Finite Group Databasebeta

← C22×A4⋊#1044 #72231295

Subgroups — C22×A4⋊#1044 #72231295

Order 4800.

1464 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 104 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
3C22×C22⋊#1044—31nomaximal, 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⋊D5D311yescharacteristic
11C23×#1044—31no—
12C23×#1044—31no—
13C2×C22⋊#1044—36no—
14C2×C22⋊#1044—36no—
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×C52Dic311yescharacteristic, complemented
24C23×C5⋊D5—31no—
25C23×C5⋊D5—33no—
26C23×C5⋊D5—31no—
27C23×C5⋊D5—33no—
28C23×C5⋊D5Dic313yescomplemented
29C23×C5⋊D5C2×D313yes—
30C22×#1044—36no—
31C22×#1044—36no—
32C22×#1044—61no—
33C22×#1044—63no—
34C22×#1044—63no—
35C22×#1044—61no—
36C22⋊#1044—312no—
37C22⋊#1044—34no—
38C22×C22⋊AGL(1,𝔽5)—152no—
39C22×C3×C52—41no—
40C2×C3×C5⋊D5—43no—
41C2×C3×C5⋊D5—43no—
42C52×A4C22×C411yescharacteristic, complemented
43C3⋊#1044—44no—
44C22×C5×A4—22no—
45C22×C5×A4AGL(1,𝔽5)12yescomplemented
46C22×ΓL(1,𝔽16)—202no—
47C2×D5×A4—56no—
48C2×D5×A4—56no—
49C2×D5×A4—106no—
50C2×D5×A4—106no—
51A4⋊AGL(1,𝔽5)—58no—
52C23×C52—31no—
53C23×C52—33no—
54C23×C52C2×Dic313yescomplemented
55C22×C5⋊D5—31no—
56C22×C5⋊D5—33no—
57C22×C5⋊D5—33no—
58C22×C5⋊D5—33no—
59C22×C5⋊D5—33no—
60C22×C5⋊D5—33no—
61C22×C5⋊D5—33no—
62C22×C5⋊D5—36no—
63C22×C5⋊D5—33no—
64C22×C5⋊D5—33no—
65C22×C5⋊D5—63no—
66C22×C5⋊D5—61no—
67C22×C5⋊D5—63no—
68C22×C5⋊D5C2×Dic313yescomplemented
69C22×C5⋊D5S411yescharacteristic
70C22×C5⋊D5C22×D311yescharacteristic
71C2×#1044—34no—
72C2×#1044—34no—
73C2×#1044—66no—
74C2×#1044—66no—
75C2×#1044—66no—
76C2×#1044—66no—
77C22×SL(2,ℤ/4)—251noHall
78C24×D5—52no—
79C24×D5—102no—
80C23×AGL(1,𝔽5)—152no—
81C23×AGL(1,𝔽5)—152no—
82C2×C22⋊AGL(1,𝔽5)—1512no—
83C2×C22⋊AGL(1,𝔽5)—1512no—
84C2×C3×C52—43no—
85C3×C5⋊D5—43no—
86C3×C5⋊D5—41no—
87C22×C3×D5—202no—
88C22×C3×D5—402no—
89C2×C5×A4—26no—
90C2×C5×A4C2×AGL(1,𝔽5)16yescomplemented
91C2×ΓL(1,𝔽16)—2012no—
92D5×A4—56no—
93D5×A4—52no—
94D5×A4—106no—
95D5×A4—102no—
96C22×C52—33no—
97C22×C52—33no—
98C22×C52—33no—
99C22×C52—61no—
100C22×C52SL(2,ℤ/4)11yescharacteristic, complemented
101C22×C52C22×Dic311yescharacteristic, complemented
102C2×C5⋊D5—33no—
103C2×C5⋊D5—33no—
104C2×C5⋊D5—33no—
105C2×C5⋊D5—33no—
106C2×C5⋊D5—36no—
107C2×C5⋊D5—31no—
108C2×C5⋊D5—33no—
109C2×C5⋊D5—33no—
110C2×C5⋊D5—31no—
111C2×C5⋊D5—63no—
112C2×C5⋊D5—63no—
113C2×C5⋊D5SL(2,ℤ/4)13yescomplemented
114C2×C5⋊D5C2×S413yes—
115#1044—64no—
116#1044—64no—
117C23×A4—251no—
118C2×SL(2,ℤ/4)—256no—
119C24×C5—22no—
120C24×C5ΓL(1,𝔽16)12yescomplemented
121C23×D5—56no—
122C23×D5—56no—
123C23×D5—106no—
124C23×D5—106no—
125C23×D5—152no—
126C23×D5—156no—
127C23×D5—156no—
128C23×D5—152no—
129C23×D5—302no—
130C23×D5—306no—
131C23×D5—306no—
132C23×D5—302no—
133C22×AGL(1,𝔽5)—1512no—
134C22×AGL(1,𝔽5)—1512no—
135C22×AGL(1,𝔽5)—302no—
136C22×AGL(1,𝔽5)—306no—
137C22×AGL(1,𝔽5)—306no—
138C22×AGL(1,𝔽5)—302no—
139C22⋊AGL(1,𝔽5)—158no—
140C22⋊AGL(1,𝔽5)—1524no—
141C3×C52—41noHall
142C22×C22⋊C4—751noSylow
143C22×C3×C5—42no—
144C22×C3×C5—82no—
145C2×C3×D5—206no—
146C2×C3×D5—206no—
147C2×C3×D5—406no—
148C2×C3×D5—406no—
149C5×A4—22no—
150C5×A4C22×AGL(1,𝔽5)12yescomplemented
151ΓL(1,𝔽16)—208no—
152C2×C52—31no—
153C2×C52—33no—
154C2×C52C2×SL(2,ℤ/4)13yescomplemented
155C5⋊D5—31no—
156C5⋊D5—33no—
157C5⋊D5C2×SL(2,ℤ/4)13yescomplemented
158C5⋊D5C22×S411yescharacteristic
159C22×A4—253no—
160C22×A4—253no—
161C22×A4#104411yescharacteristic, complemented
162C22×Dic3—1001no—
163SL(2,ℤ/4)—254no—
164C23×C5—26no—
165C23×C5—32no—
166C23×C5—36no—
167C23×C5—62no—
168C23×C5—66no—
169C23×C5C2×ΓL(1,𝔽16)16yescomplemented
170C22×D5—52no—
171C22×D5—56no—
172C22×D5—52no—
173C22×D5—102no—
174C22×D5—106no—
175C22×D5—102no—
176C22×D5—156no—
177C22×D5—156no—
178C22×D5—156no—
179C22×D5—156no—
180C22×D5—152no—
181C22×D5—156no—
182C22×D5—156no—
183C22×D5—1512no—
184C22×D5—156no—
185C22×D5—156no—
186C22×D5—306no—
187C22×D5—306no—
188C22×D5—306no—
189C22×D5—306no—
190C22×D5—302no—
191C22×D5—306no—
192C22×D5—3012no—
193C22×D5—306no—
194C22×D5—306no—
195C22×D5—306no—
196C22×D5—302no—
197C22×D5—306no—
198C22×D5—306no—
199C22×D5—3012no—
200C22×D5—304no—
201C22×D5—3012no—
202C2×AGL(1,𝔽5)—158no—
203C2×AGL(1,𝔽5)—158no—
204C2×AGL(1,𝔽5)—3012no—
205C2×AGL(1,𝔽5)—3012no—
206C2×AGL(1,𝔽5)—3012no—
207C2×AGL(1,𝔽5)—3012no—
208C25—251no—
209C23×C4—751no—
210C23×C4—751no—
211C2×C22⋊C4—756no—
212C2×C22⋊C4—756no—
213C2×C3×C5—46no—
214C2×C3×C5—86no—
215C3×D5—202no—
216C3×D5—206no—
217C3×D5—402no—
218C3×D5—406no—
219C52C22×SL(2,ℤ/4)11yescharacteristic, complemented, Sylow
220C23×C3—1001no—
221C2×A4—251no—
222C2×A4—253no—
223C2×A4C2×#104413yescomplemented
224C2×Dic3—1006no—
225C22×C5—22no—
226C22×C5—22no—
227C22×C5—36no—
228C22×C5—36no—
229C22×C5—36no—
230C22×C5—66no—
231C22×C5—66no—
232C22×C5—66no—
233C22×C5—62no—
234C22×C5—64no—
235C22×C5A4⋊AGL(1,𝔽5)12yescomplemented
236C22×C5C22×ΓL(1,𝔽16)12yescomplemented
237C2×D5—56no—
238C2×D5—56no—
239C2×D5—106no—
240C2×D5—106no—
241C2×D5—152no—
242C2×D5—156no—
243C2×D5—156no—
244C2×D5—156no—
245C2×D5—1512no—
246C2×D5—156no—
247C2×D5—156no—
248C2×D5—152no—
249C2×D5—156no—
250C2×D5—302no—
251C2×D5—306no—
252C2×D5—306no—
253C2×D5—306no—
254C2×D5—3012no—
255C2×D5—306no—
256C2×D5—306no—
257C2×D5—306no—
258C2×D5—302no—
259C2×D5—306no—
260C2×D5—306no—
261C2×D5—3012no—
262C2×D5—3012no—
263AGL(1,𝔽5)—308no—
264AGL(1,𝔽5)—308no—
265C24—253no—
266C24—253no—
267C24—751no—
268C24—753no—
269C24—751no—
270C24—753no—
271C24C3⋊#104411yescharacteristic, complemented
272C22×C4—756no—
273C22×C4—756no—
274C22×C4—1501no—
275C22×C4—1503no—
276C22×C4—1503no—
277C22×C4—1501no—
278C22⋊C4—754no—
279C22⋊C4—7512no—
280C3×C5—42no—
281C3×C5—82no—
282C22×C3—41no—
283C22×C3—1003no—
284C22×C3—1003no—
285A4C22×#104411yescharacteristic, complemented
286Dic3—1004no—
287C2×C5—26no—
288C2×C5—32no—
289C2×C5—36no—
290C2×C5—62no—
291C2×C5—66no—
292C2×C5C2×A4⋊AGL(1,𝔽5)16yescomplemented
293D5—56no—
294D5—52no—
295D5—106no—
296D5—102no—
297D5—152no—
298D5—156no—
299D5—302no—
300D5—306no—
301C23—31no—
302C23—33no—
303C23—251no—
304C23—251no—
305C23—253no—
306C23—753no—
307C23—753no—
308C23—753no—
309C23—753no—
310C23—753no—
311C23—753no—
312C23—756no—
313C23—753no—
314C23—753no—
315C23—751no—
316C23—1503no—
317C23—1501no—
318C23—1503no—
319C23C2×C3⋊#104413yescomplemented
320C2×C4—754no—
321C2×C4—754no—
322C2×C4—1506no—
323C2×C4—1506no—
324C2×C4—1506no—
325C2×C4—1506no—
326C2×C3—43no—
327C2×C3—1001no—
328C2×C3—1003no—
329C5—22no—
330C5C22×A4⋊AGL(1,𝔽5)12yescomplemented
331C22—33no—
332C22—33no—
333C22—33no—
334C22—61no—
335C22—253no—
336C22—253no—
337C22—751no—
338C22—753no—
339C22—753no—
340C22—753no—
341C22—756no—
342C22—751no—
343C22—753no—
344C22—753no—
345C22—753no—
346C22—1503no—
347C22—1503no—
348C22A4⋊#104411yescharacteristic, direct product factor
349C22C22×C3⋊#104411yescharacteristic, complemented
350C4—1504no—
351C4—1504no—
352C3—41noSylow
353C2—31no—
354C2—33no—
355C2—251no—
356C2—253no—
357C2—751no—
358C2—753no—
359C2C2×A4⋊#104413yesdirect product factor
subgroupG/H
center, hypercenterC22A4⋊#1044
commutator, hypocenterC52×A4C22×C4
FrattiniC1G
Fitting, socleC24×C52Dic3
prime radicalC22×A4×C5⋊D5C2
cosocleA4×C5⋊D5C23
LCSG ▷ C52×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×A4⋊#1044C2direct (N × H)
4. C22×C52×A4C4semidirect
6. C2×A4×C5⋊D5C4semidirect
8. A4⋊#1044C22direct (N × H)
16. C2×C52×A4C2×C4semidirect
18. A4×C5⋊D5C2×C4semidirect
23. C24×C52Dic3semidirect
28. C23×C5⋊D5Dic3semidirect
42. C52×A4C22×C4semidirect
45. C22×C5×A4AGL(1,𝔽5)semidirect
54. C23×C52C2×Dic3semidirect
68. C22×C5⋊D5C2×Dic3semidirect
90. C2×C5×A4C2×AGL(1,𝔽5)semidirect
100. C22×C52SL(2,ℤ/4)semidirect
101. C22×C52C22×Dic3semidirect
113. C2×C5⋊D5SL(2,ℤ/4)semidirect
120. C24×C5ΓL(1,𝔽16)semidirect
150. C5×A4C22×AGL(1,𝔽5)semidirect
154. C2×C52C2×SL(2,ℤ/4)semidirect
157. C5⋊D5C2×SL(2,ℤ/4)semidirect
161. C22×A4#1044semidirect
169. C23×C5C2×ΓL(1,𝔽16)semidirect
219. C52C22×SL(2,ℤ/4)semidirect
223. C2×A4C2×#1044semidirect
235. C22×C5A4⋊AGL(1,𝔽5)semidirect
236. C22×C5C22×ΓL(1,𝔽16)semidirect
271. C24C3⋊#1044semidirect
285. A4C22×#1044semidirect
292. C2×C5C2×A4⋊AGL(1,𝔽5)semidirect
319. C23C2×C3⋊#1044semidirect
330. C5C22×A4⋊AGL(1,𝔽5)semidirect
348. C22A4⋊#1044direct (N × H)
349. C22C22×C3⋊#1044semidirect
359. 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
C22×C22⋊#10443
C22×C3⋊#10444
C22×A4⋊AGL(1,𝔽5)5