Finite Group Databasebeta

← C3×C5×#2192 #14275226

Subgroups — C3×C5×#2192 #14275226

Order 1920.

440 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#576C211yesmaximal, characteristic, complemented
2C3×C5×#509C211yesmaximal, characteristic, complemented
3C2×C3×C5×D16C211yesmaximal, characteristic, complemented
4C3×C5×#472C211yesmaximal, characteristic, complemented
5C3×C5×#361C211yesmaximal, characteristic, complemented
6C3×C5×#359C211yesmaximal, characteristic, complemented
7C3×C5×#350C211yesmaximal, characteristic, complemented
8C5×#2192C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×H(5,𝔽2)—21no—
10C3×C5×AGL(1,ℤ/8)—22no—
11C3×C5×AGL(1,ℤ/8)—21no—
12C3×C5×AGL(1,ℤ/8)—21no—
13C3×C5×#135—21no—
14C3×C5×#135C2211yescharacteristic
15C2×C3×C5×D8—21no—
16C2×C3×C5×D8C2211yescharacteristic
17C2×C3×C5×D8C2211yescharacteristic, complemented
18C3×C5×#131C2211yescharacteristic
19C3×C5×SD32—21no—
20C3×C5×D16—22no—
21C3×C5×D16—21no—
22C3×C5×M32C2211yescharacteristic, complemented
23C2×C3×C5×C16C2211yescharacteristic, complemented
24C3×C5×#108C2211yescharacteristic, complemented
25C3×C5×#100—21no—
26C3×#2192C511yesmaximal, characteristic, direct product factor, Hall
27C5×#576C2×C311yescharacteristic, complemented
28C5×#509C2×C311yescharacteristic, complemented
29C2×C5×D16C2×C311yescharacteristic, complemented
30C5×#472C2×C311yescharacteristic, complemented
31C5×#361C2×C311yescharacteristic, complemented
32C5×#359C2×C311yescharacteristic, complemented
33C5×#350C2×C311yescharacteristic, complemented
34C3×C5×#41—22no—
35C3×C5×#41—21no—
36C3×C5×#41—41no—
37C3×C5×#41D411yescharacteristic
38C2×C3×C5×D4—22no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—42no—
42C2×C3×C5×D4—41no—
43C3×C5×Q16D411yescharacteristic
44C3×C5×SD16—22no—
45C3×C5×SD16—21no—
46C3×C5×SD16—41no—
47C3×C5×D8—22no—
48C3×C5×D8—21no—
49C3×C5×D8—21no—
50C3×C5×D8—21no—
51C3×C5×D8—41no—
52C3×C5×D8D411yescharacteristic
53C3×C5×D8D412yes—
54C3×C5×M16—21no—
55C3×C5×M16—21no—
56C3×C5×M16D411yescharacteristic
57C2×C3×C5×C8—21no—
58C2×C3×C5×C8C2311yescharacteristic
59C3×C5×C16—21no—
60C3×C5×C16—21no—
61C3×#576C2×C511yescharacteristic, complemented
62C3×#509C2×C511yescharacteristic, complemented
63C2×C3×D16C2×C511yescharacteristic, complemented
64C3×#472C2×C511yescharacteristic, complemented
65C3×#361C2×C511yescharacteristic, complemented
66C3×#359C2×C511yescharacteristic, complemented
67C3×#350C2×C511yescharacteristic, complemented
68C5×H(5,𝔽2)—21no—
69C5×AGL(1,ℤ/8)—22no—
70C5×AGL(1,ℤ/8)—21no—
71C5×AGL(1,ℤ/8)—21no—
72C5×#135—21no—
73C5×#135C22×C311yescharacteristic
74C2×C5×D8—21no—
75C2×C5×D8C22×C311yescharacteristic
76C2×C5×D8C22×C311yescharacteristic, complemented
77C5×#131C22×C311yescharacteristic
78C5×SD32—21no—
79C5×D16—22no—
80C5×D16—21no—
81C5×M32C22×C311yescharacteristic, complemented
82C2×C5×C16C22×C311yescharacteristic, complemented
83C5×#108C22×C311yescharacteristic, complemented
84C5×#100—21no—
85#2192C3×C511yescharacteristic, direct product factor, Sylow
86C23×C3×C5—42no—
87C23×C3×C5—41no—
88C23×C3×C5—41no—
89C3×C5×Q8—21no—
90C3×C5×Q8D811yescharacteristic
91C3×C5×D4—21no—
92C3×C5×D4—21no—
93C3×C5×D4—22no—
94C3×C5×D4—42no—
95C3×C5×D4—42no—
96C3×C5×D4—41no—
97C3×C5×D4—41no—
98C3×C5×D4—41no—
99C3×C5×D4—41no—
100C3×C5×D4D811yescharacteristic
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—42no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5C2×D411yescharacteristic
105C3×C5×C8—21no—
106C3×C5×C8—41no—
107C3×C5×C8C2×D411yescharacteristic
108C3×C5×C8C2×D411yescharacteristic
109C3×H(5,𝔽2)—21no—
110C3×AGL(1,ℤ/8)—22no—
111C3×AGL(1,ℤ/8)—21no—
112C3×AGL(1,ℤ/8)—21no—
113C3×#135—21no—
114C3×#135C22×C511yescharacteristic
115C2×C3×D8—21no—
116C2×C3×D8C22×C511yescharacteristic
117C2×C3×D8C22×C511yescharacteristic, complemented
118C3×#131C22×C511yescharacteristic
119C3×SD32—21no—
120C3×D16—22no—
121C3×D16—21no—
122C3×M32C22×C511yescharacteristic, complemented
123C2×C3×C16C22×C511yescharacteristic, complemented
124C3×#108C22×C511yescharacteristic, complemented
125C3×#100—21no—
126C5×#41—22no—
127C5×#41—21no—
128C5×#41—41no—
129C5×#41C3×D411yescharacteristic
130C2×C5×D4—22no—
131C2×C5×D4—21no—
132C2×C5×D4—21no—
133C2×C5×D4—42no—
134C2×C5×D4—41no—
135C5×Q16C3×D411yescharacteristic
136C5×SD16—22no—
137C5×SD16—21no—
138C5×SD16—41no—
139C5×D8—22no—
140C5×D8—21no—
141C5×D8—21no—
142C5×D8—21no—
143C5×D8—41no—
144C5×D8C3×D411yescharacteristic
145C5×D8C3×D412yes—
146C5×M16—21no—
147C5×M16—21no—
148C5×M16C3×D411yescharacteristic
149C2×C5×C8—21no—
150C2×C5×C8C23×C311yescharacteristic
151C5×C16—21no—
152C5×C16—21no—
153#576C2×C3×C511yescharacteristic, complemented
154#509C2×C3×C511yescharacteristic, complemented
155C2×D16C2×C3×C511yescharacteristic, complemented
156#472C2×C3×C511yescharacteristic, complemented
157#361C2×C3×C511yescharacteristic, complemented
158#359C2×C3×C511yescharacteristic, complemented
159#350C2×C3×C511yescharacteristic, complemented
160C22×C3×C5—21no—
161C22×C3×C5—42no—
162C22×C3×C5—41no—
163C22×C3×C5—81no—
164C22×C3×C5—161no—
165C22×C3×C5—161no—
166C22×C3×C5—162no—
167C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
168C3×C4×C5—21no—
169C3×C4×C5—41no—
170C3×C4×C5#12011yescharacteristic
171C3×C4×C5C2×D811yescharacteristic
172C3×#41—22no—
173C3×#41—21no—
174C3×#41—41no—
175C3×#41C5×D411yescharacteristic
176C2×C3×D4—22no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—42no—
180C2×C3×D4—41no—
181C3×Q16C5×D411yescharacteristic
182C3×SD16—22no—
183C3×SD16—21no—
184C3×SD16—41no—
185C3×D8—22no—
186C3×D8—21no—
187C3×D8—21no—
188C3×D8—21no—
189C3×D8—41no—
190C3×D8C5×D411yescharacteristic
191C3×D8C5×D412yes—
192C3×M16—21no—
193C3×M16—21no—
194C3×M16C5×D411yescharacteristic
195C2×C3×C8—21no—
196C2×C3×C8C23×C511yescharacteristic
197C3×C16—21no—
198C3×C16—21no—
199C23×C5—42no—
200C23×C5—41no—
201C23×C5—41no—
202C5×Q8—21no—
203C5×Q8C3×D811yescharacteristic
204C5×D4—21no—
205C5×D4—21no—
206C5×D4—22no—
207C5×D4—42no—
208C5×D4—42no—
209C5×D4—41no—
210C5×D4—41no—
211C5×D4—41no—
212C5×D4—41no—
213C5×D4C3×D811yescharacteristic
214C2×C4×C5—21no—
215C2×C4×C5—42no—
216C2×C4×C5—41no—
217C2×C4×C5C2×C3×D411yescharacteristic
218C5×C8—21no—
219C5×C8—41no—
220C5×C8C2×C3×D411yescharacteristic
221C5×C8C2×C3×D411yescharacteristic
222H(5,𝔽2)—21no—
223AGL(1,ℤ/8)—22no—
224AGL(1,ℤ/8)—21no—
225AGL(1,ℤ/8)—21no—
226#135—21no—
227#135C22×C3×C511yescharacteristic
228C2×D8—21no—
229C2×D8C22×C3×C511yescharacteristic
230C2×D8C22×C3×C511yescharacteristic, complemented
231#131C22×C3×C511yescharacteristic
232SD32—21no—
233D16—22no—
234D16—21no—
235M32C22×C3×C511yescharacteristic, complemented
236C2×C16C22×C3×C511yescharacteristic, complemented
237#108C22×C3×C511yescharacteristic, complemented
238#100—21no—
239C2×C3×C5—21no—
240C2×C3×C5—41no—
241C2×C3×C5—82no—
242C2×C3×C5—81no—
243C2×C3×C5—161no—
244C2×C3×C5#44711yescharacteristic
245C23×C3—42no—
246C23×C3—41no—
247C23×C3—41no—
248C3×Q8—21no—
249C3×Q8C5×D811yescharacteristic
250C3×D4—21no—
251C3×D4—21no—
252C3×D4—22no—
253C3×D4—42no—
254C3×D4—42no—
255C3×D4—41no—
256C3×D4—41no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4C5×D811yescharacteristic
260C2×C3×C4—21no—
261C2×C3×C4—42no—
262C2×C3×C4—41no—
263C2×C3×C4C2×C5×D411yescharacteristic
264C3×C8—21no—
265C3×C8—41no—
266C3×C8C2×C5×D411yescharacteristic
267C3×C8C2×C5×D411yescharacteristic
268C22×C5—21no—
269C22×C5—42no—
270C22×C5—41no—
271C22×C5—81no—
272C22×C5—161no—
273C22×C5—161no—
274C22×C5—162no—
275C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
276C4×C5—21no—
277C4×C5—41no—
278C4×C5C3×#12011yescharacteristic
279C4×C5C2×C3×D811yescharacteristic
280#41—22no—
281#41—21no—
282#41—41no—
283#41C3×C5×D411yescharacteristic
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—42no—
288C2×D4—41no—
289Q16C3×C5×D411yescharacteristic
290SD16—22no—
291SD16—21no—
292SD16—41no—
293D8—22no—
294D8—21no—
295D8—21no—
296D8—21no—
297D8—41no—
298D8C3×C5×D411yescharacteristic
299D8C3×C5×D412yes—
300M16—21no—
301M16—21no—
302M16C3×C5×D411yescharacteristic
303C2×C8—21no—
304C2×C8C23×C3×C511yescharacteristic
305C16—21no—
306C16—21no—
307C3×C5#219211yescharacteristic, direct product factor, Hall
308C22×C3—21no—
309C22×C3—42no—
310C22×C3—41no—
311C22×C3—81no—
312C22×C3—161no—
313C22×C3—161no—
314C22×C3—162no—
315C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
316C3×C4—21no—
317C3×C4—41no—
318C3×C4C5×#12011yescharacteristic
319C3×C4C2×C5×D811yescharacteristic
320C2×C5—21no—
321C2×C5—41no—
322C2×C5—82no—
323C2×C5—81no—
324C2×C5—161no—
325C2×C5C3×#44711yescharacteristic
326C23—42no—
327C23—41no—
328C23—41no—
329Q8—21no—
330Q8C3×C5×D811yescharacteristic
331D4—21no—
332D4—21no—
333D4—22no—
334D4—42no—
335D4—42no—
336D4—41no—
337D4—41no—
338D4—41no—
339D4—41no—
340D4C3×C5×D811yescharacteristic
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4C2×C3×C5×D411yescharacteristic
345C8—21no—
346C8—41no—
347C8C2×C3×C5×D411yescharacteristic
348C8C2×C3×C5×D411yescharacteristic
349C2×C3—21no—
350C2×C3—41no—
351C2×C3—82no—
352C2×C3—81no—
353C2×C3—161no—
354C2×C3C5×#44711yescharacteristic
355C5C3×#219211yescharacteristic, direct product factor, Sylow
356C22—21no—
357C22—42no—
358C22—41no—
359C22—81no—
360C22—161no—
361C22—161no—
362C22—162no—
363C22C3×C5×AGL(1,ℤ/8)11yescharacteristic
364C4—21no—
365C4—41no—
366C4C3×C5×#12011yescharacteristic
367C4C2×C3×C5×D811yescharacteristic
368C3C5×#219211yescharacteristic, direct product factor, Sylow
369C2—21no—
370C2—41no—
371C2—82no—
372C2—81no—
373C2—161no—
374C2C3×C5×#44711yescharacteristic
subgroupG/H
center, socleC2×C3×C5#447
commutator, Frattini, cosocleC2×C8C23×C3×C5
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. C3×C5×#576C2semidirect
2. C3×C5×#509C2semidirect
3. C2×C3×C5×D16C2semidirect
4. C3×C5×#472C2semidirect
5. C3×C5×#361C2semidirect
6. C3×C5×#359C2semidirect
7. C3×C5×#350C2semidirect
8. C5×#2192C3direct (N × H)
17. C2×C3×C5×D8C22semidirect
22. C3×C5×M32C22semidirect
23. C2×C3×C5×C16C22semidirect
24. C3×C5×#108C22semidirect
26. C3×#2192C5direct (N × H)
27. C5×#576C2×C3semidirect
28. C5×#509C2×C3semidirect
29. C2×C5×D16C2×C3semidirect
30. C5×#472C2×C3semidirect
31. C5×#361C2×C3semidirect
32. C5×#359C2×C3semidirect
33. C5×#350C2×C3semidirect
61. C3×#576C2×C5semidirect
62. C3×#509C2×C5semidirect
63. C2×C3×D16C2×C5semidirect
64. C3×#472C2×C5semidirect
65. C3×#361C2×C5semidirect
66. C3×#359C2×C5semidirect
67. C3×#350C2×C5semidirect
76. C2×C5×D8C22×C3semidirect
81. C5×M32C22×C3semidirect
82. C2×C5×C16C22×C3semidirect
83. C5×#108C22×C3semidirect
85. #2192C3×C5direct (N × H)
117. C2×C3×D8C22×C5semidirect
122. C3×M32C22×C5semidirect
123. C2×C3×C16C22×C5semidirect
124. C3×#108C22×C5semidirect
153. #576C2×C3×C5semidirect
154. #509C2×C3×C5semidirect
155. C2×D16C2×C3×C5semidirect
156. #472C2×C3×C5semidirect
157. #361C2×C3×C5semidirect
158. #359C2×C3×C5semidirect
159. #350C2×C3×C5semidirect
230. C2×D8C22×C3×C5semidirect
235. M32C22×C3×C5semidirect
236. C2×C16C22×C3×C5semidirect
237. #108C22×C3×C5semidirect
307. C3×C5#2192direct (N × H)
355. C5C3×#2192direct (N × H)
368. C3C5×#2192direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5762
C3×C5×#5092
C2×C3×C5×D162
C3×C5×#4722
C3×C5×#3612
C3×C5×#3592
C3×C5×#3502
C5×#21923
C3×#21925