Finite Group Databasebeta

← C3×#91785 #13087985

Subgroups — C3×#91785 #13087985

Order 1440.

322 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×C3×C4×D15C211yesmaximal, characteristic
2C2×C3×Dic30C211yesmaximal, characteristic, complemented
3C32×C5×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#6714C211yesmaximal, characteristic
5C3×#6714C211yesmaximal, characteristic
6C3×#6712C211yesmaximal, characteristic, complemented
7C3×#6712C211yesmaximal, characteristic, complemented
8#91785C311yesmaximal, characteristic, direct product factor
9C3×#4042—31nomaximal
10C22×C3×D15C2211yescharacteristic
11C2×C32×C4×C5C2211yescharacteristic
12C2×C32×C4×C5C2211yescharacteristic
13C2×C32×C4×C5C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×Dic15C2211yescharacteristic
17C3×C4×D15—22no—
18C3×Dic30—22no—
19C3×#896—51nomaximal, Hall
20C2×C4×D15C2×C311yescharacteristic
21C2×Dic30C2×C311yescharacteristic, complemented
22C2×C3×C4×D5—31no—
23C2×C3×Dic10—31no—
24C3×C5×C4⋊C4—21no—
25C3×C5×C4⋊C4D311yescharacteristic, complemented
26C3×C5×C4⋊C4C2×C311yescharacteristic, complemented
27#6714C2×C311yescharacteristic
28#6714C2×C311yescharacteristic
29#6712C2×C311yescharacteristic, complemented
30#6712C2×C311yescharacteristic, complemented
31C3×#685—31no—
32C3×#685—31no—
33C3×#683—31no—
34C3×#683—31no—
35C22×C32×C5C2311yescharacteristic
36C2×C3×D15—21no—
37C2×C3×D15—21no—
38C2×C3×D15Q812yescomplemented
39C32×C4×C5—21no—
40C32×C4×C5—21no—
41C32×C4×C5—21no—
42C3×Dic15—21no—
43C3×Dic15—21no—
44C3×Dic15D412yes—
45#4042—31noHall
46C2×C3×C4×D3—51no—
47C2×C3×Dic6—51no—
48C32×C4⋊C4D511yescharacteristic, complemented
49C3×(C2×D3)⋊C4—51no—
50C3×(C2×D3)⋊C4—51no—
51C3×Dic3⋊C4—51no—
52C3×Dic3⋊C4—51no—
53C22×D15C22×C311yescharacteristic
54C22×C3×D5—31no—
55C2×C3×C4×C5—21no—
56C2×C3×C4×C5—21no—
57C2×C3×C4×C5—21no—
58C2×C3×C4×C5C2×D311yescharacteristic
59C2×C3×C4×C5C2×D311yescharacteristic
60C2×C3×C4×C5C2×D311yescharacteristic
61C2×C3×C4×C5C22×C311yescharacteristic
62C2×C3×C4×C5C22×C311yescharacteristic
63C2×C3×C4×C5C22×C311yescharacteristic
64C2×Dic15C22×C311yescharacteristic
65C2×Dic15C22×C311yescharacteristic
66C2×Dic15C22×C311yescharacteristic
67C4×D15—22no—
68Dic30—22no—
69C2×C3×Dic5—31no—
70C2×C3×Dic5—31no—
71C2×C3×Dic5—31no—
72C3×C4×D5—62no—
73C3×Dic10—62no—
74C3×C22⋊Q8—151no—
75#896—51no—
76C2×C32×C5C2×D411yescharacteristic
77C2×C32×C5C2×Q811yescharacteristic
78C2×C32×C5#4111yescharacteristic
79C3×D15—22no—
80C2×C4×D5—31no—
81C2×Dic10—31no—
82C5×C4⋊C4C3×D311yescharacteristic, complemented
83#685—31no—
84#685—31no—
85#683—31no—
86#683—31no—
87C22×C3×D3—51no—
88C2×C32×C4C2×D511yescharacteristic
89C2×C32×C4C2×D511yescharacteristic
90C2×C32×C4C2×D511yescharacteristic
91C2×C3×Dic3—51no—
92C2×C3×Dic3—51no—
93C2×C3×Dic3—51no—
94C3×C4×D3—102no—
95C3×Dic6—102no—
96C22×C3×C5—21no—
97C22×C3×C5C22×D311yescharacteristic
98C22×C3×C5C23×C311yescharacteristic
99C2×D15—21no—
100C2×D15—21no—
101C2×D15C3×Q812yescomplemented
102C2×C3×D5—32no—
103C2×C3×D5—61no—
104C2×C3×D5—61no—
105C3×C4×C5—22no—
106C3×C4×C5—21no—
107C3×C4×C5—21no—
108C3×C4×C5—21no—
109C3×C4×C5—21no—
110C3×C4×C5—21no—
111C3×C4×C5—21no—
112C3×C4×C5—41no—
113C3×C4×C5—41no—
114Dic15—21no—
115Dic15—21no—
116Dic15C3×D412yes—
117C3×Dic5—32no—
118C3×Dic5—61no—
119C3×Dic5—61no—
120C2×C3×Q8—151no—
121C22×C3×C4—151no—
122C2×C4×D3—51no—
123C2×Dic6—51no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—151no—
126C3×C4⋊C4—151no—
127C3×C4⋊C4C3×D511yescharacteristic, complemented
128C3×C4⋊C4D1511yescharacteristic, complemented
129C3×C22⋊C4—151no—
130C3×C22⋊C4—151no—
131(C2×D3)⋊C4—51no—
132(C2×D3)⋊C4—51no—
133Dic3⋊C4—51no—
134Dic3⋊C4—51no—
135C32×C5C22⋊Q811yescharacteristic, complemented, Hall
136C22×D5—31no—
137C2×C4×C5C2×C3×D311yescharacteristic
138C2×C4×C5C2×C3×D311yescharacteristic
139C2×C4×C5C2×C3×D311yescharacteristic
140C2×Dic5—31no—
141C2×Dic5—31no—
142C2×Dic5—31no—
143C4×D5—62no—
144Dic10—62no—
145C22×C32C22×D511yescharacteristic
146C2×C3×D3—52no—
147C2×C3×D3—101no—
148C2×C3×D3—101no—
149C32×C4—21no—
150C32×C4—21no—
151C32×C4—21no—
152C3×Dic3—52no—
153C3×Dic3—101no—
154C3×Dic3—101no—
155C22⋊Q8—151noSylow
156C2×C3×C5—21no—
157C2×C3×C5—21no—
158C2×C3×C5—21no—
159C2×C3×C5#23511yescharacteristic
160C2×C3×C5D3×D411yescharacteristic
161C2×C3×C5D3×Q811yescharacteristic
162C2×C3×C5C2×C3×D411yescharacteristic
163C2×C3×C5C2×C3×Q811yescharacteristic
164C2×C3×C5C3×#4111yescharacteristic
165D15—22no—
166C3×D5—62no—
167C23×C3—151no—
168C22×D3—51no—
169C3×Q8—302no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—151no—
174C2×C3×C4—151no—
175C2×C3×C4—151no—
176C2×C3×C4—302no—
177C2×C3×C4C2×C3×D511yescharacteristic
178C2×C3×C4C2×C3×D511yescharacteristic
179C2×C3×C4C2×C3×D511yescharacteristic
180C2×C3×C4C2×D1511yescharacteristic
181C2×C3×C4C2×D1511yescharacteristic
182C2×C3×C4C2×D1511yescharacteristic
183C2×Dic3—51no—
184C2×Dic3—51no—
185C2×Dic3—51no—
186C4×D3—102no—
187Dic6—102no—
188C22×C5C22×C3×D311yescharacteristic
189C2×D5—32no—
190C2×D5—61no—
191C2×D5—61no—
192C4×C5—21no—
193C4×C5—21no—
194C4×C5—21no—
195Dic5—32no—
196Dic5—61no—
197Dic5—61no—
198C2×C32#70911yescharacteristic
199C2×C32D4×D511yescharacteristic
200C2×C32Q8×D511yescharacteristic
201C3×D3—102no—
202C2×Q8—151no—
203C22×C4—151no—
204C4⋊C4—151no—
205C4⋊C4—151no—
206C4⋊C4C3×D1511yescharacteristic, complemented
207C22⋊C4—151no—
208C22⋊C4—151no—
209C3×C5—21no—
210C3×C5#89611yescharacteristic, complemented
211C3×C5C3×C22⋊Q811yescharacteristic, complemented
212C22×C3—21no—
213C22×C3—152no—
214C22×C3—301no—
215C22×C3—301no—
216C22×C3C22×C3×D511yescharacteristic
217C22×C3C22×D1511yescharacteristic
218C2×D3—52no—
219C2×D3—101no—
220C2×D3—101no—
221C3×C4—22no—
222C3×C4—21no—
223C3×C4—21no—
224C3×C4—21no—
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—21no—
228C3×C4—41no—
229C3×C4—41no—
230C3×C4—152no—
231C3×C4—301no—
232C3×C4—301no—
233Dic3—52no—
234Dic3—101no—
235Dic3—101no—
236C2×C5C3×#23511yescharacteristic
237C2×C5C3×D3×D411yescharacteristic
238C2×C5C3×D3×Q811yescharacteristic
239D5—62no—
240C32#404211yescharacteristic, complemented, Sylow
241C23—151no—
242Q8—302no—
243C2×C4—151no—
244C2×C4—151no—
245C2×C4—151no—
246C2×C4—302no—
247C2×C4C2×C3×D1511yescharacteristic
248C2×C4C2×C3×D1511yescharacteristic
249C2×C4C2×C3×D1511yescharacteristic
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—302no—
254C2×C3C3×#70911yescharacteristic
255C2×C3C3×D4×D511yescharacteristic
256C2×C3C3×Q8×D511yescharacteristic
257C2×C3#681711yescharacteristic
258C2×C3D4×D1511yescharacteristic
259C2×C3Q8×D1511yescharacteristic
260D3—102no—
261C5C3×#89611yescharacteristic, complemented, Sylow
262C22—152no—
263C22—301no—
264C22—301no—
265C22C22×C3×D1511yescharacteristic
266C4—21no—
267C4—21no—
268C4—21no—
269C4—152no—
270C4—301no—
271C4—301no—
272C3—21no—
273C3C3×#404211yescharacteristic, complemented
274C3#9178511yescharacteristic, direct product factor
275C2—302no—
276C2C3×#681711yescharacteristic
277C2C3×D4×D1511yescharacteristic
278C2C3×Q8×D1511yescharacteristic
subgroupG/H
centerC22×C3C22×D15
commutator, cosocleC22×C3×C5C23×C3
FrattiniC22C22×C3×D15
FittingC32×C5×C4⋊C4C2
socleC22×C32×C5C23
prime radicalC22×C3×D15C22
hypercenterC3×C4⋊C4D15
hypocenterC3×C5C3×C22⋊Q8
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×Dic30C2semidirect
3. C32×C5×C4⋊C4C2semidirect
6. C3×#6712C2semidirect
7. C3×#6712C2semidirect
8. #91785C3direct (N × H)
21. C2×Dic30C2×C3semidirect
25. C3×C5×C4⋊C4D3semidirect
26. C3×C5×C4⋊C4C2×C3semidirect
29. #6712C2×C3semidirect
30. #6712C2×C3semidirect
38. C2×C3×D15Q8semidirect
48. C32×C4⋊C4D5semidirect
82. C5×C4⋊C4C3×D3semidirect
101. C2×D15C3×Q8semidirect
127. C3×C4⋊C4C3×D5semidirect
128. C3×C4⋊C4D15semidirect
135. C32×C5C22⋊Q8semidirect
206. C4⋊C4C3×D15semidirect
210. C3×C5#896semidirect
211. C3×C5C3×C22⋊Q8semidirect
240. C32#4042semidirect
261. C5C3×#896semidirect
273. C3C3×#4042semidirect
274. C3#91785direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×D152
C2×C3×Dic302
C32×C5×C4⋊C42
C3×#67142
C3×#67122
#917853
C3×#40423
C3×#8965