Finite Group Databasebeta

← C2×C3×#65693 #14414701

Subgroups — C2×C3×#65693 #14414701

Order 1920.

1128 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×#170C211yesmaximal, characteristic, complemented
2C22×C3×#700C211yesmaximal, characteristic, complemented
3C2×C3×#4027C211yesmaximal, characteristic
4C2×C3×C22⋊#170C211yesmaximal, characteristic, complemented
5C2×C3×#4003C211yesmaximal, characteristic, complemented
6C2×C3×#4002C211yesmaximal, characteristic, complemented
7C2×C3×C4⋊#170C211yesmaximal, characteristic, complemented
8C3×#65693C218yesmaximal, direct product factor
9C2×#65693C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C22⋊D5C411yescharacteristic
11C23×C3×Dic5C2211yescharacteristic
12C22×C3×C4×D5C2211yescharacteristic
13C22×C3×#170—24no—
14C22×C3×#170C2211yescharacteristic
15C22×C3×#170C2211yescharacteristic
16C22×C3×#170C2211yescharacteristic
17C22×C3×#170C2211yescharacteristic
18C22×C3×#170C2214yescomplemented
19C2×C3×#700—24no—
20C2×C3×#700C2214yescomplemented
21C2×C3×C5×C22⋊C4C411yescharacteristic
22C2×C3×#685C411yescharacteristic
23C2×C3×#683C411yescharacteristic
24C2×C3×C4×Dic5C2211yescharacteristic
25C3×#4027C414yes—
26C3×#4027C2214yes—
27C3×C22⋊#170C2214yescomplemented
28C3×#4003C2214yescomplemented
29C3×#4002C2214yescomplemented
30C3×C4⋊#170C2214yescomplemented
31C2×C3×#435—51nomaximal, Hall
32C23×#170C2×C311yescharacteristic, complemented
33C22×#700C2×C311yescharacteristic, complemented
34C2×#4027C2×C311yescharacteristic
35C2×C22⋊#170C2×C311yescharacteristic, complemented
36C2×#4003C2×C311yescharacteristic, complemented
37C2×#4002C2×C311yescharacteristic, complemented
38C2×C4⋊#170C2×C311yescharacteristic, complemented
39#65693C2×C318yesdirect product factor
40C24×C3×C5C2×C411yescharacteristic
41C23×C3×D5C2×C411yescharacteristic
42C22×C3×C4×C5C2×C411yescharacteristic
43C22×C3×C4×C5C2×C411yescharacteristic
44C2×C3×C22⋊D5—24no—
45C2×C3×C22⋊D5C2×C414yes—
46C22×C3×Dic5—22no—
47C22×C3×Dic5—22no—
48C22×C3×Dic5C2×C412yes—
49C22×C3×Dic5C2×C411yescharacteristic
50C22×C3×Dic5C2×C411yescharacteristic
51C22×C3×Dic5C2312yes—
52C22×C3×Dic5C2311yescharacteristic
53C2×C3×C4×D5—22no—
54C2×C3×C4×D5—22no—
55C2×C3×C4×D5C2×C412yes—
56C2×C3×C4×D5C2312yes—
57C2×C3×#170—22no—
58C2×C3×#170—22no—
59C2×C3×#170—22no—
60C2×C3×#170—28no—
61C2×C3×#170D414yescomplemented
62C2×C3×#170C2312yes—
63C2×C3×#170C2312yes—
64C2×C3×#170C2312yes—
65C2×C3×#170C2312yes—
66C3×#700—28no—
67C3×C5×C22⋊C4C2×C414yes—
68C3×#685C2×C414yes—
69C3×#683C2×C414yes—
70C3×C4×Dic5C2×C412yes—
71C3×C4×Dic5C2312yes—
72C22×C3×M16—51no—
73C23×C3×C8—51no—
74C2×C3×C4×D4—51no—
75C3×#435—58no—
76C2×C3×C4⋊C8—51no—
77C2×C3×C22⋊C8—51no—
78C2×C3×C22⋊C8—51no—
79C2×C3×#97—51no—
80C22×C22⋊D5C3×C411yescharacteristic
81C23×Dic5C22×C311yescharacteristic
82C22×C4×D5C22×C311yescharacteristic
83C22×#170—24no—
84C22×#170C22×C311yescharacteristic
85C22×#170C22×C311yescharacteristic
86C22×#170C22×C311yescharacteristic
87C22×#170C22×C311yescharacteristic
88C22×#170C22×C314yescomplemented
89C2×#700—24no—
90C2×#700C22×C314yescomplemented
91C2×C5×C22⋊C4C3×C411yescharacteristic
92C2×#685C3×C411yescharacteristic
93C2×#683C3×C411yescharacteristic
94C2×C4×Dic5C22×C311yescharacteristic
95#4027C3×C414yes—
96#4027C22×C314yes—
97C22⋊#170C22×C314yescomplemented
98#4003C22×C314yescomplemented
99#4002C22×C314yescomplemented
100C4⋊#170C22×C314yescomplemented
101C2×#435—51noSylow
102C23×C3×C5—22no—
103C23×C3×C5—22no—
104C23×C3×C5M1614yescomplemented
105C23×C3×C5C22×C412yes—
106C23×C3×C5C22×C411yescharacteristic
107C22×C3×D5—22no—
108C22×C3×D5—22no—
109C22×C3×D5—22no—
110C22×C3×D5C22×C412yes—
111C2×C3×C4×C5—22no—
112C2×C3×C4×C5—22no—
113C2×C3×C4×C5C22×C412yes—
114C2×C3×C4×C5C22×C412yes—
115C3×C22⋊D5—28no—
116C2×C3×Dic5—22no—
117C2×C3×Dic5—22no—
118C2×C3×Dic5—24no—
119C2×C3×Dic5—22no—
120C2×C3×Dic5—22no—
121C2×C3×Dic5C22×C412yes—
122C2×C3×Dic5C22×C412yes—
123C2×C3×Dic5C22×C411yescharacteristic
124C2×C3×Dic5C2×D412yes—
125C2×C3×Dic5#4112yes—
126C2×C3×Dic5C2411yescharacteristic
127C3×C4×D5—24no—
128C3×C4×D5—22no—
129C3×C4×D5—22no—
130C3×#170—22no—
131C3×#170—22no—
132C3×#170—22no—
133C3×#170C2×D414yescomplemented
134C22×C3×D4—51no—
135C23×C3×C4—51no—
136C23×C3×C4—51no—
137C2×C3×M16—54no—
138C2×C3×M16—104no—
139C22×C3×C8—51no—
140C22×C3×C8—51no—
141C22×C3×C8—54no—
142C22×C3×C8—51no—
143C22×C3×C8—51no—
144C22×C3×C8—104no—
145C3×C4×D4—54no—
146C3×C4×D4—54no—
147C2×C3×C4⋊C4—51no—
148C2×C3×C22⋊C4—51no—
149C2×C3×C22⋊C4AGL(1,𝔽5)11yescharacteristic
150C2×C3×C42—51no—
151C3×C4⋊C8—54no—
152C3×C22⋊C8—54no—
153C3×C22⋊C8—54no—
154C3×#97—54no—
155C24×C5C2×C3×C411yescharacteristic
156C23×D5C2×C3×C411yescharacteristic
157C22×C4×C5C2×C3×C411yescharacteristic
158C22×C4×C5C2×C3×C411yescharacteristic
159C2×C22⋊D5—24no—
160C2×C22⋊D5C2×C3×C414yes—
161C22×Dic5—22no—
162C22×Dic5—22no—
163C22×Dic5C2×C3×C412yes—
164C22×Dic5C2×C3×C411yescharacteristic
165C22×Dic5C2×C3×C411yescharacteristic
166C22×Dic5C23×C312yes—
167C22×Dic5C23×C311yescharacteristic
168C2×C4×D5—22no—
169C2×C4×D5—22no—
170C2×C4×D5C2×C3×C412yes—
171C2×C4×D5C23×C312yes—
172C2×#170—22no—
173C2×#170—22no—
174C2×#170—22no—
175C2×#170—28no—
176C2×#170C3×D414yescomplemented
177C2×#170C23×C312yes—
178C2×#170C23×C312yes—
179C2×#170C23×C312yes—
180C2×#170C23×C312yes—
181#700—28no—
182C5×C22⋊C4C2×C3×C414yes—
183#685C2×C3×C414yes—
184#683C2×C3×C414yes—
185C4×Dic5C2×C3×C412yes—
186C4×Dic5C23×C312yes—
187C22×M16—51no—
188C23×C8—51no—
189C2×C4×D4—51no—
190#435—58no—
191C2×C4⋊C8—51no—
192C2×C22⋊C8—51no—
193C2×C22⋊C8—51no—
194C2×#97—51no—
195C22×C3×C5—22no—
196C22×C3×C5—22no—
197C22×C3×C5—28no—
198C22×C3×C5C4×D412yes—
199C22×C3×C5C2×M1612yes—
200C22×C3×C5C2×M1614yescomplemented
201C22×C3×C5#13112yes—
202C22×C3×C5C23×C411yescharacteristic
203C2×C3×D5—22no—
204C2×C3×D5—22no—
205C2×C3×D5—22no—
206C2×C3×D5—44no—
207C3×C4×C5—22no—
208C3×C4×C5—22no—
209C3×Dic5—22no—
210C3×Dic5—22no—
211C3×Dic5C4×D412yes—
212C3×Dic5C22×D411yescharacteristic
213C3×Dic5C2×#4111yescharacteristic
214C24×C3—51no—
215C24×C3C2×AGL(1,𝔽5)11yescharacteristic
216C2×C3×D4—54no—
217C2×C3×D4—104no—
218C22×C3×C4—51no—
219C22×C3×C4—52no—
220C22×C3×C4—52no—
221C22×C3×C4—51no—
222C22×C3×C4—51no—
223C22×C3×C4—52no—
224C22×C3×C4—52no—
225C22×C3×C4—102no—
226C22×C3×C4—102no—
227C22×C3×C4—102no—
228C22×C3×C4—102no—
229C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
230C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
231C3×M16—108no—
232C2×C3×C8—52no—
233C2×C3×C8—54no—
234C2×C3×C8—52no—
235C2×C3×C8—52no—
236C2×C3×C8—52no—
237C2×C3×C8—108no—
238C2×C3×C8—102no—
239C2×C3×C8—102no—
240C2×C3×C8—102no—
241C3×C4⋊C4—54no—
242C3×C22⋊C4—54no—
243C3×C22⋊C4C2×AGL(1,𝔽5)14yes—
244C3×C42—52no—
245C3×C42—52no—
246C23×C5—22no—
247C23×C5—22no—
248C23×C5C3×M1614yescomplemented
249C23×C5C22×C3×C412yes—
250C23×C5C22×C3×C411yescharacteristic
251C22×D5—22no—
252C22×D5—22no—
253C22×D5—22no—
254C22×D5C22×C3×C412yes—
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5C22×C3×C412yes—
258C2×C4×C5C22×C3×C412yes—
259C22⋊D5—28no—
260C2×Dic5—22no—
261C2×Dic5—22no—
262C2×Dic5—24no—
263C2×Dic5—22no—
264C2×Dic5—22no—
265C2×Dic5C22×C3×C412yes—
266C2×Dic5C22×C3×C412yes—
267C2×Dic5C22×C3×C411yescharacteristic
268C2×Dic5C2×C3×D412yes—
269C2×Dic5C3×#4112yes—
270C2×Dic5C24×C311yescharacteristic
271C4×D5—24no—
272C4×D5—22no—
273C4×D5—22no—
274#170—22no—
275#170—22no—
276#170—22no—
277#170C2×C3×D414yescomplemented
278C22×D4—51no—
279C23×C4—51no—
280C23×C4—51no—
281C2×M16—54no—
282C2×M16—104no—
283C22×C8—51no—
284C22×C8—51no—
285C22×C8—51no—
286C22×C8—51no—
287C22×C8—54no—
288C22×C8—104no—
289C4×D4—54no—
290C4×D4—54no—
291C2×C4⋊C4—51no—
292C2×C22⋊C4—51no—
293C2×C22⋊C4C3×AGL(1,𝔽5)11yescharacteristic
294C2×C42—51no—
295C4⋊C8—54no—
296C22⋊C8—54no—
297C22⋊C8—54no—
298#97—54no—
299C2×C3×C5—24no—
300C2×C3×C5#43514yescomplemented
301C2×C3×C5C2×C4×D411yescharacteristic
302C2×C3×C5C22×M1611yescharacteristic
303C2×C3×C5C2×#13111yescharacteristic
304C3×D5—42no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—52no—
308C23×C3—102no—
309C23×C3—102no—
310C23×C3—102no—
311C23×C3#70014yescomplemented
312C23×C3C22×AGL(1,𝔽5)12yes—
313C23×C3C22×AGL(1,𝔽5)11yescharacteristic
314C3×D4—108no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—52no—
318C2×C3×C4—52no—
319C2×C3×C4—51no—
320C2×C3×C4—51no—
321C2×C3×C4—52no—
322C2×C3×C4—52no—
323C2×C3×C4—102no—
324C2×C3×C4—104no—
325C2×C3×C4—102no—
326C2×C3×C4—102no—
327C2×C3×C4—104no—
328C2×C3×C4—102no—
329C2×C3×C4—102no—
330C2×C3×C4—102no—
331C2×C3×C4C22×AGL(1,𝔽5)12yes—
332C2×C3×C4C22×AGL(1,𝔽5)12yes—
333C3×C8—54no—
334C3×C8—102no—
335C3×C8—102no—
336C3×C8—102no—
337C22×C5—22no—
338C22×C5—28no—
339C22×C5—22no—
340C22×C5C3×C4×D412yes—
341C22×C5C2×C3×M1612yes—
342C22×C5C2×C3×M1614yescomplemented
343C22×C5C3×#13112yes—
344C22×C5C23×C3×C411yescharacteristic
345C2×D5—22no—
346C2×D5—22no—
347C2×D5—22no—
348C2×D5—44no—
349C4×C5—22no—
350C4×C5—22no—
351Dic5—22no—
352Dic5—22no—
353Dic5C3×C4×D412yes—
354Dic5C22×C3×D411yescharacteristic
355Dic5C2×C3×#4111yescharacteristic
356C24—51no—
357C24C2×C3×AGL(1,𝔽5)11yescharacteristic
358C2×D4—54no—
359C2×D4—104no—
360C22×C4—51no—
361C22×C4—52no—
362C22×C4—51no—
363C22×C4—51no—
364C22×C4—52no—
365C22×C4—52no—
366C22×C4—52no—
367C22×C4—102no—
368C22×C4—102no—
369C22×C4—102no—
370C22×C4—102no—
371C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
372C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
373M16—108no—
374C2×C8—52no—
375C2×C8—54no—
376C2×C8—52no—
377C2×C8—52no—
378C2×C8—52no—
379C2×C8—108no—
380C2×C8—102no—
381C2×C8—102no—
382C2×C8—102no—
383C4⋊C4—54no—
384C22⋊C4—54no—
385C22⋊C4C2×C3×AGL(1,𝔽5)14yes—
386C42—52no—
387C42—52no—
388C3×C5C2×#43511yescharacteristic, complemented, Hall
389C22×C3—28no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—102no—
393C22×C3—102no—
394C22×C3—102no—
395C22×C3—204no—
396C22×C3C2×#70012yes—
397C22×C3C2×#70014yescomplemented
398C22×C3#413112yes—
399C22×C3D4×AGL(1,𝔽5)12yes—
400C22×C3C23×AGL(1,𝔽5)11yescharacteristic
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—52no—
404C3×C4—51no—
405C3×C4—51no—
406C3×C4—102no—
407C3×C4—102no—
408C2×C5—24no—
409C2×C5C3×#43514yescomplemented
410C2×C5C2×C3×C4×D411yescharacteristic
411C2×C5C22×C3×M1611yescharacteristic
412C2×C5C2×C3×#13111yescharacteristic
413D5—42no—
414C23—22no—
415C23—22no—
416C23—52no—
417C23—102no—
418C23—102no—
419C23—102no—
420C23C3×#70014yescomplemented
421C23C22×C3×AGL(1,𝔽5)12yes—
422C23C22×C3×AGL(1,𝔽5)11yescharacteristic
423D4—108no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—52no—
427C2×C4—52no—
428C2×C4—51no—
429C2×C4—51no—
430C2×C4—52no—
431C2×C4—52no—
432C2×C4—102no—
433C2×C4—102no—
434C2×C4—104no—
435C2×C4—102no—
436C2×C4—102no—
437C2×C4—104no—
438C2×C4—102no—
439C2×C4—102no—
440C2×C4C22×C3×AGL(1,𝔽5)12yes—
441C2×C4C22×C3×AGL(1,𝔽5)12yes—
442C8—54no—
443C8—102no—
444C8—102no—
445C8—102no—
446C2×C3—24no—
447C2×C3—202no—
448C2×C3#6569314yesdirect product factor
449C2×C3C22×#70011yescharacteristic
450C2×C3C2×#413111yescharacteristic
451C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
452C5C2×C3×#43511yescharacteristic, complemented, Sylow
453C22—28no—
454C22—22no—
455C22—22no—
456C22—102no—
457C22—102no—
458C22—102no—
459C22—204no—
460C22C2×C3×#70012yes—
461C22C2×C3×#70014yescomplemented
462C22C3×#413112yes—
463C22C3×D4×AGL(1,𝔽5)12yes—
464C22C23×C3×AGL(1,𝔽5)11yescharacteristic
465C4—22no—
466C4—22no—
467C4—51no—
468C4—51no—
469C4—52no—
470C4—102no—
471C4—102no—
472C3C2×#6569311yescharacteristic, direct product factor, Sylow
473C2—24no—
474C2—202no—
475C2C3×#6569314yesdirect product factor
476C2C22×C3×#70011yescharacteristic
477C2C2×C3×#413111yescharacteristic
478C2C2×C3×D4×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC23×C3C22×AGL(1,𝔽5)
commutatorC22×C5C23×C3×C4
FrattiniC22C23×C3×AGL(1,𝔽5)
FittingC2×C3×C5×C22⋊C4C4
socleC23×C3×C5C22×C4
prime radicalC22×C3×C22⋊D5C4
cosocleC2×Dic5C24×C3
hypercenterC2×C3×C22⋊C4AGL(1,𝔽5)
hypocenterC5C2×C3×#435
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C3×#170C2semidirect
2. C22×C3×#700C2semidirect
4. C2×C3×C22⋊#170C2semidirect
5. C2×C3×#4003C2semidirect
6. C2×C3×#4002C2semidirect
7. C2×C3×C4⋊#170C2semidirect
8. C3×#65693C2direct (N × H)
9. C2×#65693C3direct (N × H)
18. C22×C3×#170C22semidirect
20. C2×C3×#700C22semidirect
27. C3×C22⋊#170C22semidirect
28. C3×#4003C22semidirect
29. C3×#4002C22semidirect
30. C3×C4⋊#170C22semidirect
32. C23×#170C2×C3semidirect
33. C22×#700C2×C3semidirect
35. C2×C22⋊#170C2×C3semidirect
36. C2×#4003C2×C3semidirect
37. C2×#4002C2×C3semidirect
38. C2×C4⋊#170C2×C3semidirect
39. #65693C2×C3direct (N × H)
61. C2×C3×#170D4semidirect
88. C22×#170C22×C3semidirect
90. C2×#700C22×C3semidirect
97. C22⋊#170C22×C3semidirect
98. #4003C22×C3semidirect
99. #4002C22×C3semidirect
100. C4⋊#170C22×C3semidirect
104. C23×C3×C5M16semidirect
133. C3×#170C2×D4semidirect
176. C2×#170C3×D4semidirect
200. C22×C3×C5C2×M16semidirect
248. C23×C5C3×M16semidirect
277. #170C2×C3×D4semidirect
300. C2×C3×C5#435semidirect
311. C23×C3#700semidirect
342. C22×C5C2×C3×M16semidirect
388. C3×C5C2×#435semidirect
397. C22×C3C2×#700semidirect
409. C2×C5C3×#435semidirect
420. C23C3×#700semidirect
448. C2×C3#65693direct (N × H)
452. C5C2×C3×#435semidirect
461. C22C2×C3×#700semidirect
472. C3C2×#65693direct (N × H)
475. C2C3×#65693direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×#1702
C22×C3×#7002
C2×C3×#40272
C2×C3×C22⋊#1702
C2×C3×#40032
C2×C3×#40022
C2×C3×C4⋊#1702
C3×#656932
C2×#656933
C2×C3×#4355