Finite Group Databasebeta

← C2×C3×#65179 #14429812

Subgroups — C2×C3×#65179 #14429812

Order 1920.

960 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4039C211yesmaximal, characteristic
2C2×C3×#4034C211yesmaximal, characteristic, complemented
3C2×C3×C5×#107C211yesmaximal, characteristic, complemented
4C2×C3×#3953C211yesmaximal, characteristic
5C2×C3×#3946C211yesmaximal, characteristic, complemented
6C2×C3×#3941C211yesmaximal, characteristic
7C2×C3×#3939C211yesmaximal, characteristic, complemented
8C3×#65179C218yesmaximal, direct product factor
9C2×#65179C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×D20C2211yescharacteristic
11C22×C3×C4×D5—21no—
12C22×C3×Dic10—21no—
13C22×C3×C5×C8C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×#685—21no—
17C2×C3×C4⋊Dic5C2211yescharacteristic
18C2×C3×#683—21no—
19C2×C3×C4×Dic5C2211yescharacteristic
20C22×C3×#168C2211yescharacteristic
21C3×#4039—22no—
22C3×#4039C2214yes—
23C3×#4034—22no—
24C3×#4034C2214yescomplemented
25C3×C5×#107C2214yescomplemented
26C3×#3953C2214yes—
27C3×#3946C2214yescomplemented
28C3×#3941C2214yes—
29C3×#3939C2214yescomplemented
30C2×C3×#474—51nomaximal, Hall
31C2×#4039C2×C311yescharacteristic
32C2×#4034C2×C311yescharacteristic, complemented
33C2×C5×#107C2×C311yescharacteristic, complemented
34C2×#3953C2×C311yescharacteristic
35C2×#3946C2×C311yescharacteristic, complemented
36C2×#3941C2×C311yescharacteristic
37C2×#3939C2×C311yescharacteristic, complemented
38#65179C2×C318yesdirect product factor
39C23×C3×D5—21no—
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5—21no—
42C22×C3×C4×C5C2311yescharacteristic
43C22×C3×Dic5—21no—
44C22×C3×Dic5D411yescharacteristic
45C22×C3×Dic5D411yescharacteristic
46C2×C3×D20—21no—
47C2×C3×D20—22no—
48C2×C3×D20Q814yescomplemented
49C2×C3×D20C2312yes—
50C2×C3×C4×D5—24no—
51C2×C3×C4×D5—44no—
52C2×C3×Dic10—24no—
53C2×C3×Dic10—44no—
54C2×C3×C5×C8—22no—
55C2×C3×C5×C8C2312yes—
56C3×C5×C4⋊C4—21no—
57C3×C5×C4⋊C4—21no—
58C3×C5×C4⋊C4C2312yes—
59C3×C5×C4⋊C4C2312yes—
60C3×#685—24no—
61C3×C4⋊Dic5—21no—
62C3×C4⋊Dic5C2312yes—
63C3×#683—24no—
64C3×C4×Dic5D412yes—
65C3×C4×Dic5C2312yes—
66C2×C3×#168—22no—
67C2×C3×#168C2312yes—
68C2×C3×C4⋊Q8—51no—
69C2×C3×C4×D4—51no—
70C3×#474—58no—
71C2×C3×#107—51no—
72C2×C3×#107D511yescharacteristic, complemented
73C2×C3×C4⋊C8—51no—
74C2×C3×#102—51no—
75C2×C3×#102—51no—
76C22×D20C22×C311yescharacteristic
77C22×C4×D5—21no—
78C22×Dic10—21no—
79C22×C5×C8C22×C311yescharacteristic
80C2×C5×C4⋊C4C22×C311yescharacteristic
81C2×C5×C4⋊C4C22×C311yescharacteristic
82C2×#685—21no—
83C2×C4⋊Dic5C22×C311yescharacteristic
84C2×#683—21no—
85C2×C4×Dic5C22×C311yescharacteristic
86C22×#168C22×C311yescharacteristic
87#4039—22no—
88#4039C22×C314yes—
89#4034—22no—
90#4034C22×C314yescomplemented
91C5×#107C22×C314yescomplemented
92#3953C22×C314yes—
93#3946C22×C314yescomplemented
94#3941C22×C314yes—
95#3939C22×C314yescomplemented
96C2×#474—51noSylow
97C23×C3×C5C2×D411yescharacteristic
98C22×C3×D5—24no—
99C22×C3×D5—22no—
100C22×C3×D5—42no—
101C22×C3×D5—42no—
102C2×C3×C4×C5—22no—
103C2×C3×C4×C5—22no—
104C2×C3×C4×C5—42no—
105C2×C3×C4×C5—42no—
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×Q812yes—
108C2×C3×C4×C5#4112yes—
109C2×C3×C4×C5C2411yescharacteristic
110C2×C3×Dic5—22no—
111C2×C3×Dic5—22no—
112C2×C3×Dic5—42no—
113C2×C3×Dic5D814yes—
114C2×C3×Dic5C2×D412yes—
115C2×C3×Dic5C2×D412yes—
116C3×D20—24no—
117C3×D20—22no—
118C3×D20C2×Q814yescomplemented
119C3×C4×D5—48no—
120C3×Dic10—48no—
121C3×C5×C8—22no—
122C3×#168—22no—
123C22×C3×Q8—101no—
124C22×C3×D4—51no—
125C23×C3×C4—101no—
126C22×C3×C8—51no—
127C22×C3×C8C2×D511yescharacteristic
128C3×C4⋊Q8—54no—
129C3×C4⋊Q8—102no—
130C3×C4×D4—54no—
131C3×C4×D4—102no—
132C2×C3×C4⋊C4—51no—
133C2×C3×C4⋊C4—101no—
134C2×C3×C4⋊C4C2×D511yescharacteristic
135C2×C3×C4⋊C4C2×D511yescharacteristic
136C2×C3×C22⋊C4—101no—
137C2×C3×C42—51no—
138C3×#107—54no—
139C3×#107C2×D514yescomplemented
140C3×C4⋊C8—54no—
141C3×#102—54no—
142C3×#102—54no—
143C23×D5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5C23×C311yescharacteristic
147C22×Dic5—21no—
148C22×Dic5C3×D411yescharacteristic
149C22×Dic5C3×D411yescharacteristic
150C2×D20—21no—
151C2×D20—22no—
152C2×D20C3×Q814yescomplemented
153C2×D20C23×C312yes—
154C2×C4×D5—24no—
155C2×C4×D5—44no—
156C2×Dic10—24no—
157C2×Dic10—44no—
158C2×C5×C8—22no—
159C2×C5×C8C23×C312yes—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—21no—
162C5×C4⋊C4C23×C312yes—
163C5×C4⋊C4C23×C312yes—
164#685—24no—
165C4⋊Dic5—21no—
166C4⋊Dic5C23×C312yes—
167#683—24no—
168C4×Dic5C3×D412yes—
169C4×Dic5C23×C312yes—
170C2×#168—22no—
171C2×#168C23×C312yes—
172C2×C4⋊Q8—51no—
173C2×C4×D4—51no—
174#474—58no—
175C2×#107—51no—
176C2×#107C3×D511yescharacteristic, complemented
177C2×C4⋊C8—51no—
178C2×#102—51no—
179C2×#102—51no—
180C22×C3×C5C22⋊Q812yes—
181C22×C3×C5C2×D812yes—
182C22×C3×C5#13712yes—
183C22×C3×C5C22×D411yescharacteristic
184C2×C3×D5—24no—
185C2×C3×D5—42no—
186C2×C3×D5—42no—
187C2×C3×D5—48no—
188C3×C4×C5—42no—
189C3×C4×C5—42no—
190C3×C4×C5C22⋊Q812yes—
191C3×C4×C5C22×Q811yescharacteristic
192C3×C4×C5C2×#4111yescharacteristic
193C3×Dic5—22no—
194C3×Dic5—42no—
195C3×Dic5C2×D814yes—
196C24×C3—101no—
197C2×C3×Q8—104no—
198C2×C3×Q8—204no—
199C2×C3×D4—54no—
200C2×C3×D4—52no—
201C2×C3×D4—102no—
202C2×C3×D4—101no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—51no—
206C22×C3×C4—51no—
207C22×C3×C4—104no—
208C22×C3×C4—101no—
209C22×C3×C4—204no—
210C22×C3×C4C22×D511yescharacteristic
211C2×C3×C8—22no—
212C2×C3×C8—52no—
213C2×C3×C8—102no—
214C2×C3×C8C22×D512yes—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—52no—
218C3×C4⋊C4—104no—
219C3×C4⋊C4—101no—
220C3×C4⋊C4C22×D512yes—
221C3×C4⋊C4C22×D512yes—
222C3×C22⋊C4—104no—
223C3×C42—52no—
224C3×C42—52no—
225C23×C5C2×C3×D411yescharacteristic
226C22×D5—24no—
227C22×D5—22no—
228C22×D5—42no—
229C22×D5—42no—
230C2×C4×C5—22no—
231C2×C4×C5—22no—
232C2×C4×C5—42no—
233C2×C4×C5—42no—
234C2×C4×C5C2×C3×D411yescharacteristic
235C2×C4×C5C2×C3×Q812yes—
236C2×C4×C5C3×#4112yes—
237C2×C4×C5C24×C311yescharacteristic
238C2×Dic5—22no—
239C2×Dic5—22no—
240C2×Dic5—42no—
241C2×Dic5C3×D814yes—
242C2×Dic5C2×C3×D412yes—
243C2×Dic5C2×C3×D412yes—
244D20—24no—
245D20—22no—
246D20C2×C3×Q814yescomplemented
247C4×D5—48no—
248Dic10—48no—
249C5×C8—22no—
250#168—22no—
251C22×Q8—101no—
252C22×D4—51no—
253C23×C4—101no—
254C22×C8—51no—
255C22×C8C2×C3×D511yescharacteristic
256C4⋊Q8—54no—
257C4⋊Q8—102no—
258C4×D4—54no—
259C4×D4—102no—
260C2×C4⋊C4—51no—
261C2×C4⋊C4—101no—
262C2×C4⋊C4C2×C3×D511yescharacteristic
263C2×C4⋊C4C2×C3×D511yescharacteristic
264C2×C22⋊C4—101no—
265C2×C42—51no—
266#107—54no—
267#107C2×C3×D514yescomplemented
268C4⋊C8—54no—
269#102—54no—
270#102—54no—
271C2×C3×C5#47414yescomplemented
272C2×C3×C5C2×C22⋊Q811yescharacteristic
273C2×C3×C5C22×D811yescharacteristic
274C2×C3×C5C2×#13711yescharacteristic
275C3×D5—44no—
276C23×C3—102no—
277C23×C3—104no—
278C23×C3—202no—
279C23×C3—202no—
280C23×C3D4×D511yescharacteristic
281C3×Q8—208no—
282C3×D4—54no—
283C3×D4—102no—
284C3×D4—104no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—52no—
290C2×C3×C4—54no—
291C2×C3×C4—52no—
292C2×C3×C4—102no—
293C2×C3×C4—102no—
294C2×C3×C4—208no—
295C2×C3×C4—202no—
296C2×C3×C4#70912yes—
297C2×C3×C4D4×D511yescharacteristic
298C2×C3×C4Q8×D512yes—
299C2×C3×C4C23×D511yescharacteristic
300C3×C8—22no—
301C3×C8—102no—
302C22×C5C3×C22⋊Q812yes—
303C22×C5C2×C3×D812yes—
304C22×C5C3×#13712yes—
305C22×C5C22×C3×D411yescharacteristic
306C2×D5—24no—
307C2×D5—42no—
308C2×D5—42no—
309C2×D5—48no—
310C4×C5—42no—
311C4×C5—42no—
312C4×C5C3×C22⋊Q812yes—
313C4×C5C22×C3×Q811yescharacteristic
314C4×C5C2×C3×#4111yescharacteristic
315Dic5—22no—
316Dic5—42no—
317Dic5C2×C3×D814yes—
318C24—101no—
319C2×Q8—104no—
320C2×Q8—204no—
321C2×D4—54no—
322C2×D4—52no—
323C2×D4—102no—
324C2×D4—101no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—51no—
328C22×C4—51no—
329C22×C4—104no—
330C22×C4—101no—
331C22×C4—204no—
332C22×C4C22×C3×D511yescharacteristic
333C2×C8—22no—
334C2×C8—52no—
335C2×C8—102no—
336C2×C8C22×C3×D512yes—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—52no—
340C4⋊C4—104no—
341C4⋊C4—101no—
342C4⋊C4C22×C3×D512yes—
343C4⋊C4C22×C3×D512yes—
344C22⋊C4—104no—
345C42—52no—
346C42—52no—
347C3×C5C2×#47411yescharacteristic, complemented, Hall
348C22×C3—104no—
349C22×C3—202no—
350C22×C3—202no—
351C22×C3—208no—
352C22×C3#404212yes—
353C22×C3D5×D812yes—
354C22×C3#406412yes—
355C22×C3C2×D4×D511yescharacteristic
356C3×C4—42no—
357C3×C4—42no—
358C3×C4—54no—
359C3×C4—102no—
360C3×C4—202no—
361C3×C4#404212yes—
362C3×C4C2×#70911yescharacteristic
363C3×C4C2×Q8×D511yescharacteristic
364C2×C5C3×#47414yescomplemented
365C2×C5C2×C3×C22⋊Q811yescharacteristic
366C2×C5C22×C3×D811yescharacteristic
367C2×C5C2×C3×#13711yescharacteristic
368D5—44no—
369C23—102no—
370C23—104no—
371C23—202no—
372C23—202no—
373C23C3×D4×D511yescharacteristic
374Q8—208no—
375D4—54no—
376D4—102no—
377D4—104no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—52no—
383C2×C4—54no—
384C2×C4—52no—
385C2×C4—102no—
386C2×C4—102no—
387C2×C4—208no—
388C2×C4—202no—
389C2×C4C3×#70912yes—
390C2×C4C3×D4×D511yescharacteristic
391C2×C4C3×Q8×D512yes—
392C2×C4C23×C3×D511yescharacteristic
393C8—22no—
394C8—102no—
395C2×C3—204no—
396C2×C3#6517914yesdirect product factor
397C2×C3C2×#404211yescharacteristic
398C2×C3C2×D5×D811yescharacteristic
399C2×C3C2×#406411yescharacteristic
400C5C2×C3×#47411yescharacteristic, complemented, Sylow
401C22—104no—
402C22—208no—
403C22—202no—
404C22—202no—
405C22C3×#404212yes—
406C22C3×D5×D812yes—
407C22C3×#406412yes—
408C22C2×C3×D4×D511yescharacteristic
409C4—42no—
410C4—42no—
411C4—54no—
412C4—102no—
413C4—202no—
414C4C3×#404212yes—
415C4C2×C3×#70911yescharacteristic
416C4C2×C3×Q8×D511yescharacteristic
417C3C2×#6517911yescharacteristic, direct product factor, Sylow
418C2—204no—
419C2C3×#6517914yesdirect product factor
420C2C2×C3×#404211yescharacteristic
421C2C2×C3×D5×D811yescharacteristic
422C2C2×C3×#406411yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#107C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×D20C22
hypercenterC2×C3×#107D5
hypocenterC5C2×C3×#474
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#107

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4034C2semidirect
3. C2×C3×C5×#107C2semidirect
5. C2×C3×#3946C2semidirect
7. C2×C3×#3939C2semidirect
8. C3×#65179C2direct (N × H)
9. C2×#65179C3direct (N × H)
24. C3×#4034C22semidirect
25. C3×C5×#107C22semidirect
27. C3×#3946C22semidirect
29. C3×#3939C22semidirect
32. C2×#4034C2×C3semidirect
33. C2×C5×#107C2×C3semidirect
35. C2×#3946C2×C3semidirect
37. C2×#3939C2×C3semidirect
38. #65179C2×C3direct (N × H)
48. C2×C3×D20Q8semidirect
72. C2×C3×#107D5semidirect
90. #4034C22×C3semidirect
91. C5×#107C22×C3semidirect
93. #3946C22×C3semidirect
95. #3939C22×C3semidirect
118. C3×D20C2×Q8semidirect
139. C3×#107C2×D5semidirect
152. C2×D20C3×Q8semidirect
176. C2×#107C3×D5semidirect
246. D20C2×C3×Q8semidirect
267. #107C2×C3×D5semidirect
271. C2×C3×C5#474semidirect
347. C3×C5C2×#474semidirect
364. C2×C5C3×#474semidirect
396. C2×C3#65179direct (N × H)
400. C5C2×C3×#474semidirect
417. C3C2×#65179direct (N × H)
419. C2C3×#65179direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40392
C2×C3×#40342
C2×C3×C5×#1072
C2×C3×#39532
C2×C3×#39462
C2×C3×#39412
C2×C3×#39392
C3×#651792
C2×#651793
C2×C3×#4745