Finite Group Databasebeta

← C3×C5×#1973 #14275007

Subgroups — C3×C5×#1973 #14275007

Order 1920.

680 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×SD16C211yesmaximal, characteristic
2C2×C3×C5×#107C211yesmaximal, characteristic, complemented
3C2×C3×C5×#103C211yesmaximal, characteristic, complemented
4C2×C3×C5×#102C211yesmaximal, characteristic
5C2×C3×C5×#97C211yesmaximal, characteristic, complemented
6C3×C5×#391C211yesmaximal, characteristic, complemented
7C3×C5×#390C211yesmaximal, characteristic
8C5×#1973C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C22×C3×C5×D4C2211yescharacteristic
11C2×C3×C5×SD16—22no—
12C2×C3×C5×SD16—22no—
13C2×C3×C5×SD16C414yescomplemented
14C22×C3×C5×C8C2211yescharacteristic
15C22×C3×C5×C8C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C22⋊C4—22no—
19C2×C3×C42×C5C2211yescharacteristic
20C3×C5×#107—22no—
21C3×C5×#103—22no—
22C3×C5×#102—22no—
23C3×C5×#97—22no—
24C3×C5×#95—22no—
25C3×#1973C511yesmaximal, characteristic, direct product factor, Hall
26C22×C5×SD16C2×C311yescharacteristic
27C2×C5×#107C2×C311yescharacteristic, complemented
28C2×C5×#103C2×C311yescharacteristic, complemented
29C2×C5×#102C2×C311yescharacteristic
30C2×C5×#97C2×C311yescharacteristic, complemented
31C5×#391C2×C311yescharacteristic, complemented
32C5×#390C2×C311yescharacteristic
33C24×C3×C5—21no—
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8—21no—
36C2×C3×C5×Q8—21no—
37C2×C3×C5×Q8—21no—
38C2×C3×C5×Q8—21no—
39C2×C3×C5×Q8C2×C412yes—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—21no—
42C2×C3×C5×D4—21no—
43C2×C3×C5×D4—21no—
44C2×C3×C5×D4—21no—
45C2×C3×C5×D4C2×C412yes—
46C22×C3×C4×C5—21no—
47C22×C3×C4×C5—21no—
48C22×C3×C4×C5—21no—
49C22×C3×C4×C5D412yes—
50C22×C3×C4×C5C2311yescharacteristic
51C3×C5×SD16—28no—
52C2×C3×C5×C8—21no—
53C2×C3×C5×C8—21no—
54C2×C3×C5×C8C2×C412yes—
55C2×C3×C5×C8D412yes—
56C2×C3×C5×C8D412yes—
57C2×C3×C5×C8D412yes—
58C3×C5×C4⋊C4—21no—
59C3×C5×C4⋊C4—21no—
60C3×C5×C4⋊C4—21no—
61C3×C5×C4⋊C4—21no—
62C3×C5×C22⋊C4—44no—
63C3×C42×C5—21no—
64C3×C42×C5—21no—
65C22×C3×SD16C2×C511yescharacteristic
66C2×C3×#107C2×C511yescharacteristic, complemented
67C2×C3×#103C2×C511yescharacteristic, complemented
68C2×C3×#102C2×C511yescharacteristic
69C2×C3×#97C2×C511yescharacteristic, complemented
70C3×#391C2×C511yescharacteristic, complemented
71C3×#390C2×C511yescharacteristic
72C22×C5×Q8C22×C311yescharacteristic
73C22×C5×D4C22×C311yescharacteristic
74C2×C5×SD16—22no—
75C2×C5×SD16—22no—
76C2×C5×SD16C3×C414yescomplemented
77C22×C5×C8C22×C311yescharacteristic
78C22×C5×C8C22×C311yescharacteristic
79C2×C5×C4⋊C4C22×C311yescharacteristic
80C2×C5×C4⋊C4C22×C311yescharacteristic
81C2×C5×C22⋊C4—22no—
82C2×C42×C5C22×C311yescharacteristic
83C5×#107—22no—
84C5×#103—22no—
85C5×#102—22no—
86C5×#97—22no—
87C5×#95—22no—
88#1973C3×C511yescharacteristic, direct product factor, Sylow
89C23×C3×C5—22no—
90C23×C3×C5—41no—
91C23×C3×C5—41no—
92C23×C3×C5—42no—
93C23×C3×C5—42no—
94C23×C3×C5C2×D411yescharacteristic
95C3×C5×Q8—22no—
96C3×C5×Q8—22no—
97C3×C5×Q8—41no—
98C3×C5×Q8—41no—
99C3×C5×D4—22no—
100C3×C5×D4—22no—
101C3×C5×D4—41no—
102C3×C5×D4—41no—
103C2×C3×C4×C5—22no—
104C2×C3×C4×C5—22no—
105C2×C3×C4×C5—22no—
106C2×C3×C4×C5—22no—
107C2×C3×C4×C5—41no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5—41no—
111C2×C3×C4×C5—42no—
112C2×C3×C4×C5—42no—
113C2×C3×C4×C5C22×C411yescharacteristic
114C2×C3×C4×C5C2×D411yescharacteristic
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5C2×D411yescharacteristic
117C2×C3×C4×C5#4111yescharacteristic
118C2×C3×C4×C5#4111yescharacteristic
119C3×C5×C8—22no—
120C3×C5×C8C22⋊C414yescomplemented
121C22×C3×Q8C22×C511yescharacteristic
122C22×C3×D4C22×C511yescharacteristic
123C2×C3×SD16—22no—
124C2×C3×SD16—22no—
125C2×C3×SD16C4×C514yescomplemented
126C22×C3×C8C22×C511yescharacteristic
127C22×C3×C8C22×C511yescharacteristic
128C2×C3×C4⋊C4C22×C511yescharacteristic
129C2×C3×C4⋊C4C22×C511yescharacteristic
130C2×C3×C22⋊C4—22no—
131C2×C3×C42C22×C511yescharacteristic
132C3×#107—22no—
133C3×#103—22no—
134C3×#102—22no—
135C3×#97—22no—
136C3×#95—22no—
137C24×C5—21no—
138C2×C5×Q8—21no—
139C2×C5×Q8—21no—
140C2×C5×Q8—21no—
141C2×C5×Q8—21no—
142C2×C5×Q8—21no—
143C2×C5×Q8C2×C3×C412yes—
144C2×C5×D4—21no—
145C2×C5×D4—21no—
146C2×C5×D4—21no—
147C2×C5×D4—21no—
148C2×C5×D4—21no—
149C2×C5×D4C2×C3×C412yes—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5—21no—
153C22×C4×C5C3×D412yes—
154C22×C4×C5C23×C311yescharacteristic
155C5×SD16—28no—
156C2×C5×C8—21no—
157C2×C5×C8—21no—
158C2×C5×C8C2×C3×C412yes—
159C2×C5×C8C3×D412yes—
160C2×C5×C8C3×D412yes—
161C2×C5×C8C3×D412yes—
162C5×C4⋊C4—21no—
163C5×C4⋊C4—21no—
164C5×C4⋊C4—21no—
165C5×C4⋊C4—21no—
166C5×C22⋊C4—44no—
167C42×C5—21no—
168C42×C5—21no—
169C22×SD16C2×C3×C511yescharacteristic
170C2×#107C2×C3×C511yescharacteristic, complemented
171C2×#103C2×C3×C511yescharacteristic, complemented
172C2×#102C2×C3×C511yescharacteristic
173C2×#97C2×C3×C511yescharacteristic, complemented
174#391C2×C3×C511yescharacteristic, complemented
175#390C2×C3×C511yescharacteristic
176C22×C3×C5—42no—
177C22×C3×C5—44no—
178C22×C3×C5—82no—
179C22×C3×C5—81no—
180C22×C3×C5—81no—
181C22×C3×C5C4×D411yescharacteristic
182C22×C3×C5#12111yescharacteristic
183C22×C3×C5#12711yescharacteristic
184C22×C3×C5AGL(1,ℤ/8)12yes—
185C22×C3×C5#13712yes—
186C3×C4×C5—42no—
187C3×C4×C5—42no—
188C3×C4×C5—81no—
189C3×C4×C5—81no—
190C3×C4×C5C2×C22⋊C411yescharacteristic
191C3×C4×C5C4×D411yescharacteristic
192C3×C4×C5#12111yescharacteristic
193C3×C4×C5#12411yescharacteristic
194C24×C3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8C2×C4×C512yes—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4C2×C4×C512yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C5×D412yes—
211C22×C3×C4C23×C511yescharacteristic
212C3×SD16—28no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8C2×C4×C512yes—
216C2×C3×C8C5×D412yes—
217C2×C3×C8C5×D412yes—
218C2×C3×C8C5×D412yes—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C22⋊C4—44no—
224C3×C42—21no—
225C3×C42—21no—
226C23×C5—22no—
227C23×C5—41no—
228C23×C5—41no—
229C23×C5—42no—
230C23×C5—42no—
231C23×C5C2×C3×D411yescharacteristic
232C5×Q8—22no—
233C5×Q8—22no—
234C5×Q8—41no—
235C5×Q8—41no—
236C5×D4—22no—
237C5×D4—22no—
238C5×D4—41no—
239C5×D4—41no—
240C2×C4×C5—22no—
241C2×C4×C5—22no—
242C2×C4×C5—22no—
243C2×C4×C5—22no—
244C2×C4×C5—41no—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5—41no—
248C2×C4×C5—42no—
249C2×C4×C5—42no—
250C2×C4×C5C22×C3×C411yescharacteristic
251C2×C4×C5C2×C3×D411yescharacteristic
252C2×C4×C5C2×C3×D411yescharacteristic
253C2×C4×C5C2×C3×D411yescharacteristic
254C2×C4×C5C3×#4111yescharacteristic
255C2×C4×C5C3×#4111yescharacteristic
256C5×C8—22no—
257C5×C8C3×C22⋊C414yescomplemented
258C22×Q8C22×C3×C511yescharacteristic
259C22×D4C22×C3×C511yescharacteristic
260C2×SD16—22no—
261C2×SD16—22no—
262C2×SD16C3×C4×C514yescomplemented
263C22×C8C22×C3×C511yescharacteristic
264C22×C8C22×C3×C511yescharacteristic
265C2×C4⋊C4C22×C3×C511yescharacteristic
266C2×C4⋊C4C22×C3×C511yescharacteristic
267C2×C22⋊C4—22no—
268C2×C42C22×C3×C511yescharacteristic
269#107—22no—
270#103—22no—
271#102—22no—
272#97—22no—
273#95—22no—
274C2×C3×C5—82no—
275C2×C3×C5#39011yescharacteristic
276C2×C3×C5#44012yes—
277C2×C3×C5#46812yes—
278C2×C3×C5#49611yescharacteristic
279C2×C3×C5#49711yescharacteristic
280C23×C3—22no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3C2×C5×D411yescharacteristic
286C3×Q8—22no—
287C3×Q8—22no—
288C3×Q8—41no—
289C3×Q8—41no—
290C3×D4—22no—
291C3×D4—22no—
292C3×D4—41no—
293C3×D4—41no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4C22×C4×C511yescharacteristic
305C2×C3×C4C2×C5×D411yescharacteristic
306C2×C3×C4C2×C5×D411yescharacteristic
307C2×C3×C4C2×C5×D411yescharacteristic
308C2×C3×C4C5×#4111yescharacteristic
309C2×C3×C4C5×#4111yescharacteristic
310C3×C8—22no—
311C3×C8C5×C22⋊C414yescomplemented
312C22×C5—42no—
313C22×C5—44no—
314C22×C5—82no—
315C22×C5—81no—
316C22×C5—81no—
317C22×C5C3×C4×D411yescharacteristic
318C22×C5C3×#12111yescharacteristic
319C22×C5C3×#12711yescharacteristic
320C22×C5C3×AGL(1,ℤ/8)12yes—
321C22×C5C3×#13712yes—
322C4×C5—42no—
323C4×C5—42no—
324C4×C5—81no—
325C4×C5—81no—
326C4×C5C2×C3×C22⋊C411yescharacteristic
327C4×C5C3×C4×D411yescharacteristic
328C4×C5C3×#12111yescharacteristic
329C4×C5C3×#12411yescharacteristic
330C24—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8C2×C3×C4×C512yes—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4C2×C3×C4×C512yes—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4C3×C5×D412yes—
347C22×C4C23×C3×C511yescharacteristic
348SD16—28no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8C2×C3×C4×C512yes—
352C2×C8C3×C5×D412yes—
353C2×C8C3×C5×D412yes—
354C2×C8C3×C5×D412yes—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C22⋊C4—44no—
360C42—21no—
361C42—21no—
362C3×C5#197311yescharacteristic, direct product factor, Hall
363C22×C3—42no—
364C22×C3—44no—
365C22×C3—82no—
366C22×C3—81no—
367C22×C3—81no—
368C22×C3C4×C5×D411yescharacteristic
369C22×C3C5×#12111yescharacteristic
370C22×C3C5×#12711yescharacteristic
371C22×C3C5×AGL(1,ℤ/8)12yes—
372C22×C3C5×#13712yes—
373C3×C4—42no—
374C3×C4—42no—
375C3×C4—81no—
376C3×C4—81no—
377C3×C4C2×C5×C22⋊C411yescharacteristic
378C3×C4C4×C5×D411yescharacteristic
379C3×C4C5×#12111yescharacteristic
380C3×C4C5×#12411yescharacteristic
381C2×C5—82no—
382C2×C5C3×#39011yescharacteristic
383C2×C5C3×#44012yes—
384C2×C5C3×#46812yes—
385C2×C5C3×#49611yescharacteristic
386C2×C5C3×#49711yescharacteristic
387C23—22no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—42no—
392C23C2×C3×C5×D411yescharacteristic
393Q8—22no—
394Q8—22no—
395Q8—41no—
396Q8—41no—
397D4—22no—
398D4—22no—
399D4—41no—
400D4—41no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4C22×C3×C4×C511yescharacteristic
412C2×C4C2×C3×C5×D411yescharacteristic
413C2×C4C2×C3×C5×D411yescharacteristic
414C2×C4C2×C3×C5×D411yescharacteristic
415C2×C4C3×C5×#4111yescharacteristic
416C2×C4C3×C5×#4111yescharacteristic
417C8—22no—
418C8C3×C5×C22⋊C414yescomplemented
419C2×C3—82no—
420C2×C3C5×#39011yescharacteristic
421C2×C3C5×#44012yes—
422C2×C3C5×#46812yes—
423C2×C3C5×#49611yescharacteristic
424C2×C3C5×#49711yescharacteristic
425C5C3×#197311yescharacteristic, direct product factor, Sylow
426C22—42no—
427C22—44no—
428C22—82no—
429C22—81no—
430C22—81no—
431C22C3×C4×C5×D411yescharacteristic
432C22C3×C5×#12111yescharacteristic
433C22C3×C5×#12711yescharacteristic
434C22C3×C5×AGL(1,ℤ/8)12yes—
435C22C3×C5×#13712yes—
436C4—42no—
437C4—42no—
438C4—81no—
439C4—81no—
440C4C2×C3×C5×C22⋊C411yescharacteristic
441C4C3×C4×C5×D411yescharacteristic
442C4C3×C5×#12111yescharacteristic
443C4C3×C5×#12411yescharacteristic
444C3C5×#197311yescharacteristic, direct product factor, Sylow
445C2—82no—
446C2C3×C5×#39011yescharacteristic
447C2C3×C5×#44012yes—
448C2C3×C5×#46812yes—
449C2C3×C5×#49611yescharacteristic
450C2C3×C5×#49711yescharacteristic
subgroupG/H
center, socleC23×C3×C5C2×D4
commutatorC2×C4C22×C3×C4×C5
Frattini, cosocleC22×C4C23×C3×C5
prime radicalC22×C3×C5×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3×C5 ◁ C2×C3×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×#107C2semidirect
3. C2×C3×C5×#103C2semidirect
5. C2×C3×C5×#97C2semidirect
6. C3×C5×#391C2semidirect
8. C5×#1973C3direct (N × H)
13. C2×C3×C5×SD16C4semidirect
25. C3×#1973C5direct (N × H)
27. C2×C5×#107C2×C3semidirect
28. C2×C5×#103C2×C3semidirect
30. C2×C5×#97C2×C3semidirect
31. C5×#391C2×C3semidirect
66. C2×C3×#107C2×C5semidirect
67. C2×C3×#103C2×C5semidirect
69. C2×C3×#97C2×C5semidirect
70. C3×#391C2×C5semidirect
76. C2×C5×SD16C3×C4semidirect
88. #1973C3×C5direct (N × H)
120. C3×C5×C8C22⋊C4semidirect
125. C2×C3×SD16C4×C5semidirect
170. C2×#107C2×C3×C5semidirect
171. C2×#103C2×C3×C5semidirect
173. C2×#97C2×C3×C5semidirect
174. #391C2×C3×C5semidirect
257. C5×C8C3×C22⋊C4semidirect
262. C2×SD16C3×C4×C5semidirect
311. C3×C8C5×C22⋊C4semidirect
362. C3×C5#1973direct (N × H)
418. C8C3×C5×C22⋊C4semidirect
425. C5C3×#1973direct (N × H)
444. C3C5×#1973direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C5×SD162
C2×C3×C5×#1072
C2×C3×C5×#1032
C2×C3×C5×#1022
C2×C3×C5×#972
C3×C5×#3912
C3×C5×#3902
C5×#19733
C3×#19735