Finite Group Databasebeta

← #13089377

Subgroups — #13089377

Order 1440.

622 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 154 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
1Q8×C3⋊D15C211yesmaximal, characteristic, complemented
2C3×Q8×D15C211yesmaximal, characteristic, complemented
3C3×C5×#239C211yesmaximal, characteristic, complemented
4#10624200C213yesmaximal, complemented
5#10624199C213yesmaximal, complemented
6#10624196C213yesmaximal, complemented
7D3×Dic30C213yesmaximal, complemented
8#92101—31nomaximal
9#92030—31nomaximal
10C32×C5×Q8C2211yescharacteristic
11C4×C3⋊D15C2213yes—
12#67180C2213yes—
13C3×C4×D15C2213yes—
14C3×Dic30C2213yes—
15C3×C5×D12C2213yes—
16C3×C4×C5×D3C2213yes—
17#67153C2213yes—
18#67152C2213yes—
19#67151C2211yescharacteristic
20#67150C2213yes—
21D3×Dic15C2216yes—
22#64315—51nomaximal, Hall
23C3×C5×#41—31no—
24Q8×D15—31no—
25Q8×D15—31no—
26Q8×D15—61no—
27Q8×D15D311yescharacteristic, complemented
28#6819—33no—
29#6819—33no—
30#6817—33no—
31C2×Dic30—33no—
32C5×#239D311yescharacteristic, complemented
33C3×Q8×D5—31no—
34#6772—33no—
35#6771—33no—
36#6768—33no—
37D3×Dic10—33no—
38C2×C3⋊D15C2311yescharacteristic
39C2×C3×D15C2311yescharacteristic
40C2×C3×C5×D3C2313yes—
41C32×C4×C5C2313yes—
42C3⋊Dic15C2313yes—
43C3×Dic15C2313yes—
44C3×C5×Dic3C2311yescharacteristic
45#4150—91noHall
46Q8×C3⋊D3—51no—
47C3×#239D511yescharacteristic, complemented
48C3×D3×Q8—51no—
49#3798—53no—
50#3796—53no—
51#3795—53no—
52D3×Dic6—53no—
53C3×C5×Q8—21no—
54C3×C5×Q8C2×D311yescharacteristic
55C3×C5×Q8C2×D311yescharacteristic
56C3×C5×D4—33no—
57C2×C3×C4×C5—33no—
58#1220—33no—
59#1220—33no—
60C2×Dic15—36no—
61D60—31no—
62C4×D15—33no—
63C4×D15—33no—
64C4×D15—63no—
65C4×D15C2×D313yes—
66Dic30—33no—
67Dic30—33no—
68Dic30—33no—
69Dic30—63no—
70Dic30C2×D313yes—
71C5×D12C2×D313yes—
72C4×C5×D3C2×D313yes—
73C3×C4×D5—33no—
74C3×Dic10—33no—
75#1204—33no—
76#1203—33no—
77#1202—31no—
78#1201—33no—
79D3×Dic5—36no—
80#1010—151no—
81#1007—151no—
82C2×C32×C5C2411yescharacteristic
83C3⋊D15—21no—
84C3×D15—21no—
85C3×C5×D3—23no—
86C5×#41—31no—
87Q8×D5—31no—
88Q8×D5—91no—
89#711—93no—
90#711—93no—
91#709—93no—
92C2×Dic10—93no—
93C32×Q8C2×D511yescharacteristic
94C4×C3⋊D3—53no—
95#634—53no—
96C3×D12C2×D513yes—
97C3×C4×D3—53no—
98C3×C4×D3C2×D513yes—
99C3×Dic6—53no—
100#627—53no—
101#626—53no—
102#626—51no—
103#625—53no—
104D3×Dic3—56no—
105C22×C3×C5—33no—
106C2×D15—31no—
107C2×D15—31no—
108C2×D15—61no—
109C2×D15C22×D311yescharacteristic
110C2×C5×D3C22×D313yes—
111C2×C3×D5—31no—
112C3×C4×C5—23no—
113C3×C4×C5—31no—
114C3×C4×C5C22×D313yes—
115C3×C4×C5C22×D313yes—
116Dic15—33no—
117Dic15—33no—
118Dic15—63no—
119Dic15C22×D313yes—
120C3×Dic5—33no—
121C5×Dic3C22×D311yescharacteristic
122C3×#41—31no—
123C2×C3×Q8—151no—
124#239—153no—
125#239D1511yescharacteristic, complemented
126D3×Q8—51no—
127D3×Q8—151no—
128D3×Q8—151no—
129D3×Q8—153no—
130D3×Q8—301no—
131#237—153no—
132#237—153no—
133#235—153no—
134#235—153no—
135#235—153no—
136C2×Dic6—153no—
137C32×C52−1+411yescharacteristic, complemented, Hall
138C5×Q8D3211yescharacteristic
139C5×D4—33no—
140C2×C4×C5—33no—
141C22⋊D5—93no—
142C22⋊D5—93no—
143C2×Dic5—96no—
144D20—91no—
145C4×D5—33no—
146C4×D5—93no—
147Dic10—33no—
148Dic10—93no—
149Dic10—93no—
150C2×C3⋊D3—51no—
151C2×C3×D3—51no—
152C2×C3×D3C22×D513yes—
153C32×C4C22×D513yes—
154C3⋊Dic3—53no—
155C3×Dic3—53no—
156C3×Dic3C22×D511yescharacteristic
1572−1+4—451noSylow
158C2×C3×C5—21no—
159C2×C3×C5—63no—
160C2×C3×C5C23×D311yescharacteristic
161C2×C3×C5C23×D311yescharacteristic
162D15—21no—
163D15—62no—
164D15—61no—
165D15—61no—
166C3×D5—61no—
167C5×D3—23no—
168C3×Q8—21no—
169C3×Q8—153no—
170C3×Q8D3×D511yescharacteristic
171C3×Q8C2×D1511yescharacteristic
172C3×D4—33no—
173C2×C3×C4—33no—
174C2×C3×C4—153no—
175#67—153no—
176#67—153no—
177#67—153no—
178#67—151no—
179C2×Dic3—156no—
180D12—151no—
181D12—153no—
182D12C2×D1513yes—
183C4×D3—53no—
184C4×D3—156no—
185C4×D3—153no—
186C4×D3—153no—
187C4×D3—303no—
188C4×D3C2×D1513yes—
189Dic6—53no—
190Dic6—153no—
191Dic6—153no—
192Dic6—153no—
193Dic6—153no—
194Dic6—303no—
195C22×C5—33no—
196C2×D5—31no—
197C2×D5—91no—
198C4×C5—31no—
199C4×C5C2×D3213yes—
200Dic5—33no—
201Dic5—93no—
202C2×C32C23×D511yescharacteristic
203C3⋊D3—101no—
204C3×D3—23no—
205C3×D3—101no—
206#41—31no—
207#41—453no—
208#41—453no—
209#41—453no—
210C2×Q8—151no—
211C2×Q8—453no—
212C2×Q8—451no—
213C3×C5—21no—
214C3×C5#100711yescharacteristic, complemented
215C3×C5#101011yescharacteristic, complemented
216C22×C3—33no—
217C22×C3—151no—
218C2×D3—51no—
219C2×D3—151no—
220C2×D3—151no—
221C2×D3—301no—
222C2×D3C22×D1513yes—
223C3×C4—23no—
224C3×C4—31no—
225C3×C4—153no—
226C3×C4C2×D3×D513yes—
227C3×C4C22×D1513yes—
228Dic3—53no—
229Dic3—153no—
230Dic3—153no—
231Dic3—303no—
232Dic3C22×D1511yescharacteristic
233C2×C5—63no—
234C2×C5C22×D3211yescharacteristic
235D5—61no—
236D5—181no—
237C32#415011yescharacteristic, complemented, Sylow
238Q8—153no—
239Q8—453no—
240Q8—453no—
241Q8D3×D1511yescharacteristic
242D4—33no—
243D4—451no—
244D4—453no—
245D4—453no—
246C2×C4—33no—
247C2×C4—153no—
248C2×C4—456no—
249C2×C4—453no—
250C2×C3—21no—
251C2×C3—63no—
252C2×C3—301no—
253C2×C3C22×D3×D511yescharacteristic
254C2×C3C23×D1511yescharacteristic
255D3—23no—
256D3—101no—
257D3—302no—
258D3—301no—
259D3—301no—
260C5#6431511yescharacteristic, complemented, Sylow
261C22—33no—
262C22—151no—
263C22—451no—
264C4—31no—
265C4—153no—
266C4—453no—
267C4C2×D3×D1513yes—
268C3—21no—
269C3#9203011yescharacteristic, complemented
270C3#9210111yescharacteristic, complemented
271C2—63no—
272C2—301no—
273C2—901no—
274C2C22×D3×D1511yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×D15
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C5×Q8C22
hypercenterQ8D3×D15
hypocenterC32×C52−1+4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C3⋊D15C2semidirect
2. C3×Q8×D15C2semidirect
3. C3×C5×#239C2semidirect
4. #10624200C2semidirect
5. #10624199C2semidirect
6. #10624196C2semidirect
7. D3×Dic30C2semidirect
27. Q8×D15D3semidirect
32. C5×#239D3semidirect
47. C3×#239D5semidirect
125. #239D15semidirect
137. C32×C52−1+4semidirect
214. C3×C5#1007semidirect
215. C3×C5#1010semidirect
237. C32#4150semidirect
260. C5#64315semidirect
269. C3#92030semidirect
270. C3#92101semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊D152
C3×Q8×D152
C3×C5×#2392
#106242002
#106241992
#106241962
D3×Dic302
#921013
#920303
#643155