Finite Group Databasebeta

← #10608429

Subgroups — #10608429

Order 640.

728 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#4086C211yesmaximal, characteristic, complemented
2C2×#4085C212yesmaximal, complemented
3C2×#4023C211yesmaximal, characteristic, complemented
4C5×#392C211yesmaximal, characteristic, complemented
5#65512C212yesmaximal, complemented
6#65511C211yesmaximal, characteristic, complemented
7#65249C211yesmaximal, characteristic, complemented
8#65248C212yesmaximal, complemented
9#65246C212yesmaximal, complemented
10#64961C211yesmaximal, characteristic, complemented
11#64939C211yesmaximal, characteristic, complemented
12C22×C5×D4C2212yes—
13C22×C5×D4C2211yescharacteristic
14C22×C22⋊D5C2212yes—
15C22×C22⋊D5C2212yes—
16C22×C22⋊D5C2212yes—
17C23×Dic5C2211yescharacteristic
18C22×D20C2211yescharacteristic
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2212yes—
21C2×C22⋊Dic5C2211yescharacteristic
22C2×C22⋊Dic5C2212yes—
23C2×C22⋊Dic5C2212yes—
24#4086—24no—
25#4085—28no—
26C2×#685C2212yes—
27C2×#685C2212yes—
28C2×#685C2212yes—
29C2×#683C2211yescharacteristic
30C2×#683C2211yescharacteristic
31C2×#683C2212yes—
32C2×C4×Dic5C2211yescharacteristic
33C2×C4×Dic5C2211yescharacteristic
34#4023—24no—
35C5×#95C2211yescharacteristic
36#3963C2212yes—
37#3963C2211yescharacteristic
38#2619—51nomaximal, Sylow
39C24×C5C2312yes—
40C24×C5C2311yescharacteristic
41C23×D5C2312yes—
42C2×C5×D4—24no—
43C2×C5×D4—22no—
44C2×C5×D4—42no—
45C2×C5×D4—42no—
46C2×C5×D4—42no—
47C22×C4×C5C2311yescharacteristic
48C22×C4×C5C2312yes—
49C22×C4×C5C2311yescharacteristic
50C2×C22⋊D5—24no—
51C2×C22⋊D5—24no—
52C2×C22⋊D5—24no—
53C2×C22⋊D5—24no—
54C2×C22⋊D5—24no—
55C2×C22⋊D5—42no—
56C2×C22⋊D5—42no—
57C2×C22⋊D5—42no—
58C2×C22⋊D5—42no—
59C22×Dic5—22no—
60C22×Dic5—22no—
61C22×Dic5D414yescomplemented
62C22×Dic5C2311yescharacteristic
63C22×Dic5C2312yes—
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C2×D20—22no—
68C2×D20—42no—
69C5×C22⋊C4—22no—
70C5×C22⋊C4—42no—
71C22⋊Dic5—22no—
72C22⋊Dic5—24no—
73C22⋊Dic5—42no—
74#685—24no—
75#685—42no—
76#685—42no—
77#683—22no—
78#683—22no—
79#683—24no—
80C4×Dic5—22no—
81C4×Dic5D414yescomplemented
82C2×#127—51no—
83C2×#121—52no—
84C2×#117—51no—
85#394—51no—
86#394—52no—
87#394—51no—
88#392—52no—
89#392—51no—
90#392D511yescharacteristic, complemented
91#390—52no—
92#382—51no—
93C23×C5—21no—
94C23×C5—21no—
95C23×C5—22no—
96C23×C5—22no—
97C23×C5—22no—
98C23×C5—22no—
99C23×C5—22no—
100C23×C5—22no—
101C23×C5—22no—
102C23×C5—22no—
103C23×C5—22no—
104C23×C5C2×D412yes—
105C23×C5C2411yescharacteristic
106C22×D5—22no—
107C22×D5—22no—
108C22×D5—22no—
109C22×D5—22no—
110C22×D5—22no—
111C22×D5—22no—
112C22×D5—22no—
113C5×D4—44no—
114C5×D4—48no—
115C2×C4×C5—22no—
116C2×C4×C5—21no—
117C2×C4×C5—22no—
118C2×C4×C5—22no—
119C2×C4×C5—22no—
120C2×C4×C5—22no—
121C2×C4×C5C2×D412yes—
122C22⋊D5—48no—
123C22⋊D5—48no—
124C22⋊D5—48no—
125C2×Dic5—21no—
126C2×Dic5—21no—
127C2×Dic5—22no—
128C2×Dic5—28no—
129C2×Dic5—21no—
130C2×Dic5—22no—
131C2×Dic5—22no—
132C2×Dic5—22no—
133C2×Dic5—22no—
134C2×Dic5C2×D412yes—
135C2×Dic5C2×D412yes—
136C2×Dic5C2×D412yes—
137C2×Dic5C2×D412yes—
138C2×Dic5#4112yes—
139D20—44no—
140C22×D4—52no—
141C22×D4—52no—
142C22×D4—52no—
143C22×D4—51no—
144C22×D4C2×D512yes—
145C22×D4C2×D511yescharacteristic
146C23×C4—51no—
147#127—104no—
148#121—108no—
149#117—104no—
150C2×C4⋊C4—51no—
151C2×C4⋊C4—51no—
152C2×C4⋊C4—52no—
153C2×C22⋊C4—51no—
154C2×C22⋊C4—52no—
155C2×C22⋊C4—52no—
156C2×C22⋊C4—52no—
157C2×C22⋊C4—52no—
158C2×C22⋊C4—52no—
159C2×C22⋊C4C2×D511yescharacteristic
160C2×C22⋊C4C2×D512yes—
161C2×C42—51no—
162C2×C42—51no—
163#95—52no—
164#95—51no—
165#95C2×D511yescharacteristic
166C22×C5—22no—
167C22×C5—24no—
168C22×C5—41no—
169C22×C5—41no—
170C22×C5—42no—
171C22×C5—42no—
172C22×C5—42no—
173C22×C5—42no—
174C22×C5—42no—
175C22×C5—42no—
176C22×C5—42no—
177C22×C5—42no—
178C22×C5C22×D411yescharacteristic
179C22×C5C22×D411yescharacteristic
180C22×C5C2×#4111yescharacteristic
181C22×C5H(5,𝔽2)12yes—
182C22×C5H(5,𝔽2)12yes—
183C2×D5—42no—
184C2×D5—42no—
185C2×D5—42no—
186C2×D5—42no—
187C2×D5—42no—
188C2×D5—42no—
189C2×D5—42no—
190C4×C5—22no—
191C4×C5—41no—
192C4×C5—42no—
193Dic5—22no—
194Dic5—22no—
195Dic5—41no—
196Dic5—42no—
197Dic5#12114yes—
198C24—52no—
199C24C22×D512yes—
200C24C22×D511yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—104no—
207C2×D4—104no—
208C2×D4—104no—
209C2×D4—104no—
210C2×D4—104no—
211C2×D4—102no—
212C2×D4—202no—
213C2×D4—202no—
214C2×D4—202no—
215C2×D4—202no—
216C2×D4—202no—
217C22×C4—51no—
218C22×C4—54no—
219C22×C4—51no—
220C22×C4—51no—
221C22×C4—52no—
222C22×C4—51no—
223C22×C4—102no—
224C22×C4—102no—
225C22×C4C22×D512yes—
226C22×C4C22×D511yescharacteristic
227C22×C4C22×D511yescharacteristic
228C4⋊C4—102no—
229C4⋊C4—104no—
230C4⋊C4—102no—
231C22⋊C4—22no—
232C22⋊C4—42no—
233C22⋊C4—104no—
234C22⋊C4—102no—
235C22⋊C4—104no—
236C22⋊C4—202no—
237C22⋊C4—202no—
238C22⋊C4—202no—
239C42—54no—
240C42—102no—
241C2×C5—42no—
242C2×C5—82no—
243C2×C5C2×#12111yescharacteristic
244C2×C5#53511yescharacteristic
245C2×C5#53511yescharacteristic
246C2×C5#53512yes—
247C2×C5#54012yes—
248D5—82no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—21no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—102no—
261C23—102no—
262C23—102no—
263C23—102no—
264C23—102no—
265C23—102no—
266C23—102no—
267C23D4×D512yes—
268C23C23×D511yescharacteristic
269D4—44no—
270D4—48no—
271D4—204no—
272D4—208no—
273D4—208no—
274D4—208no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—52no—
282C2×C4—52no—
283C2×C4—52no—
284C2×C4—52no—
285C2×C4—52no—
286C2×C4—108no—
287C2×C4—101no—
288C2×C4—102no—
289C2×C4—101no—
290C2×C4—101no—
291C2×C4—102no—
292C2×C4—102no—
293C2×C4—102no—
294C2×C4—102no—
295C2×C4D4×D512yes—
296C5#261911yescharacteristic, complemented, Sylow
297C22—24no—
298C22—22no—
299C22—41no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—202no—
310C22—202no—
311C22—202no—
312C22—202no—
313C22—202no—
314C22—202no—
315C22—202no—
316C22C2×D4×D511yescharacteristic
317C22C2×D4×D511yescharacteristic
318C22#414412yes—
319C22#414412yes—
320C22D5×#4111yescharacteristic
321C4—22no—
322C4—41no—
323C4—42no—
324C4—54no—
325C4—102no—
326C4—102no—
327C4—202no—
328C4—201no—
329C2—42no—
330C2—82no—
331C2—402no—
332C2#6592611yescharacteristic
333C2#6592612yes—
334C2D5×#12111yescharacteristic
335C2#6594912yes—
336C2#6604911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#392C2
hypercenter#392D5
hypocenterC5#2619
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#4086C2semidirect
2. C2×#4085C2semidirect
3. C2×#4023C2semidirect
4. C5×#392C2semidirect
5. #65512C2semidirect
6. #65511C2semidirect
7. #65249C2semidirect
8. #65248C2semidirect
9. #65246C2semidirect
10. #64961C2semidirect
11. #64939C2semidirect
61. C22×Dic5D4semidirect
81. C4×Dic5D4semidirect
90. #392D5semidirect
296. C5#2619semidirect

Maximal subgroups

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

subgroupindex
C2×#40862
C2×#40852
C2×#40232
C5×#3922
#655122
#655112
#652492
#652482
#652462
#649612
#649392
#26195