Finite Group Databasebeta

← C3×C5×#1702 #14274736

Subgroups — C3×C5×#1702 #14274736

Order 1920.

324 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#486C211yesmaximal, characteristic
2C3×C5×#462C211yesmaximal, characteristic, complemented
3C3×C5×#460C211yesmaximal, characteristic
4C3×C5×#446C211yesmaximal, characteristic, complemented
5C3×C5×#334C211yesmaximal, characteristic, complemented
6C3×C5×#332C211yesmaximal, characteristic, complemented
7C3×C5×#331C211yesmaximal, characteristic
8C5×#1702C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×Q16—21no—
10C2×C3×C5×SD16—21no—
11C3×C5×C4⋊Q8C2211yescharacteristic
12C3×C5×#127C2211yescharacteristic
13C3×C4×C5×Q8C2211yescharacteristic
14C3×C5×C4⋊C8—21no—
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×#97—21no—
23C3×C4×C5×C8C2211yescharacteristic
24C3×#1702C511yesmaximal, characteristic, direct product factor, Hall
25C5×#486C2×C311yescharacteristic
26C5×#462C2×C311yescharacteristic, complemented
27C5×#460C2×C311yescharacteristic
28C5×#446C2×C311yescharacteristic, complemented
29C5×#334C2×C311yescharacteristic, complemented
30C5×#332C2×C311yescharacteristic, complemented
31C5×#331C2×C311yescharacteristic
32C2×C3×C5×Q8—21no—
33C2×C3×C5×Q8D411yescharacteristic
34C2×C3×C5×D4—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4—21no—
37C3×C5×Q16—42no—
38C3×C5×SD16—42no—
39C2×C3×C5×C8—21no—
40C2×C3×C5×C8—21no—
41C2×C3×C5×C8—21no—
42C2×C3×C5×C8D411yescharacteristic
43C2×C3×C5×C8D411yescharacteristic
44C3×C5×C4⋊C4—21no—
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4D411yescharacteristic
48C3×C42×C5—21no—
49C3×C42×C5C2311yescharacteristic
50C3×#486C2×C511yescharacteristic
51C3×#462C2×C511yescharacteristic, complemented
52C3×#460C2×C511yescharacteristic
53C3×#446C2×C511yescharacteristic, complemented
54C3×#334C2×C511yescharacteristic, complemented
55C3×#332C2×C511yescharacteristic, complemented
56C3×#331C2×C511yescharacteristic
57C2×C5×Q16—21no—
58C2×C5×SD16—21no—
59C5×C4⋊Q8C22×C311yescharacteristic
60C5×#127C22×C311yescharacteristic
61C4×C5×Q8C22×C311yescharacteristic
62C5×C4⋊C8—21no—
63C5×C4⋊C8C22×C311yescharacteristic
64C5×C4⋊C8C22×C311yescharacteristic
65C5×C4⋊C8C22×C311yescharacteristic
66C5×#103—21no—
67C5×#102—21no—
68C5×#102—21no—
69C5×#102—21no—
70C5×#97—21no—
71C4×C5×C8C22×C311yescharacteristic
72#1702C3×C511yescharacteristic, direct product factor, Sylow
73C23×C3×C5—41no—
74C3×C5×Q8—21no—
75C3×C5×Q8—21no—
76C3×C5×Q8—41no—
77C3×C5×Q8—41no—
78C3×C5×D4—41no—
79C3×C5×D4—41no—
80C3×C5×D4—41no—
81C3×C5×D4—41no—
82C3×C5×D4—81no—
83C2×C3×C4×C5—21no—
84C2×C3×C4×C5—21no—
85C2×C3×C4×C5—41no—
86C2×C3×C4×C5C2×D411yescharacteristic
87C2×C3×C4×C5C2×D411yescharacteristic
88C2×C3×C4×C5#4111yescharacteristic
89C3×C5×C8—21no—
90C3×C5×C8—21no—
91C3×C5×C8—41no—
92C3×C5×C8—41no—
93C3×C5×C8—41no—
94C2×C3×Q16—21no—
95C2×C3×SD16—21no—
96C3×C4⋊Q8C22×C511yescharacteristic
97C3×#127C22×C511yescharacteristic
98C3×C4×Q8C22×C511yescharacteristic
99C3×C4⋊C8—21no—
100C3×C4⋊C8C22×C511yescharacteristic
101C3×C4⋊C8C22×C511yescharacteristic
102C3×C4⋊C8C22×C511yescharacteristic
103C3×#103—21no—
104C3×#102—21no—
105C3×#102—21no—
106C3×#102—21no—
107C3×#97—21no—
108C3×C4×C8C22×C511yescharacteristic
109C2×C5×Q8—21no—
110C2×C5×Q8C3×D411yescharacteristic
111C2×C5×D4—21no—
112C2×C5×D4—21no—
113C2×C5×D4—21no—
114C5×Q16—42no—
115C5×SD16—42no—
116C2×C5×C8—21no—
117C2×C5×C8—21no—
118C2×C5×C8—21no—
119C2×C5×C8C3×D411yescharacteristic
120C2×C5×C8C3×D411yescharacteristic
121C5×C4⋊C4—21no—
122C5×C4⋊C4—21no—
123C5×C4⋊C4—21no—
124C5×C4⋊C4C3×D411yescharacteristic
125C42×C5—21no—
126C42×C5C23×C311yescharacteristic
127#486C2×C3×C511yescharacteristic
128#462C2×C3×C511yescharacteristic, complemented
129#460C2×C3×C511yescharacteristic
130#446C2×C3×C511yescharacteristic, complemented
131#334C2×C3×C511yescharacteristic, complemented
132#332C2×C3×C511yescharacteristic, complemented
133#331C2×C3×C511yescharacteristic
134C22×C3×C5—81no—
135C22×C3×C5—81no—
136C22×C3×C5—81no—
137C22×C3×C5#12111yescharacteristic
138C3×C4×C5—21no—
139C3×C4×C5—41no—
140C3×C4×C5—41no—
141C3×C4×C5—81no—
142C3×C4×C5#13511yescharacteristic
143C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
144C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
145C3×C4×C5#13711yescharacteristic
146C2×C3×Q8—21no—
147C2×C3×Q8C5×D411yescharacteristic
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C3×Q16—42no—
152C3×SD16—42no—
153C2×C3×C8—21no—
154C2×C3×C8—21no—
155C2×C3×C8—21no—
156C2×C3×C8C5×D411yescharacteristic
157C2×C3×C8C5×D411yescharacteristic
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4C5×D411yescharacteristic
162C3×C42—21no—
163C3×C42C23×C511yescharacteristic
164C23×C5—41no—
165C5×Q8—21no—
166C5×Q8—21no—
167C5×Q8—41no—
168C5×Q8—41no—
169C5×D4—41no—
170C5×D4—41no—
171C5×D4—41no—
172C5×D4—41no—
173C5×D4—81no—
174C2×C4×C5—21no—
175C2×C4×C5—21no—
176C2×C4×C5—41no—
177C2×C4×C5C2×C3×D411yescharacteristic
178C2×C4×C5C2×C3×D411yescharacteristic
179C2×C4×C5C3×#4111yescharacteristic
180C5×C8—21no—
181C5×C8—21no—
182C5×C8—41no—
183C5×C8—41no—
184C5×C8—41no—
185C2×Q16—21no—
186C2×SD16—21no—
187C4⋊Q8C22×C3×C511yescharacteristic
188#127C22×C3×C511yescharacteristic
189C4×Q8C22×C3×C511yescharacteristic
190C4⋊C8—21no—
191C4⋊C8C22×C3×C511yescharacteristic
192C4⋊C8C22×C3×C511yescharacteristic
193C4⋊C8C22×C3×C511yescharacteristic
194#103—21no—
195#102—21no—
196#102—21no—
197#102—21no—
198#97—21no—
199C4×C8C22×C3×C511yescharacteristic
200C2×C3×C5—161no—
201C2×C3×C5#46411yescharacteristic
202C2×C3×C5#46911yescharacteristic
203C2×C3×C5#47111yescharacteristic
204C23×C3—41no—
205C3×Q8—21no—
206C3×Q8—21no—
207C3×Q8—41no—
208C3×Q8—41no—
209C3×D4—41no—
210C3×D4—41no—
211C3×D4—41no—
212C3×D4—41no—
213C3×D4—81no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—41no—
217C2×C3×C4C2×C5×D411yescharacteristic
218C2×C3×C4C2×C5×D411yescharacteristic
219C2×C3×C4C5×#4111yescharacteristic
220C3×C8—21no—
221C3×C8—21no—
222C3×C8—41no—
223C3×C8—41no—
224C3×C8—41no—
225C22×C5—81no—
226C22×C5—81no—
227C22×C5—81no—
228C22×C5C3×#12111yescharacteristic
229C4×C5—21no—
230C4×C5—41no—
231C4×C5—41no—
232C4×C5—81no—
233C4×C5C3×#13511yescharacteristic
234C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
235C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
236C4×C5C3×#13711yescharacteristic
237C2×Q8—21no—
238C2×Q8C3×C5×D411yescharacteristic
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242Q16—42no—
243SD16—42no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8C3×C5×D411yescharacteristic
248C2×C8C3×C5×D411yescharacteristic
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4C3×C5×D411yescharacteristic
253C42—21no—
254C42C23×C3×C511yescharacteristic
255C3×C5#170211yescharacteristic, direct product factor, Hall
256C22×C3—81no—
257C22×C3—81no—
258C22×C3—81no—
259C22×C3C5×#12111yescharacteristic
260C3×C4—21no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—81no—
264C3×C4C5×#13511yescharacteristic
265C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
266C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
267C3×C4C5×#13711yescharacteristic
268C2×C5—161no—
269C2×C5C3×#46411yescharacteristic
270C2×C5C3×#46911yescharacteristic
271C2×C5C3×#47111yescharacteristic
272C23—41no—
273Q8—21no—
274Q8—21no—
275Q8—41no—
276Q8—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—81no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4C2×C3×C5×D411yescharacteristic
286C2×C4C2×C3×C5×D411yescharacteristic
287C2×C4C3×C5×#4111yescharacteristic
288C8—21no—
289C8—21no—
290C8—41no—
291C8—41no—
292C8—41no—
293C2×C3—161no—
294C2×C3C5×#46411yescharacteristic
295C2×C3C5×#46911yescharacteristic
296C2×C3C5×#47111yescharacteristic
297C5C3×#170211yescharacteristic, direct product factor, Sylow
298C22—81no—
299C22—81no—
300C22—81no—
301C22C3×C5×#12111yescharacteristic
302C4—21no—
303C4—41no—
304C4—41no—
305C4—81no—
306C4C3×C5×#13511yescharacteristic
307C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
308C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
309C4C3×C5×#13711yescharacteristic
310C3C5×#170211yescharacteristic, direct product factor, Sylow
311C2—161no—
312C2C3×C5×#46411yescharacteristic
313C2C3×C5×#46911yescharacteristic
314C2C3×C5×#47111yescharacteristic
subgroupG/H
center, socleC22×C3×C5#121
commutator, Frattini, cosocleC42C23×C3×C5
prime radicalC3×C5×#127C22
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
2. C3×C5×#462C2semidirect
4. C3×C5×#446C2semidirect
5. C3×C5×#334C2semidirect
6. C3×C5×#332C2semidirect
8. C5×#1702C3direct (N × H)
24. C3×#1702C5direct (N × H)
26. C5×#462C2×C3semidirect
28. C5×#446C2×C3semidirect
29. C5×#334C2×C3semidirect
30. C5×#332C2×C3semidirect
51. C3×#462C2×C5semidirect
53. C3×#446C2×C5semidirect
54. C3×#334C2×C5semidirect
55. C3×#332C2×C5semidirect
72. #1702C3×C5direct (N × H)
128. #462C2×C3×C5semidirect
130. #446C2×C3×C5semidirect
131. #334C2×C3×C5semidirect
132. #332C2×C3×C5semidirect
255. C3×C5#1702direct (N × H)
297. C5C3×#1702direct (N × H)
310. C3C5×#1702direct (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×#4622
C3×C5×#4602
C3×C5×#4462
C3×C5×#3342
C3×C5×#3322
C3×C5×#3312
C5×#17023
C3×#17025