Finite Group Databasebeta

← #10611545

Subgroups — #10611545

Order 640.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#65906C211yesmaximal, characteristic, complemented
2#65895C211yesmaximal, characteristic
3C5×#440C211yesmaximal, characteristic, complemented
4#65459C211yesmaximal, characteristic
5#65455C211yesmaximal, characteristic, complemented
6C4×#689C211yesmaximal, characteristic, complemented
7C4×#687C211yesmaximal, characteristic, complemented
8#65177C211yesmaximal, characteristic, complemented
9#65164C211yesmaximal, characteristic, complemented
10#65097C211yesmaximal, characteristic, complemented
11#65080C211yesmaximal, characteristic, complemented
12#65068C211yesmaximal, characteristic
13#65056C211yesmaximal, characteristic, complemented
14#65005C211yesmaximal, characteristic, complemented
15#64998C211yesmaximal, characteristic, complemented
16C2×Q8×D5C2211yescharacteristic
17C2×C5×SD16C2211yescharacteristic
18C4×C5×Q8C2211yescharacteristic
19C4×C5×D4C2211yescharacteristic
20Q8×Dic5C2211yescharacteristic
21C2×#689C2211yescharacteristic
22#4084C2211yescharacteristic
23#4082C2211yescharacteristic
24C2×#687C2211yescharacteristic
25#4075—21no—
26#4070—21no—
27C2×Dic20C2211yescharacteristic
28#4043C2211yescharacteristic
29#4038—21no—
30#4038C2211yescharacteristic
31#4034C2211yescharacteristic
32#4033—21no—
33#4033C2211yescharacteristic
34#4031—21no—
35#4025—21no—
36#4018—21no—
37#4018C2211yescharacteristic
38C4×Dic10—21no—
39C4×Dic10C2211yescharacteristic
40C4×Dic10C2211yescharacteristic
41C5×#107C2211yescharacteristic
42C5×#103C2211yescharacteristic
43C5×#102C2211yescharacteristic
44C5×#97C2211yescharacteristic
45#3967C2211yescharacteristic
46#3964C2211yescharacteristic
47#3952C2211yescharacteristic
48#3952C2211yescharacteristic
49#3949C2211yescharacteristic
50#3948C2211yescharacteristic
51#3948C2211yescharacteristic
52#3946C2211yescharacteristic
53#3946C2211yescharacteristic
54#3942C2211yescharacteristic
55#3942C2211yescharacteristic
56#3940C2211yescharacteristic
57#3939C2211yescharacteristic
58C4×#168C2211yescharacteristic
59#3121—51nomaximal, Sylow
60C2×C5×Q8C2311yescharacteristic
61C2×C5×D4C2311yescharacteristic
62C22×C4×C5—21no—
63C2×C22⋊D5—21no—
64Q8×D5—22no—
65Q8×D5—22no—
66C2×C4×D5—21no—
67C2×C4×D5C2311yescharacteristic
68C2×Dic10—21no—
69C2×Dic10—21no—
70C2×Dic10C2311yescharacteristic
71C2×Dic10C2311yescharacteristic
72C5×SD16—22no—
73C2×C5×C8C2311yescharacteristic
74C2×C5×C8C2311yescharacteristic
75C5×C4⋊C4—21no—
76C5×C4⋊C4C2311yescharacteristic
77C5×C4⋊C4C2311yescharacteristic
78C5×C22⋊C4—21no—
79C42×C5—21no—
80C42×C5C2311yescharacteristic
81C22⋊Dic5—21no—
82C22⋊Dic5—21no—
83C22⋊Dic5—21no—
84#689—22no—
85#687—22no—
86#685—21no—
87#685—21no—
88#685D411yescharacteristic
89#685D411yescharacteristic
90C4⋊Dic5—21no—
91C4⋊Dic5—21no—
92C4⋊Dic5C2311yescharacteristic
93C4⋊Dic5C2311yescharacteristic
94#683—21no—
95#683—21no—
96#683—21no—
97#683—21no—
98#683D411yescharacteristic
99#683D411yescharacteristic
100C4×Dic5—21no—
101C4×Dic5—21no—
102C4×Dic5—21no—
103C4×Dic5—21no—
104C4×Dic5—21no—
105C4×Dic5C2311yescharacteristic
106C2×#168C2311yescharacteristic
107C2×#168C2311yescharacteristic
108Dic20—22no—
109#542—51no—
110#519—51no—
111#485—51no—
112#482—51no—
113#477—51no—
114#475—51no—
115#464—51no—
116#464—51no—
117#451—51no—
118#448—51no—
119#441—51no—
120#440D511yescharacteristic, complemented
121C4×Q16—51no—
122C4×SD16—51no—
123#433—51no—
124C23×C5—21no—
125C22×D5C2×D411yescharacteristic
126C5×Q8—21no—
127C5×Q8#4112yes—
128C5×D4—21no—
129C5×D4—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—81no—
134C2×C4×C5C2×D411yescharacteristic
135C2×C4×C5C2×D411yescharacteristic
136C2×C4×C5C2411yescharacteristic
137C22⋊D5—81no—
138C2×Dic5—21no—
139C2×Dic5—21no—
140C2×Dic5—21no—
141C2×Dic5—21no—
142C2×Dic5C2×D411yescharacteristic
143C2×Dic5C2×D411yescharacteristic
144C2×Dic5C2×D411yescharacteristic
145C4×D5—24no—
146C4×D5—21no—
147C4×D5—21no—
148Dic10—24no—
149Dic10—21no—
150Dic10—21no—
151Dic10—21no—
152Dic10—81no—
153Dic10#4112yes—
154C5×C8—21no—
155C5×C8—21no—
156#168—21no—
157#168—21no—
158C22×Q8—51no—
159C2×Q16—51no—
160C2×Q16—51no—
161C2×SD16—51no—
162C2×SD16C2×D511yescharacteristic
163C4⋊Q8—51no—
164#126—101no—
165#124—51no—
166#124—101no—
167#123—101no—
168C22⋊Q8—51no—
169C22⋊Q8—101no—
170#121—51no—
171C4×Q8—51no—
172C4×Q8—51no—
173C4×Q8—51no—
174C4×Q8—51no—
175C4×Q8—101no—
176C4×Q8—101no—
177C4×Q8C2×D511yescharacteristic
178C4×D4C2×D511yescharacteristic
179#117—51no—
180#117—51no—
181#117—101no—
182#117—101no—
183#107—51no—
184#107C2×D511yescharacteristic
185#106—51no—
186#106—51no—
187C4⋊C8—51no—
188C4⋊C8—51no—
189#103—51no—
190#103—51no—
191#103—51no—
192#103—51no—
193#103—51no—
194#103C2×D511yescharacteristic
195#102—51no—
196#102C2×D511yescharacteristic
197C22⋊C8—51no—
198C22⋊C8—51no—
199#97C2×D511yescharacteristic
200C4×C8—51no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—41no—
204C22×C5C22×D411yescharacteristic
205C2×D5—21no—
206C2×D5—21no—
207C2×D5—21no—
208C4×C5—22no—
209C4×C5—21no—
210C4×C5—21no—
211C4×C5—41no—
212C4×C5—41no—
213C4×C5C2×#4111yescharacteristic
214C4×C5C2×#4111yescharacteristic
215Dic5—21no—
216Dic5—22no—
217Dic5—21no—
218Dic5—21no—
219Dic5—41no—
220Dic5—41no—
221Dic5—41no—
222C2×Q8—51no—
223C2×Q8—51no—
224C2×Q8—101no—
225C2×Q8—101no—
226C2×Q8—102no—
227C2×Q8—102no—
228C2×Q8C22×D511yescharacteristic
229C2×D4—101no—
230C2×D4C22×D511yescharacteristic
231C22×C4—21no—
232C22×C4—51no—
233C22×C4—101no—
234Q16—102no—
235Q16—102no—
236SD16—22no—
237SD16—102no—
238C2×C8—51no—
239C2×C8—51no—
240C2×C8C22×D511yescharacteristic
241C2×C8C22×D511yescharacteristic
242C4⋊C4—21no—
243C4⋊C4—51no—
244C4⋊C4—51no—
245C4⋊C4—51no—
246C4⋊C4—51no—
247C4⋊C4—101no—
248C4⋊C4—101no—
249C4⋊C4—101no—
250C4⋊C4—101no—
251C4⋊C4—101no—
252C4⋊C4—101no—
253C4⋊C4C22×D511yescharacteristic
254C4⋊C4C22×D511yescharacteristic
255C22⋊C4—21no—
256C22⋊C4—51no—
257C22⋊C4—51no—
258C22⋊C4—101no—
259C22⋊C4—101no—
260C22⋊C4—101no—
261C22⋊C4—101no—
262C22⋊C4—101no—
263C42—21no—
264C42—51no—
265C42—101no—
266C42—101no—
267C42—101no—
268C42—101no—
269C42—101no—
270C42C22×D511yescharacteristic
271C2×C5—81no—
272C2×C5#52511yescharacteristic
273C2×C5#57711yescharacteristic
274C2×C5#57811yescharacteristic
275D5—41no—
276C23—21no—
277C23—51no—
278Q8—21no—
279Q8—52no—
280Q8—104no—
281Q8—101no—
282Q8—101no—
283Q8—101no—
284Q8—401no—
285Q8#70912yes—
286D4—21no—
287D4—21no—
288D4—401no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—51no—
293C2×C4—51no—
294C2×C4—51no—
295C2×C4—81no—
296C2×C4—104no—
297C2×C4—101no—
298C2×C4—101no—
299C2×C4—101no—
300C2×C4—101no—
301C2×C4—101no—
302C2×C4—101no—
303C2×C4D4×D511yescharacteristic
304C2×C4D4×D511yescharacteristic
305C2×C4C23×D511yescharacteristic
306C8—21no—
307C8—21no—
308C8—101no—
309C8—101no—
310C5#312111yescharacteristic, complemented, Sylow
311C22—41no—
312C22—41no—
313C22—41no—
314C22—101no—
315C22—101no—
316C22—101no—
317C22C2×D4×D511yescharacteristic
318C4—21no—
319C4—21no—
320C4—22no—
321C4—41no—
322C4—41no—
323C4—101no—
324C4—101no—
325C4—101no—
326C4—102no—
327C4—201no—
328C4—201no—
329C4—201no—
330C4C2×#70911yescharacteristic
331C4D5×#4111yescharacteristic
332C2—81no—
333C2—201no—
334C2#6588111yescharacteristic
335C2#6610911yescharacteristic
336C2#6611311yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#440C2
socleC22×C5C22×D4
prime radical#4084C22
hypercenter#440D5
hypocenterC5#3121
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #65906C2semidirect
3. C5×#440C2semidirect
5. #65455C2semidirect
6. C4×#689C2semidirect
7. C4×#687C2semidirect
8. #65177C2semidirect
9. #65164C2semidirect
10. #65097C2semidirect
11. #65080C2semidirect
13. #65056C2semidirect
14. #65005C2semidirect
15. #64998C2semidirect
120. #440D5semidirect
310. C5#3121semidirect

Maximal subgroups

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

subgroupindex
#659062
#658952
C5×#4402
#654592
#654552
C4×#6892
C4×#6872
#651772
#651642
#650972
#650802
#650682
#650562
#650052
#649982
#31215