Finite Group Databasebeta

← #12014840

Subgroups — #12014840

Order 1280.

986 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C5×#3485C211yesmaximal, characteristic
2#10602930C212yesmaximal, complemented
3#10602665C214yesmaximal, complemented
4C2×C5×H(5,𝔽2)C411yescharacteristic
5C22×C5×#41C2211yescharacteristic
6C5×#547—24no—
7C5×#546—24no—
8C5×#536C2212yes—
9C5×#534—22no—
10C5×#525C412yes—
11C2×C5×#123—22no—
12C5×#518C411yescharacteristic
13C2×#3968C2214yescomplemented
14C2×#3965—22no—
15#65496—22no—
16#64768—24no—
17#64762—24no—
18#12709—51nomaximal, Sylow
19C5×H(5,𝔽2)—44no—
20C2×C5×#41—22no—
21C2×C5×#41—24no—
22C2×C5×#41—44no—
23C2×C5×#41C2×C412yes—
24C22×C5×Q8C2311yescharacteristic
25C22×C5×D4—22no—
26C22×C5×D4—22no—
27C22×C5×D4C2×C411yescharacteristic
28C22×C5×D4D412yes—
29C23×C4×C5C2×C411yescharacteristic
30C23×C4×C5D412yes—
31C5×C4⋊Q8D414yes—
32C5×#124D414yes—
33C5×#123—28no—
34C5×#123—24no—
35C5×#123—22no—
36C5×#123—24no—
37C5×C22⋊Q8—22no—
38C5×C22⋊Q8—24no—
39C5×#121—28no—
40C5×#121—22no—
41C5×#120—24no—
42C5×#120—22no—
43C4×C5×D4—24no—
44C4×C5×D4—24no—
45C5×#117C2×C412yes—
46C2×C5×C4⋊C4—24no—
47C2×C5×C22⋊C4—24no—
48C2×C5×C22⋊C4—22no—
49C2×#681—24no—
50#3968—28no—
51#3965—44no—
52C22⋊#168—44no—
53#3485D511yescharacteristic
54#2133—54no—
55#1785—52no—
56C24×C5—22no—
57C24×C5C22⋊C411yescharacteristic
58C24×C5C22⋊C412yes—
59C5×#41—44no—
60C5×#41—416no—
61C5×#41—48no—
62C2×C5×Q8—42no—
63C2×C5×Q8C22⋊C412yes—
64C2×C5×Q8C2×D414yes—
65C2×C5×D4—24no—
66C2×C5×D4—24no—
67C2×C5×D4—24no—
68C2×C5×D4—22no—
69C2×C5×D4—24no—
70C2×C5×D4—22no—
71C2×C5×D4—24no—
72C2×C5×D4—42no—
73C2×C5×D4—48no—
74C2×C5×D4—44no—
75C2×C5×D4—44no—
76C2×C5×D4C22⋊C412yes—
77C22×C4×C5—22no—
78C22×C4×C5—24no—
79C22×C4×C5—24no—
80C22×C4×C5—22no—
81C22×C4×C5—22no—
82C22×C4×C5—24no—
83C22×C4×C5—42no—
84C22×C4×C5—42no—
85C22×C4×C5C22⋊C411yescharacteristic
86C22×C4×C5C22×C411yescharacteristic
87C22×C4×C5C2×D412yes—
88C5×C4⋊C4—24no—
89C5×C4⋊C4—22no—
90C5×C4⋊C4—28no—
91C5×C22⋊C4—24no—
92C5×C22⋊C4—24no—
93C5×C22⋊C4—24no—
94C5×C22⋊C4—22no—
95C5×C22⋊C4—28no—
96C42×C5C22⋊C414yes—
97#681—44no—
98#681—44no—
99C2×#168—44no—
100C2×H(5,𝔽2)Dic511yescharacteristic
101C22×#41C2×D511yescharacteristic
102#547—24no—
103#546—24no—
104#536C2×D512yes—
105#534—22no—
106#525Dic512yes—
107C2×#123—22no—
108#518Dic511yescharacteristic
109C2×#101—54no—
110C2×#100—102no—
111#407—102no—
112#356—104no—
113#355—104no—
114C23×C5—22no—
115C23×C5—22no—
116C23×C5—22no—
117C23×C5—21no—
118C23×C5—22no—
119C23×C5—44no—
120C23×C5—82no—
121C23×C5—81no—
122C23×C5—84no—
123C23×C5#9914yes—
124C23×C5C2×C22⋊C411yescharacteristic
125C5×Q8—44no—
126C5×Q8—42no—
127C5×D4—28no—
128C5×D4—48no—
129C5×D4—42no—
130C5×D4—48no—
131C5×D4—44no—
132C5×D4—84no—
133C2×C4×C5—22no—
134C2×C4×C5—22no—
135C2×C4×C5—22no—
136C2×C4×C5—24no—
137C2×C4×C5—21no—
138C2×C4×C5—22no—
139C2×C4×C5—22no—
140C2×C4×C5—48no—
141C2×C4×C5—44no—
142C2×C4×C5—44no—
143C2×C4×C5—44no—
144C2×C4×C5—84no—
145C2×C4×C5C2×C22⋊C412yes—
146C2×C4×C5#12014yes—
147#168—84no—
148H(5,𝔽2)—44no—
149C2×#41—22no—
150C2×#41—24no—
151C2×#41—44no—
152C2×#41C2×Dic512yes—
153C22×Q8C22×D511yescharacteristic
154C22×D4—22no—
155C22×D4—22no—
156C22×D4C2×Dic511yescharacteristic
157C22×D4C22⋊D512yes—
158C23×C4C2×Dic511yescharacteristic
159C23×C4C22⋊D512yes—
160C2×M16—104no—
161C4⋊Q8C22⋊D514yes—
162#124C22⋊D514yes—
163#123—28no—
164#123—24no—
165#123—22no—
166#123—24no—
167C22⋊Q8—22no—
168C22⋊Q8—24no—
169#121—28no—
170#121—22no—
171#120—24no—
172#120—22no—
173C4×D4—24no—
174C4×D4—24no—
175#117C2×Dic512yes—
176C2×C4⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179#101—108no—
180#100—204no—
181C22⋊C8—204no—
182C22×C5—24no—
183C22×C5—41no—
184C22×C5—44no—
185C22×C5—44no—
186C22×C5—42no—
187C22×C5—44no—
188C22×C5—41no—
189C22×C5—42no—
190C22×C5—82no—
191C22×C5—82no—
192C22×C5—164no—
193C22×C5#37911yescharacteristic
194C22×C5C2×#9912yes—
195C4×C5—24no—
196C4×C5—22no—
197C4×C5—44no—
198C4×C5—41no—
199C4×C5—42no—
200C24—22no—
201C24C22⋊Dic511yescharacteristic
202C24C22⋊Dic512yes—
203#41—44no—
204#41—416no—
205#41—48no—
206C2×Q8—42no—
207C2×Q8C22⋊Dic512yes—
208C2×Q8C2×C22⋊D514yes—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—22no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—22no—
215C2×D4—24no—
216C2×D4—42no—
217C2×D4—48no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4C22⋊Dic512yes—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4C22⋊Dic511yescharacteristic
230C22×C4C22×Dic511yescharacteristic
231C22×C4C2×C22⋊D512yes—
232M16—204no—
233M16—204no—
234C2×C8—204no—
235C4⋊C4—24no—
236C4⋊C4—22no—
237C4⋊C4—28no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—22no—
242C22⋊C4—28no—
243C42C22⋊Dic514yes—
244C2×C5—21no—
245C2×C5—22no—
246C2×C5—44no—
247C2×C5—81no—
248C2×C5—82no—
249C2×C5#178111yescharacteristic
250C23—22no—
251C23—22no—
252C23—22no—
253C23—21no—
254C23—22no—
255C23—44no—
256C23—82no—
257C23—81no—
258C23—84no—
259C23#396614yes—
260C23C2×C22⋊Dic511yescharacteristic
261Q8—44no—
262Q8—42no—
263D4—28no—
264D4—48no—
265D4—42no—
266D4—48no—
267D4—44no—
268D4—84no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—48no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—84no—
281C2×C4C2×C22⋊Dic512yes—
282C2×C4#409914yes—
283C8—404no—
284C5#1270911yescharacteristic, complemented, Sylow
285C22—24no—
286C22—41no—
287C22—44no—
288C22—44no—
289C22—42no—
290C22—44no—
291C22—41no—
292C22—42no—
293C22—82no—
294C22—82no—
295C22—164no—
296C22C2×#396612yes—
297C22#6553611yescharacteristic
298C4—24no—
299C4—22no—
300C4—44no—
301C4—41no—
302C4—42no—
303C2—21no—
304C2—22no—
305C2—44no—
306C2—81no—
307C2—82no—
308C2#1060292311yescharacteristic
subgroupG/H
centerC2#10602923
commutatorC22×C4×C5C22×C4
FrattiniC22×Q8C22×D5
Fitting, prime radicalC5×#3485C2
socleC2×C5#1781
cosocleC22×C5×Q8C23
hypercenter#3485D5
hypocenterC5#12709
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ #3485

Complements

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

NH ≅ G/Ntype
2. #10602930C2semidirect
3. #10602665C2semidirect
13. C2×#3968C22semidirect
284. C5#12709semidirect

Maximal subgroups

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

subgroupindex
C5×#34852
#106029302
#106026652
#127095