Finite Group Databasebeta

← C3×#10602752 #14316936

Subgroups — C3×#10602752 #14316936

Order 1920.

432 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 90 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×C5×#563C211yesmaximal, characteristic, complemented
2C3×#65384C212yesmaximal, complemented
3C3×#65351C211yesmaximal, characteristic
4C3×#64768C212yesmaximal, complemented
5C3×#64762C211yesmaximal, characteristic, complemented
6#10602752C311yesmaximal, characteristic, direct product factor, Hall
7C3×#4147C2211yescharacteristic
8C3×C5×C4⋊Q8C2212yes—
9C3×C5×#126—22no—
10C3×C5×#125—21no—
11C3×C5×#124C2211yescharacteristic
12C3×C5×#123—21no—
13C3×C5×C22⋊Q8—22no—
14C3×#4088—22no—
15C3×#4078—21no—
16C3×#3968C2212yes—
17C3×#3968C2211yescharacteristic
18C3×#3937—21no—
19C3×#3937—22no—
20C3×#2203—51nomaximal, Hall
21C5×#563C2×C311yescharacteristic, complemented
22#65384C2×C312yescomplemented
23#65351C2×C311yescharacteristic
24#64768C2×C312yescomplemented
25#64762C2×C311yescharacteristic, complemented
26C2×C3×C5×Q8D412yes—
27C2×C3×C5×Q8C2311yescharacteristic
28C2×C3×C5×D4D411yescharacteristic
29C22×C3×C4×C5—21no—
30C3×#713—41no—
31C3×Q8×D5—41no—
32C3×#709—21no—
33C3×#709—22no—
34C3×C5×C4⋊C4—21no—
35C3×C5×C4⋊C4—22no—
36C3×C5×C4⋊C4—21no—
37C3×C5×C4⋊C4—22no—
38C3×C5×C4⋊C4—22no—
39C3×C5×C22⋊C4—22no—
40C3×C5×C22⋊C4—21no—
41C3×C5×C22⋊C4—22no—
42C3×C42×C5D411yescharacteristic
43C3×C42×C5D412yes—
44C3×#689—42no—
45C3×#688—42no—
46C3×#687—41no—
47C3×#686—41no—
48C3×#681—21no—
49C3×#681—22no—
50C3×#563D511yescharacteristic, complemented
51C3×#456—52no—
52C3×#454—51no—
53C3×#356—52no—
54C3×#355—51no—
55#4147C22×C311yescharacteristic
56C5×C4⋊Q8C22×C312yes—
57C5×#126—22no—
58C5×#125—21no—
59C5×#124C22×C311yescharacteristic
60C5×#123—21no—
61C5×C22⋊Q8—22no—
62#4088—22no—
63#4078—21no—
64#3968C22×C311yescharacteristic
65#3968C22×C312yes—
66#3937—21no—
67#3937—22no—
68#2203—51noSylow
69C23×C3×C5—21no—
70C3×C5×Q8—41no—
71C3×C5×Q8—42no—
72C3×C5×D4—41no—
73C2×C3×C4×C5—21no—
74C2×C3×C4×C5—22no—
75C2×C3×C4×C5—22no—
76C2×C3×C4×C5—81no—
77C2×C3×C4×C5C2×D412yes—
78C2×C3×C4×C5C2×D411yescharacteristic
79C3×C22⋊D5—41no—
80C3×D20—21no—
81C3×D20—22no—
82C3×C4×D5—41no—
83C3×C4×D5—42no—
84C3×Dic10—21no—
85C3×Dic10—22no—
86C3×#168—41no—
87C3×#168—42no—
88C3×2−1+4—51no—
89C3×#137—102no—
90C3×AGL(1,ℤ/8)—101no—
91C3×C4⋊Q8C2×D512yes—
92C3×#126—22no—
93C3×#125—21no—
94C3×#124C2×D511yescharacteristic
95C3×#123—21no—
96C3×C22⋊Q8—22no—
97C3×#104—101no—
98C3×#104—102no—
99C3×#101—51no—
100C3×#101—52no—
101C2×C5×Q8C3×D412yes—
102C2×C5×Q8C23×C311yescharacteristic
103C2×C5×D4C3×D411yescharacteristic
104C22×C4×C5—21no—
105#713—41no—
106Q8×D5—41no—
107#709—21no—
108#709—22no—
109C5×C4⋊C4—21no—
110C5×C4⋊C4—22no—
111C5×C4⋊C4—21no—
112C5×C4⋊C4—22no—
113C5×C4⋊C4—22no—
114C5×C22⋊C4—22no—
115C5×C22⋊C4—21no—
116C5×C22⋊C4—22no—
117C42×C5C3×D411yescharacteristic
118C42×C5C3×D412yes—
119#689—42no—
120#688—42no—
121#687—41no—
122#686—41no—
123#681—21no—
124#681—22no—
125#563C3×D511yescharacteristic, complemented
126#456—52no—
127#454—51no—
128#356—52no—
129#355—51no—
130C22×C3×C5—42no—
131C22×C3×C5—41no—
132C22×C3×C5#12011yescharacteristic
133C2×C3×D5—41no—
134C3×C4×C5—21no—
135C3×C4×C5—22no—
136C3×C4×C5—41no—
137C3×C4×C5—42no—
138C3×C4×C5—42no—
139C3×Dic5—41no—
140C3×#41—101no—
141C3×#41—102no—
142C3×#41—201no—
143C2×C3×Q8—201no—
144C2×C3×Q8C22⋊D512yes—
145C2×C3×Q8C22×D511yescharacteristic
146C2×C3×D4C22⋊D511yescharacteristic
147C22×C3×C4—21no—
148C3×Q16—202no—
149C3×SD16—201no—
150C3×SD16—202no—
151C3×D8—201no—
152C3×M16—101no—
153C3×M16—102no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—21no—
161C3×C22⋊C4—22no—
162C3×C42C22⋊D511yescharacteristic
163C3×C42C22⋊D512yes—
164C23×C5—21no—
165C5×Q8—41no—
166C5×Q8—42no—
167C5×D4—41no—
168C2×C4×C5—21no—
169C2×C4×C5—22no—
170C2×C4×C5—22no—
171C2×C4×C5—81no—
172C2×C4×C5C2×C3×D412yes—
173C2×C4×C5C2×C3×D411yescharacteristic
174C22⋊D5—41no—
175D20—21no—
176D20—22no—
177C4×D5—41no—
178C4×D5—42no—
179Dic10—21no—
180Dic10—22no—
181#168—41no—
182#168—42no—
1832−1+4—51no—
184#137—102no—
185AGL(1,ℤ/8)—101no—
186C4⋊Q8C2×C3×D512yes—
187#126—22no—
188#125—21no—
189#124C2×C3×D511yescharacteristic
190#123—21no—
191C22⋊Q8—22no—
192#104—101no—
193#104—102no—
194#101—52no—
195#101—51no—
196C2×C3×C5—21no—
197C2×C3×C5—81no—
198C2×C3×C5B(4,𝔽2)11yescharacteristic
199C3×D5—81no—
200C23×C3—21no—
201C3×Q8—41no—
202C3×Q8—42no—
203C3×Q8—101no—
204C3×Q8—102no—
205C3×D4—41no—
206C3×D4—101no—
207C3×D4—102no—
208C3×D4—201no—
209C2×C3×C4—21no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—81no—
213C2×C3×C4—201no—
214C2×C3×C4—202no—
215C2×C3×C4C2×C22⋊D512yes—
216C2×C3×C4C2×C22⋊D511yescharacteristic
217C3×C8—201no—
218C3×C8—202no—
219C22×C5—42no—
220C22×C5—41no—
221C22×C5C3×#12011yescharacteristic
222C2×D5—41no—
223C4×C5—21no—
224C4×C5—22no—
225C4×C5—41no—
226C4×C5—42no—
227C4×C5—42no—
228Dic5—41no—
229#41—101no—
230#41—102no—
231#41—201no—
232C2×Q8—201no—
233C2×Q8C3×C22⋊D512yes—
234C2×Q8C22×C3×D511yescharacteristic
235C2×D4C3×C22⋊D511yescharacteristic
236C22×C4—21no—
237Q16—202no—
238SD16—201no—
239SD16—202no—
240D8—201no—
241M16—101no—
242M16—102no—
243C4⋊C4—21no—
244C4⋊C4—22no—
245C4⋊C4—21no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C22⋊C4—22no—
249C22⋊C4—21no—
250C22⋊C4—22no—
251C42C3×C22⋊D511yescharacteristic
252C42C3×C22⋊D512yes—
253C3×C5#220311yescharacteristic, complemented, Hall
254C22×C3—42no—
255C22×C3—41no—
256C22×C3—201no—
257C22×C3#409911yescharacteristic
258C3×C4—21no—
259C3×C4—22no—
260C3×C4—41no—
261C3×C4—42no—
262C3×C4—42no—
263C3×C4—201no—
264C2×C5—21no—
265C2×C5—81no—
266C2×C5C3×B(4,𝔽2)11yescharacteristic
267D5—81no—
268C23—21no—
269Q8—41no—
270Q8—42no—
271Q8—101no—
272Q8—102no—
273D4—41no—
274D4—101no—
275D4—102no—
276D4—201no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—81no—
281C2×C4—201no—
282C2×C4—202no—
283C2×C4C2×C3×C22⋊D512yes—
284C2×C4C2×C3×C22⋊D511yescharacteristic
285C8—201no—
286C8—202no—
287C2×C3—21no—
288C2×C3—81no—
289C2×C3—401no—
290C2×C3#6553311yescharacteristic
291C5C3×#220311yescharacteristic, complemented, Sylow
292C22—42no—
293C22—41no—
294C22—201no—
295C22C3×#409911yescharacteristic
296C4—21no—
297C4—22no—
298C4—41no—
299C4—42no—
300C4—42no—
301C4—201no—
302C3#1060275211yescharacteristic, direct product factor, Sylow
303C2—21no—
304C2—81no—
305C2—401no—
306C2C3×#6553311yescharacteristic
subgroupG/H
centerC2×C3#65533
commutator, cosocleC2×C5×Q8C23×C3
FrattiniC2×Q8C22×C3×D5
FittingC3×C5×#563C2
socleC2×C3×C5B(4,𝔽2)
prime radicalC3×#65351C2
hypercenterC3×#563D5
hypocenterC5C3×#2203
LCSG ▷ C2×C5×Q8 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×#563

Complements

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

NH ≅ G/Ntype
1. C3×C5×#563C2semidirect
2. C3×#65384C2semidirect
4. C3×#64768C2semidirect
5. C3×#64762C2semidirect
6. #10602752C3direct (N × H)
21. C5×#563C2×C3semidirect
22. #65384C2×C3semidirect
24. #64768C2×C3semidirect
25. #64762C2×C3semidirect
50. C3×#563D5semidirect
125. #563C3×D5semidirect
253. C3×C5#2203semidirect
291. C5C3×#2203semidirect
302. C3#10602752direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5632
C3×#653842
C3×#653512
C3×#647682
C3×#647622
#106027523
C3×#22035