Finite Group Databasebeta

← C3×#10601787 #14316972

Subgroups — C3×#10601787 #14316972

Order 1920.

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

normal subgroups only

This group has 472 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×#66068C211yesmaximal, characteristic, complemented
2C3×C5×#456C211yesmaximal, characteristic, complemented
3C3×#65534C212yesmaximal, complemented
4C3×#64768C211yesmaximal, characteristic, complemented
5C3×#64697C212yesmaximal, complemented
6#10601787C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×2−1+4C2211yescharacteristic
8C3×C5×#137—21no—
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×#4095—22no—
11C3×#4093—21no—
12C3×#4093C2212yes—
13C3×#4041—22no—
14C3×#4038—22no—
15C3×C4×D20—21no—
16C3×C5×#104—21no—
17C3×C5×#101C2211yescharacteristic
18C3×#3969—22no—
19C3×#3968C2212yes—
20C3×#2204—51nomaximal, Hall
21#66068C2×C311yescharacteristic, complemented
22C5×#456C2×C311yescharacteristic, complemented
23#65534C2×C312yescomplemented
24#64768C2×C311yescharacteristic, complemented
25#64697C2×C312yescomplemented
26C3×C5×#41—22no—
27C3×C5×#41—21no—
28C3×C5×#41—41no—
29C2×C3×C5×Q8—41no—
30C2×C3×C5×Q8D411yescharacteristic
31C2×C3×C5×Q8C2311yescharacteristic
32C2×C3×D20—21no—
33C2×C3×D20—21no—
34C2×C3×D20D412yes—
35C2×C3×C4×D5—22no—
36C3×C5×Q16—41no—
37C3×C5×SD16—41no—
38C3×C5×M16—21no—
39C3×C5×C4⋊C4—22no—
40C3×C42×C5D411yescharacteristic
41C3×#688—42no—
42C3×#686—42no—
43C3×#685—22no—
44C3×#685—22no—
45C3×#685—22no—
46C3×C4⋊Dic5—21no—
47C3×C4×Dic5—21no—
48C3×C4×Dic5D412yes—
49C3×#681—22no—
50C3×#555—51no—
51C3×#456D511yescharacteristic, complemented
52C3×#454—52no—
53C3×#356—51no—
54C3×#355—52no—
55C5×2−1+4C22×C311yescharacteristic
56C5×#137—21no—
57C5×C4⋊Q8C22×C311yescharacteristic
58#4095—22no—
59#4093—21no—
60#4093C22×C312yes—
61#4041—22no—
62#4038—22no—
63C4×D20—21no—
64C5×#104—21no—
65C5×#101C22×C311yescharacteristic
66#3969—22no—
67#3968C22×C312yes—
68#2204—51noSylow
69C22×C3×D5—22no—
70C3×C5×Q8—22no—
71C3×C5×Q8—21no—
72C3×C5×Q8—41no—
73C3×C5×Q8—41no—
74C3×C5×D4—22no—
75C3×C5×D4—21no—
76C3×C5×D4—41no—
77C2×C3×C4×C5—21no—
78C2×C3×C4×C5—21no—
79C2×C3×C4×C5—42no—
80C2×C3×C4×C5—41no—
81C2×C3×C4×C5C2×D412yes—
82C2×C3×C4×C5C2×D411yescharacteristic
83C2×C3×Dic5—22no—
84C3×D20—42no—
85C3×D20—42no—
86C3×D20—81no—
87C3×C4×D5—44no—
88C3×C5×C8—41no—
89C3×#168—42no—
90C3×2−1+4C2×D511yescharacteristic
91C3×#137—21no—
92C3×AGL(1,ℤ/8)—102no—
93C3×C4⋊Q8C2×D511yescharacteristic
94C3×#124—52no—
95C3×#124—101no—
96C3×#121—102no—
97C3×C4×D4—101no—
98C3×#117—102no—
99C3×#104—21no—
100C3×#104—102no—
101C3×#101—52no—
102C3×#101C2×D511yescharacteristic
103C5×#41—21no—
104C5×#41—22no—
105C5×#41—41no—
106C2×C5×Q8—41no—
107C2×C5×Q8C3×D411yescharacteristic
108C2×C5×Q8C23×C311yescharacteristic
109C2×D20—21no—
110C2×D20—21no—
111C2×D20C3×D412yes—
112C2×C4×D5—22no—
113C5×Q16—41no—
114C5×SD16—41no—
115C5×M16—21no—
116C5×C4⋊C4—22no—
117C42×C5C3×D411yescharacteristic
118#688—42no—
119#686—42no—
120#685—22no—
121#685—22no—
122#685—22no—
123C4⋊Dic5—21no—
124C4×Dic5—21no—
125C4×Dic5C3×D412yes—
126#681—22no—
127#555—51no—
128#456C3×D511yescharacteristic, complemented
129#454—52no—
130#356—51no—
131#355—52no—
132C22×C3×C5—41no—
133C22×C3×C5#12011yescharacteristic
134C2×C3×D5—42no—
135C2×C3×D5—42no—
136C2×C3×D5—42no—
137C3×C4×C5—22no—
138C3×C4×C5—22no—
139C3×C4×C5—21no—
140C3×C4×C5—41no—
141C3×C4×C5—41no—
142C3×Dic5—42no—
143C3×#41—22no—
144C3×#41—21no—
145C3×#41—41no—
146C2×C3×Q8—41no—
147C2×C3×Q8C22⋊D511yescharacteristic
148C2×C3×Q8C22×D511yescharacteristic
149C2×C3×D4—52no—
150C2×C3×D4—101no—
151C2×C3×D4—101no—
152C22×C3×C4—102no—
153C3×Q16—41no—
154C3×SD16—41no—
155C3×SD16—202no—
156C3×D8—202no—
157C3×M16—21no—
158C3×M16—102no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—101no—
161C3×C22⋊C4—102no—
162C3×C22⋊C4—102no—
163C3×C22⋊C4—102no—
164C3×C42—52no—
165C3×C42—101no—
166C3×C42C22⋊D511yescharacteristic
167C22×D5—22no—
168C5×Q8—22no—
169C5×Q8—21no—
170C5×Q8—41no—
171C5×Q8—41no—
172C5×D4—22no—
173C5×D4—21no—
174C5×D4—41no—
175C2×C4×C5—21no—
176C2×C4×C5—21no—
177C2×C4×C5—42no—
178C2×C4×C5—41no—
179C2×C4×C5C2×C3×D412yes—
180C2×C4×C5C2×C3×D411yescharacteristic
181C2×Dic5—22no—
182D20—42no—
183D20—42no—
184D20—81no—
185C4×D5—44no—
186C5×C8—41no—
187#168—42no—
1882−1+4C2×C3×D511yescharacteristic
189#137—21no—
190AGL(1,ℤ/8)—102no—
191C4⋊Q8C2×C3×D511yescharacteristic
192#124—52no—
193#124—101no—
194#121—102no—
195C4×D4—101no—
196#117—102no—
197#104—21no—
198#104—102no—
199#101—52no—
200#101C2×C3×D511yescharacteristic
201C2×C3×C5—21no—
202C2×C3×C5—81no—
203C2×C3×C5B(4,𝔽2)11yescharacteristic
204C3×D5—82no—
205C23×C3—102no—
206C3×Q8—22no—
207C3×Q8—21no—
208C3×Q8—41no—
209C3×Q8—41no—
210C3×D4—22no—
211C3×D4—21no—
212C3×D4—41no—
213C3×D4—202no—
214C3×D4—202no—
215C3×D4—401no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—102no—
221C2×C3×C4—204no—
222C2×C3×C4D4×D512yes—
223C2×C3×C4C2×C22⋊D511yescharacteristic
224C3×C8—41no—
225C3×C8—202no—
226C22×C5—41no—
227C22×C5C3×#12011yescharacteristic
228C2×D5—42no—
229C2×D5—42no—
230C2×D5—42no—
231C4×C5—22no—
232C4×C5—22no—
233C4×C5—21no—
234C4×C5—41no—
235C4×C5—41no—
236Dic5—42no—
237#41—22no—
238#41—21no—
239#41—41no—
240C2×Q8—41no—
241C2×Q8C3×C22⋊D511yescharacteristic
242C2×Q8C22×C3×D511yescharacteristic
243C2×D4—52no—
244C2×D4—101no—
245C2×D4—101no—
246C22×C4—102no—
247Q16—41no—
248SD16—41no—
249SD16—202no—
250D8—202no—
251M16—21no—
252M16—102no—
253C4⋊C4—22no—
254C4⋊C4—101no—
255C22⋊C4—102no—
256C22⋊C4—102no—
257C22⋊C4—102no—
258C42—52no—
259C42—101no—
260C42C3×C22⋊D511yescharacteristic
261C3×C5#220411yescharacteristic, complemented, Hall
262C22×C3—41no—
263C22×C3—202no—
264C22×C3—202no—
265C22×C3—202no—
266C22×C3#408311yescharacteristic
267C3×C4—22no—
268C3×C4—21no—
269C3×C4—22no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—202no—
273C2×C5—21no—
274C2×C5—81no—
275C2×C5C3×B(4,𝔽2)11yescharacteristic
276D5—82no—
277C23—102no—
278Q8—22no—
279Q8—21no—
280Q8—41no—
281Q8—41no—
282D4—22no—
283D4—21no—
284D4—41no—
285D4—202no—
286D4—202no—
287D4—401no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—102no—
293C2×C4—204no—
294C2×C4C3×D4×D512yes—
295C2×C4C2×C3×C22⋊D511yescharacteristic
296C8—41no—
297C8—202no—
298C2×C3—21no—
299C2×C3—81no—
300C2×C3—402no—
301C2×C3#6532111yescharacteristic
302C5C3×#220411yescharacteristic, complemented, Sylow
303C22—41no—
304C22—202no—
305C22—202no—
306C22—202no—
307C22C3×#408311yescharacteristic
308C4—22no—
309C4—21no—
310C4—22no—
311C4—41no—
312C4—41no—
313C4—202no—
314C3#1060178711yescharacteristic, direct product factor, Sylow
315C2—21no—
316C2—81no—
317C2—402no—
318C2C3×#6532111yescharacteristic
subgroupG/H
centerC2×C3#65321
commutator, cosocleC2×C5×Q8C23×C3
FrattiniC2×Q8C22×C3×D5
FittingC3×C5×#456C2
socleC2×C3×C5B(4,𝔽2)
hypercenterC3×#456D5
hypocenterC5C3×#2204
LCSG ▷ C2×C5×Q8 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×#456

Complements

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

NH ≅ G/Ntype
1. C3×#66068C2semidirect
2. C3×C5×#456C2semidirect
3. C3×#65534C2semidirect
4. C3×#64768C2semidirect
5. C3×#64697C2semidirect
6. #10601787C3direct (N × H)
21. #66068C2×C3semidirect
22. C5×#456C2×C3semidirect
23. #65534C2×C3semidirect
24. #64768C2×C3semidirect
25. #64697C2×C3semidirect
51. C3×#456D5semidirect
128. #456C3×D5semidirect
261. C3×C5#2204semidirect
302. C5C3×#2204semidirect
314. C3#10601787direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660682
C3×C5×#4562
C3×#655342
C3×#647682
C3×#646972
#106017873
C3×#22045