Finite Group Databasebeta

← #13106055

Subgroups — #13106055

Order 1600.

528 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 128 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×C4⋊#1039C211yesmaximal, characteristic, complemented
2C5×#4075C212yesmaximal, complemented
3#11717100C211yesmaximal, characteristic, complemented
4#11717098C211yesmaximal, characteristic
5#11717091C212yesmaximal, complemented
6#11717007C212yesmaximal, complemented
7#11717003C212yesmaximal, complemented
8#11716977C212yesmaximal, complemented
9#11716963C212yesmaximal, complemented
10C22×C4×C52C2211yescharacteristic
11C22×#1039C2211yescharacteristic
12C22×#1039C2211yescharacteristic
13C2×C5×C22⋊D5C2212yes—
14C2×#6039C2211yescharacteristic
15C2×#6037C2211yescharacteristic
16C2×D5×Dic5C2212yes—
17C4⋊#1039C2212yes—
18C4⋊#1039C2212yes—
19C5×C22⋊Dic5C2212yes—
20C5×#685C2212yes—
21C5×#685C2212yes—
22C5×#683C2212yes—
23C5×#683C2212yes—
24#87781C2211yescharacteristic
25#87781C2211yescharacteristic
26#87780C2212yes—
27#87780C2212yes—
28#87778C2212yes—
29#87778C2212yes—
30Dic52C2212yes—
31#65999—52nomaximal
32C23×C52C2311yescharacteristic
33C22×C5×D5C2312yes—
34C2×C4×C52—22no—
35C2×C4×C52C2312yes—
36C2×#1039—41no—
37C2×#1039—41no—
38C2×#1039C2312yes—
39C2×#1039C2311yescharacteristic
40C2×#1039C2311yescharacteristic
41C5×C22⋊D5—24no—
42C2×C5×Dic5C2312yes—
43C2×C5×Dic5C2312yes—
44C2×C5×Dic5C2312yes—
45#6039—41no—
46#6037—41no—
47D5×Dic5—42no—
48C5×#123—52no—
49#4085—52no—
50D4×Dic5—52no—
51#4075D512yescomplemented
52C2×C4⋊Dic5—52no—
53C2×C4⋊Dic5—102no—
54#4044—52no—
55#4043—52no—
56#4036—52no—
57#4035—52no—
58#4032—52no—
59#4029—52no—
60#4027—52no—
61#4025—52no—
62#4024—52no—
63#4024—52no—
64C22×C52—21no—
65C22×C52—21no—
66C22×C52#4112yes—
67C22×C52C2411yescharacteristic
68C2×C5×D5—22no—
69C2×C5×D5—22no—
70C2×C5×D5—22no—
71C4×C52—22no—
72#1039—22no—
73#1039—21no—
74#1039—21no—
75C5×Dic5—22no—
76C5×Dic5—22no—
77C5×Dic5—22no—
78C2×C5×D4—52no—
79C22×C4×C5—22no—
80C22×C4×C5C2×D512yes—
81C2×C22⋊D5—52no—
82C2×C22⋊D5C2×D512yes—
83C22×Dic5—52no—
84C22×Dic5—52no—
85C22×Dic5—52no—
86C22×Dic5—102no—
87C22×Dic5—102no—
88C2×C4×D5—52no—
89C2×Dic10—52no—
90C5×C4⋊C4—52no—
91C5×C4⋊C4—52no—
92C5×C22⋊C4—52no—
93C5×C22⋊C4—52no—
94C5×C22⋊C4—52no—
95C22⋊Dic5—52no—
96C22⋊Dic5—52no—
97C22⋊Dic5C2×D512yes—
98#685—52no—
99#685—52no—
100#685C2×D512yes—
101#685C2×D512yes—
102C4⋊Dic5—52no—
103C4⋊Dic5—52no—
104C4⋊Dic5—52no—
105C4⋊Dic5—52no—
106C4⋊Dic5—52no—
107C4⋊Dic5—52no—
108C4⋊Dic5—104no—
109C4⋊Dic5—104no—
110#683—52no—
111#683—52no—
112#683—52no—
113#683—52no—
114#683C2×D512yes—
115#683C2×D512yes—
116C4×Dic5—52no—
117C4×Dic5—52no—
118#539—251noSylow
119C2×C52—22no—
120C2×C52C2×#4111yescharacteristic
121C2×C52H(5,𝔽2)11yescharacteristic
122C2×C522−1+411yescharacteristic
123C5×D5—42no—
124C23×C5—22no—
125C23×C5—52no—
126C23×C5C22×D512yes—
127C22×D5C22×D512yes—
128C5×D4—104no—
129C2×C4×C5—28no—
130C2×C4×C5—24no—
131C2×C4×C5—24no—
132C2×C4×C5—52no—
133C2×C4×C5—52no—
134C2×C4×C5—52no—
135C2×C4×C5C22×D512yes—
136C2×C4×C5C22×D512yes—
137C22⋊D5—24no—
138C22⋊D5—202no—
139C2×Dic5—52no—
140C2×Dic5—52no—
141C2×Dic5—52no—
142C2×Dic5—52no—
143C2×Dic5—104no—
144C2×Dic5—102no—
145C2×Dic5—102no—
146C2×Dic5—202no—
147C2×Dic5—202no—
148C2×Dic5—202no—
149C2×Dic5—204no—
150C2×Dic5—204no—
151C2×Dic5C22×D512yes—
152C2×Dic5C22×D512yes—
153C2×Dic5C22×D512yes—
154C4×D5—202no—
155Dic10—202no—
156#126—252no—
157#125—252no—
158#123—52no—
159#123—252no—
160C22⋊Q8—251no—
161C22⋊Q8—252no—
162#121—251no—
163C4×D4—252no—
164C2×C4⋊C4—251no—
165C52#53911yescharacteristic, complemented, Sylow
166C22×C5—24no—
167C22×C5—24no—
168C22×C5—22no—
169C22×C5—22no—
170C22×C5—24no—
171C22×C5—22no—
172C22×C5—102no—
173C22×C5—102no—
174C22×C5—102no—
175C22×C5#71114yes—
176C22×C5C23×D512yes—
177C2×D5—22no—
178C2×D5—22no—
179C2×D5—22no—
180C4×C5—28no—
181C4×C5—22no—
182C4×C5—22no—
183C4×C5—102no—
184C4×C5—102no—
185C4×C5—102no—
186Dic5—22no—
187Dic5—22no—
188Dic5—22no—
189Dic5—102no—
190Dic5—102no—
191Dic5—102no—
192Dic5—102no—
193Dic5—204no—
194Dic5—202no—
195Dic5—202no—
196C2×Q8—251no—
197C2×D4—52no—
198C2×D4—251no—
199C22×C4—252no—
200C22×C4—251no—
201C22×C4—251no—
202C22×C4D5211yescharacteristic
203C4⋊C4—52no—
204C4⋊C4—52no—
205C4⋊C4—252no—
206C4⋊C4—251no—
207C4⋊C4—252no—
208C4⋊C4—251no—
209C4⋊C4—252no—
210C4⋊C4—252no—
211C22⋊C4—52no—
212C22⋊C4—52no—
213C22⋊C4—52no—
214C22⋊C4—252no—
215C22⋊C4—252no—
216C42—252no—
217C2×C5—28no—
218C2×C5—24no—
219C2×C5—22no—
220C2×C5—22no—
221C2×C5—22no—
222C2×C5—202no—
223C2×C5C2×#71112yes—
224C2×C5#414912yes—
225C2×C5#415012yes—
226D5—42no—
227C23—52no—
228C23C2×D5211yescharacteristic
229Q8—1001no—
230D4—104no—
231D4—1001no—
232C2×C4—22no—
233C2×C4—52no—
234C2×C4—52no—
235C2×C4—52no—
236C2×C4—252no—
237C2×C4—251no—
238C2×C4—251no—
239C2×C4—1002no—
240C2×C4—1001no—
241C2×C4—1001no—
242C2×C4C2×D5212yes—
243C5—22no—
244C5#6599912yescomplemented
245C22—21no—
246C22—21no—
247C22—102no—
248C22—102no—
249C22—102no—
250C22#8788012yes—
251C22C22×D5211yescharacteristic
252C4—22no—
253C4—102no—
254C4—102no—
255C4—102no—
256C4—502no—
257C4—501no—
258C4—501no—
259C2—22no—
260C2—202no—
261C2#1171746311yescharacteristic
262C2#1171747311yescharacteristic
263C2C2×#8788011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC22×C4×C52C22
prime radical#11717098C2
hypercenterC22×C4D52
hypocenterC52#539
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#1039C2semidirect
2. C5×#4075C2semidirect
3. #11717100C2semidirect
5. #11717091C2semidirect
6. #11717007C2semidirect
7. #11717003C2semidirect
8. #11716977C2semidirect
9. #11716963C2semidirect
51. #4075D5semidirect
165. C52#539semidirect
244. C5#65999semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10392
C5×#40752
#117171002
#117170982
#117170912
#117170072
#117170032
#117169772
#117169632
#659995