Finite Group Databasebeta

← #24916700

Subgroups — #24916700

Order 10624.

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

normal subgroups only

This group has 738 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
1C23×Dic166C211yesmaximal, characteristic, complemented
2C2×#14581825C212yesmaximal, complemented
3#19128212C211yesmaximal, characteristic
4#19127859C214yesmaximal, complemented
5#19127842C214yesmaximal, complemented
6#19127711C212yesmaximal, complemented
7#407×C83C211yesmaximal, characteristic, complemented
8C2×#13050590C2211yescharacteristic
9C22×Dic166—28no—
10C22×Dic166—21no—
11C22×Dic166—22no—
12C22×Dic166C2214yes—
13C22×Dic166C2212yes—
14C23×Dic83—21no—
15#14581825—28no—
16C2×Dic332C2214yescomplemented
17C2×#13050579C2214yescomplemented
18#14581775—21no—
19#14581771C2212yes—
20#14581766C2212yes—
21#14581761—21no—
22C4×C22⋊D83—22no—
23#14581751C2211yescharacteristic
24#14581717C2218yescomplemented
25C23×C4×C83C2211yescharacteristic
26C2×M16×C83C2212yes—
27C22⋊C8×C83C2214yes—
28#13050590—24no—
29C2×Dic166—28no—
30C2×Dic166—216no—
31C2×Dic166—24no—
32C2×Dic166—24no—
33C2×Dic166—22no—
34C2×Dic166—216no—
35C2×Dic166D418yescomplemented
36C2×Dic166C2311yescharacteristic
37C2×Dic166C2314yes—
38C2×D332C2311yescharacteristic
39C2×C22⋊D83—22no—
40C2×C22⋊D83—21no—
41C22×Dic83—28no—
42C22×Dic83—24no—
43C22×Dic83—22no—
44C2×C4×D83—21no—
45Dic332—28no—
46#13050579—28no—
47C4⋊Dic83C2312yes—
48Dic83⋊C4—22no—
49C22⋊Dic83—22no—
50C22⋊Dic83—21no—
51(C2×D83)⋊C4—22no—
52C4×Dic83—21no—
53C24×C83D411yescharacteristic
54C22×C4×C83—24no—
55C22×C4×C83D411yescharacteristic
56C22×C4×C83D412yes—
57C22×C4×C83C2312yes—
58C22×C4×C83C2311yescharacteristic
59M16×C83—24no—
60C2×C8×C83C2314yes—
61C23×C83—22no—
62C23×C83—24no—
63C23×C83C2×D411yescharacteristic
64C23×C83C2×D412yes—
65C22×D83—21no—
66C2×C4×C83—22no—
67C2×C4×C83—24no—
68C2×C4×C83—24no—
69C2×C4×C83C2×D414yes—
70C2×C4×C83C2×D412yes—
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83C2411yescharacteristic
73C22⋊D83—42no—
74C22⋊D83—44no—
75C2×Dic83—216no—
76C2×Dic83—28no—
77C2×Dic83—24no—
78C2×Dic83—22no—
79C2×Dic83—21no—
80D332—22no—
81C4×D83—42no—
82Dic166—216no—
83Dic166—28no—
84Dic166—24no—
85Dic166—22no—
86Dic166C2×D418yes—
87C8×C83—24no—
88C22×C83—28no—
89C22×C83—22no—
90C22×C83—22no—
91C22×C83—22no—
92C22×C83#12012yes—
93C22×C83#13714yescomplemented
94C22×C83C22×D411yescharacteristic
95C2×D83—41no—
96C2×D83—42no—
97C4×C83—22no—
98C4×C83#12012yes—
99C4×C83C22×D412yes—
100Dic83—28no—
101Dic83—41no—
102Dic83—42no—
103C2×C83—22no—
104C2×C83—24no—
105C2×C83C2×#12011yescharacteristic
106C2×C83C2×#13712yes—
107D83—81no—
108#3004—831nomaximal, Sylow
109C83#300411yescharacteristic, complemented, Sylow
110C23×Q8—831no—
111C2×#137—832no—
112#525—831no—
113#451—834no—
114#448—834no—
115#419—832no—
116#407D8311yescharacteristic, complemented
117C2×#41—831no—
118C22×Q8—832no—
119C22×Q8—834no—
120C22×Q8—1668no—
121C22×Q8—1662no—
122C22×Q8—1661no—
123C23×C4—1661no—
124C23×C4C2×D8311yescharacteristic
125#137—1668no—
126C2×Q16—834no—
127C2×SD16—834no—
128C2×M16C2×D8312yes—
129#123—1661no—
130C22⋊Q8—832no—
131#121—832no—
132#120—1661no—
133C4×D4—1662no—
134#117—831no—
135#103—838no—
136C22⋊C8C2×D8314yes—
137C24D33211yescharacteristic
138#41—1664no—
139C2×Q8—838no—
140C2×Q8—834no—
141C2×Q8—831no—
142C2×Q8—16616no—
143C2×Q8—1668no—
144C2×Q8—16616no—
145C2×Q8—1664no—
146C2×Q8—1664no—
147C2×Q8—1662no—
148C2×D4—831no—
149C2×D4—1662no—
150C2×D4—1661no—
151C22×C4—24no—
152C22×C4—1661no—
153C22×C4—1668no—
154C22×C4—1664no—
155C22×C4—1662no—
156C22×C4D33212yes—
157C22×C4D33211yescharacteristic
158C22×C4C22×D8312yes—
159C22×C4C22×D8311yescharacteristic
160Q16—1668no—
161SD16—1668no—
162M16—24no—
163C2×C8C22×D8314yes—
164C4⋊C4—832no—
165C4⋊C4—1662no—
166C22⋊C4—1661no—
167C22⋊C4—1662no—
168C22⋊C4—1662no—
169C42—1661no—
170C23—22no—
171C23—24no—
172C23—1661no—
173C23C2×D33211yescharacteristic
174C23C2×D33212yes—
175Q8—838no—
176Q8—16616no—
177Q8—1662no—
178Q8—1668no—
179Q8—1664no—
180D4—3322no—
181D4—3324no—
182D4—1662no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—3322no—
187C2×C4—1662no—
188C2×C4—1661no—
189C2×C4—16616no—
190C2×C4—1668no—
191C2×C4—1664no—
192C2×C4D4×D8314yes—
193C2×C4C2×D33212yes—
194C2×C4C2×D33211yescharacteristic
195C2×C4C23×D8311yescharacteristic
196C8—24no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—3321no—
202C22—3322no—
203C22#1458176212yes—
204C22#1458182514yescomplemented
205C22C22×D33211yescharacteristic
206C4—22no—
207C4—3321no—
208C4—3322no—
209C4—1668no—
210C4#1458176212yes—
211C4C2×D4×D8312yes—
212C2—22no—
213C2—24no—
214C2—6641no—
215C2C2×#1458176211yescharacteristic
216C2C2×#1458182512yes—
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#407×C83C2
socleC22×C83C22×D4
prime radical#19128212C2
hypercenter#407D83
hypocenterC83#3004
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C23×Dic166C2semidirect
2. C2×#14581825C2semidirect
4. #19127859C2semidirect
5. #19127842C2semidirect
6. #19127711C2semidirect
7. #407×C83C2semidirect
16. C2×Dic332C22semidirect
17. C2×#13050579C22semidirect
24. #14581717C22semidirect
35. C2×Dic166D4semidirect
93. C22×C83#137semidirect
109. C83#3004semidirect
116. #407D83semidirect
204. C22#14581825semidirect

Maximal subgroups

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

subgroupindex
C23×Dic1662
C2×#145818252
#191282122
#191278592
#191278422
#191277112
#407×C832
#300483