Finite Group Databasebeta

← #10608711

Subgroups — #10608711

Order 640.

620 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 131 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×#4084C211yesmaximal, characteristic, complemented
2C2×#4039C211yesmaximal, characteristic, complemented
3C5×#394C211yesmaximal, characteristic, complemented
4#65512C211yesmaximal, characteristic, complemented
5#65275C211yesmaximal, characteristic, complemented
6#65250C212yesmaximal, complemented
7#65249C212yesmaximal, complemented
8#65245C212yesmaximal, complemented
9#65240C212yesmaximal, complemented
10#64966C211yesmaximal, characteristic, complemented
11#64953C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C22⋊D5C2212yes—
14C22×C22⋊D5C2212yes—
15C22×D20C2211yescharacteristic
16C22×C4×D5C2211yescharacteristic
17C22×C4×D5C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2212yes—
20C2×C5×C22⋊C4C2212yes—
21C2×C22⋊Dic5C2212yes—
22C2×C22⋊Dic5C2212yes—
23#4084—24no—
24C2×#685C2211yescharacteristic
25C2×#685C2211yescharacteristic
26C2×#685C2212yes—
27C2×#685C2212yes—
28C2×C4⋊Dic5C2211yescharacteristic
29C2×C4⋊Dic5C2212yes—
30C2×C4×Dic5C2211yescharacteristic
31#4039—24no—
32C5×#95C2211yescharacteristic
33#3963C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2635—51nomaximal, Sylow
38C24×C5C2312yes—
39C23×D5C2311yescharacteristic
40C23×D5C2311yescharacteristic
41C2×C5×D4—22no—
42C2×C5×D4—42no—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2312yes—
45C22×C4×C5C2311yescharacteristic
46C22×C4×C5C2311yescharacteristic
47C2×C22⋊D5—24no—
48C2×C22⋊D5—24no—
49C2×C22⋊D5—42no—
50C2×C22⋊D5—42no—
51C2×C22⋊D5—42no—
52C2×C22⋊D5—42no—
53C22×Dic5C2311yescharacteristic
54C22×Dic5C2312yes—
55C22×Dic5C2312yes—
56C22×Dic5C2311yescharacteristic
57C2×D20—22no—
58C2×D20—22no—
59C2×D20—22no—
60C2×C4×D5—22no—
61C2×C4×D5—42no—
62C2×C4×D5—42no—
63C2×C4×D5D414yescomplemented
64C5×C4⋊C4—22no—
65C5×C22⋊C4—24no—
66C5×C22⋊C4—42no—
67C22⋊Dic5—24no—
68C22⋊Dic5—42no—
69#685—22no—
70#685—22no—
71#685—42no—
72#685—42no—
73C4⋊Dic5—22no—
74C4⋊Dic5—42no—
75C4×Dic5—22no—
76C2×#121—51no—
77C2×C4×D4—51no—
78#400—51no—
79#399—51no—
80#394—52no—
81#394D511yescharacteristic, complemented
82#392—51no—
83#392—52no—
84#388—52no—
85#386—52no—
86#381—51no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5—22no—
91C23×C5—22no—
92C23×C5—22no—
93C23×C5—22no—
94C23×C5C2411yescharacteristic
95C22×D5—21no—
96C22×D5—21no—
97C22×D5—21no—
98C22×D5—21no—
99C22×D5—22no—
100C22×D5—22no—
101C22×D5—22no—
102C22×D5—22no—
103C22×D5C2×D412yes—
104C22×D5#4112yes—
105C5×D4—44no—
106C2×C4×C5—21no—
107C2×C4×C5—21no—
108C2×C4×C5—22no—
109C2×C4×C5—21no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5C2×D412yes—
115C2×C4×C5#4112yes—
116C22⋊D5—48no—
117C22⋊D5—48no—
118C2×Dic5—22no—
119C2×Dic5—21no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—22no—
124C2×Dic5—22no—
125C2×Dic5—22no—
126C2×Dic5—22no—
127C2×Dic5C2×D412yes—
128D20—28no—
129C4×D5—44no—
130C4×D5—44no—
131C22×D4—52no—
132C22×D4—52no—
133C22×D4—51no—
134C22×D4C2×D511yescharacteristic
135C23×C4—51no—
136C23×C4—51no—
137#121—104no—
138C4×D4—104no—
139C2×C4⋊C4—51no—
140C2×C4⋊C4—52no—
141C2×C4⋊C4C2×D511yescharacteristic
142C2×C22⋊C4—52no—
143C2×C22⋊C4—51no—
144C2×C22⋊C4—51no—
145C2×C22⋊C4—52no—
146C2×C22⋊C4—52no—
147C2×C22⋊C4—52no—
148C2×C22⋊C4C2×D512yes—
149C2×C22⋊C4C2×D512yes—
150C2×C42—51no—
151#95—52no—
152#95—51no—
153#95—52no—
154#95—52no—
155#95C2×D511yescharacteristic
156C22×C5—42no—
157C22×C5—42no—
158C22×C5—42no—
159C22×C5—42no—
160C22×C5—42no—
161C22×C5—42no—
162C22×C5—42no—
163C22×C5C22×D411yescharacteristic
164C22×C5C2×#4111yescharacteristic
165C22×C5C2×#4111yescharacteristic
166C22×C5H(5,𝔽2)12yes—
167C22×C5H(5,𝔽2)12yes—
168C2×D5—22no—
169C2×D5—22no—
170C2×D5—22no—
171C2×D5—22no—
172C2×D5—24no—
173C2×D5—41no—
174C2×D5—41no—
175C2×D5—42no—
176C2×D5—42no—
177C2×D5—42no—
178C4×C5—22no—
179C4×C5—22no—
180C4×C5—41no—
181C4×C5—42no—
182Dic5—22no—
183Dic5—41no—
184Dic5—42no—
185Dic5—42no—
186C24—51no—
187C24—51no—
188C24C22×D512yes—
189C2×D4—22no—
190C2×D4—42no—
191C2×D4—102no—
192C2×D4—102no—
193C2×D4—104no—
194C2×D4—104no—
195C2×D4—102no—
196C2×D4—202no—
197C2×D4—202no—
198C2×D4—202no—
199C2×D4—202no—
200C22×C4—51no—
201C22×C4—54no—
202C22×C4—52no—
203C22×C4—52no—
204C22×C4—51no—
205C22×C4—102no—
206C22×C4—202no—
207C22×C4—202no—
208C22×C4C22×D511yescharacteristic
209C22×C4C22×D512yes—
210C22×C4C22×D511yescharacteristic
211C22×C4C22×D511yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—102no—
214C4⋊C4—202no—
215C22⋊C4—24no—
216C22⋊C4—42no—
217C22⋊C4—102no—
218C22⋊C4—104no—
219C22⋊C4—102no—
220C22⋊C4—202no—
221C22⋊C4—202no—
222C22⋊C4—202no—
223C42—102no—
224C2×C5—82no—
225C2×C5#52511yescharacteristic
226C2×C5#53411yescharacteristic
227C2×C5#53511yescharacteristic
228C2×C5#53812yes—
229C2×C5#54012yes—
230D5—42no—
231D5—42no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—52no—
240C23—52no—
241C23—101no—
242C23—101no—
243C23—101no—
244C23—102no—
245C23—102no—
246C23—101no—
247C23—102no—
248C23—102no—
249C23C23×D511yescharacteristic
250D4—44no—
251D4—108no—
252D4—208no—
253D4—208no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—52no—
263C2×C4—102no—
264C2×C4—101no—
265C2×C4—102no—
266C2×C4—102no—
267C2×C4—102no—
268C2×C4—102no—
269C2×C4—102no—
270C2×C4—102no—
271C2×C4—102no—
272C2×C4—204no—
273C2×C4—204no—
274C2×C4D4×D512yes—
275C2×C4#71112yes—
276C5#263511yescharacteristic, complemented, Sylow
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—102no—
285C22—102no—
286C22—102no—
287C22—104no—
288C22—102no—
289C22—201no—
290C22—202no—
291C22—202no—
292C22—202no—
293C22—201no—
294C22C2×D4×D511yescharacteristic
295C22C2×#71111yescharacteristic
296C22#414412yes—
297C22D5×#4111yescharacteristic
298C22#414912yes—
299C4—22no—
300C4—22no—
301C4—41no—
302C4—42no—
303C4—102no—
304C4—202no—
305C4—202no—
306C4—201no—
307C2—82no—
308C2—202no—
309C2—202no—
310C2#6592311yescharacteristic
311C2#6594011yescharacteristic
312C2#6595312yes—
313C2#6599112yes—
314C2#6600811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
hypercenter#394D5
hypocenterC5#2635
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#4084C2semidirect
2. C2×#4039C2semidirect
3. C5×#394C2semidirect
4. #65512C2semidirect
5. #65275C2semidirect
6. #65250C2semidirect
7. #65249C2semidirect
8. #65245C2semidirect
9. #65240C2semidirect
10. #64966C2semidirect
11. #64953C2semidirect
63. C2×C4×D5D4semidirect
81. #394D5semidirect
276. C5#2635semidirect

Maximal subgroups

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

subgroupindex
C2×#40842
C2×#40392
C5×#3942
#655122
#652752
#652502
#652492
#652452
#652402
#649662
#649532
#26355