Finite Group Databasebeta

← #72221574

Subgroups — #72221574

Order 3200.

1362 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1362 conjugacy classes of subgroups — too many to draw. Only its 251 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×#11717355C212yesmaximal
2#13104978C211yesmaximal, characteristic
3C42×#1044C211yesmaximal, characteristic
4#13104700C211yesmaximal, characteristic
5#13104006C214yesmaximal
6#13103985C214yesmaximal
7#13103883C212yesmaximal
8C2×#87872C2212yes—
9C2×C4×#1044C2211yescharacteristic
10C2×C4×#1044C2212yes—
11C2×C4×#1044C2211yescharacteristic
12C2×C4×#1044C2212yes—
13C2×C4⋊#1043C2212yes—
14C2×C4⋊#1043C2214yes—
15C2×#87832C2214yes—
16C2×#87830C2214yes—
17#11717355—28no—
18C42×C5⋊D5C2211yescharacteristic
19C5×#4015C412yescomplemented
20#11717037C414yescomplemented
21#11717018C2214yes—
22#11716985C412yescomplemented
23#11716776C2218yes—
24AGL(1,𝔽5)×C4⋊Q8—52nomaximal
25C22×#1044C2311yescharacteristic
26C22×#1044C2312yes—
27C22×#1044C2311yescharacteristic
28C22×#1043C2314yes—
29C2×C4×C5⋊D5C2311yescharacteristic
30C2×C4×C5⋊D5C2312yes—
31C2×C5×Dic10C2×C414yes—
32C2×#6039C2×C414yes—
33C2×#6036C2314yes—
34#87872—24no—
35#87872—24no—
36C4×#1044—22no—
37C4×#1044—24no—
38C4×#1044D414yes—
39C4×#1044Q818yescomplemented
40C4⋊#1043—24no—
41C4⋊#1043—28no—
42#87832—28no—
43#87830—28no—
44C42×C52C2×C411yescharacteristic
45C4×#1039C2×C411yescharacteristic
46C4×#1039C2×C412yes—
47C5×C4⋊Dic5C2×C418yes—
48#87781C2×C418yes—
49C2×Q8×AGL(1,𝔽5)—54no—
50D5×C4⋊Q8—52no—
51(C2×AGL(1,𝔽5))⋊Q8—54no—
52#65712—58no—
53C4⋊C4×AGL(1,𝔽5)—58no—
54#65688—52no—
55#65688—102no—
56C42×AGL(1,𝔽5)—52no—
57C22×C5⋊D5C2411yescharacteristic
58C2×#1044—22no—
59C2×#1044—22no—
60C2×#1044—22no—
61C2×#1044C2×D412yes—
62C2×#1044C2×D412yes—
63C2×#1044C2×Q814yes—
64C2×#1044C2×Q814yes—
65C2×#1043—24no—
66C2×#1043—28no—
67C2×C4×C52C22×C411yescharacteristic
68C2×C4×C52C22×C412yes—
69C2×#1039C22×C411yescharacteristic
70C2×#1039C22×C412yes—
71C4×C5⋊D5C2×D412yes—
72C4×C5⋊D5C2×Q814yes—
73C4×C5⋊D5#4112yes—
74C4×C5⋊D5#4114yes—
75C2×C5×Dic5C22×C418yes—
76C5×Dic10—28no—
77#6039—28no—
78#6036—24no—
79#6036—24no—
80C2×Q8×D5—54no—
81Q8×AGL(1,𝔽5)—1016no—
82C2×C4⋊AGL(1,𝔽5)—58no—
83C2×C4⋊AGL(1,𝔽5)—54no—
84C2×C4⋊AGL(1,𝔽5)—58no—
85C2×C4⋊AGL(1,𝔽5)—104no—
86C2×C4⋊AGL(1,𝔽5)—108no—
87C2×C4×AGL(1,𝔽5)—54no—
88C2×C4×AGL(1,𝔽5)—52no—
89C2×C4×AGL(1,𝔽5)—54no—
90C2×C4×AGL(1,𝔽5)—52no—
91C2×C4×AGL(1,𝔽5)—58no—
92C5×C4⋊Q8—52no—
93#4090—54no—
94D5×C4⋊C4—58no—
95#4034—58no—
96C42×D5—52no—
97C42×D5—102no—
98#4015AGL(1,𝔽5)12yescomplemented
99(C2×AGL(1,𝔽5))⋊C4—516no—
100C4×C4⋊Q8—251noSylow
101C22×C52C23×C411yescharacteristic
102C2×C5⋊D5C22×D411yescharacteristic
103C2×C5⋊D5C22×Q812yes—
104C2×C5⋊D5C2×#4112yes—
105C2×C5⋊D5C2×#4111yescharacteristic
106#1044—22no—
107#1044—24no—
108#1044C4⋊Q814yescomplemented
109#1043—44no—
110C4×C52C4×D412yes—
111C4×C52C4×Q814yescomplemented
112#1039C4×D412yes—
113#1039C4×Q814yescomplemented
114C5×Dic5—28no—
115C22×AGL(1,𝔽5)—52no—
116C22×AGL(1,𝔽5)—52no—
117C22×AGL(1,𝔽5)—54no—
118C22×AGL(1,𝔽5)—58no—
119C22×AGL(1,𝔽5)—108no—
120C2×C5×Q8—54no—
121Q8×D5—108no—
122Q8×D5—108no—
123C2×C4×D5—52no—
124C2×C4×D5—54no—
125C2×C4×D5—58no—
126C2×C4×D5—104no—
127C2×C4×D5—102no—
128C2×Dic10—58no—
129C2×Dic10C2×AGL(1,𝔽5)14yes—
130C4⋊AGL(1,𝔽5)—108no—
131C4⋊AGL(1,𝔽5)—1016no—
132C4⋊AGL(1,𝔽5)—1016no—
133C4⋊AGL(1,𝔽5)—208no—
134C4⋊AGL(1,𝔽5)—2016no—
135C4×AGL(1,𝔽5)—58no—
136C4×AGL(1,𝔽5)—516no—
137C4×AGL(1,𝔽5)—104no—
138C4×AGL(1,𝔽5)—108no—
139C4×AGL(1,𝔽5)—1016no—
140C5×C4⋊C4—58no—
141C42×C5—22no—
142C42×C5C2×AGL(1,𝔽5)12yes—
143C4⋊Dic5C2×AGL(1,𝔽5)18yes—
144#683—516no—
145C4×Dic5—54no—
146C4×Dic5—52no—
147C4×Dic5—102no—
148C4×Dic5—104no—
149C2×C4⋊Q8—251no—
150C2×C4×Q8—252no—
151#391—252no—
152#389—254no—
153#384—251no—
154C4×C4⋊C4—254no—
155C43—251no—
156C2×C52C2×C4×D411yescharacteristic
157C2×C52C2×C4×Q812yes—
158C5⋊D5C2×C4⋊Q811yescharacteristic
159C5⋊D5#53211yescharacteristic
160C5⋊D5#53312yes—
161C22×D5—52no—
162C22×D5—102no—
163C2×AGL(1,𝔽5)—54no—
164C2×AGL(1,𝔽5)—58no—
165C2×AGL(1,𝔽5)—54no—
166C2×AGL(1,𝔽5)—58no—
167C2×AGL(1,𝔽5)—104no—
168C2×AGL(1,𝔽5)—104no—
169C2×AGL(1,𝔽5)—104no—
170C2×AGL(1,𝔽5)—108no—
171C2×AGL(1,𝔽5)—1016no—
172C2×AGL(1,𝔽5)—208no—
173C2×AGL(1,𝔽5)—2016no—
174C5×Q8—108no—
175C2×C4×C5—24no—
176C2×C4×C5—22no—
177C2×C4×C5—58no—
178C2×C4×C5C22×AGL(1,𝔽5)12yes—
179C2×C4×C5C22×AGL(1,𝔽5)14yes—
180C2×Dic5—54no—
181C2×Dic5—52no—
182C2×Dic5—104no—
183C2×Dic5—102no—
184C2×Dic5C22×AGL(1,𝔽5)18yes—
185C4×D5—54no—
186C4×D5—54no—
187C4×D5—58no—
188C4×D5—58no—
189C4×D5—104no—
190C4×D5—108no—
191C4×D5—108no—
192C4×D5—104no—
193C4×D5—108no—
194C4×D5—108no—
195Dic10—28no—
196Dic10—1016no—
197C22×Q8—252no—
198C4⋊Q8—52no—
199C4⋊Q8—254no—
200C4⋊Q8—252no—
201C4×Q8—508no—
202C2×C4⋊C4—254no—
203C2×C4⋊C4—252no—
204C2×C4⋊C4—254no—
205C2×C4⋊C4—254no—
206C2×C42—251no—
207C2×C42—252no—
208C2×C42—251no—
209C2×C42—254no—
210C2×C42—252no—
211C2×C42—251no—
212#95—258no—
213C52C4×C4⋊Q811yescharacteristic, complemented, Sylow
214C22×C5—22no—
215C22×C5C23×AGL(1,𝔽5)12yes—
216C2×D5—52no—
217C2×D5—52no—
218C2×D5—54no—
219C2×D5—54no—
220C2×D5—102no—
221C2×D5—104no—
222C2×D5—104no—
223C2×D5—102no—
224AGL(1,𝔽5)—58no—
225AGL(1,𝔽5)—104no—
226AGL(1,𝔽5)—108no—
227AGL(1,𝔽5)—208no—
228AGL(1,𝔽5)—408no—
229C4×C5—28no—
230C4×C5—24no—
231C4×C5—108no—
232C4×C5D4×AGL(1,𝔽5)14yes—
233C4×C5Q8×AGL(1,𝔽5)18yescomplemented
234Dic5—28no—
235Dic5—54no—
236Dic5—58no—
237Dic5—104no—
238Dic5—108no—
239C2×Q8—54no—
240C2×Q8—254no—
241C2×Q8—504no—
242C2×Q8—504no—
243C22×C4—254no—
244C22×C4—251no—
245C22×C4—252no—
246C22×C4—251no—
247C22×C4—252no—
248C22×C4—251no—
249C22×C4—254no—
250C4⋊C4—58no—
251C4⋊C4—258no—
252C4⋊C4—508no—
253C4⋊C4—504no—
254C4⋊C4—508no—
255C42—254no—
256C42—258no—
257C42—251no—
258C42—252no—
259C42—508no—
260C42—504no—
261C42—502no—
262C42D5⋊AGL(1,𝔽5)11yescharacteristic
263C2×C5—24no—
264C2×C5—22no—
265C2×C5C2×D4×AGL(1,𝔽5)12yes—
266C2×C5C2×Q8×AGL(1,𝔽5)14yes—
267D5—52no—
268D5—52no—
269D5—54no—
270D5—102no—
271D5—102no—
272D5—104no—
273C23—251no—
274Q8—108no—
275Q8—508no—
276C2×C4—58no—
277C2×C4—252no—
278C2×C4—254no—
279C2×C4—252no—
280C2×C4—251no—
281C2×C4—254no—
282C2×C4—254no—
283C2×C4—252no—
284C2×C4—252no—
285C2×C4—252no—
286C2×C4—254no—
287C2×C4—504no—
288C2×C4—504no—
289C2×C4—502no—
290C2×C4—502no—
291C2×C4—502no—
292C2×C4—504no—
293C2×C4—508no—
294C2×C4C2×D5⋊AGL(1,𝔽5)12yes—
295C2×C4C2×D5⋊AGL(1,𝔽5)11yescharacteristic
296C5—22no—
297C5AGL(1,𝔽5)×C4⋊Q812yescomplemented
298C22—252no—
299C22—252no—
300C22—251no—
301C22—251no—
302C22C22×D5⋊AGL(1,𝔽5)11yescharacteristic
303C4—108no—
304C4—254no—
305C4—254no—
306C4—252no—
307C4—502no—
308C4—504no—
309C4—1004no—
310C4#1171735514yescomplemented
311C4#1171736512yes—
312C2—251no—
313C2—251no—
314C2—252no—
315C2C2×#1171735512yes—
316C2C2×#1171736511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5⋊AGL(1,𝔽5)
commutator, socleC22×C52C23×C4
FittingC42×C52C2×C4
prime radical, cosocleC22×C5⋊D5C24
hypercenterC42D5⋊AGL(1,𝔽5)
hypocenterC52C4×C4⋊Q8
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
19. C5×#4015C4semidirect
20. #11717037C4semidirect
22. #11716985C4semidirect
39. C4×#1044Q8semidirect
98. #4015AGL(1,𝔽5)semidirect
108. #1044C4⋊Q8semidirect
111. C4×C52C4×Q8semidirect
113. #1039C4×Q8semidirect
213. C52C4×C4⋊Q8semidirect
233. C4×C5Q8×AGL(1,𝔽5)semidirect
297. C5AGL(1,𝔽5)×C4⋊Q8semidirect
310. C4#11717355semidirect

Maximal subgroups

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

subgroupindex
C2×#117173552
#131049782
C42×#10442
#131047002
#131040062
#131039852
#131038832
AGL(1,𝔽5)×C4⋊Q85