Finite Group Databasebeta

← #72230992

Subgroups — #72230992

Order 2400.

332 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 67 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#11925650C211yesmaximal, characteristic
2C4×C3⋊#1044C211yesmaximal, characteristic
3C3×C4⋊#1043C211yesmaximal, characteristic
4#11925356C212yesmaximal
5#11925354C212yesmaximal
6#11717355—31nomaximal, Hall
7C2×C3⋊#1044C2211yescharacteristic
8C2×C3⋊#1044C2211yescharacteristic
9C2×C3×#1043C2212yes—
10C5×Dic30C412yescomplemented
11C3×C4×C5⋊D5C2211yescharacteristic
12#10597129C412yescomplemented
13#10597126C2212yes—
14ΓL(1,𝔽16)⋊Q8—52nomaximal
15#87872—31no—
16C4×#1044—31no—
17C4⋊#1043D311yescharacteristic
18#87832—32no—
19#87830—32no—
20C2×C3×C5⋊D5C2311yescharacteristic
21C3⋊#1044—21no—
22C3⋊#1044Q812yescomplemented
23C3×#1043—22no—
24C3×C4×C52C2×C411yescharacteristic
25C5×Dic15C2×C414yes—
26C3×#1039C2×C411yescharacteristic
27D5×Dic6—52no—
28C4×ΓL(1,𝔽16)—52no—
29C3×C4⋊AGL(1,𝔽5)—52no—
30C3×C4⋊AGL(1,𝔽5)—102no—
31Dic3⋊AGL(1,𝔽5)—54no—
32Dic3×AGL(1,𝔽5)—54no—
33C2×#1044—31no—
34C2×#1044—31no—
35C2×#1043C2×D312yes—
36C4×C5⋊D5C2×D311yescharacteristic
37C5×Dic10—32no—
38#6039—32no—
39#6036—32no—
40Q8×AGL(1,𝔽5)—152no—
41C2×C3×C52C22×C411yescharacteristic
42C3×C5⋊D5C2×Q811yescharacteristic
43C3×C5⋊D5#4111yescharacteristic
44C2×ΓL(1,𝔽16)—52no—
45C2×ΓL(1,𝔽16)—52no—
46C2×C3×AGL(1,𝔽5)—54no—
47C2×C3×AGL(1,𝔽5)—104no—
48Dic30AGL(1,𝔽5)12yescomplemented
49C5×Dic6—52no—
50C3×C4×D5—52no—
51C3×C4×D5—102no—
52#1204—54no—
53D5×Dic3—54no—
54C2×C5⋊D5C22×D311yescharacteristic
55#1044—32no—
56#1044—61no—
57#1043—22no—
58C4×C52C4×D311yescharacteristic
59#1039C4×D311yescharacteristic
60C5×Dic5—34no—
61#887—251noHall
62Q8×D5—152no—
63C4⋊AGL(1,𝔽5)—52no—
64C4⋊AGL(1,𝔽5)—102no—
65C4⋊AGL(1,𝔽5)—154no—
66C4×AGL(1,𝔽5)—152no—
67C4×AGL(1,𝔽5)—154no—
68C3×C52C4×Q811yescharacteristic, complemented, Hall
69C2×C3×D5—52no—
70C2×C3×D5—102no—
71ΓL(1,𝔽16)—54no—
72ΓL(1,𝔽16)—102no—
73C3×AGL(1,𝔽5)—104no—
74C3×AGL(1,𝔽5)—204no—
75C3×C4×C5—22no—
76C3×C4×C5C2×AGL(1,𝔽5)12yes—
77Dic15C2×AGL(1,𝔽5)14yes—
78C3×Dic5—52no—
79C3×Dic5—102no—
80C5×Dic3—54no—
81C2×C52C2×C4×D311yescharacteristic
82C5⋊D5#23711yescharacteristic
83C5⋊D5D3×Q811yescharacteristic
84C2×Dic6—251no—
85C3×C4⋊C4—251no—
86Dic3⋊C4—252no—
87C4×Dic3—252no—
88C4×Dic3—251no—
89C2×AGL(1,𝔽5)—54no—
90C2×AGL(1,𝔽5)—104no—
91C2×AGL(1,𝔽5)—152no—
92C2×AGL(1,𝔽5)—152no—
93C5×Q8—152no—
94C4×D5—52no—
95C4×D5—102no—
96C4×D5—154no—
97Dic10—32no—
98Dic10—154no—
99C4×Q8—751noSylow
100C2×C3×C5—22no—
101C2×C3×C5C22×AGL(1,𝔽5)12yes—
102C3×D5—52no—
103C3×D5—52no—
104C3×D5—102no—
105C3×D5—102no—
106C52#88711yescharacteristic, complemented, Sylow
107C2×C3×C4—251no—
108C2×C3×C4—252no—
109C2×Dic3—252no—
110C2×Dic3—251no—
111C2×Dic3—251no—
112Dic6—52no—
113Dic6—252no—
114C2×D5—52no—
115C2×D5—102no—
116AGL(1,𝔽5)—104no—
117AGL(1,𝔽5)—154no—
118AGL(1,𝔽5)—204no—
119AGL(1,𝔽5)—302no—
120C4×C5—22no—
121C4×C5—154no—
122C4×C5D3×AGL(1,𝔽5)12yes—
123Dic5—34no—
124Dic5—52no—
125Dic5—102no—
126C2×Q8—751no—
127C4⋊C4—251no—
128C4⋊C4—752no—
129C42—751no—
130C42—752no—
131C3×C5—22no—
132C3×C5Q8×AGL(1,𝔽5)12yescomplemented
133C22×C3—251no—
134C3×C4—251no—
135C3×C4—502no—
136C3×C4D5⋊AGL(1,𝔽5)11yescharacteristic
137Dic3—54no—
138Dic3—252no—
139Dic3—501no—
140C2×C5—22no—
141C2×C5C2×D3×AGL(1,𝔽5)12yes—
142D5—52no—
143D5—52no—
144D5—102no—
145D5—102no—
146Q8—152no—
147Q8—752no—
148C2×C4—251no—
149C2×C4—252no—
150C2×C4—752no—
151C2×C4—751no—
152C2×C4—751no—
153C2×C3—251no—
154C2×C3—251no—
155C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
156C5—22no—
157C5ΓL(1,𝔽16)⋊Q812yescomplemented
158C22—251no—
159C4—154no—
160C4—251no—
161C4—502no—
162C4—752no—
163C4—1501no—
164C4#1059718211yescharacteristic
165C3#1171735511yescharacteristic, complemented, Sylow
166C2—251no—
167C2—251no—
168C2C2×#1059718211yescharacteristic
subgroupG/H
center, FrattiniC2C2×#10597182
commutator, socleC2×C3×C52C22×C4
FittingC3×C4×C52C2×C4
prime radical, cosocleC2×C3×C5⋊D5C23
hypercenterC4#10597182
hypocenterC3×C52C4×Q8
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
10. C5×Dic30C4semidirect
12. #10597129C4semidirect
22. C3⋊#1044Q8semidirect
48. Dic30AGL(1,𝔽5)semidirect
68. C3×C52C4×Q8semidirect
106. C52#887semidirect
132. C3×C5Q8×AGL(1,𝔽5)semidirect
157. C5ΓL(1,𝔽16)⋊Q8semidirect
165. C3#11717355semidirect

Maximal subgroups

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

subgroupindex
#119256502
C4×C3⋊#10442
C3×C4⋊#10432
#119253562
#119253542
#117173553
ΓL(1,𝔽16)⋊Q85