Finite Group Databasebeta

← C3×C5×#1685 #14274719

Subgroups — C3×C5×#1685 #14274719

Order 1920.

364 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#492C211yesmaximal, characteristic, complemented
2C3×C5×#474C211yesmaximal, characteristic, complemented
3C3×C5×#459C211yesmaximal, characteristic
4C3×C5×#436C211yesmaximal, characteristic, complemented
5C3×C5×#335C211yesmaximal, characteristic, complemented
6C3×C5×#332C211yesmaximal, characteristic, complemented
7C3×C5×#331C211yesmaximal, characteristic, complemented
8C5×#1685C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×SD16—21no—
10C2×C3×C5×SD16—21no—
11C2×C3×C5×D8—21no—
12C2×C3×C5×M16—21no—
13C3×C5×C4⋊Q8C2211yescharacteristic
14C3×C5×#127C2211yescharacteristic
15C3×C4×C5×D4C2211yescharacteristic
16C3×C5×#107—21no—
17C3×C5×C4⋊C8C2211yescharacteristic
18C3×C5×C4⋊C8C2211yescharacteristic
19C3×C5×C4⋊C8C2211yescharacteristic
20C3×C5×#102—21no—
21C3×C5×#102—21no—
22C3×C5×C22⋊C8—21no—
23C3×C4×C5×C8C2211yescharacteristic
24C3×#1685C511yesmaximal, characteristic, direct product factor, Hall
25C5×#492C2×C311yescharacteristic, complemented
26C5×#474C2×C311yescharacteristic, complemented
27C5×#459C2×C311yescharacteristic
28C5×#436C2×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×SD16—42no—
40C3×C5×D8—42no—
41C3×C5×M16—42no—
42C2×C3×C5×C8—21no—
43C2×C3×C5×C8—21no—
44C2×C3×C5×C8—21no—
45C2×C3×C5×C8D411yescharacteristic
46C2×C3×C5×C8D411yescharacteristic
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4—21no—
49C3×C5×C4⋊C4D411yescharacteristic
50C3×C5×C22⋊C4—21no—
51C3×C42×C5C2311yescharacteristic
52C3×#492C2×C511yescharacteristic, complemented
53C3×#474C2×C511yescharacteristic, complemented
54C3×#459C2×C511yescharacteristic
55C3×#436C2×C511yescharacteristic, complemented
56C3×#335C2×C511yescharacteristic, complemented
57C3×#332C2×C511yescharacteristic, complemented
58C3×#331C2×C511yescharacteristic, complemented
59C2×C5×SD16—21no—
60C2×C5×SD16—21no—
61C2×C5×D8—21no—
62C2×C5×M16—21no—
63C5×C4⋊Q8C22×C311yescharacteristic
64C5×#127C22×C311yescharacteristic
65C4×C5×D4C22×C311yescharacteristic
66C5×#107—21no—
67C5×C4⋊C8C22×C311yescharacteristic
68C5×C4⋊C8C22×C311yescharacteristic
69C5×C4⋊C8C22×C311yescharacteristic
70C5×#102—21no—
71C5×#102—21no—
72C5×C22⋊C8—21no—
73C4×C5×C8C22×C311yescharacteristic
74#1685C3×C511yescharacteristic, direct product factor, Sylow
75C23×C3×C5—21no—
76C23×C3×C5—41no—
77C3×C5×Q8—41no—
78C3×C5×Q8—41no—
79C3×C5×D4—21no—
80C3×C5×D4—21no—
81C3×C5×D4—41no—
82C3×C5×D4—41no—
83C3×C5×D4—41no—
84C3×C5×D4—41no—
85C3×C5×D4—81no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—41no—
88C2×C3×C4×C5—41no—
89C2×C3×C4×C5—41no—
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C3×C4×C5#4111yescharacteristic
93C3×C5×C8—21no—
94C3×C5×C8—41no—
95C3×C5×C8—41no—
96C3×C5×C8—41no—
97C3×C5×C8D812yescomplemented
98C2×C3×SD16—21no—
99C2×C3×SD16—21no—
100C2×C3×D8—21no—
101C2×C3×M16—21no—
102C3×C4⋊Q8C22×C511yescharacteristic
103C3×#127C22×C511yescharacteristic
104C3×C4×D4C22×C511yescharacteristic
105C3×#107—21no—
106C3×C4⋊C8C22×C511yescharacteristic
107C3×C4⋊C8C22×C511yescharacteristic
108C3×C4⋊C8C22×C511yescharacteristic
109C3×#102—21no—
110C3×#102—21no—
111C3×C22⋊C8—21no—
112C3×C4×C8C22×C511yescharacteristic
113C2×C5×Q8—21no—
114C2×C5×D4—21no—
115C2×C5×D4—21no—
116C2×C5×D4—21no—
117C2×C5×D4C3×D411yescharacteristic
118C22×C4×C5—21no—
119C5×SD16—42no—
120C5×SD16—42no—
121C5×D8—42no—
122C5×M16—42no—
123C2×C5×C8—21no—
124C2×C5×C8—21no—
125C2×C5×C8—21no—
126C2×C5×C8C3×D411yescharacteristic
127C2×C5×C8C3×D411yescharacteristic
128C5×C4⋊C4—21no—
129C5×C4⋊C4—21no—
130C5×C4⋊C4C3×D411yescharacteristic
131C5×C22⋊C4—21no—
132C42×C5C23×C311yescharacteristic
133#492C2×C3×C511yescharacteristic, complemented
134#474C2×C3×C511yescharacteristic, complemented
135#459C2×C3×C511yescharacteristic
136#436C2×C3×C511yescharacteristic, complemented
137#335C2×C3×C511yescharacteristic, complemented
138#332C2×C3×C511yescharacteristic, complemented
139#331C2×C3×C511yescharacteristic, complemented
140C22×C3×C5—41no—
141C22×C3×C5—41no—
142C22×C3×C5—41no—
143C22×C3×C5—81no—
144C22×C3×C5—81no—
145C22×C3×C5—81no—
146C22×C3×C5#12111yescharacteristic
147C3×C4×C5—21no—
148C3×C4×C5—41no—
149C3×C4×C5—81no—
150C3×C4×C5C2×D811yescharacteristic
151C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
152C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
153C3×C4×C5#13711yescharacteristic
154C2×C3×Q8—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4C5×D411yescharacteristic
159C22×C3×C4—21no—
160C3×SD16—42no—
161C3×SD16—42no—
162C3×D8—42no—
163C3×M16—42no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8C5×D411yescharacteristic
168C2×C3×C8C5×D411yescharacteristic
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4C5×D411yescharacteristic
172C3×C22⋊C4—21no—
173C3×C42C23×C511yescharacteristic
174C23×C5—21no—
175C23×C5—41no—
176C5×Q8—41no—
177C5×Q8—41no—
178C5×D4—21no—
179C5×D4—21no—
180C5×D4—41no—
181C5×D4—41no—
182C5×D4—41no—
183C5×D4—41no—
184C5×D4—81no—
185C2×C4×C5—21no—
186C2×C4×C5—41no—
187C2×C4×C5—41no—
188C2×C4×C5—41no—
189C2×C4×C5C2×C3×D411yescharacteristic
190C2×C4×C5C2×C3×D411yescharacteristic
191C2×C4×C5C3×#4111yescharacteristic
192C5×C8—21no—
193C5×C8—41no—
194C5×C8—41no—
195C5×C8—41no—
196C5×C8C3×D812yescomplemented
197C2×SD16—21no—
198C2×SD16—21no—
199C2×D8—21no—
200C2×M16—21no—
201C4⋊Q8C22×C3×C511yescharacteristic
202#127C22×C3×C511yescharacteristic
203C4×D4C22×C3×C511yescharacteristic
204#107—21no—
205C4⋊C8C22×C3×C511yescharacteristic
206C4⋊C8C22×C3×C511yescharacteristic
207C4⋊C8C22×C3×C511yescharacteristic
208#102—21no—
209#102—21no—
210C22⋊C8—21no—
211C4×C8C22×C3×C511yescharacteristic
212C2×C3×C5—81no—
213C2×C3×C5—161no—
214C2×C3×C5#45911yescharacteristic
215C2×C3×C5#46811yescharacteristic
216C2×C3×C5#47111yescharacteristic
217C23×C3—21no—
218C23×C3—41no—
219C3×Q8—41no—
220C3×Q8—41no—
221C3×D4—21no—
222C3×D4—21no—
223C3×D4—41no—
224C3×D4—41no—
225C3×D4—41no—
226C3×D4—41no—
227C3×D4—81no—
228C2×C3×C4—21no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4C2×C5×D411yescharacteristic
233C2×C3×C4C2×C5×D411yescharacteristic
234C2×C3×C4C5×#4111yescharacteristic
235C3×C8—21no—
236C3×C8—41no—
237C3×C8—41no—
238C3×C8—41no—
239C3×C8C5×D812yescomplemented
240C22×C5—41no—
241C22×C5—41no—
242C22×C5—41no—
243C22×C5—81no—
244C22×C5—81no—
245C22×C5—81no—
246C22×C5C3×#12111yescharacteristic
247C4×C5—21no—
248C4×C5—41no—
249C4×C5—81no—
250C4×C5C2×C3×D811yescharacteristic
251C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
252C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
253C4×C5C3×#13711yescharacteristic
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4C3×C5×D411yescharacteristic
259C22×C4—21no—
260SD16—42no—
261SD16—42no—
262D8—42no—
263M16—42no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8C3×C5×D411yescharacteristic
268C2×C8C3×C5×D411yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4C3×C5×D411yescharacteristic
272C22⋊C4—21no—
273C42C23×C3×C511yescharacteristic
274C3×C5#168511yescharacteristic, direct product factor, Hall
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—81no—
279C22×C3—81no—
280C22×C3—81no—
281C22×C3C5×#12111yescharacteristic
282C3×C4—21no—
283C3×C4—41no—
284C3×C4—81no—
285C3×C4C2×C5×D811yescharacteristic
286C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
287C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
288C3×C4C5×#13711yescharacteristic
289C2×C5—81no—
290C2×C5—161no—
291C2×C5C3×#45911yescharacteristic
292C2×C5C3×#46811yescharacteristic
293C2×C5C3×#47111yescharacteristic
294C23—21no—
295C23—41no—
296Q8—41no—
297Q8—41no—
298D4—21no—
299D4—21no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—41no—
304D4—81no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4C2×C3×C5×D411yescharacteristic
310C2×C4C2×C3×C5×D411yescharacteristic
311C2×C4C3×C5×#4111yescharacteristic
312C8—21no—
313C8—41no—
314C8—41no—
315C8—41no—
316C8C3×C5×D812yescomplemented
317C2×C3—81no—
318C2×C3—161no—
319C2×C3C5×#45911yescharacteristic
320C2×C3C5×#46811yescharacteristic
321C2×C3C5×#47111yescharacteristic
322C5C3×#168511yescharacteristic, direct product factor, Sylow
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22C3×C5×#12111yescharacteristic
330C4—21no—
331C4—41no—
332C4—81no—
333C4C2×C3×C5×D811yescharacteristic
334C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
335C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
336C4C3×C5×#13711yescharacteristic
337C3C5×#168511yescharacteristic, direct product factor, Sylow
338C2—81no—
339C2—161no—
340C2C3×C5×#45911yescharacteristic
341C2C3×C5×#46811yescharacteristic
342C2C3×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×#492C2semidirect
2. C3×C5×#474C2semidirect
4. C3×C5×#436C2semidirect
5. C3×C5×#335C2semidirect
6. C3×C5×#332C2semidirect
7. C3×C5×#331C2semidirect
8. C5×#1685C3direct (N × H)
24. C3×#1685C5direct (N × H)
25. C5×#492C2×C3semidirect
26. C5×#474C2×C3semidirect
28. C5×#436C2×C3semidirect
29. C5×#335C2×C3semidirect
30. C5×#332C2×C3semidirect
31. C5×#331C2×C3semidirect
52. C3×#492C2×C5semidirect
53. C3×#474C2×C5semidirect
55. C3×#436C2×C5semidirect
56. C3×#335C2×C5semidirect
57. C3×#332C2×C5semidirect
58. C3×#331C2×C5semidirect
74. #1685C3×C5direct (N × H)
97. C3×C5×C8D8semidirect
133. #492C2×C3×C5semidirect
134. #474C2×C3×C5semidirect
136. #436C2×C3×C5semidirect
137. #335C2×C3×C5semidirect
138. #332C2×C3×C5semidirect
139. #331C2×C3×C5semidirect
196. C5×C8C3×D8semidirect
239. C3×C8C5×D8semidirect
274. C3×C5#1685direct (N × H)
316. C8C3×C5×D8semidirect
322. C5C3×#1685direct (N × H)
337. C3C5×#1685direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#4922
C3×C5×#4742
C3×C5×#4592
C3×C5×#4362
C3×C5×#3352
C3×C5×#3322
C3×C5×#3312
C5×#16853
C3×#16855