Finite Group Databasebeta

← #12009510

Subgroups — #12009510

Order 1280.

552 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 107 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#10614017C211yesmaximal, characteristic
2C5×#1742C211yesmaximal, characteristic, complemented
3#10602364C211yesmaximal, characteristic, complemented
4#10597984C214yesmaximal, complemented
5C22×C5×M16—21no—
6C2×C5×#117C2211yescharacteristic
7C2×#4076—21no—
8C2×#4020C411yescharacteristic
9#65280—21no—
10#65280C2212yes—
11#65249—21no—
12#65249C412yes—
13#65228—21no—
14#65225C411yescharacteristic
15#65214—21no—
16C5×#343—21no—
17C5×#343C2212yes—
18#64748C2212yes—
19#64745—21no—
20#11699—51nomaximal, Sylow
21C23×C4×C5D411yescharacteristic
22C22×C22⋊D5—22no—
23C22×D20—21no—
24C22×D20—21no—
25C22×D20C2×C412yes—
26C2×C5×M16—24no—
27C2×C5×M16—22no—
28C2×C5×M16—41no—
29C2×C5×M16—42no—
30C22×C5×C8—41no—
31C5×#117—41no—
32C5×#117—41no—
33C2×C5×C4⋊C4D411yescharacteristic
34C2×C5×C4⋊C4C2311yescharacteristic
35C2×C5×C22⋊C4C2×C411yescharacteristic
36C2×C5×C22⋊C4D411yescharacteristic
37C2×C42×C5C2×C411yescharacteristic
38C2×C42×C5D411yescharacteristic
39#4076—42no—
40#4076—42no—
41C2×#685—22no—
42C2×#685—22no—
43C2×#685—22no—
44C2×C4⋊Dic5—21no—
45#4020—41no—
46#4020—41no—
47#3963—21no—
48#3963C2×C412yes—
49C22⋊#168—22no—
50C4⋊#168—41no—
51C4×#168—41no—
52#2619—51no—
53#1742D511yescharacteristic, complemented
54#1458—51no—
55#1345—54no—
56C24×C5C22⋊C411yescharacteristic
57C23×D5—22no—
58C22×C4×C5—21no—
59C22×C4×C5—21no—
60C22×C4×C5—22no—
61C22×C4×C5C22⋊C411yescharacteristic
62C22×C4×C5C22⋊C411yescharacteristic
63C22×C4×C5C22⋊C411yescharacteristic
64C22×C4×C5D812yes—
65C22×C4×C5SD1612yes—
66C22×C4×C5C22×C411yescharacteristic
67C22×C4×C5C2×D411yescharacteristic
68C22×C4×C5C2×D411yescharacteristic
69C2×C22⋊D5—44no—
70C2×C22⋊D5—82no—
71C2×C22⋊D5—82no—
72C22×Dic5—22no—
73C2×D20—42no—
74C2×D20—42no—
75C2×D20—41no—
76C2×D20—41no—
77C2×D20—42no—
78C2×D20—42no—
79C2×D20—82no—
80C2×D20—82no—
81C5×M16—24no—
82C5×M16—44no—
83C5×M16—42no—
84C2×C5×C8—42no—
85C2×C5×C8—44no—
86C5×C4⋊C4—41no—
87C5×C4⋊C4—41no—
88C5×C22⋊C4—41no—
89C5×C22⋊C4—41no—
90C42×C5—21no—
91C42×C5—21no—
92C42×C5—21no—
93C42×C5—21no—
94#685—42no—
95#685—42no—
96#685—82no—
97#685—82no—
98C4⋊Dic5—42no—
99C2×#168—42no—
100C22×M16—21no—
101C2×#127—51no—
102C2×#121—101no—
103C2×#117C2×D511yescharacteristic
104#431—101no—
105#394—52no—
106#394—101no—
107#392—52no—
108#392—101no—
109#390—101no—
110#382—51no—
111#343—21no—
112#343C2×D512yes—
113#324—52no—
114#323—101no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—41no—
118C23×C5—41no—
119C23×C5#10212yes—
120C23×C5C2×C22⋊C411yescharacteristic
121C22×D5—42no—
122C22×D5—42no—
123C22×D5—42no—
124C22×D5—42no—
125C22×D5—42no—
126C22×D5—42no—
127C22×D5—42no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—22no—
131C2×C4×C5—22no—
132C2×C4×C5—22no—
133C2×C4×C5—22no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—41no—
139C2×C4×C5—41no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5#9911yescharacteristic
143C2×C4×C5#9911yescharacteristic
144C2×C4×C5#10212yes—
145C2×C4×C5#10412yes—
146C2×C4×C5C2×D811yescharacteristic
147C2×C4×C5C2×SD1611yescharacteristic
148C22⋊D5—88no—
149C2×Dic5—42no—
150C2×Dic5—42no—
151C2×Dic5—42no—
152D20—84no—
153D20—84no—
154D20—82no—
155D20—82no—
156C5×C8—44no—
157#168—82no—
158C22×D4—52no—
159C22×D4—102no—
160C22×D4—101no—
161C22×D4—101no—
162C23×C4C22⋊D511yescharacteristic
163C2×M16—24no—
164C2×M16—22no—
165C2×M16—41no—
166C2×M16—42no—
167C22×C8—41no—
168#127—201no—
169#127—201no—
170#121—202no—
171#121—202no—
172#117—41no—
173#117—41no—
174C2×C4⋊C4—101no—
175C2×C4⋊C4C22⋊D511yescharacteristic
176C2×C4⋊C4C22×D511yescharacteristic
177C2×C22⋊C4—102no—
178C2×C22⋊C4—102no—
179C2×C22⋊C4—102no—
180C2×C22⋊C4C4×D511yescharacteristic
181C2×C22⋊C4D2011yescharacteristic
182C2×C42C4×D511yescharacteristic
183C2×C42D2011yescharacteristic
184C4⋊C8—201no—
185C22⋊C8—102no—
186C4×C8—201no—
187#95—52no—
188#95—101no—
189C22×C5—21no—
190C22×C5—21no—
191C22×C5—22no—
192C22×C5—42no—
193C22×C5—41no—
194C22×C5—41no—
195C22×C5—81no—
196C22×C5C2×#9911yescharacteristic
197C22×C5C2×#10211yescharacteristic
198C22×C5C2×#10411yescharacteristic
199C2×D5—82no—
200C2×D5—82no—
201C2×D5—82no—
202C2×D5—82no—
203C2×D5—82no—
204C2×D5—82no—
205C2×D5—82no—
206C4×C5—21no—
207C4×C5—21no—
208C4×C5—41no—
209C4×C5—41no—
210C4×C5—41no—
211C4×C5—41no—
212C4×C5#32712yes—
213C4×C5#32712yes—
214Dic5—82no—
215C24—102no—
216C24#68511yescharacteristic
217C2×D4—202no—
218C2×D4—202no—
219C2×D4—204no—
220C2×D4—202no—
221C2×D4—201no—
222C2×D4—201no—
223C2×D4—202no—
224C2×D4—402no—
225C2×D4—402no—
226C2×D4—402no—
227C2×D4—402no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—102no—
232C22×C4#68511yescharacteristic
233C22×C4#68511yescharacteristic
234C22×C4#68511yescharacteristic
235C22×C4#68612yes—
236C22×C4#68812yes—
237C22×C4C2×C4×D511yescharacteristic
238C22×C4C2×D2011yescharacteristic
239C22×C4C2×C22⋊D511yescharacteristic
240M16—24no—
241M16—44no—
242M16—42no—
243C2×C8—42no—
244C2×C8—44no—
245C2×C8—202no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—202no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—202no—
252C22⋊C4—202no—
253C22⋊C4—402no—
254C22⋊C4—402no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C2×C5—21no—
260C2×C5—21no—
261C2×C5—42no—
262C2×C5C2×#32711yescharacteristic
263C2×C5#212211yescharacteristic
264C2×C5#213411yescharacteristic
265D5—162no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—202no—
271C23—202no—
272C23—202no—
273C23—202no—
274C23—202no—
275C23—202no—
276C23—202no—
277C23#394112yes—
278C23C2×#68511yescharacteristic
279D4—404no—
280D4—404no—
281D4—408no—
282D4—402no—
283D4—402no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—202no—
299C2×C4—202no—
300C2×C4—202no—
301C2×C4#393712yes—
302C2×C4#393811yescharacteristic
303C2×C4#393811yescharacteristic
304C2×C4#394112yes—
305C2×C4C2×#68611yescharacteristic
306C2×C4C2×#68811yescharacteristic
307C8—44no—
308C8—402no—
309C5#1169911yescharacteristic, complemented, Sylow
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—402no—
318C22—402no—
319C22—402no—
320C22—402no—
321C22—402no—
322C22—402no—
323C22—402no—
324C22C2×#393711yescharacteristic
325C22C2×#393811yescharacteristic
326C22C2×#394111yescharacteristic
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—402no—
334C4#6467112yes—
335C4#6467112yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—802no—
340C2C2×#6467111yescharacteristic
341C2#1060093911yescharacteristic
342C2#1060094211yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#1742C2
socleC22×C5C2×#99
prime radical#10614017C2
cosocleC2×C5×C4⋊C4C23
hypercenter#1742D5
hypocenterC5#11699
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
2. C5×#1742C2semidirect
3. #10602364C2semidirect
4. #10597984C2semidirect
53. #1742D5semidirect
309. C5#11699semidirect

Maximal subgroups

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

subgroupindex
#106140172
C5×#17422
#106023642
#105979842
#116995