Finite Group Databasebeta

← C2×A4×#676 #14514131

Subgroups — C2×A4×#676 #14514131

Order 1920.

800 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D5×A4C211yesmaximal, characteristic
2C2×C5×C8×A4C211yesmaximal, characteristic, complemented
3C2×A4×#168C211yesmaximal, characteristic, complemented
4A4×#676C214yesmaximal, direct product factor
5C23×#676C311yesmaximal, characteristic, complemented, Hall
6C22×D5×A4C411yescharacteristic
7C2×C4×C5×A4C2211yescharacteristic
8C2×A4×Dic5C411yescharacteristic
9C4×D5×A4C412yes—
10C4×D5×A4C2212yes—
11C2×C3×#676—41nomaximal
12C5×C8×A4C2212yescomplemented
13A4×#168C2212yescomplemented
14C2×A4×M16—51nomaximal, Hall
15C23×C4×D5C2×C311yescharacteristic
16C23×C5×C8C2×C311yescharacteristic, complemented
17C23×#168C2×C311yescharacteristic, complemented
18C22×#676—31no—
19C22×#676—31no—
20C22×#676—31no—
21C22×#676—31no—
22C22×#676—34no—
23C22×#676C2×C314yescomplemented
24C22×C5×A4C2×C411yescharacteristic
25C2×D5×A4—22no—
26C2×D5×A4C2×C412yes—
27C2×C3×C4×D5—41no—
28C4×C5×A4C2×C411yescharacteristic
29C4×C5×A4C2311yescharacteristic
30A4×Dic5C2×C412yes—
31C2×C3×C5×C8—41no—
32C2×C3×#168—41no—
33C3×#676—44no—
34C22×C4×A4—51no—
35A4×M16—54no—
36C2×C8×A4—51no—
37C2×C8×A4D511yescharacteristic, complemented
38C24×D5C3×C411yescharacteristic
39C23×C4×C5C22×C311yescharacteristic
40C23×Dic5C3×C411yescharacteristic
41C22×C4×D5—31no—
42C22×C4×D5—31no—
43C22×C4×D5—32no—
44C22×C4×D5—31no—
45C22×C4×D5—31no—
46C22×C4×D5—32no—
47C22×C4×D5C3×C412yes—
48C22×C4×D5C22×C312yes—
49C22×C5×C8—31no—
50C22×C5×C8—31no—
51C22×C5×C8—32no—
52C22×C5×C8C22×C312yescomplemented
53C22×#168—31no—
54C22×#168—31no—
55C22×#168—32no—
56C22×#168C22×C312yescomplemented
57C2×#676—32no—
58C2×#676—34no—
59C2×#676—34no—
60C2×#676—34no—
61C2×#676—31no—
62C2×#676—31no—
63C2×#676—31no—
64C2×#676—34no—
65C2×#676—34no—
66C2×#676—34no—
67C2×#676—34no—
68C2×#676—34no—
69C2×#676A411yescharacteristic, direct product factor
70C23×M16—51noSylow
71C22×C3×D5—41no—
72C2×C5×A4M1612yescomplemented
73C2×C5×A4C22×C411yescharacteristic
74D5×A4—22no—
75C2×C3×C4×C5—41no—
76C2×C3×Dic5—41no—
77C3×C4×D5—42no—
78C3×C4×D5—42no—
79C3×C5×C8—42no—
80C3×#168—42no—
81C23×A4—51no—
82C2×C4×A4—51no—
83C2×C4×A4—52no—
84C2×C4×A4—52no—
85C2×C4×A4C2×D511yescharacteristic
86C2×C3×M16—201no—
87C8×A4—52no—
88C8×A4C2×D512yescomplemented
89C24×C5C2×C3×C411yescharacteristic
90C23×D5—22no—
91C23×D5—31no—
92C23×D5—31no—
93C23×D5—32no—
94C23×D5—62no—
95C23×D5C2×C3×C412yes—
96C22×C4×C5—31no—
97C22×C4×C5—31no—
98C22×C4×C5—31no—
99C22×C4×C5—31no—
100C22×C4×C5C2×C3×C411yescharacteristic
101C22×C4×C5C23×C311yescharacteristic
102C22×Dic5—31no—
103C22×Dic5—31no—
104C22×Dic5—32no—
105C22×Dic5C2×C3×C412yes—
106C2×C4×D5—32no—
107C2×C4×D5—34no—
108C2×C4×D5—32no—
109C2×C4×D5—34no—
110C2×C4×D5—32no—
111C2×C4×D5—31no—
112C2×C4×D5—31no—
113C2×C4×D5—31no—
114C2×C4×D5—32no—
115C2×C4×D5—32no—
116C2×C4×D5—32no—
117C2×C4×D5—32no—
118C2×C4×D5—32no—
119C2×C4×D5—32no—
120C2×C4×D5—32no—
121C2×C4×D5—32no—
122C2×C4×D5—32no—
123C2×C4×D5—32no—
124C2×C4×D5C2×A411yescharacteristic
125C2×C5×C8—32no—
126C2×C5×C8—31no—
127C2×C5×C8—32no—
128C2×C5×C8—32no—
129C2×C5×C8—32no—
130C2×C5×C8C2×A411yescharacteristic, complemented
131C2×#168—32no—
132C2×#168—31no—
133C2×#168—32no—
134C2×#168—32no—
135C2×#168—32no—
136C2×#168C2×A411yescharacteristic, complemented
137#676—38no—
138#676—34no—
139#676—34no—
140#676—34no—
141#676C2×A414yesdirect product factor
142C24×C4—51no—
143C22×M16—54no—
144C22×M16—151no—
145C22×M16—151no—
146C22×M16—151no—
147C22×M16—154no—
148C22×M16—151no—
149C23×C8—51no—
150C23×C8C3×D511yescharacteristic, complemented
151C22×C3×C5—41no—
152C2×C3×D5—42no—
153C2×C3×D5—82no—
154C5×A4C2×M1611yescharacteristic, complemented
155C3×C4×C5—41no—
156C3×C4×C5—41no—
157C3×Dic5—42no—
158C22×A4—52no—
159C22×A4—102no—
160C22×A4C4×D511yescharacteristic
161C22×C3×C4—201no—
162C4×A4—52no—
163C4×A4C4×D511yescharacteristic
164C4×A4C22×D511yescharacteristic
165C3×M16—204no—
166C2×C3×C8—41no—
167C2×C3×C8—201no—
168C23×C5—31no—
169C23×C5—32no—
170C23×C5—31no—
171C23×C5C3×M1612yescomplemented
172C23×C5C22×C3×C411yescharacteristic
173C22×D5—22no—
174C22×D5—32no—
175C22×D5—31no—
176C22×D5—32no—
177C22×D5—32no—
178C22×D5—32no—
179C22×D5—62no—
180C22×D5—62no—
181C22×D5—64no—
182C22×D5—64no—
183C22×D5—62no—
184C22×D5—62no—
185C22×D5—62no—
186C22×D5C4×A411yescharacteristic
187C2×C4×C5—32no—
188C2×C4×C5—31no—
189C2×C4×C5—31no—
190C2×C4×C5—31no—
191C2×C4×C5—31no—
192C2×C4×C5—31no—
193C2×C4×C5—31no—
194C2×C4×C5—31no—
195C2×C4×C5C22×A411yescharacteristic
196C2×Dic5—32no—
197C2×Dic5—31no—
198C2×Dic5—32no—
199C2×Dic5—32no—
200C2×Dic5—32no—
201C2×Dic5C4×A411yescharacteristic
202C4×D5—34no—
203C4×D5—34no—
204C4×D5—32no—
205C4×D5—32no—
206C4×D5—32no—
207C4×D5—32no—
208C4×D5—32no—
209C4×D5—32no—
210C4×D5C4×A412yes—
211C4×D5C22×A412yes—
212C5×C8—32no—
213C5×C8C22×A412yescomplemented
214#168—32no—
215#168C22×A412yescomplemented
216C25—51no—
217C23×C4—51no—
218C23×C4—52no—
219C23×C4—52no—
220C23×C4—151no—
221C23×C4—151no—
222C23×C4—152no—
223C23×C4—152no—
224C23×C4—151no—
225C23×C4—151no—
226C23×C4C2×C3×D511yescharacteristic
227C2×M16—51no—
228C2×M16—151no—
229C2×M16—154no—
230C2×M16—154no—
231C2×M16—151no—
232C2×M16—154no—
233C2×M16—154no—
234C2×M16—152no—
235C2×M16—154no—
236C2×M16—154no—
237C2×M16—154no—
238C2×M16—154no—
239C2×M16—151no—
240C22×C8—31no—
241C22×C8—31no—
242C22×C8—32no—
243C22×C8—52no—
244C22×C8—151no—
245C22×C8—152no—
246C22×C8—151no—
247C22×C8C2×C3×D512yescomplemented
248C2×C3×C5—41no—
249C2×C3×C5—42no—
250C3×D5—82no—
251C23×C3—201no—
252C2×A4—102no—
253C2×A4#67612yesdirect product factor
254C2×A4C2×C4×D511yescharacteristic
255C2×C3×C4—41no—
256C2×C3×C4—202no—
257C2×C3×C4—202no—
258C2×C3×C4—201no—
259C3×C8—42no—
260C3×C8—202no—
261C22×C5—32no—
262C22×C5—31no—
263C22×C5—32no—
264C22×C5—31no—
265C22×C5—31no—
266C22×C5—32no—
267C22×C5—32no—
268C22×C5C2×C3×M1611yescharacteristic, complemented
269C22×C5C2×C4×A411yescharacteristic
270C2×D5—22no—
271C2×D5—32no—
272C2×D5—62no—
273C2×D5—64no—
274C2×D5—62no—
275C2×D5—64no—
276C2×D5—62no—
277C2×D5—62no—
278C2×D5—62no—
279C2×D5C2×C4×A412yes—
280C4×C5—31no—
281C4×C5—31no—
282C4×C5C2×C4×A411yescharacteristic
283C4×C5C23×A411yescharacteristic
284Dic5—32no—
285Dic5C2×C4×A412yes—
286C24—52no—
287C24—102no—
288C24—151no—
289C24—152no—
290C24—151no—
291C24—302no—
292C24C3×C4×D511yescharacteristic
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—51no—
298C22×C4—52no—
299C22×C4—152no—
300C22×C4—152no—
301C22×C4—152no—
302C22×C4—151no—
303C22×C4—152no—
304C22×C4—151no—
305C22×C4—152no—
306C22×C4—154no—
307C22×C4—154no—
308C22×C4—152no—
309C22×C4—152no—
310C22×C4—152no—
311C22×C4—152no—
312C22×C4—152no—
313C22×C4—152no—
314C22×C4—152no—
315C22×C4—152no—
316C22×C4—151no—
317C22×C4—151no—
318C22×C4—152no—
319C22×C4—151no—
320C22×C4C3×C4×D511yescharacteristic
321C22×C4C22×C3×D511yescharacteristic
322M16—54no—
323M16—154no—
324M16—154no—
325M16—158no—
326M16—154no—
327C2×C8—32no—
328C2×C8—31no—
329C2×C8—32no—
330C2×C8—32no—
331C2×C8—32no—
332C2×C8—51no—
333C2×C8—152no—
334C2×C8—152no—
335C2×C8—152no—
336C2×C8—151no—
337C2×C8—152no—
338C2×C8D5×A411yescharacteristic, complemented
339C3×C5—41noHall
340C22×C3—41no—
341C22×C3—202no—
342C22×C3—402no—
343A4C2×#67611yescharacteristic, direct product factor
344C3×C4—41no—
345C3×C4—41no—
346C3×C4—202no—
347C2×C5—32no—
348C2×C5—31no—
349C2×C5—31no—
350C2×C5A4×M1612yescomplemented
351C2×C5C22×C4×A411yescharacteristic
352D5—22no—
353D5—62no—
354C23—31no—
355C23—32no—
356C23—31no—
357C23—51no—
358C23—102no—
359C23—152no—
360C23—152no—
361C23—152no—
362C23—152no—
363C23—151no—
364C23—302no—
365C23—302no—
366C23—302no—
367C23—304no—
368C23—302no—
369C23—302no—
370C23—304no—
371C23C3×#67612yescomplemented
372C23C2×C3×C4×D511yescharacteristic
373C2×C4—32no—
374C2×C4—31no—
375C2×C4—31no—
376C2×C4—31no—
377C2×C4—31no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—51no—
382C2×C4—52no—
383C2×C4—52no—
384C2×C4—152no—
385C2×C4—152no—
386C2×C4—152no—
387C2×C4—152no—
388C2×C4—154no—
389C2×C4—154no—
390C2×C4—152no—
391C2×C4—152no—
392C2×C4—152no—
393C2×C4—151no—
394C2×C4—152no—
395C2×C4—152no—
396C2×C4—152no—
397C2×C4C2×D5×A411yescharacteristic
398C8—32no—
399C8—52no—
400C8—152no—
401C8C2×D5×A412yescomplemented
402C2×C3—41no—
403C2×C3—42no—
404C2×C3—402no—
405C5C2×A4×M1611yescharacteristic, complemented, Sylow
406C22—32no—
407C22—31no—
408C22—32no—
409C22—31no—
410C22—31no—
411C22—32no—
412C22—32no—
413C22—52no—
414C22—102no—
415C22—152no—
416C22—302no—
417C22—302no—
418C22—304no—
419C22—302no—
420C22—302no—
421C22—304no—
422C22—302no—
423C22C2×C3×#67611yescharacteristic, complemented
424C22C4×D5×A411yescharacteristic
425C4—31no—
426C4—31no—
427C4—52no—
428C4—152no—
429C4C4×D5×A411yescharacteristic
430C4C22×D5×A411yescharacteristic
431C3—41noSylow
432C2—31no—
433C2—32no—
434C2—31no—
435C2—102no—
436C2—302no—
437C2A4×#67612yesdirect product factor
438C2C2×C4×D5×A411yescharacteristic
subgroupG/H
centerC2×C4C2×D5×A4
commutatorC23×C5C22×C3×C4
FrattiniC4C22×D5×A4
FittingC23×C5×C8C2×C3
socleC24×C5C2×C3×C4
prime radicalC22×D5×A4C4
cosocleC22×C4×C5C23×C3
hypercenterC2×C8D5×A4
hypocenterC22×C5C2×C3×M16
LCSG ▷ C23×C5 ▷ C22×C5
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C5×C8×A4C2semidirect
3. C2×A4×#168C2semidirect
4. A4×#676C2direct (N × H)
5. C23×#676C3semidirect
12. C5×C8×A4C22semidirect
13. A4×#168C22semidirect
16. C23×C5×C8C2×C3semidirect
17. C23×#168C2×C3semidirect
23. C22×#676C2×C3semidirect
37. C2×C8×A4D5semidirect
52. C22×C5×C8C22×C3semidirect
56. C22×#168C22×C3semidirect
69. C2×#676A4direct (N × H)
72. C2×C5×A4M16semidirect
88. C8×A4C2×D5semidirect
130. C2×C5×C8C2×A4semidirect
136. C2×#168C2×A4semidirect
141. #676C2×A4direct (N × H)
150. C23×C8C3×D5semidirect
154. C5×A4C2×M16semidirect
171. C23×C5C3×M16semidirect
213. C5×C8C22×A4semidirect
215. #168C22×A4semidirect
247. C22×C8C2×C3×D5semidirect
253. C2×A4#676direct (N × H)
268. C22×C5C2×C3×M16semidirect
338. C2×C8D5×A4semidirect
343. A4C2×#676direct (N × H)
350. C2×C5A4×M16semidirect
371. C23C3×#676semidirect
401. C8C2×D5×A4semidirect
405. C5C2×A4×M16semidirect
423. C22C2×C3×#676semidirect
437. C2A4×#676direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×D5×A42
C2×C5×C8×A42
C2×A4×#1682
A4×#6762
C23×#6763
C2×C3×#6764
C2×A4×M165