Finite Group Databasebeta

← #11754993

Subgroups — #11754993

Order 960.

520 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 120 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#91826C211yesmaximal, characteristic, complemented
2C5×#936C211yesmaximal, characteristic, complemented
3C3×#4083C211yesmaximal, characteristic, complemented
4#91572C212yesmaximal, complemented
5#91564C212yesmaximal, complemented
6#91534C212yesmaximal, complemented
7#91526C212yesmaximal, complemented
8#91455C212yesmaximal, complemented
9#91353C212yesmaximal, complemented
10#66017—31nomaximal, Hall
11C23×D15C2211yescharacteristic, complemented
12C23×C3×D5C2211yescharacteristic, complemented
13C2×C3×C5×D4C2211yescharacteristic
14C2×#1220C2212yes—
15C2×C5×Dic6C2211yescharacteristic, complemented
16C2×C3×C22⋊D5C2212yes—
17C2×#1204C2212yescomplemented
18C2×#1202C2212yes—
19C2×#1202C2212yes—
20#6719C2211yescharacteristic, complemented
21#6714C2212yescomplemented
22C5×#217C2212yescomplemented
23C5×#217C2212yescomplemented
24C4×C5×Dic3C2211yescharacteristic, complemented
25C3×C22⋊Dic5C2211yescharacteristic, complemented
26C3×#685C2212yescomplemented
27#6667C2212yescomplemented
28#6667C2212yescomplemented
29#6665C2212yescomplemented
30#6665C2212yescomplemented
31Dic3×Dic5C2212yescomplemented
32#5594—51nomaximal, Hall
33C5×#124—31no—
34#4093—32no—
35#4083—31no—
36#4083D311yescharacteristic, complemented
37#4031—32no—
38#4031—32no—
39#4028—32no—
40#4028—32no—
41#4021—32no—
42C23×C3×C5C2311yescharacteristic
43C23×C3×C5C2311yescharacteristic
44C22×D15—41no—
45C22×D15—42no—
46C22×D15C2312yes—
47C22×C3×D5—41no—
48C22×C3×D5—42no—
49C22×C3×D5C2312yes—
50C3×C5×D4—41no—
51C2×C3×C4×C5C2311yescharacteristic
52#1220—42no—
53C2×Dic15C2312yes—
54C2×C5×Dic3C2312yes—
55C2×C5×Dic3C2312yes—
56C5×Dic6—41no—
57C3×C22⋊D5—42no—
58C2×C3×Dic5C2312yes—
59#1204—42no—
60#1202—42no—
61#1202—42no—
62C3×#120—51no—
63#953—52no—
64#937—51no—
65#937—52no—
66#936—52no—
67#936D511yescharacteristic, complemented
68#885—52no—
69#885—52no—
70#885—52no—
71C23×D5—31no—
72C23×D5C2×D311yescharacteristic, complemented
73C2×C5×Q8—31no—
74C2×C5×D4C2×D311yescharacteristic
75C2×C22⋊D5—32no—
76C2×C22⋊D5C2×D312yes—
77C2×D20—32no—
78C2×D20—32no—
79C2×Dic10—32no—
80C5×C22⋊C4—32no—
81C5×C22⋊C4—32no—
82C42×C5—31no—
83C22⋊Dic5—31no—
84C22⋊Dic5C2×D311yescharacteristic, complemented
85#685—32no—
86#685—32no—
87#685—32no—
88#685—32no—
89#685—32no—
90#685C2×D312yescomplemented
91C4×Dic5—32no—
92H(3,𝔽4)—151noSylow
93C22×C3×C5—22no—
94C22×C3×C5—22no—
95C22×C3×C5—21no—
96C22×C3×C5—21no—
97C22×C3×C5C2411yescharacteristic
98C2×D15—22no—
99C2×D15—22no—
100C2×D15—22no—
101C2×D15—44no—
102C2×C3×D5—22no—
103C2×C3×D5—22no—
104C2×C3×D5—22no—
105C2×C3×D5—44no—
106C3×C4×C5—21no—
107Dic15—22no—
108C3×Dic5—22no—
109C5×Dic3—22no—
110C5×Dic3—22no—
111C24×C3—51no—
112C23×D3—51no—
113C2×C3×D4—52no—
114C2×C3×D4C2×D511yescharacteristic
115C2×#67—52no—
116C2×#67—52no—
117C2×#67—52no—
118C2×Dic6—52no—
119C2×Dic6C2×D511yescharacteristic, complemented
120C3×C22⋊C4—51no—
121C3×C22⋊C4—52no—
122#217—51no—
123#217—52no—
124#217—52no—
125#217C2×D512yescomplemented
126#217C2×D512yescomplemented
127(C2×D3)⋊C4—52no—
128(C2×D3)⋊C4—52no—
129(C2×D3)⋊C4—52no—
130C4×Dic3—52no—
131C4×Dic3C2×D511yescharacteristic, complemented
132C23×C5C22×D311yescharacteristic
133C23×C5C22×D311yescharacteristic
134C22×D5—32no—
135C22×D5—41no—
136C22×D5—42no—
137C22×D5—121no—
138C22×D5—122no—
139C22×D5C22×D312yes—
140C5×Q8—121no—
141C5×D4—41no—
142C2×C4×C5—32no—
143C2×C4×C5—32no—
144C2×C4×C5C22×D311yescharacteristic
145C22⋊D5—42no—
146C22⋊D5—122no—
147C2×Dic5—32no—
148C2×Dic5C22×D312yes—
149D20—122no—
150D20—122no—
151Dic10—122no—
152#124—31no—
153#124—152no—
154#124—152no—
155#124—152no—
156#124—152no—
157#120—51no—
158#120—152no—
159#120—152no—
160#120—151no—
161C2×C3×C5—41no—
162C2×C3×C5—41no—
163C2×C3×C5H(5,𝔽2)11yescharacteristic
164C2×C3×C5H(5,𝔽2)12yes—
165D15—42no—
166C3×D5—42no—
167C23×C3—52no—
168C23×C3—201no—
169C23×C3—202no—
170C23×C3C22×D511yescharacteristic
171C23×C3C22×D511yescharacteristic
172C22×D3—52no—
173C22×D3—201no—
174C22×D3—202no—
175C3×D4—41no—
176C3×D4—202no—
177C2×C3×C4—52no—
178C2×C3×C4C22×D511yescharacteristic
179#67—202no—
180#67—202no—
181#67—202no—
182C2×Dic3—52no—
183C2×Dic3C22×D512yes—
184C2×Dic3C22×D512yes—
185Dic6—41no—
186Dic6—202no—
187C22×C5—22no—
188C22×C5—22no—
189C22×C5—21no—
190C22×C5—21no—
191C22×C5C23×D311yescharacteristic
192C2×D5—22no—
193C2×D5—22no—
194C2×D5—22no—
195C2×D5—44no—
196C2×D5—62no—
197C2×D5—62no—
198C2×D5—62no—
199C2×D5—124no—
200C4×C5—21no—
201C4×C5—62no—
202C4×C5—62no—
203Dic5—22no—
204Dic5—62no—
205C24—51no—
206C24—151no—
207C2×Q8—31no—
208C2×Q8—152no—
209C2×D4—52no—
210C2×D4—152no—
211C2×D4—152no—
212C2×D4—152no—
213C2×D4D3×D511yescharacteristic
214C22⋊C4—32no—
215C22⋊C4—32no—
216C22⋊C4—51no—
217C22⋊C4—52no—
218C22⋊C4—152no—
219C22⋊C4—152no—
220C22⋊C4—152no—
221C22⋊C4—152no—
222C22⋊C4—151no—
223C22⋊C4—152no—
224C42—31no—
225C42—152no—
226C3×C5H(3,𝔽4)11yescharacteristic, complemented, Hall
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—102no—
232C22×C3—102no—
233C22×C3—102no—
234C22×C3—204no—
235C22×C3C23×D511yescharacteristic
236C2×D3—102no—
237C2×D3—102no—
238C2×D3—102no—
239C2×D3—204no—
240C3×C4—21no—
241C3×C4—102no—
242Dic3—22no—
243Dic3—22no—
244Dic3—102no—
245C2×C5—41no—
246C2×C5—41no—
247C2×C5#100412yes—
248C2×C5#100411yescharacteristic
249D5—42no—
250D5—122no—
251C23—52no—
252C23—152no—
253C23—202no—
254C23—201no—
255C23—602no—
256C23—601no—
257C23C2×D3×D511yescharacteristic
258C23C2×D3×D511yescharacteristic
259Q8—121no—
260Q8—602no—
261D4—41no—
262D4—202no—
263D4—602no—
264D4—602no—
265D4—602no—
266C2×C4—32no—
267C2×C4—32no—
268C2×C4—52no—
269C2×C4—152no—
270C2×C4C2×D3×D511yescharacteristic
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—202no—
274C2×C3#414411yescharacteristic
275C2×C3#414912yes—
276D3—202no—
277C5#559411yescharacteristic, complemented, Sylow
278C22—22no—
279C22—22no—
280C22—21no—
281C22—21no—
282C22—102no—
283C22—102no—
284C22—102no—
285C22—204no—
286C22—302no—
287C22—302no—
288C22—302no—
289C22—604no—
290C22C22×D3×D511yescharacteristic
291C4—21no—
292C4—62no—
293C4—62no—
294C4—102no—
295C4—302no—
296C3#6601711yescharacteristic, complemented, Sylow
297C2—41no—
298C2—41no—
299C2—202no—
300C2—602no—
301C2#9202612yes—
302C2#9204911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5H(3,𝔽4)
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #91826C2semidirect
2. C5×#936C2semidirect
3. C3×#4083C2semidirect
4. #91572C2semidirect
5. #91564C2semidirect
6. #91534C2semidirect
7. #91526C2semidirect
8. #91455C2semidirect
9. #91353C2semidirect
11. C23×D15C22semidirect
12. C23×C3×D5C22semidirect
15. C2×C5×Dic6C22semidirect
17. C2×#1204C22semidirect
20. #6719C22semidirect
21. #6714C22semidirect
22. C5×#217C22semidirect
23. C5×#217C22semidirect
24. C4×C5×Dic3C22semidirect
25. C3×C22⋊Dic5C22semidirect
26. C3×#685C22semidirect
27. #6667C22semidirect
28. #6667C22semidirect
29. #6665C22semidirect
30. #6665C22semidirect
31. Dic3×Dic5C22semidirect
36. #4083D3semidirect
67. #936D5semidirect
72. C23×D5C2×D3semidirect
84. C22⋊Dic5C2×D3semidirect
90. #685C2×D3semidirect
119. C2×Dic6C2×D5semidirect
125. #217C2×D5semidirect
126. #217C2×D5semidirect
131. C4×Dic3C2×D5semidirect
226. C3×C5H(3,𝔽4)semidirect
277. C5#5594semidirect
296. C3#66017semidirect

Maximal subgroups

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

subgroupindex
#918262
C5×#9362
C3×#40832
#915722
#915642
#915342
#915262
#914552
#913532
#660173
#55945