Finite Group Databasebeta

← #10606418

Subgroups — #10606418

Order 640.

864 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#711C211yesmaximal, characteristic
2C2×#4081C211yesmaximal, characteristic, complemented
3C22×C4⋊Dic5C211yesmaximal, characteristic, complemented
4C2×#4032C211yesmaximal, characteristic, complemented
5C5×#386C211yesmaximal, characteristic, complemented
6#65502C211yesmaximal, characteristic, complemented
7#65250C212yesmaximal, complemented
8#65239C212yesmaximal, complemented
9#65232C211yesmaximal, characteristic, complemented
10#64957C212yesmaximal, complemented
11#64945C212yesmaximal, complemented
12C22×C5×D4C2211yescharacteristic
13C23×C4×C5C2211yescharacteristic
14C22×C22⋊D5C2211yescharacteristic
15C22×C22⋊D5C2211yescharacteristic
16C23×Dic5C2211yescharacteristic
17C23×Dic5C2211yescharacteristic
18C23×Dic5C2211yescharacteristic
19C2×#711—24no—
20C2×#711—24no—
21C2×#711—24no—
22C22×C4×D5C2211yescharacteristic
23C22×Dic10C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2212yes—
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2212yes—
28#4081—24no—
29C2×#685C2212yes—
30C2×#685C2212yes—
31C2×C4⋊Dic5—24no—
32C2×C4⋊Dic5C2212yes—
33C2×C4⋊Dic5C2212yes—
34C2×C4⋊Dic5C2212yes—
35C2×#683C2212yes—
36#4032—24no—
37C5×#95C2212yes—
38#3963C2212yes—
39#3963C2212yes—
40#3963C2212yes—
41#2406—51nomaximal, Sylow
42C24×C5C2311yescharacteristic
43C24×C5C2311yescharacteristic
44C23×D5C2311yescharacteristic
45C2×C5×D4—22no—
46C2×C5×D4—22no—
47C2×C5×D4D414yes—
48C22×C4×C5—22no—
49C22×C4×C5—24no—
50C22×C4×C5C2312yes—
51C22×C4×C5C2311yescharacteristic
52C22×C4×C5C2312yes—
53C2×C22⋊D5—22no—
54C2×C22⋊D5—22no—
55C2×C22⋊D5—22no—
56C2×C22⋊D5—22no—
57C2×C22⋊D5—22no—
58C2×C22⋊D5D414yes—
59C22×Dic5—22no—
60C22×Dic5—22no—
61C22×Dic5—22no—
62C22×Dic5—22no—
63C22×Dic5—22no—
64C22×Dic5—22no—
65C22×Dic5—22no—
66C22×Dic5—22no—
67C22×Dic5D414yes—
68C22×Dic5C2312yes—
69C22×Dic5C2312yes—
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73#711—232no—
74C2×C4×D5—22no—
75C2×C4×D5—22no—
76C2×C4×D5—22no—
77C2×Dic10—22no—
78C2×Dic10—22no—
79C2×Dic10—22no—
80C5×C22⋊C4—22no—
81C5×C22⋊C4—24no—
82C22⋊Dic5—22no—
83C22⋊Dic5—24no—
84#685—24no—
85#685—24no—
86C4⋊Dic5—24no—
87C4⋊Dic5—24no—
88C4⋊Dic5—24no—
89#683—24no—
90C22×#41—51no—
91C2×#123—51no—
92C2×#123—51no—
93C22×C4⋊C4—51no—
94#395—52no—
95#394—52no—
96#386—51no—
97#386D511yescharacteristic, complemented
98#385—52no—
99#385—52no—
100#381—51no—
101C23×C5—22no—
102C23×C5—22no—
103C23×C5—21no—
104C23×C5—21no—
105C23×C5—22no—
106C23×C5—22no—
107C23×C5C2×D412yes—
108C23×C5C2×D412yes—
109C23×C5C2×D412yes—
110C23×C5C2×D412yes—
111C23×C5C2411yescharacteristic
112C22×D5—21no—
113C22×D5—21no—
114C22×D5—22no—
115C22×D5—22no—
116C22×D5C2×D412yes—
117C5×D4—28no—
118C2×C4×C5—28no—
119C2×C4×C5—22no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—22no—
123C2×C4×C5—22no—
124C2×C4×C5—22no—
125C2×C4×C5—22no—
126C2×C4×C5C2×D412yes—
127C2×C4×C5#4114yescomplemented
128C22⋊D5—28no—
129C22⋊D5—28no—
130C2×Dic5—28no—
131C2×Dic5—28no—
132C2×Dic5—21no—
133C2×Dic5—21no—
134C2×Dic5—21no—
135C2×Dic5—21no—
136C2×Dic5—21no—
137C2×Dic5—21no—
138C2×Dic5—22no—
139C2×Dic5—22no—
140C2×Dic5—22no—
141C2×Dic5—22no—
142C2×Dic5—22no—
143C2×Dic5—22no—
144C2×Dic5—44no—
145C2×Dic5C2×D412yes—
146C2×Dic5C2×D412yes—
147C2×Dic5C2×D412yes—
148C4×D5—28no—
149Dic10—28no—
150C2×#41—104no—
151C2×#41—104no—
152C2×#41—104no—
153C22×Q8—51no—
154C22×D4—51no—
155C22×D4—51no—
156C22×D4C2×D511yescharacteristic
157C23×C4—51no—
158C23×C4—51no—
159C23×C4—51no—
160C23×C4—51no—
161C23×C4C2×D511yescharacteristic
162#123—104no—
163#123—104no—
164C2×C4⋊C4—52no—
165C2×C4⋊C4—52no—
166C2×C4⋊C4—52no—
167C2×C4⋊C4—52no—
168C2×C4⋊C4—104no—
169C2×C22⋊C4—51no—
170C2×C22⋊C4—52no—
171C2×C22⋊C4—52no—
172C2×C22⋊C4—52no—
173C2×C22⋊C4C2×D511yescharacteristic
174C2×C22⋊C4C2×D512yes—
175#95—52no—
176#95—52no—
177#95—52no—
178#95C2×D512yes—
179C22×C5—22no—
180C22×C5—22no—
181C22×C5—24no—
182C22×C5—24no—
183C22×C5—22no—
184C22×C5—24no—
185C22×C5—41no—
186C22×C5—41no—
187C22×C5—42no—
188C22×C5#12014yes—
189C22×C5C22×D411yescharacteristic
190C22×C5C22×D411yescharacteristic
191C22×C5C22×D411yescharacteristic
192C22×C5C2×#4112yes—
193C22×C52−1+412yes—
194C2×D5—22no—
195C2×D5—24no—
196C2×D5—41no—
197C2×D5—41no—
198C2×D5—42no—
199C4×C5—22no—
200C4×C5—24no—
201C4×C5—42no—
202Dic5—22no—
203Dic5—22no—
204Dic5—22no—
205Dic5—42no—
206Dic5—42no—
207C24—51no—
208C24C22×D511yescharacteristic
209C24C22×D511yescharacteristic
210#41—1032no—
211C2×Q8—102no—
212C2×Q8—102no—
213C2×Q8—102no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—54no—
217C2×D4—102no—
218C2×D4—102no—
219C2×D4—102no—
220C2×D4—102no—
221C2×D4—102no—
222C2×D4D2014yes—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—54no—
226C22×C4—51no—
227C22×C4—51no—
228C22×C4—51no—
229C22×C4—52no—
230C22×C4—52no—
231C22×C4—102no—
232C22×C4—102no—
233C22×C4—102no—
234C22×C4—102no—
235C22×C4—102no—
236C22×C4—102no—
237C22×C4—102no—
238C22×C4—102no—
239C22×C4—102no—
240C22×C4—102no—
241C22×C4—102no—
242C22×C4C22×D512yes—
243C22×C4C22×D511yescharacteristic
244C22×C4C22×D512yes—
245C4⋊C4—104no—
246C4⋊C4—104no—
247C4⋊C4—104no—
248C4⋊C4—104no—
249C22⋊C4—22no—
250C22⋊C4—24no—
251C22⋊C4—104no—
252C22⋊C4—102no—
253C22⋊C4—104no—
254C22⋊C4—104no—
255C2×C5—24no—
256C2×C5—42no—
257C2×C5C2×#12011yescharacteristic
258C2×C5#52511yescharacteristic
259C2×C5#53611yescharacteristic
260C2×C5#54712yes—
261C2×C5#54712yes—
262D5—42no—
263C23—22no—
264C23—22no—
265C23—21no—
266C23—21no—
267C23—22no—
268C23—22no—
269C23—52no—
270C23—101no—
271C23—101no—
272C23—102no—
273C23—102no—
274C23C2×D2012yes—
275C23C2×D2012yes—
276C23D4×D512yes—
277C23D4×D512yes—
278C23C23×D511yescharacteristic
279Q8—108no—
280D4—28no—
281D4—108no—
282D4—108no—
283C2×C4—28no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—52no—
292C2×C4—52no—
293C2×C4—52no—
294C2×C4—108no—
295C2×C4—108no—
296C2×C4—101no—
297C2×C4—101no—
298C2×C4—101no—
299C2×C4—101no—
300C2×C4—101no—
301C2×C4—101no—
302C2×C4—108no—
303C2×C4—102no—
304C2×C4—102no—
305C2×C4—102no—
306C2×C4—102no—
307C2×C4—102no—
308C2×C4—102no—
309C2×C4—204no—
310C2×C4C2×D2012yes—
311C2×C4#71114yescomplemented
312C5#240611yescharacteristic, complemented, Sylow
313C22—24no—
314C22—22no—
315C22—24no—
316C22—22no—
317C22—22no—
318C22—24no—
319C22—41no—
320C22—42no—
321C22—41no—
322C22—102no—
323C22—104no—
324C22—202no—
325C22—201no—
326C22—201no—
327C22#402814yes—
328C22C22×D2011yescharacteristic
329C22C2×D4×D511yescharacteristic
330C22C2×D4×D511yescharacteristic
331C22C2×#71112yes—
332C22#415012yes—
333C4—22no—
334C4—24no—
335C4—42no—
336C4—102no—
337C4—102no—
338C4—102no—
339C4—202no—
340C4—202no—
341C2—24no—
342C2—42no—
343C2—202no—
344C2C2×#402811yescharacteristic
345C2#6588912yes—
346C2#6592011yescharacteristic
347C2#6594512yes—
348C2#6598211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#386C2
prime radicalC22×#711C2
hypercenter#386D5
hypocenterC5#2406
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#4081C2semidirect
3. C22×C4⋊Dic5C2semidirect
4. C2×#4032C2semidirect
5. C5×#386C2semidirect
6. #65502C2semidirect
7. #65250C2semidirect
8. #65239C2semidirect
9. #65232C2semidirect
10. #64957C2semidirect
11. #64945C2semidirect
97. #386D5semidirect
127. C2×C4×C5#41semidirect
311. C2×C4#711semidirect
312. C5#2406semidirect

Maximal subgroups

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

subgroupindex
C22×#7112
C2×#40812
C22×C4⋊Dic52
C2×#40322
C5×#3862
#655022
#652502
#652392
#652322
#649572
#649452
#24065