Finite Group Databasebeta

← #10614404

Subgroups — #10614404

Order 640.

574 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#121C211yesmaximal, characteristic
2C2×#4025C211yesmaximal, characteristic, complemented
3#65511C211yesmaximal, characteristic, complemented
4#65511C211yesmaximal, characteristic, complemented
5#65511C211yesmaximal, characteristic, complemented
6#65510C211yesmaximal, characteristic, complemented
7#65509C211yesmaximal, characteristic, complemented
8#65509C211yesmaximal, characteristic, complemented
9#65499C211yesmaximal, characteristic, complemented
10C4×C22⋊Dic5C211yesmaximal, characteristic, complemented
11#65275C211yesmaximal, characteristic, complemented
12#65261C211yesmaximal, characteristic, complemented
13#65234C211yesmaximal, characteristic, complemented
14#65233C211yesmaximal, characteristic, complemented
15C22⋊C4×Dic5C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C22×C5×D4C2211yescharacteristic
19C23×C4×C5C2211yescharacteristic
20C23×Dic5C2211yescharacteristic
21C5×#121—24no—
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C22⋊Dic5C2211yescharacteristic
33C2×C22⋊Dic5C2211yescharacteristic
34C2×C22⋊Dic5C2211yescharacteristic
35C2×C22⋊Dic5C2211yescharacteristic
36C2×C4⋊Dic5C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×C4×Dic5C2211yescharacteristic
42C2×C4×Dic5C2211yescharacteristic
43C2×C4×Dic5C2211yescharacteristic
44#4025—24no—
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2559—51nomaximal, Sylow
54C24×C5C2311yescharacteristic
55C24×C5C2311yescharacteristic
56C24×C5C2311yescharacteristic
57C2×C5×D4—22no—
58C2×C5×D4—22no—
59C2×C5×D4—22no—
60C2×C5×D4—22no—
61C2×C5×D4—22no—
62C2×C5×D4—22no—
63C2×C5×D4—41no—
64C2×C5×D4—41no—
65C2×C5×D4—41no—
66C22×C4×C5—22no—
67C22×C4×C5—22no—
68C22×C4×C5—41no—
69C22×C4×C5C2311yescharacteristic
70C22×C4×C5C2311yescharacteristic
71C22×C4×C5C2311yescharacteristic
72C22×C4×C5C2311yescharacteristic
73C22×Dic5—22no—
74C22×Dic5—41no—
75C22×Dic5—41no—
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C22×Dic5C2311yescharacteristic
81C22×Dic5C2311yescharacteristic
82C22×Dic5C2311yescharacteristic
83C22×Dic5C2311yescharacteristic
84C5×C4⋊C4—22no—
85C5×C22⋊C4—22no—
86C5×C22⋊C4D414yes—
87C22⋊Dic5—22no—
88C22⋊Dic5—22no—
89C22⋊Dic5—22no—
90C22⋊Dic5—22no—
91C22⋊Dic5—22no—
92C22⋊Dic5—22no—
93C22⋊Dic5—22no—
94C22⋊Dic5—22no—
95C22⋊Dic5—41no—
96C22⋊Dic5—41no—
97C22⋊Dic5—41no—
98C4⋊Dic5—22no—
99#683—22no—
100#683—22no—
101#683—22no—
102#683—41no—
103C4×Dic5—22no—
104C4×Dic5—41no—
105C4×Dic5—41no—
106C2×#126—51no—
107C2×#121D511yescharacteristic
108#400—51no—
109#394—51no—
110#394—51no—
111#394—51no—
112#390—51no—
113#389—51no—
114#388—51no—
115#388—51no—
116#386—51no—
117#386—51no—
118#385—51no—
119C4×C22⋊C4—51no—
120C4×C22⋊C4—51no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5—21no—
127C23×C5—21no—
128C23×C5—21no—
129C23×C5—21no—
130C23×C5—21no—
131C23×C5—21no—
132C23×C5—21no—
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—21no—
137C23×C5—21no—
138C23×C5—21no—
139C23×C5—21no—
140C23×C5C2×D412yes—
141C23×C5#4112yes—
142C23×C5C2411yescharacteristic
143C5×D4—44no—
144C5×D4—44no—
145C5×D4—44no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5—21no—
155C2×C4×C5—44no—
156C2×C4×C5C2×D412yes—
157C2×C4×C5C2×D412yes—
158C2×C4×C5#4112yes—
159C2×Dic5—21no—
160C2×Dic5—21no—
161C2×Dic5—21no—
162C2×Dic5—21no—
163C2×Dic5—21no—
164C2×Dic5—21no—
165C2×Dic5—21no—
166C2×Dic5—21no—
167C2×Dic5—21no—
168C2×Dic5—21no—
169C2×Dic5—21no—
170C2×Dic5—21no—
171C2×Dic5—21no—
172C2×Dic5—21no—
173C2×Dic5—21no—
174C2×Dic5—21no—
175C2×Dic5—21no—
176C2×Dic5—21no—
177C2×Dic5—21no—
178C2×Dic5—21no—
179C2×Dic5—21no—
180C2×Dic5—21no—
181C2×Dic5—44no—
182C2×Dic5#4112yes—
183C2×Dic5#4112yes—
184C22×D4C2×D511yescharacteristic
185C22×D4C2×D511yescharacteristic
186C22×D4C2×D511yescharacteristic
187C23×C4—51no—
188C23×C4C2×D511yescharacteristic
189#126—104no—
190#121—24no—
191C2×C4⋊C4—51no—
192C2×C4⋊C4—51no—
193C2×C4⋊C4—51no—
194C2×C4⋊C4—51no—
195C2×C4⋊C4—51no—
196C2×C4⋊C4C2×D511yescharacteristic
197C2×C22⋊C4—51no—
198C2×C22⋊C4—51no—
199C2×C22⋊C4—51no—
200C2×C22⋊C4—51no—
201C2×C22⋊C4—51no—
202C2×C22⋊C4—51no—
203C2×C22⋊C4—51no—
204C2×C22⋊C4—51no—
205C2×C22⋊C4—51no—
206C2×C22⋊C4—51no—
207C2×C22⋊C4—51no—
208C2×C22⋊C4C2×D511yescharacteristic
209C2×C22⋊C4C2×D511yescharacteristic
210C2×C42—51no—
211C2×C42—51no—
212C2×C42—51no—
213#95—51no—
214#95—51no—
215#95—51no—
216#95—51no—
217#95—51no—
218#95—51no—
219#95—51no—
220#95—51no—
221C22×C5—22no—
222C22×C5—22no—
223C22×C5—22no—
224C22×C5—22no—
225C22×C5—22no—
226C22×C5—22no—
227C22×C5—41no—
228C22×C5—41no—
229C22×C5—41no—
230C22×C5—41no—
231C22×C5—41no—
232C22×C5—41no—
233C22×C5—41no—
234C22×C5—41no—
235C22×C5—41no—
236C22×C5—41no—
237C22×C5—41no—
238C22×C5—41no—
239C22×C5—41no—
240C22×C5—41no—
241C22×C5—41no—
242C22×C5C22×D411yescharacteristic
243C22×C5C2×#4111yescharacteristic
244C22×C5C2×#4111yescharacteristic
245C22×C5C2×#4111yescharacteristic
246C22×C5C2×#4111yescharacteristic
247C22×C5H(5,𝔽2)11yescharacteristic
248C22×C5H(5,𝔽2)11yescharacteristic
249C4×C5—22no—
250C4×C5—22no—
251C4×C5—22no—
252C4×C5—41no—
253Dic5—22no—
254Dic5—22no—
255Dic5—41no—
256Dic5—41no—
257Dic5—41no—
258Dic5—41no—
259Dic5—41no—
260Dic5—41no—
261C24C22×D511yescharacteristic
262C24C22×D511yescharacteristic
263C24C22×D511yescharacteristic
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—51no—
277C22×C4—51no—
278C22×C4—51no—
279C22×C4—51no—
280C22×C4—51no—
281C22×C4—51no—
282C22×C4—51no—
283C22×C4—51no—
284C22×C4—102no—
285C22×C4—201no—
286C22×C4—201no—
287C22×C4C22×D511yescharacteristic
288C22×C4C22×D511yescharacteristic
289C22×C4C22×D511yescharacteristic
290C22×C4C22×D511yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—102no—
293C4⋊C4—102no—
294C4⋊C4—102no—
295C4⋊C4—102no—
296C4⋊C4—201no—
297C22⋊C4—22no—
298C22⋊C4—102no—
299C22⋊C4—102no—
300C22⋊C4—102no—
301C22⋊C4—102no—
302C22⋊C4—102no—
303C22⋊C4—102no—
304C22⋊C4—102no—
305C22⋊C4—102no—
306C22⋊C4—201no—
307C22⋊C4—201no—
308C22⋊C4—201no—
309C22⋊C4C22⋊D514yes—
310C42—102no—
311C42—201no—
312C42—201no—
313C2×C5—42no—
314C2×C5—42no—
315C2×C5—81no—
316C2×C5#52911yescharacteristic
317C2×C5#53211yescharacteristic
318C2×C5#53811yescharacteristic
319C2×C5#54611yescharacteristic
320C2×C5#54611yescharacteristic
321C2×C5#55311yescharacteristic
322C2×C5#55311yescharacteristic
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23#71112yes—
343C23C2×C22⋊D512yes—
344C23C23×D511yescharacteristic
345D4—44no—
346D4—44no—
347D4—44no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—44no—
358C2×C4—52no—
359C2×C4—52no—
360C2×C4—101no—
361C2×C4—101no—
362C2×C4—101no—
363C2×C4—101no—
364C2×C4—101no—
365C2×C4—101no—
366C2×C4—101no—
367C2×C4—101no—
368C2×C4—101no—
369C2×C4—101no—
370C2×C4—101no—
371C2×C4—101no—
372C2×C4—101no—
373C2×C4—101no—
374C2×C4—101no—
375C2×C4—101no—
376C2×C4—101no—
377C2×C4—101no—
378C2×C4—101no—
379C2×C4—101no—
380C2×C4—101no—
381C2×C4—101no—
382C2×C4—204no—
383C2×C4#71112yes—
384C2×C4C2×C22⋊D512yes—
385C2×C4C2×C22⋊D512yes—
386C5#255911yescharacteristic, complemented, Sylow
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22C2×#71111yescharacteristic
409C22C2×#71111yescharacteristic
410C22#414411yescharacteristic
411C22#414411yescharacteristic
412C22D5×#4111yescharacteristic
413C22D5×#4111yescharacteristic
414C22C22×C22⋊D511yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—102no—
420C4—102no—
421C4—201no—
422C4—201no—
423C4—201no—
424C4—201no—
425C4—201no—
426C4—201no—
427C2—42no—
428C2—42no—
429C2—81no—
430C2#6593411yescharacteristic
431C2#6593511yescharacteristic
432C2#6594711yescharacteristic
433C2#6594811yescharacteristic
434C2#6614011yescharacteristic
435C2#6614011yescharacteristic
436C2#6616611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
Fitting, prime radicalC2×C5×#121C2
hypercenterC2×#121D5
hypocenterC5#2559
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#4025C2semidirect
3. #65511C2semidirect
4. #65511C2semidirect
5. #65511C2semidirect
6. #65510C2semidirect
7. #65509C2semidirect
8. #65509C2semidirect
9. #65499C2semidirect
10. C4×C22⋊Dic5C2semidirect
11. #65275C2semidirect
12. #65261C2semidirect
13. #65234C2semidirect
14. #65233C2semidirect
15. C22⋊C4×Dic5C2semidirect
386. C5#2559semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1212
C2×#40252
#655112
#655102
#655092
#654992
C4×C22⋊Dic52
#652752
#652612
#652342
#652332
C22⋊C4×Dic52
#25595