Finite Group Databasebeta

← C2×A4×#689 #14514200

Subgroups — C2×A4×#689 #14514200

Order 1920.

688 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 99 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×C5×Q8×A4C211yesmaximal, characteristic
2C2×A4×Dic10C211yesmaximal, characteristic
3A4×#689C214yesmaximal, direct product factor
4C2×A4×#168C211yesmaximal, characteristic
5C23×#689C311yesmaximal, characteristic, complemented, Hall
6C5×Q8×A4—21no—
7C5×Q8×A4C2212yes—
8C2×C4×C5×A4—21no—
9C2×C4×C5×A4C2211yescharacteristic
10C2×A4×Dic5—21no—
11A4×Dic10—21no—
12A4×Dic10C2212yes—
13C2×C3×#689—41nomaximal
14A4×#168C2212yes—
15C2×A4×Q16—51nomaximal, Hall
16C23×C5×Q8C2×C311yescharacteristic
17C23×Dic10C2×C311yescharacteristic
18C22×#689—31no—
19C22×#689—31no—
20C22×#689—34no—
21C22×#689—31no—
22C22×#689—31no—
23C22×#689C2×C314yescomplemented
24C23×#168C2×C311yescharacteristic
25C22×C5×A4D411yescharacteristic
26C2×C3×C5×Q8—41no—
27C2×C3×Dic10—41no—
28C4×C5×A4—22no—
29C4×C5×A4D411yescharacteristic
30C4×C5×A4C2311yescharacteristic
31A4×Dic5—22no—
32C3×#689—44no—
33C2×C3×#168—41no—
34C2×Q8×A4—51no—
35C2×Q8×A4D511yescharacteristic
36A4×Q16—54no—
37C2×C8×A4—51no—
38C22×C5×Q8—21no—
39C22×C5×Q8—31no—
40C22×C5×Q8—32no—
41C22×C5×Q8—31no—
42C22×C5×Q8—61no—
43C22×C5×Q8—61no—
44C22×C5×Q8C22×C312yes—
45C23×C4×C5—21no—
46C23×C4×C5C22×C311yescharacteristic
47C23×Dic5—21no—
48C22×Dic10—21no—
49C22×Dic10—31no—
50C22×Dic10—31no—
51C22×Dic10—32no—
52C22×Dic10—61no—
53C22×Dic10—61no—
54C22×Dic10C22×C312yes—
55C2×#689—34no—
56C2×#689—34no—
57C2×#689—32no—
58C2×#689—34no—
59C2×#689—31no—
60C2×#689—31no—
61C2×#689—34no—
62C2×#689—34no—
63C2×#689—34no—
64C2×#689—34no—
65C2×#689—34no—
66C2×#689—31no—
67C2×#689A411yescharacteristic, direct product factor
68C22×#168—31no—
69C22×#168—31no—
70C22×#168—32no—
71C22×#168C22×C312yes—
72C23×Q16—51noSylow
73C2×C5×A4Q1612yescomplemented
74C2×C5×A4C2×D411yescharacteristic
75C3×C5×Q8—42no—
76C3×C5×Q8—81no—
77C2×C3×C4×C5—41no—
78C2×C3×C4×C5—81no—
79C2×C3×Dic5—81no—
80C3×Dic10—42no—
81C3×Dic10—81no—
82C3×#168—42no—
83Q8×A4—21no—
84Q8×A4—52no—
85Q8×A4—101no—
86Q8×A4C2×D512yes—
87C2×C4×A4—21no—
88C2×C4×A4—101no—
89C2×C4×A4C2×D511yescharacteristic
90C2×C3×Q16—201no—
91C8×A4—52no—
92C24×C5C3×D411yescharacteristic
93C2×C5×Q8—32no—
94C2×C5×Q8—32no—
95C2×C5×Q8—32no—
96C2×C5×Q8—32no—
97C2×C5×Q8—31no—
98C2×C5×Q8—61no—
99C2×C5×Q8—61no—
100C2×C5×Q8—61no—
101C2×C5×Q8—62no—
102C2×C5×Q8—61no—
103C2×C5×Q8—62no—
104C2×C5×Q8—61no—
105C2×C5×Q8—64no—
106C2×C5×Q8—61no—
107C2×C5×Q8C2×A411yescharacteristic
108C22×C4×C5—22no—
109C22×C4×C5—31no—
110C22×C4×C5—31no—
111C22×C4×C5—31no—
112C22×C4×C5—31no—
113C22×C4×C5—61no—
114C22×C4×C5—62no—
115C22×C4×C5—61no—
116C22×C4×C5C3×D411yescharacteristic
117C22×C4×C5C23×C311yescharacteristic
118C22×Dic5—22no—
119C22×Dic5—61no—
120C22×Dic5—62no—
121C22×Dic5—61no—
122C2×Dic10—32no—
123C2×Dic10—31no—
124C2×Dic10—32no—
125C2×Dic10—32no—
126C2×Dic10—32no—
127C2×Dic10—61no—
128C2×Dic10—61no—
129C2×Dic10—61no—
130C2×Dic10—64no—
131C2×Dic10—61no—
132C2×Dic10—62no—
133C2×Dic10—61no—
134C2×Dic10—61no—
135C2×Dic10—62no—
136C2×Dic10C2×A411yescharacteristic
137#689—38no—
138#689—34no—
139#689—34no—
140#689—34no—
141#689C2×A414yesdirect product factor
142C2×#168—32no—
143C2×#168—31no—
144C2×#168—32no—
145C2×#168—32no—
146C2×#168—32no—
147C2×#168C2×A411yescharacteristic
148C23×Q8—51no—
149C23×Q8C3×D511yescharacteristic
150C22×Q16—54no—
151C22×Q16—151no—
152C22×Q16—151no—
153C22×Q16—154no—
154C22×Q16—151no—
155C22×Q16—151no—
156C23×C8—51no—
157C22×C3×C5—41no—
158C5×A4C2×Q1611yescharacteristic, complemented
159C3×C4×C5—41no—
160C3×C4×C5—41no—
161C3×C4×C5—82no—
162C3×Dic5—82no—
163C22×A4C22⋊D511yescharacteristic
164C2×C3×Q8—41no—
165C2×C3×Q8—201no—
166C4×A4—22no—
167C4×A4—102no—
168C4×A4C22⋊D511yescharacteristic
169C4×A4C22×D511yescharacteristic
170C3×Q16—204no—
171C2×C3×C8—201no—
172C23×C5—31no—
173C23×C5—32no—
174C23×C5—31no—
175C23×C5C3×Q1612yescomplemented
176C23×C5C2×C3×D411yescharacteristic
177C5×Q8—21no—
178C5×Q8—32no—
179C5×Q8—62no—
180C5×Q8—62no—
181C5×Q8—62no—
182C5×Q8—62no—
183C5×Q8—61no—
184C5×Q8C22×A412yes—
185C2×C4×C5—21no—
186C2×C4×C5—32no—
187C2×C4×C5—31no—
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—62no—
195C2×C4×C5—62no—
196C2×C4×C5—62no—
197C2×C4×C5—61no—
198C2×C4×C5—62no—
199C2×C4×C5C22×A411yescharacteristic
200C2×Dic5—21no—
201C2×Dic5—62no—
202C2×Dic5—62no—
203C2×Dic5—62no—
204C2×Dic5—61no—
205C2×Dic5—62no—
206Dic10—21no—
207Dic10—32no—
208Dic10—62no—
209Dic10—62no—
210Dic10—62no—
211Dic10—61no—
212Dic10—62no—
213Dic10C22×A412yes—
214#168—32no—
215#168C22×A412yes—
216C22×Q8—21no—
217C22×Q8—31no—
218C22×Q8—32no—
219C22×Q8—31no—
220C22×Q8—52no—
221C22×Q8—61no—
222C22×Q8—61no—
223C22×Q8—101no—
224C22×Q8—151no—
225C22×Q8—151no—
226C22×Q8—152no—
227C22×Q8—301no—
228C22×Q8—301no—
229C22×Q8C2×C3×D512yes—
230C23×C4—21no—
231C23×C4—101no—
232C23×C4C2×C3×D511yescharacteristic
233C2×Q16—51no—
234C2×Q16—151no—
235C2×Q16—154no—
236C2×Q16—154no—
237C2×Q16—151no—
238C2×Q16—154no—
239C2×Q16—154no—
240C2×Q16—154no—
241C2×Q16—152no—
242C2×Q16—151no—
243C2×Q16—154no—
244C2×Q16—154no—
245C2×Q16—154no—
246C22×C8—52no—
247C22×C8—151no—
248C22×C8—152no—
249C22×C8—151no—
250C2×C3×C5—41no—
251C2×C3×C5—42no—
252C2×A4#68912yesdirect product factor
253C2×A4C2×C22⋊D511yescharacteristic
254C3×Q8—42no—
255C3×Q8—81no—
256C3×Q8—202no—
257C3×Q8—401no—
258C2×C3×C4—41no—
259C2×C3×C4—81no—
260C2×C3×C4—401no—
261C3×C8—202no—
262C22×C5—32no—
263C22×C5—31no—
264C22×C5—32no—
265C22×C5—31no—
266C22×C5—31no—
267C22×C5—32no—
268C22×C5—32no—
269C22×C5C2×C3×Q1611yescharacteristic, complemented
270C22×C5D4×A411yescharacteristic
271C4×C5—22no—
272C4×C5—31no—
273C4×C5—31no—
274C4×C5—62no—
275C4×C5D4×A411yescharacteristic
276C4×C5C23×A411yescharacteristic
277Dic5—22no—
278Dic5—62no—
279C24C3×C22⋊D511yescharacteristic
280C2×Q8—32no—
281C2×Q8—32no—
282C2×Q8—32no—
283C2×Q8—32no—
284C2×Q8—31no—
285C2×Q8—51no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8—61no—
289C2×Q8—62no—
290C2×Q8—61no—
291C2×Q8—62no—
292C2×Q8—61no—
293C2×Q8—64no—
294C2×Q8—61no—
295C2×Q8—152no—
296C2×Q8—152no—
297C2×Q8—152no—
298C2×Q8—152no—
299C2×Q8—151no—
300C2×Q8—301no—
301C2×Q8—301no—
302C2×Q8—301no—
303C2×Q8—302no—
304C2×Q8—301no—
305C2×Q8—302no—
306C2×Q8—301no—
307C2×Q8—304no—
308C2×Q8—301no—
309C2×Q8D5×A411yescharacteristic
310C22×C4—22no—
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—31no—
314C22×C4—31no—
315C22×C4—61no—
316C22×C4—62no—
317C22×C4—61no—
318C22×C4—102no—
319C22×C4—301no—
320C22×C4—302no—
321C22×C4—301no—
322C22×C4C3×C22⋊D511yescharacteristic
323C22×C4C22×C3×D511yescharacteristic
324Q16—54no—
325Q16—154no—
326Q16—154no—
327Q16—158no—
328Q16—154no—
329C2×C8—51no—
330C2×C8—152no—
331C2×C8—152no—
332C2×C8—152no—
333C2×C8—151no—
334C2×C8—152no—
335C3×C5—41noHall
336C22×C3—41no—
337A4C2×#68911yescharacteristic, direct product factor
338C3×C4—41no—
339C3×C4—41no—
340C3×C4—82no—
341C3×C4—402no—
342C2×C5—32no—
343C2×C5—31no—
344C2×C5—31no—
345C2×C5A4×Q1612yescomplemented
346C2×C5C2×D4×A411yescharacteristic
347C23—31no—
348C23—32no—
349C23—31no—
350C23C3×#68912yescomplemented
351C23C2×C3×C22⋊D511yescharacteristic
352Q8—21no—
353Q8—32no—
354Q8—52no—
355Q8—62no—
356Q8—62no—
357Q8—62no—
358Q8—62no—
359Q8—61no—
360Q8—101no—
361Q8—152no—
362Q8—302no—
363Q8—302no—
364Q8—302no—
365Q8—302no—
366Q8—301no—
367Q8C2×D5×A412yes—
368C2×C4—21no—
369C2×C4—32no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—31no—
376C2×C4—31no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—61no—
381C2×C4—62no—
382C2×C4—101no—
383C2×C4—302no—
384C2×C4—302no—
385C2×C4—302no—
386C2×C4—301no—
387C2×C4—302no—
388C2×C4C2×D5×A411yescharacteristic
389C8—52no—
390C8—152no—
391C2×C3—41no—
392C2×C3—42no—
393C5C2×A4×Q1611yescharacteristic, complemented, Sylow
394C22—32no—
395C22—31no—
396C22—32no—
397C22—31no—
398C22—31no—
399C22—32no—
400C22—32no—
401C22C2×C3×#68911yescharacteristic, complemented
402C22A4×C22⋊D511yescharacteristic
403C4—22no—
404C4—31no—
405C4—31no—
406C4—62no—
407C4—102no—
408C4—302no—
409C4A4×C22⋊D511yescharacteristic
410C4C22×D5×A411yescharacteristic
411C3—41noSylow
412C2—31no—
413C2—32no—
414C2—31no—
415C2A4×#68912yesdirect product factor
416C2C2×A4×C22⋊D511yescharacteristic
subgroupG/H
centerC22A4×C22⋊D5
commutator, cosocleC22×C4×C5C23×C3
FrattiniC4C22×D5×A4
FittingC23×C5×Q8C2×C3
socleC24×C5C3×D4
prime radicalC22×C5×A4D4
hypercenterC2×Q8D5×A4
hypocenterC22×C5C2×C3×Q16
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C22×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. A4×#689C2direct (N × H)
5. C23×#689C3semidirect
23. C22×#689C2×C3semidirect
67. C2×#689A4direct (N × H)
73. C2×C5×A4Q16semidirect
141. #689C2×A4direct (N × H)
158. C5×A4C2×Q16semidirect
175. C23×C5C3×Q16semidirect
252. C2×A4#689direct (N × H)
269. C22×C5C2×C3×Q16semidirect
337. A4C2×#689direct (N × H)
345. C2×C5A4×Q16semidirect
350. C23C3×#689semidirect
393. C5C2×A4×Q16semidirect
401. C22C2×C3×#689semidirect
415. C2A4×#689direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×Q8×A42
C2×A4×Dic102
A4×#6892
C2×A4×#1682
C23×#6893
C2×C3×#6894
C2×A4×Q165