Finite Group Databasebeta

← #72223237

Subgroups — #72223237

Order 3200.

3488 conjugacy classes of subgroups; only the 394 normal subgroups are shown.

normal subgroups only

This group has 3488 conjugacy classes of subgroups — too many to draw. Only its 394 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×#41×C5⋊D5C211yesmaximal, characteristic
2C2×Q8×#1043C212yesmaximal, complemented
3#13104792C216yesmaximal, complemented
4#13104777C213yesmaximal, complemented
5C4×C22⋊#1043C213yesmaximal, complemented
6C2×C52×#41C411yescharacteristic, complemented
7#41×C5⋊D5—28no—
8C2×#87904C411yescharacteristic, complemented
9C2×Q8×C5⋊D5C2211yescharacteristic
10C2×#87902C413yescomplemented
11C2×D4×C5⋊D5C2213yes—
12C2×#87900C413yescomplemented
13C22×C4×C5⋊D5C2213yes—
14C2×C22⋊#1043C2216yes—
15C2×C22⋊#1043C2216yes—
16Q8×#1043—28no—
17C2×C4⋊#1043C2216yes—
18C2×C4×#1043C2216yes—
19#11716874C2211yescharacteristic
20#11716874C2213yes—
21#10617558—56nomaximal
22C23×C5⋊D5C2313yes—
23C22×#1043C2316yes—
24C22×#1043C2312yes—
25C52×#41—24no—
26C2×C52×Q8C2×C411yescharacteristic
27C2×C52×D4C2×C413yes—
28C22×C4×C52C2×C413yes—
29C2×C22⋊(C5⋊D5)C2×C416yes—
30C22×#1039C2×C413yes—
31#87904—24no—
32Q8×C5⋊D5D418yes—
33#87902—212no—
34D4×C5⋊D5—212no—
35#87900—212no—
36C2×C4⋊(C5⋊D5)C2×C413yes—
37C2×C4×C5⋊D5—212no—
38C2×C4×C5⋊D5C2×C413yes—
39C2×C4×C5⋊D5C2×C413yes—
40C2×C4×C5⋊D5C2311yescharacteristic
41C2×C4×C5⋊D5C2313yes—
42C2×#6045C2×C413yes—
43C22⋊#1043—212no—
44C22⋊#1043—46no—
45C4⋊#1043—212no—
46C4×#1043—212no—
47C2×D5×#41—56no—
48C2×Q8×AGL(1,𝔽5)—512no—
49#65780—536no—
50#65765—518no—
51C4×C22⋊AGL(1,𝔽5)—518no—
52C23×C52C22×C413yes—
53C22×C5⋊D5—26no—
54C22×C5⋊D5—26no—
55C22×C5⋊D5—23no—
56C22×C5⋊D5—23no—
57C22×C5⋊D5C22×C413yes—
58C22×C5⋊D5C2411yescharacteristic
59C2×#1043—22no—
60C2×#1043—22no—
61C2×#1043—26no—
62C2×#1043—26no—
63C2×#1043—26no—
64C2×#1043#4114yescomplemented
65C52×Q8C22⋊C414yescomplemented
66C52×D4—26no—
67C2×C4×C52—26no—
68C2×C4×C52C22×C411yescharacteristic
69C2×C4×C52C22×C413yes—
70C22⋊(C5⋊D5)—212no—
71C2×#1039—26no—
72C2×#1039C22×C411yescharacteristic
73C2×#1039C22×C413yes—
74C4⋊(C5⋊D5)—26no—
75C4×C5⋊D5—22no—
76C4×C5⋊D5—26no—
77C4×C5⋊D5—26no—
78C4×C5⋊D5C2×D4112yes—
79#6045C22⋊C4112yescomplemented
80C2×C5×#41AGL(1,𝔽5)16yescomplemented
81D5×#41—1048no—
82C2×#713—56no—
83C2×Q8×D5—56no—
84C2×#711—518no—
85C2×D4×D5—518no—
86C2×#709—518no—
87C22×C4×D5—518no—
88C2×C22⋊AGL(1,𝔽5)—536no—
89C2×C22⋊AGL(1,𝔽5)—536no—
90Q8×AGL(1,𝔽5)—1048no—
91C2×C4⋊AGL(1,𝔽5)—536no—
92C2×C4×AGL(1,𝔽5)—536no—
93(C2×AGL(1,𝔽5))⋊C4—56no—
94(C2×AGL(1,𝔽5))⋊C4—518no—
95#2341—251noSylow
96C22×C52—23no—
97C22×C52—23no—
98C22×C52—23no—
99C22×C52C23×C411yescharacteristic
100C2×C5⋊D5—23no—
101C2×C5⋊D5—23no—
102C2×C5⋊D5—23no—
103C2×C5⋊D5—46no—
104C2×C5⋊D5—46no—
105C2×C5⋊D5C22×D412yes—
106C2×C5⋊D5C2×#4112yes—
107C2×C5⋊D5H(5,𝔽2)11yescharacteristic
108C2×C5⋊D52−1+411yescharacteristic
109#1043—24no—
110#1043—46no—
111C4×C52—21no—
112C4×C52C2×C22⋊C416yes—
113#1039—21no—
114#1039C2×C22⋊C416yes—
115C23×D5—518no—
116C22×AGL(1,𝔽5)—512no—
117C22×AGL(1,𝔽5)—536no—
118C5×#41—224no—
119C2×C5×Q8C2×AGL(1,𝔽5)16yes—
120C2×C5×D4C2×AGL(1,𝔽5)118yes—
121C22×C4×C5C2×AGL(1,𝔽5)118yes—
122C2×C22⋊D5—536no—
123C22×Dic5—518no—
124#713—1024no—
125Q8×D5—548no—
126#711—1072no—
127D4×D5—1072no—
128#709—1072no—
129C2×D20—518no—
130C2×C4×D5—518no—
131C2×C4×D5—56no—
132C2×C4×D5—518no—
133C2×C4×D5—518no—
134C2×C4×D5—1072no—
135C2×Dic10—518no—
136C22⋊AGL(1,𝔽5)—1072no—
137C22⋊AGL(1,𝔽5)—2036no—
138C4⋊AGL(1,𝔽5)—1072no—
139C4×AGL(1,𝔽5)—1072no—
140C22×#41—251no—
141C2×C4×Q8—252no—
142#390—256no—
143#380—253no—
144C4×C22⋊C4—253no—
145C2×C52—43no—
146C2×C52C22×C22⋊C411yescharacteristic
147C2×C52C4×#4111yescharacteristic
148C2×C52#52011yescharacteristic
149C5⋊D5—43no—
150C5⋊D5#54812yes—
151C5⋊D5#55012yes—
152C23×C5C22×AGL(1,𝔽5)118yes—
153C22×D5—56no—
154C22×D5—518no—
155C22×D5—1036no—
156C22×D5—1018no—
157C22×D5—1036no—
158C22×D5—1018no—
159C2×AGL(1,𝔽5)—524no—
160C2×AGL(1,𝔽5)—1012no—
161C2×AGL(1,𝔽5)—1012no—
162C2×AGL(1,𝔽5)—1036no—
163C2×AGL(1,𝔽5)—1036no—
164C2×AGL(1,𝔽5)—1036no—
165C5×Q8C22⋊AGL(1,𝔽5)124yescomplemented
166C5×D4—236no—
167C2×C4×C5—236no—
168C2×C4×C5C22×AGL(1,𝔽5)16yes—
169C2×C4×C5C22×AGL(1,𝔽5)118yes—
170C22⋊D5—1072no—
171C2×Dic5—518no—
172C2×Dic5—56no—
173C2×Dic5—1036no—
174D20—1036no—
175C4×D5—572no—
176C4×D5—1036no—
177C4×D5—1012no—
178C4×D5—1036no—
179Dic10—572no—
180C2×#41—251no—
181C2×#41—253no—
182C2×#41—253no—
183C2×#41—508no—
184C2×#41#104311yescharacteristic, complemented
185C22×Q8—251no—
186C22×D4—253no—
187C23×C4—253no—
188C4×Q8—508no—
189C2×C4⋊C4—256no—
190C2×C22⋊C4—256no—
191C2×C22⋊C4—256no—
192C2×C42—256no—
193#95—251no—
194#95—253no—
195C52#234111yescharacteristic, complemented, Sylow
196C22×C5—218no—
197C22×C5—218no—
198C22×C5—218no—
199C22×C5C23×AGL(1,𝔽5)16yes—
200C2×D5—512no—
201C2×D5—56no—
202C2×D5—56no—
203C2×D5—512no—
204C2×D5—1018no—
205C2×D5—1018no—
206C2×D5—1018no—
207C2×D5—2036no—
208C2×D5—2036no—
209AGL(1,𝔽5)—1024no—
210AGL(1,𝔽5)—2036no—
211C4×C5—26no—
212C4×C5C2×C22⋊AGL(1,𝔽5)136yes—
213Dic5—536no—
214Dic5—106no—
215C24—253no—
216#41—24no—
217#41—5012no—
218#41—5012no—
219#41—504no—
220C2×Q8—258no—
221C2×Q8—253no—
222C2×Q8C2×#104311yescharacteristic
223C2×D4—256no—
224C2×D4—253no—
225C2×D4—5012no—
226C2×D4C2×#104313yes—
227C22×C4—253no—
228C22×C4—252no—
229C22×C4—253no—
230C22×C4—251no—
231C22×C4—253no—
232C22×C4—253no—
233C22×C4—256no—
234C22×C4—5012no—
235C22×C4C2×#104313yes—
236C4⋊C4—5012no—
237C22⋊C4—5012no—
238C22⋊C4—1006no—
239C42—5012no—
240C2×C5—418no—
241C2×C5#41×AGL(1,𝔽5)16yes—
242C2×C5#6626616yes—
243C2×C5C22×C22⋊AGL(1,𝔽5)16yes—
244D5—512no—
245D5—512no—
246D5—2018no—
247C23—251no—
248C23—253no—
249C23—506no—
250C23—503no—
251C23—506no—
252C23—503no—
253C23C22×#104313yes—
254Q8—2512no—
255Q8C22⋊#104314yescomplemented
256D4—26no—
257D4—5012no—
258D4—506no—
259C2×C4—26no—
260C2×C4—253no—
261C2×C4—2512no—
262C2×C4—254no—
263C2×C4—251no—
264C2×C4—506no—
265C2×C4—502no—
266C2×C4—506no—
267C2×C4—502no—
268C2×C4—502no—
269C2×C4—506no—
270C2×C4—506no—
271C2×C4—506no—
272C2×C4—506no—
273C2×C4C22×#104311yescharacteristic
274C2×C4C22×#104313yes—
275C5#1061755816yescomplemented
276C22—23no—
277C22—23no—
278C22—23no—
279C22—252no—
280C22—251no—
281C22—251no—
282C22—252no—
283C22—503no—
284C22—503no—
285C22—503no—
286C22—1006no—
287C22—1006no—
288C22C23×#104311yescharacteristic
289C4—21no—
290C4—256no—
291C4—504no—
292C4—501no—
293C4—1006no—
294C4C2×C22⋊#104316yes—
295C2—43no—
296C2—252no—
297C2—252no—
298C2—1003no—
299C2#41×#104311yescharacteristic
300C2#1310775411yescharacteristic
301C2C22×C22⋊#104311yescharacteristic
subgroupG/H
center, FrattiniC22C23×#1043
commutator, socleC22×C52C23×C4
FittingC2×C52×#41C4
prime radicalC2×#41×C5⋊D5C2
cosocleC22×C5⋊D5C24
hypercenterC2×#41#1043
hypocenterC52#2341
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#1043C2semidirect
3. #13104792C2semidirect
4. #13104777C2semidirect
5. C4×C22⋊#1043C2semidirect
6. C2×C52×#41C4semidirect
8. C2×#87904C4semidirect
10. C2×#87902C4semidirect
12. C2×#87900C4semidirect
64. C2×#1043#41semidirect
65. C52×Q8C22⋊C4semidirect
79. #6045C22⋊C4semidirect
80. C2×C5×#41AGL(1,𝔽5)semidirect
165. C5×Q8C22⋊AGL(1,𝔽5)semidirect
184. C2×#41#1043semidirect
195. C52#2341semidirect
255. Q8C22⋊#1043semidirect
275. C5#10617558semidirect

Maximal subgroups

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

subgroupindex
C2×#41×C5⋊D52
C2×Q8×#10432
#131047922
#131047772
C4×C22⋊#10432
#106175585