Finite Group Databasebeta

← C3×C5×#2080 #14275114

Subgroups — C3×C5×#2080 #14275114

Order 1920.

380 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#424C211yesmaximal, characteristic, complemented
2C3×C5×#421C211yesmaximal, characteristic
3C3×C5×#416C211yesmaximal, characteristic
4C3×C5×#413C211yesmaximal, characteristic
5C3×C5×#344C211yesmaximal, characteristic, complemented
6C3×C5×#341C211yesmaximal, characteristic, complemented
7C3×C5×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C5×#2080C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C3×C5×M16C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C2×C3×C5×M16C2211yescharacteristic
13C22×C3×C5×C8C2211yescharacteristic
14C3×C5×#117C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16C3×C5×C4⋊C8—22no—
17C3×C5×#104—22no—
18C3×C5×#103—21no—
19C3×C5×#102—21no—
20C3×C5×#101—21no—
21C3×C5×#100—21no—
22C3×#2080C511yesmaximal, characteristic, direct product factor, Hall
23C5×#424C2×C311yescharacteristic, complemented
24C5×#421C2×C311yescharacteristic
25C5×#416C2×C311yescharacteristic
26C5×#413C2×C311yescharacteristic
27C5×#344C2×C311yescharacteristic, complemented
28C5×#341C2×C311yescharacteristic, complemented
29C5×H(3,ℤ/4)C2×C311yescharacteristic, complemented
30C3×C5×#41—21no—
31C3×C5×#41—21no—
32C3×C5×#41—21no—
33C3×C5×#41—21no—
34C2×C3×C5×Q8D411yescharacteristic
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4D411yescharacteristic
37C22×C3×C4×C5—21no—
38C22×C3×C4×C5C2311yescharacteristic
39C3×C5×M16—21no—
40C3×C5×M16—21no—
41C3×C5×M16—21no—
42C3×C5×M16—21no—
43C3×C5×M16—21no—
44C3×C5×M16—21no—
45C2×C3×C5×C8—21no—
46C2×C3×C5×C8—21no—
47C2×C3×C5×C8—21no—
48C2×C3×C5×C8—21no—
49C2×C3×C5×C8—21no—
50C2×C3×C5×C8—21no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C4⋊C4—21no—
53C3×C5×C22⋊C4—21no—
54C3×C5×C22⋊C4—21no—
55C3×C42×C5—21no—
56C3×C42×C5D412yes—
57C3×#424C2×C511yescharacteristic, complemented
58C3×#421C2×C511yescharacteristic
59C3×#416C2×C511yescharacteristic
60C3×#413C2×C511yescharacteristic
61C3×#344C2×C511yescharacteristic, complemented
62C3×#341C2×C511yescharacteristic, complemented
63C3×H(3,ℤ/4)C2×C511yescharacteristic, complemented
64C2×C5×#41C22×C311yescharacteristic
65C2×C5×M16C22×C311yescharacteristic
66C2×C5×M16C22×C311yescharacteristic
67C2×C5×M16C22×C311yescharacteristic
68C22×C5×C8C22×C311yescharacteristic
69C5×#117C22×C311yescharacteristic
70C5×#117C22×C311yescharacteristic
71C5×C4⋊C8—22no—
72C5×#104—22no—
73C5×#103—21no—
74C5×#102—21no—
75C5×#101—21no—
76C5×#100—21no—
77#2080C3×C511yescharacteristic, direct product factor, Sylow
78C23×C3×C5—21no—
79C23×C3×C5#4111yescharacteristic
80C3×C5×Q8—21no—
81C3×C5×Q8—21no—
82C3×C5×D4—21no—
83C3×C5×D4—21no—
84C3×C5×D4—41no—
85C3×C5×D4—41no—
86C2×C3×C4×C5—22no—
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5—41no—
89C2×C3×C4×C5—41no—
90C2×C3×C4×C5—41no—
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C3×C4×C5C2×D411yescharacteristic
93C2×C3×C4×C5#4111yescharacteristic
94C2×C3×C4×C5#4111yescharacteristic
95C2×C3×C4×C5#4111yescharacteristic
96C2×C3×C4×C5#4111yescharacteristic
97C3×C5×C8—41no—
98C3×C5×C8—41no—
99C3×C5×C8—41no—
100C3×C5×C8—41no—
101C2×C3×#41C22×C511yescharacteristic
102C2×C3×M16C22×C511yescharacteristic
103C2×C3×M16C22×C511yescharacteristic
104C2×C3×M16C22×C511yescharacteristic
105C22×C3×C8C22×C511yescharacteristic
106C3×#117C22×C511yescharacteristic
107C3×#117C22×C511yescharacteristic
108C3×C4⋊C8—22no—
109C3×#104—22no—
110C3×#103—21no—
111C3×#102—21no—
112C3×#101—21no—
113C3×#100—21no—
114C5×#41—21no—
115C5×#41—21no—
116C5×#41—21no—
117C5×#41—21no—
118C2×C5×Q8C3×D411yescharacteristic
119C2×C5×D4—21no—
120C2×C5×D4C3×D411yescharacteristic
121C22×C4×C5—21no—
122C22×C4×C5C23×C311yescharacteristic
123C5×M16—21no—
124C5×M16—21no—
125C5×M16—21no—
126C5×M16—21no—
127C5×M16—21no—
128C5×M16—21no—
129C2×C5×C8—21no—
130C2×C5×C8—21no—
131C2×C5×C8—21no—
132C2×C5×C8—21no—
133C2×C5×C8—21no—
134C2×C5×C8—21no—
135C5×C4⋊C4—21no—
136C5×C4⋊C4—21no—
137C5×C22⋊C4—21no—
138C5×C22⋊C4—21no—
139C42×C5—21no—
140C42×C5C3×D412yes—
141#424C2×C3×C511yescharacteristic, complemented
142#421C2×C3×C511yescharacteristic
143#416C2×C3×C511yescharacteristic
144#413C2×C3×C511yescharacteristic
145#344C2×C3×C511yescharacteristic, complemented
146#341C2×C3×C511yescharacteristic, complemented
147H(3,ℤ/4)C2×C3×C511yescharacteristic, complemented
148C22×C3×C5—41no—
149C22×C3×C5—41no—
150C22×C3×C5—81no—
151C22×C3×C5#12311yescharacteristic
152C22×C3×C5#12311yescharacteristic
153C22×C3×C5#12611yescharacteristic
154C3×C4×C5—42no—
155C3×C4×C5—41no—
156C3×C4×C5—81no—
157C3×C4×C5#12111yescharacteristic
158C3×C4×C5#12311yescharacteristic
159C3×C4×C5#12411yescharacteristic
160C3×C4×C5#12611yescharacteristic
161C3×#41—21no—
162C3×#41—21no—
163C3×#41—21no—
164C3×#41—21no—
165C2×C3×Q8C5×D411yescharacteristic
166C2×C3×D4—21no—
167C2×C3×D4C5×D411yescharacteristic
168C22×C3×C4—21no—
169C22×C3×C4C23×C511yescharacteristic
170C3×M16—21no—
171C3×M16—21no—
172C3×M16—21no—
173C3×M16—21no—
174C3×M16—21no—
175C3×M16—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C22⋊C4—21no—
185C3×C22⋊C4—21no—
186C3×C42—21no—
187C3×C42C5×D412yes—
188C23×C5—21no—
189C23×C5C3×#4111yescharacteristic
190C5×Q8—21no—
191C5×Q8—21no—
192C5×D4—21no—
193C5×D4—21no—
194C5×D4—41no—
195C5×D4—41no—
196C2×C4×C5—22no—
197C2×C4×C5—21no—
198C2×C4×C5—41no—
199C2×C4×C5—41no—
200C2×C4×C5—41no—
201C2×C4×C5C2×C3×D411yescharacteristic
202C2×C4×C5C2×C3×D411yescharacteristic
203C2×C4×C5C3×#4111yescharacteristic
204C2×C4×C5C3×#4111yescharacteristic
205C2×C4×C5C3×#4111yescharacteristic
206C2×C4×C5C3×#4111yescharacteristic
207C5×C8—41no—
208C5×C8—41no—
209C5×C8—41no—
210C5×C8—41no—
211C2×#41C22×C3×C511yescharacteristic
212C2×M16C22×C3×C511yescharacteristic
213C2×M16C22×C3×C511yescharacteristic
214C2×M16C22×C3×C511yescharacteristic
215C22×C8C22×C3×C511yescharacteristic
216#117C22×C3×C511yescharacteristic
217#117C22×C3×C511yescharacteristic
218C4⋊C8—22no—
219#104—22no—
220#103—21no—
221#102—21no—
222#101—21no—
223#100—21no—
224C2×C3×C5—21no—
225C2×C3×C5—21no—
226C2×C3×C5—21no—
227C2×C3×C5—81no—
228C2×C3×C5#39711yescharacteristic
229C23×C3—21no—
230C23×C3C5×#4111yescharacteristic
231C3×Q8—21no—
232C3×Q8—21no—
233C3×D4—21no—
234C3×D4—21no—
235C3×D4—41no—
236C3×D4—41no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4C2×C5×D411yescharacteristic
243C2×C3×C4C2×C5×D411yescharacteristic
244C2×C3×C4C5×#4111yescharacteristic
245C2×C3×C4C5×#4111yescharacteristic
246C2×C3×C4C5×#4111yescharacteristic
247C2×C3×C4C5×#4111yescharacteristic
248C3×C8—41no—
249C3×C8—41no—
250C3×C8—41no—
251C3×C8—41no—
252C22×C5—41no—
253C22×C5—41no—
254C22×C5—81no—
255C22×C5C3×#12311yescharacteristic
256C22×C5C3×#12311yescharacteristic
257C22×C5C3×#12611yescharacteristic
258C4×C5—42no—
259C4×C5—41no—
260C4×C5—81no—
261C4×C5C3×#12111yescharacteristic
262C4×C5C3×#12311yescharacteristic
263C4×C5C3×#12411yescharacteristic
264C4×C5C3×#12611yescharacteristic
265#41—21no—
266#41—21no—
267#41—21no—
268#41—21no—
269C2×Q8C3×C5×D411yescharacteristic
270C2×D4—21no—
271C2×D4C3×C5×D411yescharacteristic
272C22×C4—21no—
273C22×C4C23×C3×C511yescharacteristic
274M16—21no—
275M16—21no—
276M16—21no—
277M16—21no—
278M16—21no—
279M16—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C42—21no—
291C42C3×C5×D412yes—
292C3×C5#208011yescharacteristic, direct product factor, Hall
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—81no—
296C22×C3C5×#12311yescharacteristic
297C22×C3C5×#12311yescharacteristic
298C22×C3C5×#12611yescharacteristic
299C3×C4—42no—
300C3×C4—41no—
301C3×C4—81no—
302C3×C4C5×#12111yescharacteristic
303C3×C4C5×#12311yescharacteristic
304C3×C4C5×#12411yescharacteristic
305C3×C4C5×#12611yescharacteristic
306C2×C5—21no—
307C2×C5—21no—
308C2×C5—21no—
309C2×C5—81no—
310C2×C5C3×#39711yescharacteristic
311C23—21no—
312C23C3×C5×#4111yescharacteristic
313Q8—21no—
314Q8—21no—
315D4—21no—
316D4—21no—
317D4—41no—
318D4—41no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4C2×C3×C5×D411yescharacteristic
325C2×C4C2×C3×C5×D411yescharacteristic
326C2×C4C3×C5×#4111yescharacteristic
327C2×C4C3×C5×#4111yescharacteristic
328C2×C4C3×C5×#4111yescharacteristic
329C2×C4C3×C5×#4111yescharacteristic
330C8—41no—
331C8—41no—
332C8—41no—
333C8—41no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—81no—
338C2×C3C5×#39711yescharacteristic
339C5C3×#208011yescharacteristic, direct product factor, Sylow
340C22—41no—
341C22—41no—
342C22—81no—
343C22C3×C5×#12311yescharacteristic
344C22C3×C5×#12311yescharacteristic
345C22C3×C5×#12611yescharacteristic
346C4—42no—
347C4—41no—
348C4—81no—
349C4C3×C5×#12111yescharacteristic
350C4C3×C5×#12311yescharacteristic
351C4C3×C5×#12411yescharacteristic
352C4C3×C5×#12611yescharacteristic
353C3C5×#208011yescharacteristic, direct product factor, Sylow
354C2—21no—
355C2—21no—
356C2—21no—
357C2—81no—
358C2C3×C5×#39711yescharacteristic
subgroupG/H
centerC3×C4×C5#124
commutator, Frattini, cosocleC22×C4C23×C3×C5
socleC2×C3×C5#397
prime radicalC2×C3×C5×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4×C5 ◁ C22×C3×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C5×#424C2semidirect
5. C3×C5×#344C2semidirect
6. C3×C5×#341C2semidirect
7. C3×C5×H(3,ℤ/4)C2semidirect
8. C5×#2080C3direct (N × H)
22. C3×#2080C5direct (N × H)
23. C5×#424C2×C3semidirect
27. C5×#344C2×C3semidirect
28. C5×#341C2×C3semidirect
29. C5×H(3,ℤ/4)C2×C3semidirect
57. C3×#424C2×C5semidirect
61. C3×#344C2×C5semidirect
62. C3×#341C2×C5semidirect
63. C3×H(3,ℤ/4)C2×C5semidirect
77. #2080C3×C5direct (N × H)
141. #424C2×C3×C5semidirect
145. #344C2×C3×C5semidirect
146. #341C2×C3×C5semidirect
147. H(3,ℤ/4)C2×C3×C5semidirect
292. C3×C5#2080direct (N × H)
339. C5C3×#2080direct (N × H)
353. C3C5×#2080direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#4242
C3×C5×#4212
C3×C5×#4162
C3×C5×#4132
C3×C5×#3442
C3×C5×#3412
C3×C5×H(3,ℤ/4)2
C5×#20803
C3×#20805