Finite Group Databasebeta

← #12010355

Subgroups — #12010355

Order 1280.

476 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 99 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#10614207C211yesmaximal, characteristic
2C5×#1461C211yesmaximal, characteristic, complemented
3#10602430C211yesmaximal, characteristic, complemented
4#10597947C214yesmaximal, complemented
5C2×C4×C5×Q8C2211yescharacteristic
6C2×#4093—21no—
7C2×#4020C411yescharacteristic
8C5×#431—21no—
9C2×#3968—21no—
10#65280C412yes—
11#65280C2212yes—
12#65277—21no—
13#65228—21no—
14#65228—21no—
15C4×C4⋊Dic5C411yescharacteristic
16#65213—21no—
17C5×#324—21no—
18C5×#324C2212yes—
19#64750C2212yes—
20#11882—51nomaximal, Sylow
21C22×C5×Q8D411yescharacteristic
22C22×D20—21no—
23C22×D20—21no—
24C22×D20C2×C412yes—
25C4×C5×Q8—41no—
26C4×C5×Q8—41no—
27C2×C5×C4⋊C4C2×C411yescharacteristic
28C2×C5×C4⋊C4D411yescharacteristic
29C2×C5×C4⋊C4C2311yescharacteristic
30C2×C42×C5C2×C411yescharacteristic
31C2×C42×C5D411yescharacteristic
32C2×C42×C5D411yescharacteristic
33#4093—42no—
34#4093—42no—
35C2×#685—22no—
36C2×#685—22no—
37C2×#685—22no—
38C2×#685—22no—
39C2×C4⋊Dic5—21no—
40C2×C4×Dic5—21no—
41C2×#681—22no—
42#4020—41no—
43#4020—41no—
44C5×C4⋊C8—41no—
45C5×C22⋊C8—22no—
46C4×C5×C8—41no—
47#3968—42no—
48#3963C2×C412yes—
49C4⋊#168—41no—
50C4⋊#168—41no—
51#2548—51no—
52#1801—51no—
53#1461D511yescharacteristic, complemented
54#1345—54no—
55C23×D5—22no—
56C2×C5×Q8—41no—
57C2×C5×Q8—41no—
58C2×C5×Q8D812yes—
59C2×C5×Q8SD1612yes—
60C22×C4×C5C22⋊C411yescharacteristic
61C22×C4×C5C22⋊C411yescharacteristic
62C22×C4×C5C22⋊C411yescharacteristic
63C22×C4×C5C22⋊C411yescharacteristic
64C22×C4×C5C22×C411yescharacteristic
65C22×C4×C5C2×D411yescharacteristic
66C22×C4×C5C2×D411yescharacteristic
67C22×Dic5—22no—
68C2×D20—42no—
69C2×D20—42no—
70C2×D20—42no—
71C2×D20—42no—
72C2×D20—42no—
73C2×D20—41no—
74C2×D20—41no—
75C2×D20—42no—
76C2×D20—82no—
77C2×C5×C8—42no—
78C5×C4⋊C4—41no—
79C5×C4⋊C4—41no—
80C5×C4⋊C4—41no—
81C42×C5—21no—
82C42×C5—21no—
83C42×C5—21no—
84C42×C5—21no—
85C42×C5—21no—
86C42×C5—21no—
87#685—44no—
88#685—42no—
89#685—42no—
90#685—82no—
91#685—82no—
92C4⋊Dic5—42no—
93C4×Dic5—42no—
94#681—42no—
95#681—42no—
96C2×#168—42no—
97C2×#127—51no—
98C2×#124—101no—
99C2×C4×Q8C2×D511yescharacteristic
100#431—21no—
101#423—101no—
102C2×#101—101no—
103#394—52no—
104#394—52no—
105#390—101no—
106#390—101no—
107#390—101no—
108C4×C4⋊C4—51no—
109#343—52no—
110#324—21no—
111#324C2×D512yes—
112C23×C5C2×C22⋊C411yescharacteristic
113C22×D5—42no—
114C22×D5—42no—
115C22×D5—42no—
116C22×D5—42no—
117C22×D5—42no—
118C22×D5—42no—
119C22×D5—42no—
120C5×Q8—42no—
121C5×Q8—42no—
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—22no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—41no—
133C2×C4×C5—41no—
134C2×C4×C5—41no—
135C2×C4×C5#9911yescharacteristic
136C2×C4×C5#9911yescharacteristic
137C2×C4×C5#10212yes—
138C2×C4×C5#10212yes—
139C2×C4×C5C2×D811yescharacteristic
140C2×C4×C5C2×SD1611yescharacteristic
141C2×Dic5—42no—
142C2×Dic5—42no—
143C2×Dic5—42no—
144D20—82no—
145D20—84no—
146D20—84no—
147D20—82no—
148C5×C8—82no—
149#168—82no—
150C22×Q8D2011yescharacteristic
151C22×D4—52no—
152C22×D4—101no—
153C22×D4—101no—
154C2×M16—102no—
155#127—201no—
156#127—201no—
157#124—202no—
158#124—202no—
159C4×Q8—41no—
160C4×Q8—41no—
161C2×C4⋊C4—101no—
162C2×C4⋊C4C4×D511yescharacteristic
163C2×C4⋊C4C22⋊D511yescharacteristic
164C2×C4⋊C4C22×D511yescharacteristic
165C2×C22⋊C4—102no—
166C2×C22⋊C4—102no—
167C2×C22⋊C4—102no—
168C2×C22⋊C4—102no—
169C2×C42—101no—
170C2×C42C4×D511yescharacteristic
171C2×C42D2011yescharacteristic
172C2×C42C22⋊D511yescharacteristic
173C4⋊C8—41no—
174C4⋊C8—201no—
175C4⋊C8—201no—
176#101—202no—
177C22⋊C8—22no—
178C4×C8—41no—
179#95—52no—
180C22×C5—21no—
181C22×C5—21no—
182C22×C5C2×#9911yescharacteristic
183C22×C5C2×#10211yescharacteristic
184C22×C5#42111yescharacteristic
185C2×D5—82no—
186C2×D5—82no—
187C2×D5—82no—
188C2×D5—82no—
189C2×D5—82no—
190C2×D5—82no—
191C2×D5—82no—
192C4×C5—21no—
193C4×C5—21no—
194C4×C5—22no—
195C4×C5—22no—
196C4×C5—21no—
197C4×C5—41no—
198C4×C5—41no—
199C4×C5—41no—
200C4×C5#35512yes—
201Dic5—82no—
202C24—102no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8#67712yes—
206C2×Q8D4012yes—
207C2×D4—202no—
208C2×D4—202no—
209C2×D4—202no—
210C2×D4—202no—
211C2×D4—202no—
212C2×D4—201no—
213C2×D4—201no—
214C2×D4—202no—
215C2×D4—402no—
216C22×C4—102no—
217C22×C4#68511yescharacteristic
218C22×C4#68511yescharacteristic
219C22×C4#68511yescharacteristic
220C22×C4#68511yescharacteristic
221C22×C4C2×C4×D511yescharacteristic
222C22×C4C2×D2011yescharacteristic
223C22×C4C2×C22⋊D511yescharacteristic
224M16—202no—
225M16—202no—
226C2×C8—42no—
227C2×C8—202no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—202no—
232C22⋊C4—204no—
233C22⋊C4—202no—
234C22⋊C4—202no—
235C22⋊C4—402no—
236C22⋊C4—402no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—202no—
244C2×C5—21no—
245C2×C5—21no—
246C2×C5#151911yescharacteristic
247C2×C5#212211yescharacteristic
248C2×C5C2×#35511yescharacteristic
249D5—162no—
250C23—202no—
251C23—202no—
252C23—202no—
253C23—202no—
254C23—202no—
255C23—202no—
256C23—202no—
257C23C2×#68511yescharacteristic
258Q8—42no—
259Q8—42no—
260D4—404no—
261D4—402no—
262D4—404no—
263D4—402no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—202no—
278C2×C4—202no—
279C2×C4—202no—
280C2×C4#393811yescharacteristic
281C2×C4#393811yescharacteristic
282C2×C4#395312yes—
283C2×C4#395312yes—
284C2×C4C2×#67711yescharacteristic
285C2×C4C2×D4011yescharacteristic
286C8—82no—
287C8—402no—
288C5#1188211yescharacteristic, complemented, Sylow
289C22—21no—
290C22—21no—
291C22—402no—
292C22—402no—
293C22—402no—
294C22—402no—
295C22—402no—
296C22—402no—
297C22—402no—
298C22C2×#393811yescharacteristic
299C22C2×#395311yescharacteristic
300C22#6542811yescharacteristic
301C4—21no—
302C4—22no—
303C4—22no—
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—402no—
310C4#6469712yes—
311C2—21no—
312C2—21no—
313C2—802no—
314C2#1060073711yescharacteristic
315C2#1060085011yescharacteristic
316C2C2×#6469711yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#1461C2
socleC22×C5C2×#99
prime radicalC2×#4020C4
cosocleC2×C5×C4⋊C4C23
hypercenter#1461D5
hypocenterC5#11882
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. C5×#1461C2semidirect
3. #10602430C2semidirect
4. #10597947C2semidirect
53. #1461D5semidirect
288. C5#11882semidirect

Maximal subgroups

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

subgroupindex
#106142072
C5×#14612
#106024302
#105979472
#118825