Finite Group Databasebeta

← C3×#10602723 #14316975

Subgroups — C3×#10602723 #14316975

Order 1920.

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

normal subgroups only

This group has 472 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×#555C211yesmaximal, characteristic, complemented
2C3×#65384C211yesmaximal, characteristic, complemented
3C3×#65351C212yesmaximal, complemented
4C3×#64768C211yesmaximal, characteristic, complemented
5C3×#64762C212yesmaximal, complemented
6#10602723C311yesmaximal, characteristic, direct product factor, Hall
7C3×#4147C2211yescharacteristic
8C3×C5×C4⋊Q8C2211yescharacteristic
9C3×C5×#124—21no—
10C3×C5×#124C2212yes—
11C3×C5×#121—22no—
12C3×C4×C5×D4—21no—
13C3×C5×#117—22no—
14C3×#4088—21no—
15C3×#4078—22no—
16C3×#3968C2211yescharacteristic
17C3×#3968C2212yes—
18C3×#3937—22no—
19C3×#3937—21no—
20C3×#2204—51nomaximal, Hall
21C5×#555C2×C311yescharacteristic, complemented
22#65384C2×C311yescharacteristic, complemented
23#65351C2×C312yescomplemented
24#64768C2×C311yescharacteristic, complemented
25#64762C2×C312yescomplemented
26C2×C3×C5×Q8D411yescharacteristic
27C2×C3×C5×Q8C2311yescharacteristic
28C2×C3×C5×D4—21no—
29C2×C3×C5×D4—21no—
30C2×C3×C5×D4D412yes—
31C22×C3×C4×C5—22no—
32C3×#713—41no—
33C3×Q8×D5—41no—
34C3×#709—22no—
35C3×#709—21no—
36C3×C5×C4⋊C4—21no—
37C3×C5×C4⋊C4—22no—
38C3×C5×C22⋊C4—22no—
39C3×C5×C22⋊C4—22no—
40C3×C5×C22⋊C4—22no—
41C3×C42×C5—21no—
42C3×C42×C5D412yes—
43C3×C42×C5D411yescharacteristic
44C3×#689—41no—
45C3×#688—41no—
46C3×#687—42no—
47C3×#686—42no—
48C3×#681—22no—
49C3×#681—21no—
50C3×#555D511yescharacteristic, complemented
51C3×#456—51no—
52C3×#454—52no—
53C3×#356—51no—
54C3×#355—52no—
55#4147C22×C311yescharacteristic
56C5×C4⋊Q8C22×C311yescharacteristic
57C5×#124—21no—
58C5×#124C22×C312yes—
59C5×#121—22no—
60C4×C5×D4—21no—
61C5×#117—22no—
62#4088—21no—
63#4078—22no—
64#3968C22×C312yes—
65#3968C22×C311yescharacteristic
66#3937—22no—
67#3937—21no—
68#2204—51noSylow
69C23×C3×C5—22no—
70C3×C5×Q8—41no—
71C3×C5×Q8—41no—
72C3×C5×D4—42no—
73C3×C5×D4—42no—
74C3×C5×D4—81no—
75C2×C3×C4×C5—21no—
76C2×C3×C4×C5—22no—
77C2×C3×C4×C5—21no—
78C2×C3×C4×C5—44no—
79C2×C3×C4×C5C2×D411yescharacteristic
80C2×C3×C4×C5C2×D412yes—
81C3×C22⋊D5—41no—
82C3×D20—22no—
83C3×D20—21no—
84C3×C4×D5—42no—
85C3×C4×D5—41no—
86C3×Dic10—22no—
87C3×Dic10—21no—
88C3×#168—42no—
89C3×#168—41no—
90C3×2−1+4—51no—
91C3×#137—101no—
92C3×AGL(1,ℤ/8)—102no—
93C3×C4⋊Q8C2×D511yescharacteristic
94C3×#124—21no—
95C3×#124C2×D512yes—
96C3×#121—22no—
97C3×C4×D4—21no—
98C3×#117—22no—
99C3×#104—102no—
100C3×#104—101no—
101C3×#101—52no—
102C3×#101—51no—
103C2×C5×Q8C3×D411yescharacteristic
104C2×C5×Q8C23×C311yescharacteristic
105C2×C5×D4—21no—
106C2×C5×D4—21no—
107C2×C5×D4C3×D412yes—
108C22×C4×C5—22no—
109#713—41no—
110Q8×D5—41no—
111#709—22no—
112#709—21no—
113C5×C4⋊C4—21no—
114C5×C4⋊C4—22no—
115C5×C22⋊C4—22no—
116C5×C22⋊C4—22no—
117C5×C22⋊C4—22no—
118C42×C5—21no—
119C42×C5C3×D412yes—
120C42×C5C3×D411yescharacteristic
121#689—41no—
122#688—41no—
123#687—42no—
124#686—42no—
125#681—22no—
126#681—21no—
127#555C3×D511yescharacteristic, complemented
128#456—51no—
129#454—52no—
130#356—51no—
131#355—52no—
132C22×C3×C5—42no—
133C22×C3×C5—42no—
134C22×C3×C5—42no—
135C22×C3×C5#12011yescharacteristic
136C2×C3×D5—41no—
137C3×C4×C5—22no—
138C3×C4×C5—22no—
139C3×C4×C5—21no—
140C3×C4×C5—42no—
141C3×C4×C5—41no—
142C3×Dic5—41no—
143C3×#41—102no—
144C3×#41—101no—
145C3×#41—201no—
146C2×C3×Q8—201no—
147C2×C3×Q8C22⋊D511yescharacteristic
148C2×C3×Q8C22×D511yescharacteristic
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4C22⋊D512yes—
152C22×C3×C4—22no—
153C3×Q16—201no—
154C3×SD16—202no—
155C3×SD16—201no—
156C3×D8—202no—
157C3×M16—102no—
158C3×M16—101no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—22no—
161C3×C22⋊C4—22no—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—22no—
164C3×C42—21no—
165C3×C42C22⋊D512yes—
166C3×C42C22⋊D511yescharacteristic
167C23×C5—22no—
168C5×Q8—41no—
169C5×Q8—41no—
170C5×D4—42no—
171C5×D4—42no—
172C5×D4—81no—
173C2×C4×C5—21no—
174C2×C4×C5—22no—
175C2×C4×C5—21no—
176C2×C4×C5—44no—
177C2×C4×C5C2×C3×D411yescharacteristic
178C2×C4×C5C2×C3×D412yes—
179C22⋊D5—41no—
180D20—22no—
181D20—21no—
182C4×D5—42no—
183C4×D5—41no—
184Dic10—22no—
185Dic10—21no—
186#168—42no—
187#168—41no—
1882−1+4—51no—
189#137—101no—
190AGL(1,ℤ/8)—102no—
191C4⋊Q8C2×C3×D511yescharacteristic
192#124—21no—
193#124C2×C3×D512yes—
194#121—22no—
195C4×D4—21no—
196#117—22no—
197#104—102no—
198#104—101no—
199#101—51no—
200#101—52no—
201C2×C3×C5—21no—
202C2×C3×C5—82no—
203C2×C3×C5B(4,𝔽2)11yescharacteristic
204C3×D5—81no—
205C23×C3—22no—
206C3×Q8—41no—
207C3×Q8—41no—
208C3×Q8—102no—
209C3×Q8—101no—
210C3×D4—42no—
211C3×D4—42no—
212C3×D4—81no—
213C3×D4—102no—
214C3×D4—101no—
215C3×D4—201no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—21no—
219C2×C3×C4—44no—
220C2×C3×C4—202no—
221C2×C3×C4—201no—
222C2×C3×C4C2×C22⋊D511yescharacteristic
223C2×C3×C4C2×C22⋊D512yes—
224C3×C8—202no—
225C3×C8—201no—
226C22×C5—42no—
227C22×C5—42no—
228C22×C5—42no—
229C22×C5C3×#12011yescharacteristic
230C2×D5—41no—
231C4×C5—22no—
232C4×C5—22no—
233C4×C5—21no—
234C4×C5—42no—
235C4×C5—41no—
236Dic5—41no—
237#41—102no—
238#41—101no—
239#41—201no—
240C2×Q8—201no—
241C2×Q8C3×C22⋊D511yescharacteristic
242C2×Q8C22×C3×D511yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4C3×C22⋊D512yes—
246C22×C4—22no—
247Q16—201no—
248SD16—202no—
249SD16—201no—
250D8—202no—
251M16—102no—
252M16—101no—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C42—21no—
259C42C3×C22⋊D512yes—
260C42C3×C22⋊D511yescharacteristic
261C3×C5#220411yescharacteristic, complemented, Hall
262C22×C3—42no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3—201no—
266C22×C3#409911yescharacteristic
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—42no—
271C3×C4—41no—
272C3×C4—201no—
273C2×C5—21no—
274C2×C5—82no—
275C2×C5C3×B(4,𝔽2)11yescharacteristic
276D5—81no—
277C23—22no—
278Q8—41no—
279Q8—41no—
280Q8—102no—
281Q8—101no—
282D4—42no—
283D4—42no—
284D4—81no—
285D4—102no—
286D4—101no—
287D4—201no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—44no—
292C2×C4—202no—
293C2×C4—201no—
294C2×C4C2×C3×C22⋊D511yescharacteristic
295C2×C4C2×C3×C22⋊D512yes—
296C8—202no—
297C8—201no—
298C2×C3—21no—
299C2×C3—82no—
300C2×C3—401no—
301C2×C3#6553311yescharacteristic
302C5C3×#220411yescharacteristic, complemented, Sylow
303C22—42no—
304C22—42no—
305C22—42no—
306C22—201no—
307C22C3×#409911yescharacteristic
308C4—22no—
309C4—22no—
310C4—21no—
311C4—42no—
312C4—41no—
313C4—201no—
314C3#1060272311yescharacteristic, direct product factor, Sylow
315C2—21no—
316C2—82no—
317C2—401no—
318C2C3×#6553311yescharacteristic
subgroupG/H
centerC2×C3#65533
commutator, cosocleC2×C5×Q8C23×C3
FrattiniC2×Q8C22×C3×D5
FittingC3×C5×#555C2
socleC2×C3×C5B(4,𝔽2)
hypercenterC3×#555D5
hypocenterC5C3×#2204
LCSG ▷ C2×C5×Q8 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×#555

Complements

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

NH ≅ G/Ntype
1. C3×C5×#555C2semidirect
2. C3×#65384C2semidirect
3. C3×#65351C2semidirect
4. C3×#64768C2semidirect
5. C3×#64762C2semidirect
6. #10602723C3direct (N × H)
21. C5×#555C2×C3semidirect
22. #65384C2×C3semidirect
23. #65351C2×C3semidirect
24. #64768C2×C3semidirect
25. #64762C2×C3semidirect
50. C3×#555D5semidirect
127. #555C3×D5semidirect
261. C3×C5#2204semidirect
302. C5C3×#2204semidirect
314. C3#10602723direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5552
C3×#653842
C3×#653512
C3×#647682
C3×#647622
#106027233
C3×#22045