Finite Group Databasebeta

← #11755003

Subgroups — #11755003

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×#937C211yesmaximal, characteristic, complemented
3C3×#4082C211yesmaximal, characteristic, complemented
4#91573C212yesmaximal, complemented
5#91570C212yesmaximal, complemented
6#91526C212yesmaximal, complemented
7#91525C212yesmaximal, complemented
8#91350C212yesmaximal, complemented
9#91326C212yesmaximal, complemented
10#65928—31nomaximal, Hall
11C23×D15C2211yescharacteristic, complemented
12C23×C5×D3C2211yescharacteristic, complemented
13C2×C3×C5×D4C2211yescharacteristic
14C2×#1220C2212yes—
15C2×C5×#67C2212yes—
16C2×C3×Dic10C2211yescharacteristic, complemented
17C2×#1204C2212yescomplemented
18C2×#1203C2212yes—
19C2×#1203C2212yes—
20#6719C2211yescharacteristic, complemented
21#6714C2212yescomplemented
22C5×#217C2211yescharacteristic, complemented
23C5×(C2×D3)⋊C4C2212yescomplemented
24C3×C22⋊Dic5C2212yescomplemented
25C3×C22⋊Dic5C2212yescomplemented
26C3×C4×Dic5C2211yescharacteristic, complemented
27#6667C2212yescomplemented
28#6667C2212yescomplemented
29(C2×D3)⋊Dic5C2212yescomplemented
30(C2×D3)⋊Dic5C2212yescomplemented
31Dic3×Dic5C2212yescomplemented
32#5683—51nomaximal, Hall
33C5×#120—31no—
34#4099—32no—
35#4083—31no—
36#4083—32no—
37#4082—32no—
38#4082D311yescharacteristic, complemented
39#4031—32no—
40#4031—32no—
41#4031—32no—
42C23×C3×C5C2311yescharacteristic
43C23×C3×C5C2311yescharacteristic
44C22×D15—41no—
45C22×D15—42no—
46C22×D15C2312yes—
47C22×C5×D3—41no—
48C22×C5×D3—42no—
49C22×C5×D3C2312yes—
50C3×C5×D4—41no—
51C2×C3×C4×C5C2311yescharacteristic
52#1220—42no—
53C2×Dic15C2312yes—
54C5×#67—42no—
55C2×C5×Dic3C2312yes—
56C2×C3×Dic5C2312yes—
57C2×C3×Dic5C2312yes—
58C3×Dic10—41no—
59#1204—42no—
60#1203—42no—
61#1203—42no—
62C3×#124—51no—
63#947—52no—
64#937—51no—
65#937D511yescharacteristic, complemented
66#885—52no—
67#885—52no—
68#882—52no—
69#882—52no—
70#875—52no—
71C24×C5—31no—
72C23×D5—31no—
73C2×C5×D4—32no—
74C2×C5×D4C2×D311yescharacteristic
75C2×C22⋊D5—32no—
76C2×C22⋊D5—32no—
77C2×C22⋊D5—32no—
78C2×Dic10—32no—
79C2×Dic10C2×D311yescharacteristic, complemented
80C5×C22⋊C4—31no—
81C5×C22⋊C4—32no—
82C22⋊Dic5—31no—
83C22⋊Dic5—32no—
84C22⋊Dic5—32no—
85C22⋊Dic5C2×D312yescomplemented
86C22⋊Dic5C2×D312yescomplemented
87#685—32no—
88#685—32no—
89#685—32no—
90C4×Dic5—32no—
91C4×Dic5C2×D311yescharacteristic, complemented
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×C5×D3—22no—
103C2×C5×D3—22no—
104C2×C5×D3—22no—
105C2×C5×D3—44no—
106C3×C4×C5—21no—
107Dic15—22no—
108C3×Dic5—22no—
109C3×Dic5—22no—
110C5×Dic3—22no—
111C23×D3—51no—
112C23×D3C2×D511yescharacteristic, complemented
113C2×C3×Q8—51no—
114C2×C3×D4C2×D511yescharacteristic
115C2×#67—52no—
116C2×#67C2×D512yes—
117C2×D12—52no—
118C2×D12—52no—
119C2×Dic6—52no—
120C3×C22⋊C4—52no—
121C3×C22⋊C4—52no—
122C3×C42—51no—
123#217—51no—
124#217C2×D511yescharacteristic, complemented
125(C2×D3)⋊C4—52no—
126(C2×D3)⋊C4—52no—
127(C2×D3)⋊C4—52no—
128(C2×D3)⋊C4—52no—
129(C2×D3)⋊C4—52no—
130(C2×D3)⋊C4C2×D512yescomplemented
131C4×Dic3—52no—
132C23×C5—32no—
133C23×C5—121no—
134C23×C5—122no—
135C23×C5C22×D311yescharacteristic
136C23×C5C22×D311yescharacteristic
137C22×D5—32no—
138C22×D5—121no—
139C22×D5—122no—
140C5×D4—41no—
141C5×D4—122no—
142C2×C4×C5—32no—
143C2×C4×C5C22×D311yescharacteristic
144C22⋊D5—122no—
145C22⋊D5—122no—
146C22⋊D5—122no—
147C2×Dic5—32no—
148C2×Dic5C22×D312yes—
149C2×Dic5C22×D312yes—
150Dic10—41no—
151Dic10—122no—
152#124—51no—
153#124—152no—
154#124—152no—
155#124—152no—
156#124—152no—
157#120—31no—
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—
166C5×D3—42no—
167C23×C3C22×D511yescharacteristic
168C23×C3C22×D511yescharacteristic
169C22×D3—41no—
170C22×D3—42no—
171C22×D3—52no—
172C22×D3—201no—
173C22×D3—202no—
174C22×D3C22×D512yes—
175C3×Q8—201no—
176C3×D4—41no—
177C2×C3×C4—52no—
178C2×C3×C4—52no—
179C2×C3×C4C22×D511yescharacteristic
180#67—42no—
181#67—202no—
182C2×Dic3—52no—
183C2×Dic3C22×D512yes—
184D12—202no—
185D12—202no—
186Dic6—202no—
187C22×C5—21no—
188C22×C5—21no—
189C22×C5—22no—
190C22×C5—22no—
191C22×C5—62no—
192C22×C5—62no—
193C22×C5—62no—
194C22×C5—124no—
195C22×C5C23×D311yescharacteristic
196C2×D5—62no—
197C2×D5—62no—
198C2×D5—62no—
199C2×D5—124no—
200C4×C5—21no—
201C4×C5—62no—
202Dic5—22no—
203Dic5—22no—
204Dic5—62no—
205C24—31no—
206C24—151no—
207C2×Q8—51no—
208C2×Q8—152no—
209C2×D4—32no—
210C2×D4—152no—
211C2×D4—152no—
212C2×D4—152no—
213C2×D4D3×D511yescharacteristic
214C22⋊C4—31no—
215C22⋊C4—32no—
216C22⋊C4—52no—
217C22⋊C4—52no—
218C22⋊C4—152no—
219C22⋊C4—152no—
220C22⋊C4—152no—
221C22⋊C4—152no—
222C22⋊C4—151no—
223C22⋊C4—152no—
224C42—51no—
225C42—152no—
226C3×C5H(3,𝔽4)11yescharacteristic, complemented, Hall
227C22×C3—22no—
228C22×C3—22no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3C23×D511yescharacteristic
232C2×D3—22no—
233C2×D3—22no—
234C2×D3—22no—
235C2×D3—44no—
236C2×D3—102no—
237C2×D3—102no—
238C2×D3—102no—
239C2×D3—204no—
240C3×C4—21no—
241C3×C4—102no—
242C3×C4—102no—
243Dic3—22no—
244Dic3—102no—
245C2×C5—41no—
246C2×C5—41no—
247C2×C5—122no—
248C2×C5#100411yescharacteristic
249C2×C5#100912yes—
250D5—122no—
251C23—32no—
252C23—122no—
253C23—121no—
254C23—152no—
255C23—602no—
256C23—601no—
257C23C2×D3×D511yescharacteristic
258C23C2×D3×D511yescharacteristic
259Q8—201no—
260Q8—602no—
261D4—41no—
262D4—122no—
263D4—602no—
264D4—602no—
265D4—602no—
266C2×C4—32no—
267C2×C4—52no—
268C2×C4—52no—
269C2×C4—152no—
270C2×C4C2×D3×D511yescharacteristic
271C2×C3—41no—
272C2×C3—41no—
273C2×C3#414412yes—
274C2×C3#414411yescharacteristic
275D3—42no—
276D3—202no—
277C5#568311yescharacteristic, complemented, Sylow
278C22—22no—
279C22—22no—
280C22—21no—
281C22—21no—
282C22—62no—
283C22—62no—
284C22—62no—
285C22—124no—
286C22—302no—
287C22—302no—
288C22—302no—
289C22—604no—
290C22C22×D3×D511yescharacteristic
291C4—21no—
292C4—62no—
293C4—102no—
294C4—102no—
295C4—302no—
296C3#6592811yescharacteristic, complemented, Sylow
297C2—41no—
298C2—41no—
299C2—122no—
300C2—602no—
301C2#9202712yes—
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×#937C2semidirect
3. C3×#4082C2semidirect
4. #91573C2semidirect
5. #91570C2semidirect
6. #91526C2semidirect
7. #91525C2semidirect
8. #91350C2semidirect
9. #91326C2semidirect
11. C23×D15C22semidirect
12. C23×C5×D3C22semidirect
16. C2×C3×Dic10C22semidirect
17. C2×#1204C22semidirect
20. #6719C22semidirect
21. #6714C22semidirect
22. C5×#217C22semidirect
23. C5×(C2×D3)⋊C4C22semidirect
24. C3×C22⋊Dic5C22semidirect
25. C3×C22⋊Dic5C22semidirect
26. C3×C4×Dic5C22semidirect
27. #6667C22semidirect
28. #6667C22semidirect
29. (C2×D3)⋊Dic5C22semidirect
30. (C2×D3)⋊Dic5C22semidirect
31. Dic3×Dic5C22semidirect
38. #4082D3semidirect
65. #937D5semidirect
79. C2×Dic10C2×D3semidirect
85. C22⋊Dic5C2×D3semidirect
86. C22⋊Dic5C2×D3semidirect
91. C4×Dic5C2×D3semidirect
112. C23×D3C2×D5semidirect
124. #217C2×D5semidirect
130. (C2×D3)⋊C4C2×D5semidirect
226. C3×C5H(3,𝔽4)semidirect
277. C5#5683semidirect
296. C3#65928semidirect

Maximal subgroups

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

subgroupindex
#918262
C5×#9372
C3×#40822
#915732
#915702
#915262
#915252
#913502
#913262
#659283
#56835