Finite Group Databasebeta

← C3×#72214019 #72231351

Subgroups — C3×#72214019 #72231351

Order 7200.

572 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 50 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
1C3×A4⋊#1044C211yesmaximal, characteristic, complemented
2C3×(C5⋊S4)⋊D5C211yesmaximal, characteristic
3C3×A4×#1043C211yesmaximal, characteristic, complemented
4C3×#11717375—31nomaximal
5#72214019C311yesmaximal, characteristic, direct product factor
6C3×A4×C5⋊D5C2211yescharacteristic
7C3×C5×C5⋊S4C412yescomplemented
8C3×#10597182—41nomaximal
9C3×AGL(1,𝔽5)×S4—52nomaximal
10C22×C3×#1043D311yescharacteristic, complemented
11C2×C3×D5⋊AGL(1,𝔽5)—31no—
12(C5⋊S4)⋊D5C2×C311yescharacteristic
13A4×#1043—21no—
14A4×#1043C2×C311yescharacteristic, complemented
15A4⋊#1044C2×C311yescharacteristic, complemented
16C3×#87886—31no—
17C3×C22⋊#1044—31no—
18C3×#87832—31no—
19C3×#87831—31no—
20C3×#87830—31no—
21C3×C52×A4C2×C411yescharacteristic, complemented
22C3×D15⋊D5—41no—
23C3×C3⋊#1044—41no—
24C32×#1043—41no—
25#11717375—31noHall
26C3×D5×S4—52no—
27C3×A4×AGL(1,𝔽5)—52no—
28C3×A4×AGL(1,𝔽5)—102no—
29C3×A4⋊AGL(1,𝔽5)—52no—
30C22×C3×C5⋊D5C2×D311yescharacteristic
31C2×C3×D52—31no—
32C2×C3×#1044—31no—
33C2×C3×#1044—31no—
34C2×C3×#1043—31no—
35C2×C3×#1043—31no—
36A4×C5⋊D5—21no—
37A4×C5⋊D5C22×C311yescharacteristic
38C5×C5⋊S4C3×C412yescomplemented
39#10597182—41no—
40C3×D5⋊AGL(1,𝔽5)—61no—
41C3×D5⋊AGL(1,𝔽5)—61no—
42C3×C5×C22⋊D5—32no—
43C3×#6038—32no—
44C3×#6036—31no—
45AGL(1,𝔽5)×S4—52no—
46C3×D4×AGL(1,𝔽5)—152no—
47C3×C5×D15—42no—
48C32×C5⋊D5—41no—
49C22×#1043C3×D311yescharacteristic, complemented
50C2×D5⋊AGL(1,𝔽5)—31no—
51#87886—31no—
52C22⋊#1044—31no—
53#87832—31no—
54#87831—31no—
55#87830—31no—
56C3×D5×A4—52no—
57C3×D5×A4—102no—
58C3×C5⋊S4AGL(1,𝔽5)12yescomplemented
59C3×C5×S4—52no—
60C3×D3×AGL(1,𝔽5)—202no—
61C22×C3×C52C4×D311yescharacteristic, complemented
62C2×C3×C5⋊D5—31no—
63C2×C3×C5⋊D5—31no—
64C2×C3×C5×D5—32no—
65C52×A4—21no—
66C52×A4C2×C3×C411yescharacteristic, complemented
67D15⋊D5—41no—
68C3×D52—61no—
69C3×D52—61no—
70C3⋊#1044—41no—
71C3×#1044—61no—
72C3×#1044—61no—
73C3×#1043—31no—
74C3×#1043—41no—
75C3×#1043—81no—
76C3×#1043S411yescharacteristic, complemented
77C3×C5×Dic5—32no—
78C3×C4×S4—251noHall
79C22×C3×AGL(1,𝔽5)—52no—
80C22×C3×AGL(1,𝔽5)—102no—
81C22×C3×AGL(1,𝔽5)—152no—
82D5×S4—52no—
83A4×AGL(1,𝔽5)—52no—
84A4×AGL(1,𝔽5)—102no—
85A4×AGL(1,𝔽5)—102no—
86A4×AGL(1,𝔽5)—104no—
87A4⋊AGL(1,𝔽5)—52no—
88C3×D4×D5—152no—
89C3×C22⋊AGL(1,𝔽5)—152no—
90C3×C22⋊AGL(1,𝔽5)—152no—
91C3×C4⋊AGL(1,𝔽5)—152no—
92C3×C4×AGL(1,𝔽5)—152no—
93C32×C52—41noHall
94C22×C5⋊D5C2×C3×D311yescharacteristic
95C2×D52—31no—
96C2×#1044—31no—
97C2×#1044—31no—
98C2×#1043—31no—
99C2×#1043—31no—
100D5⋊AGL(1,𝔽5)—61no—
101D5⋊AGL(1,𝔽5)—61no—
102C5×C22⋊D5—32no—
103#6038—32no—
104#6036—31no—
105C3×C5×A4—22no—
106C3×C5×A4C2×AGL(1,𝔽5)12yescomplemented
107C3×D3×D5—202no—
108C3×ΓL(1,𝔽16)—202no—
109C32×AGL(1,𝔽5)—202no—
110C32×AGL(1,𝔽5)—402no—
111D4×AGL(1,𝔽5)—152no—
112C2×C3×C52—31no—
113C5×D15—42no—
114C3×C5⋊D5—31no—
115C3×C5⋊D5—41no—
116C3×C5⋊D5—81no—
117C3×C5⋊D5C2×S411yescharacteristic
118C3×C5×D5—62no—
119C2×C3×S4—251no—
120C3×C4×A4—251no—
121C3×SL(2,ℤ/4)—251no—
122C22×C3×D5—52no—
123C22×C3×D5—102no—
124C22×C3×D5—152no—
125C2×C3×AGL(1,𝔽5)—152no—
126C2×C3×AGL(1,𝔽5)—152no—
127C2×C3×AGL(1,𝔽5)—152no—
128C2×C3×AGL(1,𝔽5)—152no—
129C2×C3×AGL(1,𝔽5)—302no—
130C2×C3×AGL(1,𝔽5)—302no—
131C2×C3×AGL(1,𝔽5)—302no—
132C2×C3×AGL(1,𝔽5)—302no—
133D5×A4—52no—
134D5×A4—102no—
135D5×A4—102no—
136D5×A4—104no—
137C5⋊S4C3×AGL(1,𝔽5)12yescomplemented
138C5×S4—52no—
139D3×AGL(1,𝔽5)—202no—
140C3×C5×D4—152no—
141C3×C22⋊D5—32no—
142C3×C22⋊D5—152no—
143C3×D20—152no—
144C3×C4×D5—152no—
145C22×C52C3×C4×D311yescharacteristic, complemented
146C2×C5⋊D5—31no—
147C2×C5⋊D5—31no—
148C2×C5×D5—32no—
149D52—61no—
150D52—61no—
151#1044—61no—
152#1044—61no—
153#1043—31no—
154#1043C3×S411yescharacteristic, complemented
155C5×Dic5—32no—
156C4×S4—251no—
157C3×C4×D4—751no—
158C3×D15—42no—
159C3×C5×D3—202no—
160C32×D5—202no—
161C32×D5—402no—
162C22×AGL(1,𝔽5)—52no—
163C22×AGL(1,𝔽5)—102no—
164C22×AGL(1,𝔽5)—152no—
165D4×D5—152no—
166C22⋊AGL(1,𝔽5)—152no—
167C22⋊AGL(1,𝔽5)—152no—
168C4⋊AGL(1,𝔽5)—152no—
169C4×AGL(1,𝔽5)—152no—
170C3×C52—41no—
171C3×C52—81no—
172C3×C52C4×S411yescharacteristic, complemented
173C2×C3×A4—251no—
174C3×S4—52no—
175C3×C4×D3—1001no—
176C22×C3×C5—22no—
177C22×C3×C5—152no—
178C22×C3×C5D3×AGL(1,𝔽5)12yescomplemented
179C2×C3×D5—32no—
180C2×C3×D5—152no—
181C2×C3×D5—152no—
182C2×C3×D5—302no—
183C2×C3×D5—302no—
184C2×C3×D5—302no—
185C2×C3×D5—302no—
186C5×A4—24no—
187C5×A4—22no—
188C5×A4—22no—
189C5×A4C2×C3×AGL(1,𝔽5)12yescomplemented
190D3×D5—202no—
191ΓL(1,𝔽16)—202no—
192C3×AGL(1,𝔽5)—52no—
193C3×AGL(1,𝔽5)—102no—
194C3×AGL(1,𝔽5)—152no—
195C3×AGL(1,𝔽5)—202no—
196C3×AGL(1,𝔽5)—302no—
197C3×AGL(1,𝔽5)—302no—
198C3×AGL(1,𝔽5)—302no—
199C3×AGL(1,𝔽5)—402no—
200C3×AGL(1,𝔽5)—402no—
201C3×AGL(1,𝔽5)—404no—
202C3×C4×C5—152no—
203C3×Dic5—32no—
204C2×C52—31no—
205C5⋊D5—31no—
206C5⋊D5C2×C3×S411yescharacteristic
207C5×D5—62no—
208C2×S4—251no—
209C2×C3×D4—751no—
210C22×C3×C4—251no—
211C22×C3×C4—751no—
212C4×A4—251no—
213C4×A4—501no—
214SL(2,ℤ/4)—251no—
215C3×C4⋊C4—751no—
216C3×C22⋊C4—751no—
217C3×C22⋊C4—751no—
218C3×C42—751no—
219C32×C5—42no—
220C32×C5—82no—
221C22×D5—52no—
222C22×D5—102no—
223C22×D5—152no—
224C2×AGL(1,𝔽5)—152no—
225C2×AGL(1,𝔽5)—152no—
226C2×AGL(1,𝔽5)—152no—
227C2×AGL(1,𝔽5)—152no—
228C2×AGL(1,𝔽5)—302no—
229C2×AGL(1,𝔽5)—302no—
230C2×AGL(1,𝔽5)—302no—
231C2×AGL(1,𝔽5)—302no—
232C5×D4—152no—
233C22⋊D5—32no—
234C22⋊D5—152no—
235D20—152no—
236C4×D5—152no—
237C2×C3×D3—1001no—
238C3×A4D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
239C32×C4—1001no—
240C3×Dic3—1001no—
241C4×D4—751noSylow
242C2×C3×C5—32no—
243C2×C3×C5—62no—
244C2×C3×C5—302no—
245D15—42no—
246C3×D5—52no—
247C3×D5—62no—
248C3×D5—102no—
249C3×D5—152no—
250C3×D5—202no—
251C3×D5—302no—
252C3×D5—402no—
253C3×D5—402no—
254C3×D5—404no—
255C5×D3—202no—
256C52C3×C4×S411yescharacteristic, complemented, Sylow
257C23×C3—251no—
258C23×C3—751no—
259C2×A4—251no—
260C2×A4—501no—
261S4—52no—
262C3×D4—152no—
263C3×D4—752no—
264C2×C3×C4—751no—
265C2×C3×C4—751no—
266C2×C3×C4—751no—
267C2×C3×C4—751no—
268C2×C3×C4—751no—
269C2×C3×C4—1501no—
270C2×C3×C4—1501no—
271C4×D3—1001no—
272C22×C5—22no—
273C22×C5—152no—
274C22×C5C3×D3×AGL(1,𝔽5)12yescomplemented
275C2×D5—32no—
276C2×D5—152no—
277C2×D5—152no—
278C2×D5—302no—
279C2×D5—302no—
280C2×D5—302no—
281C2×D5—302no—
282AGL(1,𝔽5)—52no—
283AGL(1,𝔽5)—102no—
284AGL(1,𝔽5)—152no—
285AGL(1,𝔽5)—302no—
286AGL(1,𝔽5)—302no—
287AGL(1,𝔽5)—302no—
288C4×C5—152no—
289Dic5—32no—
290C2×C32—1001no—
291C3×D3—202no—
292C2×D4—751no—
293C22×C4—251no—
294C22×C4—751no—
295C4⋊C4—751no—
296C22⋊C4—751no—
297C22⋊C4—751no—
298C42—751no—
299C3×C5—22no—
300C3×C5—42no—
301C3×C5—82no—
302C3×C5—84no—
303C3×C5—82no—
304C3×C5AGL(1,𝔽5)×S412yescomplemented
305C22×C3—152no—
306C22×C3—751no—
307C22×C3—751no—
308C22×C3—1501no—
309C22×C3—1501no—
310C22×C3#1059718211yescharacteristic, complemented
311C2×D3—1001no—
312A4—21no—
313A4C3×D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
314C3×C4—152no—
315C3×C4—251no—
316C3×C4—751no—
317C3×C4—1001no—
318C3×C4—1501no—
319C3×C4—1501no—
320C3×C4—2001no—
321Dic3—1001no—
322C2×C5—32no—
323C2×C5—62no—
324C2×C5—302no—
325D5—52no—
326D5—62no—
327D5—102no—
328D5—152no—
329D5—302no—
330C32—41noSylow
331C23—251no—
332C23—751no—
333D4—152no—
334D4—752no—
335C2×C4—751no—
336C2×C4—751no—
337C2×C4—751no—
338C2×C4—751no—
339C2×C4—751no—
340C2×C4—1501no—
341C2×C4—1501no—
342C2×C3—31no—
343C2×C3—251no—
344C2×C3—302no—
345C2×C3—751no—
346C2×C3—1001no—
347C2×C3—2001no—
348D3—202no—
349C5—22no—
350C5C3×AGL(1,𝔽5)×S412yescomplemented
351C22—152no—
352C22—751no—
353C22—751no—
354C22—1501no—
355C22—1501no—
356C22C3×#1059718211yescharacteristic, complemented
357C4—152no—
358C4—251no—
359C4—751no—
360C4—1501no—
361C4—1501no—
362C3—41no—
363C3—81no—
364C3#7221401911yescharacteristic, direct product factor
365C2—31no—
366C2—251no—
367C2—302no—
368C2—751no—
subgroupG/H
center, hypercenterC3#72214019
commutator, hypocenterC52×A4C2×C3×C4
FrattiniC1G
Fitting, socleC22×C3×C52C4×D3
prime radicalC3×(C5⋊S4)⋊D5C2
cosocleA4×C5⋊D5C22×C3
LCSG ▷ C52×A4
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×A4⋊#1044C2semidirect
3. C3×A4×#1043C2semidirect
5. #72214019C3direct (N × H)
7. C3×C5×C5⋊S4C4semidirect
10. C22×C3×#1043D3semidirect
14. A4×#1043C2×C3semidirect
15. A4⋊#1044C2×C3semidirect
21. C3×C52×A4C2×C4semidirect
38. C5×C5⋊S4C3×C4semidirect
49. C22×#1043C3×D3semidirect
58. C3×C5⋊S4AGL(1,𝔽5)semidirect
61. C22×C3×C52C4×D3semidirect
66. C52×A4C2×C3×C4semidirect
76. C3×#1043S4semidirect
106. C3×C5×A4C2×AGL(1,𝔽5)semidirect
137. C5⋊S4C3×AGL(1,𝔽5)semidirect
145. C22×C52C3×C4×D3semidirect
154. #1043C3×S4semidirect
172. C3×C52C4×S4semidirect
178. C22×C3×C5D3×AGL(1,𝔽5)semidirect
189. C5×A4C2×C3×AGL(1,𝔽5)semidirect
238. C3×A4D5⋊AGL(1,𝔽5)semidirect
256. C52C3×C4×S4semidirect
274. C22×C5C3×D3×AGL(1,𝔽5)semidirect
304. C3×C5AGL(1,𝔽5)×S4semidirect
310. C22×C3#10597182semidirect
313. A4C3×D5⋊AGL(1,𝔽5)semidirect
350. C5C3×AGL(1,𝔽5)×S4semidirect
356. C22C3×#10597182semidirect
364. C3#72214019direct (N × H)

Maximal subgroups

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

subgroupindex
C3×A4⋊#10442
C3×(C5⋊S4)⋊D52
C3×A4×#10432
C3×#117173753
#722140193
C3×#105971824
C3×AGL(1,𝔽5)×S45