Finite Group Databasebeta

← #24907901

Subgroups — #24907901

Order 10624.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#14581766C211yesmaximal, characteristic
2C2×#14581714C212yesmaximal, complemented
3#19127561C211yesmaximal, characteristic, complemented
4#19127464C212yesmaximal, complemented
5C22×M16×C83C211yesmaximal, characteristic, complemented
6C22×D332C2211yescharacteristic
7C22×C22⋊D83—21no—
8#14581766—22no—
9#14581766C414yescomplemented
10C2×C4⋊Dic83C2212yes—
11C2×C4⋊Dic83C2211yescharacteristic
12C2×C22⋊Dic83—21no—
13C2×(C2×D83)⋊C4—21no—
14#14581714—24no—
15C83⋊#95—21no—
16C23×C4×C83C2211yescharacteristic
17C2×M16×C83—22no—
18C2×M16×C83—24no—
19C22×C8×C83C2212yes—
20C23×D83—21no—
21C2×D332—22no—
22C2×D332—22no—
23C2×D332—21no—
24C2×D332C2×C412yes—
25C2×C22⋊D83—24no—
26C2×C22⋊D83—44no—
27C22×Dic83—21no—
28C22×Dic83—22no—
29C4⋊Dic83—21no—
30C4⋊Dic83—22no—
31C4⋊Dic83—22no—
32C4⋊Dic83C2×C412yes—
33C22⋊Dic83—42no—
34(C2×D83)⋊C4—24no—
35C24×C83D411yescharacteristic
36C22×C4×C83—22no—
37C22×C4×C83—22no—
38C22×C4×C83C2×C412yes—
39C22×C4×C83D411yescharacteristic
40C22×C4×C83D412yes—
41C22×C4×C83C2311yescharacteristic
42M16×C83—28no—
43C2×C8×C83—22no—
44C2×C8×C83—22no—
45C2×C8×C83D414yescomplemented
46C23×C83—22no—
47C23×C83—21no—
48C23×C83—22no—
49C23×C83—21no—
50C23×C83C22⋊C412yes—
51C23×C83C2×D411yescharacteristic
52C22×D83—22no—
53C22×D83—42no—
54C22×D83—42no—
55C22×D83—41no—
56C22×D83—41no—
57C2×C4×C83—22no—
58C2×C4×C83—24no—
59C2×C4×C83—22no—
60C2×C4×C83—22no—
61C2×C4×C83C22⋊C412yes—
62C2×C4×C83C22×C411yescharacteristic
63C2×C4×C83C2×D412yes—
64C2×C4×C83C2×D411yescharacteristic
65C2×C4×C83#4112yes—
66C22⋊D83—48no—
67C2×Dic83—22no—
68C2×Dic83—42no—
69C2×Dic83—42no—
70C2×Dic83—42no—
71C2×Dic83—42no—
72D332—24no—
73D332—41no—
74D332—41no—
75C8×C83—24no—
76C22×C83—22no—
77C22×C83—22no—
78C22×C83—22no—
79C22×C83—42no—
80C22×C83—42no—
81C22×C83C2×C22⋊C411yescharacteristic
82C22×C83#12112yes—
83C22×C83AGL(1,ℤ/8)11yescharacteristic
84C22×C83AGL(1,ℤ/8)12yes—
85C22×C83#13711yescharacteristic
86C2×D83—42no—
87C2×D83—42no—
88C2×D83—42no—
89C2×D83—82no—
90C2×D83—82no—
91C4×C83—22no—
92C4×C83C4×D412yes—
93C4×C83#12011yescharacteristic
94C4×C83#12311yescharacteristic
95Dic83—42no—
96Dic83—82no—
97C2×C83—42no—
98C2×C83#38611yescharacteristic
99C2×C83#41711yescharacteristic
100C2×C83#41811yescharacteristic
101C2×C83#46812yes—
102C2×C83#46912yes—
103D83—82no—
104#1895—831nomaximal, Sylow
105C83#189511yescharacteristic, complemented, Sylow
106C22×M16D8311yescharacteristic, complemented
107C2×#121—831no—
108C2×#102—832no—
109#380—831no—
110#340—832no—
111C22×D4—831no—
112C22×D4—1661no—
113C23×C4C2×D8311yescharacteristic
114C2×M16—22no—
115C2×M16—24no—
116C22×C8C2×D8312yes—
117#121—834no—
118#121—1662no—
119C2×C4⋊C4—831no—
120C2×C4⋊C4—832no—
121C2×C22⋊C4—1661no—
122C2×C22⋊C4—1661no—
123#102—1664no—
124#95—1661no—
125C24—1661no—
126C24D33211yescharacteristic
127C2×D4—3324no—
128C2×D4—832no—
129C2×D4—1664no—
130C2×D4—1661no—
131C2×D4—1662no—
132C2×D4—1662no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—1662no—
136C22×C4—1661no—
137C22×C4C4×D8312yes—
138C22×C4D33211yescharacteristic
139C22×C4C22⋊D8312yes—
140C22×C4C22×D8311yescharacteristic
141M16—28no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8C22⋊D8314yescomplemented
145C4⋊C4—832no—
146C4⋊C4—1662no—
147C4⋊C4—1662no—
148C4⋊C4—1661no—
149C22⋊C4—3322no—
150C22⋊C4—1664no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—21no—
155C23—3322no—
156C23—3322no—
157C23—3321no—
158C23—3321no—
159C23—1662no—
160C23(C2×D83)⋊C412yes—
161C23C2×D33211yescharacteristic
162D4—3328no—
163D4—3321no—
164D4—3321no—
165D4—1664no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—3322no—
171C2×C4—3322no—
172C2×C4—3322no—
173C2×C4—3322no—
174C2×C4—1662no—
175C2×C4(C2×D83)⋊C412yes—
176C2×C4C2×C4×D8311yescharacteristic
177C2×C4C2×C22⋊D8311yescharacteristic
178C2×C4C2×C22⋊D8312yes—
179C2×C4#1305059012yes—
180C8—24no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—3322no—
187C22—3322no—
188C22—3322no—
189C22—6642no—
190C22—6642no—
191C22C2×(C2×D83)⋊C411yescharacteristic
192C22#1458176612yes—
193C22#1458182012yes—
194C22#1458182011yescharacteristic
195C22#1458182511yescharacteristic
196C4—22no—
197C4—3322no—
198C4—6642no—
199C4C4×C22⋊D8312yes—
200C4#1458176111yescharacteristic
201C4#1458177511yescharacteristic
202C2—42no—
203C2—6642no—
204C2#1912758011yescharacteristic
205C2#1912770311yescharacteristic
206C2#1912770711yescharacteristic
207C2#1912794712yes—
208C2#1912795412yes—
subgroupG/H
centerC23C2×D332
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
FittingC22×M16×C83C2
socleC23×C83C2×D4
prime radicalC2×#14581766C2
cosocleC22×C4×C83C23
hypercenterC22×M16D83
hypocenterC83#1895
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#14581714C2semidirect
3. #19127561C2semidirect
4. #19127464C2semidirect
5. C22×M16×C83C2semidirect
9. #14581766C4semidirect
45. C2×C8×C83D4semidirect
105. C83#1895semidirect
106. C22×M16D83semidirect
144. C2×C8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817662
C2×#145817142
#191275612
#191274642
C22×M16×C832
#189583