Finite Group Databasebeta

← #10607722

Subgroups — #10607722

Order 640.

496 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4044C211yesmaximal, characteristic
2C5×#389C211yesmaximal, characteristic, complemented
3#65279C211yesmaximal, characteristic
4#65279C211yesmaximal, characteristic
5#65279C211yesmaximal, characteristic
6#65270C211yesmaximal, characteristic, complemented
7#65267C211yesmaximal, characteristic, complemented
8C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
10C4×#685C211yesmaximal, characteristic
11C4×#683C211yesmaximal, characteristic, complemented
12#64957C211yesmaximal, characteristic
13#64955C211yesmaximal, characteristic
14#64942C211yesmaximal, characteristic, complemented
15#64937C211yesmaximal, characteristic, complemented
16C22×C4×D5C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C42×C5C2211yescharacteristic
22C2×#685C2211yescharacteristic
23C2×#685C2211yescharacteristic
24C2×#685C2211yescharacteristic
25C2×#685C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×#685C2211yescharacteristic
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
37C2×C4×Dic5C2211yescharacteristic
38C2×C4×Dic5C2211yescharacteristic
39#4044C418yescomplemented
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#2373—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—41no—
71C5×C4⋊C4—22no—
72C5×C4⋊C4—22no—
73C5×C4⋊C4—41no—
74C5×C4⋊C4C2×C414yescomplemented
75C42×C5—41no—
76#685—22no—
77#685—22no—
78#685—41no—
79#685C2×C414yes—
80#685C2×C414yes—
81#685C2×C414yes—
82C4⋊Dic5—22no—
83C4⋊Dic5C2×C414yescomplemented
84#683—22no—
85#683C2×C414yescomplemented
86C4×Dic5—22no—
87C4×Dic5—22no—
88C4×Dic5—22no—
89C4×Dic5—22no—
90C4×Dic5—41no—
91C4×Dic5—41no—
92C4×Dic5C2×C414yescomplemented
93C2×#126—51no—
94#389D511yescharacteristic, complemented
95#388—51no—
96#388—51no—
97#388—51no—
98#387—51no—
99#387—51no—
100#387—51no—
101#385—51no—
102C4×C4⋊C4—51no—
103C4×C4⋊C4—51no—
104C4×C4⋊C4—51no—
105C4×C22⋊C4—51no—
106C4×C22⋊C4—51no—
107#376—51no—
108C23×C5C2411yescharacteristic
109C22×D5—21no—
110C22×D5—21no—
111C22×D5—21no—
112C22×D5—21no—
113C22×D5—21no—
114C22×D5—21no—
115C22×D5C22×C412yes—
116C2×C4×C5—21no—
117C2×C4×C5—21no—
118C2×C4×C5—21no—
119C2×C4×C5—21no—
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×C5C22×C412yes—
133C2×C4×C5C22×C412yes—
134C2×C4×C5C22×C412yes—
135C2×C4×C5#4112yes—
136C2×C4×C5#4112yes—
137C2×Dic5—21no—
138C2×Dic5—21no—
139C2×Dic5—21no—
140C2×Dic5—21no—
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×Dic5C22×C412yes—
154C2×Dic5C22×C412yes—
155C2×Dic5C22×C412yes—
156C2×Dic5#4112yes—
157C2×Dic5#4112yes—
158C4×D5—44no—
159C23×C4—51no—
160#126—58no—
161C2×C4⋊C4—51no—
162C2×C4⋊C4—51no—
163C2×C4⋊C4—51no—
164C2×C4⋊C4—51no—
165C2×C4⋊C4C2×D511yescharacteristic
166C2×C4⋊C4C2×D511yescharacteristic
167C2×C4⋊C4C2×D511yescharacteristic
168C2×C4⋊C4C2×D511yescharacteristic
169C2×C22⋊C4—51no—
170C2×C22⋊C4—51no—
171C2×C22⋊C4—51no—
172C2×C22⋊C4—51no—
173C2×C22⋊C4—51no—
174C2×C22⋊C4—51no—
175C2×C42—51no—
176C2×C42—51no—
177C2×C42—51no—
178C2×C42—51no—
179C2×C42—51no—
180C2×C42—51no—
181C2×C42—51no—
182C2×C42C2×D511yescharacteristic
183#95—51no—
184#95—51no—
185#95—51no—
186#95—51no—
187#95—51no—
188#95—51no—
189#95—51no—
190#95—51no—
191#95—51no—
192#95—51no—
193#95C2×D511yescharacteristic
194#95C2×D511yescharacteristic
195C22×C5C23×C411yescharacteristic
196C22×C5C2×#4111yescharacteristic
197C22×C5C2×#4111yescharacteristic
198C22×C5C2×#4111yescharacteristic
199C22×C5C2×#4111yescharacteristic
200C22×C5H(5,𝔽2)11yescharacteristic
201C22×C52−1+411yescharacteristic
202C2×D5—22no—
203C2×D5—22no—
204C2×D5—22no—
205C2×D5—41no—
206C2×D5—41no—
207C2×D5—41no—
208C2×D5—41no—
209C4×C5—22no—
210C4×C5—22no—
211C4×C5—22no—
212C4×C5—22no—
213C4×C5—22no—
214C4×C5—41no—
215C4×C5—41no—
216Dic5—22no—
217Dic5—22no—
218Dic5—22no—
219Dic5—22no—
220Dic5—22no—
221Dic5—41no—
222Dic5—41no—
223C24—51no—
224C22×C4—51no—
225C22×C4—51no—
226C22×C4—51no—
227C22×C4—51no—
228C22×C4—51no—
229C22×C4—51no—
230C22×C4—51no—
231C22×C4—102no—
232C22×C4—102no—
233C22×C4—201no—
234C22×C4C22×D511yescharacteristic
235C22×C4C22×D511yescharacteristic
236C22×C4C22×D511yescharacteristic
237C22×C4C22×D511yescharacteristic
238C22×C4C22×D511yescharacteristic
239C22×C4C22×D511yescharacteristic
240C22×C4C22×D511yescharacteristic
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—41no—
244C4⋊C4—54no—
245C4⋊C4—54no—
246C4⋊C4—102no—
247C4⋊C4—102no—
248C4⋊C4C4×D514yescomplemented
249C22⋊C4—54no—
250C22⋊C4—54no—
251C22⋊C4—54no—
252C22⋊C4—102no—
253C22⋊C4—102no—
254C22⋊C4—201no—
255C42—41no—
256C42—54no—
257C42—102no—
258C42—102no—
259C42—102no—
260C42—102no—
261C42—201no—
262C42—201no—
263C2×C5C4×#4111yescharacteristic
264C2×C5C4×#4111yescharacteristic
265C2×C5#52011yescharacteristic
266C2×C5#55111yescharacteristic
267C2×C5#55211yescharacteristic
268C2×C5#55311yescharacteristic
269C2×C5#55611yescharacteristic
270D5—42no—
271C23—52no—
272C23—101no—
273C23—101no—
274C23—101no—
275C23—101no—
276C23—101no—
277C23—101no—
278C23C23×D511yescharacteristic
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—52no—
296C2×C4—52no—
297C2×C4—52no—
298C2×C4—52no—
299C2×C4—52no—
300C2×C4—101no—
301C2×C4—101no—
302C2×C4—101no—
303C2×C4—101no—
304C2×C4—101no—
305C2×C4—101no—
306C2×C4—101no—
307C2×C4—101no—
308C2×C4—101no—
309C2×C4—101no—
310C2×C4—101no—
311C2×C4—101no—
312C2×C4—101no—
313C2×C4—101no—
314C2×C4—101no—
315C2×C4—101no—
316C2×C4—204no—
317C2×C4C2×C4×D512yes—
318C2×C4C2×C4×D512yes—
319C2×C4C2×C4×D512yes—
320C2×C4#71112yes—
321C2×C4#71312yes—
322C5#237311yescharacteristic, complemented, Sylow
323C22—102no—
324C22—102no—
325C22—102no—
326C22—201no—
327C22—201no—
328C22—201no—
329C22—201no—
330C22C22×C4×D511yescharacteristic
331C22C2×#71111yescharacteristic
332C22#414411yescharacteristic
333C22C2×#71311yescharacteristic
334C22#414711yescharacteristic
335C22D5×#4111yescharacteristic
336C22D5×#4111yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—102no—
345C4—102no—
346C4—102no—
347C4—102no—
348C4—102no—
349C4—201no—
350C4—201no—
351C2—202no—
352C2#6583811yescharacteristic
353C2C4×#71111yescharacteristic
354C2C4×#71311yescharacteristic
355C2#6593511yescharacteristic
356C2#6597111yescharacteristic
357C2#6602711yescharacteristic
358C2#6606611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutatorC22×C5C23×C4
FittingC5×#389C2
socle, cosocleC23×C5C24
prime radicalC23×D5C23
hypercenter#389D5
hypocenterC5#2373
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C5×#389C2semidirect
6. #65270C2semidirect
7. #65267C2semidirect
8. C4⋊C4×Dic5C2semidirect
9. C4⋊C4×Dic5C2semidirect
11. C4×#683C2semidirect
14. #64942C2semidirect
15. #64937C2semidirect
39. #4044C4semidirect
74. C5×C4⋊C4C2×C4semidirect
83. C4⋊Dic5C2×C4semidirect
85. #683C2×C4semidirect
92. C4×Dic5C2×C4semidirect
94. #389D5semidirect
248. C4⋊C4C4×D5semidirect
322. C5#2373semidirect

Maximal subgroups

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

subgroupindex
C2×#40442
C5×#3892
#652792
#652702
#652672
C4⋊C4×Dic52
C4×#6852
C4×#6832
#649572
#649552
#649422
#649372
#23735