Finite Group Databasebeta

← #72230634

Subgroups — #72230634

Order 2400.

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

normal subgroups only

This group has 412 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
1C2×Dic3×C5⋊D5C211yesmaximal, characteristic
2#11925540C211yesmaximal, characteristic
3C3×C22⋊#1044C211yesmaximal, characteristic
4Dic3⋊#1044C212yesmaximal, complemented
5Dic3×#1044C212yesmaximal, complemented
6#11717451—31nomaximal, Hall
7C22×C3×C5⋊D5C2211yescharacteristic
8C2×C3⋊#1044C2212yes—
9C2×C3×#1044C2212yes—
10C2×C3⋊#1039C411yescharacteristic, complemented
11C2×C52×Dic3C411yescharacteristic, complemented
12Dic3×C5⋊D5C411yescharacteristic, complemented
13Dic3×C5⋊D5C411yescharacteristic, complemented
14Dic3×C5⋊D5C2211yescharacteristic
15Dic3×C5⋊D5C2211yescharacteristic
16#91923—52nomaximal
17C2×C4×C5⋊D5—31no—
18C22⋊#1044—31no—
19C22⋊#1044D311yescharacteristic
20C4⋊#1044—32no—
21C4×#1044—32no—
22C22×C3×C52C2×C411yescharacteristic
23C2×C3×C5⋊D5—22no—
24C2×C3×C5⋊D5C2×C411yescharacteristic
25C2×C3×C5⋊D5C2311yescharacteristic
26C3⋊#1044—22no—
27C3×#1044—22no—
28C3⋊#1039C2×C411yescharacteristic
29C3⋊#1039C2×C411yescharacteristic
30C52×Dic3C2×C411yescharacteristic
31C52×Dic3C2×C411yescharacteristic
32C2×D5×Dic3—52no—
33C2×D5×Dic3—102no—
34#6763—52no—
35C3×C22⋊AGL(1,𝔽5)—52no—
36Dic3⋊AGL(1,𝔽5)—54no—
37Dic3×AGL(1,𝔽5)—54no—
38C22×C5⋊D5C2×D311yescharacteristic
39C2×#1044—32no—
40C2×#1044C2×D312yes—
41C2×C4×C52—31no—
42C2×#1039—31no—
43C4×C5⋊D5—31no—
44C4×C5⋊D5—31no—
45C4×C5⋊D5—31no—
46C4×C5⋊D5—31no—
47#4130—152no—
48C2×C3×C52—21no—
49C2×C3×C52C22×C411yescharacteristic
50C3×C5⋊D5—21no—
51C3×C5⋊D5#4112yes—
52C22×C3×D5—52no—
53C22×C3×D5—102no—
54C2×ΓL(1,𝔽16)—54no—
55C2×C3×AGL(1,𝔽5)—54no—
56C2×Dic15—52no—
57C2×Dic15—102no—
58C2×C5×Dic3—22no—
59C2×C5×Dic3AGL(1,𝔽5)12yescomplemented
60D5×Dic3—52no—
61D5×Dic3—52no—
62D5×Dic3—52no—
63D5×Dic3—52no—
64D5×Dic3—102no—
65D5×Dic3—102no—
66D5×Dic3—102no—
67D5×Dic3—102no—
68C22×C52C4×D311yescharacteristic
69C2×C5⋊D5—22no—
70C2×C5⋊D5C4×D311yescharacteristic
71C2×C5⋊D5C22×D311yescharacteristic
72#1044—22no—
73#1044—62no—
74C4×C52—31no—
75C4×C52—31no—
76#1039—31no—
77#1039—31no—
78#877—251noHall
79C2×C4×D5—152no—
80C2×C4×D5—302no—
81C22⋊AGL(1,𝔽5)—52no—
82C22⋊AGL(1,𝔽5)—152no—
83C4⋊AGL(1,𝔽5)—154no—
84C4×AGL(1,𝔽5)—154no—
85C3×C52#11711yescharacteristic, complemented, Hall
86C22×C3×C5—22no—
87C22×C3×C5C2×AGL(1,𝔽5)12yes—
88C2×C3×D5—52no—
89C2×C3×D5—52no—
90C2×C3×D5—102no—
91C2×C3×D5—102no—
92C2×C3×D5—104no—
93C2×C3×D5—104no—
94C2×C3×D5—104no—
95ΓL(1,𝔽16)—104no—
96C3×AGL(1,𝔽5)—104no—
97Dic15—52no—
98Dic15—52no—
99Dic15—102no—
100Dic15—102no—
101C5×Dic3—22no—
102C5×Dic3—22no—
103C5×Dic3C2×AGL(1,𝔽5)12yes—
104C5×Dic3C2×AGL(1,𝔽5)12yes—
105C2×C52—21no—
106C2×C52C2×C4×D311yescharacteristic
107C5⋊D5—21no—
108C5⋊D5#23712yes—
109C22×Dic3—251no—
110C3×C22⋊C4—251no—
111#217—251no—
112Dic3⋊C4—252no—
113C4×Dic3—252no—
114C22×D5—52no—
115C22×D5—102no—
116C2×AGL(1,𝔽5)—54no—
117C2×AGL(1,𝔽5)—154no—
118C2×C4×C5—32no—
119C2×C4×C5—62no—
120C2×Dic5—152no—
121C2×Dic5—302no—
122C4×D5—152no—
123C4×D5—152no—
124C4×D5—152no—
125C4×D5—152no—
126C4×D5—302no—
127C4×D5—302no—
128C4×D5—302no—
129C4×D5—302no—
130#117—751noSylow
131C2×C3×C5—22no—
132C2×C3×C5—22no—
133C2×C3×C5—22no—
134C2×C3×C5—22no—
135C2×C3×C5C22×AGL(1,𝔽5)12yes—
136C3×D5—54no—
137C3×D5—104no—
138C3×D5—102no—
139C3×D5—102no—
140C3×D5—102no—
141C52#87711yescharacteristic, complemented, Sylow
142C23×C3—251no—
143C2×C3×C4—252no—
144C2×Dic3—251no—
145C2×Dic3—251no—
146C2×Dic3—251no—
147C2×Dic3—251no—
148C2×Dic3—251no—
149C2×Dic3—252no—
150C2×Dic3#104411yescharacteristic, complemented
151C22×C5—22no—
152C22×C5D3×AGL(1,𝔽5)12yes—
153C2×D5—52no—
154C2×D5—52no—
155C2×D5—102no—
156C2×D5—102no—
157C2×D5—104no—
158C2×D5—104no—
159C2×D5—104no—
160AGL(1,𝔽5)—104no—
161AGL(1,𝔽5)—304no—
162C4×C5—32no—
163C4×C5—32no—
164C4×C5—62no—
165C4×C5—62no—
166Dic5—152no—
167Dic5—152no—
168Dic5—302no—
169Dic5—302no—
170C22×C4—751no—
171C4⋊C4—752no—
172C22⋊C4—251no—
173C22⋊C4—751no—
174C42—752no—
175C3×C5—22no—
176C3×C5#413012yescomplemented
177C22×C3—251no—
178C22×C3—251no—
179C22×C3—502no—
180C22×C3C2×#104411yescharacteristic
181C3×C4—502no—
182Dic3—251no—
183Dic3—251no—
184Dic3—502no—
185Dic3C2×#104411yescharacteristic
186Dic3C2×#104411yescharacteristic
187C2×C5—22no—
188C2×C5—22no—
189C2×C5—22no—
190C2×C5—22no—
191C2×C5C2×D3×AGL(1,𝔽5)12yes—
192D5—54no—
193D5—104no—
194D5—102no—
195D5—102no—
196D5—102no—
197C23—251no—
198C2×C4—31no—
199C2×C4—252no—
200C2×C4—751no—
201C2×C4—751no—
202C2×C4—751no—
203C2×C4—751no—
204C2×C4—751no—
205C2×C4—752no—
206C2×C3—21no—
207C2×C3—252no—
208C2×C3—501no—
209C2×C3C22×#104411yescharacteristic
210C5—22no—
211C5#9192312yescomplemented
212C22—251no—
213C22—251no—
214C22—502no—
215C22D3×#104411yescharacteristic
216C4—31no—
217C4—31no—
218C4—502no—
219C4—751no—
220C4—751no—
221C4—1502no—
222C3#1171745111yescharacteristic, complemented, Sylow
223C2—21no—
224C2—252no—
225C2—501no—
226C2C2×D3×#104411yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×#1044
commutator, socleC2×C3×C52C22×C4
FittingC22×C3×C52C2×C4
prime radicalC22×C3×C5⋊D5C22
cosocleC2×C3×C5⋊D5C23
hypercenterC22D3×#1044
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. Dic3⋊#1044C2semidirect
5. Dic3×#1044C2semidirect
10. C2×C3⋊#1039C4semidirect
11. C2×C52×Dic3C4semidirect
12. Dic3×C5⋊D5C4semidirect
13. Dic3×C5⋊D5C4semidirect
59. C2×C5×Dic3AGL(1,𝔽5)semidirect
85. C3×C52#117semidirect
141. C52#877semidirect
150. C2×Dic3#1044semidirect
176. C3×C5#4130semidirect
211. C5#91923semidirect
222. C3#11717451semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×C5⋊D52
#119255402
C3×C22⋊#10442
Dic3⋊#10442
Dic3×#10442
#117174513
#919235