Finite Group Databasebeta

← #72231100

Subgroups — #72231100

Order 2400.

372 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 79 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×C3×#6036C211yesmaximal, characteristic
2#11925540C211yesmaximal, characteristic
3#11925533C211yesmaximal, characteristic
4#11925351C212yesmaximal, complemented
5#11925349C212yesmaximal, complemented
6#11717367—31nomaximal, Hall
7C22×C3×C5⋊D5C2211yescharacteristic
8C2×C3⋊#1044C2212yes—
9C2×C3⋊#1043C2212yes—
10C2×C3×C5×Dic5C412yescomplemented
11C3×#6036C412yescomplemented
12C3×#6036C2211yescharacteristic
13C3×#6036C2211yescharacteristic
14#91989—52nomaximal
15C2×#6036D311yescharacteristic
16C22⋊#1044—31no—
17C22⋊#1043—31no—
18#87832—32no—
19#87830—32no—
20C22×C3×C52C2×C411yescharacteristic
21C2×C3×C5⋊D5—22no—
22C2×C3×C5⋊D5C2×C411yescharacteristic
23C2×C3×C5⋊D5C2311yescharacteristic
24C3⋊#1044—22no—
25C3⋊#1043—22no—
26C3×C5×Dic5C2×C412yes—
27C3×C5×Dic5C2×C412yes—
28C2×C3×C4×D5—52no—
29#6763—52no—
30#6763—52no—
31#6763—102no—
32C4⋊ΓL(1,𝔽16)—54no—
33C4×ΓL(1,𝔽16)—54no—
34C22×C5⋊D5C2×D311yescharacteristic
35C2×#1044—32no—
36C2×#1043—32no—
37C2×C5×Dic5Dic312yescomplemented
38#6036Dic312yescomplemented
39#6036C2×D311yescharacteristic
40#6036C2×D311yescharacteristic
41#4130—152no—
42C2×C3×C52—21no—
43C2×C3×C52C22×C411yescharacteristic
44C3×C5⋊D5—21no—
45C3×C5⋊D5#4112yes—
46C22×C3×D5—52no—
47C22×C3×D5—102no—
48C2×ΓL(1,𝔽16)—54no—
49C2×ΓL(1,𝔽16)—54no—
50C2×ΓL(1,𝔽16)—104no—
51C2×C3×C4×C5—52no—
52C2×C3×Dic5AGL(1,𝔽5)12yescomplemented
53C3×C4×D5—52no—
54C3×C4×D5—52no—
55C3×C4×D5—52no—
56C3×C4×D5—52no—
57C22×C52C2×Dic311yescharacteristic
58C2×C5⋊D5—22no—
59C2×C5⋊D5C2×Dic311yescharacteristic
60C2×C5⋊D5C22×D311yescharacteristic
61#1044—62no—
62#1043—62no—
63C5×Dic5C2×Dic312yes—
64C5×Dic5C2×Dic312yes—
65#926—251noHall
66C2×C4×D5—52no—
67C22⋊AGL(1,𝔽5)—152no—
68C22⋊AGL(1,𝔽5)—152no—
69C22⋊AGL(1,𝔽5)—302no—
70C4⋊AGL(1,𝔽5)—154no—
71C4×AGL(1,𝔽5)—154no—
72C3×C52#11711yescharacteristic, complemented, Hall
73C22×C3×C5—22no—
74C22×C3×C5C2×AGL(1,𝔽5)12yes—
75C2×C3×D5—52no—
76C2×C3×D5—52no—
77C2×C3×D5—102no—
78C2×C3×D5—102no—
79C2×C3×D5—104no—
80C2×C3×D5—204no—
81ΓL(1,𝔽16)—104no—
82ΓL(1,𝔽16)—104no—
83ΓL(1,𝔽16)—204no—
84C3×C4×C5—52no—
85C3×C4×C5—52no—
86C3×Dic5C2×AGL(1,𝔽5)12yes—
87C3×Dic5C2×AGL(1,𝔽5)12yes—
88C2×C52—21no—
89C2×C52C22×Dic311yescharacteristic
90C5⋊D5—21no—
91C5⋊D5#23512yes—
92C22×C3×C4—251no—
93#217—251no—
94#217—251no—
95C4⋊Dic3—252no—
96C4×Dic3—252no—
97C22×D5—52no—
98C22×D5—102no—
99C2×AGL(1,𝔽5)—154no—
100C2×AGL(1,𝔽5)—154no—
101C2×AGL(1,𝔽5)—304no—
102C2×C4×C5—52no—
103C2×Dic5ΓL(1,𝔽16)12yescomplemented
104C4×D5—52no—
105C4×D5—52no—
106C4×D5—52no—
107C4×D5—52no—
108#117—751noSylow
109C2×C3×C5—22no—
110C2×C3×C5—22no—
111C2×C3×C5—42no—
112C2×C3×C5C22×AGL(1,𝔽5)12yes—
113C3×D5—54no—
114C3×D5—104no—
115C3×D5—102no—
116C3×D5—202no—
117C52#92611yescharacteristic, complemented, Sylow
118C23×C3—251no—
119C2×C3×C4—52no—
120C2×C3×C4—251no—
121C2×C3×C4—251no—
122C2×C3×C4—252no—
123C2×Dic3—252no—
124C2×Dic3—252no—
125C22×C5—22no—
126C22×C5C2×ΓL(1,𝔽16)12yes—
127C2×D5—52no—
128C2×D5—52no—
129C2×D5—102no—
130C2×D5—102no—
131C2×D5—104no—
132C2×D5—204no—
133AGL(1,𝔽5)—304no—
134AGL(1,𝔽5)—304no—
135AGL(1,𝔽5)—604no—
136C4×C5—52no—
137C4×C5—52no—
138Dic5C2×ΓL(1,𝔽16)12yes—
139Dic5C2×ΓL(1,𝔽16)12yes—
140C22×C4—251no—
141C4⋊C4—752no—
142C22⋊C4—751no—
143C22⋊C4—751no—
144C42—752no—
145C3×C5—22no—
146C3×C5#413012yescomplemented
147C22×C3—251no—
148C22×C3—251no—
149C22×C3—502no—
150C22×C3D5⋊AGL(1,𝔽5)11yescharacteristic
151C3×C4—52no—
152C3×C4—52no—
153Dic3—502no—
154Dic3—502no—
155C2×C5—22no—
156C2×C5—22no—
157C2×C5—42no—
158C2×C5C22×ΓL(1,𝔽16)12yes—
159D5—54no—
160D5—104no—
161D5—102no—
162D5—202no—
163C23—251no—
164C2×C4—52no—
165C2×C4—251no—
166C2×C4—251no—
167C2×C4—252no—
168C2×C4—752no—
169C2×C4—752no—
170C2×C3—21no—
171C2×C3—252no—
172C2×C3—501no—
173C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
174C5—22no—
175C5#9198912yescomplemented
176C22—251no—
177C22—251no—
178C22—502no—
179C22D5⋊ΓL(1,𝔽16)11yescharacteristic
180C4—52no—
181C4—52no—
182C4—1502no—
183C4—1502no—
184C3#1171736711yescharacteristic, complemented, Sylow
185C2—21no—
186C2—252no—
187C2—501no—
188C2C2×D5⋊ΓL(1,𝔽16)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D5⋊ΓL(1,𝔽16)
commutator, socleC2×C3×C52C22×C4
FittingC22×C3×C52C2×C4
prime radicalC22×C3×C5⋊D5C22
cosocleC2×C3×C5⋊D5C23
hypercenterC22D5⋊ΓL(1,𝔽16)
hypocenterC3×C52#117
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
4. #11925351C2semidirect
5. #11925349C2semidirect
10. C2×C3×C5×Dic5C4semidirect
11. C3×#6036C4semidirect
37. C2×C5×Dic5Dic3semidirect
38. #6036Dic3semidirect
52. C2×C3×Dic5AGL(1,𝔽5)semidirect
72. C3×C52#117semidirect
103. C2×Dic5ΓL(1,𝔽16)semidirect
117. C52#926semidirect
146. C3×C5#4130semidirect
175. C5#91989semidirect
184. C3#11717367semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#60362
#119255402
#119255332
#119253512
#119253492
#117173673
#919895