Finite Group Databasebeta

← #13087518

Subgroups — #13087518

Order 1440.

310 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 62 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
2#10624205C211yesmaximal, characteristic
3C3×#6717C211yesmaximal, characteristic, complemented
4C3×C5×#215C211yesmaximal, characteristic, complemented
5#10623805C211yesmaximal, characteristic, complemented
6#10623803C211yesmaximal, characteristic, complemented
7#10623794C211yesmaximal, characteristic, complemented
8SD16×D15—31nomaximal
9#91505—31nomaximal
10C2×D3×D15—21no—
11C32×C5×Q8C2211yescharacteristic, complemented
12C4×C3⋊D15—21no—
13C4×C3⋊D15C2211yescharacteristic
14#67180—21no—
15#67180C2211yescharacteristic, complemented
16C3×D60C2211yescharacteristic
17C3×C5×D12C2211yescharacteristic
18#67150—21no—
19C3×C3⋊#168C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21#63940—51nomaximal, Hall
22Q8×D15—31no—
23Q8×D15—31no—
24Q8×D15—61no—
25D4×D15—31no—
26#6777—31no—
27C3×C5×SD16—31no—
28#6717D311yescharacteristic, complemented
29#6716—31no—
30#6706—31no—
31C8×D15—31no—
32C5×#215D311yescharacteristic, complemented
33C3×#688—31no—
34#6659—31no—
35#6657—31no—
36#6648—31no—
37C2×C3⋊D15D411yescharacteristic
38C2×C3×D15—21no—
39C2×C3×C5×D3—21no—
40D3×D15—42no—
41C32×C4×C5—21no—
42C32×C4×C5C2311yescharacteristic
43C3⋊Dic15—21no—
44C3⋊Dic15D411yescharacteristic
45D5×SD16—91noHall
46Q8×C3⋊D3—51no—
47#3802—51no—
48C3×#215—51no—
49C3×#215D511yescharacteristic, complemented
50#3716—51no—
51#3716—51no—
52#3710—51no—
53C22×D15—61no—
54C2×D3×D5—61no—
55C3×C5×Q8—21no—
56C3×C5×Q8C2×D311yescharacteristic, complemented
57C3×C5×Q8C2×D311yescharacteristic, complemented
58C3×C5×D4—31no—
59#1220—61no—
60D60C2×D311yescharacteristic
61C4×D15—31no—
62C4×D15—31no—
63C4×D15—61no—
64C4×D15—61no—
65C4×D15—61no—
66C4×D15—62no—
67Dic30—31no—
68Dic30—31no—
69Dic30—61no—
70Dic30—61no—
71Dic30—61no—
72Dic30—62no—
73C5×D12C2×D311yescharacteristic
74C3×D20—31no—
75#1201—61no—
76C3×C5×C8—31no—
77C3⋊#168C2×D311yescharacteristic, complemented
78C3×#168—31no—
79C5×C3⋊C8C2×D311yescharacteristic, complemented
80D3×SD16—151no—
81D3×SD16—151no—
82C2×C32×C5C2×D411yescharacteristic
83C3⋊D15SD1612yescomplemented
84C3×D15—41no—
85C3×C5×D3—41no—
86Q8×D5—91no—
87D4×D5—91no—
88C5×SD16—31no—
89#688—31no—
90#687—91no—
91#677—91no—
92C8×D5—91no—
93C2×D32—101no—
94C32×Q8C2×D511yescharacteristic, complemented
95C4×C3⋊D3—51no—
96C4×C3⋊D3—101no—
97#634—51no—
98#634—101no—
99C3×D12—51no—
100C3×D12C2×D511yescharacteristic
101#625—101no—
102C3×C3⋊C8—51no—
103C3×C3⋊C8C2×D511yescharacteristic, complemented
104C22×C3×C5—61no—
105C2×D15—21no—
106C2×D15—31no—
107C2×D15—31no—
108C2×D15—61no—
109C2×D15—122no—
110C2×C5×D3—21no—
111C2×C3×D5—61no—
112D3×D5—122no—
113C3×C4×C5—22no—
114C3×C4×C5—21no—
115C3×C4×C5—21no—
116C3×C4×C5—21no—
117C3×C4×C5C22×D311yescharacteristic
118C3×C4×C5C22×D311yescharacteristic
119Dic15—31no—
120Dic15—31no—
121Dic15—61no—
122Dic15—62no—
123Dic15—61no—
124Dic15—61no—
125D3×Q8—151no—
126D3×Q8—151no—
127D3×Q8—301no—
128D3×D4—151no—
129D3×D4—151no—
130C3×SD16—31no—
131C3×SD16—151no—
132#215—51no—
133#215D1511yescharacteristic, complemented
134#214—151no—
135#214—151no—
136#204—151no—
137#204—151no—
138D3×C8—151no—
139D3×C8—151no—
140C32×C5C2×SD1611yescharacteristic, complemented, Hall
141C22×D5—181no—
142C5×Q8D3211yescharacteristic, complemented
143C5×D4—31no—
144C22⋊D5—181no—
145D20—31no—
146C4×D5—91no—
147C4×D5—181no—
148Dic10—91no—
149Dic10—181no—
150C5×C8—31no—
151#168—31no—
152C2×C3⋊D3—51no—
153C2×C3×D3—21no—
154C2×C3×D3—101no—
155D32—202no—
156C32×C4—21no—
157C32×C4C22×D511yescharacteristic
158C3⋊Dic3—51no—
159C3⋊Dic3—101no—
160C2×SD16—451noSylow
161C2×C3×C5—21no—
162C2×C3×C5—121no—
163C2×C3×C5D3×D411yescharacteristic
164C2×C3×C5D3×D411yescharacteristic
165D15—32no—
166D15—32no—
167D15—41no—
168D15—62no—
169C3×D5—121no—
170C5×D3—41no—
171C22×D3—301no—
172C22×D3—301no—
173C3×Q8—21no—
174C3×Q8D3×D511yescharacteristic, complemented
175C3×Q8C2×D1511yescharacteristic, complemented
176C3×D4—31no—
177C3×D4—151no—
178#67—301no—
179#67—301no—
180D12—51no—
181D12C2×D1511yescharacteristic
182C4×D3—151no—
183C4×D3—151no—
184C4×D3—301no—
185C4×D3—301no—
186C4×D3—301no—
187C4×D3—302no—
188Dic6—151no—
189Dic6—151no—
190Dic6—301no—
191Dic6—301no—
192Dic6—301no—
193Dic6—302no—
194C3×C8—31no—
195C3×C8—151no—
196C3⋊C8—51no—
197C3⋊C8C2×D1511yescharacteristic, complemented
198C22×C5—61no—
199C2×D5—61no—
200C2×D5—91no—
201C2×D5—362no—
202C4×C5—21no—
203C4×C5C2×D3211yescharacteristic
204Dic5—91no—
205Dic5—181no—
206C2×C32D4×D511yescharacteristic
207C3⋊D3—52no—
208C3×D3—41no—
209C3×D3—201no—
210C2×Q8—451no—
211C2×D4—451no—
212SD16—31no—
213SD16—151no—
214SD16—451no—
215SD16—451no—
216C2×C8—451no—
217C3×C5—21no—
218C3×C5D3×SD1611yescharacteristic, complemented
219C3×C5D3×SD1611yescharacteristic, complemented
220C22×C3—61no—
221C22×C3—301no—
222C2×D3—21no—
223C2×D3—101no—
224C2×D3—151no—
225C2×D3—151no—
226C2×D3—301no—
227C2×D3—602no—
228C2×D3—602no—
229C3×C4—22no—
230C3×C4—21no—
231C3×C4—21no—
232C3×C4—21no—
233C3×C4C2×D3×D511yescharacteristic
234C3×C4C22×D1511yescharacteristic
235Dic3—151no—
236Dic3—151no—
237Dic3—301no—
238Dic3—302no—
239Dic3—301no—
240Dic3—301no—
241C2×C5—121no—
242C2×C5#381111yescharacteristic
243D5—92no—
244D5—121no—
245C32D5×SD1611yescharacteristic, complemented, Sylow
246C23—901no—
247Q8—451no—
248Q8—901no—
249Q8D3×D1511yescharacteristic, complemented
250D4—31no—
251D4—151no—
252D4—901no—
253C2×C4—451no—
254C2×C4—901no—
255C8—31no—
256C8—151no—
257C2×C3—21no—
258C2×C3—121no—
259C2×C3—601no—
260C2×C3#679011yescharacteristic
261C2×C3D4×D1511yescharacteristic
262D3—41no—
263D3—152no—
264D3—152no—
265D3—201no—
266D3—302no—
267C5#6394011yescharacteristic, complemented, Sylow
268C22—61no—
269C22—301no—
270C22—451no—
271C22—1802no—
272C4—21no—
273C4—451no—
274C4—901no—
275C4C2×D3×D1511yescharacteristic
276C3—21no—
277C3#9150511yescharacteristic, complemented
278C3SD16×D1511yescharacteristic, complemented
279C2—121no—
280C2—452no—
281C2—601no—
282C2#1062421811yescharacteristic
subgroupG/H
centerC2#10624218
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×D3×D15
FittingC32×C5×Q8C22
socleC2×C32×C5C2×D4
prime radical#10624205C2
hypercenterQ8D3×D15
hypocenterC32×C5C2×SD16
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C3⋊D15C2semidirect
3. C3×#6717C2semidirect
4. C3×C5×#215C2semidirect
5. #10623805C2semidirect
6. #10623803C2semidirect
7. #10623794C2semidirect
11. C32×C5×Q8C22semidirect
15. #67180C22semidirect
19. C3×C3⋊#168C22semidirect
20. C3×C5×C3⋊C8C22semidirect
28. #6717D3semidirect
32. C5×#215D3semidirect
49. C3×#215D5semidirect
56. C3×C5×Q8C2×D3semidirect
57. C3×C5×Q8C2×D3semidirect
77. C3⋊#168C2×D3semidirect
79. C5×C3⋊C8C2×D3semidirect
83. C3⋊D15SD16semidirect
94. C32×Q8C2×D5semidirect
103. C3×C3⋊C8C2×D5semidirect
133. #215D15semidirect
140. C32×C5C2×SD16semidirect
142. C5×Q8D32semidirect
174. C3×Q8D3×D5semidirect
175. C3×Q8C2×D15semidirect
197. C3⋊C8C2×D15semidirect
218. C3×C5D3×SD16semidirect
219. C3×C5D3×SD16semidirect
245. C32D5×SD16semidirect
249. Q8D3×D15semidirect
267. C5#63940semidirect
277. C3#91505semidirect
278. C3SD16×D15semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊D152
#106242052
C3×#67172
C3×C5×#2152
#106238052
#106238032
#106237942
SD16×D153
#915053
#639405