Finite Group Databasebeta

← #12008679

Subgroups — #12008679

Order 1280.

598 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 95 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×#65263C211yesmaximal, characteristic
2C5×#1463C211yesmaximal, characteristic
3#10602345C211yesmaximal, characteristic
4#10597946C214yesmaximal, complemented
5C23×Dic10C411yescharacteristic
6C22×C5×C4⋊C4C2211yescharacteristic
7C22×C4×Dic5C411yescharacteristic
8C5×#407—21no—
9C2×#3963—21no—
10C2×#3963—21no—
11#65496—21no—
12#65263—21no—
13#65263—21no—
14#65263C412yes—
15#65263C2212yes—
16C5×#324—21no—
17C5×#324C2212yes—
18#64748—21no—
19#64748C2212yes—
20#11506—51nomaximal, Sylow
21C23×C4×C5C2×C411yescharacteristic
22C23×C4×C5D411yescharacteristic
23C23×C4×C5D411yescharacteristic
24C23×Dic5—21no—
25C23×Dic5—21no—
26C22×Dic10—21no—
27C22×Dic10—24no—
28C22×Dic10—41no—
29C22×Dic10—41no—
30C22×Dic10—42no—
31C22×Dic10C2×C412yes—
32C2×C5×M16—41no—
33C2×C5×C4⋊C4—41no—
34C2×C5×C4⋊C4—41no—
35C2×C5×C4⋊C4C2×C411yescharacteristic
36C2×C5×C4⋊C4D411yescharacteristic
37C2×C5×C4⋊C4D411yescharacteristic
38C2×C5×C4⋊C4C2311yescharacteristic
39C2×C4×Dic5—21no—
40C2×C4×Dic5—42no—
41C2×C4×Dic5C2×C412yes—
42C2×#681—41no—
43C5×C22⋊C8—22no—
44C5×C22⋊C8—41no—
45#3963—22no—
46#3963—22no—
47#3963—22no—
48#3963—22no—
49C22⋊#168—22no—
50C22⋊#168—41no—
51C2×#391—51no—
52#1463—51no—
53#1463D511yescharacteristic
54#1319—54no—
55C24×C5C22⋊C411yescharacteristic
56C22×C4×C5—21no—
57C22×C4×C5—21no—
58C22×C4×C5—22no—
59C22×C4×C5—22no—
60C22×C4×C5—21no—
61C22×C4×C5—21no—
62C22×C4×C5—21no—
63C22×C4×C5—21no—
64C22×C4×C5—42no—
65C22×C4×C5—42no—
66C22×C4×C5C22⋊C411yescharacteristic
67C22×C4×C5C22⋊C411yescharacteristic
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C22×C411yescharacteristic
70C22×C4×C5C2×D411yescharacteristic
71C22×C4×C5C2×D411yescharacteristic
72C22×Dic5—24no—
73C22×Dic5—24no—
74C22×Dic5—22no—
75C22×Dic5—22no—
76C22×Dic5—42no—
77C22×Dic5—42no—
78C22×Dic5—42no—
79C22×Dic5—41no—
80C22×Dic5—41no—
81C2×Dic10—24no—
82C2×Dic10—22no—
83C2×Dic10—42no—
84C2×Dic10—48no—
85C2×Dic10—48no—
86C2×Dic10—42no—
87C2×Dic10—42no—
88C2×Dic10—42no—
89C2×Dic10M1614yescomplemented
90C5×M16—82no—
91C2×C5×C8—42no—
92C5×C4⋊C4—41no—
93C5×C4⋊C4—41no—
94C5×C4⋊C4—41no—
95C5×C4⋊C4—41no—
96C4×Dic5—42no—
97C4×Dic5—42no—
98#681—82no—
99C2×#168—42no—
100C23×Q8—51no—
101C22×C4⋊C4C2×D511yescharacteristic
102C22×C42—51no—
103#407—21no—
104#407—101no—
105#391—52no—
106#391—52no—
107#391—101no—
108#391—101no—
109C2×#95—101no—
110C2×#95—101no—
111#324—21no—
112#324—52no—
113#324—101no—
114#324C2×D512yes—
115C23×C5—22no—
116C23×C5—22no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5#9911yescharacteristic
120C23×C5#9911yescharacteristic
121C23×C5C2×C22⋊C411yescharacteristic
122C2×C4×C5—24no—
123C2×C4×C5—24no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
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—81no—
140C2×C4×C5#10012yes—
141C2×C4×C5C2×M1612yes—
142C2×Dic5—24no—
143C2×Dic5—24no—
144C2×Dic5—44no—
145C2×Dic5—44no—
146C2×Dic5—48no—
147C2×Dic5—42no—
148C2×Dic5—42no—
149C2×Dic5—42no—
150C2×Dic5—42no—
151Dic10—48no—
152Dic10—44no—
153Dic10—44no—
154C5×C8—82no—
155#168—82no—
156C22×Q8—52no—
157C22×Q8—104no—
158C22×Q8—101no—
159C22×Q8—202no—
160C22×Q8—201no—
161C22×Q8—201no—
162C23×C4—101no—
163C23×C4—101no—
164C23×C4C4×D511yescharacteristic
165C23×C4D2011yescharacteristic
166C23×C4C22⋊D511yescharacteristic
167C2×M16—41no—
168C2×M16—201no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4C4×D511yescharacteristic
172C2×C4⋊C4D2011yescharacteristic
173C2×C4⋊C4C22⋊D511yescharacteristic
174C2×C4⋊C4C22×D511yescharacteristic
175C2×C42—52no—
176C2×C42—101no—
177C2×C42—202no—
178C22⋊C8—22no—
179C22⋊C8—41no—
180C22⋊C8—102no—
181C22⋊C8—201no—
182#95—102no—
183#95—102no—
184#95—102no—
185#95—102no—
186C22×C5—24no—
187C22×C5—24no—
188C22×C5—21no—
189C22×C5—21no—
190C22×C5—21no—
191C22×C5—21no—
192C22×C5—21no—
193C22×C5—21no—
194C22×C5#35412yes—
195C22×C5#35612yes—
196C22×C5#40711yescharacteristic
197C22×C5C2×#9911yescharacteristic
198C22×C5C2×#10011yescharacteristic
199C4×C5—22no—
200C4×C5—22no—
201C4×C5—41no—
202C4×C5—41no—
203C4×C5—41no—
204C4×C5—41no—
205Dic5—44no—
206Dic5—44no—
207C24#68511yescharacteristic
208C2×Q8—54no—
209C2×Q8—104no—
210C2×Q8—102no—
211C2×Q8—208no—
212C2×Q8—208no—
213C2×Q8—202no—
214C2×Q8—202no—
215C2×Q8—202no—
216C2×Q8—202no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—104no—
228C22×C4—104no—
229C22×C4—102no—
230C22×C4—102no—
231C22×C4—202no—
232C22×C4—202no—
233C22×C4—202no—
234C22×C4—201no—
235C22×C4—201no—
236C22×C4#68511yescharacteristic
237C22×C4#68511yescharacteristic
238C22×C4#68511yescharacteristic
239C22×C4C2×C4×D511yescharacteristic
240C22×C4C2×D2011yescharacteristic
241C22×C4C2×C22⋊D511yescharacteristic
242M16—82no—
243M16—402no—
244C2×C8—42no—
245C2×C8—202no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C42—202no—
251C42—202no—
252C2×C5—21no—
253C2×C5—22no—
254C2×C5—22no—
255C2×C5—21no—
256C2×C5#146211yescharacteristic
257C2×C5C2×#35411yescharacteristic
258C2×C5C2×#35611yescharacteristic
259C23—21no—
260C23—21no—
261C23—22no—
262C23—22no—
263C23#393811yescharacteristic
264C23#393811yescharacteristic
265C23C2×#68511yescharacteristic
266Q8—204no—
267Q8—208no—
268Q8—204no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—104no—
288C2×C4—104no—
289C2×C4—204no—
290C2×C4—204no—
291C2×C4—208no—
292C2×C4—202no—
293C2×C4—202no—
294C2×C4—202no—
295C2×C4—202no—
296C2×C4#395512yes—
297C2×C4D5×M1612yes—
298C8—82no—
299C8—402no—
300C5#1150611yescharacteristic, complemented, Sylow
301C22—24no—
302C22—24no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22#6469312yes—
310C22#6469812yes—
311C22C2×#393811yescharacteristic
312C22#6541511yescharacteristic
313C22C2×#395511yescharacteristic
314C4—22no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—204no—
321C4—204no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—22no—
326C2#1060076011yescharacteristic
327C2C2×#6469311yescharacteristic
328C2C2×#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#11506
LCSG ▷ C22×C4×C5 ▷ C22×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. #10597946C2semidirect
89. C2×Dic10M16semidirect
300. C5#11506semidirect

Maximal subgroups

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

subgroupindex
C2×#652632
C5×#14632
#106023452
#105979462
#115065