Finite Group Databasebeta

← #10608217

Subgroups — #10608217

Order 640.

460 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4090C211yesmaximal, characteristic
2C5×#391C211yesmaximal, characteristic
3#65519C211yesmaximal, characteristic
4#65267C211yesmaximal, characteristic
5#65261C211yesmaximal, characteristic
6#65222C211yesmaximal, characteristic
7C4×#683C211yesmaximal, characteristic
8#64944C211yesmaximal, characteristic
9#64941C211yesmaximal, characteristic
10#64939C211yesmaximal, characteristic
11#64938C211yesmaximal, characteristic
12#64937C211yesmaximal, characteristic
13#64936C211yesmaximal, characteristic
14#64935C211yesmaximal, characteristic
15#64935C211yesmaximal, characteristic
16C22×C5×Q8C2211yescharacteristic
17C22×Dic10C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C42×C5C2211yescharacteristic
20#4090C418yescomplemented
21C2×C4⋊Dic5C2211yescharacteristic
22C2×#683C2211yescharacteristic
23C2×#683C2211yescharacteristic
24C2×#683C2211yescharacteristic
25C2×#683C2211yescharacteristic
26C2×#683C2211yescharacteristic
27C2×#683C2211yescharacteristic
28C2×#683C2211yescharacteristic
29C2×#683C2211yescharacteristic
30C2×C4×Dic5C2211yescharacteristic
31C2×C4×Dic5C2211yescharacteristic
32C2×C4×Dic5C2211yescharacteristic
33C2×C4×Dic5C2211yescharacteristic
34C2×C4×Dic5C2211yescharacteristic
35C2×C4×Dic5C2211yescharacteristic
36C5×#95C2211yescharacteristic
37C5×#95C2211yescharacteristic
38C5×#95C2211yescharacteristic
39C5×#95C2211yescharacteristic
40#3963C2211yescharacteristic
41#3963C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#2331—51nomaximal, Sylow
53C2×C5×Q8—41no—
54C2×C5×Q8—41no—
55C2×C5×Q8C2×C414yes—
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×Dic5C2311yescharacteristic
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C2×Dic10—41no—
72C2×Dic10—41no—
73C2×Dic10C2×C414yes—
74C5×C4⋊C4—41no—
75C42×C5—41no—
76C4⋊Dic5—41no—
77#683—22no—
78#683—22no—
79#683—41no—
80#683—41no—
81#683C2×C414yes—
82#683C2×C414yes—
83#683C2×C414yes—
84#683C2×C414yes—
85C4×Dic5—22no—
86C4×Dic5—41no—
87C4×Dic5—41no—
88C4×Dic5—41no—
89C4×Dic5C2×C414yes—
90C4×Dic5Q814yescomplemented
91C2×C4⋊Q8—51no—
92#391—51no—
93#391—51no—
94#391—51no—
95#391D511yescharacteristic
96#389—51no—
97#389—51no—
98#387—51no—
99#387—51no—
100#387—51no—
101#387—51no—
102#382—51no—
103C4×C4⋊C4—51no—
104C4×C4⋊C4—51no—
105#376—51no—
106C23×C5C2411yescharacteristic
107C5×Q8—44no—
108C2×C4×C5—21no—
109C2×C4×C5—21no—
110C2×C4×C5—21no—
111C2×C4×C5—21no—
112C2×C4×C5—21no—
113C2×C4×C5—21no—
114C2×C4×C5—21no—
115C2×C4×C5—21no—
116C2×C4×C5—21no—
117C2×C4×C5—21no—
118C2×C4×C5—21no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5C22×C412yes—
126C2×C4×C5C22×C412yes—
127C2×C4×C5C22×C412yes—
128C2×C4×C5C2×Q812yes—
129C2×Dic5—21no—
130C2×Dic5—21no—
131C2×Dic5—21no—
132C2×Dic5—21no—
133C2×Dic5—21no—
134C2×Dic5—21no—
135C2×Dic5—21no—
136C2×Dic5—21no—
137C2×Dic5—21no—
138C2×Dic5—21no—
139C2×Dic5—21no—
140C2×Dic5—21no—
141C2×Dic5—21no—
142C2×Dic5—21no—
143C2×Dic5—21no—
144C2×Dic5—21no—
145C2×Dic5—21no—
146C2×Dic5C22×C412yes—
147C2×Dic5C22×C412yes—
148C2×Dic5C22×C412yes—
149C2×Dic5C22×C412yes—
150C2×Dic5C2×Q812yes—
151C2×Dic5C2×Q812yes—
152C2×Dic5#4112yes—
153Dic10—44no—
154C22×Q8—51no—
155C22×Q8C2×D511yescharacteristic
156C4⋊Q8—58no—
157C2×C4⋊C4—51no—
158C2×C4⋊C4—51no—
159C2×C4⋊C4—51no—
160C2×C4⋊C4—51no—
161C2×C4⋊C4—51no—
162C2×C4⋊C4—51no—
163C2×C4⋊C4—51no—
164C2×C4⋊C4—51no—
165C2×C4⋊C4—51no—
166C2×C4⋊C4C2×D511yescharacteristic
167C2×C42—51no—
168C2×C42—51no—
169C2×C42—51no—
170C2×C42—51no—
171C2×C42—51no—
172C2×C42—51no—
173C2×C42C2×D511yescharacteristic
174#95—51no—
175#95—51no—
176#95—51no—
177#95—51no—
178#95—51no—
179#95—51no—
180#95—51no—
181#95—51no—
182#95—51no—
183#95—51no—
184#95—51no—
185#95—51no—
186#95C2×D511yescharacteristic
187#95C2×D511yescharacteristic
188#95C2×D511yescharacteristic
189#95C2×D511yescharacteristic
190C22×C5C23×C411yescharacteristic
191C22×C5C22×Q811yescharacteristic
192C22×C5C2×#4111yescharacteristic
193C22×C5H(5,𝔽2)11yescharacteristic
194C22×C5H(5,𝔽2)11yescharacteristic
195C22×C52−1+411yescharacteristic
196C22×C52−1+411yescharacteristic
197C4×C5—22no—
198C4×C5—22no—
199C4×C5—22no—
200C4×C5—22no—
201C4×C5—41no—
202C4×C5—41no—
203C4×C5—41no—
204Dic5—22no—
205Dic5—22no—
206Dic5—22no—
207Dic5—22no—
208Dic5—41no—
209Dic5—41no—
210Dic5—41no—
211Dic5C4×Q814yes—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—54no—
215C2×Q8—201no—
216C2×Q8—201no—
217C2×Q8C4×D514yes—
218C22×C4—51no—
219C22×C4—51no—
220C22×C4—51no—
221C22×C4—51no—
222C22×C4—51no—
223C22×C4—51no—
224C22×C4—51no—
225C22×C4—51no—
226C22×C4C22×D511yescharacteristic
227C22×C4C22×D511yescharacteristic
228C22×C4C22×D511yescharacteristic
229C22×C4C22×D511yescharacteristic
230C22×C4C22×D511yescharacteristic
231C22×C4C22×D511yescharacteristic
232C22×C4C22×D511yescharacteristic
233C4⋊C4—41no—
234C4⋊C4—54no—
235C4⋊C4—54no—
236C4⋊C4—54no—
237C4⋊C4—54no—
238C4⋊C4—102no—
239C4⋊C4—102no—
240C4⋊C4—201no—
241C4⋊C4—201no—
242C4⋊C4—201no—
243C42—41no—
244C42—54no—
245C42—54no—
246C42—102no—
247C42—201no—
248C42—201no—
249C42—201no—
250C2×C5C2×C4×Q811yescharacteristic
251C2×C5#51811yescharacteristic
252C2×C5#51911yescharacteristic
253C2×C5#54211yescharacteristic
254C2×C5#54211yescharacteristic
255C2×C5#54411yescharacteristic
256C2×C5#54411yescharacteristic
257C23C23×D511yescharacteristic
258Q8—44no—
259Q8—204no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
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—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—52no—
278C2×C4—52no—
279C2×C4—52no—
280C2×C4—52no—
281C2×C4—52no—
282C2×C4—52no—
283C2×C4—52no—
284C2×C4—101no—
285C2×C4—101no—
286C2×C4—101no—
287C2×C4—101no—
288C2×C4—101no—
289C2×C4—101no—
290C2×C4—101no—
291C2×C4—101no—
292C2×C4—101no—
293C2×C4—101no—
294C2×C4—101no—
295C2×C4—101no—
296C2×C4—101no—
297C2×C4—101no—
298C2×C4—101no—
299C2×C4—101no—
300C2×C4—101no—
301C2×C4C2×C4×D512yes—
302C2×C4C2×C4×D512yes—
303C2×C4C2×C4×D512yes—
304C2×C4Q8×D512yes—
305C5#233111yescharacteristic, complemented, Sylow
306C22C22×C4×D511yescharacteristic
307C22#414411yescharacteristic
308C22#414411yescharacteristic
309C22C2×Q8×D511yescharacteristic
310C22#414711yescharacteristic
311C22D5×#4111yescharacteristic
312C22#415011yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—54no—
321C4—102no—
322C4—102no—
323C4—102no—
324C4—102no—
325C4—201no—
326C4—201no—
327C4—201no—
328C2#6582011yescharacteristic
329C2C4×Q8×D511yescharacteristic
330C2#6590611yescharacteristic
331C2#6595711yescharacteristic
332C2#6597411yescharacteristic
333C2#6600311yescharacteristic
334C2#6606311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutatorC22×C5C23×C4
FittingC5×#391C2
socle, prime radical, cosocleC23×C5C24
hypercenter#391D5
hypocenterC5#2331
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
20. #4090C4semidirect
90. C4×Dic5Q8semidirect
305. C5#2331semidirect

Maximal subgroups

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

subgroupindex
C2×#40902
C5×#3912
#655192
#652672
#652612
#652222
C4×#6832
#649442
#649412
#649392
#649382
#649372
#649362
#649352
#23315