Finite Group Databasebeta

← C3×Q8×C22⋊AGL(1,𝔽5) #14326123

Subgroups — C3×Q8×C22⋊AGL(1,𝔽5) #14326123

Order 1920.

1640 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Q8×D5C211yesmaximal, characteristic
2C2×C3×Q8×AGL(1,𝔽5)C212yesmaximal, complemented
3C3×(C2×AGL(1,𝔽5))⋊Q8C216yesmaximal, complemented
4C3×#65765C213yesmaximal
5C3×C4×C22⋊AGL(1,𝔽5)C213yesmaximal
6Q8×C22⋊AGL(1,𝔽5)C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C5×Q8C411yescharacteristic, complemented
8C2×C3×Q8×D5—28no—
9C2×C3×Q8×D5C411yescharacteristic, complemented
10C2×C3×Q8×D5C413yescomplemented
11C2×C3×Q8×D5C2213yes—
12C2×C3×Q8×D5C2211yescharacteristic
13C22×C3×C4×D5C2213yes—
14C22×C3×Dic10C413yescomplemented
15C2×C3×C22⋊AGL(1,𝔽5)C2213yes—
16C2×C3×C22⋊AGL(1,𝔽5)C2211yescharacteristic
17C3×Q8×AGL(1,𝔽5)—28no—
18C2×C3×C4⋊AGL(1,𝔽5)C2216yes—
19C2×C3×C4×AGL(1,𝔽5)C2216yes—
20C3×(C2×AGL(1,𝔽5))⋊C4C2216yes—
21C3×(C2×AGL(1,𝔽5))⋊C4C2216yes—
22C3×Q8×C22⋊C4—51nomaximal, Hall
23C22×Q8×D5C2×C311yescharacteristic
24C2×Q8×AGL(1,𝔽5)C2×C312yescomplemented
25(C2×AGL(1,𝔽5))⋊Q8C2×C316yescomplemented
26#65765C2×C313yes—
27C4×C22⋊AGL(1,𝔽5)C2×C313yes—
28C23×C3×D5C2311yescharacteristic
29C22×C3×AGL(1,𝔽5)C2316yes—
30C22×C3×AGL(1,𝔽5)C2312yes—
31C2×C3×C5×Q8—24no—
32C2×C3×C5×Q8C2×C413yes—
33C2×C3×C5×Q8C2×C411yescharacteristic
34C22×C3×C4×C5C2×C413yes—
35C22×C3×Dic5C2×C413yes—
36C3×Q8×D5—24no—
37C3×Q8×D5—212no—
38C3×Q8×D5—212no—
39C3×Q8×D5D418yes—
40C2×C3×C4×D5—212no—
41C2×C3×C4×D5C2×C413yes—
42C2×C3×C4×D5C2×C413yes—
43C2×C3×C4×D5C2313yes—
44C2×C3×C4×D5C2313yes—
45C2×C3×Dic10—212no—
46C2×C3×Dic10C2×C413yes—
47C2×C3×Dic10C2×C416yes—
48C2×C3×Dic10C2×C413yes—
49C3×C22⋊AGL(1,𝔽5)—26no—
50C3×C22⋊AGL(1,𝔽5)Q814yesdirect product factor
51C3×C4⋊AGL(1,𝔽5)—212no—
52C3×C4×AGL(1,𝔽5)—212no—
53C23×C3×Q8—51no—
54C2×C3×C4×Q8—52no—
55C3×#391—56no—
56C3×#380—53no—
57C3×C4×C22⋊C4—53no—
58C22×C5×Q8C3×C411yescharacteristic, complemented
59C2×Q8×D5—28no—
60C2×Q8×D5C3×C411yescharacteristic, complemented
61C2×Q8×D5C3×C413yescomplemented
62C2×Q8×D5C22×C313yes—
63C2×Q8×D5C22×C311yescharacteristic
64C22×C4×D5C22×C313yes—
65C22×Dic10C3×C413yescomplemented
66C2×C22⋊AGL(1,𝔽5)C22×C313yes—
67C2×C22⋊AGL(1,𝔽5)C22×C311yescharacteristic
68Q8×AGL(1,𝔽5)—28no—
69C2×C4⋊AGL(1,𝔽5)C22×C316yes—
70C2×C4×AGL(1,𝔽5)C22×C316yes—
71(C2×AGL(1,𝔽5))⋊C4C22×C316yes—
72(C2×AGL(1,𝔽5))⋊C4C22×C316yes—
73Q8×C22⋊C4—51noSylow
74C23×C3×C5C22×C411yescharacteristic
75C22×C3×D5—22no—
76C22×C3×D5—22no—
77C22×C3×D5C22×C411yescharacteristic
78C22×C3×D5C2×Q812yes—
79C22×C3×D5#4112yes—
80C22×C3×D5C2411yescharacteristic
81C2×C3×AGL(1,𝔽5)—22no—
82C2×C3×AGL(1,𝔽5)—22no—
83C2×C3×AGL(1,𝔽5)—26no—
84C2×C3×AGL(1,𝔽5)—26no—
85C2×C3×AGL(1,𝔽5)—26no—
86C2×C3×AGL(1,𝔽5)C2×Q814yescomplemented
87C3×C5×Q8—26no—
88C3×C5×Q8C22⋊C414yescomplemented
89C2×C3×C4×C5—26no—
90C2×C3×C4×C5C22×C413yes—
91C2×C3×C4×C5C22×C413yes—
92C2×C3×Dic5—26no—
93C2×C3×Dic5C22×C413yes—
94C2×C3×Dic5C22×C413yes—
95C3×C4×D5—26no—
96C3×C4×D5—26no—
97C3×C4×D5—26no—
98C3×C4×D5C2×D4112yes—
99C3×Dic10—26no—
100C3×Dic10—212no—
101C3×Dic10C22⋊C4112yescomplemented
102C22×C3×Q8—51no—
103C22×C3×Q8—53no—
104C22×C3×Q8—53no—
105C22×C3×Q8—51no—
106C22×C3×Q8—53no—
107C22×C3×Q8—108no—
108C22×C3×Q8AGL(1,𝔽5)11yescharacteristic, complemented
109C23×C3×C4—53no—
110C3×C4×Q8—108no—
111C2×C3×C4⋊C4—56no—
112C2×C3×C22⋊C4—51no—
113C2×C3×C22⋊C4—53no—
114C2×C3×C42—56no—
115C3×#95—56no—
116C3×#95—56no—
117C23×D5C23×C311yescharacteristic
118C22×AGL(1,𝔽5)C23×C316yes—
119C22×AGL(1,𝔽5)C23×C312yes—
120C2×C5×Q8—24no—
121C2×C5×Q8C2×C3×C413yes—
122C2×C5×Q8C2×C3×C411yescharacteristic
123C22×C4×C5C2×C3×C413yes—
124C22×Dic5C2×C3×C413yes—
125Q8×D5—24no—
126Q8×D5—212no—
127Q8×D5—212no—
128Q8×D5C3×D418yes—
129C2×C4×D5—212no—
130C2×C4×D5C2×C3×C413yes—
131C2×C4×D5C2×C3×C413yes—
132C2×C4×D5C23×C313yes—
133C2×C4×D5C23×C313yes—
134C2×Dic10—212no—
135C2×Dic10C2×C3×C413yes—
136C2×Dic10C2×C3×C416yes—
137C2×Dic10C2×C3×C413yes—
138C22⋊AGL(1,𝔽5)—26no—
139C22⋊AGL(1,𝔽5)C3×Q814yesdirect product factor
140C4⋊AGL(1,𝔽5)—212no—
141C4×AGL(1,𝔽5)—212no—
142C23×Q8—51no—
143C2×C4×Q8—52no—
144#391—56no—
145#380—53no—
146C4×C22⋊C4—53no—
147C22×C3×C5—21no—
148C22×C3×C5—21no—
149C22×C3×C5C4×Q812yescomplemented
150C22×C3×C5C23×C411yescharacteristic
151C2×C3×D5—21no—
152C2×C3×D5—21no—
153C2×C3×D5—24no—
154C2×C3×D5—24no—
155C2×C3×D5C4×Q812yescomplemented
156C2×C3×D5C22×D412yes—
157C2×C3×D5C22×Q811yescharacteristic
158C2×C3×D5C2×#4111yescharacteristic
159C2×C3×D52−1+412yes—
160C3×AGL(1,𝔽5)—24no—
161C3×AGL(1,𝔽5)—46no—
162C3×C4×C5—23no—
163C3×C4×C5C2×C22⋊C416yes—
164C3×Dic5—23no—
165C3×Dic5C2×C22⋊C416yes—
166C24×C3—51no—
167C2×C3×Q8—24no—
168C2×C3×Q8—58no—
169C2×C3×Q8—53no—
170C2×C3×Q8—56no—
171C2×C3×Q8—53no—
172C2×C3×Q8—1012no—
173C2×C3×Q8—1012no—
174C2×C3×Q8—1012no—
175C2×C3×Q8—104no—
176C2×C3×Q8C2×AGL(1,𝔽5)13yes—
177C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
178C22×C3×C4—53no—
179C22×C3×C4—52no—
180C22×C3×C4—53no—
181C22×C3×C4—53no—
182C22×C3×C4—53no—
183C22×C3×C4—53no—
184C22×C3×C4—56no—
185C22×C3×C4—1012no—
186C22×C3×C4C2×AGL(1,𝔽5)13yes—
187C3×C4⋊C4—1012no—
188C3×C22⋊C4—54no—
189C3×C22⋊C4—106no—
190C3×C42—1012no—
191C23×C5C22×C3×C411yescharacteristic
192C22×D5—22no—
193C22×D5—22no—
194C22×D5C22×C3×C411yescharacteristic
195C22×D5C2×C3×Q812yes—
196C22×D5C3×#4112yes—
197C22×D5C24×C311yescharacteristic
198C2×AGL(1,𝔽5)—22no—
199C2×AGL(1,𝔽5)—22no—
200C2×AGL(1,𝔽5)—26no—
201C2×AGL(1,𝔽5)—26no—
202C2×AGL(1,𝔽5)—26no—
203C2×AGL(1,𝔽5)C2×C3×Q814yescomplemented
204C5×Q8—26no—
205C5×Q8C3×C22⋊C414yescomplemented
206C2×C4×C5—26no—
207C2×C4×C5C22×C3×C413yes—
208C2×C4×C5C22×C3×C413yes—
209C2×Dic5—26no—
210C2×Dic5C22×C3×C413yes—
211C2×Dic5C22×C3×C413yes—
212C4×D5—26no—
213C4×D5—26no—
214C4×D5—26no—
215C4×D5C2×C3×D4112yes—
216Dic10—26no—
217Dic10—212no—
218Dic10C3×C22⋊C4112yescomplemented
219C22×Q8—51no—
220C22×Q8—53no—
221C22×Q8—51no—
222C22×Q8—53no—
223C22×Q8—53no—
224C22×Q8—108no—
225C22×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
226C23×C4—53no—
227C4×Q8—108no—
228C2×C4⋊C4—56no—
229C2×C22⋊C4—51no—
230C2×C22⋊C4—53no—
231C2×C42—56no—
232#95—56no—
233#95—56no—
234C2×C3×C5—22no—
235C2×C3×C5C22×C22⋊C411yescharacteristic
236C2×C3×C5C2×C4×Q811yescharacteristic
237C2×C3×C5#51911yescharacteristic
238C3×D5—22no—
239C3×D5#54812yes—
240C3×D5D4×Q812yes—
241C23×C3—51no—
242C23×C3—52no—
243C23×C3—52no—
244C23×C3—51no—
245C23×C3—102no—
246C23×C3—102no—
247C23×C3C22×AGL(1,𝔽5)11yescharacteristic
248C3×Q8—26no—
249C3×Q8—512no—
250C3×Q8—1012no—
251C3×Q8—106no—
252C3×Q8C22⋊AGL(1,𝔽5)14yesdirect product factor
253C2×C3×C4—26no—
254C2×C3×C4—53no—
255C2×C3×C4—512no—
256C2×C3×C4—54no—
257C2×C3×C4—53no—
258C2×C3×C4—106no—
259C2×C3×C4—106no—
260C2×C3×C4—106no—
261C2×C3×C4—102no—
262C2×C3×C4—102no—
263C2×C3×C4—106no—
264C2×C3×C4—106no—
265C2×C3×C4—106no—
266C2×C3×C4—106no—
267C2×C3×C4C22×AGL(1,𝔽5)13yes—
268C2×C3×C4C22×AGL(1,𝔽5)13yes—
269C22×C5—21no—
270C22×C5—21no—
271C22×C5C3×C4×Q812yescomplemented
272C22×C5C23×C3×C411yescharacteristic
273C2×D5—21no—
274C2×D5—21no—
275C2×D5—24no—
276C2×D5—24no—
277C2×D5C3×C4×Q812yescomplemented
278C2×D5C22×C3×D412yes—
279C2×D5C22×C3×Q811yescharacteristic
280C2×D5C2×C3×#4111yescharacteristic
281C2×D5C3×2−1+412yes—
282AGL(1,𝔽5)—24no—
283AGL(1,𝔽5)—46no—
284C4×C5—23no—
285C4×C5C2×C3×C22⋊C416yes—
286Dic5—23no—
287Dic5C2×C3×C22⋊C416yes—
288C24—51no—
289C2×Q8—24no—
290C2×Q8—58no—
291C2×Q8—53no—
292C2×Q8—56no—
293C2×Q8—53no—
294C2×Q8—1012no—
295C2×Q8—1012no—
296C2×Q8—1012no—
297C2×Q8—104no—
298C2×Q8C2×C3×AGL(1,𝔽5)13yes—
299C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
300C22×C4—53no—
301C22×C4—52no—
302C22×C4—53no—
303C22×C4—53no—
304C22×C4—53no—
305C22×C4—53no—
306C22×C4—56no—
307C22×C4—1012no—
308C22×C4C2×C3×AGL(1,𝔽5)13yes—
309C4⋊C4—1012no—
310C22⋊C4—54no—
311C22⋊C4—106no—
312C42—1012no—
313C3×C5Q8×C22⋊C411yescharacteristic, complemented, Hall
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—51no—
317C22×C3—51no—
318C22×C3—52no—
319C22×C3—52no—
320C22×C3—52no—
321C22×C3—104no—
322C22×C3—104no—
323C22×C3—101no—
324C22×C3—101no—
325C22×C3Q8×AGL(1,𝔽5)12yescomplemented
326C22×C3C23×AGL(1,𝔽5)11yescharacteristic
327C3×C4—23no—
328C3×C4—56no—
329C3×C4—104no—
330C3×C4—103no—
331C3×C4—206no—
332C3×C4C2×C22⋊AGL(1,𝔽5)16yes—
333C2×C5—22no—
334C2×C5C22×C3×C22⋊C411yescharacteristic
335C2×C5C2×C3×C4×Q811yescharacteristic
336C2×C5C3×#51911yescharacteristic
337D5—22no—
338D5C3×#54812yes—
339D5C3×D4×Q812yes—
340C23—51no—
341C23—52no—
342C23—52no—
343C23—51no—
344C23—102no—
345C23—102no—
346C23C22×C3×AGL(1,𝔽5)11yescharacteristic
347Q8—26no—
348Q8—512no—
349Q8—1012no—
350Q8—106no—
351Q8C3×C22⋊AGL(1,𝔽5)14yesdirect product factor
352C2×C4—26no—
353C2×C4—53no—
354C2×C4—512no—
355C2×C4—54no—
356C2×C4—53no—
357C2×C4—106no—
358C2×C4—106no—
359C2×C4—106no—
360C2×C4—102no—
361C2×C4—102no—
362C2×C4—106no—
363C2×C4—106no—
364C2×C4—106no—
365C2×C4—106no—
366C2×C4C22×C3×AGL(1,𝔽5)13yes—
367C2×C4C22×C3×AGL(1,𝔽5)13yes—
368C2×C3—22no—
369C2×C3—52no—
370C2×C3—52no—
371C2×C3—102no—
372C2×C3C2×Q8×AGL(1,𝔽5)11yescharacteristic
373C2×C3#6626211yescharacteristic
374C2×C3C22×C22⋊AGL(1,𝔽5)11yescharacteristic
375C5C3×Q8×C22⋊C411yescharacteristic, complemented, Sylow
376C22—21no—
377C22—21no—
378C22—51no—
379C22—51no—
380C22—52no—
381C22—52no—
382C22—52no—
383C22—104no—
384C22—104no—
385C22—101no—
386C22—101no—
387C22C3×Q8×AGL(1,𝔽5)12yescomplemented
388C22C23×C3×AGL(1,𝔽5)11yescharacteristic
389C4—23no—
390C4—56no—
391C4—104no—
392C4—103no—
393C4—206no—
394C4C2×C3×C22⋊AGL(1,𝔽5)16yes—
395C3Q8×C22⋊AGL(1,𝔽5)11yescharacteristic, direct product factor, Sylow
396C2—22no—
397C2—52no—
398C2—52no—
399C2—102no—
400C2C2×C3×Q8×AGL(1,𝔽5)11yescharacteristic
401C2C3×#6626211yescharacteristic
402C2C22×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22×C3C23×AGL(1,𝔽5)
commutatorC22×C5C23×C3×C4
FrattiniC22C23×C3×AGL(1,𝔽5)
FittingC22×C3×C5×Q8C4
socleC22×C3×C5C23×C4
prime radicalC23×C3×D5C23
cosocleC22×D5C24×C3
hypercenterC22×C3×Q8AGL(1,𝔽5)
hypocenterC5C3×Q8×C22⋊C4
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×Q8×AGL(1,𝔽5)C2semidirect
3. C3×(C2×AGL(1,𝔽5))⋊Q8C2semidirect
6. Q8×C22⋊AGL(1,𝔽5)C3direct (N × H)
7. C22×C3×C5×Q8C4semidirect
9. C2×C3×Q8×D5C4semidirect
10. C2×C3×Q8×D5C4semidirect
14. C22×C3×Dic10C4semidirect
24. C2×Q8×AGL(1,𝔽5)C2×C3semidirect
25. (C2×AGL(1,𝔽5))⋊Q8C2×C3semidirect
50. C3×C22⋊AGL(1,𝔽5)Q8direct (N × H)
58. C22×C5×Q8C3×C4semidirect
60. C2×Q8×D5C3×C4semidirect
61. C2×Q8×D5C3×C4semidirect
65. C22×Dic10C3×C4semidirect
86. C2×C3×AGL(1,𝔽5)C2×Q8semidirect
88. C3×C5×Q8C22⋊C4semidirect
101. C3×Dic10C22⋊C4semidirect
108. C22×C3×Q8AGL(1,𝔽5)semidirect
139. C22⋊AGL(1,𝔽5)C3×Q8direct (N × H)
149. C22×C3×C5C4×Q8semidirect
155. C2×C3×D5C4×Q8semidirect
203. C2×AGL(1,𝔽5)C2×C3×Q8semidirect
205. C5×Q8C3×C22⋊C4semidirect
218. Dic10C3×C22⋊C4semidirect
225. C22×Q8C3×AGL(1,𝔽5)semidirect
252. C3×Q8C22⋊AGL(1,𝔽5)direct (N × H)
271. C22×C5C3×C4×Q8semidirect
277. C2×D5C3×C4×Q8semidirect
313. C3×C5Q8×C22⋊C4semidirect
325. C22×C3Q8×AGL(1,𝔽5)semidirect
351. Q8C3×C22⋊AGL(1,𝔽5)direct (N × H)
375. C5C3×Q8×C22⋊C4semidirect
387. C22C3×Q8×AGL(1,𝔽5)semidirect
395. C3Q8×C22⋊AGL(1,𝔽5)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q8×D52
C2×C3×Q8×AGL(1,𝔽5)2
C3×(C2×AGL(1,𝔽5))⋊Q82
C3×#657652
C3×C4×C22⋊AGL(1,𝔽5)2
Q8×C22⋊AGL(1,𝔽5)3
C3×Q8×C22⋊C45