Finite Group Databasebeta

← #13015188

Subgroups — #13015188

Order 1280.

388 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 98 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
1#10606086C211yesmaximal, characteristic
2#10604189C211yesmaximal, characteristic
3#10604186C211yesmaximal, characteristic
4#10598713C214yesmaximal
5C2×#4036—21no—
6C2×#4004—21no—
7C5×#384C411yescharacteristic
8#65274—21no—
9#65274C2212yes—
10#65266—21no—
11#65262—21no—
12#65262C412yes—
13#65224C411yescharacteristic
14C42×Dic5C2211yescharacteristic
15#64918—21no—
16#64918—21no—
17#64918—21no—
18#64918C2212yes—
19#64918C2212yes—
20#7720—51nomaximal, Sylow
21C22×#170—22no—
22C22×#170—22no—
23C2×C5×C4⋊C4—21no—
24C2×C5×C4⋊C4—21no—
25C2×C5×C4⋊C4C2×C412yes—
26C2×C42×C5C2×C411yescharacteristic
27C2×C4⋊Dic5—21no—
28C2×C4⋊Dic5—21no—
29C2×C4⋊Dic5—22no—
30C2×C4⋊Dic5C2×C412yes—
31C2×#683—22no—
32C2×C4×Dic5C2×C411yescharacteristic
33C2×C4×Dic5D411yescharacteristic
34C2×C4×Dic5D411yescharacteristic
35C2×C4×Dic5D411yescharacteristic
36C2×C4×Dic5D411yescharacteristic
37C2×C4×Dic5C2311yescharacteristic
38#4036—44no—
39#4004—42no—
40#3963—22no—
41#3963—22no—
42#2859—51no—
43#1839—51no—
44#1831—51no—
45#1271—54no—
46C22×C4×C5—22no—
47C22×C4×C5C22⋊C411yescharacteristic
48C22×C4×C5C22⋊C411yescharacteristic
49C22×C4×C5C22×C411yescharacteristic
50C22×Dic5—22no—
51C22×Dic5C22⋊C411yescharacteristic
52C22×Dic5C22⋊C411yescharacteristic
53C22×Dic5C2×D411yescharacteristic
54C22×Dic5C2×D411yescharacteristic
55C2×#170—42no—
56C2×#170—42no—
57C2×#170—42no—
58C2×#170—42no—
59C2×#170—42no—
60C2×#170—42no—
61C5×C4⋊C4—24no—
62C5×C4⋊C4—42no—
63C5×C4⋊C4—42no—
64C42×C5—22no—
65C4⋊Dic5—42no—
66C4⋊Dic5—42no—
67C4⋊Dic5—44no—
68C4⋊Dic5—42no—
69#683—44no—
70C4×Dic5—22no—
71C4×Dic5—22no—
72C4×Dic5—41no—
73C4×Dic5—41no—
74C4×Dic5—41no—
75C4×Dic5SD1612yes—
76C4×Dic5Q1612yes—
77C2×#125—101no—
78C2×C4⋊C8—101no—
79#389—52no—
80#389—52no—
81#389—101no—
82#389—101no—
83#384—51no—
84#384AGL(1,𝔽5)11yescharacteristic
85#382—101no—
86C43—51no—
87#336—52no—
88#336—52no—
89#336—101no—
90#336—101no—
91#336—101no—
92C23×C5C2×C22⋊C411yescharacteristic
93C2×C4×C5—21no—
94C2×C4×C5—21no—
95C2×C4×C5—21no—
96C2×C4×C5—21no—
97C2×C4×C5—21no—
98C2×C4×C5—21no—
99C2×C4×C5—42no—
100C2×C4×C5—42no—
101C2×C4×C5—42no—
102C2×C4×C5#10112yes—
103C2×C4×C5#10312yes—
104C2×C4×C5#13112yes—
105C2×Dic5—21no—
106C2×Dic5—21no—
107C2×Dic5—21no—
108C2×Dic5—21no—
109C2×Dic5—21no—
110C2×Dic5—21no—
111C2×Dic5—21no—
112C2×Dic5—21no—
113C2×Dic5—21no—
114C2×Dic5—42no—
115C2×Dic5—42no—
116C2×Dic5—42no—
117C2×Dic5#10312yes—
118C2×Dic5C2×SD1611yescharacteristic
119C2×Dic5C2×Q1611yescharacteristic
120C2×Dic5AGL(1,ℤ/8)12yes—
121#170—82no—
122#170—82no—
123C22×C8—102no—
124C22×C8—102no—
125#125—204no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—52no—
129C2×C4⋊C4—102no—
130C2×C4⋊C4—102no—
131C2×C4⋊C4—101no—
132C2×C4⋊C4—101no—
133C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
134C2×C42—51no—
135C2×C42—51no—
136C2×C42—51no—
137C2×C42—51no—
138C2×C42—51no—
139C2×C42—51no—
140C2×C42C2×AGL(1,𝔽5)11yescharacteristic
141C4⋊C8—202no—
142#95—102no—
143#95—102no—
144C22×C5#40811yescharacteristic
145C22×C5#41011yescharacteristic
146C22×C5C2×#10111yescharacteristic
147C22×C5C2×#10311yescharacteristic
148C22×C5#41811yescharacteristic
149C22×C5#42111yescharacteristic
150C22×C5#42111yescharacteristic
151C4×C5—22no—
152C4×C5—22no—
153C4×C5—22no—
154C4×C5—82no—
155Dic5—22no—
156Dic5—22no—
157Dic5—21no—
158Dic5—21no—
159Dic5—41no—
160Dic5—82no—
161C22×C4—22no—
162C22×C4—51no—
163C22×C4—51no—
164C22×C4—51no—
165C22×C4—51no—
166C22×C4—102no—
167C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
168C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
169C22×C4C22×AGL(1,𝔽5)11yescharacteristic
170C2×C8—202no—
171C2×C8—202no—
172C2×C8—202no—
173C2×C8—202no—
174C2×C8—202no—
175C2×C8—202no—
176C4⋊C4—24no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—202no—
180C4⋊C4—202no—
181C4⋊C4—202no—
182C4⋊C4—204no—
183C4⋊C4—204no—
184C42—22no—
185C42—52no—
186C42—52no—
187C42—102no—
188C42—102no—
189C42—201no—
190C42—201no—
191C42—201no—
192C2×C5#147011yescharacteristic
193C2×C5#148111yescharacteristic
194C2×C5#149611yescharacteristic
195C2×C5#151411yescharacteristic
196C2×C5#151811yescharacteristic
197C2×C5#153411yescharacteristic
198C2×C5#154911yescharacteristic
199C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—52no—
210C2×C4—51no—
211C2×C4—52no—
212C2×C4—51no—
213C2×C4—101no—
214C2×C4—101no—
215C2×C4—101no—
216C2×C4—101no—
217C2×C4—101no—
218C2×C4—101no—
219C2×C4—101no—
220C2×C4—101no—
221C2×C4—101no—
222C2×C4—202no—
223C2×C4—202no—
224C2×C4—202no—
225C2×C4#400512yes—
226C2×C4#400912yes—
227C2×C4#413112yes—
228C8—402no—
229C8—402no—
230C5#772011yescharacteristic, complemented, Sylow
231C22C2×#400511yescharacteristic
232C22#6576711yescharacteristic
233C22#6576911yescharacteristic
234C22#6577611yescharacteristic
235C22C2×#400911yescharacteristic
236C22#6578411yescharacteristic
237C22#6578511yescharacteristic
238C4—22no—
239C4—22no—
240C4—22no—
241C4—82no—
242C4—102no—
243C4—101no—
244C4—101no—
245C4—102no—
246C4—201no—
247C4—402no—
248C2#1060426711yescharacteristic
249C2#1060427811yescharacteristic
250C2#1060429811yescharacteristic
251C2#1060460711yescharacteristic
252C2#1060462711yescharacteristic
253C2#1060463111yescharacteristic
254C2#1060464511yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC22×C4×C5C22×C4
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC5×#384C4
socle, prime radicalC23×C5C2×C22⋊C4
cosocleC2×C4×Dic5C23
hypercenter#384AGL(1,𝔽5)
hypocenterC5#7720
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
230. C5#7720semidirect

Maximal subgroups

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

subgroupindex
#106060862
#106041892
#106041862
#105987132
#77205