Finite Group Databasebeta

← C3×C5×#2203 #14275237

Subgroups — C3×C5×#2203 #14275237

Order 1920.

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

normal subgroups only

This group has 432 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×#456C212yesmaximal, complemented
3C3×C5×#454C211yesmaximal, characteristic
4C3×C5×#356C212yesmaximal, complemented
5C3×C5×#355C211yesmaximal, characteristic, complemented
6C5×#2203C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×2−1+4C2211yescharacteristic
8C3×C5×#137—22no—
9C3×C5×AGL(1,ℤ/8)—21no—
10C3×C5×C4⋊Q8C2212yes—
11C3×C5×#126—22no—
12C3×C5×#125—21no—
13C3×C5×#124C2211yescharacteristic
14C3×C5×#123—21no—
15C3×C5×C22⋊Q8—22no—
16C3×C5×#104—21no—
17C3×C5×#104—22no—
18C3×C5×#101C2211yescharacteristic
19C3×C5×#101C2212yes—
20C3×#2203C511yesmaximal, characteristic, direct product factor, Hall
21C5×#563C2×C311yescharacteristic, complemented
22C5×#456C2×C312yescomplemented
23C5×#454C2×C311yescharacteristic
24C5×#356C2×C312yescomplemented
25C5×#355C2×C311yescharacteristic, complemented
26C3×C5×#41—21no—
27C3×C5×#41—22no—
28C3×C5×#41—41no—
29C2×C3×C5×Q8—41no—
30C2×C3×C5×Q8D412yes—
31C2×C3×C5×Q8C2311yescharacteristic
32C2×C3×C5×D4D411yescharacteristic
33C22×C3×C4×C5—21no—
34C3×C5×Q16—42no—
35C3×C5×SD16—42no—
36C3×C5×SD16—41no—
37C3×C5×D8—41no—
38C3×C5×M16—21no—
39C3×C5×M16—22no—
40C3×C5×C4⋊C4—21no—
41C3×C5×C4⋊C4—22no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4—22no—
44C3×C5×C4⋊C4—22no—
45C3×C5×C22⋊C4—22no—
46C3×C5×C22⋊C4—21no—
47C3×C5×C22⋊C4—22no—
48C3×C42×C5D411yescharacteristic
49C3×C42×C5D412yes—
50C3×#563C2×C511yescharacteristic, complemented
51C3×#456C2×C512yescomplemented
52C3×#454C2×C511yescharacteristic
53C3×#356C2×C512yescomplemented
54C3×#355C2×C511yescharacteristic, complemented
55C5×2−1+4C22×C311yescharacteristic
56C5×#137—22no—
57C5×AGL(1,ℤ/8)—21no—
58C5×C4⋊Q8C22×C312yes—
59C5×#126—22no—
60C5×#125—21no—
61C5×#124C22×C311yescharacteristic
62C5×#123—21no—
63C5×C22⋊Q8—22no—
64C5×#104—21no—
65C5×#104—22no—
66C5×#101C22×C312yes—
67C5×#101C22×C311yescharacteristic
68#2203C3×C511yescharacteristic, direct product factor, Sylow
69C23×C3×C5—21no—
70C3×C5×Q8—21no—
71C3×C5×Q8—22no—
72C3×C5×Q8—41no—
73C3×C5×Q8—42no—
74C3×C5×D4—21no—
75C3×C5×D4—22no—
76C3×C5×D4—41no—
77C3×C5×D4—41no—
78C2×C3×C4×C5—21no—
79C2×C3×C4×C5—22no—
80C2×C3×C4×C5—22no—
81C2×C3×C4×C5—41no—
82C2×C3×C4×C5—42no—
83C2×C3×C4×C5—81no—
84C2×C3×C4×C5C2×D412yes—
85C2×C3×C4×C5C2×D411yescharacteristic
86C3×C5×C8—41no—
87C3×C5×C8—42no—
88C3×2−1+4C22×C511yescharacteristic
89C3×#137—22no—
90C3×AGL(1,ℤ/8)—21no—
91C3×C4⋊Q8C22×C512yes—
92C3×#126—22no—
93C3×#125—21no—
94C3×#124C22×C511yescharacteristic
95C3×#123—21no—
96C3×C22⋊Q8—22no—
97C3×#104—21no—
98C3×#104—22no—
99C3×#101C22×C511yescharacteristic
100C3×#101C22×C512yes—
101C5×#41—21no—
102C5×#41—22no—
103C5×#41—41no—
104C2×C5×Q8—41no—
105C2×C5×Q8C3×D412yes—
106C2×C5×Q8C23×C311yescharacteristic
107C2×C5×D4C3×D411yescharacteristic
108C22×C4×C5—21no—
109C5×Q16—42no—
110C5×SD16—42no—
111C5×SD16—41no—
112C5×D8—41no—
113C5×M16—21no—
114C5×M16—22no—
115C5×C4⋊C4—21no—
116C5×C4⋊C4—22no—
117C5×C4⋊C4—21no—
118C5×C4⋊C4—22no—
119C5×C4⋊C4—22no—
120C5×C22⋊C4—22no—
121C5×C22⋊C4—21no—
122C5×C22⋊C4—22no—
123C42×C5C3×D411yescharacteristic
124C42×C5C3×D412yes—
125#563C2×C3×C511yescharacteristic, complemented
126#456C2×C3×C512yescomplemented
127#454C2×C3×C511yescharacteristic
128#356C2×C3×C512yescomplemented
129#355C2×C3×C511yescharacteristic, complemented
130C22×C3×C5—42no—
131C22×C3×C5—41no—
132C22×C3×C5—41no—
133C22×C3×C5#12011yescharacteristic
134C3×C4×C5—21no—
135C3×C4×C5—22no—
136C3×C4×C5—41no—
137C3×C4×C5—42no—
138C3×C4×C5—42no—
139C3×C4×C5—41no—
140C3×#41—21no—
141C3×#41—22no—
142C3×#41—41no—
143C2×C3×Q8—41no—
144C2×C3×Q8C5×D412yes—
145C2×C3×Q8C23×C511yescharacteristic
146C2×C3×D4C5×D411yescharacteristic
147C22×C3×C4—21no—
148C3×Q16—42no—
149C3×SD16—42no—
150C3×SD16—41no—
151C3×D8—41no—
152C3×M16—21no—
153C3×M16—22no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—21no—
161C3×C22⋊C4—22no—
162C3×C42C5×D411yescharacteristic
163C3×C42C5×D412yes—
164C23×C5—21no—
165C5×Q8—21no—
166C5×Q8—22no—
167C5×Q8—41no—
168C5×Q8—42no—
169C5×D4—21no—
170C5×D4—22no—
171C5×D4—41no—
172C5×D4—41no—
173C2×C4×C5—21no—
174C2×C4×C5—22no—
175C2×C4×C5—22no—
176C2×C4×C5—41no—
177C2×C4×C5—42no—
178C2×C4×C5—81no—
179C2×C4×C5C2×C3×D412yes—
180C2×C4×C5C2×C3×D411yescharacteristic
181C5×C8—41no—
182C5×C8—42no—
1832−1+4C22×C3×C511yescharacteristic
184#137—22no—
185AGL(1,ℤ/8)—21no—
186C4⋊Q8C22×C3×C512yes—
187#126—22no—
188#125—21no—
189#124C22×C3×C511yescharacteristic
190#123—21no—
191C22⋊Q8—22no—
192#104—21no—
193#104—22no—
194#101C22×C3×C512yes—
195#101C22×C3×C511yescharacteristic
196C2×C3×C5—21no—
197C2×C3×C5—81no—
198C2×C3×C5—81no—
199C2×C3×C5B(4,𝔽2)11yescharacteristic
200C23×C3—21no—
201C3×Q8—21no—
202C3×Q8—22no—
203C3×Q8—41no—
204C3×Q8—42no—
205C3×D4—21no—
206C3×D4—22no—
207C3×D4—41no—
208C3×D4—41no—
209C2×C3×C4—21no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—81no—
215C2×C3×C4C2×C5×D412yes—
216C2×C3×C4C2×C5×D411yescharacteristic
217C3×C8—41no—
218C3×C8—42no—
219C22×C5—42no—
220C22×C5—41no—
221C22×C5—41no—
222C22×C5C3×#12011yescharacteristic
223C4×C5—21no—
224C4×C5—22no—
225C4×C5—41no—
226C4×C5—42no—
227C4×C5—42no—
228C4×C5—41no—
229#41—21no—
230#41—22no—
231#41—41no—
232C2×Q8—41no—
233C2×Q8C3×C5×D412yes—
234C2×Q8C23×C3×C511yescharacteristic
235C2×D4C3×C5×D411yescharacteristic
236C22×C4—21no—
237Q16—42no—
238SD16—42no—
239SD16—41no—
240D8—41no—
241M16—21no—
242M16—22no—
243C4⋊C4—21no—
244C4⋊C4—22no—
245C4⋊C4—21no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C22⋊C4—22no—
249C22⋊C4—21no—
250C22⋊C4—22no—
251C42C3×C5×D411yescharacteristic
252C42C3×C5×D412yes—
253C3×C5#220311yescharacteristic, direct product factor, Hall
254C22×C3—42no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3C5×#12011yescharacteristic
258C3×C4—21no—
259C3×C4—22no—
260C3×C4—41no—
261C3×C4—42no—
262C3×C4—42no—
263C3×C4—41no—
264C2×C5—21no—
265C2×C5—81no—
266C2×C5—81no—
267C2×C5C3×B(4,𝔽2)11yescharacteristic
268C23—21no—
269Q8—21no—
270Q8—22no—
271Q8—41no—
272Q8—42no—
273D4—21no—
274D4—22no—
275D4—41no—
276D4—41no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4C2×C3×C5×D412yes—
284C2×C4C2×C3×C5×D411yescharacteristic
285C8—41no—
286C8—42no—
287C2×C3—21no—
288C2×C3—81no—
289C2×C3—81no—
290C2×C3C5×B(4,𝔽2)11yescharacteristic
291C5C3×#220311yescharacteristic, direct product factor, Sylow
292C22—42no—
293C22—41no—
294C22—41no—
295C22C3×C5×#12011yescharacteristic
296C4—21no—
297C4—22no—
298C4—41no—
299C4—42no—
300C4—42no—
301C4—41no—
302C3C5×#220311yescharacteristic, direct product factor, Sylow
303C2—21no—
304C2—81no—
305C2—81no—
306C2C3×C5×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C3×C5B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C23×C3×C5
prime radicalC3×C5×#454C2
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×#563C2semidirect
2. C3×C5×#456C2semidirect
4. C3×C5×#356C2semidirect
5. C3×C5×#355C2semidirect
6. C5×#2203C3direct (N × H)
20. C3×#2203C5direct (N × H)
21. C5×#563C2×C3semidirect
22. C5×#456C2×C3semidirect
24. C5×#356C2×C3semidirect
25. C5×#355C2×C3semidirect
50. C3×#563C2×C5semidirect
51. C3×#456C2×C5semidirect
53. C3×#356C2×C5semidirect
54. C3×#355C2×C5semidirect
68. #2203C3×C5direct (N × H)
125. #563C2×C3×C5semidirect
126. #456C2×C3×C5semidirect
128. #356C2×C3×C5semidirect
129. #355C2×C3×C5semidirect
253. C3×C5#2203direct (N × H)
291. C5C3×#2203direct (N × H)
302. C3C5×#2203direct (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×#4562
C3×C5×#4542
C3×C5×#3562
C3×C5×#3552
C5×#22033
C3×#22035