Finite Group Databasebeta

← #12010452

Subgroups — #12010452

Order 1280.

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

normal subgroups only

This group has 372 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
1#10614212C211yesmaximal, characteristic
2C5×#1801C211yesmaximal, characteristic
3#10602417C211yesmaximal, characteristic
4#10597987C212yesmaximal
5#10597985C212yesmaximal
6C2×C4×C5×Q8C2211yescharacteristic
7C2×C4×Dic10C411yescharacteristic
8C5×#423—21no—
9C2×C5×#101—21no—
10#65519—21no—
11#65275C411yescharacteristic
12#65275C2211yescharacteristic
13#65268C411yescharacteristic
14#65268C2211yescharacteristic
15#65262—21no—
16#65225C411yescharacteristic
17#65222—21no—
18#65222—21no—
19#65214—21no—
20C5×#343C2212yes—
21#64748—21no—
22#64748C2212yes—
23#11905—51nomaximal, Sylow
24C22×C5×Q8D411yescharacteristic
25C22×Dic10C2×C411yescharacteristic
26C2×C5×M16—22no—
27C4×C5×Q8—41no—
28C4×C5×Q8—41no—
29C2×C5×C4⋊C4C2×C411yescharacteristic
30C2×C5×C4⋊C4D411yescharacteristic
31C2×C5×C4⋊C4C2311yescharacteristic
32C2×C42×C5C2×C411yescharacteristic
33C2×C42×C5D411yescharacteristic
34C2×C42×C5D411yescharacteristic
35C2×C4⋊Dic5—21no—
36C2×C4⋊Dic5C2×C411yescharacteristic
37C2×#683—21no—
38C2×#683—21no—
39C2×#683—21no—
40C2×C4×Dic5—21no—
41C4×Dic10—41no—
42C4×Dic10—41no—
43C5×C4⋊C8—41no—
44C5×C4⋊C8—41no—
45C5×#101—42no—
46#3963—21no—
47#3963—21no—
48#3963—21no—
49#3963—21no—
50#3963—21no—
51#3963—21no—
52#3963—21no—
53#3963C2×C411yescharacteristic
54#3963C2×C411yescharacteristic
55C22⋊#168—22no—
56C4⋊#168—41no—
57C4×#168—41no—
58#2581—51no—
59#1801D511yescharacteristic
60#1461—51no—
61#1354—52no—
62#1353—52no—
63C2×C5×Q8—41no—
64C2×C5×Q8—41no—
65C2×C5×Q8SD1612yes—
66C2×C5×Q8Q1612yes—
67C22×C4×C5C22⋊C411yescharacteristic
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C22⋊C411yescharacteristic
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5C22×C411yescharacteristic
72C22×C4×C5C2×D411yescharacteristic
73C22×C4×C5C2×D411yescharacteristic
74C22×Dic5—21no—
75C22×Dic5—21no—
76C22×Dic5—21no—
77C22×Dic5—21no—
78C2×Dic10—41no—
79C2×Dic10—42no—
80C5×M16—42no—
81C5×M16—42no—
82C2×C5×C8—42no—
83C5×C4⋊C4—41no—
84C5×C4⋊C4—41no—
85C5×C4⋊C4—41no—
86C42×C5—21no—
87C42×C5—21no—
88C42×C5—21no—
89C42×C5—21no—
90C42×C5—21no—
91C42×C5—21no—
92C4⋊Dic5—42no—
93C4⋊Dic5—41no—
94#683—42no—
95#683—42no—
96#683—41no—
97#683—41no—
98C4×Dic5—42no—
99C2×#168—42no—
100C2×C4×Q8—51no—
101C2×C4×Q8C2×D511yescharacteristic
102#431—101no—
103#423—21no—
104C2×#101—21no—
105#400—51no—
106#400—51no—
107#395—51no—
108#395—51no—
109#391—101no—
110#389—101no—
111#387—101no—
112#387—101no—
113#382—51no—
114#343C2×D512yes—
115#324—52no—
116#324—101no—
117C23×C5C2×C22⋊C411yescharacteristic
118C5×Q8—42no—
119C5×Q8—42no—
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—41no—
130C2×C4×C5—41no—
131C2×C4×C5—41no—
132C2×C4×C5—41no—
133C2×C4×C5#9911yescharacteristic
134C2×C4×C5#9911yescharacteristic
135C2×C4×C5#10312yes—
136C2×C4×C5#10312yes—
137C2×C4×C5C2×SD1611yescharacteristic
138C2×C4×C5C2×Q1611yescharacteristic
139C2×Dic5—22no—
140C2×Dic5—41no—
141C2×Dic5—41no—
142C2×Dic5—41no—
143C2×Dic5—41no—
144C2×Dic5—41no—
145C2×Dic5—41no—
146C2×Dic5—42no—
147C2×Dic5—41no—
148C2×Dic5—41no—
149C2×Dic5—41no—
150Dic10—42no—
151Dic10—42no—
152C5×C8—82no—
153#168—82no—
154C22×Q8—51no—
155C22×Q8C22⋊D511yescharacteristic
156C2×M16—22no—
157C4×Q8—41no—
158C4×Q8—41no—
159C4×Q8—201no—
160C4×Q8—201no—
161C2×C4⋊C4—51no—
162C2×C4⋊C4—101no—
163C2×C4⋊C4—101no—
164C2×C4⋊C4—101no—
165C2×C4⋊C4—101no—
166C2×C4⋊C4C4×D511yescharacteristic
167C2×C4⋊C4D2011yescharacteristic
168C2×C4⋊C4C22×D511yescharacteristic
169C2×C42—101no—
170C2×C42C4×D511yescharacteristic
171C2×C42D2011yescharacteristic
172C2×C42C22⋊D511yescharacteristic
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—201no—
176#101—42no—
177C22⋊C8—102no—
178C4×C8—201no—
179#95—51no—
180#95—51no—
181#95—101no—
182#95—101no—
183#95—101no—
184#95—101no—
185#95—101no—
186#95—101no—
187#95—101no—
188C22×C5—21no—
189C22×C5—21no—
190C22×C5C2×#9911yescharacteristic
191C22×C5C2×#10311yescharacteristic
192C22×C5#42111yescharacteristic
193C4×C5—21no—
194C4×C5—21no—
195C4×C5—21no—
196C4×C5—22no—
197C4×C5—22no—
198C4×C5—21no—
199C4×C5—41no—
200C4×C5—41no—
201C4×C5—41no—
202Dic5—42no—
203Dic5—81no—
204Dic5—81no—
205Dic5—81no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—201no—
209C2×Q8—202no—
210C2×Q8#68712yes—
211C2×Q8#68912yes—
212C22×C4—101no—
213C22×C4—101no—
214C22×C4—101no—
215C22×C4—101no—
216C22×C4#68511yescharacteristic
217C22×C4#68511yescharacteristic
218C22×C4#68511yescharacteristic
219C22×C4#68511yescharacteristic
220C22×C4C2×C4×D511yescharacteristic
221C22×C4C2×D2011yescharacteristic
222C22×C4C2×C22⋊D511yescharacteristic
223M16—42no—
224M16—42no—
225C2×C8—42no—
226C2×C8—202no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—201no—
231C4⋊C4—202no—
232C4⋊C4—202no—
233C4⋊C4—202no—
234C4⋊C4—201no—
235C4⋊C4—201no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—202no—
243C2×C5—21no—
244C2×C5—21no—
245C2×C5#152011yescharacteristic
246C2×C5#212011yescharacteristic
247C2×C5#213311yescharacteristic
248C23C2×#68511yescharacteristic
249Q8—42no—
250Q8—42no—
251Q8—202no—
252Q8—202no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—102no—
267C2×C4—201no—
268C2×C4—201no—
269C2×C4—201no—
270C2×C4—201no—
271C2×C4—201no—
272C2×C4—201no—
273C2×C4—202no—
274C2×C4—201no—
275C2×C4—201no—
276C2×C4—201no—
277C2×C4#393811yescharacteristic
278C2×C4#393811yescharacteristic
279C2×C4#394212yes—
280C2×C4#394212yes—
281C2×C4C2×#68711yescharacteristic
282C2×C4C2×#68911yescharacteristic
283C8—82no—
284C8—402no—
285C5#1190511yescharacteristic, complemented, Sylow
286C22—21no—
287C22—21no—
288C22C2×#393811yescharacteristic
289C22C2×#394211yescharacteristic
290C22#6529411yescharacteristic
291C4—21no—
292C4—22no—
293C4—22no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—202no—
301C4—401no—
302C4—401no—
303C4—401no—
304C2—21no—
305C2—21no—
306C2#1060030311yescharacteristic
307C2#1060092111yescharacteristic
308C2#1060093811yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#1801C2
socleC22×C5C2×#99
prime radicalC23×C5C2×C22⋊C4
cosocleC2×C5×C4⋊C4C23
hypercenter#1801D5
hypocenterC5#11905
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
285. C5#11905semidirect

Maximal subgroups

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

subgroupindex
#106142122
C5×#18012
#106024172
#105979872
#105979852
#119055