Finite Group Databasebeta

← C3×C5×#2201 #14275235

Subgroups — C3×C5×#2201 #14275235

Order 1920.

452 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#563C211yesmaximal, characteristic, complemented
2C3×C5×#458C212yesmaximal
3C3×C5×#455C211yesmaximal, characteristic
4C3×C5×#354C211yesmaximal, characteristic, complemented
5C3×C5×#352C212yesmaximal, complemented
6C5×#2201C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×H(5,𝔽2)C2211yescharacteristic
8C3×C5×#137—21no—
9C3×C5×C4⋊Q8—21no—
10C3×C5×#126—22no—
11C3×C5×#125—21no—
12C3×C5×#124C2211yescharacteristic
13C3×C5×#123—22no—
14C3×C5×#123C2212yes—
15C3×C5×C22⋊Q8—22no—
16C3×C5×#104—21no—
17C3×C5×#100C2211yescharacteristic
18C3×C5×#99—22no—
19C3×C5×#99C2212yes—
20C3×#2201C511yesmaximal, characteristic, direct product factor, Hall
21C5×#563C2×C311yescharacteristic, complemented
22C5×#458C2×C312yes—
23C5×#455C2×C311yescharacteristic
24C5×#354C2×C311yescharacteristic, complemented
25C5×#352C2×C312yescomplemented
26C3×C5×#41—21no—
27C3×C5×#41—41no—
28C2×C3×C5×Q8—21no—
29C2×C3×C5×Q8D411yescharacteristic
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4—41no—
32C2×C3×C5×D4C2311yescharacteristic
33C22×C3×C4×C5D412yes—
34C3×C5×Q16—41no—
35C3×C5×SD16—41no—
36C3×C5×M16—21no—
37C3×C5×C4⋊C4—21no—
38C3×C5×C4⋊C4—22no—
39C3×C5×C4⋊C4—22no—
40C3×C5×C4⋊C4—21no—
41C3×C5×C4⋊C4—22no—
42C3×C5×C4⋊C4—42no—
43C3×C5×C22⋊C4—22no—
44C3×C5×C22⋊C4—22no—
45C3×C5×C22⋊C4—22no—
46C3×C5×C22⋊C4—42no—
47C3×C5×C22⋊C4D412yes—
48C3×C42×C5—21no—
49C3×C42×C5D411yescharacteristic
50C3×#563C2×C511yescharacteristic, complemented
51C3×#458C2×C512yes—
52C3×#455C2×C511yescharacteristic
53C3×#354C2×C511yescharacteristic, complemented
54C3×#352C2×C512yescomplemented
55C5×H(5,𝔽2)C22×C311yescharacteristic
56C5×#137—21no—
57C5×C4⋊Q8—21no—
58C5×#126—22no—
59C5×#125—21no—
60C5×#124C22×C311yescharacteristic
61C5×#123—22no—
62C5×#123C22×C312yes—
63C5×C22⋊Q8—22no—
64C5×#104—21no—
65C5×#100C22×C311yescharacteristic
66C5×#99—22no—
67C5×#99C22×C312yes—
68#2201C3×C511yescharacteristic, direct product factor, Sylow
69C23×C3×C5—22no—
70C23×C3×C5C2×D412yes—
71C3×C5×Q8—21no—
72C3×C5×Q8—41no—
73C3×C5×Q8—81no—
74C3×C5×D4—21no—
75C3×C5×D4—41no—
76C3×C5×D4—42no—
77C3×C5×D4—41no—
78C2×C3×C4×C5—22no—
79C2×C3×C4×C5—21no—
80C2×C3×C4×C5—22no—
81C2×C3×C4×C5—21no—
82C2×C3×C4×C5—22no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5—42no—
85C2×C3×C4×C5—42no—
86C2×C3×C4×C5C2×D411yescharacteristic
87C3×C5×C8—41no—
88C3×H(5,𝔽2)C22×C511yescharacteristic
89C3×#137—21no—
90C3×C4⋊Q8—21no—
91C3×#126—22no—
92C3×#125—21no—
93C3×#124C22×C511yescharacteristic
94C3×#123—22no—
95C3×#123C22×C512yes—
96C3×C22⋊Q8—22no—
97C3×#104—21no—
98C3×#100C22×C511yescharacteristic
99C3×#99—22no—
100C3×#99C22×C512yes—
101C5×#41—21no—
102C5×#41—41no—
103C2×C5×Q8—21no—
104C2×C5×Q8C3×D411yescharacteristic
105C2×C5×D4—22no—
106C2×C5×D4—41no—
107C2×C5×D4C23×C311yescharacteristic
108C22×C4×C5C3×D412yes—
109C5×Q16—41no—
110C5×SD16—41no—
111C5×M16—21no—
112C5×C4⋊C4—21no—
113C5×C4⋊C4—22no—
114C5×C4⋊C4—22no—
115C5×C4⋊C4—21no—
116C5×C4⋊C4—22no—
117C5×C4⋊C4—42no—
118C5×C22⋊C4—22no—
119C5×C22⋊C4—22no—
120C5×C22⋊C4—22no—
121C5×C22⋊C4—42no—
122C5×C22⋊C4C3×D412yes—
123C42×C5—21no—
124C42×C5C3×D411yescharacteristic
125#563C2×C3×C511yescharacteristic, complemented
126#458C2×C3×C512yes—
127#455C2×C3×C511yescharacteristic
128#354C2×C3×C511yescharacteristic, complemented
129#352C2×C3×C512yescomplemented
130C22×C3×C5—22no—
131C22×C3×C5—42no—
132C22×C3×C5—41no—
133C22×C3×C5—82no—
134C22×C3×C5#12011yescharacteristic
135C3×C4×C5—21no—
136C3×C4×C5—42no—
137C3×C4×C5—41no—
138C3×C4×C5—41no—
139C3×C4×C5—42no—
140C3×C4×C5—41no—
141C3×C4×C5—82no—
142C3×#41—21no—
143C3×#41—41no—
144C2×C3×Q8—21no—
145C2×C3×Q8C5×D411yescharacteristic
146C2×C3×D4—22no—
147C2×C3×D4—41no—
148C2×C3×D4C23×C511yescharacteristic
149C22×C3×C4C5×D412yes—
150C3×Q16—41no—
151C3×SD16—41no—
152C3×M16—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—42no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—22no—
161C3×C22⋊C4—22no—
162C3×C22⋊C4—42no—
163C3×C22⋊C4C5×D412yes—
164C3×C42—21no—
165C3×C42C5×D411yescharacteristic
166C23×C5—22no—
167C23×C5C2×C3×D412yes—
168C5×Q8—21no—
169C5×Q8—41no—
170C5×Q8—81no—
171C5×D4—21no—
172C5×D4—41no—
173C5×D4—42no—
174C5×D4—41no—
175C2×C4×C5—22no—
176C2×C4×C5—21no—
177C2×C4×C5—22no—
178C2×C4×C5—21no—
179C2×C4×C5—22no—
180C2×C4×C5—41no—
181C2×C4×C5—42no—
182C2×C4×C5—42no—
183C2×C4×C5C2×C3×D411yescharacteristic
184C5×C8—41no—
185H(5,𝔽2)C22×C3×C511yescharacteristic
186#137—21no—
187C4⋊Q8—21no—
188#126—22no—
189#125—21no—
190#124C22×C3×C511yescharacteristic
191#123—22no—
192#123C22×C3×C512yes—
193C22⋊Q8—22no—
194#104—21no—
195#100C22×C3×C511yescharacteristic
196#99—22no—
197#99C22×C3×C512yes—
198C2×C3×C5—21no—
199C2×C3×C5—42no—
200C2×C3×C5—81no—
201C2×C3×C5B(4,𝔽2)11yescharacteristic
202C23×C3—22no—
203C23×C3C2×C5×D412yes—
204C3×Q8—21no—
205C3×Q8—41no—
206C3×Q8—81no—
207C3×D4—21no—
208C3×D4—41no—
209C3×D4—42no—
210C3×D4—41no—
211C2×C3×C4—22no—
212C2×C3×C4—21no—
213C2×C3×C4—22no—
214C2×C3×C4—21no—
215C2×C3×C4—22no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4C2×C5×D411yescharacteristic
220C3×C8—41no—
221C22×C5—22no—
222C22×C5—42no—
223C22×C5—41no—
224C22×C5—82no—
225C22×C5C3×#12011yescharacteristic
226C4×C5—21no—
227C4×C5—42no—
228C4×C5—41no—
229C4×C5—41no—
230C4×C5—42no—
231C4×C5—41no—
232C4×C5—82no—
233#41—21no—
234#41—41no—
235C2×Q8—21no—
236C2×Q8C3×C5×D411yescharacteristic
237C2×D4—22no—
238C2×D4—41no—
239C2×D4C23×C3×C511yescharacteristic
240C22×C4C3×C5×D412yes—
241Q16—41no—
242SD16—41no—
243M16—21no—
244C4⋊C4—21no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—21no—
248C4⋊C4—22no—
249C4⋊C4—42no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—22no—
253C22⋊C4—42no—
254C22⋊C4C3×C5×D412yes—
255C42—21no—
256C42C3×C5×D411yescharacteristic
257C3×C5#220111yescharacteristic, direct product factor, Hall
258C22×C3—22no—
259C22×C3—42no—
260C22×C3—41no—
261C22×C3—82no—
262C22×C3C5×#12011yescharacteristic
263C3×C4—21no—
264C3×C4—42no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—42no—
268C3×C4—41no—
269C3×C4—82no—
270C2×C5—21no—
271C2×C5—42no—
272C2×C5—81no—
273C2×C5C3×B(4,𝔽2)11yescharacteristic
274C23—22no—
275C23C2×C3×C5×D412yes—
276Q8—21no—
277Q8—41no—
278Q8—81no—
279D4—21no—
280D4—41no—
281D4—42no—
282D4—41no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4C2×C3×C5×D411yescharacteristic
292C8—41no—
293C2×C3—21no—
294C2×C3—42no—
295C2×C3—81no—
296C2×C3C5×B(4,𝔽2)11yescharacteristic
297C5C3×#220111yescharacteristic, direct product factor, Sylow
298C22—22no—
299C22—42no—
300C22—41no—
301C22—82no—
302C22C3×C5×#12011yescharacteristic
303C4—21no—
304C4—42no—
305C4—41no—
306C4—41no—
307C4—42no—
308C4—41no—
309C4—82no—
310C3C5×#220111yescharacteristic, direct product factor, Sylow
311C2—21no—
312C2—42no—
313C2—81no—
314C2C3×C5×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C3×C5B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C23×C3×C5
prime radicalC3×C5×H(5,𝔽2)C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C5 ◁ C22×C3×C5 ◁ C2×C3×C5×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C5×#563C2semidirect
4. C3×C5×#354C2semidirect
5. C3×C5×#352C2semidirect
6. C5×#2201C3direct (N × H)
20. C3×#2201C5direct (N × H)
21. C5×#563C2×C3semidirect
24. C5×#354C2×C3semidirect
25. C5×#352C2×C3semidirect
50. C3×#563C2×C5semidirect
53. C3×#354C2×C5semidirect
54. C3×#352C2×C5semidirect
68. #2201C3×C5direct (N × H)
125. #563C2×C3×C5semidirect
128. #354C2×C3×C5semidirect
129. #352C2×C3×C5semidirect
257. C3×C5#2201direct (N × H)
297. C5C3×#2201direct (N × H)
310. C3C5×#2201direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5632
C3×C5×#4582
C3×C5×#4552
C3×C5×#3542
C3×C5×#3522
C5×#22013
C3×#22015