Finite Group Databasebeta

← C3×#10598507 #14277193

Subgroups — C3×#10598507 #14277193

Order 1920.

492 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 96 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×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
2C3×#65756C211yesmaximal, characteristic
3C3×#64955C211yesmaximal, characteristic
4#10598507C311yesmaximal, characteristic, direct product factor, Hall
5C22×C3×C4×D5C2211yescharacteristic
6C2×C3×C22⋊AGL(1,𝔽5)—21no—
7C2×C3×C4×AGL(1,𝔽5)—21no—
8C2×C3×#700C412yescomplemented
9C2×C3×#685—21no—
10C2×C3×C4×Dic5—21no—
11C3×C22⋊#170—21no—
12C3×(C2×AGL(1,𝔽5))⋊C4C412yescomplemented
13C3×#4003—21no—
14C3×C5×#95C411yescharacteristic, complemented
15C3×#3963C411yescharacteristic, complemented
16C3×#1297—51nomaximal, Hall
17C4×C22⋊AGL(1,𝔽5)C2×C311yescharacteristic
18#65756C2×C311yescharacteristic
19#64955C2×C311yescharacteristic
20C23×C3×D5D411yescharacteristic
21C22×C3×AGL(1,𝔽5)—22no—
22C22×C3×C4×C5—21no—
23C22×C3×C4×C5C2×C411yescharacteristic
24C22×C3×Dic5—21no—
25C22×C3×Dic5D411yescharacteristic
26C2×C3×C4×D5—42no—
27C2×C3×C4×D5C2×C411yescharacteristic
28C2×C3×C4×D5C2×C411yescharacteristic
29C2×C3×C4×D5D411yescharacteristic
30C2×C3×C4×D5Q811yescharacteristic
31C3×C22⋊AGL(1,𝔽5)—22no—
32C3×C22⋊AGL(1,𝔽5)—22no—
33C2×C3×#170—22no—
34C3×C4×AGL(1,𝔽5)—42no—
35C3×#700—42no—
36C3×#685—24no—
37C3×C4×Dic5—42no—
38C3×#407—51no—
39C3×C4×C22⋊C4—51no—
40C3×C4×C22⋊C4—51no—
41C22×C4×D5C22×C311yescharacteristic
42C2×C22⋊AGL(1,𝔽5)—21no—
43C2×C4×AGL(1,𝔽5)—21no—
44C2×#700C3×C412yescomplemented
45C2×#685—21no—
46C2×C4×Dic5—21no—
47C22⋊#170—21no—
48(C2×AGL(1,𝔽5))⋊C4C3×C412yescomplemented
49#4003—21no—
50C5×#95C3×C411yescharacteristic, complemented
51#3963C3×C411yescharacteristic, complemented
52#1297—51noSylow
53C23×C3×C5C22⋊C411yescharacteristic
54C22×C3×D5—21no—
55C22×C3×D5—21no—
56C22×C3×D5—22no—
57C22×C3×D5—41no—
58C22×C3×D5C22⋊C411yescharacteristic
59C22×C3×D5C4⋊C411yescharacteristic
60C2×C3×AGL(1,𝔽5)—24no—
61C2×C3×AGL(1,𝔽5)—42no—
62C2×C3×AGL(1,𝔽5)—42no—
63C2×C3×C4×C5—22no—
64C2×C3×C4×C5—41no—
65C2×C3×C4×C5—41no—
66C2×C3×C4×C5—41no—
67C2×C3×C4×C5C4211yescharacteristic
68C2×C3×C4×C5C4⋊C411yescharacteristic
69C2×C3×Dic5—22no—
70C2×C3×Dic5—22no—
71C2×C3×Dic5—41no—
72C2×C3×Dic5—41no—
73C2×C3×Dic5C22⋊C411yescharacteristic
74C2×C3×Dic5C4⋊C411yescharacteristic
75C3×C4×D5—22no—
76C3×C4×D5—22no—
77C3×C4×D5—41no—
78C3×C4×D5—41no—
79C3×#170—42no—
80C23×C3×C4—51no—
81C2×C3×M16—52no—
82C2×C3×C22⋊C4—101no—
83C2×C3×C22⋊C4—101no—
84C2×C3×C42—101no—
85C2×C3×C42—101no—
86C3×C22⋊C8—101no—
87C3×C22⋊C8—101no—
88C3×#95—52no—
89C3×#95—51no—
90C3×#95AGL(1,𝔽5)11yescharacteristic, complemented
91C23×D5C3×D411yescharacteristic
92C22×AGL(1,𝔽5)—22no—
93C22×C4×C5—21no—
94C22×C4×C5C2×C3×C411yescharacteristic
95C22×Dic5—21no—
96C22×Dic5C3×D411yescharacteristic
97C2×C4×D5—42no—
98C2×C4×D5C2×C3×C411yescharacteristic
99C2×C4×D5C2×C3×C411yescharacteristic
100C2×C4×D5C3×D411yescharacteristic
101C2×C4×D5C3×Q811yescharacteristic
102C22⋊AGL(1,𝔽5)—22no—
103C22⋊AGL(1,𝔽5)—22no—
104C2×#170—22no—
105C4×AGL(1,𝔽5)—42no—
106#700—42no—
107#685—24no—
108C4×Dic5—42no—
109#407—51no—
110C4×C22⋊C4—51no—
111C4×C22⋊C4—51no—
112C22×C3×C5—21no—
113C22×C3×C5—21no—
114C22×C3×C5#9511yescharacteristic
115C22×C3×C5#10411yescharacteristic
116C22×C3×C5#10411yescharacteristic
117C2×C3×D5—21no—
118C2×C3×D5—21no—
119C2×C3×D5—21no—
120C2×C3×D5—21no—
121C2×C3×D5—21no—
122C2×C3×D5—42no—
123C2×C3×D5—42no—
124C2×C3×D5#10412yes—
125C3×AGL(1,𝔽5)—44no—
126C3×C4×C5—21no—
127C3×C4×C5—21no—
128C3×C4×C5—42no—
129C3×Dic5—21no—
130C3×Dic5—42no—
131C3×Dic5#9912yes—
132C24×C3—51no—
133C22×C3×C4—21no—
134C22×C3×C4—51no—
135C22×C3×C4—51no—
136C22×C3×C4—51no—
137C22×C3×C4—51no—
138C22×C3×C4—51no—
139C22×C3×C4—102no—
140C22×C3×C4—101no—
141C22×C3×C4—202no—
142C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
143C3×M16—202no—
144C2×C3×C8—102no—
145C3×C22⋊C4—104no—
146C3×C22⋊C4—102no—
147C3×C22⋊C4—102no—
148C3×C42—202no—
149C3×C42—202no—
150C23×C5C3×C22⋊C411yescharacteristic
151C22×D5—21no—
152C22×D5—21no—
153C22×D5—22no—
154C22×D5—41no—
155C22×D5C3×C22⋊C411yescharacteristic
156C22×D5C3×C4⋊C411yescharacteristic
157C2×AGL(1,𝔽5)—24no—
158C2×AGL(1,𝔽5)—42no—
159C2×AGL(1,𝔽5)—42no—
160C2×C4×C5—22no—
161C2×C4×C5—41no—
162C2×C4×C5—41no—
163C2×C4×C5—41no—
164C2×C4×C5C3×C4211yescharacteristic
165C2×C4×C5C3×C4⋊C411yescharacteristic
166C2×Dic5—22no—
167C2×Dic5—22no—
168C2×Dic5—41no—
169C2×Dic5—41no—
170C2×Dic5C3×C22⋊C411yescharacteristic
171C2×Dic5C3×C4⋊C411yescharacteristic
172C4×D5—22no—
173C4×D5—22no—
174C4×D5—41no—
175C4×D5—41no—
176#170—42no—
177C23×C4—51no—
178C2×M16—52no—
179C2×C22⋊C4—101no—
180C2×C22⋊C4—101no—
181C2×C42—101no—
182C2×C42—101no—
183C22⋊C8—101no—
184C22⋊C8—101no—
185#95—52no—
186#95—51no—
187#95C3×AGL(1,𝔽5)11yescharacteristic, complemented
188C2×C3×C5—21no—
189C2×C3×C5—21no—
190C2×C3×C5#33911yescharacteristic
191C2×C3×C5#33911yescharacteristic
192C2×C3×C5#34211yescharacteristic
193C3×D5—22no—
194C3×D5—41no—
195C23×C3—51no—
196C23×C3—51no—
197C23×C3—102no—
198C23×C3—101no—
199C23×C3—101no—
200C23×C3—201no—
201C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—51no—
207C2×C3×C4—51no—
208C2×C3×C4—102no—
209C2×C3×C4—102no—
210C2×C3×C4—102no—
211C2×C3×C4—102no—
212C2×C3×C4—104no—
213C2×C3×C4—201no—
214C2×C3×C4—201no—
215C2×C3×C4—201no—
216C2×C3×C4—202no—
217C2×C3×C4—202no—
218C2×C3×C4—201no—
219C2×C3×C4C4×AGL(1,𝔽5)11yescharacteristic
220C2×C3×C4C4⋊AGL(1,𝔽5)11yescharacteristic
221C3×C8—202no—
222C22×C5—21no—
223C22×C5—21no—
224C22×C5C3×#9511yescharacteristic
225C22×C5C3×#10411yescharacteristic
226C22×C5C3×#10411yescharacteristic
227C2×D5—21no—
228C2×D5—21no—
229C2×D5—21no—
230C2×D5—21no—
231C2×D5—21no—
232C2×D5—42no—
233C2×D5—42no—
234C2×D5C3×#10412yes—
235AGL(1,𝔽5)—44no—
236C4×C5—21no—
237C4×C5—21no—
238C4×C5—42no—
239Dic5—21no—
240Dic5—42no—
241Dic5C3×#9912yes—
242C24—51no—
243C22×C4—21no—
244C22×C4—51no—
245C22×C4—51no—
246C22×C4—51no—
247C22×C4—51no—
248C22×C4—51no—
249C22×C4—102no—
250C22×C4—101no—
251C22×C4—202no—
252C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
253M16—202no—
254C2×C8—102no—
255C22⋊C4—104no—
256C22⋊C4—102no—
257C22⋊C4—102no—
258C42—202no—
259C42—202no—
260C3×C5#129711yescharacteristic, complemented, Hall
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—52no—
264C22×C3—101no—
265C22×C3—101no—
266C22×C3—101no—
267C22×C3—101no—
268C22×C3—101no—
269C22×C3—202no—
270C22×C3—202no—
271C22×C3(C2×AGL(1,𝔽5))⋊C411yescharacteristic
272C22×C3#400811yescharacteristic
273C22×C3#401011yescharacteristic
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—42no—
277C3×C4—52no—
278C3×C4—101no—
279C3×C4—202no—
280C3×C4—204no—
281C2×C5—21no—
282C2×C5—21no—
283C2×C5C3×#33911yescharacteristic
284C2×C5C3×#33911yescharacteristic
285C2×C5C3×#34211yescharacteristic
286D5—22no—
287D5—41no—
288C23—51no—
289C23—51no—
290C23—102no—
291C23—101no—
292C23—101no—
293C23—201no—
294C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—51no—
300C2×C4—51no—
301C2×C4—102no—
302C2×C4—102no—
303C2×C4—102no—
304C2×C4—102no—
305C2×C4—104no—
306C2×C4—201no—
307C2×C4—201no—
308C2×C4—201no—
309C2×C4—202no—
310C2×C4—202no—
311C2×C4—201no—
312C2×C4C3×C4×AGL(1,𝔽5)11yescharacteristic
313C2×C4C3×C4⋊AGL(1,𝔽5)11yescharacteristic
314C8—202no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—102no—
318C2×C3—201no—
319C2×C3#6486211yescharacteristic
320C2×C3#6486611yescharacteristic
321C2×C3#6487511yescharacteristic
322C5C3×#129711yescharacteristic, complemented, Sylow
323C22—21no—
324C22—21no—
325C22—52no—
326C22—101no—
327C22—101no—
328C22—101no—
329C22—101no—
330C22—101no—
331C22—202no—
332C22—202no—
333C22C3×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
334C22C3×#400811yescharacteristic
335C22C3×#401011yescharacteristic
336C4—21no—
337C4—21no—
338C4—42no—
339C4—52no—
340C4—101no—
341C4—202no—
342C4—204no—
343C3#1059850711yescharacteristic, direct product factor, Sylow
344C2—21no—
345C2—21no—
346C2—102no—
347C2—201no—
348C2C3×#6486211yescharacteristic
349C2C3×#6486611yescharacteristic
350C2C3×#6487511yescharacteristic
subgroupG/H
centerC22×C3(C2×AGL(1,𝔽5))⋊C4
commutatorC2×C4×C5C3×C42
FrattiniC22×C4C2×C3×AGL(1,𝔽5)
FittingC3×C5×#95C4
socleC22×C3×C5#95
prime radicalC23×C3×D5D4
cosocleC22×C4×D5C22×C3
hypercenterC3×#95AGL(1,𝔽5)
hypocenterC5C3×#1297
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
4. #10598507C3direct (N × H)
8. C2×C3×#700C4semidirect
12. C3×(C2×AGL(1,𝔽5))⋊C4C4semidirect
14. C3×C5×#95C4semidirect
15. C3×#3963C4semidirect
44. C2×#700C3×C4semidirect
48. (C2×AGL(1,𝔽5))⋊C4C3×C4semidirect
50. C5×#95C3×C4semidirect
51. #3963C3×C4semidirect
90. C3×#95AGL(1,𝔽5)semidirect
187. #95C3×AGL(1,𝔽5)semidirect
260. C3×C5#1297semidirect
322. C5C3×#1297semidirect
343. C3#10598507direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×C22⋊AGL(1,𝔽5)2
C3×#657562
C3×#649552
#105985073
C3×#12975