Finite Group Databasebeta

← C3×#10601780 #14316911

Subgroups — C3×#10601780 #14316911

Order 1920.

496 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 82 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×#66017C211yesmaximal, characteristic, complemented
2C3×C5×#454C211yesmaximal, characteristic, complemented
3C3×#65534C212yesmaximal, complemented
4C3×#64762C211yesmaximal, characteristic, complemented
5C3×#64697C212yesmaximal, complemented
6#10601780C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×2−1+4C2211yescharacteristic, complemented
8C3×C5×AGL(1,ℤ/8)—21no—
9C3×C5×#124C2211yescharacteristic
10C3×#4095—22no—
11C3×#4093C2212yes—
12C3×#4083—21no—
13C3×#4031—22no—
14C3×#4028—22no—
15C3×#4021—21no—
16C3×C5×#104—21no—
17C3×C5×#101C2211yescharacteristic, complemented
18C3×#3969—22no—
19C3×#3968C2212yescomplemented
20C3×#2202—51nomaximal, Hall
21#66017C2×C311yescharacteristic, complemented
22C5×#454C2×C311yescharacteristic, complemented
23#65534C2×C312yescomplemented
24#64762C2×C311yescharacteristic, complemented
25#64697C2×C312yescomplemented
26C23×C3×D5—21no—
27C3×C5×#41—22no—
28C3×C5×#41—21no—
29C3×C5×#41—41no—
30C2×C3×C5×Q8—41no—
31C2×C3×C5×Q8C2311yescharacteristic
32C2×C3×C5×D4D411yescharacteristic
33C2×C3×C22⋊D5—22no—
34C2×C3×D20—21no—
35C2×C3×D20D412yes—
36C2×C3×Dic10—21no—
37C3×C5×SD16—41no—
38C3×C5×D8—41no—
39C3×C5×M16—21no—
40C3×C5×C22⋊C4—22no—
41C3×C42×C5D411yescharacteristic
42C3×C22⋊Dic5—21no—
43C3×#688—42no—
44C3×#686—42no—
45C3×#685—22no—
46C3×#685—22no—
47C3×#685—22no—
48C3×C4×Dic5D412yes—
49C3×#681—22no—
50C3×H(3,𝔽4)—51no—
51C3×#454—52no—
52C3×#454D511yescharacteristic, complemented
53C3×#355—51no—
54C3×#355—52no—
55C5×2−1+4C22×C311yescharacteristic, complemented
56C5×AGL(1,ℤ/8)—21no—
57C5×#124C22×C311yescharacteristic
58#4095—22no—
59#4093C22×C312yes—
60#4083—21no—
61#4031—22no—
62#4028—22no—
63#4021—21no—
64C5×#104—21no—
65C5×#101C22×C311yescharacteristic, complemented
66#3969—22no—
67#3968C22×C312yescomplemented
68#2202—51noSylow
69C23×C3×C5—21no—
70C22×C3×D5—22no—
71C22×C3×D5—82no—
72C22×C3×D5—81no—
73C3×C5×Q8—22no—
74C3×C5×Q8—21no—
75C3×C5×Q8—41no—
76C3×C5×D4—22no—
77C3×C5×D4—21no—
78C3×C5×D4—41no—
79C3×C5×D4—41no—
80C2×C3×C4×C5—21no—
81C2×C3×C4×C5—42no—
82C2×C3×C4×C5—41no—
83C2×C3×C4×C5C2×D412yes—
84C2×C3×C4×C5C2×D411yescharacteristic
85C3×C22⋊D5—82no—
86C2×C3×Dic5—22no—
87C3×D20—42no—
88C3×D20—81no—
89C3×Dic10—81no—
90C3×C5×C8—41no—
91C3×#168—42no—
92C3×2−1+4C2×D511yescharacteristic, complemented
93C3×AGL(1,ℤ/8)—21no—
94C3×AGL(1,ℤ/8)—102no—
95C3×#124—52no—
96C3×#124—102no—
97C3×#124—101no—
98C3×#124C2×D511yescharacteristic
99C3×#120—102no—
100C3×#120—101no—
101C3×#104—21no—
102C3×#104—102no—
103C3×#101—52no—
104C3×#101C2×D511yescharacteristic, complemented
105C23×D5—21no—
106C5×#41—21no—
107C5×#41—22no—
108C5×#41—41no—
109C2×C5×Q8—41no—
110C2×C5×Q8C23×C311yescharacteristic
111C2×C5×D4C3×D411yescharacteristic
112C2×C22⋊D5—22no—
113C2×D20—21no—
114C2×D20C3×D412yes—
115C2×Dic10—21no—
116C5×SD16—41no—
117C5×D8—41no—
118C5×M16—21no—
119C5×C22⋊C4—22no—
120C42×C5C3×D411yescharacteristic
121C22⋊Dic5—21no—
122#688—42no—
123#686—42no—
124#685—22no—
125#685—22no—
126#685—22no—
127C4×Dic5C3×D412yes—
128#681—22no—
129H(3,𝔽4)—51no—
130#454—52no—
131#454C3×D511yescharacteristic, complemented
132#355—51no—
133#355—52no—
134C22×C3×C5—42no—
135C22×C3×C5—41no—
136C22×C3×C5—41no—
137C22×C3×C5#12011yescharacteristic
138C2×C3×D5—42no—
139C2×C3×D5—42no—
140C2×C3×D5—42no—
141C2×C3×D5—84no—
142C3×C4×C5—22no—
143C3×C4×C5—21no—
144C3×C4×C5—41no—
145C3×C4×C5—41no—
146C3×Dic5—42no—
147C24×C3—101no—
148C3×#41—22no—
149C3×#41—21no—
150C3×#41—41no—
151C2×C3×Q8—41no—
152C2×C3×Q8—101no—
153C2×C3×Q8C22×D511yescharacteristic
154C2×C3×D4—52no—
155C2×C3×D4—101no—
156C2×C3×D4—102no—
157C2×C3×D4C22⋊D511yescharacteristic
158C3×SD16—41no—
159C3×SD16—202no—
160C3×D8—41no—
161C3×D8—202no—
162C3×M16—21no—
163C3×M16—102no—
164C3×C22⋊C4—22no—
165C3×C22⋊C4—102no—
166C3×C22⋊C4—101no—
167C3×C22⋊C4—102no—
168C3×C22⋊C4—102no—
169C3×C42—52no—
170C3×C42C22⋊D511yescharacteristic
171C23×C5—21no—
172C22×D5—22no—
173C22×D5—82no—
174C22×D5—81no—
175C5×Q8—22no—
176C5×Q8—21no—
177C5×Q8—41no—
178C5×D4—22no—
179C5×D4—21no—
180C5×D4—41no—
181C5×D4—41no—
182C2×C4×C5—21no—
183C2×C4×C5—42no—
184C2×C4×C5—41no—
185C2×C4×C5C2×C3×D412yes—
186C2×C4×C5C2×C3×D411yescharacteristic
187C22⋊D5—82no—
188C2×Dic5—22no—
189D20—42no—
190D20—81no—
191Dic10—81no—
192C5×C8—41no—
193#168—42no—
1942−1+4C2×C3×D511yescharacteristic, complemented
195AGL(1,ℤ/8)—21no—
196AGL(1,ℤ/8)—102no—
197#124—52no—
198#124—102no—
199#124—101no—
200#124C2×C3×D511yescharacteristic
201#120—102no—
202#120—101no—
203#104—21no—
204#104—102no—
205#101—52no—
206#101C2×C3×D511yescharacteristic, complemented
207C2×C3×C5—21no—
208C2×C3×C5—81no—
209C2×C3×C5—81no—
210C2×C3×C5B(4,𝔽2)11yescharacteristic
211C3×D5—82no—
212C23×C3—21no—
213C23×C3—102no—
214C23×C3—402no—
215C23×C3—401no—
216C3×Q8—21no—
217C3×Q8—22no—
218C3×Q8—41no—
219C3×Q8—401no—
220C3×D4—21no—
221C3×D4—22no—
222C3×D4—41no—
223C3×D4—41no—
224C3×D4—202no—
225C3×D4—401no—
226C3×D4—402no—
227C2×C3×C4—21no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—102no—
231C2×C3×C4D4×D512yes—
232C2×C3×C4C2×C22⋊D511yescharacteristic
233C3×C8—41no—
234C3×C8—202no—
235C22×C5—42no—
236C22×C5—41no—
237C22×C5—41no—
238C22×C5C3×#12011yescharacteristic
239C2×D5—42no—
240C2×D5—42no—
241C2×D5—42no—
242C2×D5—84no—
243C4×C5—22no—
244C4×C5—21no—
245C4×C5—41no—
246C4×C5—41no—
247Dic5—42no—
248C24—101no—
249#41—22no—
250#41—21no—
251#41—41no—
252C2×Q8—41no—
253C2×Q8—101no—
254C2×Q8C22×C3×D511yescharacteristic
255C2×D4—52no—
256C2×D4—101no—
257C2×D4—102no—
258C2×D4C3×C22⋊D511yescharacteristic
259SD16—41no—
260SD16—202no—
261D8—41no—
262D8—202no—
263M16—21no—
264M16—102no—
265C22⋊C4—22no—
266C22⋊C4—102no—
267C22⋊C4—101no—
268C22⋊C4—102no—
269C22⋊C4—102no—
270C42—52no—
271C42C3×C22⋊D511yescharacteristic
272C3×C5#220211yescharacteristic, complemented, Hall
273C22×C3—42no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—202no—
277C22×C3—202no—
278C22×C3—202no—
279C22×C3—404no—
280C22×C3#408311yescharacteristic
281C3×C4—21no—
282C3×C4—22no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—202no—
286C2×C5—21no—
287C2×C5—81no—
288C2×C5—81no—
289C2×C5C3×B(4,𝔽2)11yescharacteristic
290D5—82no—
291C23—21no—
292C23—102no—
293C23—402no—
294C23—401no—
295Q8—21no—
296Q8—22no—
297Q8—41no—
298Q8—401no—
299D4—21no—
300D4—22no—
301D4—41no—
302D4—41no—
303D4—202no—
304D4—401no—
305D4—402no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—102no—
310C2×C4C3×D4×D512yes—
311C2×C4C2×C3×C22⋊D511yescharacteristic
312C8—41no—
313C8—202no—
314C2×C3—21no—
315C2×C3—81no—
316C2×C3—81no—
317C2×C3—402no—
318C2×C3#6532111yescharacteristic
319C5C3×#220211yescharacteristic, complemented, Sylow
320C22—42no—
321C22—41no—
322C22—41no—
323C22—202no—
324C22—202no—
325C22—202no—
326C22—404no—
327C22C3×#408311yescharacteristic
328C4—21no—
329C4—22no—
330C4—41no—
331C4—41no—
332C4—202no—
333C3#1060178011yescharacteristic, direct product factor, Sylow
334C2—21no—
335C2—81no—
336C2—81no—
337C2—402no—
338C2C3×#6532111yescharacteristic
subgroupG/H
centerC2×C3#65321
commutator, cosocleC2×C5×Q8C23×C3
FrattiniC2×Q8C22×C3×D5
FittingC3×C5×#454C2
socleC2×C3×C5B(4,𝔽2)
hypercenterC3×#454D5
hypocenterC5C3×#2202
LCSG ▷ C2×C5×Q8 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×#454

Complements

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

NH ≅ G/Ntype
1. C3×#66017C2semidirect
2. C3×C5×#454C2semidirect
3. C3×#65534C2semidirect
4. C3×#64762C2semidirect
5. C3×#64697C2semidirect
6. #10601780C3direct (N × H)
7. C3×C5×2−1+4C22semidirect
17. C3×C5×#101C22semidirect
19. C3×#3968C22semidirect
21. #66017C2×C3semidirect
22. C5×#454C2×C3semidirect
23. #65534C2×C3semidirect
24. #64762C2×C3semidirect
25. #64697C2×C3semidirect
52. C3×#454D5semidirect
55. C5×2−1+4C22×C3semidirect
65. C5×#101C22×C3semidirect
67. #3968C22×C3semidirect
92. C3×2−1+4C2×D5semidirect
104. C3×#101C2×D5semidirect
131. #454C3×D5semidirect
194. 2−1+4C2×C3×D5semidirect
206. #101C2×C3×D5semidirect
272. C3×C5#2202semidirect
319. C5C3×#2202semidirect
333. C3#10601780direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660172
C3×C5×#4542
C3×#655342
C3×#647622
C3×#646972
#106017803
C3×#22025