Finite Group Databasebeta

← C3×C5×#2204 #14275238

Subgroups — C3×C5×#2204 #14275238

Order 1920.

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

normal subgroups only

This group has 472 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×#555C211yesmaximal, characteristic, complemented
2C3×C5×#456C211yesmaximal, characteristic, complemented
3C3×C5×#454C212yesmaximal, complemented
4C3×C5×#356C211yesmaximal, characteristic, complemented
5C3×C5×#355C212yesmaximal, complemented
6C5×#2204C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×2−1+4C2211yescharacteristic
8C3×C5×#137—21no—
9C3×C5×AGL(1,ℤ/8)—22no—
10C3×C5×C4⋊Q8C2211yescharacteristic
11C3×C5×#124—21no—
12C3×C5×#124C2212yes—
13C3×C5×#121—22no—
14C3×C4×C5×D4—21no—
15C3×C5×#117—22no—
16C3×C5×#104—22no—
17C3×C5×#104—21no—
18C3×C5×#101C2212yes—
19C3×C5×#101C2211yescharacteristic
20C3×#2204C511yesmaximal, characteristic, direct product factor, Hall
21C5×#555C2×C311yescharacteristic, complemented
22C5×#456C2×C311yescharacteristic, complemented
23C5×#454C2×C312yescomplemented
24C5×#356C2×C311yescharacteristic, complemented
25C5×#355C2×C312yescomplemented
26C3×C5×#41—22no—
27C3×C5×#41—21no—
28C3×C5×#41—41no—
29C2×C3×C5×Q8—41no—
30C2×C3×C5×Q8D411yescharacteristic
31C2×C3×C5×Q8C2311yescharacteristic
32C2×C3×C5×D4—21no—
33C2×C3×C5×D4—21no—
34C2×C3×C5×D4D412yes—
35C22×C3×C4×C5—22no—
36C3×C5×Q16—41no—
37C3×C5×SD16—41no—
38C3×C5×SD16—42no—
39C3×C5×D8—42no—
40C3×C5×M16—22no—
41C3×C5×M16—21no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4—22no—
44C3×C5×C22⋊C4—22no—
45C3×C5×C22⋊C4—22no—
46C3×C5×C22⋊C4—22no—
47C3×C42×C5—21no—
48C3×C42×C5D412yes—
49C3×C42×C5D411yescharacteristic
50C3×#555C2×C511yescharacteristic, complemented
51C3×#456C2×C511yescharacteristic, complemented
52C3×#454C2×C512yescomplemented
53C3×#356C2×C511yescharacteristic, complemented
54C3×#355C2×C512yescomplemented
55C5×2−1+4C22×C311yescharacteristic
56C5×#137—21no—
57C5×AGL(1,ℤ/8)—22no—
58C5×C4⋊Q8C22×C311yescharacteristic
59C5×#124—21no—
60C5×#124C22×C312yes—
61C5×#121—22no—
62C4×C5×D4—21no—
63C5×#117—22no—
64C5×#104—22no—
65C5×#104—21no—
66C5×#101C22×C311yescharacteristic
67C5×#101C22×C312yes—
68#2204C3×C511yescharacteristic, direct product factor, Sylow
69C23×C3×C5—22no—
70C3×C5×Q8—22no—
71C3×C5×Q8—21no—
72C3×C5×Q8—41no—
73C3×C5×Q8—41no—
74C3×C5×D4—22no—
75C3×C5×D4—21no—
76C3×C5×D4—42no—
77C3×C5×D4—42no—
78C3×C5×D4—41no—
79C3×C5×D4—81no—
80C2×C3×C4×C5—21no—
81C2×C3×C4×C5—22no—
82C2×C3×C4×C5—21no—
83C2×C3×C4×C5—44no—
84C2×C3×C4×C5—42no—
85C2×C3×C4×C5—41no—
86C2×C3×C4×C5C2×D411yescharacteristic
87C2×C3×C4×C5C2×D412yes—
88C3×C5×C8—42no—
89C3×C5×C8—41no—
90C3×2−1+4C22×C511yescharacteristic
91C3×#137—21no—
92C3×AGL(1,ℤ/8)—22no—
93C3×C4⋊Q8C22×C511yescharacteristic
94C3×#124—21no—
95C3×#124C22×C512yes—
96C3×#121—22no—
97C3×C4×D4—21no—
98C3×#117—22no—
99C3×#104—22no—
100C3×#104—21no—
101C3×#101C22×C512yes—
102C3×#101C22×C511yescharacteristic
103C5×#41—22no—
104C5×#41—21no—
105C5×#41—41no—
106C2×C5×Q8—41no—
107C2×C5×Q8C3×D411yescharacteristic
108C2×C5×Q8C23×C311yescharacteristic
109C2×C5×D4—21no—
110C2×C5×D4—21no—
111C2×C5×D4C3×D412yes—
112C22×C4×C5—22no—
113C5×Q16—41no—
114C5×SD16—41no—
115C5×SD16—42no—
116C5×D8—42no—
117C5×M16—22no—
118C5×M16—21no—
119C5×C4⋊C4—21no—
120C5×C4⋊C4—22no—
121C5×C22⋊C4—22no—
122C5×C22⋊C4—22no—
123C5×C22⋊C4—22no—
124C42×C5—21no—
125C42×C5C3×D412yes—
126C42×C5C3×D411yescharacteristic
127#555C2×C3×C511yescharacteristic, complemented
128#456C2×C3×C511yescharacteristic, complemented
129#454C2×C3×C512yescomplemented
130#356C2×C3×C511yescharacteristic, complemented
131#355C2×C3×C512yescomplemented
132C22×C3×C5—42no—
133C22×C3×C5—42no—
134C22×C3×C5—41no—
135C22×C3×C5—42no—
136C22×C3×C5#12011yescharacteristic
137C3×C4×C5—22no—
138C3×C4×C5—22no—
139C3×C4×C5—21no—
140C3×C4×C5—42no—
141C3×C4×C5—41no—
142C3×C4×C5—41no—
143C3×#41—22no—
144C3×#41—21no—
145C3×#41—41no—
146C2×C3×Q8—41no—
147C2×C3×Q8C5×D411yescharacteristic
148C2×C3×Q8C23×C511yescharacteristic
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4C5×D412yes—
152C22×C3×C4—22no—
153C3×Q16—41no—
154C3×SD16—41no—
155C3×SD16—42no—
156C3×D8—42no—
157C3×M16—22no—
158C3×M16—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—22no—
161C3×C22⋊C4—22no—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—22no—
164C3×C42—21no—
165C3×C42C5×D412yes—
166C3×C42C5×D411yescharacteristic
167C23×C5—22no—
168C5×Q8—22no—
169C5×Q8—21no—
170C5×Q8—41no—
171C5×Q8—41no—
172C5×D4—22no—
173C5×D4—21no—
174C5×D4—42no—
175C5×D4—42no—
176C5×D4—41no—
177C5×D4—81no—
178C2×C4×C5—21no—
179C2×C4×C5—22no—
180C2×C4×C5—21no—
181C2×C4×C5—44no—
182C2×C4×C5—42no—
183C2×C4×C5—41no—
184C2×C4×C5C2×C3×D411yescharacteristic
185C2×C4×C5C2×C3×D412yes—
186C5×C8—42no—
187C5×C8—41no—
1882−1+4C22×C3×C511yescharacteristic
189#137—21no—
190AGL(1,ℤ/8)—22no—
191C4⋊Q8C22×C3×C511yescharacteristic
192#124—21no—
193#124C22×C3×C512yes—
194#121—22no—
195C4×D4—21no—
196#117—22no—
197#104—22no—
198#104—21no—
199#101C22×C3×C511yescharacteristic
200#101C22×C3×C512yes—
201C2×C3×C5—21no—
202C2×C3×C5—81no—
203C2×C3×C5—82no—
204C2×C3×C5B(4,𝔽2)11yescharacteristic
205C23×C3—22no—
206C3×Q8—22no—
207C3×Q8—21no—
208C3×Q8—41no—
209C3×Q8—41no—
210C3×D4—22no—
211C3×D4—21no—
212C3×D4—42no—
213C3×D4—42no—
214C3×D4—41no—
215C3×D4—81no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—21no—
219C2×C3×C4—44no—
220C2×C3×C4—42no—
221C2×C3×C4—41no—
222C2×C3×C4C2×C5×D411yescharacteristic
223C2×C3×C4C2×C5×D412yes—
224C3×C8—42no—
225C3×C8—41no—
226C22×C5—42no—
227C22×C5—42no—
228C22×C5—41no—
229C22×C5—42no—
230C22×C5C3×#12011yescharacteristic
231C4×C5—22no—
232C4×C5—22no—
233C4×C5—21no—
234C4×C5—42no—
235C4×C5—41no—
236C4×C5—41no—
237#41—22no—
238#41—21no—
239#41—41no—
240C2×Q8—41no—
241C2×Q8C3×C5×D411yescharacteristic
242C2×Q8C23×C3×C511yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4C3×C5×D412yes—
246C22×C4—22no—
247Q16—41no—
248SD16—41no—
249SD16—42no—
250D8—42no—
251M16—22no—
252M16—21no—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C42—21no—
259C42C3×C5×D412yes—
260C42C3×C5×D411yescharacteristic
261C3×C5#220411yescharacteristic, direct product factor, Hall
262C22×C3—42no—
263C22×C3—42no—
264C22×C3—41no—
265C22×C3—42no—
266C22×C3C5×#12011yescharacteristic
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—42no—
271C3×C4—41no—
272C3×C4—41no—
273C2×C5—21no—
274C2×C5—82no—
275C2×C5—81no—
276C2×C5C3×B(4,𝔽2)11yescharacteristic
277C23—22no—
278Q8—22no—
279Q8—21no—
280Q8—41no—
281Q8—41no—
282D4—22no—
283D4—21no—
284D4—42no—
285D4—42no—
286D4—41no—
287D4—81no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4C2×C3×C5×D411yescharacteristic
295C2×C4C2×C3×C5×D412yes—
296C8—42no—
297C8—41no—
298C2×C3—21no—
299C2×C3—82no—
300C2×C3—81no—
301C2×C3C5×B(4,𝔽2)11yescharacteristic
302C5C3×#220411yescharacteristic, direct product factor, Sylow
303C22—42no—
304C22—42no—
305C22—41no—
306C22—42no—
307C22C3×C5×#12011yescharacteristic
308C4—22no—
309C4—22no—
310C4—21no—
311C4—42no—
312C4—41no—
313C4—41no—
314C3C5×#220411yescharacteristic, direct product factor, Sylow
315C2—21no—
316C2—82no—
317C2—81no—
318C2C3×C5×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C3×C5B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C23×C3×C5
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C5 ◁ C22×C3×C5 ◁ C2×C3×C5×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C5×#555C2semidirect
2. C3×C5×#456C2semidirect
3. C3×C5×#454C2semidirect
4. C3×C5×#356C2semidirect
5. C3×C5×#355C2semidirect
6. C5×#2204C3direct (N × H)
20. C3×#2204C5direct (N × H)
21. C5×#555C2×C3semidirect
22. C5×#456C2×C3semidirect
23. C5×#454C2×C3semidirect
24. C5×#356C2×C3semidirect
25. C5×#355C2×C3semidirect
50. C3×#555C2×C5semidirect
51. C3×#456C2×C5semidirect
52. C3×#454C2×C5semidirect
53. C3×#356C2×C5semidirect
54. C3×#355C2×C5semidirect
68. #2204C3×C5direct (N × H)
127. #555C2×C3×C5semidirect
128. #456C2×C3×C5semidirect
129. #454C2×C3×C5semidirect
130. #356C2×C3×C5semidirect
131. #355C2×C3×C5semidirect
261. C3×C5#2204direct (N × H)
302. C5C3×#2204direct (N × H)
314. C3C5×#2204direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5552
C3×C5×#4562
C3×C5×#4542
C3×C5×#3562
C3×C5×#3552
C5×#22043
C3×#22045