Finite Group Databasebeta

← C3×#91773 #13087982

Subgroups — C3×#91773 #13087982

Order 1440.

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

normal subgroups only

This group has 410 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×#1220C211yesmaximal, characteristic, complemented
2C2×C3×#1220C211yesmaximal, characteristic, complemented
3C2×C3×D60C211yesmaximal, characteristic, complemented
4C2×C3×C4×D15C211yesmaximal, characteristic, complemented
5C32×C5×C22⋊C4C211yesmaximal, characteristic, complemented
6C3×#6714C211yesmaximal, characteristic, complemented
7C3×#6712C211yesmaximal, characteristic, complemented
8#91773C311yesmaximal, characteristic, direct product factor
9C3×#4030—31nomaximal
10C23×C32×C5C2211yescharacteristic
11C22×C3×D15C2211yescharacteristic
12C22×C3×D15C2211yescharacteristic
13C2×C32×C4×C5C2211yescharacteristic
14C2×C32×C4×C5C2211yescharacteristic
15C3×#1220—22no—
16C3×#1220—22no—
17C2×C3×Dic15C2211yescharacteristic
18C2×C3×Dic15C2211yescharacteristic
19C3×D60—22no—
20C3×C4×D15—22no—
21C3×#884—51nomaximal, Hall
22C2×#1220C2×C311yescharacteristic, complemented
23C2×#1220C2×C311yescharacteristic, complemented
24C2×D60C2×C311yescharacteristic, complemented
25C2×C4×D15C2×C311yescharacteristic, complemented
26C2×C3×C22⋊D5—31no—
27C2×C3×C22⋊D5—31no—
28C2×C3×D20—31no—
29C2×C3×C4×D5—31no—
30C3×C5×C22⋊C4—21no—
31C3×C5×C22⋊C4D311yescharacteristic, complemented
32C3×C5×C22⋊C4C2×C311yescharacteristic, complemented
33#6714C2×C311yescharacteristic, complemented
34#6712C2×C311yescharacteristic, complemented
35C3×#685—31no—
36C3×#683—31no—
37C22×C32×C5—21no—
38C22×C32×C5—21no—
39C22×C32×C5—21no—
40C22×C32×C5C2311yescharacteristic
41C2×C3×D15—21no—
42C2×C3×D15—21no—
43C2×C3×D15—21no—
44C2×C3×D15—21no—
45C2×C3×D15—21no—
46C2×C3×D15D412yescomplemented
47C32×C4×C5—21no—
48C32×C4×C5—21no—
49C3×Dic15—21no—
50C3×Dic15D412yescomplemented
51#4030—31noHall
52C2×C3×#67—51no—
53C2×C3×#67—51no—
54C2×C3×D12—51no—
55C2×C3×C4×D3—51no—
56C32×C22⋊C4D511yescharacteristic, complemented
57C3×(C2×D3)⋊C4—51no—
58C3×Dic3⋊C4—51no—
59C23×C3×C5—21no—
60C23×C3×C5C2×D311yescharacteristic
61C23×C3×C5C22×C311yescharacteristic
62C22×D15C22×C311yescharacteristic
63C22×D15C22×C311yescharacteristic
64C22×C3×D5—31no—
65C22×C3×D5—31no—
66C2×C3×C4×C5—21no—
67C2×C3×C4×C5—21no—
68C2×C3×C4×C5C2×D311yescharacteristic
69C2×C3×C4×C5C2×D311yescharacteristic
70C2×C3×C4×C5C22×C311yescharacteristic
71C2×C3×C4×C5C22×C311yescharacteristic
72#1220—22no—
73#1220—22no—
74C2×Dic15C22×C311yescharacteristic
75C2×Dic15C22×C311yescharacteristic
76D60—22no—
77C4×D15—22no—
78C3×C22⋊D5—62no—
79C3×C22⋊D5—62no—
80C2×C3×Dic5—31no—
81C2×C3×Dic5—31no—
82C3×D20—62no—
83C3×C4×D5—62no—
84C3×#121—151no—
85#884—51no—
86C2×C32×C5—41no—
87C2×C32×C5C2×D411yescharacteristic
88C2×C32×C5C2×D411yescharacteristic
89C2×C32×C5#4111yescharacteristic
90C3×D15—22no—
91C3×D15—41no—
92C2×C22⋊D5—31no—
93C2×C22⋊D5—31no—
94C2×D20—31no—
95C2×C4×D5—31no—
96C5×C22⋊C4C3×D311yescharacteristic, complemented
97#685—31no—
98#683—31no—
99C23×C32C2×D511yescharacteristic
100C22×C3×D3—51no—
101C22×C3×D3—51no—
102C2×C32×C4C2×D511yescharacteristic
103C2×C32×C4C2×D511yescharacteristic
104C3×#67—102no—
105C3×#67—102no—
106C2×C3×Dic3—51no—
107C2×C3×Dic3—51no—
108C3×D12—102no—
109C3×C4×D3—102no—
110C22×C3×C5—22no—
111C22×C3×C5—21no—
112C22×C3×C5—21no—
113C22×C3×C5—21no—
114C22×C3×C5—21no—
115C22×C3×C5—21no—
116C22×C3×C5—21no—
117C22×C3×C5—21no—
118C22×C3×C5—41no—
119C22×C3×C5—41no—
120C22×C3×C5C22×D311yescharacteristic
121C22×C3×C5C23×C311yescharacteristic
122C2×D15—21no—
123C2×D15—21no—
124C2×D15—21no—
125C2×D15—21no—
126C2×D15—21no—
127C2×D15C3×D412yescomplemented
128C2×C3×D5—32no—
129C2×C3×D5—61no—
130C2×C3×D5—61no—
131C2×C3×D5—61no—
132C2×C3×D5—61no—
133C2×C3×D5—61no—
134C3×C4×C5—21no—
135C3×C4×C5—21no—
136C3×C4×C5—21no—
137C3×C4×C5—21no—
138C3×C4×C5—41no—
139C3×C4×C5—41no—
140Dic15—21no—
141Dic15C3×D412yescomplemented
142C3×Dic5—32no—
143C3×Dic5—61no—
144C2×C3×D4—151no—
145C2×C3×D4—151no—
146C2×C3×D4—151no—
147C22×C3×C4—151no—
148C2×#67—51no—
149C2×#67—51no—
150C2×D12—51no—
151C2×C4×D3—51no—
152C3×C4⋊C4—151no—
153C3×C22⋊C4—21no—
154C3×C22⋊C4—151no—
155C3×C22⋊C4C3×D511yescharacteristic, complemented
156C3×C22⋊C4D1511yescharacteristic, complemented
157(C2×D3)⋊C4—51no—
158Dic3⋊C4—51no—
159C32×C5#12111yescharacteristic, complemented, Hall
160C23×C5C2×C3×D311yescharacteristic
161C22×D5—31no—
162C22×D5—31no—
163C2×C4×C5C2×C3×D311yescharacteristic
164C2×C4×C5C2×C3×D311yescharacteristic
165C22⋊D5—62no—
166C22⋊D5—62no—
167C2×Dic5—31no—
168C2×Dic5—31no—
169D20—62no—
170C4×D5—62no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32C22×D511yescharacteristic
175C2×C3×D3—52no—
176C2×C3×D3—101no—
177C2×C3×D3—101no—
178C2×C3×D3—101no—
179C2×C3×D3—101no—
180C2×C3×D3—101no—
181C32×C4—21no—
182C32×C4—21no—
183C3×Dic3—52no—
184C3×Dic3—101no—
185#121—151noSylow
186C2×C3×C5—21no—
187C2×C3×C5—21no—
188C2×C3×C5—21no—
189C2×C3×C5—42no—
190C2×C3×C5—41no—
191C2×C3×C5—41no—
192C2×C3×C5#23511yescharacteristic
193C2×C3×C5D3×D411yescharacteristic
194C2×C3×C5D3×D411yescharacteristic
195C2×C3×C5C2×C3×D411yescharacteristic
196C2×C3×C5C2×C3×D411yescharacteristic
197C2×C3×C5C3×#4111yescharacteristic
198D15—22no—
199D15—41no—
200C3×D5—62no—
201C3×D5—121no—
202C23×C3—21no—
203C23×C3—151no—
204C23×C3—151no—
205C23×C3C2×C3×D511yescharacteristic
206C23×C3C2×D1511yescharacteristic
207C22×D3—51no—
208C22×D3—51no—
209C3×D4—302no—
210C3×D4—302no—
211C3×D4—302no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—151no—
215C2×C3×C4—151no—
216C2×C3×C4—302no—
217C2×C3×C4C2×C3×D511yescharacteristic
218C2×C3×C4C2×C3×D511yescharacteristic
219C2×C3×C4C2×D1511yescharacteristic
220C2×C3×C4C2×D1511yescharacteristic
221#67—102no—
222#67—102no—
223C2×Dic3—51no—
224C2×Dic3—51no—
225D12—102no—
226C4×D3—102no—
227C22×C5—21no—
228C22×C5—21no—
229C22×C5—21no—
230C22×C5C22×C3×D311yescharacteristic
231C2×D5—32no—
232C2×D5—61no—
233C2×D5—61no—
234C2×D5—61no—
235C2×D5—61no—
236C2×D5—61no—
237C4×C5—21no—
238C4×C5—21no—
239Dic5—32no—
240Dic5—61no—
241C2×C32—41no—
242C2×C32#70911yescharacteristic
243C2×C32D4×D511yescharacteristic
244C2×C32D4×D511yescharacteristic
245C3×D3—102no—
246C3×D3—201no—
247C2×D4—151no—
248C2×D4—151no—
249C2×D4—151no—
250C22×C4—151no—
251C4⋊C4—151no—
252C22⋊C4—151no—
253C22⋊C4C3×D1511yescharacteristic, complemented
254C3×C5—21no—
255C3×C5#88411yescharacteristic, complemented
256C3×C5C3×#12111yescharacteristic, complemented
257C22×C3—22no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—152no—
268C22×C3—301no—
269C22×C3—301no—
270C22×C3—301no—
271C22×C3—301no—
272C22×C3—301no—
273C22×C3C22×C3×D511yescharacteristic
274C22×C3C22×D1511yescharacteristic
275C2×D3—52no—
276C2×D3—101no—
277C2×D3—101no—
278C2×D3—101no—
279C2×D3—101no—
280C2×D3—101no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4—152no—
288C3×C4—301no—
289Dic3—52no—
290Dic3—101no—
291C2×C5—41no—
292C2×C5C3×#23511yescharacteristic
293C2×C5C3×D3×D411yescharacteristic
294C2×C5C3×D3×D411yescharacteristic
295D5—62no—
296D5—121no—
297C32#403011yescharacteristic, complemented, Sylow
298C23—151no—
299C23—151no—
300C23C2×C3×D1511yescharacteristic
301D4—302no—
302D4—302no—
303D4—302no—
304C2×C4—151no—
305C2×C4—151no—
306C2×C4—302no—
307C2×C4C2×C3×D1511yescharacteristic
308C2×C4C2×C3×D1511yescharacteristic
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—42no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—302no—
316C2×C3—601no—
317C2×C3C3×#70911yescharacteristic
318C2×C3C3×D4×D511yescharacteristic
319C2×C3C3×D4×D511yescharacteristic
320C2×C3#681711yescharacteristic
321C2×C3D4×D1511yescharacteristic
322C2×C3D4×D1511yescharacteristic
323D3—102no—
324D3—201no—
325C5C3×#88411yescharacteristic, complemented, Sylow
326C22—21no—
327C22—21no—
328C22—21no—
329C22—152no—
330C22—301no—
331C22—301no—
332C22—301no—
333C22—301no—
334C22—301no—
335C22C22×C3×D1511yescharacteristic
336C4—21no—
337C4—21no—
338C4—152no—
339C4—301no—
340C3—21no—
341C3C3×#403011yescharacteristic, complemented
342C3#9177311yescharacteristic, direct product factor
343C2—41no—
344C2—302no—
345C2—601no—
346C2C3×#681711yescharacteristic
347C2C3×D4×D1511yescharacteristic
348C2C3×D4×D1511yescharacteristic
subgroupG/H
centerC22×C3C22×D15
commutator, cosocleC22×C3×C5C23×C3
FrattiniC22C22×C3×D15
FittingC32×C5×C22⋊C4C2
socleC22×C32×C5C23
hypercenterC3×C22⋊C4D15
hypocenterC3×C5C3×#121
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
1. C2×C3×#1220C2semidirect
2. C2×C3×#1220C2semidirect
3. C2×C3×D60C2semidirect
4. C2×C3×C4×D15C2semidirect
5. C32×C5×C22⋊C4C2semidirect
6. C3×#6714C2semidirect
7. C3×#6712C2semidirect
8. #91773C3direct (N × H)
22. C2×#1220C2×C3semidirect
23. C2×#1220C2×C3semidirect
24. C2×D60C2×C3semidirect
25. C2×C4×D15C2×C3semidirect
31. C3×C5×C22⋊C4D3semidirect
32. C3×C5×C22⋊C4C2×C3semidirect
33. #6714C2×C3semidirect
34. #6712C2×C3semidirect
46. C2×C3×D15D4semidirect
50. C3×Dic15D4semidirect
56. C32×C22⋊C4D5semidirect
96. C5×C22⋊C4C3×D3semidirect
127. C2×D15C3×D4semidirect
141. Dic15C3×D4semidirect
155. C3×C22⋊C4C3×D5semidirect
156. C3×C22⋊C4D15semidirect
159. C32×C5#121semidirect
253. C22⋊C4C3×D15semidirect
255. C3×C5#884semidirect
256. C3×C5C3×#121semidirect
297. C32#4030semidirect
325. C5C3×#884semidirect
341. C3C3×#4030semidirect
342. C3#91773direct (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×D602
C2×C3×C4×D152
C32×C5×C22⋊C42
C3×#67142
C3×#67122
#917733
C3×#40303
C3×#8845