Finite Group Databasebeta

← C3×#10604353 #14313463

Subgroups — C3×#10604353 #14313463

Order 1920.

572 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 108 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×#65859C211yesmaximal, characteristic
2C2×C3×#3999C212yesmaximal, complemented
3C3×#64854C212yesmaximal, complemented
4C3×#64853C212yesmaximal, complemented
5#10604353C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×#709C411yescharacteristic, complemented
7C22×C3×D20C2211yescharacteristic
8C2×C3×C22⋊AGL(1,𝔽5)—22no—
9C3×C5×#117C411yescharacteristic, complemented
10C3×#4041—22no—
11C3×#4028—22no—
12C3×#4021C411yescharacteristic, complemented
13C3×#4021C2211yescharacteristic
14C3×#4020C411yescharacteristic, complemented
15C3×#4020C2211yescharacteristic
16C3×#3999—22no—
17C3×#3999C2214yes—
18C3×#2127—51nomaximal, Hall
19#65859C2×C311yescharacteristic
20C2×#3999C2×C312yescomplemented
21#64854C2×C312yescomplemented
22#64853C2×C312yescomplemented
23C23×C3×D5D412yes—
24C22×C3×AGL(1,𝔽5)—42no—
25C22×C3×C4×C5C2×C411yescharacteristic
26C2×C3×C22⋊D5—21no—
27C3×#709—42no—
28C2×C3×D20—22no—
29C2×C3×D20—22no—
30C2×C3×D20—42no—
31C2×C3×D20C2×C411yescharacteristic
32C2×C3×D20C2×C411yescharacteristic, complemented
33C2×C3×D20D412yes—
34C2×C3×D20C2311yescharacteristic
35C2×C3×C4×D5—21no—
36C2×C3×Dic10C2×C411yescharacteristic, complemented
37C3×C22⋊AGL(1,𝔽5)—24no—
38C3×C22⋊AGL(1,𝔽5)—42no—
39C3×C5×C4⋊C4—21no—
40C3×C5×C22⋊C4—21no—
41C3×C42×C5C2×C411yescharacteristic, complemented
42C3×C42×C5C2×C411yescharacteristic, complemented
43C3×#685—22no—
44C3×#685—22no—
45C3×#535—51no—
46C2×C3×#99—52no—
47C3×#354—52no—
48C3×#353—52no—
49C2×#709C3×C411yescharacteristic, complemented
50C22×D20C22×C311yescharacteristic
51C2×C22⋊AGL(1,𝔽5)—22no—
52C5×#117C3×C411yescharacteristic, complemented
53#4041—22no—
54#4028—22no—
55#4021C3×C411yescharacteristic, complemented
56#4021C22×C311yescharacteristic
57#4020C3×C411yescharacteristic, complemented
58#4020C22×C311yescharacteristic
59#3999—22no—
60#3999C22×C314yes—
61#2127—51noSylow
62C23×C3×C5C22⋊C411yescharacteristic
63C22×C3×D5—22no—
64C22×C3×D5—22no—
65C22×C3×D5—21no—
66C22×C3×D5—82no—
67C22×C3×D5C22⋊C412yes—
68C22×C3×D5C2×D412yes—
69C2×C3×AGL(1,𝔽5)—44no—
70C2×C3×AGL(1,𝔽5)—84no—
71C2×C3×C4×C5—22no—
72C2×C3×C4×C5—21no—
73C2×C3×C4×C5—21no—
74C2×C3×C4×C5C22⋊C411yescharacteristic
75C2×C3×C4×C5C22×C411yescharacteristic
76C3×C22⋊D5—42no—
77C2×C3×Dic5—21no—
78C3×D20—41no—
79C3×D20—42no—
80C3×D20—41no—
81C3×D20—41no—
82C3×D20—82no—
83C3×D20—82no—
84C3×C4×D5—42no—
85C3×Dic10—41no—
86C2×C3×#41—51no—
87C22×C3×D4—51no—
88C3×#127—51no—
89C3×#127—51no—
90C3×#124—51no—
91C3×#124—51no—
92C3×#121—102no—
93C3×#120—102no—
94C3×#117AGL(1,𝔽5)11yescharacteristic, complemented
95C2×C3×C22⋊C4—102no—
96C3×#99—54no—
97C3×#99—102no—
98C23×D5C3×D412yes—
99C22×AGL(1,𝔽5)—42no—
100C22×C4×C5C2×C3×C411yescharacteristic
101C2×C22⋊D5—21no—
102#709—42no—
103C2×D20—22no—
104C2×D20—22no—
105C2×D20—42no—
106C2×D20C2×C3×C411yescharacteristic
107C2×D20C2×C3×C411yescharacteristic, complemented
108C2×D20C3×D412yes—
109C2×D20C23×C311yescharacteristic
110C2×C4×D5—21no—
111C2×Dic10C2×C3×C411yescharacteristic, complemented
112C22⋊AGL(1,𝔽5)—24no—
113C22⋊AGL(1,𝔽5)—42no—
114C5×C4⋊C4—21no—
115C5×C22⋊C4—21no—
116C42×C5C2×C3×C411yescharacteristic, complemented
117C42×C5C2×C3×C411yescharacteristic, complemented
118#685—22no—
119#685—22no—
120#535—51no—
121C2×#99—52no—
122#354—52no—
123#353—52no—
124C22×C3×C5—41no—
125C22×C3×C5#9912yes—
126C22×C3×C5C2×C22⋊C411yescharacteristic
127C2×C3×D5—22no—
128C2×C3×D5—22no—
129C2×C3×D5—44no—
130C2×C3×D5—42no—
131C2×C3×D5—42no—
132C2×C3×D5—42no—
133C2×C3×D5—41no—
134C2×C3×D5—82no—
135C3×AGL(1,𝔽5)—84no—
136C3×C4×C5—21no—
137C3×C4×C5—21no—
138C3×C4×C5—41no—
139C3×C4×C5—41no—
140C3×Dic5—41no—
141C24×C3—52no—
142C3×#41—202no—
143C2×C3×Q8—51no—
144C2×C3×D4—52no—
145C2×C3×D4—51no—
146C2×C3×D4—51no—
147C2×C3×D4—51no—
148C2×C3×D4—102no—
149C2×C3×D4—102no—
150C2×C3×D4—101no—
151C2×C3×D4—202no—
152C22×C3×C4—101no—
153C22×C3×C4—202no—
154C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
155C3×C4⋊C4—21no—
156C3×C22⋊C4—21no—
157C3×C22⋊C4—102no—
158C3×C22⋊C4—102no—
159C3×C22⋊C4—104no—
160C3×C22⋊C4—202no—
161C3×C42C2×AGL(1,𝔽5)11yescharacteristic, complemented
162C3×C42C2×AGL(1,𝔽5)11yescharacteristic, complemented
163C23×C5C3×C22⋊C411yescharacteristic
164C22×D5—22no—
165C22×D5—22no—
166C22×D5—21no—
167C22×D5—82no—
168C22×D5C3×C22⋊C412yes—
169C22×D5C2×C3×D412yes—
170C2×AGL(1,𝔽5)—44no—
171C2×AGL(1,𝔽5)—84no—
172C2×C4×C5—22no—
173C2×C4×C5—21no—
174C2×C4×C5—21no—
175C2×C4×C5C3×C22⋊C411yescharacteristic
176C2×C4×C5C22×C3×C411yescharacteristic
177C22⋊D5—42no—
178C2×Dic5—21no—
179D20—41no—
180D20—42no—
181D20—41no—
182D20—41no—
183D20—82no—
184D20—82no—
185C4×D5—42no—
186Dic10—41no—
187C2×#41—51no—
188C22×D4—51no—
189#127—51no—
190#127—51no—
191#124—51no—
192#124—51no—
193#121—102no—
194#120—102no—
195#117C3×AGL(1,𝔽5)11yescharacteristic, complemented
196C2×C22⋊C4—102no—
197#99—54no—
198#99—102no—
199C2×C3×C5—21no—
200C2×C3×C5—22no—
201C2×C3×C5C2×#9911yescharacteristic
202C3×D5—42no—
203C3×D5—42no—
204C3×D5—81no—
205C23×C3—52no—
206C23×C3—52no—
207C23×C3—102no—
208C23×C3—102no—
209C23×C3—101no—
210C23×C3—402no—
211C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
212C3×Q8—201no—
213C3×D4—202no—
214C3×D4—201no—
215C3×D4—201no—
216C3×D4—201no—
217C3×D4—202no—
218C3×D4—402no—
219C3×D4—402no—
220C2×C3×C4—22no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—101no—
224C2×C3×C4—202no—
225C2×C3×C4—204no—
226C2×C3×C4—404no—
227C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
228C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
229C22×C5—41no—
230C22×C5C3×#9912yes—
231C22×C5C2×C3×C22⋊C411yescharacteristic
232C2×D5—22no—
233C2×D5—22no—
234C2×D5—44no—
235C2×D5—42no—
236C2×D5—42no—
237C2×D5—42no—
238C2×D5—41no—
239C2×D5—82no—
240AGL(1,𝔽5)—84no—
241C4×C5—21no—
242C4×C5—21no—
243C4×C5—41no—
244C4×C5—41no—
245Dic5—41no—
246C24—52no—
247#41—202no—
248C2×Q8—51no—
249C2×D4—52no—
250C2×D4—51no—
251C2×D4—51no—
252C2×D4—51no—
253C2×D4—102no—
254C2×D4—102no—
255C2×D4—101no—
256C2×D4—202no—
257C22×C4—101no—
258C22×C4—202no—
259C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
260C4⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—102no—
263C22⋊C4—102no—
264C22⋊C4—104no—
265C22⋊C4—202no—
266C42C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
267C42C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
268C3×C5#212711yescharacteristic, complemented, Hall
269C22×C3—41no—
270C22×C3—102no—
271C22×C3—102no—
272C22×C3—202no—
273C22×C3—202no—
274C22×C3—204no—
275C22×C3—202no—
276C22×C3—201no—
277C22×C3—402no—
278C22×C3#399912yes—
279C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4—201no—
285C3×C4—404no—
286C2×C5—21no—
287C2×C5—22no—
288C2×C5C2×C3×#9911yescharacteristic
289D5—42no—
290D5—42no—
291D5—81no—
292C23—52no—
293C23—52no—
294C23—102no—
295C23—102no—
296C23—101no—
297C23—402no—
298C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
299Q8—201no—
300D4—202no—
301D4—201no—
302D4—201no—
303D4—201no—
304D4—202no—
305D4—402no—
306D4—402no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—101no—
311C2×C4—202no—
312C2×C4—204no—
313C2×C4—404no—
314C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
315C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
316C2×C3—21no—
317C2×C3—22no—
318C2×C3—202no—
319C2×C3—202no—
320C2×C3—401no—
321C2×C3C2×#399911yescharacteristic
322C5C3×#212711yescharacteristic, complemented, Sylow
323C22—41no—
324C22—102no—
325C22—102no—
326C22—202no—
327C22—202no—
328C22—204no—
329C22—202no—
330C22—201no—
331C22—402no—
332C22C3×#399912yes—
333C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—201no—
339C4—404no—
340C3#1060435311yescharacteristic, direct product factor, Sylow
341C2—21no—
342C2—22no—
343C2—202no—
344C2—202no—
345C2—401no—
346C2C2×C3×#399911yescharacteristic
subgroupG/H
centerC2×C3C2×#3999
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#117C4
socleC2×C3×C5C2×#99
prime radicalC3×#65859C2
cosocleC2×D20C23×C3
hypercenterC3×#117AGL(1,𝔽5)
hypocenterC5C3×#2127
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C22×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3999C2semidirect
3. C3×#64854C2semidirect
4. C3×#64853C2semidirect
5. #10604353C3direct (N × H)
6. C2×C3×#709C4semidirect
9. C3×C5×#117C4semidirect
12. C3×#4021C4semidirect
14. C3×#4020C4semidirect
20. C2×#3999C2×C3semidirect
21. #64854C2×C3semidirect
22. #64853C2×C3semidirect
32. C2×C3×D20C2×C4semidirect
36. C2×C3×Dic10C2×C4semidirect
41. C3×C42×C5C2×C4semidirect
42. C3×C42×C5C2×C4semidirect
49. C2×#709C3×C4semidirect
52. C5×#117C3×C4semidirect
55. #4021C3×C4semidirect
57. #4020C3×C4semidirect
94. C3×#117AGL(1,𝔽5)semidirect
107. C2×D20C2×C3×C4semidirect
111. C2×Dic10C2×C3×C4semidirect
116. C42×C5C2×C3×C4semidirect
117. C42×C5C2×C3×C4semidirect
161. C3×C42C2×AGL(1,𝔽5)semidirect
162. C3×C42C2×AGL(1,𝔽5)semidirect
195. #117C3×AGL(1,𝔽5)semidirect
266. C42C2×C3×AGL(1,𝔽5)semidirect
267. C42C2×C3×AGL(1,𝔽5)semidirect
268. C3×C5#2127semidirect
322. C5C3×#2127semidirect
340. C3#10604353direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658592
C2×C3×#39992
C3×#648542
C3×#648532
#106043533
C3×#21275