Finite Group Databasebeta

← #13105205

Subgroups — #13105205

Order 1600.

624 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C22⋊C4×C5⋊D5C211yesmaximal, characteristic
2C5×#4025C212yesmaximal, complemented
3#11717092C211yesmaximal, characteristic
4#11717092C211yesmaximal, characteristic
5#11717074C212yesmaximal, complemented
6#11717018C211yesmaximal, characteristic, complemented
7#11717001C211yesmaximal, characteristic, complemented
8#11717001C211yesmaximal, characteristic, complemented
9C4×#6036C211yesmaximal, characteristic, complemented
10#11716987C212yesmaximal, complemented
11#11716966C212yesmaximal, complemented
12C23×C5⋊D5C2211yescharacteristic
13C2×C4×C5⋊D5C2211yescharacteristic
14C2×C4×C5⋊D5C2211yescharacteristic
15C2×#6036C2211yescharacteristic
16C2×#6036C2211yescharacteristic
17C2×#6036C2211yescharacteristic
18C2×#6036C2211yescharacteristic
19C52×C22⋊C4C2211yescharacteristic
20C22⋊#1039C2211yescharacteristic
21#87808C2211yescharacteristic
22#87808C2211yescharacteristic
23C5×C22⋊Dic5C2212yes—
24C5×C22⋊Dic5C2212yes—
25C5×C4⋊Dic5C2212yescomplemented
26C5×#683C2212yescomplemented
27C5×#683C2212yescomplemented
28C4×C5×Dic5C2212yescomplemented
29#87781C2212yescomplemented
30#87780C2212yescomplemented
31#87780C2212yescomplemented
32#87779C2212yes—
33#87779C2212yes—
34Dic52C2212yescomplemented
35D5×#126—52nomaximal
36C23×C52C2311yescharacteristic
37C22×C5⋊D5—22no—
38C22×C5⋊D5—22no—
39C22×C5⋊D5—22no—
40C22×C5⋊D5C2311yescharacteristic
41C22×C5⋊D5C2311yescharacteristic
42C2×C4×C52C2311yescharacteristic
43C2×C4×C52C2311yescharacteristic
44C2×#1039C2311yescharacteristic
45C2×#1039C2311yescharacteristic
46C4×C5⋊D5—22no—
47C4×C5⋊D5—22no—
48C2×C5×Dic5C2312yes—
49C2×C5×Dic5C2312yes—
50C2×C5×Dic5C2312yes—
51C2×C5×Dic5C2312yes—
52#6036—22no—
53#6036—22no—
54#6036—22no—
55#6036—22no—
56C5×#126—52no—
57#4044—52no—
58#4044—52no—
59#4044—52no—
60D5×C4⋊C4—52no—
61D5×C4⋊C4—52no—
62D5×C4⋊C4—52no—
63D5×C22⋊C4—52no—
64D5×C22⋊C4—52no—
65D5×C22⋊C4—52no—
66D5×C22⋊C4—102no—
67#4025—52no—
68#4025—52no—
69#4025D512yescomplemented
70#4022—52no—
71C42×D5—52no—
72C22×C52—21no—
73C22×C52—21no—
74C22×C52—21no—
75C22×C52C2411yescharacteristic
76C2×C5⋊D5—21no—
77C2×C5⋊D5—21no—
78C2×C5⋊D5—21no—
79C2×C5⋊D5—44no—
80C2×C5⋊D5#4112yes—
81C2×C5⋊D5#4112yes—
82C2×C5⋊D5#4112yes—
83C4×C52—21no—
84C4×C52—21no—
85#1039—21no—
86#1039—21no—
87C5×Dic5—22no—
88C5×Dic5—22no—
89C5×Dic5—22no—
90C5×Dic5—22no—
91C23×D5—52no—
92C23×D5—102no—
93C2×C4×D5—52no—
94C2×C4×D5—52no—
95C2×C4×D5—52no—
96C2×C4×D5—52no—
97C2×C4×D5—52no—
98C2×C4×D5—52no—
99C2×C4×D5—102no—
100C2×C4×D5—102no—
101C5×C4⋊C4—52no—
102C5×C4⋊C4—52no—
103C5×C4⋊C4—52no—
104C5×C22⋊C4—22no—
105C5×C22⋊C4—52no—
106C5×C22⋊C4—52no—
107C5×C22⋊C4C2×D512yes—
108C42×C5—52no—
109C22⋊Dic5—52no—
110C22⋊Dic5—102no—
111C22⋊Dic5C2×D512yes—
112C22⋊Dic5C2×D512yes—
113#685—52no—
114#685—52no—
115#685—52no—
116#685—52no—
117#685—52no—
118#685—52no—
119#685—102no—
120#685—102no—
121C4⋊Dic5—52no—
122C4⋊Dic5—52no—
123C4⋊Dic5C2×D512yescomplemented
124#683—52no—
125#683—52no—
126#683—52no—
127#683—52no—
128#683C2×D512yescomplemented
129#683C2×D512yescomplemented
130C4×Dic5—52no—
131C4×Dic5—52no—
132C4×Dic5C2×D512yescomplemented
133C2×#126—251noSylow
134C2×C52—41no—
135C2×C52C2×#4111yescharacteristic
136C2×C52C2×#4111yescharacteristic
137C2×C52C2×#4111yescharacteristic
138C5⋊D5—41no—
139C5⋊D5#12614yescomplemented
140C23×C5—22no—
141C23×C5C22×D512yes—
142C22×D5—52no—
143C22×D5—52no—
144C22×D5—102no—
145C22×D5—102no—
146C22×D5—104no—
147C22×D5—104no—
148C22×D5—104no—
149C22×D5—108no—
150C22×D5—204no—
151C22×D5—204no—
152C2×C4×C5—22no—
153C2×C4×C5—22no—
154C2×C4×C5—52no—
155C2×C4×C5—52no—
156C2×C4×C5—52no—
157C2×C4×C5—52no—
158C2×C4×C5C22×D512yes—
159C2×C4×C5C22×D512yes—
160C2×Dic5—52no—
161C2×Dic5—52no—
162C2×Dic5—102no—
163C2×Dic5—102no—
164C2×Dic5C22×D512yes—
165C2×Dic5C22×D512yes—
166C2×Dic5C22×D512yes—
167C2×Dic5C22×D512yes—
168C4×D5—104no—
169C4×D5—104no—
170C4×D5—104no—
171C4×D5—104no—
172C4×D5—104no—
173C4×D5—104no—
174C4×D5—204no—
175C4×D5—204no—
176#126—52no—
177#126—252no—
178#126—252no—
179#126—252no—
180C2×C4⋊C4—251no—
181C2×C4⋊C4—251no—
182C2×C4⋊C4—251no—
183C2×C22⋊C4—251no—
184C2×C22⋊C4—251no—
185C2×C22⋊C4—251no—
186C2×C42—251no—
187C52C2×#12611yescharacteristic, complemented, Sylow
188C22×C5—24no—
189C22×C5—22no—
190C22×C5—22no—
191C22×C5—22no—
192C22×C5—22no—
193C22×C5—42no—
194C22×C5—42no—
195C22×C5C23×D512yes—
196C2×D5—54no—
197C2×D5—54no—
198C2×D5—54no—
199C2×D5—104no—
200C2×D5—104no—
201C2×D5—104no—
202C2×D5—104no—
203C2×D5—102no—
204C2×D5—102no—
205C2×D5—102no—
206C2×D5—202no—
207C2×D5—202no—
208C2×D5—208no—
209C2×D5—2016no—
210C4×C5—22no—
211C4×C5—22no—
212C4×C5—42no—
213C4×C5—42no—
214C4×C5—102no—
215C4×C5—102no—
216C4×C5—102no—
217C4×C5—102no—
218Dic5—22no—
219Dic5—22no—
220Dic5—22no—
221Dic5—22no—
222Dic5—102no—
223Dic5—102no—
224Dic5—202no—
225Dic5—202no—
226C24—251no—
227C22×C4—251no—
228C22×C4—251no—
229C22×C4—251no—
230C22×C4—251no—
231C22×C4—251no—
232C22×C4—251no—
233C4⋊C4—52no—
234C4⋊C4—52no—
235C4⋊C4—52no—
236C4⋊C4—252no—
237C4⋊C4—252no—
238C4⋊C4—252no—
239C22⋊C4—52no—
240C22⋊C4—52no—
241C22⋊C4—252no—
242C22⋊C4—252no—
243C22⋊C4—251no—
244C22⋊C4—251no—
245C22⋊C4—251no—
246C22⋊C4D5211yescharacteristic
247C42—52no—
248C42—252no—
249C2×C5—22no—
250C2×C5—22no—
251C2×C5—22no—
252C2×C5—44no—
253C2×C5—42no—
254C2×C5D5×#4112yes—
255C2×C5D5×#4112yes—
256C2×C5D5×#4112yes—
257D5—58no—
258D5—108no—
259D5—204no—
260D5—202no—
261C23—251no—
262C23—251no—
263C23—502no—
264C23—502no—
265C23—502no—
266C23C2×D5211yescharacteristic
267C2×C4—52no—
268C2×C4—52no—
269C2×C4—52no—
270C2×C4—52no—
271C2×C4—251no—
272C2×C4—251no—
273C2×C4—502no—
274C2×C4—502no—
275C2×C4—502no—
276C2×C4—502no—
277C2×C4—502no—
278C2×C4—502no—
279C2×C4C2×D5211yescharacteristic
280C2×C4C2×D5211yescharacteristic
281C5—22no—
282C5D5×#12612yescomplemented
283C22—21no—
284C22—21no—
285C22—21no—
286C22—252no—
287C22—252no—
288C22—252no—
289C22—501no—
290C22—501no—
291C22—501no—
292C22—1004no—
293C22C22×D5211yescharacteristic
294C4—21no—
295C4—21no—
296C4—102no—
297C4—102no—
298C4—102no—
299C4—102no—
300C4—501no—
301C4—501no—
302C2—41no—
303C2—254no—
304C2—1001no—
305C2#1171747111yescharacteristic
306C2#1171747711yescharacteristic
307C2#1171747711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radicalC23×C5⋊D5C22
hypercenterC22⋊C4D52
hypocenterC52C2×#126
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#4025C2semidirect
5. #11717074C2semidirect
6. #11717018C2semidirect
7. #11717001C2semidirect
8. #11717001C2semidirect
9. C4×#6036C2semidirect
10. #11716987C2semidirect
11. #11716966C2semidirect
25. C5×C4⋊Dic5C22semidirect
26. C5×#683C22semidirect
27. C5×#683C22semidirect
28. C4×C5×Dic5C22semidirect
29. #87781C22semidirect
30. #87780C22semidirect
31. #87780C22semidirect
34. Dic52C22semidirect
69. #4025D5semidirect
123. C4⋊Dic5C2×D5semidirect
128. #683C2×D5semidirect
129. #683C2×D5semidirect
132. C4×Dic5C2×D5semidirect
139. C5⋊D5#126semidirect
187. C52C2×#126semidirect
282. C5D5×#126semidirect

Maximal subgroups

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

subgroupindex
C22⋊C4×C5⋊D52
C5×#40252
#117170922
#117170742
#117170182
#117170012
C4×#60362
#117169872
#117169662
D5×#1265