Finite Group Databasebeta

← C3×#91772 #13087981

Subgroups — C3×#91772 #13087981

Order 1440.

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

normal subgroups only

This group has 346 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×#1220C211yesmaximal, characteristic
2C2×C3×C4×D15C211yesmaximal, characteristic, complemented
3C32×C5×C22⋊C4C211yesmaximal, characteristic, complemented
4C3×#6719C211yesmaximal, characteristic, complemented
5C3×#6714C211yesmaximal, characteristic, complemented
6C3×C4⋊Dic15C211yesmaximal, characteristic, complemented
7C3×#6712C211yesmaximal, characteristic, complemented
8#91772C311yesmaximal, characteristic, direct product factor
9C3×#4029—31nomaximal
10C23×C32×C5C2211yescharacteristic
11C22×C3×D15C2211yescharacteristic
12C2×C32×C4×C5C2211yescharacteristic
13C2×C32×C4×C5C2211yescharacteristic
14C3×#1220—22no—
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×Dic15C2211yescharacteristic
17C2×C3×Dic15C2211yescharacteristic
18C3×C4×D15—22no—
19C3×#883—51nomaximal, Hall
20C2×#1220C2×C311yescharacteristic
21C2×C4×D15C2×C311yescharacteristic, complemented
22C2×C3×C22⋊D5—31no—
23C2×C3×C4×D5—31no—
24C3×C5×C22⋊C4—21no—
25C3×C5×C22⋊C4D311yescharacteristic, complemented
26C3×C5×C22⋊C4C2×C311yescharacteristic, complemented
27#6719C2×C311yescharacteristic, complemented
28#6714C2×C311yescharacteristic, complemented
29C4⋊Dic15C2×C311yescharacteristic, complemented
30#6712C2×C311yescharacteristic, complemented
31C3×C22⋊Dic5—31no—
32C3×#685—31no—
33C3×C4⋊Dic5—31no—
34C3×#683—31no—
35C22×C32×C5—21no—
36C22×C32×C5—21no—
37C22×C32×C5—21no—
38C22×C32×C5C2311yescharacteristic
39C2×C3×D15—21no—
40C2×C3×D15—21no—
41C2×C3×D15D412yes—
42C32×C4×C5—21no—
43C32×C4×C5—21no—
44C3×Dic15—21no—
45C3×Dic15—21no—
46C3×Dic15—21no—
47#4029—31noHall
48C2×C3×#67—51no—
49C2×C3×C4×D3—51no—
50C32×C22⋊C4D511yescharacteristic, complemented
51C3×#217—51no—
52C3×(C2×D3)⋊C4—51no—
53C3×C4⋊Dic3—51no—
54C3×Dic3⋊C4—51no—
55C23×C3×C5—21no—
56C23×C3×C5C2×D311yescharacteristic
57C23×C3×C5C22×C311yescharacteristic
58C22×D15C22×C311yescharacteristic
59C22×C3×D5—31no—
60C2×C3×C4×C5—21no—
61C2×C3×C4×C5—21no—
62C2×C3×C4×C5C2×D311yescharacteristic
63C2×C3×C4×C5C2×D311yescharacteristic
64C2×C3×C4×C5C22×C311yescharacteristic
65C2×C3×C4×C5C22×C311yescharacteristic
66#1220—22no—
67C2×Dic15C22×C311yescharacteristic
68C2×Dic15C22×C311yescharacteristic
69C2×Dic15C22×C311yescharacteristic
70C4×D15—22no—
71C3×C22⋊D5—62no—
72C2×C3×Dic5—31no—
73C2×C3×Dic5—31no—
74C2×C3×Dic5—31no—
75C3×C4×D5—62no—
76C3×#123—151no—
77#883—51no—
78C2×C32×C5—41no—
79C2×C32×C5C2×D411yescharacteristic
80C2×C32×C5#4111yescharacteristic
81C2×C32×C5#4111yescharacteristic
82C3×D15—22no—
83C2×C22⋊D5—31no—
84C2×C4×D5—31no—
85C5×C22⋊C4C3×D311yescharacteristic, complemented
86C22⋊Dic5—31no—
87#685—31no—
88C4⋊Dic5—31no—
89#683—31no—
90C23×C32C2×D511yescharacteristic
91C22×C3×D3—51no—
92C2×C32×C4C2×D511yescharacteristic
93C2×C32×C4C2×D511yescharacteristic
94C3×#67—102no—
95C2×C3×Dic3—51no—
96C2×C3×Dic3—51no—
97C2×C3×Dic3—51no—
98C3×C4×D3—102no—
99C22×C3×C5—22no—
100C22×C3×C5—21no—
101C22×C3×C5—21no—
102C22×C3×C5—21no—
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—21no—
106C22×C3×C5—21no—
107C22×C3×C5—41no—
108C22×C3×C5—41no—
109C22×C3×C5C22×D311yescharacteristic
110C22×C3×C5C23×C311yescharacteristic
111C2×D15—21no—
112C2×D15—21no—
113C2×D15C3×D412yes—
114C2×C3×D5—32no—
115C2×C3×D5—61no—
116C2×C3×D5—61no—
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—
125Dic15—21no—
126C3×Dic5—61no—
127C3×Dic5—61no—
128C3×Dic5—61no—
129C2×C3×D4—151no—
130C22×C3×C4—151no—
131C2×#67—51no—
132C2×C4×D3—51no—
133C3×C4⋊C4—151no—
134C3×C4⋊C4—151no—
135C3×C22⋊C4—21no—
136C3×C22⋊C4—151no—
137C3×C22⋊C4—151no—
138C3×C22⋊C4C3×D511yescharacteristic, complemented
139C3×C22⋊C4D1511yescharacteristic, complemented
140#217—51no—
141(C2×D3)⋊C4—51no—
142C4⋊Dic3—51no—
143Dic3⋊C4—51no—
144C32×C5#12311yescharacteristic, complemented, Hall
145C23×C5C2×C3×D311yescharacteristic
146C22×D5—31no—
147C2×C4×C5C2×C3×D311yescharacteristic
148C2×C4×C5C2×C3×D311yescharacteristic
149C22⋊D5—62no—
150C2×Dic5—31no—
151C2×Dic5—31no—
152C2×Dic5—31no—
153C4×D5—62no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32C22×D511yescharacteristic
158C2×C3×D3—52no—
159C2×C3×D3—101no—
160C2×C3×D3—101no—
161C32×C4—21no—
162C32×C4—21no—
163C3×Dic3—101no—
164C3×Dic3—101no—
165C3×Dic3—101no—
166#123—151noSylow
167C2×C3×C5—21no—
168C2×C3×C5—21no—
169C2×C3×C5—21no—
170C2×C3×C5—42no—
171C2×C3×C5—41no—
172C2×C3×C5—41no—
173C2×C3×C5#23511yescharacteristic
174C2×C3×C5D3×D411yescharacteristic
175C2×C3×C5#23711yescharacteristic
176C2×C3×C5C2×C3×D411yescharacteristic
177C2×C3×C5C3×#4111yescharacteristic
178C2×C3×C5C3×#4111yescharacteristic
179D15—22no—
180C3×D5—62no—
181C23×C3—21no—
182C23×C3—151no—
183C23×C3C2×C3×D511yescharacteristic
184C23×C3C2×D1511yescharacteristic
185C22×D3—51no—
186C3×D4—302no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—151no—
190C2×C3×C4—151no—
191C2×C3×C4—151no—
192C2×C3×C4—302no—
193C2×C3×C4C2×C3×D511yescharacteristic
194C2×C3×C4C2×C3×D511yescharacteristic
195C2×C3×C4C2×D1511yescharacteristic
196C2×C3×C4C2×D1511yescharacteristic
197#67—102no—
198C2×Dic3—51no—
199C2×Dic3—51no—
200C2×Dic3—51no—
201C4×D3—102no—
202C22×C5—21no—
203C22×C5—21no—
204C22×C5—21no—
205C22×C5C22×C3×D311yescharacteristic
206C2×D5—32no—
207C2×D5—61no—
208C2×D5—61no—
209C4×C5—21no—
210C4×C5—21no—
211Dic5—61no—
212Dic5—61no—
213Dic5—61no—
214C2×C32—41no—
215C2×C32#70911yescharacteristic
216C2×C32D4×D511yescharacteristic
217C2×C32#71111yescharacteristic
218C3×D3—102no—
219C2×D4—151no—
220C22×C4—151no—
221C4⋊C4—151no—
222C4⋊C4—151no—
223C22⋊C4—151no—
224C22⋊C4—151no—
225C22⋊C4C3×D1511yescharacteristic, complemented
226C3×C5—21no—
227C3×C5#88311yescharacteristic, complemented
228C3×C5C3×#12311yescharacteristic, complemented
229C22×C3—22no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—152no—
240C22×C3—301no—
241C22×C3—301no—
242C22×C3C22×C3×D511yescharacteristic
243C22×C3C22×D1511yescharacteristic
244C2×D3—52no—
245C2×D3—101no—
246C2×D3—101no—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—301no—
254C3×C4—301no—
255C3×C4—301no—
256Dic3—101no—
257Dic3—101no—
258Dic3—101no—
259C2×C5—41no—
260C2×C5C3×#23511yescharacteristic
261C2×C5C3×D3×D411yescharacteristic
262C2×C5C3×#23711yescharacteristic
263D5—62no—
264C32#402911yescharacteristic, complemented, Sylow
265C23—151no—
266C23C2×C3×D1511yescharacteristic
267D4—302no—
268C2×C4—151no—
269C2×C4—151no—
270C2×C4—151no—
271C2×C4—302no—
272C2×C4C2×C3×D1511yescharacteristic
273C2×C4C2×C3×D1511yescharacteristic
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—42no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3—302no—
281C2×C3C3×#70911yescharacteristic
282C2×C3C3×D4×D511yescharacteristic
283C2×C3C3×#71111yescharacteristic
284C2×C3#681711yescharacteristic
285C2×C3D4×D1511yescharacteristic
286C2×C3#681911yescharacteristic
287D3—102no—
288C5C3×#88311yescharacteristic, complemented, Sylow
289C22—21no—
290C22—21no—
291C22—21no—
292C22—152no—
293C22—301no—
294C22—301no—
295C22C22×C3×D1511yescharacteristic
296C4—21no—
297C4—21no—
298C4—301no—
299C4—301no—
300C4—301no—
301C3—21no—
302C3C3×#402911yescharacteristic, complemented
303C3#9177211yescharacteristic, direct product factor
304C2—41no—
305C2—302no—
306C2C3×#681711yescharacteristic
307C2C3×D4×D1511yescharacteristic
308C2C3×#681911yescharacteristic
subgroupG/H
centerC22×C3C22×D15
commutator, cosocleC22×C3×C5C23×C3
FrattiniC22C22×C3×D15
FittingC32×C5×C22⋊C4C2
socleC22×C32×C5C23
prime radicalC2×C3×#1220C2
hypercenterC3×C22⋊C4D15
hypocenterC3×C5C3×#123
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×D15C2semidirect
3. C32×C5×C22⋊C4C2semidirect
4. C3×#6719C2semidirect
5. C3×#6714C2semidirect
6. C3×C4⋊Dic15C2semidirect
7. C3×#6712C2semidirect
8. #91772C3direct (N × H)
21. C2×C4×D15C2×C3semidirect
25. C3×C5×C22⋊C4D3semidirect
26. C3×C5×C22⋊C4C2×C3semidirect
27. #6719C2×C3semidirect
28. #6714C2×C3semidirect
29. C4⋊Dic15C2×C3semidirect
30. #6712C2×C3semidirect
50. C32×C22⋊C4D5semidirect
85. C5×C22⋊C4C3×D3semidirect
138. C3×C22⋊C4C3×D5semidirect
139. C3×C22⋊C4D15semidirect
144. C32×C5#123semidirect
225. C22⋊C4C3×D15semidirect
227. C3×C5#883semidirect
228. C3×C5C3×#123semidirect
264. C32#4029semidirect
288. C5C3×#883semidirect
302. C3C3×#4029semidirect
303. C3#91772direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#12202
C2×C3×C4×D152
C32×C5×C22⋊C42
C3×#67192
C3×#67142
C3×C4⋊Dic152
C3×#67122
#917723
C3×#40293
C3×#8835