Finite Group Databasebeta

← #24908003

Subgroups — #24908003

Order 10624.

380 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 85 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#19128260C211yesmaximal, characteristic
2#19128176C211yesmaximal, characteristic, complemented
3C2×#14581706C212yesmaximal, complemented
4#19127468C212yesmaximal, complemented
5C2×#99×C83C211yesmaximal, characteristic, complemented
6C22×C22⋊D83C2211yescharacteristic
7#14581775—21no—
8#14581773C412yescomplemented
9#14581764—22no—
10#14581761—21no—
11#14581760C412yescomplemented
12D4×Dic83—24no—
13#14581751—21no—
14C2×C22⋊Dic83—21no—
15C2×C22⋊Dic83C2211yescharacteristic
16C2×C22⋊Dic83C2212yes—
17#14581706—24no—
18C22×D4×C83C2211yescharacteristic
19C2×C22⋊C4×C83C2212yes—
20#99×C83—22no—
21C23×D83C2×C411yescharacteristic, complemented
22C2×C22⋊D83—21no—
23C2×C22⋊D83—21no—
24C2×C22⋊D83—22no—
25C2×C22⋊D83—22no—
26C2×C22⋊D83—22no—
27C22×Dic83—22no—
28C22×Dic83—21no—
29C22×Dic83—22no—
30C22×Dic83C2×C411yescharacteristic, complemented
31C4⋊Dic83—22no—
32Dic83⋊C4—22no—
33C22⋊Dic83—22no—
34C22⋊Dic83—22no—
35C22⋊Dic83—21no—
36C22⋊Dic83—22no—
37C22⋊Dic83C2×C412yescomplemented
38C22⋊Dic83D414yescomplemented
39(C2×D83)⋊C4—22no—
40C4×Dic83—22no—
41C24×C83D411yescharacteristic
42C24×C83C2311yescharacteristic
43C2×D4×C83—22no—
44C2×D4×C83—22no—
45C2×D4×C83C2×C412yes—
46C2×D4×C83D412yes—
47C22×C4×C83—22no—
48C22×C4×C83D411yescharacteristic
49C22⋊C4×C83—22no—
50C22⋊C4×C83—22no—
51C23×C83—21no—
52C23×C83—21no—
53C23×C83—81no—
54C23×C83—81no—
55C23×C83C22⋊C412yes—
56C23×C83C22×C411yescharacteristic
57C23×C83C2×D412yes—
58C23×C83C2×D411yescharacteristic
59C23×C83C2×D411yescharacteristic
60C23×C83#4112yes—
61C22×D83—21no—
62C22×D83—22no—
63C22×D83—81no—
64D4×C83—24no—
65D4×C83—24no—
66C2×C4×C83—21no—
67C2×C4×C83—21no—
68C2×C4×C83—42no—
69C2×C4×C83—82no—
70C2×C4×C83C22⋊C412yes—
71C22⋊D83—41no—
72C22⋊D83—42no—
73C22⋊D83—41no—
74C22⋊D83—82no—
75C2×Dic83—22no—
76C2×Dic83—21no—
77C2×Dic83—22no—
78C2×Dic83—24no—
79C2×Dic83—82no—
80C2×Dic83—82no—
81C2×Dic83—81no—
82C22×C83—22no—
83C22×C83—42no—
84C22×C83—41no—
85C22×C83—42no—
86C22×C83—41no—
87C22×C83—41no—
88C22×C83—42no—
89C22×C83—42no—
90C22×C83—81no—
91C22×C83C2×C22⋊C411yescharacteristic
92C22×C83C4×D412yes—
93C22×C83#12011yescharacteristic
94C22×C83#12112yes—
95C22×C83#12311yescharacteristic
96C2×D83—42no—
97C2×D83—41no—
98C2×D83—82no—
99C4×C83—22no—
100C4×C83—82no—
101Dic83—44no—
102Dic83—42no—
103Dic83—41no—
104C2×C83—21no—
105C2×C83—21no—
106C2×C83—22no—
107C2×C83—22no—
108C2×C83—21no—
109C2×C83—42no—
110C2×C83#38611yescharacteristic
111D83—81no—
112#1913—831nomaximal, Sylow
113C83#191311yescharacteristic, complemented, Sylow
114#534—831no—
115#518—831no—
116C2×#99—832no—
117C2×#99D8311yescharacteristic, complemented
118#342—832no—
119C22×D4—831no—
120C22×D4C2×D8311yescharacteristic
121#123—832no—
122#123—1661no—
123#121—1662no—
124#120—832no—
125#120—1661no—
126C4×D4—1664no—
127#117—1661no—
128C2×C22⋊C4—832no—
129C2×C22⋊C4—831no—
130C2×C22⋊C4—1661no—
131C2×C22⋊C4C2×D8312yes—
132#99—22no—
133#99—1664no—
134C24—831no—
135C24D33211yescharacteristic
136C24C22×D8311yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—1662no—
140C2×D4—1661no—
141C2×D4—1661no—
142C2×D4—1662no—
143C2×D4—1662no—
144C2×D4C4×D8312yes—
145C2×D4C22⋊D8312yes—
146C22×C4—22no—
147C22×C4—831no—
148C22×C4—1662no—
149C22×C4—1662no—
150C22×C4—1661no—
151C22×C4D33211yescharacteristic
152C4⋊C4—1662no—
153C4⋊C4—1662no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—832no—
157C22⋊C4—834no—
158C22⋊C4—1662no—
159C22⋊C4—1662no—
160C22⋊C4—1662no—
161C22⋊C4—1661no—
162C22⋊C4—1662no—
163C42—1662no—
164C23—21no—
165C23—21no—
166C23—81no—
167C23—81no—
168C23—6641no—
169C23—1661no—
170C23—1662no—
171C23(C2×D83)⋊C412yes—
172C23C2×C4×D8311yescharacteristic
173C23D4×D8312yes—
174C23C2×C22⋊D8311yescharacteristic
175C23C2×D33211yescharacteristic
176C23#1305058912yes—
177D4—24no—
178D4—24no—
179D4—3321no—
180D4—3322no—
181D4—3321no—
182D4—6642no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4—6642no—
188C2×C4—6642no—
189C2×C4—6641no—
190C2×C4—1664no—
191C2×C4—1662no—
192C2×C4—1661no—
193C2×C4—1662no—
194C2×C4(C2×D83)⋊C412yes—
195C22—22no—
196C22—42no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—3322no—
205C22—3321no—
206C22—6642no—
207C22C2×(C2×D83)⋊C411yescharacteristic
208C22(C22⋊D83)⋊C412yes—
209C22#1458176211yescharacteristic
210C22#1458176412yes—
211C22#1458177611yescharacteristic
212C4—22no—
213C4—82no—
214C4—3324no—
215C4—3322no—
216C4—3321no—
217C2—22no—
218C2—21no—
219C2—21no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—6641no—
224C2#1912757811yescharacteristic
subgroupG/H
centerC2#19127578
commutatorC23×C83C22×C4
FrattiniC24C22×D83
FittingC2×#99×C83C2
socleC2×C83#386
prime radical#19128260C2
cosocleC24×C83C23
hypercenterC2×#99D83
hypocenterC83#1913
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #19128176C2semidirect
3. C2×#14581706C2semidirect
4. #19127468C2semidirect
5. C2×#99×C83C2semidirect
8. #14581773C4semidirect
11. #14581760C4semidirect
21. C23×D83C2×C4semidirect
30. C22×Dic83C2×C4semidirect
37. C22⋊Dic83C2×C4semidirect
38. C22⋊Dic83D4semidirect
113. C83#1913semidirect
117. C2×#99D83semidirect

Maximal subgroups

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

subgroupindex
#191282602
#191281762
C2×#145817062
#191274682
C2×#99×C832
#191383