Finite Group Databasebeta

← C3×C5×#2223 #14275257

Subgroups — C3×C5×#2223 #14275257

Order 1920.

288 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×SD32C211yesmaximal, characteristic, complemented
2C3×C5×#504C211yesmaximal, characteristic, complemented
3C3×C5×#473C211yesmaximal, characteristic, complemented
4C3×C5×#472C211yesmaximal, characteristic
5C3×C5×#368C211yesmaximal, characteristic, complemented
6C3×C5×#362C211yesmaximal, characteristic, complemented
7C3×C5×#361C211yesmaximal, characteristic, complemented
8C5×#2223C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×#137—21no—
10C3×C5×AGL(1,ℤ/8)—21no—
11C2×C3×C5×Q16C2211yescharacteristic
12C2×C3×C5×D8C2211yescharacteristic
13C3×C5×#131C2211yescharacteristic
14C3×C5×SD32—21no—
15C3×C5×SD32—21no—
16C3×C5×M32—21no—
17C3×C5×M32C2211yescharacteristic
18C2×C3×C5×C16—21no—
19C2×C3×C5×C16C2211yescharacteristic
20C3×C5×#108C2211yescharacteristic
21C3×C5×#108C2211yescharacteristic
22C3×C5×#101—21no—
23C3×C5×#100—21no—
24C3×#2223C511yesmaximal, characteristic, direct product factor, Hall
25C2×C5×SD32C2×C311yescharacteristic, complemented
26C5×#504C2×C311yescharacteristic, complemented
27C5×#473C2×C311yescharacteristic, complemented
28C5×#472C2×C311yescharacteristic
29C5×#368C2×C311yescharacteristic, complemented
30C5×#362C2×C311yescharacteristic, complemented
31C5×#361C2×C311yescharacteristic, complemented
32C3×C5×#41D411yescharacteristic
33C2×C3×C5×Q8—21no—
34C2×C3×C5×D4—21no—
35C3×C5×Q16—21no—
36C3×C5×Q16—21no—
37C3×C5×Q16—41no—
38C3×C5×SD16—41no—
39C3×C5×SD16—41no—
40C3×C5×D8—21no—
41C3×C5×D8—21no—
42C3×C5×D8—41no—
43C3×C5×M16—21no—
44C3×C5×M16—21no—
45C3×C5×M16—21no—
46C3×C5×M16D411yescharacteristic
47C2×C3×C5×C8—21no—
48C2×C3×C5×C8C2311yescharacteristic
49C3×C5×C16—21no—
50C3×C5×C16D411yescharacteristic
51C3×C5×C16D411yescharacteristic
52C2×C3×SD32C2×C511yescharacteristic, complemented
53C3×#504C2×C511yescharacteristic, complemented
54C3×#473C2×C511yescharacteristic, complemented
55C3×#472C2×C511yescharacteristic
56C3×#368C2×C511yescharacteristic, complemented
57C3×#362C2×C511yescharacteristic, complemented
58C3×#361C2×C511yescharacteristic, complemented
59C5×#137—21no—
60C5×AGL(1,ℤ/8)—21no—
61C2×C5×Q16C22×C311yescharacteristic
62C2×C5×D8C22×C311yescharacteristic
63C5×#131C22×C311yescharacteristic
64C5×SD32—21no—
65C5×SD32—21no—
66C5×M32—21no—
67C5×M32C22×C311yescharacteristic
68C2×C5×C16—21no—
69C2×C5×C16C22×C311yescharacteristic
70C5×#108C22×C311yescharacteristic
71C5×#108C22×C311yescharacteristic
72C5×#101—21no—
73C5×#100—21no—
74#2223C3×C511yescharacteristic, direct product factor, Sylow
75C23×C3×C5—41no—
76C3×C5×Q8—41no—
77C3×C5×Q8—41no—
78C3×C5×Q8D811yescharacteristic
79C3×C5×D4—21no—
80C3×C5×D4—41no—
81C3×C5×D4—41no—
82C3×C5×D4D811yescharacteristic
83C2×C3×C4×C5—21no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5C2×D411yescharacteristic
86C3×C5×C8—21no—
87C3×C5×C8—41no—
88C3×C5×C8—41no—
89C3×C5×C8C2×D411yescharacteristic
90C3×C5×C8#4111yescharacteristic
91C3×#137—21no—
92C3×AGL(1,ℤ/8)—21no—
93C2×C3×Q16C22×C511yescharacteristic
94C2×C3×D8C22×C511yescharacteristic
95C3×#131C22×C511yescharacteristic
96C3×SD32—21no—
97C3×SD32—21no—
98C3×M32—21no—
99C3×M32C22×C511yescharacteristic
100C2×C3×C16—21no—
101C2×C3×C16C22×C511yescharacteristic
102C3×#108C22×C511yescharacteristic
103C3×#108C22×C511yescharacteristic
104C3×#101—21no—
105C3×#100—21no—
106C5×#41C3×D411yescharacteristic
107C2×C5×Q8—21no—
108C2×C5×D4—21no—
109C5×Q16—21no—
110C5×Q16—21no—
111C5×Q16—41no—
112C5×SD16—41no—
113C5×SD16—41no—
114C5×D8—21no—
115C5×D8—21no—
116C5×D8—41no—
117C5×M16—21no—
118C5×M16—21no—
119C5×M16—21no—
120C5×M16C3×D411yescharacteristic
121C2×C5×C8—21no—
122C2×C5×C8C23×C311yescharacteristic
123C5×C16—21no—
124C5×C16C3×D411yescharacteristic
125C5×C16C3×D411yescharacteristic
126C2×SD32C2×C3×C511yescharacteristic, complemented
127#504C2×C3×C511yescharacteristic, complemented
128#473C2×C3×C511yescharacteristic, complemented
129#472C2×C3×C511yescharacteristic
130#368C2×C3×C511yescharacteristic, complemented
131#362C2×C3×C511yescharacteristic, complemented
132#361C2×C3×C511yescharacteristic, complemented
133C22×C3×C5—21no—
134C22×C3×C5—81no—
135C22×C3×C5—161no—
136C22×C3×C5#13511yescharacteristic
137C3×C4×C5—21no—
138C3×C4×C5—81no—
139C3×C4×C5#12111yescharacteristic
140C3×C4×C5C2×D811yescharacteristic
141C3×#41C5×D411yescharacteristic
142C2×C3×Q8—21no—
143C2×C3×D4—21no—
144C3×Q16—21no—
145C3×Q16—21no—
146C3×Q16—41no—
147C3×SD16—41no—
148C3×SD16—41no—
149C3×D8—21no—
150C3×D8—21no—
151C3×D8—41no—
152C3×M16—21no—
153C3×M16—21no—
154C3×M16—21no—
155C3×M16C5×D411yescharacteristic
156C2×C3×C8—21no—
157C2×C3×C8C23×C511yescharacteristic
158C3×C16—21no—
159C3×C16C5×D411yescharacteristic
160C3×C16C5×D411yescharacteristic
161C23×C5—41no—
162C5×Q8—41no—
163C5×Q8—41no—
164C5×Q8C3×D811yescharacteristic
165C5×D4—21no—
166C5×D4—41no—
167C5×D4—41no—
168C5×D4C3×D811yescharacteristic
169C2×C4×C5—21no—
170C2×C4×C5—41no—
171C2×C4×C5C2×C3×D411yescharacteristic
172C5×C8—21no—
173C5×C8—41no—
174C5×C8—41no—
175C5×C8C2×C3×D411yescharacteristic
176C5×C8C3×#4111yescharacteristic
177#137—21no—
178AGL(1,ℤ/8)—21no—
179C2×Q16C22×C3×C511yescharacteristic
180C2×D8C22×C3×C511yescharacteristic
181#131C22×C3×C511yescharacteristic
182SD32—21no—
183SD32—21no—
184M32—21no—
185M32C22×C3×C511yescharacteristic
186C2×C16—21no—
187C2×C16C22×C3×C511yescharacteristic
188#108C22×C3×C511yescharacteristic
189#108C22×C3×C511yescharacteristic
190#101—21no—
191#100—21no—
192C2×C3×C5—21no—
193C2×C3×C5—41no—
194C2×C3×C5—161no—
195C2×C3×C5#46611yescharacteristic
196C23×C3—41no—
197C3×Q8—41no—
198C3×Q8—41no—
199C3×Q8C5×D811yescharacteristic
200C3×D4—21no—
201C3×D4—41no—
202C3×D4—41no—
203C3×D4C5×D811yescharacteristic
204C2×C3×C4—21no—
205C2×C3×C4—41no—
206C2×C3×C4C2×C5×D411yescharacteristic
207C3×C8—21no—
208C3×C8—41no—
209C3×C8—41no—
210C3×C8C2×C5×D411yescharacteristic
211C3×C8C5×#4111yescharacteristic
212C22×C5—21no—
213C22×C5—81no—
214C22×C5—161no—
215C22×C5C3×#13511yescharacteristic
216C4×C5—21no—
217C4×C5—81no—
218C4×C5C3×#12111yescharacteristic
219C4×C5C2×C3×D811yescharacteristic
220#41C3×C5×D411yescharacteristic
221C2×Q8—21no—
222C2×D4—21no—
223Q16—21no—
224Q16—21no—
225Q16—41no—
226SD16—41no—
227SD16—41no—
228D8—21no—
229D8—21no—
230D8—41no—
231M16—21no—
232M16—21no—
233M16—21no—
234M16C3×C5×D411yescharacteristic
235C2×C8—21no—
236C2×C8C23×C3×C511yescharacteristic
237C16—21no—
238C16C3×C5×D411yescharacteristic
239C16C3×C5×D411yescharacteristic
240C3×C5#222311yescharacteristic, direct product factor, Hall
241C22×C3—21no—
242C22×C3—81no—
243C22×C3—161no—
244C22×C3C5×#13511yescharacteristic
245C3×C4—21no—
246C3×C4—81no—
247C3×C4C5×#12111yescharacteristic
248C3×C4C2×C5×D811yescharacteristic
249C2×C5—21no—
250C2×C5—41no—
251C2×C5—161no—
252C2×C5C3×#46611yescharacteristic
253C23—41no—
254Q8—41no—
255Q8—41no—
256Q8C3×C5×D811yescharacteristic
257D4—21no—
258D4—41no—
259D4—41no—
260D4C3×C5×D811yescharacteristic
261C2×C4—21no—
262C2×C4—41no—
263C2×C4C2×C3×C5×D411yescharacteristic
264C8—21no—
265C8—41no—
266C8—41no—
267C8C2×C3×C5×D411yescharacteristic
268C8C3×C5×#4111yescharacteristic
269C2×C3—21no—
270C2×C3—41no—
271C2×C3—161no—
272C2×C3C5×#46611yescharacteristic
273C5C3×#222311yescharacteristic, direct product factor, Sylow
274C22—21no—
275C22—81no—
276C22—161no—
277C22C3×C5×#13511yescharacteristic
278C4—21no—
279C4—81no—
280C4C3×C5×#12111yescharacteristic
281C4C2×C3×C5×D811yescharacteristic
282C3C5×#222311yescharacteristic, direct product factor, Sylow
283C2—21no—
284C2—41no—
285C2—161no—
286C2C3×C5×#46611yescharacteristic
subgroupG/H
center, socleC2×C3×C5#466
commutator, Frattini, cosocleC2×C8C23×C3×C5
prime radicalC3×C5×#472C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C5 ◁ C2×C3×C4×C5 ◁ C3×C5×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×SD32C2semidirect
2. C3×C5×#504C2semidirect
3. C3×C5×#473C2semidirect
5. C3×C5×#368C2semidirect
6. C3×C5×#362C2semidirect
7. C3×C5×#361C2semidirect
8. C5×#2223C3direct (N × H)
24. C3×#2223C5direct (N × H)
25. C2×C5×SD32C2×C3semidirect
26. C5×#504C2×C3semidirect
27. C5×#473C2×C3semidirect
29. C5×#368C2×C3semidirect
30. C5×#362C2×C3semidirect
31. C5×#361C2×C3semidirect
52. C2×C3×SD32C2×C5semidirect
53. C3×#504C2×C5semidirect
54. C3×#473C2×C5semidirect
56. C3×#368C2×C5semidirect
57. C3×#362C2×C5semidirect
58. C3×#361C2×C5semidirect
74. #2223C3×C5direct (N × H)
126. C2×SD32C2×C3×C5semidirect
127. #504C2×C3×C5semidirect
128. #473C2×C3×C5semidirect
130. #368C2×C3×C5semidirect
131. #362C2×C3×C5semidirect
132. #361C2×C3×C5semidirect
240. C3×C5#2223direct (N × H)
273. C5C3×#2223direct (N × H)
282. C3C5×#2223direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×SD322
C3×C5×#5042
C3×C5×#4732
C3×C5×#4722
C3×C5×#3682
C3×C5×#3622
C3×C5×#3612
C5×#22233
C3×#22235