Finite Group Databasebeta

← #24907906

Subgroups — #24907906

Order 10624.

442 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 101 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#19128212C211yesmaximal, characteristic
2C2×#14581720C212yesmaximal, complemented
3#19127665C211yesmaximal, characteristic, complemented
4#19127461C212yesmaximal, complemented
5C22×C42×C83C211yesmaximal, characteristic, complemented
6C2×#13050590C2211yescharacteristic
7C2×#13050574C2212yes—
8#14581775—21no—
9#14581771C412yescomplemented
10#14581766C412yescomplemented
11#14581761—21no—
12C4×C22⋊D83—22no—
13#14581751C2211yescharacteristic
14#14581720—24no—
15C22⋊(C83⋊C8)—22no—
16C23×C4×C83—21no—
17C23×C4×C83C2211yescharacteristic
18C2×C42×C83—21no—
19C2×C42×C83—24no—
20C2×C42×C83C2212yes—
21#13050590—22no—
22#13050590—22no—
23C2×Dic166C2×C411yescharacteristic, complemented
24C2×D332C2×C411yescharacteristic, complemented
25C2×C22⋊D83—22no—
26C2×C22⋊D83—21no—
27C2×C4×D83—21no—
28#13050574—22no—
29#13050574—22no—
30C2×C83⋊C8—22no—
31C4⋊Dic83C2×C412yescomplemented
32Dic83⋊C4—22no—
33C22⋊Dic83—22no—
34C22⋊Dic83—21no—
35(C2×D83)⋊C4—22no—
36C4×Dic83—21no—
37C24×C83D411yescharacteristic
38C22×C4×C83—24no—
39C22×C4×C83—28no—
40C22×C4×C83—22no—
41C22×C4×C83—22no—
42C22×C4×C83—22no—
43C22×C4×C83C2×C412yes—
44C22×C4×C83D412yes—
45C22×C4×C83D411yescharacteristic
46C22×C4×C83C2311yescharacteristic
47C42×C83—24no—
48C42×C83—22no—
49C42×C83D414yescomplemented
50C23×C83—24no—
51C23×C83—21no—
52C23×C83—21no—
53C23×C83C22⋊C412yes—
54C23×C83C2×D411yescharacteristic
55C22×D83—21no—
56C2×C4×C83—216no—
57C2×C4×C83—28no—
58C2×C4×C83—21no—
59C2×C4×C83—24no—
60C2×C4×C83—21no—
61C2×C4×C83—24no—
62C2×C4×C83—24no—
63C2×C4×C83C22⋊C412yes—
64C2×C4×C83C22×C411yescharacteristic
65C2×C4×C83C2×D412yes—
66C2×C4×C83C2×D411yescharacteristic
67C2×C4×C83#4112yes—
68C22⋊D83—44no—
69C22⋊D83—41no—
70C22⋊D83—41no—
71C2×Dic83—22no—
72C2×Dic83—21no—
73D332—22no—
74C4×D83—42no—
75Dic166—22no—
76C83⋊C8—42no—
77C22×C83—24no—
78C22×C83—24no—
79C22×C83—22no—
80C22×C83—22no—
81C22×C83—22no—
82C22×C83#10414yescomplemented
83C22×C83C2×C22⋊C411yescharacteristic
84C22×C83#12011yescharacteristic
85C22×C83#12311yescharacteristic
86C2×D83—41no—
87C2×D83—42no—
88C4×C83—28no—
89C4×C83—22no—
90C4×C83C4×D412yes—
91C4×C83#12112yes—
92Dic83—41no—
93Dic83—42no—
94C2×C83—21no—
95C2×C83—24no—
96C2×C83—21no—
97C2×C83#38611yescharacteristic
98C2×C83C2×#10412yes—
99D83—81no—
100#1896—831nomaximal, Sylow
101C83#189611yescharacteristic, complemented, Sylow
102#525—831no—
103C22×C42D8311yescharacteristic, complemented
104C2×#104—832no—
105#407—831no—
106#339—832no—
107C2×#41—831no—
108C23×C4—21no—
109C23×C4C2×D8311yescharacteristic
110C2×M16—832no—
111#123—1661no—
112C22⋊Q8—832no—
113#121—832no—
114#120—1661no—
115C4×D4—1662no—
116#117—831no—
117C2×C42—21no—
118C2×C42—24no—
119C2×C42C2×D8312yes—
120#104—1664no—
121C22⋊C8—1662no—
122C24C22⋊D8311yescharacteristic
123#41—1662no—
124#41—1662no—
125C2×Q8—831no—
126C2×D4—831no—
127C2×D4—1662no—
128C2×D4—1661no—
129C22×C4—24no—
130C22×C4—28no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—1661no—
135C22×C4C4×D8312yes—
136C22×C4D33212yes—
137C22×C4C22⋊D8311yescharacteristic
138C22×C4C22×D8311yescharacteristic
139M16—1662no—
140M16—1662no—
141C2×C8—1662no—
142C4⋊C4—832no—
143C4⋊C4—1662no—
144C22⋊C4—1662no—
145C22⋊C4—1661no—
146C22⋊C4—1662no—
147C42—24no—
148C42—22no—
149C42—1661no—
150C42C22⋊D8314yescomplemented
151C23—21no—
152C23—21no—
153C23—24no—
154C23—1661no—
155C23(C2×D83)⋊C412yes—
156C23C2×C22⋊D8311yescharacteristic
157Q8—1662no—
158D4—3321no—
159D4—3321no—
160D4—3324no—
161D4—1662no—
162C2×C4—216no—
163C2×C4—28no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—3322no—
170C2×C4—1662no—
171C2×C4—1661no—
172C2×C4(C2×D83)⋊C412yes—
173C2×C4C2×C4×D8311yescharacteristic
174C2×C4C2×C22⋊D8312yes—
175C2×C4C2×D33211yescharacteristic
176C2×C4#1305059012yes—
177C8—3322no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—3321no—
184C22—3322no—
185C22#1458172014yescomplemented
186C22C2×(C2×D83)⋊C411yescharacteristic
187C22#1458176111yescharacteristic
188C22#1458177511yescharacteristic
189C4—28no—
190C4—22no—
191C4—3321no—
192C4—3322no—
193C4C4×C22⋊D8312yes—
194C4#1458176612yes—
195C2—21no—
196C2—21no—
197C2—24no—
198C2—6641no—
199C2#1912758011yescharacteristic
200C2C2×#1458172012yes—
subgroupG/H
centerC2×C4C2×D332
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
FittingC22×C42×C83C2
socleC22×C83C2×C22⋊C4
prime radical#19128212C2
cosocleC22×C4×C83C23
hypercenterC22×C42D83
hypocenterC83#1896
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. C2×#14581720C2semidirect
3. #19127665C2semidirect
4. #19127461C2semidirect
5. C22×C42×C83C2semidirect
9. #14581771C4semidirect
10. #14581766C4semidirect
23. C2×Dic166C2×C4semidirect
24. C2×D332C2×C4semidirect
31. C4⋊Dic83C2×C4semidirect
49. C42×C83D4semidirect
82. C22×C83#104semidirect
101. C83#1896semidirect
103. C22×C42D83semidirect
150. C42C22⋊D83semidirect
185. C22#14581720semidirect

Maximal subgroups

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

subgroupindex
#191282122
C2×#145817202
#191276652
#191274612
C22×C42×C832
#189683