Finite Group Databasebeta

← #72260148

Subgroups — #72260148

Order 2400.

772 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11925657C211yesmaximal, characteristic, complemented
2#11925646C212yesmaximal, complemented
3C2×#10597116C211yesmaximal, characteristic, complemented
4C5×D3×C22⋊D5C212yesmaximal, complemented
5C3×#87880C211yesmaximal, characteristic, complemented
6#11925449C212yesmaximal, complemented
7#11925435C212yesmaximal, complemented
8#11925423C212yesmaximal, complemented
9D5×#1203C212yesmaximal, complemented
10#11717495—31nomaximal, Hall
11C22×C3⋊(C5⋊D5)C2211yescharacteristic, complemented
12C22×C52×D3C2211yescharacteristic, complemented
13C2×D5×D15C2212yescomplemented
14C2×C5×D3×D5C2212yescomplemented
15C5×#1220C2212yescomplemented
16C2×C3×#1039C2211yescharacteristic, complemented
17C3×C5×C22⋊D5C2212yescomplemented
18#10597129C2211yescharacteristic, complemented
19#10597128C2211yescharacteristic, complemented
20#10597125C2212yescomplemented
21#10597123C2212yescomplemented
22#10597122C2212yescomplemented
23#10597120C2212yescomplemented
24#10597116C2212yescomplemented
25#10597116C2212yescomplemented
26C5×#1203C2212yescomplemented
27C5×#1201C2212yescomplemented
28C5×D3×Dic5C2212yescomplemented
29C3×#6039C2211yescharacteristic, complemented
30C3×#6037C2211yescharacteristic, complemented
31C3×D5×Dic5C2212yescomplemented
32#92027—52nomaximal
33C2×C22⋊(C5⋊D5)—31no—
34C2×C5×C22⋊D5—32no—
35#87886—32no—
36D5×C22⋊D5—32no—
37D5×C22⋊D5—32no—
38#87880—31no—
39#87880D311yescharacteristic, complemented
40#87879—32no—
41#87879—32no—
42C22×C3×C52C2311yescharacteristic
43C2×C3⋊(C5⋊D5)—41no—
44C2×C3⋊(C5⋊D5)C2312yes—
45C2×C5×D15C2312yes—
46C2×C52×D3—41no—
47C2×C52×D3C2312yes—
48C2×C3×C5×D5C2312yes—
49D5×D15—42no—
50C5×D3×D5—42no—
51C5×Dic15C2312yes—
52C3×#1039C2311yescharacteristic
53C3×#1039C2311yescharacteristic
54C3×C5×Dic5C2312yes—
55D4×D15—52no—
56C5×D3×D4—52no—
57C3×#711—52no—
58#6790—52no—
59#6790—52no—
60D3×C22⋊D5—52no—
61D3×C22⋊D5D512yescomplemented
62C2×#1203—52no—
63C2×#1203—52no—
64C2×#1203—102no—
65#6782—52no—
66#6782—52no—
67D5×D12—52no—
68#6771—52no—
69#6769—52no—
70#6768—52no—
71C23×C52—31no—
72C22×C5⋊D5—31no—
73C22×C5×D5—32no—
74C2×D52—32no—
75C22⋊(C5⋊D5)—32no—
76C22⋊(C5⋊D5)—32no—
77C2×#1039C2×D311yescharacteristic, complemented
78C5×C22⋊D5—32no—
79C5×C22⋊D5—32no—
80C5×C22⋊D5—32no—
81C5×C22⋊D5C2×D312yescomplemented
82C2×C5×Dic5—32no—
83#6039—31no—
84#6039C2×D311yescharacteristic, complemented
85#6038—32no—
86#6038—32no—
87#6037—31no—
88#6037C2×D311yescharacteristic, complemented
89#6036—32no—
90D5×Dic5—32no—
91D5×Dic5C2×D312yescomplemented
92#4144—152no—
93C2×C3×C52—21no—
94C2×C3×C52C2411yescharacteristic
95C3⋊(C5⋊D5)—22no—
96C5×D15—22no—
97C52×D3—22no—
98C3×C5×D5—22no—
99C22×D15—52no—
100C22×D15—52no—
101C22×D15—102no—
102C22×C5×D3—22no—
103C22×C5×D3—52no—
104C22×C5×D3C2×D512yescomplemented
105C2×D3×D5—52no—
106C2×D3×D5C2×D512yescomplemented
107C3×C5×D4—52no—
108#1220—52no—
109#1220C2×D512yescomplemented
110D60—52no—
111C4×D15—52no—
112C5×#67—52no—
113C5×#67—52no—
114C5×D12—52no—
115C4×C5×D3—52no—
116C3×C22⋊D5—52no—
117C3×C22⋊D5C2×D512yescomplemented
118C2×C3×Dic5—52no—
119C2×C3×Dic5—52no—
120C2×C3×Dic5—102no—
121C3×C4×D5—52no—
122C3×Dic10—52no—
123#1204—52no—
124#1203—52no—
125#1203—52no—
126#1203—52no—
127#1203—52no—
128#1203—52no—
129#1203—52no—
130#1203—104no—
131#1203—104no—
132#1203C2×D512yescomplemented
133#1202—52no—
134#1201C2×D512yescomplemented
135#1200—52no—
136#1200—52no—
137D3×Dic5—52no—
138D3×Dic5C2×D512yescomplemented
139C22×C52—32no—
140C22×C52—121no—
141C22×C52C22×D311yescharacteristic
142C2×C5⋊D5—32no—
143C2×C5⋊D5—121no—
144C2×C5×D5—32no—
145C2×C5×D5—122no—
146C2×C5×D5C22×D312yes—
147D52—122no—
148#1039C22×D311yescharacteristic
149#1039C22×D311yescharacteristic
150C5×Dic5—32no—
151C5×Dic5C22×D312yes—
152#1009—251noHall
153C2×C5×D4—152no—
154C2×C22⋊D5—32no—
155C2×C22⋊D5—152no—
156C2×C22⋊D5—152no—
157C2×C22⋊D5—152no—
158C2×C22⋊D5—302no—
159#711—52no—
160#711—152no—
161#711—152no—
162#711—152no—
163D4×D5—152no—
164D4×D5—152no—
165D4×D5—152no—
166D4×D5—152no—
167#709—152no—
168#709—152no—
169C3×C52H(5,𝔽2)11yescharacteristic, complemented, Hall
170C22×C3×C5—22no—
171C22×C3×C5—52no—
172C22×C3×C5C22×D512yes—
173C2×D15—52no—
174C2×D15—52no—
175C2×D15—104no—
176C2×D15—202no—
177C2×D15—202no—
178C2×D15—204no—
179C2×D15C22×D512yes—
180C2×C5×D3—24no—
181C2×C5×D3—42no—
182C2×C5×D3—44no—
183C2×C5×D3—52no—
184C2×C5×D3—202no—
185C2×C5×D3C22×D512yes—
186C2×C5×D3C22×D512yes—
187C2×C3×D5C22×D512yes—
188D3×D5—42no—
189D3×D5—202no—
190C3×C4×C5—52no—
191Dic15C22×D512yes—
192C3×Dic5—52no—
193C3×Dic5—52no—
194C3×Dic5—102no—
195C3×Dic5—102no—
196C3×Dic5C22×D512yes—
197C5×Dic3—52no—
198C2×C52—21no—
199C2×C52—62no—
200C2×C52C23×D311yescharacteristic
201C5⋊D5—62no—
202C5×D5—22no—
203C5×D5—62no—
204C3×#41—251no—
205#239—252no—
206D3×D4—52no—
207D3×D4—252no—
208D3×D4—252no—
209#235—252no—
210#235—251no—
211C2×D12—252no—
212C2×D12—251no—
213C23×C5—32no—
214C23×C5—62no—
215C23×C5—152no—
216C22×D5—32no—
217C22×D5—152no—
218C22×D5—152no—
219C22×D5—152no—
220C22×D5—302no—
221C5×D4—52no—
222C5×D4—152no—
223C5×D4—152no—
224C5×D4—152no—
225C2×C4×C5—152no—
226C22⋊D5—32no—
227C22⋊D5—32no—
228C22⋊D5—32no—
229C22⋊D5—52no—
230C22⋊D5—152no—
231C22⋊D5—152no—
232C22⋊D5—152no—
233C22⋊D5—152no—
234C22⋊D5—152no—
235C22⋊D5—152no—
236C22⋊D5—152no—
237C22⋊D5—304no—
238C22⋊D5—304no—
239C22⋊D5D3×D512yescomplemented
240C2×Dic5—32no—
241C2×Dic5—52no—
242C2×Dic5—52no—
243C2×Dic5—102no—
244C2×Dic5—152no—
245D20—152no—
246D20—152no—
247C4×D5—52no—
248C4×D5—152no—
249C4×D5—152no—
250C4×D5—152no—
251Dic10—52no—
252Dic10—152no—
253H(5,𝔽2)—751noSylow
254C2×C3×C5—24no—
255C2×C3×C5—22no—
256C2×C3×C5—22no—
257C2×C3×C5—102no—
258C2×C3×C5C23×D512yes—
259D15—22no—
260D15—102no—
261D15—102no—
262D15—204no—
263C3×D5—22no—
264C5×D3—22no—
265C5×D3—22no—
266C5×D3—44no—
267C5×D3—102no—
268C52#100911yescharacteristic, complemented, Sylow
269C22×D3—52no—
270C22×D3—252no—
271C22×D3—251no—
272C22×D3D5211yescharacteristic, complemented
273C3×Q8—251no—
274C3×D4—52no—
275C3×D4—251no—
276C2×C3×C4—252no—
277C2×C3×C4—251no—
278#67—52no—
279#67—52no—
280#67—252no—
281D12—52no—
282D12—252no—
283D12—251no—
284D12—252no—
285D12—252no—
286C4×D3—52no—
287C4×D3—252no—
288C4×D3—252no—
289Dic6—251no—
290C22×C5—22no—
291C22×C5—32no—
292C22×C5—32no—
293C22×C5—52no—
294C22×C5—64no—
295C22×C5—122no—
296C22×C5—124no—
297C22×C5—152no—
298C22×C5—602no—
299C22×C5C2×D3×D512yes—
300C2×D5—32no—
301C2×D5—122no—
302C2×D5—152no—
303C2×D5—152no—
304C2×D5—304no—
305C2×D5—602no—
306C2×D5—602no—
307C2×D5—602no—
308C2×D5—604no—
309C2×D5C2×D3×D512yes—
310C4×C5—52no—
311C4×C5—152no—
312Dic5—32no—
313Dic5—52no—
314Dic5—52no—
315Dic5—102no—
316Dic5—102no—
317Dic5C2×D3×D512yes—
318#41—251no—
319#41—752no—
320#41—752no—
321#41—751no—
322C2×D4—152no—
323C2×D4—752no—
324C2×D4—751no—
325C2×D4—752no—
326C2×D4—752no—
327C3×C5—22no—
328C3×C5#414412yescomplemented
329C22×C3—52no—
330C22×C3C2×D5211yescharacteristic
331C2×D3—41no—
332C2×D3—52no—
333C2×D3—202no—
334C2×D3—252no—
335C2×D3—1002no—
336C2×D3—1001no—
337C2×D3C2×D5212yes—
338C3×C4—52no—
339C3×C4—251no—
340C3×C4—251no—
341Dic3—52no—
342C2×C5—24no—
343C2×C5—22no—
344C2×C5—22no—
345C2×C5—62no—
346C2×C5—62no—
347C2×C5—102no—
348C2×C5—124no—
349C2×C5—302no—
350C2×C5C22×D3×D512yes—
351D5—22no—
352D5—62no—
353D5—302no—
354D5—302no—
355D5—604no—
356C23—31no—
357C23—152no—
358C23—752no—
359C23—751no—
360Q8—251no—
361Q8—751no—
362D4—52no—
363D4—152no—
364D4—152no—
365D4—152no—
366D4—251no—
367D4—752no—
368D4—752no—
369D4—752no—
370D4—752no—
371D4—751no—
372C2×C4—152no—
373C2×C4—252no—
374C2×C4—251no—
375C2×C4—752no—
376C2×C4—752no—
377C2×C3—21no—
378C2×C3—102no—
379C2×C3C22×D5211yescharacteristic
380D3—22no—
381D3—102no—
382D3—502no—
383C5—22no—
384C5#9202712yescomplemented
385C22—32no—
386C22—52no—
387C22—121no—
388C22—152no—
389C22—3001no—
390C22—3002no—
391C22—602no—
392C22—752no—
393C22D3×D5211yescharacteristic
394C4—52no—
395C4—152no—
396C4—251no—
397C4—251no—
398C3#1171749511yescharacteristic, complemented, Sylow
399C2—21no—
400C2—62no—
401C2—102no—
402C2—302no—
403C2—1502no—
404C2C2×D3×D5211yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×D52
commutator, socle, cosocleC2×C3×C52C24
FittingC22×C3×C52C23
hypercenterC22D3×D52
hypocenterC3×C52H(5,𝔽2)
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #11925657C2semidirect
2. #11925646C2semidirect
3. C2×#10597116C2semidirect
4. C5×D3×C22⋊D5C2semidirect
5. C3×#87880C2semidirect
6. #11925449C2semidirect
7. #11925435C2semidirect
8. #11925423C2semidirect
9. D5×#1203C2semidirect
11. C22×C3⋊(C5⋊D5)C22semidirect
12. C22×C52×D3C22semidirect
13. C2×D5×D15C22semidirect
14. C2×C5×D3×D5C22semidirect
15. C5×#1220C22semidirect
16. C2×C3×#1039C22semidirect
17. C3×C5×C22⋊D5C22semidirect
18. #10597129C22semidirect
19. #10597128C22semidirect
20. #10597125C22semidirect
21. #10597123C22semidirect
22. #10597122C22semidirect
23. #10597120C22semidirect
24. #10597116C22semidirect
25. #10597116C22semidirect
26. C5×#1203C22semidirect
27. C5×#1201C22semidirect
28. C5×D3×Dic5C22semidirect
29. C3×#6039C22semidirect
30. C3×#6037C22semidirect
31. C3×D5×Dic5C22semidirect
39. #87880D3semidirect
61. D3×C22⋊D5D5semidirect
77. C2×#1039C2×D3semidirect
81. C5×C22⋊D5C2×D3semidirect
84. #6039C2×D3semidirect
88. #6037C2×D3semidirect
91. D5×Dic5C2×D3semidirect
104. C22×C5×D3C2×D5semidirect
106. C2×D3×D5C2×D5semidirect
109. #1220C2×D5semidirect
117. C3×C22⋊D5C2×D5semidirect
132. #1203C2×D5semidirect
134. #1201C2×D5semidirect
138. D3×Dic5C2×D5semidirect
169. C3×C52H(5,𝔽2)semidirect
239. C22⋊D5D3×D5semidirect
268. C52#1009semidirect
272. C22×D3D52semidirect
328. C3×C5#4144semidirect
384. C5#92027semidirect
398. C3#11717495semidirect

Maximal subgroups

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

subgroupindex
#119256572
#119256462
C2×#105971162
C5×D3×C22⋊D52
C3×#878802
#119254492
#119254352
#119254232
D5×#12032
#117174953
#920275