Finite Group Databasebeta

← C3×C5×#1637 #14274671

Subgroups — C3×C5×#1637 #14274671

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×D4×Q8C211yesmaximal, characteristic
2C3×C5×#487C211yesmaximal, characteristic, complemented
3C3×C5×#477C211yesmaximal, characteristic, complemented
4C3×C5×#476C211yesmaximal, characteristic
5C3×C5×#332C211yesmaximal, characteristic, complemented
6C3×C5×Q8⋊C8C211yesmaximal, characteristic, complemented
7C3×C5×D4⋊C8C211yesmaximal, characteristic
8C5×#1637C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C5×Q8—21no—
10C3×C5×C4⋊Q8—21no—
11C3×C5×C4⋊Q8C2211yescharacteristic
12C3×C5×C4⋊Q8C2211yescharacteristic
13C3×C5×C22⋊Q8—21no—
14C3×C5×C22⋊Q8—22no—
15C3×C4×C5×Q8C2211yescharacteristic
16C3×C4×C5×D4—21no—
17C3×C4×C5×D4C2211yescharacteristic
18C3×C5×#107—21no—
19C3×C5×#106—21no—
20C3×C5×C4⋊C8C2211yescharacteristic
21C3×C5×C4⋊C8C2211yescharacteristic
22C3×C5×#103—21no—
23C3×C5×#103—22no—
24C3×C5×#102—21no—
25C3×C4×C5×C8C2211yescharacteristic
26C3×#1637C511yesmaximal, characteristic, direct product factor, Hall
27C5×D4×Q8C2×C311yescharacteristic
28C5×#487C2×C311yescharacteristic, complemented
29C5×#477C2×C311yescharacteristic, complemented
30C5×#476C2×C311yescharacteristic
31C5×#332C2×C311yescharacteristic, complemented
32C5×Q8⋊C8C2×C311yescharacteristic, complemented
33C5×D4⋊C8C2×C311yescharacteristic
34C2×C3×C5×Q8—22no—
35C2×C3×C5×Q8—21no—
36C2×C3×C5×Q8—21no—
37C2×C3×C5×Q8—42no—
38C2×C3×C5×Q8—42no—
39C2×C3×C5×Q8D411yescharacteristic
40C2×C3×C5×D4D411yescharacteristic
41C22×C3×C4×C5—22no—
42C22×C3×C4×C5—21no—
43C2×C3×C5×C8—21no—
44C2×C3×C5×C8—21no—
45C2×C3×C5×C8—21no—
46C3×C5×C4⋊C4—22no—
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4—21no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4D411yescharacteristic
52C3×C5×C4⋊C4D411yescharacteristic
53C3×C5×C4⋊C4D412yes—
54C3×C5×C22⋊C4—22no—
55C3×C5×C22⋊C4—21no—
56C3×C42×C5—21no—
57C3×C42×C5C2311yescharacteristic
58C3×D4×Q8C2×C511yescharacteristic
59C3×#487C2×C511yescharacteristic, complemented
60C3×#477C2×C511yescharacteristic, complemented
61C3×#476C2×C511yescharacteristic
62C3×#332C2×C511yescharacteristic, complemented
63C3×Q8⋊C8C2×C511yescharacteristic, complemented
64C3×D4⋊C8C2×C511yescharacteristic
65C22×C5×Q8—21no—
66C5×C4⋊Q8—21no—
67C5×C4⋊Q8C22×C311yescharacteristic
68C5×C4⋊Q8C22×C311yescharacteristic
69C5×C22⋊Q8—21no—
70C5×C22⋊Q8—22no—
71C4×C5×Q8C22×C311yescharacteristic
72C4×C5×D4—21no—
73C4×C5×D4C22×C311yescharacteristic
74C5×#107—21no—
75C5×#106—21no—
76C5×C4⋊C8C22×C311yescharacteristic
77C5×C4⋊C8C22×C311yescharacteristic
78C5×#103—21no—
79C5×#103—22no—
80C5×#102—21no—
81C4×C5×C8C22×C311yescharacteristic
82#1637C3×C511yescharacteristic, direct product factor, Sylow
83C23×C3×C5—21no—
84C3×C5×Q8—21no—
85C3×C5×Q8—44no—
86C3×C5×Q8—42no—
87C3×C5×Q8—41no—
88C3×C5×Q8—41no—
89C3×C5×Q8SD1612yes—
90C3×C5×D4—21no—
91C3×C5×D4Q1612yes—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—22no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—44no—
97C2×C3×C4×C5—41no—
98C2×C3×C4×C5—41no—
99C2×C3×C4×C5—41no—
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C3×C4×C5C2×D411yescharacteristic
103C3×C5×C8—41no—
104C3×C5×C8—41no—
105C3×C5×C8—41no—
106C22×C3×Q8—21no—
107C3×C4⋊Q8—21no—
108C3×C4⋊Q8C22×C511yescharacteristic
109C3×C4⋊Q8C22×C511yescharacteristic
110C3×C22⋊Q8—21no—
111C3×C22⋊Q8—22no—
112C3×C4×Q8C22×C511yescharacteristic
113C3×C4×D4—21no—
114C3×C4×D4C22×C511yescharacteristic
115C3×#107—21no—
116C3×#106—21no—
117C3×C4⋊C8C22×C511yescharacteristic
118C3×C4⋊C8C22×C511yescharacteristic
119C3×#103—21no—
120C3×#103—22no—
121C3×#102—21no—
122C3×C4×C8C22×C511yescharacteristic
123C2×C5×Q8—22no—
124C2×C5×Q8—21no—
125C2×C5×Q8—21no—
126C2×C5×Q8—42no—
127C2×C5×Q8—42no—
128C2×C5×Q8C3×D411yescharacteristic
129C2×C5×D4C3×D411yescharacteristic
130C22×C4×C5—22no—
131C22×C4×C5—21no—
132C2×C5×C8—21no—
133C2×C5×C8—21no—
134C2×C5×C8—21no—
135C5×C4⋊C4—22no—
136C5×C4⋊C4—21no—
137C5×C4⋊C4—21no—
138C5×C4⋊C4—21no—
139C5×C4⋊C4—21no—
140C5×C4⋊C4C3×D411yescharacteristic
141C5×C4⋊C4C3×D411yescharacteristic
142C5×C4⋊C4C3×D412yes—
143C5×C22⋊C4—22no—
144C5×C22⋊C4—21no—
145C42×C5—21no—
146C42×C5C23×C311yescharacteristic
147D4×Q8C2×C3×C511yescharacteristic
148#487C2×C3×C511yescharacteristic, complemented
149#477C2×C3×C511yescharacteristic, complemented
150#476C2×C3×C511yescharacteristic
151#332C2×C3×C511yescharacteristic, complemented
152Q8⋊C8C2×C3×C511yescharacteristic, complemented
153D4⋊C8C2×C3×C511yescharacteristic
154C22×C3×C5—22no—
155C22×C3×C5—41no—
156C22×C3×C5—41no—
157C22×C3×C5#12011yescharacteristic
158C3×C4×C5—22no—
159C3×C4×C5—21no—
160C3×C4×C5—42no—
161C3×C4×C5—41no—
162C3×C4×C5—41no—
163C3×C4×C5—81no—
164C3×C4×C5C2×SD1611yescharacteristic
165C3×C4×C5C2×Q1611yescharacteristic
166C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
167C3×C4×C5#13711yescharacteristic
168C2×C3×Q8—22no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—42no—
172C2×C3×Q8—42no—
173C2×C3×Q8C5×D411yescharacteristic
174C2×C3×D4C5×D411yescharacteristic
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4C5×D412yes—
186C3×C4⋊C4C5×D411yescharacteristic
187C3×C4⋊C4C5×D411yescharacteristic
188C3×C22⋊C4—22no—
189C3×C22⋊C4—21no—
190C3×C42—21no—
191C3×C42C23×C511yescharacteristic
192C23×C5—21no—
193C5×Q8—21no—
194C5×Q8—44no—
195C5×Q8—42no—
196C5×Q8—41no—
197C5×Q8—41no—
198C5×Q8C3×SD1612yes—
199C5×D4—21no—
200C5×D4C3×Q1612yes—
201C2×C4×C5—21no—
202C2×C4×C5—21no—
203C2×C4×C5—22no—
204C2×C4×C5—21no—
205C2×C4×C5—44no—
206C2×C4×C5—41no—
207C2×C4×C5—41no—
208C2×C4×C5—41no—
209C2×C4×C5C2×C3×D411yescharacteristic
210C2×C4×C5C2×C3×D411yescharacteristic
211C2×C4×C5C2×C3×D411yescharacteristic
212C5×C8—41no—
213C5×C8—41no—
214C5×C8—41no—
215C22×Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8C22×C3×C511yescharacteristic
218C4⋊Q8C22×C3×C511yescharacteristic
219C22⋊Q8—21no—
220C22⋊Q8—22no—
221C4×Q8C22×C3×C511yescharacteristic
222C4×D4—21no—
223C4×D4C22×C3×C511yescharacteristic
224#107—21no—
225#106—21no—
226C4⋊C8C22×C3×C511yescharacteristic
227C4⋊C8C22×C3×C511yescharacteristic
228#103—21no—
229#103—22no—
230#102—21no—
231C4×C8C22×C3×C511yescharacteristic
232C2×C3×C5—42no—
233C2×C3×C5#45011yescharacteristic
234C2×C3×C5#45111yescharacteristic
235C2×C3×C5#45411yescharacteristic
236C23×C3—21no—
237C3×Q8—21no—
238C3×Q8—44no—
239C3×Q8—42no—
240C3×Q8—41no—
241C3×Q8—41no—
242C3×Q8C5×SD1612yes—
243C3×D4—21no—
244C3×D4C5×Q1612yes—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—22no—
248C2×C3×C4—21no—
249C2×C3×C4—44no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4C2×C5×D411yescharacteristic
254C2×C3×C4C2×C5×D411yescharacteristic
255C2×C3×C4C2×C5×D411yescharacteristic
256C3×C8—41no—
257C3×C8—41no—
258C3×C8—41no—
259C22×C5—22no—
260C22×C5—41no—
261C22×C5—41no—
262C22×C5C3×#12011yescharacteristic
263C4×C5—22no—
264C4×C5—21no—
265C4×C5—42no—
266C4×C5—41no—
267C4×C5—41no—
268C4×C5—81no—
269C4×C5C2×C3×SD1611yescharacteristic
270C4×C5C2×C3×Q1611yescharacteristic
271C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
272C4×C5C3×#13711yescharacteristic
273C2×Q8—22no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8C3×C5×D411yescharacteristic
279C2×D4C3×C5×D411yescharacteristic
280C22×C4—22no—
281C22×C4—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4C3×C5×D412yes—
291C4⋊C4C3×C5×D411yescharacteristic
292C4⋊C4C3×C5×D411yescharacteristic
293C22⋊C4—22no—
294C22⋊C4—21no—
295C42—21no—
296C42C23×C3×C511yescharacteristic
297C3×C5#163711yescharacteristic, direct product factor, Hall
298C22×C3—22no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3C5×#12011yescharacteristic
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—42no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—81no—
308C3×C4C2×C5×SD1611yescharacteristic
309C3×C4C2×C5×Q1611yescharacteristic
310C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
311C3×C4C5×#13711yescharacteristic
312C2×C5—42no—
313C2×C5C3×#45011yescharacteristic
314C2×C5C3×#45111yescharacteristic
315C2×C5C3×#45411yescharacteristic
316C23—21no—
317Q8—21no—
318Q8—44no—
319Q8—42no—
320Q8—41no—
321Q8—41no—
322Q8C3×C5×SD1612yes—
323D4—21no—
324D4C3×C5×Q1612yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4C2×C3×C5×D411yescharacteristic
334C2×C4C2×C3×C5×D411yescharacteristic
335C2×C4C2×C3×C5×D411yescharacteristic
336C8—41no—
337C8—41no—
338C8—41no—
339C2×C3—42no—
340C2×C3C5×#45011yescharacteristic
341C2×C3C5×#45111yescharacteristic
342C2×C3C5×#45411yescharacteristic
343C5C3×#163711yescharacteristic, direct product factor, Sylow
344C22—22no—
345C22—41no—
346C22—41no—
347C22C3×C5×#12011yescharacteristic
348C4—22no—
349C4—21no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—81no—
354C4C2×C3×C5×SD1611yescharacteristic
355C4C2×C3×C5×Q1611yescharacteristic
356C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
357C4C3×C5×#13711yescharacteristic
358C3C5×#163711yescharacteristic, direct product factor, Sylow
359C2—42no—
360C2C3×C5×#45011yescharacteristic
361C2C3×C5×#45111yescharacteristic
362C2C3×C5×#45411yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC42C23×C3×C5
prime radicalC2×C3×C5×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C3×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×C5×#487C2semidirect
3. C3×C5×#477C2semidirect
5. C3×C5×#332C2semidirect
6. C3×C5×Q8⋊C8C2semidirect
8. C5×#1637C3direct (N × H)
26. C3×#1637C5direct (N × H)
28. C5×#487C2×C3semidirect
29. C5×#477C2×C3semidirect
31. C5×#332C2×C3semidirect
32. C5×Q8⋊C8C2×C3semidirect
59. C3×#487C2×C5semidirect
60. C3×#477C2×C5semidirect
62. C3×#332C2×C5semidirect
63. C3×Q8⋊C8C2×C5semidirect
82. #1637C3×C5direct (N × H)
148. #487C2×C3×C5semidirect
149. #477C2×C3×C5semidirect
151. #332C2×C3×C5semidirect
152. Q8⋊C8C2×C3×C5semidirect
297. C3×C5#1637direct (N × H)
343. C5C3×#1637direct (N × H)
358. C3C5×#1637direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×D4×Q82
C3×C5×#4872
C3×C5×#4772
C3×C5×#4762
C3×C5×#3322
C3×C5×Q8⋊C82
C3×C5×D4⋊C82
C5×#16373
C3×#16375