Finite Group Databasebeta

← #10606432

Subgroups — #10606432

Order 640.

852 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×Dic10C211yesmaximal, characteristic, complemented
2C2×#4085C211yesmaximal, characteristic
3C2×#4031C212yesmaximal, complemented
4C5×#386C211yesmaximal, characteristic, complemented
5#65509C211yesmaximal, characteristic, complemented
6#65502C211yesmaximal, characteristic, complemented
7#65247C211yesmaximal, characteristic, complemented
8#65236C212yesmaximal, complemented
9C22⋊C4×Dic5C211yesmaximal, characteristic, complemented
10#64962C212yesmaximal, complemented
11#64935C212yesmaximal, complemented
12C22×C5×D4C2211yescharacteristic
13C23×C4×C5C2211yescharacteristic
14C22×C22⋊D5C2211yescharacteristic
15C22×C22⋊D5C2211yescharacteristic
16C23×Dic5C2211yescharacteristic
17C23×Dic5C2211yescharacteristic
18C22×Dic10—24no—
19C22×Dic10—28no—
20C22×Dic10C2212yes—
21C22×Dic10C2212yes—
22C2×C5×C22⋊C4C2212yes—
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C22⋊Dic5C2212yes—
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27#4085—24no—
28C2×#685C2212yes—
29C2×#685C2212yes—
30C2×#685C2211yescharacteristic
31C2×#683C2211yescharacteristic
32C2×C4×Dic5C2212yes—
33#4031—28no—
34C5×#95C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#3963C2212yes—
38#3963C2212yes—
39#2436—51nomaximal, Sylow
40C24×C5C2311yescharacteristic
41C24×C5C2311yescharacteristic
42C23×D5C2311yescharacteristic
43C2×C5×D4—22no—
44C2×C5×D4—22no—
45C2×C5×D4—22no—
46C22×C4×C5—22no—
47C22×C4×C5—24no—
48C22×C4×C5C2312yes—
49C22×C4×C5C2311yescharacteristic
50C22×C4×C5C2312yes—
51C2×C22⋊D5—22no—
52C2×C22⋊D5—22no—
53C2×C22⋊D5—22no—
54C2×C22⋊D5—24no—
55C2×C22⋊D5—22no—
56C22×Dic5—22no—
57C22×Dic5—24no—
58C22×Dic5—24no—
59C22×Dic5D414yes—
60C22×Dic5C2312yes—
61C22×Dic5C2311yescharacteristic
62C22×Dic5C2311yescharacteristic
63C22×Dic5C2312yes—
64C22×Dic5C2311yescharacteristic
65C2×Dic10—232no—
66C2×Dic10—24no—
67C2×Dic10—24no—
68C2×Dic10—24no—
69C2×Dic10—24no—
70C2×Dic10—24no—
71C2×Dic10D418yescomplemented
72C5×C22⋊C4—22no—
73C5×C22⋊C4—24no—
74C22⋊Dic5—22no—
75C22⋊Dic5—24no—
76C22⋊Dic5—22no—
77#685—24no—
78#685—24no—
79#685—22no—
80#683—22no—
81C4×Dic5—24no—
82C23×Q8—51no—
83C2×#124—52no—
84C2×#121—51no—
85#393—52no—
86#393—52no—
87#391—52no—
88#386—51no—
89#386—51no—
90#386—51no—
91#386D511yescharacteristic, complemented
92C4×C22⋊C4—51no—
93C23×C5—22no—
94C23×C5—22no—
95C23×C5—22no—
96C23×C5—21no—
97C23×C5—21no—
98C23×C5—21no—
99C23×C5—22no—
100C23×C5C2×D412yes—
101C23×C5#4112yes—
102C23×C5#4112yes—
103C23×C5C2411yescharacteristic
104C22×D5—22no—
105C22×D5—21no—
106C22×D5—21no—
107C22×D5—21no—
108C22×D5—22no—
109C5×D4—44no—
110C2×C4×C5—28no—
111C2×C4×C5—22no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5—21no—
115C2×C4×C5—22no—
116C2×C4×C5—21no—
117C2×C4×C5—21no—
118C2×C4×C5—22no—
119C2×C4×C5C2×D414yes—
120C22⋊D5—44no—
121C22⋊D5—44no—
122C2×Dic5—28no—
123C2×Dic5—22no—
124C2×Dic5—22no—
125C2×Dic5—22no—
126C2×Dic5—28no—
127C2×Dic5—22no—
128C2×Dic5—21no—
129C2×Dic5—22no—
130C2×Dic5—21no—
131C2×Dic5—21no—
132C2×Dic5—22no—
133C2×Dic5C2×D412yes—
134C2×Dic5C2×D414yes—
135C2×Dic5C2×D414yes—
136C2×Dic5C2×D412yes—
137Dic10—216no—
138Dic10—216no—
139C22×Q8—52no—
140C22×Q8—52no—
141C22×Q8—104no—
142C22×Q8—108no—
143C22×D4—51no—
144C22×D4—51no—
145C22×D4C2×D511yescharacteristic
146C23×C4—51no—
147C23×C4—51no—
148C23×C4C2×D511yescharacteristic
149#124—108no—
150#121—104no—
151C2×C4⋊C4—51no—
152C2×C22⋊C4—51no—
153C2×C22⋊C4—52no—
154C2×C22⋊C4—51no—
155C2×C22⋊C4—51no—
156C2×C22⋊C4—52no—
157C2×C22⋊C4—52no—
158C2×C22⋊C4C2×D512yes—
159C2×C22⋊C4C2×D511yescharacteristic
160C2×C42—52no—
161#95—52no—
162#95—52no—
163#95—52no—
164#95—52no—
165#95C2×D512yes—
166C22×C5—22no—
167C22×C5—22no—
168C22×C5—24no—
169C22×C5—24no—
170C22×C5—41no—
171C22×C5—41no—
172C22×C5—41no—
173C22×C5—42no—
174C22×C5—42no—
175C22×C5#12414yescomplemented
176C22×C5C22×D411yescharacteristic
177C22×C5C22×D412yes—
178C22×C5C2×#4111yescharacteristic
179C22×C5C2×#4111yescharacteristic
180C22×C52−1+412yes—
181C2×D5—41no—
182C2×D5—41no—
183C2×D5—41no—
184C2×D5—42no—
185C2×D5—42no—
186C4×C5—24no—
187C4×C5—41no—
188C4×C5—42no—
189Dic5—22no—
190Dic5—24no—
191Dic5—41no—
192Dic5—42no—
193Dic5#12014yes—
194C24—51no—
195C24C22×D511yescharacteristic
196C24C22×D511yescharacteristic
197C2×Q8—58no—
198C2×Q8—104no—
199C2×Q8—104no—
200C2×Q8—104no—
201C2×Q8—104no—
202C2×Q8—1032no—
203C2×Q8—104no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—102no—
208C2×D4—104no—
209C2×D4—102no—
210C2×D4—102no—
211C2×D4—102no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—51no—
215C22×C4—54no—
216C22×C4—51no—
217C22×C4—52no—
218C22×C4—51no—
219C22×C4—52no—
220C22×C4—104no—
221C22×C4—104no—
222C22×C4—102no—
223C22×C4C22×D511yescharacteristic
224C22×C4C22×D512yes—
225C22×C4C22×D512yes—
226C4⋊C4—102no—
227C22⋊C4—22no—
228C22⋊C4—24no—
229C22⋊C4—102no—
230C22⋊C4—104no—
231C22⋊C4—102no—
232C22⋊C4—104no—
233C22⋊C4—102no—
234C22⋊C4—104no—
235C42—104no—
236C2×C5—24no—
237C2×C5—81no—
238C2×C5C2×#12011yescharacteristic
239C2×C5C2×#12411yescharacteristic
240C2×C5#53611yescharacteristic
241C2×C5#54812yes—
242C2×C5#54812yes—
243D5—81no—
244C23—22no—
245C23—22no—
246C23—21no—
247C23—22no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—101no—
252C23—102no—
253C23—101no—
254C23—101no—
255C23—102no—
256C23#70912yes—
257C23D4×D512yes—
258C23#71112yes—
259C23C23×D511yescharacteristic
260Q8—1016no—
261Q8—1016no—
262D4—44no—
263D4—204no—
264D4—204no—
265C2×C4—28no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—54no—
275C2×C4—52no—
276C2×C4—52no—
277C2×C4—54no—
278C2×C4—108no—
279C2×C4—108no—
280C2×C4—102no—
281C2×C4—102no—
282C2×C4—102no—
283C2×C4—102no—
284C2×C4—101no—
285C2×C4—102no—
286C2×C4—101no—
287C2×C4—101no—
288C2×C4—102no—
289C2×C4D4×D514yes—
290C5#243611yescharacteristic, complemented, Sylow
291C22—24no—
292C22—22no—
293C22—24no—
294C22—22no—
295C22—41no—
296C22—42no—
297C22—41no—
298C22—42no—
299C22—41no—
300C22—201no—
301C22—202no—
302C22—201no—
303C22—202no—
304C22—201no—
305C22#403114yescomplemented
306C22C2×#70911yescharacteristic
307C22C2×D4×D511yescharacteristic
308C22C2×D4×D512yes—
309C22C2×#71111yescharacteristic
310C22#415012yes—
311C4—24no—
312C4—41no—
313C4—42no—
314C4—54no—
315C4—102no—
316C4—104no—
317C4—201no—
318C4—202no—
319C2—24no—
320C2—81no—
321C2—401no—
322C2C2×#403111yescharacteristic
323C2#6588612yes—
324C2D5×#12011yescharacteristic
325C2#6593212yes—
326C2#6598211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#386C2
prime radicalC2×#4085C2
hypercenter#386D5
hypocenterC5#2436
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C23×Dic10C2semidirect
3. C2×#4031C2semidirect
4. C5×#386C2semidirect
5. #65509C2semidirect
6. #65502C2semidirect
7. #65247C2semidirect
8. #65236C2semidirect
9. C22⋊C4×Dic5C2semidirect
10. #64962C2semidirect
11. #64935C2semidirect
71. C2×Dic10D4semidirect
91. #386D5semidirect
175. C22×C5#124semidirect
290. C5#2436semidirect
305. C22#4031semidirect

Maximal subgroups

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

subgroupindex
C23×Dic102
C2×#40852
C2×#40312
C5×#3862
#655092
#655022
#652472
#652362
C22⋊C4×Dic52
#649622
#649352
#24365