Finite Group Databasebeta

← C3×#10605022 #14280961

Subgroups — C3×#10605022 #14280961

Order 1920.

880 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 148 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
1C22×C3×C22⋊Dic5C211yesmaximal, characteristic
2C3×#65798C212yesmaximal, complemented
3C3×#64933C214yesmaximal, complemented
4#10605022C311yesmaximal, characteristic, direct product factor, Hall
5C25×C3×C5C411yescharacteristic
6C23×C3×Dic5C411yescharacteristic
7C23×C3×Dic5C2211yescharacteristic
8C2×C3×#704—22no—
9C2×C3×C22⋊Dic5—22no—
10C2×C3×C22⋊Dic5—22no—
11C2×C3×C22⋊Dic5C412yes—
12C2×C3×C22⋊Dic5C2212yes—
13C3×C22⋊#170—22no—
14C3×C22⋊#170C2214yescomplemented
15C3×#1462—51nomaximal, Hall
16C22×C22⋊Dic5C2×C311yescharacteristic
17#65798C2×C312yescomplemented
18#64933C2×C314yescomplemented
19C24×C3×C5—21no—
20C24×C3×C5—21no—
21C24×C3×C5—24no—
22C24×C3×C5—42no—
23C24×C3×C5—42no—
24C24×C3×C5C2×C411yescharacteristic
25C24×C3×C5C2×C412yes—
26C22×C3×Dic5—21no—
27C22×C3×Dic5—21no—
28C22×C3×Dic5—42no—
29C22×C3×Dic5—42no—
30C22×C3×Dic5C2×C412yes—
31C22×C3×Dic5C2×C411yescharacteristic
32C22×C3×Dic5D412yes—
33C22×C3×Dic5D412yes—
34C22×C3×Dic5C2311yescharacteristic
35C3×#704—44no—
36C2×C3×#170—24no—
37C3×C22⋊Dic5—24no—
38C3×C22⋊Dic5—24no—
39C22×C3×C22⋊C4—51no—
40C3×#407—52no—
41C3×#323—54no—
42C25×C5C3×C411yescharacteristic
43C23×Dic5C3×C411yescharacteristic
44C23×Dic5C22×C311yescharacteristic
45C2×#704—22no—
46C2×C22⋊Dic5—22no—
47C2×C22⋊Dic5—22no—
48C2×C22⋊Dic5C3×C412yes—
49C2×C22⋊Dic5C22×C312yes—
50C22⋊#170—22no—
51C22⋊#170C22×C314yescomplemented
52#1462—51noSylow
53C23×C3×C5—22no—
54C23×C3×C5—22no—
55C23×C3×C5—22no—
56C23×C3×C5—24no—
57C23×C3×C5—24no—
58C23×C3×C5—22no—
59C23×C3×C5—22no—
60C23×C3×C5—416no—
61C23×C3×C5—42no—
62C23×C3×C5—42no—
63C23×C3×C5—42no—
64C23×C3×C5—42no—
65C23×C3×C5—42no—
66C23×C3×C5—42no—
67C23×C3×C5C22⋊C411yescharacteristic
68C23×C3×C5C22⋊C411yescharacteristic
69C23×C3×C5M1614yescomplemented
70C23×C3×C5C22×C411yescharacteristic
71C2×C3×Dic5—24no—
72C2×C3×Dic5—22no—
73C2×C3×Dic5—41no—
74C2×C3×Dic5—42no—
75C2×C3×Dic5—41no—
76C2×C3×Dic5—42no—
77C2×C3×Dic5—42no—
78C2×C3×Dic5—42no—
79C2×C3×Dic5C22⋊C412yes—
80C2×C3×Dic5C2×D412yes—
81C3×#170—44no—
82C25×C3AGL(1,𝔽5)11yescharacteristic
83C23×C3×C4—51no—
84C23×C3×C4—51no—
85C2×C3×M16—102no—
86C2×C3×C22⋊C4—52no—
87C2×C3×C22⋊C4—52no—
88C2×C3×C22⋊C4—102no—
89C2×C3×C22⋊C4—102no—
90C3×C22⋊C8—54no—
91C3×C22⋊C8—102no—
92C24×C5—21no—
93C24×C5—21no—
94C24×C5—24no—
95C24×C5—42no—
96C24×C5—42no—
97C24×C5C2×C3×C411yescharacteristic
98C24×C5C2×C3×C412yes—
99C22×Dic5—21no—
100C22×Dic5—21no—
101C22×Dic5—42no—
102C22×Dic5—42no—
103C22×Dic5C2×C3×C412yes—
104C22×Dic5C2×C3×C411yescharacteristic
105C22×Dic5C3×D412yes—
106C22×Dic5C3×D412yes—
107C22×Dic5C23×C311yescharacteristic
108#704—44no—
109C2×#170—24no—
110C22⋊Dic5—24no—
111C22⋊Dic5—24no—
112C22×C22⋊C4—51no—
113#407—52no—
114#323—54no—
115C22×C3×C5—28no—
116C22×C3×C5—21no—
117C22×C3×C5—21no—
118C22×C3×C5—21no—
119C22×C3×C5—21no—
120C22×C3×C5—21no—
121C22×C3×C5—21no—
122C22×C3×C5—24no—
123C22×C3×C5—44no—
124C22×C3×C5—48no—
125C22×C3×C5—48no—
126C22×C3×C5—44no—
127C22×C3×C5—42no—
128C22×C3×C5—42no—
129C22×C3×C5#9912yes—
130C22×C3×C5#10012yes—
131C22×C3×C5C2×C22⋊C411yescharacteristic
132C22×C3×C5C2×M1612yes—
133C3×Dic5—22no—
134C3×Dic5—22no—
135C3×Dic5—42no—
136C24×C3—24no—
137C24×C3—21no—
138C24×C3—21no—
139C24×C3—42no—
140C24×C3—42no—
141C24×C3C2×AGL(1,𝔽5)12yes—
142C24×C3C2×AGL(1,𝔽5)11yescharacteristic
143C22×C3×C4—52no—
144C22×C3×C4—52no—
145C22×C3×C4—51no—
146C22×C3×C4—51no—
147C22×C3×C4—52no—
148C22×C3×C4—101no—
149C22×C3×C4—101no—
150C22×C3×C4—202no—
151C22×C3×C4—202no—
152C3×M16—204no—
153C2×C3×C8—104no—
154C3×C22⋊C4—104no—
155C3×C22⋊C4—104no—
156C23×C5—22no—
157C23×C5—22no—
158C23×C5—22no—
159C23×C5—24no—
160C23×C5—24no—
161C23×C5—22no—
162C23×C5—22no—
163C23×C5—42no—
164C23×C5—42no—
165C23×C5—416no—
166C23×C5—42no—
167C23×C5—42no—
168C23×C5—42no—
169C23×C5—42no—
170C23×C5C3×C22⋊C411yescharacteristic
171C23×C5C3×C22⋊C411yescharacteristic
172C23×C5C3×M1614yescomplemented
173C23×C5C22×C3×C411yescharacteristic
174C2×Dic5—24no—
175C2×Dic5—22no—
176C2×Dic5—41no—
177C2×Dic5—42no—
178C2×Dic5—41no—
179C2×Dic5—42no—
180C2×Dic5—42no—
181C2×Dic5—42no—
182C2×Dic5C3×C22⋊C412yes—
183C2×Dic5C2×C3×D412yes—
184#170—44no—
185C25C3×AGL(1,𝔽5)11yescharacteristic
186C23×C4—51no—
187C23×C4—51no—
188C2×M16—102no—
189C2×C22⋊C4—52no—
190C2×C22⋊C4—52no—
191C2×C22⋊C4—102no—
192C2×C22⋊C4—102no—
193C22⋊C8—54no—
194C22⋊C8—102no—
195C2×C3×C5—22no—
196C2×C3×C5—22no—
197C2×C3×C5—22no—
198C2×C3×C5—44no—
199C2×C3×C5#40711yescharacteristic
200C2×C3×C5C2×#9911yescharacteristic
201C2×C3×C5C2×#10011yescharacteristic
202C23×C3—22no—
203C23×C3—22no—
204C23×C3—22no—
205C23×C3—24no—
206C23×C3—24no—
207C23×C3—22no—
208C23×C3—22no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3—416no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3#70414yescomplemented
217C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
218C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
219C23×C3C22×AGL(1,𝔽5)11yescharacteristic
220C2×C3×C4—52no—
221C2×C3×C4—52no—
222C2×C3×C4—104no—
223C2×C3×C4—102no—
224C2×C3×C4—201no—
225C2×C3×C4—202no—
226C2×C3×C4—201no—
227C2×C3×C4—202no—
228C2×C3×C4—202no—
229C2×C3×C4—202no—
230C3×C8—204no—
231C22×C5—28no—
232C22×C5—21no—
233C22×C5—21no—
234C22×C5—21no—
235C22×C5—21no—
236C22×C5—21no—
237C22×C5—21no—
238C22×C5—24no—
239C22×C5—44no—
240C22×C5—48no—
241C22×C5—48no—
242C22×C5—44no—
243C22×C5—42no—
244C22×C5—42no—
245C22×C5C3×#9912yes—
246C22×C5C3×#10012yes—
247C22×C5C2×C3×C22⋊C411yescharacteristic
248C22×C5C2×C3×M1612yes—
249Dic5—22no—
250Dic5—22no—
251Dic5—42no—
252C24—24no—
253C24—21no—
254C24—21no—
255C24—42no—
256C24—42no—
257C24C2×C3×AGL(1,𝔽5)11yescharacteristic
258C24C2×C3×AGL(1,𝔽5)12yes—
259C22×C4—52no—
260C22×C4—52no—
261C22×C4—51no—
262C22×C4—51no—
263C22×C4—52no—
264C22×C4—101no—
265C22×C4—101no—
266C22×C4—202no—
267C22×C4—202no—
268M16—204no—
269C2×C8—104no—
270C22⋊C4—104no—
271C22⋊C4—104no—
272C3×C5#146211yescharacteristic, complemented, Hall
273C22×C3—28no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—24no—
281C22×C3—44no—
282C22×C3—44no—
283C22×C3—48no—
284C22×C3—48no—
285C22×C3—42no—
286C22×C3—42no—
287C22×C3#401112yes—
288C22×C3#401312yes—
289C22×C3C2×#70412yes—
290C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
291C3×C4—102no—
292C3×C4—102no—
293C3×C4—202no—
294C2×C5—22no—
295C2×C5—22no—
296C2×C5—22no—
297C2×C5—44no—
298C2×C5C3×#40711yescharacteristic
299C2×C5C2×C3×#9911yescharacteristic
300C2×C5C2×C3×#10011yescharacteristic
301C23—22no—
302C23—22no—
303C23—22no—
304C23—24no—
305C23—24no—
306C23—22no—
307C23—22no—
308C23—42no—
309C23—42no—
310C23—416no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23C3×#70414yescomplemented
316C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
317C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
318C23C22×C3×AGL(1,𝔽5)11yescharacteristic
319C2×C4—52no—
320C2×C4—52no—
321C2×C4—104no—
322C2×C4—102no—
323C2×C4—201no—
324C2×C4—202no—
325C2×C4—201no—
326C2×C4—202no—
327C2×C4—202no—
328C2×C4—202no—
329C8—204no—
330C2×C3—22no—
331C2×C3—22no—
332C2×C3—22no—
333C2×C3—44no—
334C2×C3C2×#401111yescharacteristic
335C2×C3#6579811yescharacteristic
336C2×C3C2×#401311yescharacteristic
337C5C3×#146211yescharacteristic, complemented, Sylow
338C22—28no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—24no—
346C22—44no—
347C22—44no—
348C22—48no—
349C22—48no—
350C22—42no—
351C22—42no—
352C22C3×#401112yes—
353C22C3×#401312yes—
354C22C2×C3×#70412yes—
355C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
356C4—102no—
357C4—102no—
358C4—202no—
359C3#1060502211yescharacteristic, direct product factor, Sylow
360C2—22no—
361C2—22no—
362C2—22no—
363C2—44no—
364C2C2×C3×#401111yescharacteristic
365C2C3×#6579811yescharacteristic
366C2C2×C3×#401311yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
Fitting, prime radicalC25×C3×C5C4
socleC22×C3×C5C2×C22⋊C4
cosocleC22×Dic5C23×C3
hypercenterC25×C3AGL(1,𝔽5)
hypocenterC5C3×#1462
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C25×C3

Complements

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

NH ≅ G/Ntype
2. C3×#65798C2semidirect
3. C3×#64933C2semidirect
4. #10605022C3direct (N × H)
14. C3×C22⋊#170C22semidirect
17. #65798C2×C3semidirect
18. #64933C2×C3semidirect
51. C22⋊#170C22×C3semidirect
69. C23×C3×C5M16semidirect
172. C23×C5C3×M16semidirect
216. C23×C3#704semidirect
272. C3×C5#1462semidirect
315. C23C3×#704semidirect
337. C5C3×#1462semidirect
359. C3#10605022direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Dic52
C3×#657982
C3×#649332
#106050223
C3×#14625