Finite Group Databasebeta

← C3×#91783 #13087992

Subgroups — C3×#91783 #13087992

Order 1440.

338 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 98 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×D60C211yesmaximal, characteristic
2C2×C3×C4×D15C211yesmaximal, characteristic, complemented
3C32×C5×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#6714C211yesmaximal, characteristic, complemented
5C3×#6714C211yesmaximal, characteristic, complemented
6C3×#6714C211yesmaximal, characteristic, complemented
7C3×#6712C211yesmaximal, characteristic, complemented
8#91783C311yesmaximal, characteristic, direct product factor
9C3×#4040—31nomaximal
10C22×C3×D15C2211yescharacteristic
11C22×C3×D15C2211yescharacteristic
12C2×C32×C4×C5C2211yescharacteristic
13C2×C32×C4×C5C2211yescharacteristic
14C2×C32×C4×C5C2211yescharacteristic
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×Dic15C2211yescharacteristic
17C3×D60—22no—
18C3×C4×D15—22no—
19C3×#894—51nomaximal, Hall
20C2×D60C2×C311yescharacteristic
21C2×C4×D15C2×C311yescharacteristic, complemented
22C2×C3×D20—31no—
23C2×C3×C4×D5—31no—
24C3×C5×C4⋊C4—21no—
25C3×C5×C4⋊C4D311yescharacteristic, complemented
26C3×C5×C4⋊C4C2×C311yescharacteristic, complemented
27#6714C2×C311yescharacteristic, complemented
28#6714C2×C311yescharacteristic, complemented
29#6714C2×C311yescharacteristic, complemented
30#6712C2×C311yescharacteristic, complemented
31C3×#685—31no—
32C3×#685—31no—
33C3×#685—31no—
34C3×#683—31no—
35C22×C32×C5C2311yescharacteristic
36C2×C3×D15—21no—
37C2×C3×D15—21no—
38C2×C3×D15—21no—
39C2×C3×D15—21no—
40C2×C3×D15—21no—
41C2×C3×D15D412yes—
42C32×C4×C5—21no—
43C32×C4×C5—21no—
44C32×C4×C5—21no—
45C3×Dic15—21no—
46C3×Dic15—21no—
47#4040—31noHall
48C2×C3×D12—51no—
49C2×C3×C4×D3—51no—
50C32×C4⋊C4D511yescharacteristic, complemented
51C3×(C2×D3)⋊C4—51no—
52C3×(C2×D3)⋊C4—51no—
53C3×(C2×D3)⋊C4—51no—
54C3×Dic3⋊C4—51no—
55C22×D15C22×C311yescharacteristic
56C22×D15C22×C311yescharacteristic
57C22×C3×D5—31no—
58C22×C3×D5—31no—
59C2×C3×C4×C5—21no—
60C2×C3×C4×C5—21no—
61C2×C3×C4×C5—21no—
62C2×C3×C4×C5C2×D311yescharacteristic
63C2×C3×C4×C5C2×D311yescharacteristic
64C2×C3×C4×C5C2×D311yescharacteristic
65C2×C3×C4×C5C22×C311yescharacteristic
66C2×C3×C4×C5C22×C311yescharacteristic
67C2×C3×C4×C5C22×C311yescharacteristic
68C2×Dic15C22×C311yescharacteristic
69C2×Dic15C22×C311yescharacteristic
70D60—22no—
71C4×D15—22no—
72C2×C3×Dic5—31no—
73C2×C3×Dic5—31no—
74C3×D20—62no—
75C3×C4×D5—62no—
76C3×#123—151no—
77#894—51no—
78C2×C32×C5C2×D411yescharacteristic
79C2×C32×C5#4111yescharacteristic
80C2×C32×C5#4111yescharacteristic
81C3×D15—22no—
82C3×D15—41no—
83C2×D20—31no—
84C2×C4×D5—31no—
85C5×C4⋊C4C3×D311yescharacteristic, complemented
86#685—31no—
87#685—31no—
88#685—31no—
89#683—31no—
90C22×C3×D3—51no—
91C22×C3×D3—51no—
92C2×C32×C4C2×D511yescharacteristic
93C2×C32×C4C2×D511yescharacteristic
94C2×C32×C4C2×D511yescharacteristic
95C2×C3×Dic3—51no—
96C2×C3×Dic3—51no—
97C3×D12—102no—
98C3×C4×D3—102no—
99C22×C3×C5—21no—
100C22×C3×C5C22×D311yescharacteristic
101C22×C3×C5C23×C311yescharacteristic
102C2×D15—21no—
103C2×D15—21no—
104C2×D15—21no—
105C2×D15—21no—
106C2×D15—21no—
107C2×D15C3×D412yes—
108C2×C3×D5—32no—
109C2×C3×D5—61no—
110C2×C3×D5—61no—
111C2×C3×D5—61no—
112C2×C3×D5—61no—
113C2×C3×D5—61no—
114C3×C4×C5—22no—
115C3×C4×C5—21no—
116C3×C4×C5—21no—
117C3×C4×C5—21no—
118C3×C4×C5—21no—
119C3×C4×C5—21no—
120C3×C4×C5—21no—
121C3×C4×C5—41no—
122C3×C4×C5—41no—
123Dic15—21no—
124Dic15—21no—
125C3×Dic5—61no—
126C3×Dic5—61no—
127C2×C3×D4—151no—
128C22×C3×C4—151no—
129C2×D12—51no—
130C2×C4×D3—51no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—151no—
133C3×C4⋊C4C3×D511yescharacteristic, complemented
134C3×C4⋊C4D1511yescharacteristic, complemented
135C3×C22⋊C4—151no—
136C3×C22⋊C4—151no—
137C3×C22⋊C4—151no—
138(C2×D3)⋊C4—51no—
139(C2×D3)⋊C4—51no—
140(C2×D3)⋊C4—51no—
141Dic3⋊C4—51no—
142C32×C5#12311yescharacteristic, complemented, Hall
143C22×D5—31no—
144C22×D5—31no—
145C2×C4×C5C2×C3×D311yescharacteristic
146C2×C4×C5C2×C3×D311yescharacteristic
147C2×C4×C5C2×C3×D311yescharacteristic
148C2×Dic5—31no—
149C2×Dic5—31no—
150D20—62no—
151C4×D5—62no—
152C22×C32C22×D511yescharacteristic
153C2×C3×D3—52no—
154C2×C3×D3—101no—
155C2×C3×D3—101no—
156C2×C3×D3—101no—
157C2×C3×D3—101no—
158C2×C3×D3—101no—
159C32×C4—21no—
160C32×C4—21no—
161C32×C4—21no—
162C3×Dic3—101no—
163C3×Dic3—101no—
164#123—151noSylow
165C2×C3×C5—21no—
166C2×C3×C5—21no—
167C2×C3×C5—21no—
168C2×C3×C5#23511yescharacteristic
169C2×C3×C5D3×D411yescharacteristic
170C2×C3×C5#23911yescharacteristic
171C2×C3×C5C2×C3×D411yescharacteristic
172C2×C3×C5C3×#4111yescharacteristic
173C2×C3×C5C3×#4111yescharacteristic
174D15—22no—
175D15—41no—
176C3×D5—62no—
177C3×D5—121no—
178C23×C3—151no—
179C23×C3—151no—
180C22×D3—51no—
181C22×D3—51no—
182C3×D4—302no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—151no—
187C2×C3×C4—151no—
188C2×C3×C4—302no—
189C2×C3×C4C2×C3×D511yescharacteristic
190C2×C3×C4C2×C3×D511yescharacteristic
191C2×C3×C4C2×C3×D511yescharacteristic
192C2×C3×C4C2×D1511yescharacteristic
193C2×C3×C4C2×D1511yescharacteristic
194C2×C3×C4C2×D1511yescharacteristic
195C2×Dic3—51no—
196C2×Dic3—51no—
197D12—102no—
198C4×D3—102no—
199C22×C5C22×C3×D311yescharacteristic
200C2×D5—32no—
201C2×D5—61no—
202C2×D5—61no—
203C2×D5—61no—
204C2×D5—61no—
205C2×D5—61no—
206C4×C5—21no—
207C4×C5—21no—
208C4×C5—21no—
209Dic5—61no—
210Dic5—61no—
211C2×C32#70911yescharacteristic
212C2×C32D4×D511yescharacteristic
213C2×C32#71311yescharacteristic
214C3×D3—102no—
215C3×D3—201no—
216C2×D4—151no—
217C22×C4—151no—
218C4⋊C4—151no—
219C4⋊C4C3×D1511yescharacteristic, complemented
220C22⋊C4—151no—
221C22⋊C4—151no—
222C22⋊C4—151no—
223C3×C5—21no—
224C3×C5#89411yescharacteristic, complemented
225C3×C5C3×#12311yescharacteristic, complemented
226C22×C3—21no—
227C22×C3—152no—
228C22×C3—301no—
229C22×C3—301no—
230C22×C3—301no—
231C22×C3—301no—
232C22×C3—301no—
233C22×C3C22×C3×D511yescharacteristic
234C22×C3C22×D1511yescharacteristic
235C2×D3—52no—
236C2×D3—101no—
237C2×D3—101no—
238C2×D3—101no—
239C2×D3—101no—
240C2×D3—101no—
241C3×C4—22no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—21no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C4—301no—
251C3×C4—301no—
252Dic3—101no—
253Dic3—101no—
254C2×C5C3×#23511yescharacteristic
255C2×C5C3×D3×D411yescharacteristic
256C2×C5C3×#23911yescharacteristic
257D5—62no—
258D5—121no—
259C32#404011yescharacteristic, complemented, Sylow
260C23—151no—
261C23—151no—
262D4—302no—
263C2×C4—151no—
264C2×C4—151no—
265C2×C4—302no—
266C2×C4C2×C3×D1511yescharacteristic
267C2×C4C2×C3×D1511yescharacteristic
268C2×C4C2×C3×D1511yescharacteristic
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—21no—
272C2×C3—302no—
273C2×C3—601no—
274C2×C3C3×#70911yescharacteristic
275C2×C3C3×D4×D511yescharacteristic
276C2×C3C3×#71311yescharacteristic
277C2×C3#681711yescharacteristic
278C2×C3D4×D1511yescharacteristic
279C2×C3#682111yescharacteristic
280D3—102no—
281D3—201no—
282C5C3×#89411yescharacteristic, complemented, Sylow
283C22—152no—
284C22—301no—
285C22—301no—
286C22—301no—
287C22—301no—
288C22—301no—
289C22C22×C3×D1511yescharacteristic
290C4—21no—
291C4—21no—
292C4—21no—
293C4—301no—
294C4—301no—
295C3—21no—
296C3C3×#404011yescharacteristic, complemented
297C3#9178311yescharacteristic, direct product factor
298C2—302no—
299C2—601no—
300C2C3×#681711yescharacteristic
301C2C3×D4×D1511yescharacteristic
302C2C3×#682111yescharacteristic
subgroupG/H
centerC22×C3C22×D15
commutator, cosocleC22×C3×C5C23×C3
FrattiniC22C22×C3×D15
FittingC32×C5×C4⋊C4C2
socleC22×C32×C5C23
prime radicalC2×C3×D60C2
hypercenterC3×C4⋊C4D15
hypocenterC3×C5C3×#123
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×C4×D15C2semidirect
3. C32×C5×C4⋊C4C2semidirect
4. C3×#6714C2semidirect
5. C3×#6714C2semidirect
6. C3×#6714C2semidirect
7. C3×#6712C2semidirect
8. #91783C3direct (N × H)
21. C2×C4×D15C2×C3semidirect
25. C3×C5×C4⋊C4D3semidirect
26. C3×C5×C4⋊C4C2×C3semidirect
27. #6714C2×C3semidirect
28. #6714C2×C3semidirect
29. #6714C2×C3semidirect
30. #6712C2×C3semidirect
50. C32×C4⋊C4D5semidirect
85. C5×C4⋊C4C3×D3semidirect
133. C3×C4⋊C4C3×D5semidirect
134. C3×C4⋊C4D15semidirect
142. C32×C5#123semidirect
219. C4⋊C4C3×D15semidirect
224. C3×C5#894semidirect
225. C3×C5C3×#123semidirect
259. C32#4040semidirect
282. C5C3×#894semidirect
296. C3C3×#4040semidirect
297. C3#91783direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D602
C2×C3×C4×D152
C32×C5×C4⋊C42
C3×#67142
C3×#67122
#917833
C3×#40403
C3×#8945