Finite Group Databasebeta

← #13103630

Subgroups — #13103630

Order 1600.

353 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 94 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×C5×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
2C2×#87825C211yesmaximal, characteristic
3C2×#87825C211yesmaximal, characteristic
4C2×Dic5×AGL(1,𝔽5)C211yesmaximal, characteristic
5D5×#683C211yesmaximal, characteristic
6#11716849C211yesmaximal, characteristic
7(C2×AGL(1,𝔽5))⋊Dic5C211yesmaximal, characteristic
8C22×#1042C2211yescharacteristic
9C22×#1042C2211yescharacteristic
10C22×C5×AGL(1,𝔽5)C2211yescharacteristic
11C22×C5×AGL(1,𝔽5)C2211yescharacteristic
12C2×C4×C5×D5C2211yescharacteristic
13C2×D5×Dic5C2211yescharacteristic
14C2×D5×Dic5C2211yescharacteristic
15C5×C4⋊AGL(1,𝔽5)—22no—
16#87825—22no—
17#87825—22no—
18Dic5×AGL(1,𝔽5)—22no—
19#87806C411yescharacteristic, complemented
20C5×#683C411yescharacteristic, complemented
21#87781C411yescharacteristic, complemented
22#87780C411yescharacteristic, complemented
23#65712—51nomaximal
24#65261—51nomaximal
25C22×C5×D5C2311yescharacteristic
26C2×#1042—21no—
27C2×#1042—21no—
28C2×#1042—21no—
29C2×#1042—21no—
30C2×#1042—21no—
31C2×#1042Q812yes—
32C2×C5×AGL(1,𝔽5)—21no—
33C2×C5×AGL(1,𝔽5)—21no—
34C2×C5×AGL(1,𝔽5)—21no—
35C2×C5×AGL(1,𝔽5)—21no—
36C2×C5×AGL(1,𝔽5)—21no—
37C2×C5×AGL(1,𝔽5)D412yes—
38C2×C4×C52C2×C411yescharacteristic
39C2×#1039C2×C411yescharacteristic
40C2×#1039C2×C411yescharacteristic
41C2×C5×Dic5C2×C411yescharacteristic
42C2×C5×Dic5C2×C411yescharacteristic
43C2×C5×Dic5C2×C411yescharacteristic
44C4×C5×D5—21no—
45C4×C5×D5—21no—
46D5×Dic5—21no—
47D5×Dic5—21no—
48D5×Dic5D412yes—
49D5×Dic5Q812yes—
50C2×C4⋊AGL(1,𝔽5)—51no—
51C2×C4⋊AGL(1,𝔽5)—51no—
52C2×C4⋊AGL(1,𝔽5)D511yescharacteristic
53C2×C4×AGL(1,𝔽5)—51no—
54C2×C5×C4⋊C4—51no—
55C2×#683—51no—
56C2×#683—51no—
57C2×#683—51no—
58C2×C4×Dic5—51no—
59D5×C4⋊C4—51no—
60(C2×AGL(1,𝔽5))⋊C4—51no—
61(C2×AGL(1,𝔽5))⋊C4—51no—
62#3963—51no—
63#3963—51no—
64C22×C52C22×C411yescharacteristic
65C2×C5×D5C2×D411yescharacteristic
66C2×C5×D5C2×D411yescharacteristic
67C2×C5×D5C2×Q811yescharacteristic
68C2×C5×D5C2×Q811yescharacteristic
69C2×C5×D5#4111yescharacteristic
70C2×C5×D5#4111yescharacteristic
71#1042—22no—
72#1042—41no—
73C5×AGL(1,𝔽5)—22no—
74C5×AGL(1,𝔽5)—41no—
75C4×C52—21no—
76#1039—21no—
77#1039C4⋊C412yescomplemented
78C5×Dic5—21no—
79C5×Dic5—21no—
80C5×Dic5C4⋊C412yescomplemented
81C22×AGL(1,𝔽5)—51no—
82C22×AGL(1,𝔽5)—51no—
83C22×AGL(1,𝔽5)C2×D511yescharacteristic
84C22×AGL(1,𝔽5)C2×D511yescharacteristic
85C22×C4×C5—51no—
86C22×C4×C5—51no—
87C22×C4×C5—51no—
88C22×Dic5—51no—
89C22×Dic5—51no—
90C22×Dic5—51no—
91C22×Dic5—51no—
92C2×C4×D5—51no—
93C2×C4×D5—51no—
94C2×C4×D5C2×D511yescharacteristic
95C4⋊AGL(1,𝔽5)—22no—
96C4⋊AGL(1,𝔽5)—102no—
97C4⋊AGL(1,𝔽5)—102no—
98C4×AGL(1,𝔽5)—102no—
99C5×C4⋊C4—51no—
100C5×C4⋊C4—102no—
101C4⋊Dic5—51no—
102#683—51no—
103#683—51no—
104#683—51no—
105#683—51no—
106#683—51no—
107#683—102no—
108#683—102no—
109#683—201no—
110#683AGL(1,𝔽5)11yescharacteristic, complemented
111C4×Dic5—102no—
112#389—251noSylow
113C2×C52C2×C4⋊C411yescharacteristic
114C2×C52C4×D411yescharacteristic
115C2×C52C4×Q811yescharacteristic
116C5×D5#12111yescharacteristic
117C5×D5C22⋊Q811yescharacteristic
118C5×D5#12511yescharacteristic
119C5×D5C4⋊Q811yescharacteristic
120C23×C5—51no—
121C22×D5C22×D511yescharacteristic
122C2×AGL(1,𝔽5)—21no—
123C2×AGL(1,𝔽5)—21no—
124C2×AGL(1,𝔽5)—21no—
125C2×AGL(1,𝔽5)—21no—
126C2×AGL(1,𝔽5)—21no—
127C2×AGL(1,𝔽5)—52no—
128C2×AGL(1,𝔽5)—101no—
129C2×AGL(1,𝔽5)—101no—
130C2×AGL(1,𝔽5)—101no—
131C2×AGL(1,𝔽5)—101no—
132C2×AGL(1,𝔽5)—101no—
133C2×AGL(1,𝔽5)C22⋊D512yes—
134C2×C4×C5—41no—
135C2×C4×C5—51no—
136C2×C4×C5—52no—
137C2×C4×C5—51no—
138C2×C4×C5—51no—
139C2×C4×C5—101no—
140C2×C4×C5—101no—
141C2×C4×C5—101no—
142C2×C4×C5—101no—
143C2×C4×C5—101no—
144C2×C4×C5—101no—
145C2×C4×C5—101no—
146C2×C4×C5C4×D511yescharacteristic
147C2×C4×C5C2×AGL(1,𝔽5)11yescharacteristic
148C2×Dic5—51no—
149C2×Dic5—51no—
150C2×Dic5—52no—
151C2×Dic5—52no—
152C2×Dic5—52no—
153C2×Dic5—51no—
154C2×Dic5—51no—
155C2×Dic5—101no—
156C2×Dic5—101no—
157C2×Dic5—101no—
158C2×Dic5—101no—
159C2×Dic5—101no—
160C2×Dic5—101no—
161C2×Dic5—101no—
162C2×Dic5—201no—
163C2×Dic5—201no—
164C2×Dic5C4×D511yescharacteristic
165C2×Dic5C2×AGL(1,𝔽5)11yescharacteristic
166C2×Dic5C2×AGL(1,𝔽5)11yescharacteristic
167C4×D5—21no—
168C4×D5—21no—
169C4×D5—52no—
170C4×D5—52no—
171C4×D5—101no—
172C4×D5—101no—
173C2×C4⋊C4—51no—
174C2×C4⋊C4—251no—
175C2×C4⋊C4—251no—
176C2×C4⋊C4—251no—
177C2×C42—251no—
178#95—251no—
179#95—251no—
180C52#38911yescharacteristic, complemented, Sylow
181C22×C5—41no—
182C22×C5—51no—
183C22×C5—51no—
184C22×C5—51no—
185C22×C5—51no—
186C22×C5—51no—
187C22×C5—51no—
188C22×C5C2×C4×D511yescharacteristic
189C22×C5C22×AGL(1,𝔽5)11yescharacteristic
190C2×D5#70911yescharacteristic
191C2×D5D4×D511yescharacteristic
192C2×D5#71111yescharacteristic
193C2×D5Q8×D511yescharacteristic
194C2×D5Q8×D511yescharacteristic
195C2×D5C2×C22⋊D511yescharacteristic
196AGL(1,𝔽5)—22no—
197AGL(1,𝔽5)—41no—
198AGL(1,𝔽5)—102no—
199AGL(1,𝔽5)—201no—
200C4×C5—21no—
201C4×C5—21no—
202C4×C5—52no—
203C4×C5—81no—
204C4×C5—102no—
205C4×C5—101no—
206C4×C5—101no—
207C4×C5—201no—
208Dic5—21no—
209Dic5—21no—
210Dic5—52no—
211Dic5—52no—
212Dic5—102no—
213Dic5—101no—
214Dic5—101no—
215Dic5—202no—
216Dic5—201no—
217Dic5—401no—
218Dic5C4⋊AGL(1,𝔽5)12yescomplemented
219C22×C4—51no—
220C22×C4—51no—
221C22×C4—51no—
222C22×C4—251no—
223C22×C4—251no—
224C22×C4—251no—
225C22×C4—251no—
226C4⋊C4—51no—
227C4⋊C4—102no—
228C4⋊C4—251no—
229C4⋊C4—251no—
230C4⋊C4—251no—
231C4⋊C4—502no—
232C4⋊C4—502no—
233C42—502no—
234C2×C5—41no—
235C2×C5—41no—
236C2×C5—41no—
237C2×C5—51no—
238C2×C5—51no—
239C2×C5—51no—
240C2×C5—51no—
241C2×C5#403311yescharacteristic
242C2×C5D5×C4⋊C411yescharacteristic
243C2×C5C4×C22⋊D511yescharacteristic
244C2×C5C2×C4⋊AGL(1,𝔽5)11yescharacteristic
245C2×C5D4×AGL(1,𝔽5)11yescharacteristic
246C2×C5Q8×AGL(1,𝔽5)11yescharacteristic
247D5#403511yescharacteristic
248D5#404211yescharacteristic
249D5#408511yescharacteristic
250D5#409011yescharacteristic
251C23—51no—
252C2×C4—51no—
253C2×C4—52no—
254C2×C4—51no—
255C2×C4—51no—
256C2×C4—101no—
257C2×C4—101no—
258C2×C4—101no—
259C2×C4—101no—
260C2×C4—101no—
261C2×C4—101no—
262C2×C4—101no—
263C2×C4—252no—
264C2×C4—252no—
265C2×C4—251no—
266C2×C4—252no—
267C2×C4—251no—
268C2×C4—501no—
269C2×C4—501no—
270C2×C4—501no—
271C2×C4—501no—
272C2×C4—501no—
273C2×C4—501no—
274C2×C4—501no—
275C2×C4D5×AGL(1,𝔽5)11yescharacteristic
276C5—41no—
277C5#6526111yescharacteristic, complemented
278C5#6571211yescharacteristic, complemented
279C22—51no—
280C22—51no—
281C22—51no—
282C22—51no—
283C22—51no—
284C22—51no—
285C22C2×D5×AGL(1,𝔽5)11yescharacteristic
286C4—21no—
287C4—52no—
288C4—102no—
289C4—101no—
290C4—101no—
291C4—201no—
292C4—252no—
293C4—502no—
294C4—501no—
295C4—1001no—
296C2—51no—
297C2—51no—
298C2—51no—
299C2—51no—
300C2#1042⋊Q811yescharacteristic
301C2D5×C4⋊AGL(1,𝔽5)11yescharacteristic
302C2AGL(1,𝔽5)×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×AGL(1,𝔽5)
commutator, socleC22×C52C22×C4
FittingC2×C4×C52C2×C4
prime radical, cosocleC22×C5×D5C23
hypercenterC2×C4D5×AGL(1,𝔽5)
hypocenterC52#389
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
19. #87806C4semidirect
20. C5×#683C4semidirect
21. #87781C4semidirect
22. #87780C4semidirect
77. #1039C4⋊C4semidirect
80. C5×Dic5C4⋊C4semidirect
110. #683AGL(1,𝔽5)semidirect
180. C52#389semidirect
218. Dic5C4⋊AGL(1,𝔽5)semidirect
277. C5#65261semidirect
278. C5#65712semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C4⋊AGL(1,𝔽5)2
C2×#878252
C2×Dic5×AGL(1,𝔽5)2
D5×#6832
#117168492
(C2×AGL(1,𝔽5))⋊Dic52
#657125
#652615