Finite Group Databasebeta

← #72231084

Subgroups — #72231084

Order 2400.

332 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 68 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#11925431C211yesmaximal, characteristic
2#11925359C211yesmaximal, characteristic
3#11925356C211yesmaximal, characteristic
4#11925351C211yesmaximal, characteristic
5Dic3×#1044C211yesmaximal, characteristic
6Dic3⋊#1043C211yesmaximal, characteristic
7C3×#87830C211yesmaximal, characteristic
8#11717355—31nomaximal, Hall
9C2×C3⋊#1044C2211yescharacteristic
10C2×C3⋊#1043C2211yescharacteristic
11C2×C3×#1044C2211yescharacteristic
12C2×C3×#1043C2211yescharacteristic
13#10597126C2211yescharacteristic
14#10597124C412yescomplemented
15Dic3×C5⋊D5C2211yescharacteristic
16C5×#1204C412yescomplemented
17C3×#6036C2211yescharacteristic
18AGL(1,𝔽5)×Dic6—52nomaximal
19#87872—31no—
20C4×#1044—31no—
21C4⋊#1043—31no—
22#87832—31no—
23#87832—31no—
24#87830—31no—
25#87830D311yescharacteristic
26C2×C3×C5⋊D5C2311yescharacteristic
27C3⋊#1044—21no—
28C3⋊#1043—21no—
29C3×#1044Q812yescomplemented
30C3×#1043—21no—
31C3⋊#1039C2×C411yescharacteristic
32C5×Dic15C2×C412yes—
33C52×Dic3C2×C411yescharacteristic
34C3×C5×Dic5C2×C412yes—
35D5×Dic6—52no—
36C4⋊ΓL(1,𝔽16)—52no—
37C3×C4×AGL(1,𝔽5)—52no—
38Dic3⋊AGL(1,𝔽5)—52no—
39Dic3⋊AGL(1,𝔽5)—52no—
40Dic3⋊AGL(1,𝔽5)—102no—
41Dic3×AGL(1,𝔽5)—52no—
42Dic3×AGL(1,𝔽5)—52no—
43C2×#1044—31no—
44C2×#1044C2×D311yescharacteristic
45C2×#1043—31no—
46C2×#1043C2×D311yescharacteristic
47C4×C5⋊D5—31no—
48C5×Dic10—32no—
49#6039—32no—
50#6036—31no—
51#6036C2×D311yescharacteristic
52Q8×AGL(1,𝔽5)—152no—
53C2×C3×C52C22×C411yescharacteristic
54C3×C5⋊D5C2×Q811yescharacteristic
55C3×C5⋊D5#4111yescharacteristic
56C2×ΓL(1,𝔽16)—52no—
57C2×ΓL(1,𝔽16)—52no—
58C2×ΓL(1,𝔽16)—102no—
59C2×C3×AGL(1,𝔽5)—52no—
60C2×C3×AGL(1,𝔽5)—52no—
61C2×C3×AGL(1,𝔽5)—102no—
62Dic30—52no—
63C5×Dic6—52no—
64C3×C4×D5—52no—
65#1204—52no—
66#1204AGL(1,𝔽5)12yescomplemented
67D5×Dic3—52no—
68D5×Dic3—52no—
69D5×Dic3—102no—
70C2×C5⋊D5C22×D311yescharacteristic
71#1044—61no—
72#1044Dic612yescomplemented
73#1043—21no—
74#1043—61no—
75C4×C52—31no—
76#1039—31no—
77C5×Dic5—32no—
78C5×Dic5C4×D312yes—
79C4×Dic6—251noHall
80Q8×D5—152no—
81C4⋊AGL(1,𝔽5)—152no—
82C4⋊AGL(1,𝔽5)—152no—
83C4⋊AGL(1,𝔽5)—152no—
84C4⋊AGL(1,𝔽5)—302no—
85C4×AGL(1,𝔽5)—52no—
86C4×AGL(1,𝔽5)—152no—
87C4×AGL(1,𝔽5)—152no—
88C3×C52C4×Q811yescharacteristic, complemented, Hall
89C2×C3×D5—52no—
90C2×C3×D5—102no—
91ΓL(1,𝔽16)—102no—
92ΓL(1,𝔽16)—102no—
93ΓL(1,𝔽16)—202no—
94C3×AGL(1,𝔽5)—54no—
95C3×AGL(1,𝔽5)—102no—
96C3×AGL(1,𝔽5)—202no—
97C3×C4×C5—52no—
98Dic15—52no—
99Dic15—102no—
100Dic15C2×AGL(1,𝔽5)12yes—
101C3×Dic5C2×AGL(1,𝔽5)12yes—
102C5×Dic3—22no—
103C5×Dic3—52no—
104C5×Dic3C2×AGL(1,𝔽5)12yes—
105C2×C52C2×C4×D311yescharacteristic
106C5⋊D5C2×Dic611yescharacteristic
107C5⋊D5#23511yescharacteristic
108C2×Dic6—251no—
109C3×C42—251no—
110C4⋊Dic3—251no—
111Dic3⋊C4—251no—
112Dic3⋊C4—251no—
113C4×Dic3—251no—
114C4×Dic3—251no—
115C2×AGL(1,𝔽5)—52no—
116C2×AGL(1,𝔽5)—52no—
117C2×AGL(1,𝔽5)—102no—
118C2×AGL(1,𝔽5)—152no—
119C2×AGL(1,𝔽5)—152no—
120C2×AGL(1,𝔽5)—302no—
121C5×Q8—152no—
122C4×D5—52no—
123C4×D5—152no—
124C4×D5—152no—
125C4×D5—302no—
126Dic10—32no—
127Dic10—152no—
128Dic10—152no—
129C4×Q8—751noSylow
130C2×C3×C5—22no—
131C2×C3×C5C22×AGL(1,𝔽5)12yes—
132C3×D5—52no—
133C3×D5—52no—
134C3×D5—102no—
135C3×D5—102no—
136C52C4×Dic611yescharacteristic, complemented, Sylow
137C2×C3×C4—251no—
138C2×C3×C4—251no—
139C2×C3×C4—251no—
140C2×Dic3—251no—
141C2×Dic3—251no—
142C2×Dic3—251no—
143C2×Dic3—251no—
144Dic6—52no—
145Dic6—252no—
146C2×D5—52no—
147C2×D5—102no—
148AGL(1,𝔽5)—54no—
149AGL(1,𝔽5)—102no—
150AGL(1,𝔽5)—202no—
151AGL(1,𝔽5)—302no—
152AGL(1,𝔽5)—302no—
153AGL(1,𝔽5)—602no—
154C4×C5—32no—
155C4×C5—52no—
156C4×C5—62no—
157C4×C5—152no—
158Dic5—32no—
159Dic5—152no—
160Dic5—302no—
161Dic5D3×AGL(1,𝔽5)12yes—
162C2×Q8—751no—
163C4⋊C4—751no—
164C4⋊C4—751no—
165C4⋊C4—751no—
166C42—251no—
167C42—751no—
168C42—751no—
169C3×C5—22no—
170C3×C5Q8×AGL(1,𝔽5)12yescomplemented
171C22×C3—251no—
172C3×C4—52no—
173C3×C4—252no—
174C3×C4—501no—
175Dic3—52no—
176Dic3—251no—
177Dic3—501no—
178Dic3—501no—
179Dic3D5⋊AGL(1,𝔽5)11yescharacteristic
180C2×C5—22no—
181C2×C5C2×D3×AGL(1,𝔽5)12yes—
182D5—52no—
183D5—52no—
184D5—102no—
185D5—102no—
186Q8—152no—
187Q8—752no—
188C2×C4—251no—
189C2×C4—251no—
190C2×C4—251no—
191C2×C4—751no—
192C2×C4—751no—
193C2×C4—751no—
194C2×C4—751no—
195C2×C3—251no—
196C2×C3—251no—
197C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
198C5—22no—
199C5AGL(1,𝔽5)×Dic612yescomplemented
200C22—251no—
201C4—31no—
202C4—52no—
203C4—152no—
204C4—252no—
205C4—501no—
206C4—751no—
207C4—1501no—
208C4—1501no—
209C3#1171735511yescharacteristic, complemented, Sylow
210C2—251no—
211C2—251no—
212C2D3×D5⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×D5⋊AGL(1,𝔽5)
commutator, Fitting, socleC2×C3×C52C22×C4
prime radical, cosocleC2×C3×C5⋊D5C23
hypocenterC3×C52C4×Q8
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
14. #10597124C4semidirect
16. C5×#1204C4semidirect
29. C3×#1044Q8semidirect
66. #1204AGL(1,𝔽5)semidirect
72. #1044Dic6semidirect
88. C3×C52C4×Q8semidirect
136. C52C4×Dic6semidirect
170. C3×C5Q8×AGL(1,𝔽5)semidirect
199. C5AGL(1,𝔽5)×Dic6semidirect
209. C3#11717355semidirect

Maximal subgroups

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

subgroupindex
#119254312
#119253592
#119253562
#119253512
Dic3×#10442
Dic3⋊#10432
C3×#878302
#117173553
AGL(1,𝔽5)×Dic65