Finite Group Databasebeta

← #10606696

Subgroups — #10606696

Order 640.

620 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4042C211yesmaximal, characteristic
2C2×D5×C4⋊C4C211yesmaximal, characteristic
3C2×C4×Dic10C211yesmaximal, characteristic, complemented
4C5×#387C211yesmaximal, characteristic, complemented
5#65274C211yesmaximal, characteristic, complemented
6#65261C211yesmaximal, characteristic, complemented
7C4×#685C211yesmaximal, characteristic
8#64957C211yesmaximal, characteristic
9#64957C211yesmaximal, characteristic
10#64955C211yesmaximal, characteristic
11#64955C211yesmaximal, characteristic
12#64944C211yesmaximal, characteristic, complemented
13#64938C211yesmaximal, characteristic, complemented
14#64936C211yesmaximal, characteristic, complemented
15#64935C211yesmaximal, characteristic, complemented
16C22×C4×D5C2211yescharacteristic
17C22×C4×D5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C22×Dic10C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C42×C5C2211yescharacteristic
23C2×#685C2211yescharacteristic
24C2×#685C2211yescharacteristic
25C2×#685C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×C4⋊Dic5C2211yescharacteristic
28C2×C4⋊Dic5C2211yescharacteristic
29C2×C4⋊Dic5C2211yescharacteristic
30C2×#683C2211yescharacteristic
31C2×#683C2211yescharacteristic
32C2×#683C2211yescharacteristic
33C2×#683C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×C4×Dic5C2211yescharacteristic
37C2×C4×Dic5C2211yescharacteristic
38C2×C4×Dic5C2211yescharacteristic
39C2×C4×Dic5C2211yescharacteristic
40C2×C4×Dic5C2211yescharacteristic
41#4042C418yescomplemented
42D5×C4⋊C4—24no—
43C4×Dic10—24no—
44C5×#95C2211yescharacteristic
45C5×#95C2211yescharacteristic
46C5×#95C2211yescharacteristic
47C5×#95C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#2360—51nomaximal, Sylow
55C23×D5C2311yescharacteristic
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
70C2×C4×D5—22no—
71C2×C4×D5—22no—
72C2×C4×D5—22no—
73C2×C4×D5—22no—
74C2×C4×D5—22no—
75C2×C4×D5—22no—
76C2×C4×D5—41no—
77C2×C4×D5—41no—
78C2×C4×D5C2×C414yes—
79C2×Dic10—22no—
80C2×Dic10—22no—
81C2×Dic10C2×C414yescomplemented
82C5×C4⋊C4—22no—
83C5×C4⋊C4C2×C414yescomplemented
84C42×C5—22no—
85#685—22no—
86#685C2×C414yes—
87#685C2×C414yes—
88#685Q814yescomplemented
89C4⋊Dic5—22no—
90C4⋊Dic5—22no—
91C4⋊Dic5—41no—
92#683—22no—
93#683—22no—
94#683—41no—
95#683C2×C414yescomplemented
96#683C2×C414yescomplemented
97#683D414yes—
98C4×Dic5—22no—
99C4×Dic5—22no—
100C4×Dic5—22no—
101C4×Dic5—41no—
102C4×Dic5—41no—
103C2×C22⋊Q8—51no—
104C2×C4×Q8—51no—
105C22×C4⋊C4—51no—
106#391—51no—
107#389—51no—
108#389—51no—
109#389—51no—
110#387—51no—
111#387D511yescharacteristic, complemented
112#385—51no—
113#385—51no—
114C4×C4⋊C4—51no—
115C4×C22⋊C4—51no—
116C4×C22⋊C4—51no—
117C4×C22⋊C4—51no—
118C23×C5C2411yescharacteristic
119C22×D5—21no—
120C22×D5—21no—
121C22×D5—21no—
122C22×D5—21no—
123C22×D5C22×C412yes—
124C22×D5C2×Q812yes—
125C22×D5#4112yes—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5C22×C412yes—
143C2×C4×C5C22×C412yes—
144C2×C4×C5C22×C412yes—
145C2×C4×C5C2×D412yes—
146C2×C4×C5C2×Q812yes—
147C2×Dic5—21no—
148C2×Dic5—21no—
149C2×Dic5—21no—
150C2×Dic5—21no—
151C2×Dic5—21no—
152C2×Dic5—21no—
153C2×Dic5—21no—
154C2×Dic5—21no—
155C2×Dic5—21no—
156C2×Dic5—21no—
157C2×Dic5—21no—
158C2×Dic5—21no—
159C2×Dic5—21no—
160C2×Dic5—21no—
161C2×Dic5C22×C412yes—
162C2×Dic5C22×C412yes—
163C2×Dic5C22×C412yes—
164C2×Dic5C2×D412yes—
165C2×Dic5C2×D412yes—
166C2×Dic5C2×Q812yes—
167C2×Dic5#4112yes—
168C4×D5—28no—
169C4×D5—44no—
170C4×D5—44no—
171Dic10—28no—
172C22×Q8—51no—
173C23×C4—51no—
174C23×C4—51no—
175C23×C4—51no—
176C22⋊Q8—58no—
177C4×Q8—104no—
178C2×C4⋊C4—51no—
179C2×C4⋊C4—51no—
180C2×C4⋊C4—51no—
181C2×C4⋊C4—51no—
182C2×C4⋊C4—51no—
183C2×C4⋊C4—51no—
184C2×C4⋊C4—51no—
185C2×C4⋊C4—51no—
186C2×C4⋊C4—51no—
187C2×C4⋊C4—104no—
188C2×C4⋊C4C2×D511yescharacteristic
189C2×C4⋊C4C2×D511yescharacteristic
190C2×C22⋊C4—51no—
191C2×C22⋊C4—51no—
192C2×C22⋊C4—51no—
193C2×C22⋊C4—51no—
194C2×C42—51no—
195C2×C42—51no—
196C2×C42—51no—
197C2×C42—51no—
198C2×C42—51no—
199C2×C42C2×D511yescharacteristic
200#95—51no—
201#95—51no—
202#95—51no—
203#95—51no—
204#95—51no—
205#95—51no—
206#95C2×D511yescharacteristic
207#95C2×D511yescharacteristic
208#95C2×D511yescharacteristic
209#95C2×D511yescharacteristic
210C22×C5C23×C411yescharacteristic
211C22×C5C22×D411yescharacteristic
212C22×C5C22×Q811yescharacteristic
213C22×C5C2×#4111yescharacteristic
214C22×C5C2×#4111yescharacteristic
215C22×C5H(5,𝔽2)11yescharacteristic
216C22×C52−1+411yescharacteristic
217C2×D5—22no—
218C2×D5—22no—
219C2×D5—22no—
220C2×D5—22no—
221C2×D5—22no—
222C2×D5—22no—
223C2×D5C4×Q814yescomplemented
224C4×C5—22no—
225C4×C5—22no—
226C4×C5—22no—
227C4×C5—22no—
228C4×C5—22no—
229C4×C5—41no—
230C4×C5—41no—
231Dic5—22no—
232Dic5—22no—
233Dic5—22no—
234Dic5—22no—
235Dic5—41no—
236Dic5—41no—
237Dic5C4×D414yes—
238C24—51no—
239C2×Q8—54no—
240C2×Q8—102no—
241C2×Q8—102no—
242C22×C4—51no—
243C22×C4—51no—
244C22×C4—51no—
245C22×C4—51no—
246C22×C4—51no—
247C22×C4—54no—
248C22×C4—51no—
249C22×C4—51no—
250C22×C4—102no—
251C22×C4—102no—
252C22×C4—102no—
253C22×C4—102no—
254C22×C4—102no—
255C22×C4—102no—
256C22×C4—201no—
257C22×C4—201no—
258C22×C4C22×D511yescharacteristic
259C22×C4C22×D511yescharacteristic
260C22×C4C22×D511yescharacteristic
261C22×C4C22×D511yescharacteristic
262C22×C4C22×D511yescharacteristic
263C22×C4C22×D511yescharacteristic
264C22×C4C22×D511yescharacteristic
265C4⋊C4—22no—
266C4⋊C4—54no—
267C4⋊C4—54no—
268C4⋊C4—54no—
269C4⋊C4—102no—
270C4⋊C4—102no—
271C4⋊C4—102no—
272C4⋊C4—102no—
273C4⋊C4—201no—
274C4⋊C4—201no—
275C4⋊C4C4×D514yescomplemented
276C22⋊C4—54no—
277C22⋊C4—54no—
278C22⋊C4—54no—
279C22⋊C4—102no—
280C42—22no—
281C42—102no—
282C42—102no—
283C42—102no—
284C42—201no—
285C42—201no—
286C2×C5C2×C4×D411yescharacteristic
287C2×C5C2×C4×Q811yescharacteristic
288C2×C5#52011yescharacteristic
289C2×C5#54611yescharacteristic
290C2×C5D4×Q811yescharacteristic
291C2×C5#55211yescharacteristic
292C2×C5#55411yescharacteristic
293D5—24no—
294C23—52no—
295C23—52no—
296C23—52no—
297C23—101no—
298C23—101no—
299C23—101no—
300C23—101no—
301C23C23×D511yescharacteristic
302Q8—108no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—52no—
320C2×C4—52no—
321C2×C4—52no—
322C2×C4—52no—
323C2×C4—52no—
324C2×C4—52no—
325C2×C4—52no—
326C2×C4—108no—
327C2×C4—101no—
328C2×C4—101no—
329C2×C4—101no—
330C2×C4—101no—
331C2×C4—101no—
332C2×C4—101no—
333C2×C4—101no—
334C2×C4—101no—
335C2×C4—101no—
336C2×C4—101no—
337C2×C4—101no—
338C2×C4—101no—
339C2×C4—101no—
340C2×C4—101no—
341C2×C4—204no—
342C2×C4—204no—
343C2×C4C2×C4×D512yes—
344C2×C4C2×C4×D512yes—
345C2×C4C2×C4×D512yes—
346C2×C4D4×D512yes—
347C2×C4Q8×D512yes—
348C5#236011yescharacteristic, complemented, Sylow
349C22—54no—
350C22—102no—
351C22—102no—
352C22—102no—
353C22—102no—
354C22—102no—
355C22—102no—
356C22C22×C4×D511yescharacteristic
357C22C2×D4×D511yescharacteristic
358C22C2×Q8×D511yescharacteristic
359C22D5×#4111yescharacteristic
360C22D5×#4111yescharacteristic
361C22#414911yescharacteristic
362C22#415011yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—22no—
368C4—41no—
369C4—41no—
370C4—54no—
371C4—102no—
372C4—102no—
373C4—102no—
374C4—102no—
375C4—201no—
376C4—201no—
377C2—104no—
378C2#6585711yescharacteristic
379C2C4×D4×D511yescharacteristic
380C2C4×Q8×D511yescharacteristic
381C2#6596611yescharacteristic
382C2#6598811yescharacteristic
383C2#6602411yescharacteristic
384C2#6604211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutatorC22×C5C23×C4
FittingC5×#387C2
socle, cosocleC23×C5C24
prime radicalC23×D5C23
hypercenter#387D5
hypocenterC5#2360
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. C2×C4×Dic10C2semidirect
4. C5×#387C2semidirect
5. #65274C2semidirect
6. #65261C2semidirect
12. #64944C2semidirect
13. #64938C2semidirect
14. #64936C2semidirect
15. #64935C2semidirect
41. #4042C4semidirect
81. C2×Dic10C2×C4semidirect
83. C5×C4⋊C4C2×C4semidirect
88. #685Q8semidirect
95. #683C2×C4semidirect
96. #683C2×C4semidirect
111. #387D5semidirect
223. C2×D5C4×Q8semidirect
275. C4⋊C4C4×D5semidirect
348. C5#2360semidirect

Maximal subgroups

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

subgroupindex
C2×#40422
C2×D5×C4⋊C42
C2×C4×Dic102
C5×#3872
#652742
#652612
C4×#6852
#649572
#649552
#649442
#649382
#649362
#649352
#23605