Finite Group Databasebeta

← C3×C5×#1681 #14274715

Subgroups — C3×C5×#1681 #14274715

Order 1920.

364 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#486C211yesmaximal, characteristic, complemented
2C3×C5×#461C211yesmaximal, characteristic, complemented
3C3×C5×#459C211yesmaximal, characteristic
4C3×C5×C8×D4C211yesmaximal, characteristic, complemented
5C3×C5×#335C211yesmaximal, characteristic, complemented
6C3×C5×#332C211yesmaximal, characteristic, complemented
7C3×C5×#331C211yesmaximal, characteristic, complemented
8C5×#1681C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×SD16—21no—
10C2×C3×C5×D8—21no—
11C22×C3×C5×C8—21no—
12C3×C5×C4⋊Q8C2211yescharacteristic
13C3×C5×#127C2211yescharacteristic
14C3×C4×C5×D4C2211yescharacteristic
15C3×C5×C4⋊C8C2211yescharacteristic
16C3×C5×C4⋊C8C2211yescharacteristic
17C3×C5×C4⋊C8C2211yescharacteristic
18C3×C5×#103—21no—
19C3×C5×#102—21no—
20C3×C5×#102—21no—
21C3×C5×#102—21no—
22C3×C5×C22⋊C8—21no—
23C3×C4×C5×C8C2211yescharacteristic
24C3×#1681C511yesmaximal, characteristic, direct product factor, Hall
25C5×#486C2×C311yescharacteristic, complemented
26C5×#461C2×C311yescharacteristic, complemented
27C5×#459C2×C311yescharacteristic
28C5×C8×D4C2×C311yescharacteristic, complemented
29C5×#335C2×C311yescharacteristic, complemented
30C5×#332C2×C311yescharacteristic, complemented
31C5×#331C2×C311yescharacteristic, complemented
32C2×C3×C5×Q8—21no—
33C2×C3×C5×D4—21no—
34C2×C3×C5×D4—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4D411yescharacteristic
37C22×C3×C4×C5—21no—
38C3×C5×SD16—42no—
39C3×C5×D8—42no—
40C2×C3×C5×C8—21no—
41C2×C3×C5×C8—21no—
42C2×C3×C5×C8—21no—
43C2×C3×C5×C8—42no—
44C2×C3×C5×C8D411yescharacteristic
45C2×C3×C5×C8D411yescharacteristic
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4D411yescharacteristic
49C3×C5×C22⋊C4—21no—
50C3×C42×C5C2311yescharacteristic
51C3×#486C2×C511yescharacteristic, complemented
52C3×#461C2×C511yescharacteristic, complemented
53C3×#459C2×C511yescharacteristic
54C3×C8×D4C2×C511yescharacteristic, complemented
55C3×#335C2×C511yescharacteristic, complemented
56C3×#332C2×C511yescharacteristic, complemented
57C3×#331C2×C511yescharacteristic, complemented
58C2×C5×SD16—21no—
59C2×C5×D8—21no—
60C22×C5×C8—21no—
61C5×C4⋊Q8C22×C311yescharacteristic
62C5×#127C22×C311yescharacteristic
63C4×C5×D4C22×C311yescharacteristic
64C5×C4⋊C8C22×C311yescharacteristic
65C5×C4⋊C8C22×C311yescharacteristic
66C5×C4⋊C8C22×C311yescharacteristic
67C5×#103—21no—
68C5×#102—21no—
69C5×#102—21no—
70C5×#102—21no—
71C5×C22⋊C8—21no—
72C4×C5×C8C22×C311yescharacteristic
73#1681C3×C511yescharacteristic, direct product factor, Sylow
74C23×C3×C5—21no—
75C23×C3×C5—41no—
76C3×C5×Q8—41no—
77C3×C5×Q8—41no—
78C3×C5×D4—21no—
79C3×C5×D4—41no—
80C3×C5×D4—41no—
81C3×C5×D4—41no—
82C3×C5×D4—41no—
83C3×C5×D4—81no—
84C3×C5×D4D812yes—
85C2×C3×C4×C5—21no—
86C2×C3×C4×C5—41no—
87C2×C3×C4×C5—41no—
88C2×C3×C4×C5—41no—
89C2×C3×C4×C5C2×D411yescharacteristic
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5#4111yescharacteristic
92C3×C5×C8—21no—
93C3×C5×C8—21no—
94C3×C5×C8—41no—
95C3×C5×C8—41no—
96C3×C5×C8—41no—
97C2×C3×SD16—21no—
98C2×C3×D8—21no—
99C22×C3×C8—21no—
100C3×C4⋊Q8C22×C511yescharacteristic
101C3×#127C22×C511yescharacteristic
102C3×C4×D4C22×C511yescharacteristic
103C3×C4⋊C8C22×C511yescharacteristic
104C3×C4⋊C8C22×C511yescharacteristic
105C3×C4⋊C8C22×C511yescharacteristic
106C3×#103—21no—
107C3×#102—21no—
108C3×#102—21no—
109C3×#102—21no—
110C3×C22⋊C8—21no—
111C3×C4×C8C22×C511yescharacteristic
112C2×C5×Q8—21no—
113C2×C5×D4—21no—
114C2×C5×D4—21no—
115C2×C5×D4—21no—
116C2×C5×D4C3×D411yescharacteristic
117C22×C4×C5—21no—
118C5×SD16—42no—
119C5×D8—42no—
120C2×C5×C8—21no—
121C2×C5×C8—21no—
122C2×C5×C8—21no—
123C2×C5×C8—42no—
124C2×C5×C8C3×D411yescharacteristic
125C2×C5×C8C3×D411yescharacteristic
126C5×C4⋊C4—21no—
127C5×C4⋊C4—21no—
128C5×C4⋊C4C3×D411yescharacteristic
129C5×C22⋊C4—21no—
130C42×C5C23×C311yescharacteristic
131#486C2×C3×C511yescharacteristic, complemented
132#461C2×C3×C511yescharacteristic, complemented
133#459C2×C3×C511yescharacteristic
134C8×D4C2×C3×C511yescharacteristic, complemented
135#335C2×C3×C511yescharacteristic, complemented
136#332C2×C3×C511yescharacteristic, complemented
137#331C2×C3×C511yescharacteristic, complemented
138C22×C3×C5—22no—
139C22×C3×C5—41no—
140C22×C3×C5—41no—
141C22×C3×C5—81no—
142C22×C3×C5—81no—
143C22×C3×C5—81no—
144C22×C3×C5#12111yescharacteristic
145C3×C4×C5—21no—
146C3×C4×C5—41no—
147C3×C4×C5—81no—
148C3×C4×C5C2×D811yescharacteristic
149C3×C4×C5#13511yescharacteristic
150C3×C4×C5#13511yescharacteristic
151C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
152C2×C3×Q8—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4C5×D411yescharacteristic
157C22×C3×C4—21no—
158C3×SD16—42no—
159C3×D8—42no—
160C2×C3×C8—21no—
161C2×C3×C8—21no—
162C2×C3×C8—21no—
163C2×C3×C8—42no—
164C2×C3×C8C5×D411yescharacteristic
165C2×C3×C8C5×D411yescharacteristic
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4C5×D411yescharacteristic
169C3×C22⋊C4—21no—
170C3×C42C23×C511yescharacteristic
171C23×C5—21no—
172C23×C5—41no—
173C5×Q8—41no—
174C5×Q8—41no—
175C5×D4—21no—
176C5×D4—41no—
177C5×D4—41no—
178C5×D4—41no—
179C5×D4—41no—
180C5×D4—81no—
181C5×D4C3×D812yes—
182C2×C4×C5—21no—
183C2×C4×C5—41no—
184C2×C4×C5—41no—
185C2×C4×C5—41no—
186C2×C4×C5C2×C3×D411yescharacteristic
187C2×C4×C5C2×C3×D411yescharacteristic
188C2×C4×C5C3×#4111yescharacteristic
189C5×C8—21no—
190C5×C8—21no—
191C5×C8—41no—
192C5×C8—41no—
193C5×C8—41no—
194C2×SD16—21no—
195C2×D8—21no—
196C22×C8—21no—
197C4⋊Q8C22×C3×C511yescharacteristic
198#127C22×C3×C511yescharacteristic
199C4×D4C22×C3×C511yescharacteristic
200C4⋊C8C22×C3×C511yescharacteristic
201C4⋊C8C22×C3×C511yescharacteristic
202C4⋊C8C22×C3×C511yescharacteristic
203#103—21no—
204#102—21no—
205#102—21no—
206#102—21no—
207C22⋊C8—21no—
208C4×C8C22×C3×C511yescharacteristic
209C2×C3×C5—42no—
210C2×C3×C5—161no—
211C2×C3×C5#46311yescharacteristic
212C2×C3×C5#46611yescharacteristic
213C2×C3×C5#47111yescharacteristic
214C23×C3—21no—
215C23×C3—41no—
216C3×Q8—41no—
217C3×Q8—41no—
218C3×D4—21no—
219C3×D4—41no—
220C3×D4—41no—
221C3×D4—41no—
222C3×D4—41no—
223C3×D4—81no—
224C3×D4C5×D812yes—
225C2×C3×C4—21no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4C2×C5×D411yescharacteristic
230C2×C3×C4C2×C5×D411yescharacteristic
231C2×C3×C4C5×#4111yescharacteristic
232C3×C8—21no—
233C3×C8—21no—
234C3×C8—41no—
235C3×C8—41no—
236C3×C8—41no—
237C22×C5—22no—
238C22×C5—41no—
239C22×C5—41no—
240C22×C5—81no—
241C22×C5—81no—
242C22×C5—81no—
243C22×C5C3×#12111yescharacteristic
244C4×C5—21no—
245C4×C5—41no—
246C4×C5—81no—
247C4×C5C2×C3×D811yescharacteristic
248C4×C5C3×#13511yescharacteristic
249C4×C5C3×#13511yescharacteristic
250C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
251C2×Q8—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4C3×C5×D411yescharacteristic
256C22×C4—21no—
257SD16—42no—
258D8—42no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—42no—
263C2×C8C3×C5×D411yescharacteristic
264C2×C8C3×C5×D411yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4C3×C5×D411yescharacteristic
268C22⋊C4—21no—
269C42C23×C3×C511yescharacteristic
270C3×C5#168111yescharacteristic, direct product factor, Hall
271C22×C3—22no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—81no—
275C22×C3—81no—
276C22×C3—81no—
277C22×C3C5×#12111yescharacteristic
278C3×C4—21no—
279C3×C4—41no—
280C3×C4—81no—
281C3×C4C2×C5×D811yescharacteristic
282C3×C4C5×#13511yescharacteristic
283C3×C4C5×#13511yescharacteristic
284C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
285C2×C5—42no—
286C2×C5—161no—
287C2×C5C3×#46311yescharacteristic
288C2×C5C3×#46611yescharacteristic
289C2×C5C3×#47111yescharacteristic
290C23—21no—
291C23—41no—
292Q8—41no—
293Q8—41no—
294D4—21no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—81no—
300D4C3×C5×D812yes—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4C2×C3×C5×D411yescharacteristic
306C2×C4C2×C3×C5×D411yescharacteristic
307C2×C4C3×C5×#4111yescharacteristic
308C8—21no—
309C8—21no—
310C8—41no—
311C8—41no—
312C8—41no—
313C2×C3—42no—
314C2×C3—161no—
315C2×C3C5×#46311yescharacteristic
316C2×C3C5×#46611yescharacteristic
317C2×C3C5×#47111yescharacteristic
318C5C3×#168111yescharacteristic, direct product factor, Sylow
319C22—22no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22C3×C5×#12111yescharacteristic
326C4—21no—
327C4—41no—
328C4—81no—
329C4C2×C3×C5×D811yescharacteristic
330C4C3×C5×#13511yescharacteristic
331C4C3×C5×#13511yescharacteristic
332C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
333C3C5×#168111yescharacteristic, direct product factor, Sylow
334C2—42no—
335C2—161no—
336C2C3×C5×#46311yescharacteristic
337C2C3×C5×#46611yescharacteristic
338C2C3×C5×#47111yescharacteristic
subgroupG/H
center, socleC22×C3×C5#121
commutator, Frattini, cosocleC42C23×C3×C5
prime radicalC3×C5×#459C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C3×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C5×#486C2semidirect
2. C3×C5×#461C2semidirect
4. C3×C5×C8×D4C2semidirect
5. C3×C5×#335C2semidirect
6. C3×C5×#332C2semidirect
7. C3×C5×#331C2semidirect
8. C5×#1681C3direct (N × H)
24. C3×#1681C5direct (N × H)
25. C5×#486C2×C3semidirect
26. C5×#461C2×C3semidirect
28. C5×C8×D4C2×C3semidirect
29. C5×#335C2×C3semidirect
30. C5×#332C2×C3semidirect
31. C5×#331C2×C3semidirect
51. C3×#486C2×C5semidirect
52. C3×#461C2×C5semidirect
54. C3×C8×D4C2×C5semidirect
55. C3×#335C2×C5semidirect
56. C3×#332C2×C5semidirect
57. C3×#331C2×C5semidirect
73. #1681C3×C5direct (N × H)
131. #486C2×C3×C5semidirect
132. #461C2×C3×C5semidirect
134. C8×D4C2×C3×C5semidirect
135. #335C2×C3×C5semidirect
136. #332C2×C3×C5semidirect
137. #331C2×C3×C5semidirect
270. C3×C5#1681direct (N × H)
318. C5C3×#1681direct (N × H)
333. C3C5×#1681direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#4862
C3×C5×#4612
C3×C5×#4592
C3×C5×C8×D42
C3×C5×#3352
C3×C5×#3322
C3×C5×#3312
C5×#16813
C3×#16815