Finite Group Databasebeta

← H(5,𝔽2)×#1044 #72221791

Subgroups — H(5,𝔽2)×#1044 #72221791

Order 3200.

4218 conjugacy classes of subgroups; only the 890 normal subgroups are shown.

normal subgroups only

This group has 4218 conjugacy classes of subgroups — too many to draw. Only its 890 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
1H(5,𝔽2)×C5⋊D5C211yesmaximal, characteristic
2#13107775C216yesmaximal, complemented
3#41×#1044C216yesmaximal, complemented
4#13107767C219yesmaximal, complemented
5C2×D4×#1044C219yesmaximal, complemented
6C23×#1044C2216yescomplemented
7C52×H(5,𝔽2)C411yescharacteristic, complemented
8#11717522C416yescomplemented
9#41×C5⋊D5C2216yes—
10#11717517C419yescomplemented
11C2×D4×C5⋊D5C2219yes—
12C2×C22⋊#1044C22118yescomplemented
13Q8×#1044C2212yescomplemented
14Q8×#1044C2216yescomplemented
15D4×#1044C22118yescomplemented
16D4×#1044C22136yescomplemented
17D4×#1044C22118yescomplemented
18#11717451C22118yescomplemented
19C2×C4⋊#1044C2219yescomplemented
20C2×C4×#1044C2219yescomplemented
21AGL(1,𝔽5)×H(5,𝔽2)—52nomaximal
22C23×C5⋊D5C2316yes—
23C22×#1044—412no—
24C22×#1044C2319yes—
25C22×#1044C2319yes—
26C22×#1044C23118yes—
27C52×#41C2×C416yescomplemented
28C2×C52×D4C2×C419yescomplemented
29C2×C22⋊(C5⋊D5)C2×C4118yescomplemented
30#87904C2×C416yescomplemented
31Q8×C5⋊D5C2312yes—
32#87902C2×C4118yescomplemented
33D4×C5⋊D5C2×C4136yescomplemented
34D4×C5⋊D5C23118yes—
35#87900C2×C4118yescomplemented
36C2×C4⋊(C5⋊D5)C2×C419yescomplemented
37C2×C4×C5⋊D5C2319yes—
38C22⋊#1044C23136yes—
39C4⋊#1044C23118yes—
40C4⋊#1044C2316yes—
41C4×#1044C2316yes—
42C4×#1044C23118yes—
43D5×H(5,𝔽2)—52no—
44D5×H(5,𝔽2)—102no—
45#66266—512no—
46#41×AGL(1,𝔽5)—512no—
47#66258—518no—
48C2×D4×AGL(1,𝔽5)—518no—
49C23×C52C22×C416yescomplemented
50C22×C5⋊D5—46no—
51C22×C5⋊D5—46no—
52C22×C5⋊D5C22×C4118yescomplemented
53C22×C5⋊D5C2419yes—
54C2×#1044—218no—
55C2×#1044—418no—
56C2×#1044—49no—
57C2×#1044C2416yes—
58C2×#1044C2419yes—
59C2×#1044C2411yescharacteristic
60C52×Q8C22×C412yescomplemented
61C52×D4C22×C4118yescomplemented
62C2×C4×C52C22×C419yescomplemented
63C22⋊(C5⋊D5)C22×C4136yescomplemented
64C2×#1039C22×C419yescomplemented
65C4⋊(C5⋊D5)C22×C4118yescomplemented
66C4×C5⋊D5C22×C4118yescomplemented
67C4×C5⋊D5C2416yes—
68#6045C22×C416yescomplemented
69C23×AGL(1,𝔽5)—512no—
70C5×H(5,𝔽2)—22no—
71C5×H(5,𝔽2)AGL(1,𝔽5)12yescomplemented
72#4149—512no—
73#4149—1012no—
74D5×#41—512no—
75D5×#41—1012no—
76#4144—518no—
77#4144—1018no—
78C2×D4×D5—518no—
79C2×D4×D5—1018no—
80C2×C22⋊AGL(1,𝔽5)—536no—
81Q8×AGL(1,𝔽5)—54no—
82Q8×AGL(1,𝔽5)—512no—
83D4×AGL(1,𝔽5)—536no—
84D4×AGL(1,𝔽5)—572no—
85D4×AGL(1,𝔽5)—536no—
86#4130—536no—
87C2×C4⋊AGL(1,𝔽5)—518no—
88C2×C4×AGL(1,𝔽5)—518no—
89C4×H(5,𝔽2)—251noSylow
90C22×C52—46no—
91C22×C52C23×C419yes—
92C2×C5⋊D5—29no—
93C2×C5⋊D5—29no—
94C2×C5⋊D5—418no—
95C2×C5⋊D5C23×C419yes—
96C2×C5⋊D5C2511yescharacteristic
97#1044—29no—
98#1044—26no—
99#1044H(5,𝔽2)12yesdirect product factor
100C4×C52C23×C416yes—
101#1039C23×C416yes—
102C23×D5—512no—
103C23×D5—1012no—
104C22×AGL(1,𝔽5)—518no—
105C22×AGL(1,𝔽5)—518no—
106C22×AGL(1,𝔽5)—536no—
107C22×AGL(1,𝔽5)—2024no—
108C5×#41—212no—
109C5×#41C2×AGL(1,𝔽5)112yescomplemented
110C2×C5×D4—218no—
111C2×C5×D4C2×AGL(1,𝔽5)118yescomplemented
112C2×C22⋊D5—536no—
113C2×C22⋊D5—1036no—
114#713—512no—
115#713—1012no—
116Q8×D5—54no—
117Q8×D5—104no—
118#711—536no—
119#711—1036no—
120D4×D5—536no—
121D4×D5—572no—
122D4×D5—1036no—
123D4×D5—1072no—
124#709—536no—
125#709—1036no—
126C2×D20—518no—
127C2×D20—1018no—
128C2×C4×D5—518no—
129C2×C4×D5—1018no—
130C22⋊AGL(1,𝔽5)—572no—
131C4⋊AGL(1,𝔽5)—536no—
132C4⋊AGL(1,𝔽5)—512no—
133C4×AGL(1,𝔽5)—512no—
134C4×AGL(1,𝔽5)—536no—
135C2×H(5,𝔽2)—251no—
136#520—256no—
137#518—259no—
138C4×#41—256no—
139C2×C4×D4—259no—
140C2×C52—29no—
141C2×C52C24×C411yescharacteristic
142C5⋊D5—29no—
143C5⋊D5C2×H(5,𝔽2)11yescharacteristic
144C5⋊D5#58511yescharacteristic
145C23×C5—212no—
146C23×C5C22×AGL(1,𝔽5)112yescomplemented
147C22×D5—518no—
148C22×D5—536no—
149C22×D5—1018no—
150C22×D5—1036no—
151C22×D5—2012no—
152C22×D5—2012no—
153C22×D5—4012no—
154C22×D5—4012no—
155C2×AGL(1,𝔽5)—52no—
156C2×AGL(1,𝔽5)—512no—
157C2×AGL(1,𝔽5)—518no—
158C2×AGL(1,𝔽5)—1036no—
159C2×AGL(1,𝔽5)—2018no—
160C2×AGL(1,𝔽5)—2036no—
161C5×Q8—24no—
162C5×Q8C22×AGL(1,𝔽5)14yescomplemented
163C5×D4—236no—
164C5×D4C22×AGL(1,𝔽5)136yescomplemented
165C2×C4×C5—218no—
166C2×C4×C5C22×AGL(1,𝔽5)118yescomplemented
167C22⋊D5—572no—
168C22⋊D5—1072no—
169C2×Dic5—518no—
170C2×Dic5—1018no—
171D20—536no—
172D20—1036no—
173C4×D5—536no—
174C4×D5—512no—
175C4×D5—1036no—
176C4×D5—1012no—
177Dic10—512no—
178Dic10—1012no—
179H(5,𝔽2)—256no—
180H(5,𝔽2)—259no—
181H(5,𝔽2)#104411yescharacteristic, direct product factor
182C2×#41—256no—
183C22×D4—259no—
184C23×C4—256no—
185C4×Q8—252no—
186C4×Q8—256no—
187C4×D4—2518no—
188C4×D4—2536no—
189C4×D4—2518no—
190#117—2518no—
191C2×C4⋊C4—259no—
192C2×C22⋊C4—2518no—
193C2×C42—259no—
194C52C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
195C22×C5—218no—
196C22×C5—412no—
197C22×C5—812no—
198C22×C5C23×AGL(1,𝔽5)118yes—
199C2×D5—52no—
200C2×D5—518no—
201C2×D5—102no—
202C2×D5—1018no—
203C2×D5—1018no—
204C2×D5—1018no—
205C2×D5—2018no—
206C2×D5—2018no—
207C2×D5—2036no—
208C2×D5—4036no—
209AGL(1,𝔽5)—54no—
210AGL(1,𝔽5)—1018no—
211AGL(1,𝔽5)—1012no—
212C4×C5—212no—
213C4×C5C23×AGL(1,𝔽5)112yes—
214Dic5—512no—
215Dic5—1012no—
216C24—256no—
217#41—256no—
218#41—2518no—
219#41—2518no—
220#41C2×#104416yescomplemented
221C2×Q8—252no—
222C2×D4—2518no—
223C2×D4—2518no—
224C2×D4—2536no—
225C2×D4—259no—
226C2×D4C2×#104419yescomplemented
227C22×C4—259no—
228C22×C4—259no—
229C22×C4—259no—
230C22×C4—2518no—
231C22×C4—10012no—
232C4⋊C4—2518no—
233C4⋊C4—256no—
234C22⋊C4—2536no—
235C42—256no—
236C42—2518no—
237C2×C5—218no—
238C2×C5—22no—
239C2×C5—418no—
240C2×C5C24×AGL(1,𝔽5)12yes—
241D5—52no—
242D5—52no—
243D5—102no—
244D5—102no—
245D5—1018no—
246D5—2018no—
247C23—259no—
248C23—2518no—
249C23—1006no—
250C23—1006no—
251C23C22×#104416yescomplemented
252Q8—256no—
253Q8C22×#104412yescomplemented
254D4—2536no—
255D4—2518no—
256D4C22×#1044118yescomplemented
257C2×C4—251no—
258C2×C4—259no—
259C2×C4—256no—
260C2×C4—2518no—
261C2×C4—256no—
262C2×C4—259no—
263C2×C4—5018no—
264C2×C4—10018no—
265C2×C4—1009no—
266C2×C4C22×#104419yescomplemented
267C5—22no—
268C5AGL(1,𝔽5)×H(5,𝔽2)12yescomplemented
269C22—46no—
270C22—251no—
271C22—259no—
272C22—509no—
273C22—509no—
274C22—10018no—
275C22C23×#104419yes—
276C4—252no—
277C4—256no—
278C4—509no—
279C4—506no—
280C4C23×#104416yes—
281C2—29no—
282C2—251no—
283C2—251no—
284C2—509no—
285C2C24×#104411yescharacteristic
subgroupG/H
center, FrattiniC2C24×#1044
commutator, socleC2×C52C24×C4
FittingC52×H(5,𝔽2)C4
prime radicalH(5,𝔽2)×C5⋊D5C2
cosocleC2×C5⋊D5C25
hypercenterH(5,𝔽2)#1044
hypocenterC52C4×H(5,𝔽2)
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. #13107775C2semidirect
3. #41×#1044C2semidirect
4. #13107767C2semidirect
5. C2×D4×#1044C2semidirect
6. C23×#1044C22semidirect
7. C52×H(5,𝔽2)C4semidirect
8. #11717522C4semidirect
10. #11717517C4semidirect
12. C2×C22⋊#1044C22semidirect
13. Q8×#1044C22semidirect
14. Q8×#1044C22semidirect
15. D4×#1044C22semidirect
16. D4×#1044C22semidirect
17. D4×#1044C22semidirect
18. #11717451C22semidirect
19. C2×C4⋊#1044C22semidirect
20. C2×C4×#1044C22semidirect
27. C52×#41C2×C4semidirect
28. C2×C52×D4C2×C4semidirect
29. C2×C22⋊(C5⋊D5)C2×C4semidirect
30. #87904C2×C4semidirect
32. #87902C2×C4semidirect
33. D4×C5⋊D5C2×C4semidirect
35. #87900C2×C4semidirect
36. C2×C4⋊(C5⋊D5)C2×C4semidirect
49. C23×C52C22×C4semidirect
52. C22×C5⋊D5C22×C4semidirect
60. C52×Q8C22×C4semidirect
61. C52×D4C22×C4semidirect
62. C2×C4×C52C22×C4semidirect
63. C22⋊(C5⋊D5)C22×C4semidirect
64. C2×#1039C22×C4semidirect
65. C4⋊(C5⋊D5)C22×C4semidirect
66. C4×C5⋊D5C22×C4semidirect
68. #6045C22×C4semidirect
71. C5×H(5,𝔽2)AGL(1,𝔽5)semidirect
99. #1044H(5,𝔽2)direct (N × H)
109. C5×#41C2×AGL(1,𝔽5)semidirect
111. C2×C5×D4C2×AGL(1,𝔽5)semidirect
146. C23×C5C22×AGL(1,𝔽5)semidirect
162. C5×Q8C22×AGL(1,𝔽5)semidirect
164. C5×D4C22×AGL(1,𝔽5)semidirect
166. C2×C4×C5C22×AGL(1,𝔽5)semidirect
181. H(5,𝔽2)#1044direct (N × H)
194. C52C4×H(5,𝔽2)semidirect
220. #41C2×#1044semidirect
226. C2×D4C2×#1044semidirect
251. C23C22×#1044semidirect
253. Q8C22×#1044semidirect
256. D4C22×#1044semidirect
266. C2×C4C22×#1044semidirect
268. C5AGL(1,𝔽5)×H(5,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
H(5,𝔽2)×C5⋊D52
#131077752
#41×#10442
#131077672
C2×D4×#10442
AGL(1,𝔽5)×H(5,𝔽2)5