Finite Group Databasebeta

← C22×A4×#1044 #72260249

Subgroups — C22×A4×#1044 #72260249

Order 4800.

1624 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 1624 conjugacy classes of subgroups — too many to draw. Only its 126 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×A4×#1044C216yesmaximal, direct product factor
2C22×A4×C5⋊D5C211yesmaximal, characteristic
3C24×#1044C311yesmaximal, characteristic, complemented, Hall
4C22×C52×A4C411yescharacteristic, complemented
5C22×C3×#1044—41nomaximal
6C2×A4×C5⋊D5C413yescomplemented
7C2×A4×C5⋊D5C2213yes—
8A4×#1044C2214yesdirect product factor
9C22×A4×AGL(1,𝔽5)—52nomaximal
10C24×C5⋊D5C2×C311yescharacteristic
11C23×#1044—31no—
12C23×#1044—31no—
13C23×#1044—36no—
14C23×#1044C2×C316yescomplemented
15C22×C3×C5⋊D5—41no—
16C2×C52×A4C2×C413yescomplemented
17C2×C3×#1044—46no—
18A4×C5⋊D5C2×C413yescomplemented
19A4×C5⋊D5C2311yescharacteristic
20C22×D5×A4—52no—
21C22×D5×A4—102no—
22C2×A4×AGL(1,𝔽5)—512no—
23C24×C52C3×C411yescharacteristic, complemented
24C23×C5⋊D5—31no—
25C23×C5⋊D5—33no—
26C23×C5⋊D5—31no—
27C23×C5⋊D5—33no—
28C23×C5⋊D5C3×C413yescomplemented
29C23×C5⋊D5C22×C313yes—
30C22×#1044—36no—
31C22×#1044—38no—
32C22×#1044—31no—
33C22×#1044—36no—
34C22×#1044—36no—
35C22×#1044—36no—
36C22×#1044—312no—
37C22×#1044A411yescharacteristic, direct product factor
38C22×#1044C22×C314yescomplemented
39C24×AGL(1,𝔽5)—52no—
40C22×C3×C52—41no—
41C2×C3×C5⋊D5—43no—
42C2×C3×C5⋊D5—43no—
43C52×A4C22×C411yescharacteristic, complemented
44C3×#1044—44no—
45C22×C5×A4—22no—
46C22×C5×A4AGL(1,𝔽5)12yescomplemented
47C22×C3×AGL(1,𝔽5)—202no—
48C2×D5×A4—56no—
49C2×D5×A4—56no—
50C2×D5×A4—106no—
51C2×D5×A4—106no—
52A4×AGL(1,𝔽5)—58no—
53C23×C52—31no—
54C23×C52—33no—
55C23×C52C2×C3×C413yescomplemented
56C22×C5⋊D5—32no—
57C22×C5⋊D5—36no—
58C22×C5⋊D5—36no—
59C22×C5⋊D5—31no—
60C22×C5⋊D5—33no—
61C22×C5⋊D5—33no—
62C22×C5⋊D5—33no—
63C22×C5⋊D5—33no—
64C22×C5⋊D5—33no—
65C22×C5⋊D5—33no—
66C22×C5⋊D5—36no—
67C22×C5⋊D5—33no—
68C22×C5⋊D5—33no—
69C22×C5⋊D5C2×C3×C413yescomplemented
70C22×C5⋊D5C2×A411yescharacteristic
71C22×C5⋊D5C23×C311yescharacteristic
72C2×#1044—312no—
73C2×#1044—36no—
74C2×#1044—312no—
75C2×#1044—34no—
76C2×#1044—34no—
77C2×#1044C2×A416yesdirect product factor
78C22×C4×A4—251noHall
79C24×D5—52no—
80C24×D5—102no—
81C23×AGL(1,𝔽5)—512no—
82C23×AGL(1,𝔽5)—152no—
83C23×AGL(1,𝔽5)—1512no—
84C23×AGL(1,𝔽5)—152no—
85C2×C3×C52—43no—
86C3×C5⋊D5—43no—
87C3×C5⋊D5—41no—
88C22×C3×D5—202no—
89C22×C3×D5—402no—
90C2×C5×A4—26no—
91C2×C5×A4C2×AGL(1,𝔽5)16yescomplemented
92C2×C3×AGL(1,𝔽5)—2012no—
93D5×A4—56no—
94D5×A4—52no—
95D5×A4—106no—
96D5×A4—102no—
97C22×C52—32no—
98C22×C52—33no—
99C22×C52—33no—
100C22×C52—33no—
101C22×C52C4×A411yescharacteristic, complemented
102C22×C52C22×C3×C411yescharacteristic, complemented
103C2×C5⋊D5—36no—
104C2×C5⋊D5—36no—
105C2×C5⋊D5—33no—
106C2×C5⋊D5—33no—
107C2×C5⋊D5—33no—
108C2×C5⋊D5—36no—
109C2×C5⋊D5—33no—
110C2×C5⋊D5—33no—
111C2×C5⋊D5—31no—
112C2×C5⋊D5—33no—
113C2×C5⋊D5—31no—
114C2×C5⋊D5C4×A413yescomplemented
115C2×C5⋊D5C22×A413yes—
116#1044—34no—
117#1044C22×A414yesdirect product factor
118C23×A4—251no—
119C2×C4×A4—256no—
120C24×C5—22no—
121C24×C5C3×AGL(1,𝔽5)12yescomplemented
122C23×D5—56no—
123C23×D5—56no—
124C23×D5—106no—
125C23×D5—106no—
126C23×D5—152no—
127C23×D5—156no—
128C23×D5—156no—
129C23×D5—152no—
130C23×D5—302no—
131C23×D5—306no—
132C23×D5—306no—
133C23×D5—302no—
134C22×AGL(1,𝔽5)—52no—
135C22×AGL(1,𝔽5)—58no—
136C22×AGL(1,𝔽5)—1512no—
137C22×AGL(1,𝔽5)—1512no—
138C22×AGL(1,𝔽5)—152no—
139C22×AGL(1,𝔽5)—1516no—
140C22×AGL(1,𝔽5)—1512no—
141C22×AGL(1,𝔽5)—1524no—
142C22×AGL(1,𝔽5)—1512no—
143C3×C52—41noHall
144C24×C4—251noSylow
145C22×C3×C5—42no—
146C22×C3×C5—82no—
147C2×C3×D5—206no—
148C2×C3×D5—206no—
149C2×C3×D5—406no—
150C2×C3×D5—406no—
151C5×A4—22no—
152C5×A4C22×AGL(1,𝔽5)12yescomplemented
153C3×AGL(1,𝔽5)—208no—
154C2×C52—33no—
155C2×C52—31no—
156C2×C52C2×C4×A413yescomplemented
157C5⋊D5—31no—
158C5⋊D5—33no—
159C5⋊D5C2×C4×A413yescomplemented
160C5⋊D5C23×A411yescharacteristic
161C22×A4—253no—
162C22×A4—253no—
163C22×A4#104411yescharacteristic, direct product factor
164C22×C3×C4—1001no—
165C4×A4—254no—
166C23×C5—26no—
167C23×C5—36no—
168C23×C5—32no—
169C23×C5—62no—
170C23×C5—66no—
171C23×C5C2×C3×AGL(1,𝔽5)16yescomplemented
172C22×D5—52no—
173C22×D5—56no—
174C22×D5—52no—
175C22×D5—102no—
176C22×D5—106no—
177C22×D5—102no—
178C22×D5—156no—
179C22×D5—156no—
180C22×D5—156no—
181C22×D5—156no—
182C22×D5—154no—
183C22×D5—1512no—
184C22×D5—1512no—
185C22×D5—152no—
186C22×D5—156no—
187C22×D5—1512no—
188C22×D5—156no—
189C22×D5—156no—
190C22×D5—156no—
191C22×D5—306no—
192C22×D5—306no—
193C22×D5—306no—
194C22×D5—306no—
195C22×D5—302no—
196C22×D5—304no—
197C22×D5—3012no—
198C22×D5—3012no—
199C22×D5—306no—
200C22×D5—3012no—
201C22×D5—306no—
202C22×D5—306no—
203C22×D5—306no—
204C2×AGL(1,𝔽5)—512no—
205C2×AGL(1,𝔽5)—158no—
206C2×AGL(1,𝔽5)—1524no—
207C2×AGL(1,𝔽5)—1512no—
208C2×AGL(1,𝔽5)—1524no—
209C2×AGL(1,𝔽5)—158no—
210C25—251no—
211C23×C4—256no—
212C23×C4—751no—
213C23×C4—756no—
214C23×C4—751no—
215C2×C3×C5—46no—
216C2×C3×C5—86no—
217C3×D5—202no—
218C3×D5—206no—
219C3×D5—402no—
220C3×D5—406no—
221C52C22×C4×A411yescharacteristic, complemented, Sylow
222C23×C3—1001no—
223C2×A4—251no—
224C2×A4—253no—
225C2×A4C2×#104413yesdirect product factor
226C2×C3×C4—1006no—
227C22×C5—22no—
228C22×C5—22no—
229C22×C5—36no—
230C22×C5—34no—
231C22×C5—36no—
232C22×C5—36no—
233C22×C5—66no—
234C22×C5—66no—
235C22×C5—66no—
236C22×C5—64no—
237C22×C5A4×AGL(1,𝔽5)12yescomplemented
238C22×C5C22×C3×AGL(1,𝔽5)12yescomplemented
239C2×D5—56no—
240C2×D5—56no—
241C2×D5—106no—
242C2×D5—106no—
243C2×D5—152no—
244C2×D5—156no—
245C2×D5—156no—
246C2×D5—156no—
247C2×D5—1512no—
248C2×D5—1512no—
249C2×D5—1512no—
250C2×D5—156no—
251C2×D5—156no—
252C2×D5—156no—
253C2×D5—152no—
254C2×D5—302no—
255C2×D5—306no—
256C2×D5—306no—
257C2×D5—306no—
258C2×D5—3012no—
259C2×D5—306no—
260C2×D5—306no—
261C2×D5—3012no—
262C2×D5—3012no—
263C2×D5—306no—
264C2×D5—302no—
265AGL(1,𝔽5)—58no—
266AGL(1,𝔽5)—158no—
267C24—253no—
268C24—253no—
269C24—751no—
270C24—753no—
271C24—753no—
272C24—751no—
273C24C3×#104411yescharacteristic, complemented
274C22×C4—251no—
275C22×C4—254no—
276C22×C4—756no—
277C22×C4—756no—
278C22×C4—751no—
279C22×C4—758no—
280C22×C4—756no—
281C22×C4—7512no—
282C22×C4—756no—
283C3×C5—42no—
284C3×C5—82no—
285C22×C3—41no—
286C22×C3—1003no—
287C22×C3—1003no—
288A4C22×#104411yescharacteristic, direct product factor
289C3×C4—1004no—
290C2×C5—26no—
291C2×C5—36no—
292C2×C5—32no—
293C2×C5—62no—
294C2×C5—66no—
295C2×C5C2×A4×AGL(1,𝔽5)16yescomplemented
296D5—56no—
297D5—52no—
298D5—106no—
299D5—102no—
300D5—152no—
301D5—156no—
302D5—302no—
303D5—306no—
304C23—33no—
305C23—31no—
306C23—251no—
307C23—251no—
308C23—253no—
309C23—753no—
310C23—753no—
311C23—753no—
312C23—753no—
313C23—753no—
314C23—753no—
315C23—756no—
316C23—753no—
317C23—753no—
318C23—752no—
319C23—756no—
320C23—756no—
321C23—751no—
322C23C2×C3×#104413yescomplemented
323C2×C4—256no—
324C2×C4—754no—
325C2×C4—7512no—
326C2×C4—756no—
327C2×C4—7512no—
328C2×C4—754no—
329C2×C3—43no—
330C2×C3—1001no—
331C2×C3—1003no—
332C5—22no—
333C5C22×A4×AGL(1,𝔽5)12yescomplemented
334C22—33no—
335C22—32no—
336C22—33no—
337C22—33no—
338C22—253no—
339C22—253no—
340C22—751no—
341C22—753no—
342C22—753no—
343C22—753no—
344C22—756no—
345C22—751no—
346C22—753no—
347C22—756no—
348C22—756no—
349C22—753no—
350C22—753no—
351C22A4×#104411yescharacteristic, direct product factor
352C22C22×C3×#104411yescharacteristic, complemented
353C4—254no—
354C4—754no—
355C3—41noSylow
356C2—31no—
357C2—33no—
358C2—251no—
359C2—253no—
360C2—751no—
361C2—753no—
362C2C2×A4×#104413yesdirect product factor
subgroupG/H
center, hypercenterC22A4×#1044
commutator, hypocenterC22×C52C22×C3×C4
FrattiniC1G
Fitting, socleC24×C52C3×C4
prime radicalC22×A4×C5⋊D5C2
cosocleC22×C5⋊D5C23×C3
LCSG ▷ C22×C52
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×A4×#1044C2direct (N × H)
3. C24×#1044C3semidirect
4. C22×C52×A4C4semidirect
6. C2×A4×C5⋊D5C4semidirect
8. A4×#1044C22direct (N × H)
14. C23×#1044C2×C3semidirect
16. C2×C52×A4C2×C4semidirect
18. A4×C5⋊D5C2×C4semidirect
23. C24×C52C3×C4semidirect
28. C23×C5⋊D5C3×C4semidirect
37. C22×#1044A4direct (N × H)
38. C22×#1044C22×C3semidirect
43. C52×A4C22×C4semidirect
46. C22×C5×A4AGL(1,𝔽5)semidirect
55. C23×C52C2×C3×C4semidirect
69. C22×C5⋊D5C2×C3×C4semidirect
77. C2×#1044C2×A4direct (N × H)
91. C2×C5×A4C2×AGL(1,𝔽5)semidirect
101. C22×C52C4×A4semidirect
102. C22×C52C22×C3×C4semidirect
114. C2×C5⋊D5C4×A4semidirect
117. #1044C22×A4direct (N × H)
121. C24×C5C3×AGL(1,𝔽5)semidirect
152. C5×A4C22×AGL(1,𝔽5)semidirect
156. C2×C52C2×C4×A4semidirect
159. C5⋊D5C2×C4×A4semidirect
163. C22×A4#1044direct (N × H)
171. C23×C5C2×C3×AGL(1,𝔽5)semidirect
221. C52C22×C4×A4semidirect
225. C2×A4C2×#1044direct (N × H)
237. C22×C5A4×AGL(1,𝔽5)semidirect
238. C22×C5C22×C3×AGL(1,𝔽5)semidirect
273. C24C3×#1044semidirect
288. A4C22×#1044direct (N × H)
295. C2×C5C2×A4×AGL(1,𝔽5)semidirect
322. C23C2×C3×#1044semidirect
333. C5C22×A4×AGL(1,𝔽5)semidirect
351. C22A4×#1044direct (N × H)
352. C22C22×C3×#1044semidirect
362. C2C2×A4×#1044direct (N × H)

Maximal subgroups

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

subgroupindex
C2×A4×#10442
C22×A4×C5⋊D52
C24×#10443
C22×C3×#10444
C22×A4×AGL(1,𝔽5)5