Finite Group Databasebeta

← C3×C5×#1617 #14274651

Subgroups — C3×C5×#1617 #14274651

Order 1920.

404 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#544C211yesmaximal, characteristic
2C3×C5×#484C212yesmaximal
3C3×C5×#400C211yesmaximal, characteristic
4C3×C5×#328C212yesmaximal
5C3×C5×#324C211yesmaximal, characteristic
6C5×#1617C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×C4⋊Q8—22no—
8C3×C5×#125—22no—
9C3×C5×C22⋊Q8—21no—
10C3×C5×C22⋊Q8C2212yes—
11C3×C5×#117—21no—
12C2×C3×C5×C4⋊C4C2211yescharacteristic
13C2×C3×C5×C4⋊C4C2211yescharacteristic
14C3×C5×#107—22no—
15C3×C5×#103—22no—
16C3×C5×C22⋊C8C2212yes—
17C3×C5×#95—21no—
18C3×C5×#95—21no—
19C3×C5×#95C2211yescharacteristic
20C3×#1617C511yesmaximal, characteristic, direct product factor, Hall
21C5×#544C2×C311yescharacteristic
22C5×#484C2×C312yes—
23C5×#400C2×C311yescharacteristic
24C5×#328C2×C312yes—
25C5×#324C2×C311yescharacteristic
26C2×C3×C5×Q8—21no—
27C2×C3×C5×Q8D412yes—
28C22×C3×C4×C5—21no—
29C22×C3×C4×C5—21no—
30C22×C3×C4×C5D411yescharacteristic
31C22×C3×C4×C5D411yescharacteristic
32C22×C3×C4×C5C2311yescharacteristic
33C2×C3×C5×C8—22no—
34C3×C5×C4⋊C4—21no—
35C3×C5×C4⋊C4—21no—
36C3×C5×C4⋊C4—21no—
37C3×C5×C4⋊C4—22no—
38C3×C5×C4⋊C4—22no—
39C3×C5×C4⋊C4—22no—
40C3×C5×C4⋊C4—22no—
41C3×C5×C4⋊C4D412yes—
42C3×C5×C22⋊C4—22no—
43C3×C42×C5—22no—
44C3×#544C2×C511yescharacteristic
45C3×#484C2×C512yes—
46C3×#400C2×C511yescharacteristic
47C3×#328C2×C512yes—
48C3×#324C2×C511yescharacteristic
49C5×C4⋊Q8—22no—
50C5×#125—22no—
51C5×C22⋊Q8—21no—
52C5×C22⋊Q8C22×C312yes—
53C5×#117—21no—
54C2×C5×C4⋊C4C22×C311yescharacteristic
55C2×C5×C4⋊C4C22×C311yescharacteristic
56C5×#107—22no—
57C5×#103—22no—
58C5×C22⋊C8C22×C312yes—
59C5×#95—21no—
60C5×#95—21no—
61C5×#95C22×C311yescharacteristic
62#1617C3×C511yescharacteristic, direct product factor, Sylow
63C23×C3×C5C2×D411yescharacteristic
64C3×C5×Q8—42no—
65C3×C5×Q8—81no—
66C2×C3×C4×C5—21no—
67C2×C3×C4×C5—21no—
68C2×C3×C4×C5—22no—
69C2×C3×C4×C5—22no—
70C2×C3×C4×C5—21no—
71C2×C3×C4×C5—41no—
72C2×C3×C4×C5—41no—
73C2×C3×C4×C5—41no—
74C2×C3×C4×C5—41no—
75C2×C3×C4×C5—42no—
76C2×C3×C4×C5—41no—
77C2×C3×C4×C5—41no—
78C2×C3×C4×C5Q1612yes—
79C2×C3×C4×C5C2×D412yes—
80C3×C5×C8—42no—
81C3×C4⋊Q8—22no—
82C3×#125—22no—
83C3×C22⋊Q8—21no—
84C3×C22⋊Q8C22×C512yes—
85C3×#117—21no—
86C2×C3×C4⋊C4C22×C511yescharacteristic
87C2×C3×C4⋊C4C22×C511yescharacteristic
88C3×#107—22no—
89C3×#103—22no—
90C3×C22⋊C8C22×C512yes—
91C3×#95—21no—
92C3×#95—21no—
93C3×#95C22×C511yescharacteristic
94C2×C5×Q8—21no—
95C2×C5×Q8C3×D412yes—
96C22×C4×C5—21no—
97C22×C4×C5—21no—
98C22×C4×C5C3×D411yescharacteristic
99C22×C4×C5C3×D411yescharacteristic
100C22×C4×C5C23×C311yescharacteristic
101C2×C5×C8—22no—
102C5×C4⋊C4—21no—
103C5×C4⋊C4—21no—
104C5×C4⋊C4—21no—
105C5×C4⋊C4—22no—
106C5×C4⋊C4—22no—
107C5×C4⋊C4—22no—
108C5×C4⋊C4—22no—
109C5×C4⋊C4C3×D412yes—
110C5×C22⋊C4—22no—
111C42×C5—22no—
112#544C2×C3×C511yescharacteristic
113#484C2×C3×C512yes—
114#400C2×C3×C511yescharacteristic
115#328C2×C3×C512yes—
116#324C2×C3×C511yescharacteristic
117C22×C3×C5—21no—
118C22×C3×C5—21no—
119C22×C3×C5#12011yescharacteristic
120C22×C3×C5C2×Q1611yescharacteristic
121C22×C3×C5#13711yescharacteristic
122C3×C4×C5—22no—
123C3×C4×C5—22no—
124C3×C4×C5—42no—
125C3×C4×C5—41no—
126C3×C4×C5—42no—
127C3×C4×C5—81no—
128C3×C4×C5—81no—
129C2×C3×Q8—21no—
130C2×C3×Q8C5×D412yes—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4C5×D411yescharacteristic
134C22×C3×C4C5×D411yescharacteristic
135C22×C3×C4C23×C511yescharacteristic
136C2×C3×C8—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4C5×D412yes—
145C3×C22⋊C4—22no—
146C3×C42—22no—
147C23×C5C2×C3×D411yescharacteristic
148C5×Q8—42no—
149C5×Q8—81no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—22no—
153C2×C4×C5—22no—
154C2×C4×C5—21no—
155C2×C4×C5—41no—
156C2×C4×C5—41no—
157C2×C4×C5—41no—
158C2×C4×C5—41no—
159C2×C4×C5—42no—
160C2×C4×C5—41no—
161C2×C4×C5—41no—
162C2×C4×C5C3×Q1612yes—
163C2×C4×C5C2×C3×D412yes—
164C5×C8—42no—
165C4⋊Q8—22no—
166#125—22no—
167C22⋊Q8—21no—
168C22⋊Q8C22×C3×C512yes—
169#117—21no—
170C2×C4⋊C4C22×C3×C511yescharacteristic
171C2×C4⋊C4C22×C3×C511yescharacteristic
172#107—22no—
173#103—22no—
174C22⋊C8C22×C3×C512yes—
175#95—21no—
176#95—21no—
177#95C22×C3×C511yescharacteristic
178C2×C3×C5—21no—
179C2×C3×C5—21no—
180C2×C3×C5#45111yescharacteristic
181C2×C3×C5#45411yescharacteristic
182C2×C3×C5#45811yescharacteristic
183C23×C3C2×C5×D411yescharacteristic
184C3×Q8—42no—
185C3×Q8—81no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—21no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—42no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4C5×Q1612yes—
199C2×C3×C4C2×C5×D412yes—
200C3×C8—42no—
201C22×C5—21no—
202C22×C5—21no—
203C22×C5C3×#12011yescharacteristic
204C22×C5C2×C3×Q1611yescharacteristic
205C22×C5C3×#13711yescharacteristic
206C4×C5—22no—
207C4×C5—22no—
208C4×C5—42no—
209C4×C5—41no—
210C4×C5—42no—
211C4×C5—81no—
212C4×C5—81no—
213C2×Q8—21no—
214C2×Q8C3×C5×D412yes—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4C3×C5×D411yescharacteristic
218C22×C4C3×C5×D411yescharacteristic
219C22×C4C23×C3×C511yescharacteristic
220C2×C8—22no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4C3×C5×D412yes—
229C22⋊C4—22no—
230C42—22no—
231C3×C5#161711yescharacteristic, direct product factor, Hall
232C22×C3—21no—
233C22×C3—21no—
234C22×C3C5×#12011yescharacteristic
235C22×C3C2×C5×Q1611yescharacteristic
236C22×C3C5×#13711yescharacteristic
237C3×C4—22no—
238C3×C4—22no—
239C3×C4—42no—
240C3×C4—41no—
241C3×C4—42no—
242C3×C4—81no—
243C3×C4—81no—
244C2×C5—21no—
245C2×C5—21no—
246C2×C5C3×#45111yescharacteristic
247C2×C5C3×#45411yescharacteristic
248C2×C5C3×#45811yescharacteristic
249C23C2×C3×C5×D411yescharacteristic
250Q8—42no—
251Q8—81no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4C3×C5×Q1612yes—
265C2×C4C2×C3×C5×D412yes—
266C8—42no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3C5×#45111yescharacteristic
270C2×C3C5×#45411yescharacteristic
271C2×C3C5×#45811yescharacteristic
272C5C3×#161711yescharacteristic, direct product factor, Sylow
273C22—21no—
274C22—21no—
275C22C3×C5×#12011yescharacteristic
276C22C2×C3×C5×Q1611yescharacteristic
277C22C3×C5×#13711yescharacteristic
278C4—22no—
279C4—22no—
280C4—42no—
281C4—41no—
282C4—42no—
283C4—81no—
284C4—81no—
285C3C5×#161711yescharacteristic, direct product factor, Sylow
286C2—21no—
287C2—21no—
288C2C3×C5×#45111yescharacteristic
289C2C3×C5×#45411yescharacteristic
290C2C3×C5×#45811yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC22×C4C23×C3×C5
prime radicalC23×C3×C5C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C22×C3×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
6. C5×#1617C3direct (N × H)
20. C3×#1617C5direct (N × H)
62. #1617C3×C5direct (N × H)
231. C3×C5#1617direct (N × H)
272. C5C3×#1617direct (N × H)
285. C3C5×#1617direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5442
C3×C5×#4842
C3×C5×#4002
C3×C5×#3282
C3×C5×#3242
C5×#16173
C3×#16175