Finite Group Databasebeta

← C3×C5×#2186 #14275220

Subgroups — C3×C5×#2186 #14275220

Order 1920.

460 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#135C211yesmaximal, characteristic, complemented
2C2×C3×C5×SD32C211yesmaximal, characteristic, complemented
3C2×C3×C5×D16C211yesmaximal, characteristic, complemented
4C3×C5×#466C211yesmaximal, characteristic, complemented
5C3×C5×#358C211yesmaximal, characteristic, complemented
6C3×C5×#357C211yesmaximal, characteristic, complemented
7C3×C5×C22⋊C16C211yesmaximal, characteristic, complemented
8C5×#2186C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41—21no—
10C3×C5×#135—22no—
11C3×C5×#135—22no—
12C2×C3×C5×Q16C2211yescharacteristic
13C2×C3×C5×SD16—21no—
14C2×C3×C5×D8C2211yescharacteristic
15C2×C3×C5×D8C2211yescharacteristic
16C22×C3×C5×C8C2211yescharacteristic
17C3×C5×#121—21no—
18C3×C5×SD32—22no—
19C3×C5×D16—22no—
20C2×C3×C5×C16C2211yescharacteristic
21C2×C3×C5×C16C2211yescharacteristic
22C3×C5×#107C2211yescharacteristic
23C3×C5×#102—21no—
24C3×#2186C511yesmaximal, characteristic, direct product factor, Hall
25C2×C5×#135C2×C311yescharacteristic, complemented
26C2×C5×SD32C2×C311yescharacteristic, complemented
27C2×C5×D16C2×C311yescharacteristic, complemented
28C5×#466C2×C311yescharacteristic, complemented
29C5×#358C2×C311yescharacteristic, complemented
30C5×#357C2×C311yescharacteristic, complemented
31C5×C22⋊C16C2×C311yescharacteristic, complemented
32C3×C5×#41—24no—
33C3×C5×#41—42no—
34C2×C3×C5×Q8—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4—41no—
38C2×C3×C5×D4—41no—
39C22×C3×C4×C5—41no—
40C22×C3×C4×C5D411yescharacteristic
41C3×C5×Q16—21no—
42C3×C5×Q16D412yescomplemented
43C3×C5×SD16—24no—
44C3×C5×D8—21no—
45C3×C5×D8—21no—
46C3×C5×D8—21no—
47C3×C5×D8D412yescomplemented
48C2×C3×C5×C8—21no—
49C2×C3×C5×C8—21no—
50C2×C3×C5×C8D411yescharacteristic
51C2×C3×C5×C8C2311yescharacteristic
52C3×C5×C4⋊C4—21no—
53C3×C5×C22⋊C4—41no—
54C3×C5×C16—21no—
55C3×C5×C16—21no—
56C2×C3×#135C2×C511yescharacteristic, complemented
57C2×C3×SD32C2×C511yescharacteristic, complemented
58C2×C3×D16C2×C511yescharacteristic, complemented
59C3×#466C2×C511yescharacteristic, complemented
60C3×#358C2×C511yescharacteristic, complemented
61C3×#357C2×C511yescharacteristic, complemented
62C3×C22⋊C16C2×C511yescharacteristic, complemented
63C2×C5×#41—21no—
64C5×#135—22no—
65C5×#135—22no—
66C2×C5×Q16C22×C311yescharacteristic
67C2×C5×SD16—21no—
68C2×C5×D8C22×C311yescharacteristic
69C2×C5×D8C22×C311yescharacteristic
70C22×C5×C8C22×C311yescharacteristic
71C5×#121—21no—
72C5×SD32—22no—
73C5×D16—22no—
74C2×C5×C16C22×C311yescharacteristic
75C2×C5×C16C22×C311yescharacteristic
76C5×#107C22×C311yescharacteristic
77C5×#102—21no—
78#2186C3×C511yescharacteristic, direct product factor, Sylow
79C23×C3×C5—41no—
80C23×C3×C5—41no—
81C23×C3×C5D811yescharacteristic
82C3×C5×Q8—22no—
83C3×C5×Q8—41no—
84C3×C5×D4—22no—
85C3×C5×D4—44no—
86C3×C5×D4—41no—
87C3×C5×D4—41no—
88C3×C5×D4—41no—
89C3×C5×D4—82no—
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5—21no—
92C2×C3×C4×C5—44no—
93C2×C3×C4×C5—41no—
94C2×C3×C4×C5—41no—
95C2×C3×C4×C5D811yescharacteristic
96C2×C3×C4×C5C2×D411yescharacteristic
97C3×C5×C8—21no—
98C3×C5×C8C2×D411yescharacteristic
99C3×C5×C8C2×D411yescharacteristic
100C2×C3×#41—21no—
101C3×#135—22no—
102C3×#135—22no—
103C2×C3×Q16C22×C511yescharacteristic
104C2×C3×SD16—21no—
105C2×C3×D8C22×C511yescharacteristic
106C2×C3×D8C22×C511yescharacteristic
107C22×C3×C8C22×C511yescharacteristic
108C3×#121—21no—
109C3×SD32—22no—
110C3×D16—22no—
111C2×C3×C16C22×C511yescharacteristic
112C2×C3×C16C22×C511yescharacteristic
113C3×#107C22×C511yescharacteristic
114C3×#102—21no—
115C5×#41—24no—
116C5×#41—42no—
117C2×C5×Q8—21no—
118C2×C5×D4—21no—
119C2×C5×D4—21no—
120C2×C5×D4—41no—
121C2×C5×D4—41no—
122C22×C4×C5—41no—
123C22×C4×C5C3×D411yescharacteristic
124C5×Q16—21no—
125C5×Q16C3×D412yescomplemented
126C5×SD16—24no—
127C5×D8—21no—
128C5×D8—21no—
129C5×D8—21no—
130C5×D8C3×D412yescomplemented
131C2×C5×C8—21no—
132C2×C5×C8—21no—
133C2×C5×C8C3×D411yescharacteristic
134C2×C5×C8C23×C311yescharacteristic
135C5×C4⋊C4—21no—
136C5×C22⋊C4—41no—
137C5×C16—21no—
138C5×C16—21no—
139C2×#135C2×C3×C511yescharacteristic, complemented
140C2×SD32C2×C3×C511yescharacteristic, complemented
141C2×D16C2×C3×C511yescharacteristic, complemented
142#466C2×C3×C511yescharacteristic, complemented
143#358C2×C3×C511yescharacteristic, complemented
144#357C2×C3×C511yescharacteristic, complemented
145C22⋊C16C2×C3×C511yescharacteristic, complemented
146C22×C3×C5—21no—
147C22×C3×C5—21no—
148C22×C3×C5—21no—
149C22×C3×C5—42no—
150C22×C3×C5—81no—
151C22×C3×C5—81no—
152C22×C3×C5—81no—
153C22×C3×C5—81no—
154C22×C3×C5—81no—
155C22×C3×C5C2×D811yescharacteristic
156C3×C4×C5—21no—
157C3×C4×C5—42no—
158C3×C4×C5—81no—
159C3×C4×C5#12011yescharacteristic
160C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
161C3×#41—24no—
162C3×#41—42no—
163C2×C3×Q8—21no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—41no—
167C2×C3×D4—41no—
168C22×C3×C4—41no—
169C22×C3×C4C5×D411yescharacteristic
170C3×Q16—21no—
171C3×Q16C5×D412yescomplemented
172C3×SD16—24no—
173C3×D8—21no—
174C3×D8—21no—
175C3×D8—21no—
176C3×D8C5×D412yescomplemented
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8C5×D411yescharacteristic
180C2×C3×C8C23×C511yescharacteristic
181C3×C4⋊C4—21no—
182C3×C22⋊C4—41no—
183C3×C16—21no—
184C3×C16—21no—
185C23×C5—41no—
186C23×C5—41no—
187C23×C5C3×D811yescharacteristic
188C5×Q8—22no—
189C5×Q8—41no—
190C5×D4—22no—
191C5×D4—44no—
192C5×D4—41no—
193C5×D4—41no—
194C5×D4—41no—
195C5×D4—82no—
196C2×C4×C5—21no—
197C2×C4×C5—21no—
198C2×C4×C5—44no—
199C2×C4×C5—41no—
200C2×C4×C5—41no—
201C2×C4×C5C3×D811yescharacteristic
202C2×C4×C5C2×C3×D411yescharacteristic
203C5×C8—21no—
204C5×C8C2×C3×D411yescharacteristic
205C5×C8C2×C3×D411yescharacteristic
206C2×#41—21no—
207#135—22no—
208#135—22no—
209C2×Q16C22×C3×C511yescharacteristic
210C2×SD16—21no—
211C2×D8C22×C3×C511yescharacteristic
212C2×D8C22×C3×C511yescharacteristic
213C22×C8C22×C3×C511yescharacteristic
214#121—21no—
215SD32—22no—
216D16—22no—
217C2×C16C22×C3×C511yescharacteristic
218C2×C16C22×C3×C511yescharacteristic
219#107C22×C3×C511yescharacteristic
220#102—21no—
221C2×C3×C5—41no—
222C2×C3×C5—82no—
223C2×C3×C5—161no—
224C2×C3×C5#44711yescharacteristic
225C2×C3×C5#50811yescharacteristic
226C2×C3×C5#50911yescharacteristic
227C23×C3—41no—
228C23×C3—41no—
229C23×C3C5×D811yescharacteristic
230C3×Q8—22no—
231C3×Q8—41no—
232C3×D4—22no—
233C3×D4—44no—
234C3×D4—41no—
235C3×D4—41no—
236C3×D4—41no—
237C3×D4—82no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—44no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4C5×D811yescharacteristic
244C2×C3×C4C2×C5×D411yescharacteristic
245C3×C8—21no—
246C3×C8C2×C5×D411yescharacteristic
247C3×C8C2×C5×D411yescharacteristic
248C22×C5—21no—
249C22×C5—21no—
250C22×C5—21no—
251C22×C5—42no—
252C22×C5—81no—
253C22×C5—81no—
254C22×C5—81no—
255C22×C5—81no—
256C22×C5—81no—
257C22×C5C2×C3×D811yescharacteristic
258C4×C5—21no—
259C4×C5—42no—
260C4×C5—81no—
261C4×C5C3×#12011yescharacteristic
262C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
263#41—24no—
264#41—42no—
265C2×Q8—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—41no—
270C22×C4—41no—
271C22×C4C3×C5×D411yescharacteristic
272Q16—21no—
273Q16C3×C5×D412yescomplemented
274SD16—24no—
275D8—21no—
276D8—21no—
277D8—21no—
278D8C3×C5×D412yescomplemented
279C2×C8—21no—
280C2×C8—21no—
281C2×C8C3×C5×D411yescharacteristic
282C2×C8C23×C3×C511yescharacteristic
283C4⋊C4—21no—
284C22⋊C4—41no—
285C16—21no—
286C16—21no—
287C3×C5#218611yescharacteristic, direct product factor, Hall
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—42no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3C2×C5×D811yescharacteristic
298C3×C4—21no—
299C3×C4—42no—
300C3×C4—81no—
301C3×C4C5×#12011yescharacteristic
302C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
303C2×C5—41no—
304C2×C5—82no—
305C2×C5—161no—
306C2×C5C3×#44711yescharacteristic
307C2×C5C3×#50811yescharacteristic
308C2×C5C3×#50911yescharacteristic
309C23—41no—
310C23—41no—
311C23C3×C5×D811yescharacteristic
312Q8—22no—
313Q8—41no—
314D4—22no—
315D4—44no—
316D4—41no—
317D4—41no—
318D4—41no—
319D4—82no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—44no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4C3×C5×D811yescharacteristic
326C2×C4C2×C3×C5×D411yescharacteristic
327C8—21no—
328C8C2×C3×C5×D411yescharacteristic
329C8C2×C3×C5×D411yescharacteristic
330C2×C3—41no—
331C2×C3—82no—
332C2×C3—161no—
333C2×C3C5×#44711yescharacteristic
334C2×C3C5×#50811yescharacteristic
335C2×C3C5×#50911yescharacteristic
336C5C3×#218611yescharacteristic, direct product factor, Sylow
337C22—21no—
338C22—21no—
339C22—21no—
340C22—42no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22C2×C3×C5×D811yescharacteristic
347C4—21no—
348C4—42no—
349C4—81no—
350C4C3×C5×#12011yescharacteristic
351C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
352C3C5×#218611yescharacteristic, direct product factor, Sylow
353C2—41no—
354C2—82no—
355C2—161no—
356C2C3×C5×#44711yescharacteristic
357C2C3×C5×#50811yescharacteristic
358C2C3×C5×#50911yescharacteristic
subgroupG/H
center, socleC22×C3×C5C2×D8
commutator, Frattini, cosocleC2×C8C23×C3×C5
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C22×C3×C4×C5 ◁ C22×C3×C5×C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#135C2semidirect
2. C2×C3×C5×SD32C2semidirect
3. C2×C3×C5×D16C2semidirect
4. C3×C5×#466C2semidirect
5. C3×C5×#358C2semidirect
6. C3×C5×#357C2semidirect
7. C3×C5×C22⋊C16C2semidirect
8. C5×#2186C3direct (N × H)
24. C3×#2186C5direct (N × H)
25. C2×C5×#135C2×C3semidirect
26. C2×C5×SD32C2×C3semidirect
27. C2×C5×D16C2×C3semidirect
28. C5×#466C2×C3semidirect
29. C5×#358C2×C3semidirect
30. C5×#357C2×C3semidirect
31. C5×C22⋊C16C2×C3semidirect
42. C3×C5×Q16D4semidirect
47. C3×C5×D8D4semidirect
56. C2×C3×#135C2×C5semidirect
57. C2×C3×SD32C2×C5semidirect
58. C2×C3×D16C2×C5semidirect
59. C3×#466C2×C5semidirect
60. C3×#358C2×C5semidirect
61. C3×#357C2×C5semidirect
62. C3×C22⋊C16C2×C5semidirect
78. #2186C3×C5direct (N × H)
125. C5×Q16C3×D4semidirect
130. C5×D8C3×D4semidirect
139. C2×#135C2×C3×C5semidirect
140. C2×SD32C2×C3×C5semidirect
141. C2×D16C2×C3×C5semidirect
142. #466C2×C3×C5semidirect
143. #358C2×C3×C5semidirect
144. #357C2×C3×C5semidirect
145. C22⋊C16C2×C3×C5semidirect
171. C3×Q16C5×D4semidirect
176. C3×D8C5×D4semidirect
273. Q16C3×C5×D4semidirect
278. D8C3×C5×D4semidirect
287. C3×C5#2186direct (N × H)
336. C5C3×#2186direct (N × H)
352. C3C5×#2186direct (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×#1352
C2×C3×C5×SD322
C2×C3×C5×D162
C3×C5×#4662
C3×C5×#3582
C3×C5×#3572
C3×C5×C22⋊C162
C5×#21863
C3×#21865