Finite Group Databasebeta

← C3×(C2×D3)⋊#1044 #72231320

Subgroups — C3×(C2×D3)⋊#1044 #72231320

Order 3600.

442 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 64 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×D3×C5⋊D5C211yesmaximal, characteristic
2C2×C3×C3⋊#1044C211yesmaximal, characteristic, complemented
3C2×C32×#1044C211yesmaximal, characteristic, complemented
4C3×C22⋊#1044—31nomaximal
5(C2×D3)⋊#1044C311yesmaximal, characteristic, direct product factor
6C2×C3×C3⋊(C5⋊D5)C411yescharacteristic, complemented
7C2×C3×C52×D3C411yescharacteristic, complemented
8C2×C32×C5⋊D5C2211yescharacteristic
9C3×D3×C5⋊D5—21no—
10C3×D3×C5⋊D5—21no—
11C3×C3⋊#1044—21no—
12C32×#1044—21no—
13C3×(C2×D3)⋊AGL(1,𝔽5)—52nomaximal
14C22×C3×C5⋊D5—31no—
15C2×D3×C5⋊D5C2×C311yescharacteristic
16C2×C3⋊#1044C2×C311yescharacteristic, complemented
17C2×C3×#1044—21no—
18C2×C3×#1044—31no—
19C2×C3×#1044D311yescharacteristic, complemented
20C2×C3×#1044C2×C311yescharacteristic, complemented
21C2×C32×C52C2×C411yescharacteristic
22C3×C3⋊(C5⋊D5)—21no—
23C3×C52×D3—21no—
24C32×C5⋊D5D411yescharacteristic
25C32×C5⋊D5D411yescharacteristic
26C22⋊#1044—31noHall
27C2×C3×D3×D5—52no—
28C2×C3×D3×D5—102no—
29C2×C3×ΓL(1,𝔽16)—52no—
30C2×C32×AGL(1,𝔽5)—52no—
31C22×C3×C52—31no—
32C2×C3⋊(C5⋊D5)C3×C411yescharacteristic, complemented
33C2×C52×D3C3×C411yescharacteristic, complemented
34C2×C3×C5⋊D5—21no—
35C2×C3×C5⋊D5—31no—
36C2×C3×C5⋊D5—61no—
37C2×C3×C5⋊D5—61no—
38C2×C3×C5⋊D5C2×D311yescharacteristic
39C2×C3×C5⋊D5C22×C311yescharacteristic
40D3×C5⋊D5—21no—
41D3×C5⋊D5—21no—
42C3⋊#1044—21no—
43C3×#1044—21no—
44C3×#1044—21no—
45C3×#1044—21no—
46C3×#1044—21no—
47C3×#1044—61no—
48C3×C22⋊AGL(1,𝔽5)—152no—
49(C2×D3)⋊AGL(1,𝔽5)—52no—
50C32×C52C22⋊C411yescharacteristic, complemented, Hall
51C22×C5⋊D5—31no—
52C2×#1044—31no—
53C2×#1044C3×D311yescharacteristic, complemented
54C2×C3×D15—52no—
55C2×C3×D15—102no—
56C2×C3×C5×D3—22no—
57C2×C3×C5×D3AGL(1,𝔽5)12yescomplemented
58C2×C32×D5—52no—
59C2×C32×D5—102no—
60C3×D3×D5—102no—
61C3×D3×D5—102no—
62C3×D3×D5—102no—
63C3×D3×D5—102no—
64C3×D3×D5—102no—
65C3×D3×D5—102no—
66C3×ΓL(1,𝔽16)—102no—
67C32×AGL(1,𝔽5)—102no—
68C2×C3×C52—21no—
69C2×C3×C52—61no—
70C2×C3×C52C4×D311yescharacteristic
71C2×C3×C52C2×C3×C411yescharacteristic
72C3⋊(C5⋊D5)—21no—
73C52×D3—21no—
74C3×C5⋊D5—21no—
75C3×C5⋊D5—21no—
76C3×C5⋊D5—61no—
77C3×C5⋊D5D1211yescharacteristic
78C3×C5⋊D5#6711yescharacteristic
79C3×C5⋊D5C3×D411yescharacteristic
80C3×C5⋊D5C3×D411yescharacteristic
81C3×(C2×D3)⋊C4—251noHall
82C22×C3×D5—152no—
83C22×C3×D5—302no—
84C2×D3×D5—52no—
85C2×D3×D5—102no—
86C2×ΓL(1,𝔽16)—52no—
87C2×C3×AGL(1,𝔽5)—52no—
88C2×C3×AGL(1,𝔽5)—52no—
89C2×C3×AGL(1,𝔽5)—102no—
90C2×C3×AGL(1,𝔽5)—152no—
91C22×C52—31no—
92C2×C5⋊D5—31no—
93C2×C5⋊D5—61no—
94C2×C5⋊D5—61no—
95C2×C5⋊D5C2×C3×D311yescharacteristic
96#1044—21no—
97#1044—61no—
98C2×C32×C5—22no—
99C2×C32×C5C2×AGL(1,𝔽5)12yes—
100C3×D15—102no—
101C3×D15—102no—
102C3×D15—102no—
103C3×C5×D3—22no—
104C3×C5×D3—22no—
105C3×C5×D3—22no—
106C32×D5—52no—
107C32×D5—52no—
108C32×D5—102no—
109C32×D5—102no—
110C22⋊AGL(1,𝔽5)—152no—
111C3×C52—21no—
112C3×C52(C2×D3)⋊C411yescharacteristic, complemented
113C3×C52C3×C22⋊C411yescharacteristic, complemented
114C22×C3×D3—251no—
115C2×C32×C4—251no—
116C2×C3×Dic3—251no—
117C22×C3×C5—32no—
118C22×C3×C5—62no—
119C2×D15—52no—
120C2×D15—102no—
121C2×C5×D3—22no—
122C2×C5×D3C3×AGL(1,𝔽5)12yescomplemented
123C2×C3×D5—52no—
124C2×C3×D5—52no—
125C2×C3×D5—102no—
126C2×C3×D5—102no—
127C2×C3×D5—102no—
128C2×C3×D5—152no—
129C2×C3×D5—202no—
130C2×C3×D5—302no—
131C2×C3×D5—302no—
132C2×C3×D5—302no—
133C2×C3×D5—302no—
134C2×C3×D5—302no—
135C2×C3×D5—302no—
136C2×C3×D5—302no—
137D3×D5—102no—
138D3×D5—102no—
139D3×D5—102no—
140D3×D5—102no—
141D3×D5—102no—
142D3×D5—102no—
143ΓL(1,𝔽16)—102no—
144C3×AGL(1,𝔽5)—102no—
145C3×AGL(1,𝔽5)—102no—
146C3×AGL(1,𝔽5)—102no—
147C3×AGL(1,𝔽5)—102no—
148C3×AGL(1,𝔽5)—302no—
149C2×C52—61no—
150C2×C52C3×C4×D311yescharacteristic
151C5⋊D5—61no—
152C5⋊D5C3×D1211yescharacteristic
153C5⋊D5C3×#6711yescharacteristic
154C3×C22⋊C4—751no—
155(C2×D3)⋊C4—251no—
156C32×C5—22no—
157C32×C5C22⋊AGL(1,𝔽5)12yescomplemented
158C22×D5—152no—
159C22×D5—302no—
160C2×AGL(1,𝔽5)—52no—
161C2×AGL(1,𝔽5)—152no—
162C22×C32—251no—
163C2×C3×D3—251no—
164C2×C3×D3—501no—
165C2×C3×D3—501no—
166C2×C3×D3#104411yescharacteristic, complemented
167C32×C4—501no—
168C3×Dic3—501no—
169C2×C3×C5—22no—
170C2×C3×C5—22no—
171C2×C3×C5—22no—
172C2×C3×C5—42no—
173C2×C3×C5—62no—
174C2×C3×C5—62no—
175C2×C3×C5—62no—
176C2×C3×C5D3×AGL(1,𝔽5)12yes—
177C2×C3×C5C2×C3×AGL(1,𝔽5)12yes—
178D15—102no—
179D15—102no—
180D15—102no—
181C3×D5—52no—
182C3×D5—52no—
183C3×D5—52no—
184C3×D5—52no—
185C3×D5—102no—
186C3×D5—102no—
187C3×D5—102no—
188C3×D5—102no—
189C3×D5—102no—
190C3×D5—102no—
191C3×D5—202no—
192C3×D5—202no—
193C3×D5—302no—
194C3×D5—302no—
195C3×D5—302no—
196C5×D3—22no—
197C5×D3—22no—
198C5×D3—22no—
199C52C3×(C2×D3)⋊C411yescharacteristic, complemented, Sylow
200C23×C3—751no—
201C22×D3—251no—
202C2×C3×C4—251no—
203C2×C3×C4—251no—
204C2×C3×C4—501no—
205C2×C3×C4—751no—
206C2×Dic3—251no—
207C22×C5—32no—
208C22×C5—62no—
209C2×D5—52no—
210C2×D5—102no—
211C2×D5—152no—
212C2×D5—302no—
213C2×D5—302no—
214C2×D5—302no—
215C2×D5—302no—
216C2×D5—302no—
217C2×D5—302no—
218C2×D5—302no—
219AGL(1,𝔽5)—102no—
220AGL(1,𝔽5)—302no—
221C2×C32—251no—
222C2×C32—251no—
223C2×C32C2×#104411yescharacteristic
224C3×D3—21no—
225C3×D3—501no—
226C22⋊C4—751noSylow
227C3×C5—22no—
228C3×C5—22no—
229C3×C5—22no—
230C3×C5—42no—
231C3×C5(C2×D3)⋊AGL(1,𝔽5)12yescomplemented
232C3×C5C3×C22⋊AGL(1,𝔽5)12yescomplemented
233C22×C3—31no—
234C22×C3—251no—
235C22×C3—251no—
236C22×C3—501no—
237C22×C3—751no—
238C22×C3—1501no—
239C22×C3—1501no—
240C2×D3—251no—
241C2×D3—501no—
242C2×D3—501no—
243C2×D3C3×#104411yescharacteristic, complemented
244C3×C4—501no—
245C3×C4—501no—
246C3×C4—501no—
247C3×C4—501no—
248C3×C4—1501no—
249Dic3—501no—
250C2×C5—22no—
251C2×C5—62no—
252C2×C5—62no—
253C2×C5—62no—
254C2×C5C3×D3×AGL(1,𝔽5)12yes—
255D5—52no—
256D5—52no—
257D5—102no—
258D5—102no—
259D5—302no—
260D5—302no—
261D5—302no—
262C32C22⋊#104411yescharacteristic, complemented, Sylow
263C23—751no—
264C2×C4—251no—
265C2×C4—751no—
266C2×C3—21no—
267C2×C3—61no—
268C2×C3—251no—
269C2×C3—251no—
270C2×C3—251no—
271C2×C3—251no—
272C2×C3—501no—
273C2×C3—501no—
274C2×C3—1501no—
275C2×C3D3×#104411yescharacteristic
276C2×C3C2×C3×#104411yescharacteristic
277D3—21no—
278D3—501no—
279C5—22no—
280C5C3×(C2×D3)⋊AGL(1,𝔽5)12yescomplemented
281C22—31no—
282C22—251no—
283C22—751no—
284C22—1501no—
285C22—1501no—
286C4—501no—
287C4—1501no—
288C3—21no—
289C3(C2×D3)⋊#104411yescharacteristic, direct product factor
290C3C3×C22⋊#104411yescharacteristic, complemented
291C2—61no—
292C2—251no—
293C2—251no—
294C2—1501no—
295C2C3×D3×#104411yescharacteristic
subgroupG/H
center, hypercenterC2×C3D3×#1044
commutatorC2×C3×C52C2×C3×C4
FrattiniC2C3×D3×#1044
Fitting, socleC2×C32×C52C2×C4
prime radicalC2×C3×D3×C5⋊D5C2
cosocleC2×C3×C5⋊D5C22×C3
hypocenterC3×C52C3×C22⋊C4
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×C3⋊#1044C2semidirect
3. C2×C32×#1044C2semidirect
5. (C2×D3)⋊#1044C3direct (N × H)
6. C2×C3×C3⋊(C5⋊D5)C4semidirect
7. C2×C3×C52×D3C4semidirect
16. C2×C3⋊#1044C2×C3semidirect
19. C2×C3×#1044D3semidirect
20. C2×C3×#1044C2×C3semidirect
32. C2×C3⋊(C5⋊D5)C3×C4semidirect
33. C2×C52×D3C3×C4semidirect
50. C32×C52C22⋊C4semidirect
53. C2×#1044C3×D3semidirect
57. C2×C3×C5×D3AGL(1,𝔽5)semidirect
112. C3×C52(C2×D3)⋊C4semidirect
113. C3×C52C3×C22⋊C4semidirect
122. C2×C5×D3C3×AGL(1,𝔽5)semidirect
157. C32×C5C22⋊AGL(1,𝔽5)semidirect
166. C2×C3×D3#1044semidirect
199. C52C3×(C2×D3)⋊C4semidirect
231. C3×C5(C2×D3)⋊AGL(1,𝔽5)semidirect
232. C3×C5C3×C22⋊AGL(1,𝔽5)semidirect
243. C2×D3C3×#1044semidirect
262. C32C22⋊#1044semidirect
280. C5C3×(C2×D3)⋊AGL(1,𝔽5)semidirect
289. C3(C2×D3)⋊#1044direct (N × H)
290. C3C3×C22⋊#1044semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×C5⋊D52
C2×C3×C3⋊#10442
C2×C32×#10442
C3×C22⋊#10443
(C2×D3)⋊#10443
C3×(C2×D3)⋊AGL(1,𝔽5)5