Finite Group Databasebeta

← #12010039

Subgroups — #12010039

Order 1280.

586 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 87 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×#65268C211yesmaximal, characteristic
2C5×#1463C211yesmaximal, characteristic
3#10602355C211yesmaximal, characteristic
4#10597948C214yesmaximal, complemented
5C23×Dic10C411yescharacteristic
6C22×C5×C4⋊C4C2211yescharacteristic
7C22×#683—21no—
8C5×#407—21no—
9C2×#3963—21no—
10C2×#3963C411yescharacteristic
11#65496—21no—
12#65268—21no—
13#65268—21no—
14#65268C412yes—
15#65268C2212yes—
16C5×#324—21no—
17C5×#324C2212yes—
18#64750—21no—
19#64750C2212yes—
20#11815—51nomaximal, Sylow
21C23×C4×C5C2×C411yescharacteristic
22C23×C4×C5D411yescharacteristic
23C23×C4×C5D411yescharacteristic
24C23×Dic5—21no—
25C23×Dic5—21no—
26C22×Dic10—21no—
27C22×Dic10—41no—
28C22×Dic10—41no—
29C22×Dic10—44no—
30C22×Dic10C2×C412yes—
31C2×C5×M16—41no—
32C2×C5×C4⋊C4—41no—
33C2×C5×C4⋊C4—41no—
34C2×C5×C4⋊C4C2×C411yescharacteristic
35C2×C5×C4⋊C4D411yescharacteristic
36C2×C5×C4⋊C4D411yescharacteristic
37C2×C5×C4⋊C4C2311yescharacteristic
38C2×#683—22no—
39C2×#683—22no—
40C2×#683—44no—
41C2×#681—22no—
42C5×C22⋊C8—22no—
43C5×C22⋊C8—41no—
44#3963—22no—
45#3963—21no—
46#3963—22no—
47#3963C2×C412yes—
48C22⋊#168—41no—
49C22⋊#168—41no—
50C2×#395—51no—
51#1821—51no—
52#1463D511yescharacteristic
53#1354—54no—
54C24×C5C22⋊C411yescharacteristic
55C22×C4×C5—21no—
56C22×C4×C5—21no—
57C22×C4×C5—21no—
58C22×C4×C5—21no—
59C22×C4×C5—21no—
60C22×C4×C5—21no—
61C22×C4×C5—22no—
62C22×C4×C5—41no—
63C22×C4×C5—42no—
64C22×C4×C5—42no—
65C22×C4×C5C22⋊C411yescharacteristic
66C22×C4×C5C22⋊C411yescharacteristic
67C22×C4×C5C22⋊C411yescharacteristic
68C22×C4×C5C22×C411yescharacteristic
69C22×C4×C5C2×D411yescharacteristic
70C22×C4×C5C2×D411yescharacteristic
71C22×Dic5—24no—
72C22×Dic5—24no—
73C22×Dic5—22no—
74C22×Dic5—22no—
75C22×Dic5—42no—
76C22×Dic5—42no—
77C22×Dic5—42no—
78C22×Dic5—41no—
79C22×Dic5—41no—
80C2×Dic10—44no—
81C2×Dic10—42no—
82C2×Dic10—44no—
83C2×Dic10—48no—
84C2×Dic10—42no—
85C2×Dic10—44no—
86C2×Dic10—44no—
87C5×M16—82no—
88C2×C5×C8—42no—
89C5×C4⋊C4—41no—
90C5×C4⋊C4—41no—
91C5×C4⋊C4—41no—
92C5×C4⋊C4—41no—
93#683—44no—
94#683—44no—
95#681—42no—
96#681—42no—
97C2×#168—42no—
98C23×Q8—51no—
99C22×C4⋊C4—101no—
100C22×C4⋊C4C2×D511yescharacteristic
101#407—21no—
102#407—101no—
103#395—52no—
104#395—52no—
105#395—101no—
106#395—101no—
107C2×#95—51no—
108C2×#95—101no—
109#343—52no—
110#343—101no—
111#324—21no—
112#324C2×D512yes—
113C23×C5—22no—
114C23×C5—22no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5#9911yescharacteristic
118C23×C5#9911yescharacteristic
119C23×C5C2×C22⋊C411yescharacteristic
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—42no—
130C2×C4×C5—42no—
131C2×C4×C5—42no—
132C2×C4×C5—42no—
133C2×C4×C5—42no—
134C2×C4×C5—41no—
135C2×C4×C5—41no—
136C2×C4×C5—41no—
137C2×C4×C5—41no—
138C2×C4×C5—81no—
139C2×C4×C5#13712yes—
140C2×Dic5—24no—
141C2×Dic5—24no—
142C2×Dic5—42no—
143C2×Dic5—42no—
144C2×Dic5—44no—
145C2×Dic5—48no—
146C2×Dic5—44no—
147C2×Dic5—42no—
148C2×Dic5—42no—
149Dic10—48no—
150Dic10—44no—
151Dic10—44no—
152C5×C8—82no—
153#168—82no—
154C22×Q8—52no—
155C22×Q8—101no—
156C22×Q8—201no—
157C22×Q8—201no—
158C22×Q8—204no—
159C23×C4—101no—
160C23×C4—101no—
161C23×C4C4×D511yescharacteristic
162C23×C4D2011yescharacteristic
163C23×C4C22⋊D511yescharacteristic
164C2×M16—41no—
165C2×M16—102no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—102no—
169C2×C4⋊C4—102no—
170C2×C4⋊C4—204no—
171C2×C4⋊C4C4×D511yescharacteristic
172C2×C4⋊C4D2011yescharacteristic
173C2×C4⋊C4C22⋊D511yescharacteristic
174C2×C4⋊C4C22×D511yescharacteristic
175C22⋊C8—22no—
176C22⋊C8—41no—
177C22⋊C8—201no—
178C22⋊C8—201no—
179#95—52no—
180#95—101no—
181#95—102no—
182#95—102no—
183C22×C5—24no—
184C22×C5—24no—
185C22×C5—21no—
186C22×C5—21no—
187C22×C5—21no—
188C22×C5—21no—
189C22×C5—21no—
190C22×C5—21no—
191C22×C5AΣL(1,𝔽16)12yes—
192C22×C5#35612yes—
193C22×C5C2×#9911yescharacteristic
194C22×C5#41911yescharacteristic
195C22×C5#42111yescharacteristic
196C4×C5—22no—
197C4×C5—41no—
198C4×C5—41no—
199C4×C5—41no—
200C4×C5—41no—
201C4×C5—41no—
202Dic5—44no—
203Dic5—44no—
204C24#68511yescharacteristic
205C2×Q8—202no—
206C2×Q8—202no—
207C2×Q8—204no—
208C2×Q8—204no—
209C2×Q8—208no—
210C2×Q8—204no—
211C2×Q8—204no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—104no—
223C22×C4—104no—
224C22×C4—102no—
225C22×C4—102no—
226C22×C4—202no—
227C22×C4—202no—
228C22×C4—202no—
229C22×C4—201no—
230C22×C4—201no—
231C22×C4#68511yescharacteristic
232C22×C4#68511yescharacteristic
233C22×C4#68511yescharacteristic
234C22×C4C2×C4×D511yescharacteristic
235C22×C4C2×D2011yescharacteristic
236C22×C4C2×C22⋊D511yescharacteristic
237M16—82no—
238M16—202no—
239M16—202no—
240C2×C8—42no—
241C2×C8—202no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—204no—
247C4⋊C4—204no—
248C2×C5—21no—
249C2×C5—22no—
250C2×C5—22no—
251C2×C5—21no—
252C2×C5#151111yescharacteristic
253C2×C5C2×AΣL(1,𝔽16)11yescharacteristic
254C2×C5C2×#35611yescharacteristic
255C23—21no—
256C23—21no—
257C23—22no—
258C23—22no—
259C23#393811yescharacteristic
260C23#393811yescharacteristic
261C23C2×#68511yescharacteristic
262Q8—204no—
263Q8—208no—
264Q8—204no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—104no—
285C2×C4—104no—
286C2×C4—202no—
287C2×C4—202no—
288C2×C4—204no—
289C2×C4—208no—
290C2×C4—202no—
291C2×C4—204no—
292C2×C4—202no—
293C2×C4#405512yes—
294C8—82no—
295C8—402no—
296C5#1181511yescharacteristic, complemented, Sylow
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#6469112yes—
306C22#6469812yes—
307C22C2×#393811yescharacteristic
308C22#6540911yescharacteristic
309C22#6542811yescharacteristic
310C4—22no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—204no—
317C4—204no—
318C2—21no—
319C2—21no—
320C2—22no—
321C2—22no—
322C2#1060077411yescharacteristic
323C2C2×#6469111yescharacteristic
324C2C2×#6469811yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#1463C2
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC2×C5×C4⋊C4C23
hypercenter#1463D5
hypocenterC5#11815
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
4. #10597948C2semidirect
296. C5#11815semidirect

Maximal subgroups

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

subgroupindex
C2×#652682
C5×#14632
#106023552
#105979482
#118155