, complemented
2#19127626C212yesmaximal, complemented
3#19127613C212yesmaximal, complemented
4#19127595C214yesmaximal, complemented
5#19127564C212yesmaximal, complemented
6#19127557C212yesmaximal
7C4×C22⋊Dic83C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C83C211yesmaximal, characteristic
9C22×Dic166C2211yescharacteristic, complemented
10#14581771—24no—
11C2×C4⋊Dic83C2211yescharacteristic, complemented
12C2×Dic83⋊C4C2212yescomplemented
13C2×C22⋊Dic83C2212yes—
14C2×C22⋊Dic83C2212yes—
15C2×C22⋊Dic83C2214yes—
16C2×C22⋊Dic83C2214yes—
17C2×C4×Dic83C2212yescomplemented
18C83⋊#95C2214yescomplemented
19C83⋊#95C2214yescomplemented
20C83⋊#95C2212yescomplemented
21C25×C83C2211yescharacteristic
22C23×C4×C83C2211yescharacteristic
23C23×C4×C83C2211yescharacteristic
24C2×C22⋊C4×C83—24no—
25C2×C22⋊C4×C83C2212yes—
26C2×C22⋊C4×C83C2212yes—
27C2×Dic166—22no—
28C2×Dic166—42no—
29C22×Dic83C2312yes—
30C22×Dic83C2314yes—
31C22×Dic83C2312yes—
32C4⋊Dic83—22no—
33Dic83⋊C4—24no—
34C22⋊Dic83—24no—
35C22⋊Dic83—24no—
36C22⋊Dic83—28no—
37C22⋊Dic83—44no—
38C4×Dic83—42no—
39C24×C83—22no—
40C24×C83—24no—
41C24×C83—24no—
42C24×C83—42no—
43C24×C83C2312yes—
44C24×C83C2311yescharacteristic
45C22×C4×C83—22no—
46C22×C4×C83—24no—
47C22×C4×C83—42no—
48C22×C4×C83D414yes—
49C22×C4×C83C2312yes—
50C22×C4×C83C2311yescharacteristic
51C22×C4×C83C2311yescharacteristic
52C22⋊C4×C83—24no—
53C22⋊C4×C83—24no—
54C23×C83—22no—
55C23×C83—22no—
56C23×C83—28no—
57C23×C83—22no—
58C23×C83—24no—
59C23×C83—22no—
60C23×C83—24no—
61C23×C83—416no—
62C23×C83—44no—
63C23×C83—44no—
64C23×C83C2×D412yes—
65C23×C83#4114yes—
66C23×C83#4114yes—
67C23×C83C2411yescharacteristic
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—24no—
72C2×C4×C83—44no—
73C2×C4×C83—44no—
74C2×C4×C83C2×D412yes—
75C2×C4×C83C2×D412yes—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—24no—
79C2×Dic83—24no—
80C2×Dic83—24no—
81C2×Dic83—24no—
82C2×Dic83—24no—
83Dic166—44no—
84C22×C83—22no—
85C22×C83—22no—
86C22×C83—28no—
87C22×C83—24no—
88C22×C83—28no—
89C22×C83—416no—
90C22×C83—42no—
91C22×C83—42no—
92C22×C83—44no—
93C22×C83#12414yes—
94C22×C83C22×D411yescharacteristic
95C22×C83C2×#4112yes—
96C22×C83C2×#4112yes—
97C22×C83H(5,𝔽2)12yes—
98C4×C83—22no—
99C4×C83—22no—
100C4×C83—42no—
101Dic83—42no—
102Dic83—42no—
103Dic83—44no—
104C2×C83—24no—
105C2×C83—44no—
106C2×C83#52511yescharacteristic
107C2×C83C2×#12411yescharacteristic
108C2×C83#53411yescharacteristic
109C2×C83#55114yes—
110#2561—831nomaximal, Sylow
111C83#256111yescharacteristic, complemented, Sylow
112C2×C22⋊Q8—831no—
113C22×C22⋊C4D8311yescharacteristic
114#397—832no—
115#393—832no—
116#388—834no—
117#381—832no—
118#379—832no—
119C4×C22⋊C4—831no—
120C25C2×D8311yescharacteristic
121C22×Q8—831no—
122C23×C4C2×D8311yescharacteristic
123C23×C4C2×D8311yescharacteristic
124C22⋊Q8—1664no—
125C2×C4⋊C4—832no—
126C2×C4⋊C4—831no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—834no—
129C2×C22⋊C4—832no—
130C2×C22⋊C4—832no—
131C2×C22⋊C4—834no—
132C2×C22⋊C4C2×D8312yes—
133C2×C22⋊C4C2×D8312yes—
134C2×C42—832no—
135#95—832no—
136#95—834no—
137#95—834no—
138C24—22no—
139C24—24no—
140C24—24no—
141C24—42no—
142C24C22×D8312yes—
143C24C22×D8311yescharacteristic
144C2×Q8—3322no—
145C2×Q8—1662no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—42no—
149C22×C4—834no—
150C22×C4—832no—
151C22×C4—832no—
152C22×C4C22⋊D8314yes—
153C22×C4C22×D8312yes—
154C22×C4C22×D8311yescharacteristic
155C22×C4C22×D8311yescharacteristic
156C4⋊C4—1664no—
157C4⋊C4—1662no—
158C22⋊C4—24no—
159C22⋊C4—24no—
160C22⋊C4—3324no—
161C22⋊C4—1664no—
162C22⋊C4—1664no—
163C22⋊C4—1668no—
164C42—3322no—
165C23—22no—
166C23—22no—
167C23—28no—
168C23—22no—
169C23—24no—
170C23—22no—
171C23—24no—
172C23—416no—
173C23—44no—
174C23—44no—
175C23C2×C22⋊D8312yes—
176C23#1305058914yes—
177C23#1305059014yes—
178C23C23×D8311yescharacteristic
179Q8—3324no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—1662no—
187C2×C4—1662no—
188C2×C4—1664no—
189C2×C4—1664no—
190C2×C4—1664no—
191C2×C4—1664no—
192C2×C4—1664no—
193C2×C4C2×C22⋊D8312yes—
194C2×C4C2×C22⋊D8312yes—
195C22—28no—
196C22—22no—
197C22—22no—
198C22—28no—
199C22—24no—
200C22—416no—
201C22—44no—
202C22—42no—
203C22—42no—
204C22#1458177914yes—
205C22C22×C22⋊D8311yescharacteristic
206C22C2×#1305058912yes—
207C22C2×#1305059012yes—
208C22#1458183712yes—
209C4—22no—
210C4—22no—
211C4—42no—
212C4—3324no—
213C4—3322no—
214C4—3322no—
215C2—24no—
216C2—44no—
217C2#1912821211yescharacteristic
218C2C2×#1458177911yescharacteristic
219C2#1912826011yescharacteristic
220C2#1912837714yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC22×C22⋊C4×C83C2
prime radicalC25×C83C22
hypercenterC22×C22⋊C4D83
hypocenterC83#2561
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#14581771C2semidirect
2. #19127626C2semidirect
3. #19127613C2semidirect
4. #19127595C2semidirect
5. #19127564C2semidirect
7. C4×C22⋊Dic83C2semidirect
9. C22×Dic166C22semidirect
11. C2×C4⋊Dic83C22semidirect
12. C2×Dic83⋊C4C22semidirect
17. C2×C4×Dic83C22semidirect
18. C83⋊#95C22semidirect
19. C83⋊#95C22semidirect
20. C83⋊#95C22semidirect
111. C83#2561semidirect

Maximal subgroups

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

subgroupindex
C2×#145817712
#191276262
#191276132
#191275952
#191275642
#191275572
C4×C22⋊Dic832
C22×C22⋊C4×C832
#256183