Finite Group Databasebeta

← #10607042

Subgroups — #10607042

Order 640.

548 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4043C211yesmaximal, characteristic
2C5×#387C211yesmaximal, characteristic, complemented
3#65278C211yesmaximal, characteristic
4#65270C211yesmaximal, characteristic, complemented
5C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
6C4×#685C211yesmaximal, characteristic
7C4×C4⋊Dic5C211yesmaximal, characteristic, complemented
8#64957C211yesmaximal, characteristic
9#64955C211yesmaximal, characteristic
10#64955C211yesmaximal, characteristic
11D5×#95C211yesmaximal, characteristic
12#64943C211yesmaximal, characteristic, complemented
13#64942C211yesmaximal, characteristic, complemented
14#64935C211yesmaximal, 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×C4×Dic5C2211yescharacteristic
33C2×C4×Dic5C2211yescharacteristic
34C2×C4×Dic5C2211yescharacteristic
35C2×C4×Dic5C2211yescharacteristic
36C2×C4×Dic5C2211yescharacteristic
37#4043C418yescomplemented
38C5×#95C2211yescharacteristic
39C5×#95C2211yescharacteristic
40C5×#95C2211yescharacteristic
41C5×#95C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#2368—51nomaximal, Sylow
53C23×D5C2311yescharacteristic
54C22×C4×C5C2311yescharacteristic
55C22×C4×C5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×Dic5C2311yescharacteristic
62C22×Dic5C2311yescharacteristic
63C22×Dic5C2311yescharacteristic
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C2×C4×D5—22no—
69C2×C4×D5—22no—
70C2×C4×D5—22no—
71C2×C4×D5—22no—
72C2×C4×D5—41no—
73C2×C4×D5—41no—
74C2×C4×D5—41no—
75C2×C4×D5—41no—
76C2×C4×D5C2×C414yes—
77C2×Dic10—41no—
78C2×Dic10—41no—
79C2×Dic10C2×C414yescomplemented
80C5×C4⋊C4—22no—
81C5×C4⋊C4C2×C414yescomplemented
82C42×C5—22no—
83#685—22no—
84#685C2×C414yes—
85#685C2×C414yes—
86#685Q814yescomplemented
87C4⋊Dic5—22no—
88C4⋊Dic5C2×C414yescomplemented
89C4⋊Dic5C2×C414yescomplemented
90#683—41no—
91#683—41no—
92C4×Dic5—22no—
93C4×Dic5—41no—
94C4×Dic5—41no—
95C4×Dic5—41no—
96C4×Dic5—41no—
97C2×C22⋊Q8—51no—
98#391—51no—
99#391—51no—
100#387—51no—
101#387—51no—
102#387—51no—
103#387D511yescharacteristic, complemented
104#385—51no—
105#385—51no—
106C4×C4⋊C4—51no—
107C4×C4⋊C4—51no—
108C4×C22⋊C4—51no—
109C4×C22⋊C4—51no—
110C4×C22⋊C4—51no—
111C2×#95—51no—
112C23×C5C2411yescharacteristic
113C22×D5—21no—
114C22×D5—21no—
115C22×D5—21no—
116C22×D5—21no—
117C22×D5C22×C412yes—
118C22×D5C2×Q812yes—
119C22×D5#4112yes—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
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×C5C22×C412yes—
137C2×C4×C5C22×C412yes—
138C2×C4×C5C22×C412yes—
139C2×C4×C5C2×Q812yes—
140C2×C4×C5#4112yes—
141C2×Dic5—21no—
142C2×Dic5—21no—
143C2×Dic5—21no—
144C2×Dic5—21no—
145C2×Dic5—21no—
146C2×Dic5—21no—
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×Dic5C22×C412yes—
158C2×Dic5C22×C412yes—
159C2×Dic5C22×C412yes—
160C2×Dic5C2×Q812yes—
161C2×Dic5#4112yes—
162C4×D5—44no—
163C4×D5—44no—
164C4×D5—44no—
165Dic10—44no—
166C22×Q8—51no—
167C23×C4—51no—
168C23×C4—51no—
169C23×C4—51no—
170C22⋊Q8—58no—
171C2×C4⋊C4—51no—
172C2×C4⋊C4—51no—
173C2×C4⋊C4—51no—
174C2×C4⋊C4—51no—
175C2×C4⋊C4—51no—
176C2×C4⋊C4C2×D511yescharacteristic
177C2×C4⋊C4C2×D511yescharacteristic
178C2×C22⋊C4—51no—
179C2×C22⋊C4—51no—
180C2×C22⋊C4—51no—
181C2×C22⋊C4—51no—
182C2×C42—51no—
183C2×C42—51no—
184C2×C42—51no—
185C2×C42—51no—
186C2×C42—51no—
187C2×C42C2×D511yescharacteristic
188#95—51no—
189#95—51no—
190#95—51no—
191#95—51no—
192#95—51no—
193#95—51no—
194#95—51no—
195#95—51no—
196#95—51no—
197#95—51no—
198#95C2×D511yescharacteristic
199#95C2×D511yescharacteristic
200#95C2×D511yescharacteristic
201#95C2×D511yescharacteristic
202C22×C5C23×C411yescharacteristic
203C22×C5C22×Q811yescharacteristic
204C22×C5C2×#4111yescharacteristic
205C22×C5C2×#4111yescharacteristic
206C22×C5C2×#4111yescharacteristic
207C22×C5H(5,𝔽2)11yescharacteristic
208C22×C5H(5,𝔽2)11yescharacteristic
209C2×D5—22no—
210C2×D5—22no—
211C2×D5—22no—
212C2×D5—22no—
213C2×D5—22no—
214C2×D5—22no—
215C2×D5C4×Q814yes—
216C4×C5—22no—
217C4×C5—22no—
218C4×C5—22no—
219C4×C5—22no—
220C4×C5—22no—
221C4×C5—41no—
222C4×C5—41no—
223Dic5—22no—
224Dic5—22no—
225Dic5—22no—
226Dic5—22no—
227Dic5—22no—
228Dic5—41no—
229Dic5—41no—
230C24—51no—
231C2×Q8—54no—
232C2×Q8—201no—
233C2×Q8—201no—
234C22×C4—51no—
235C22×C4—51no—
236C22×C4—51no—
237C22×C4—51no—
238C22×C4—51no—
239C22×C4—54no—
240C22×C4—51no—
241C22×C4—51no—
242C22×C4—102no—
243C22×C4—102no—
244C22×C4—102no—
245C22×C4—102no—
246C22×C4—201no—
247C22×C4—201no—
248C22×C4—201no—
249C22×C4—201no—
250C22×C4C22×D511yescharacteristic
251C22×C4C22×D511yescharacteristic
252C22×C4C22×D511yescharacteristic
253C22×C4C22×D511yescharacteristic
254C22×C4C22×D511yescharacteristic
255C22×C4C22×D511yescharacteristic
256C22×C4C22×D511yescharacteristic
257C4⋊C4—22no—
258C4⋊C4—54no—
259C4⋊C4—54no—
260C4⋊C4—102no—
261C4⋊C4—201no—
262C4⋊C4—201no—
263C4⋊C4C4×D514yescomplemented
264C22⋊C4—54no—
265C22⋊C4—54no—
266C22⋊C4—54no—
267C22⋊C4—102no—
268C42—22no—
269C42—102no—
270C42—201no—
271C42—201no—
272C42—201no—
273C42—201no—
274C2×C5C2×C4×Q811yescharacteristic
275C2×C5C4×#4111yescharacteristic
276C2×C5#51811yescharacteristic
277C2×C5#55111yescharacteristic
278C2×C5#55111yescharacteristic
279C2×C5#55411yescharacteristic
280C2×C5#55411yescharacteristic
281D5—24no—
282C23—52no—
283C23—52no—
284C23—52no—
285C23—101no—
286C23—101no—
287C23—101no—
288C23—101no—
289C23C23×D511yescharacteristic
290Q8—204no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—52no—
308C2×C4—52no—
309C2×C4—52no—
310C2×C4—52no—
311C2×C4—52no—
312C2×C4—101no—
313C2×C4—101no—
314C2×C4—101no—
315C2×C4—101no—
316C2×C4—101no—
317C2×C4—101no—
318C2×C4—101no—
319C2×C4—101no—
320C2×C4—101no—
321C2×C4—101no—
322C2×C4—101no—
323C2×C4—101no—
324C2×C4—101no—
325C2×C4—101no—
326C2×C4—101no—
327C2×C4—101no—
328C2×C4—204no—
329C2×C4—204no—
330C2×C4—204no—
331C2×C4C2×C4×D512yes—
332C2×C4C2×C4×D512yes—
333C2×C4C2×C4×D512yes—
334C2×C4#71112yes—
335C2×C4Q8×D512yes—
336C5#236811yescharacteristic, complemented, Sylow
337C22—54no—
338C22—102no—
339C22—102no—
340C22—102no—
341C22—102no—
342C22—102no—
343C22—102no—
344C22C22×C4×D511yescharacteristic
345C22C2×#71111yescharacteristic
346C22C2×Q8×D511yescharacteristic
347C22D5×#4111yescharacteristic
348C22D5×#4111yescharacteristic
349C22#414911yescharacteristic
350C22#414911yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—102no—
359C4—102no—
360C4—102no—
361C4—102no—
362C4—102no—
363C4—201no—
364C4—201no—
365C2—104no—
366C2#6585511yescharacteristic
367C2C4×#71111yescharacteristic
368C2C4×Q8×D511yescharacteristic
369C2#6596911yescharacteristic
370C2#6599211yescharacteristic
371C2#6602411yescharacteristic
372C2#6603911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutatorC22×C5C23×C4
FittingC5×#387C2
socle, cosocleC23×C5C24
prime radicalC23×D5C23
hypercenter#387D5
hypocenterC5#2368
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C5×#387C2semidirect
4. #65270C2semidirect
5. C4⋊C4×Dic5C2semidirect
7. C4×C4⋊Dic5C2semidirect
12. #64943C2semidirect
13. #64942C2semidirect
14. #64935C2semidirect
15. #64935C2semidirect
37. #4043C4semidirect
79. C2×Dic10C2×C4semidirect
81. C5×C4⋊C4C2×C4semidirect
86. #685Q8semidirect
88. C4⋊Dic5C2×C4semidirect
89. C4⋊Dic5C2×C4semidirect
103. #387D5semidirect
263. C4⋊C4C4×D5semidirect
336. C5#2368semidirect

Maximal subgroups

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

subgroupindex
C2×#40432
C5×#3872
#652782
#652702
C4⋊C4×Dic52
C4×#6852
C4×C4⋊Dic52
#649572
#649552
D5×#952
#649432
#649422
#649352
#23685