Finite Group Databasebeta

← #11754747

Subgroups — #11754747

Order 960.

480 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 128 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
1D4×Dic15C211yesmaximal, characteristic, complemented
2C5×#940C211yesmaximal, characteristic, complemented
3C3×#4084C211yesmaximal, characteristic, complemented
4#91571C212yesmaximal, complemented
5#91565C212yesmaximal, complemented
6#91555C212yesmaximal, complemented
7#91543C212yesmaximal, complemented
8#91449C211yesmaximal, characteristic, complemented
9#91435C211yesmaximal, characteristic, complemented
10#91381C211yesmaximal, characteristic, complemented
11#91357C211yesmaximal, characteristic, complemented
12#66053—31nomaximal, Hall
13C2×C3×C5×D4C2211yescharacteristic
14C22×Dic15C2212yes—
15C2×C5×#67C2212yes—
16C2×C5×#67C2212yes—
17C2×C5×D12C2211yescharacteristic
18C2×C3×C22⋊D5C2212yes—
19C2×C3×C4×D5C2211yescharacteristic
20C2×#1201C2212yes—
21C2×D3×Dic5C2212yes—
22#6719C2212yes—
23C4⋊Dic15C2211yescharacteristic
24C4×Dic15C2211yescharacteristic
25C4×C5×Dic3C2211yescharacteristic
26C3×C22⋊Dic5C2212yes—
27C3×C4⋊Dic5C2211yescharacteristic
28#6669C2212yes—
29#6668C2211yescharacteristic
30#6668C2212yes—
31#6668C2211yescharacteristic
32(C2×D3)⋊Dic5C2212yes—
33(C2×D3)⋊Dic5C2212yes—
34#6665C2211yescharacteristic
35#6665C2211yescharacteristic
36#5619—51nomaximal, Hall
37C5×#127—31no—
38#4085—32no—
39#4085—32no—
40#4084—31no—
41#4084D311yescharacteristic, complemented
42#4081—32no—
43#4081—32no—
44D4×Dic5—31no—
45D4×Dic5—31no—
46#4018—31no—
47#4016—31no—
48C23×C3×C5C2312yes—
49C22×C5×D3C2312yes—
50C22×C3×D5C2311yescharacteristic
51C3×C5×D4—22no—
52C2×C3×C4×C5C2311yescharacteristic
53C2×Dic15—42no—
54C2×Dic15C2311yescharacteristic
55C2×Dic15C2311yescharacteristic
56C2×Dic15C2312yes—
57C5×#67—42no—
58C5×#67—42no—
59C2×C5×Dic3C2311yescharacteristic
60C2×C5×Dic3C2311yescharacteristic
61C5×D12—22no—
62C3×C22⋊D5—42no—
63C2×C3×Dic5C2311yescharacteristic
64C2×C3×Dic5C2312yes—
65C3×C4×D5—22no—
66#1201—42no—
67D3×Dic5—42no—
68C3×#121—51no—
69#940D511yescharacteristic, complemented
70#939—52no—
71#938—52no—
72D4×Dic3—51no—
73#930—51no—
74#926—51no—
75#893—51no—
76#890—51no—
77#886—52no—
78#883—52no—
79C2×C5×D4—32no—
80C2×C5×D4—32no—
81C2×C5×D4—31no—
82C2×C5×D4C2×D311yescharacteristic
83C2×C22⋊D5—32no—
84C2×C22⋊D5C2×D312yes—
85C22×Dic5—32no—
86C22×Dic5—32no—
87C2×C4×D5C2×D311yescharacteristic
88C42×C5—31no—
89C22⋊Dic5—32no—
90C22⋊Dic5—32no—
91C22⋊Dic5—32no—
92C22⋊Dic5C2×D312yes—
93#685—31no—
94#685—31no—
95C4⋊Dic5—31no—
96C4⋊Dic5C2×D311yescharacteristic
97#683—32no—
98#683—31no—
99#683—31no—
100#683—32no—
101C4×Dic5—31no—
102#540—151noSylow
103C22×C3×C5—22no—
104C22×C3×C5—22no—
105C22×C3×C5—22no—
106C22×C3×C5C2411yescharacteristic
107C2×C5×D3—22no—
108C2×C5×D3—22no—
109C2×C5×D3—22no—
110C2×C3×D5—21no—
111C2×C3×D5—21no—
112C2×C3×D5—21no—
113C3×C4×C5#4112yes—
114Dic15—21no—
115Dic15—22no—
116Dic15—21no—
117C3×Dic5—21no—
118C3×Dic5—22no—
119C5×Dic3—21no—
120C5×Dic3—21no—
121C2×C3×D4—52no—
122C2×C3×D4C2×D511yescharacteristic
123C22×C3×C4—51no—
124C2×#67—52no—
125C2×#67C2×D512yes—
126C2×#67C2×D512yes—
127C22×Dic3—52no—
128C2×D12C2×D511yescharacteristic
129C2×C4×D3—52no—
130C3×C4⋊C4—51no—
131C3×C22⋊C4—52no—
132#217—52no—
133#217—51no—
134#217—51no—
135(C2×D3)⋊C4—52no—
136(C2×D3)⋊C4—52no—
137C4⋊Dic3—51no—
138C4⋊Dic3—51no—
139C4⋊Dic3—51no—
140C4⋊Dic3—52no—
141Dic3⋊C4—52no—
142C4×Dic3—51no—
143C4×Dic3C2×D511yescharacteristic
144C23×C5—32no—
145C23×C5C22×D312yes—
146C22×D5C22×D311yescharacteristic
147C5×D4—22no—
148C5×D4—62no—
149C5×D4—122no—
150C5×D4—122no—
151C2×C4×C5—31no—
152C2×C4×C5—31no—
153C2×C4×C5C22×D311yescharacteristic
154C22⋊D5—42no—
155C22⋊D5—122no—
156C2×Dic5—31no—
157C2×Dic5—31no—
158C2×Dic5—32no—
159C2×Dic5—122no—
160C2×Dic5—122no—
161C2×Dic5C22×D312yes—
162C2×Dic5C22×D311yescharacteristic
163C4×D5—22no—
164#127—31no—
165#125—151no—
166#123—152no—
167#123—152no—
168#121—51no—
169#121—151no—
170#121—152no—
171#121—152no—
172C4×D4—151no—
173C4×D4—151no—
174#117—151no—
175C2×C3×C5—42no—
176C2×C3×C5C2×#4111yescharacteristic
177C2×C3×C5H(5,𝔽2)11yescharacteristic
178C2×C3×C5H(5,𝔽2)11yescharacteristic
179C3×D5—41no—
180C5×D3—42no—
181C23×C3—51no—
182C23×C3C22×D512yes—
183C22×D3C22×D512yes—
184C3×D4—22no—
185C3×D4—202no—
186C2×C3×C4—52no—
187C2×C3×C4—51no—
188C2×C3×C4—102no—
189C2×C3×C4C22×D511yescharacteristic
190#67—42no—
191#67—42no—
192#67—202no—
193C2×Dic3—51no—
194C2×Dic3—51no—
195C2×Dic3—52no—
196C2×Dic3—202no—
197C2×Dic3C22×D511yescharacteristic
198C2×Dic3C22×D511yescharacteristic
199D12—22no—
200C4×D3—202no—
201C22×C5—22no—
202C22×C5—22no—
203C22×C5—22no—
204C22×C5—62no—
205C22×C5—62no—
206C22×C5—62no—
207C22×C5C23×D311yescharacteristic
208C2×D5—21no—
209C2×D5—21no—
210C2×D5—21no—
211C4×C5—61no—
212C4×C5—61no—
213C4×C5#23712yes—
214Dic5—21no—
215Dic5—22no—
216Dic5—61no—
217Dic5—61no—
218Dic5—62no—
219C2×D4—32no—
220C2×D4—32no—
221C2×D4—31no—
222C2×D4—52no—
223C2×D4—152no—
224C2×D4D3×D511yescharacteristic
225C22×C4—51no—
226C22×C4—152no—
227C22×C4—152no—
228C4⋊C4—51no—
229C4⋊C4—151no—
230C4⋊C4—151no—
231C4⋊C4—152no—
232C4⋊C4—152no—
233C4⋊C4—151no—
234C22⋊C4—52no—
235C22⋊C4—151no—
236C22⋊C4—151no—
237C22⋊C4—152no—
238C22⋊C4—152no—
239C22⋊C4—152no—
240C42—31no—
241C42—151no—
242C3×C5#54011yescharacteristic, complemented, Hall
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—101no—
247C22×C3—101no—
248C22×C3—101no—
249C22×C3C23×D511yescharacteristic
250C2×D3—22no—
251C2×D3—22no—
252C2×D3—22no—
253C3×C4—101no—
254C3×C4—102no—
255C3×C4#71112yes—
256Dic3—21no—
257Dic3—21no—
258Dic3—101no—
259Dic3—101no—
260Dic3—102no—
261C2×C5—42no—
262C2×C5—122no—
263C2×C5C2×#23711yescharacteristic
264C2×C5#100411yescharacteristic
265C2×C5#100911yescharacteristic
266D5—41no—
267C23—32no—
268C23—51no—
269C23C2×D3×D512yes—
270D4—22no—
271D4—62no—
272D4—122no—
273D4—122no—
274D4—202no—
275D4—602no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—51no—
279C2×C4—52no—
280C2×C4—102no—
281C2×C4—151no—
282C2×C4—151no—
283C2×C4—152no—
284C2×C4—602no—
285C2×C4—602no—
286C2×C4C2×D3×D511yescharacteristic
287C2×C3—42no—
288C2×C3—201no—
289C2×C3C2×#71111yescharacteristic
290C2×C3#414411yescharacteristic
291C2×C3#414411yescharacteristic
292D3—42no—
293C5#561911yescharacteristic, complemented, Sylow
294C22—22no—
295C22—22no—
296C22—22no—
297C22—62no—
298C22—62no—
299C22—62no—
300C22—101no—
301C22—101no—
302C22—101no—
303C22C22×D3×D511yescharacteristic
304C4—61no—
305C4—61no—
306C4—102no—
307C4—101no—
308C4—301no—
309C4—301no—
310C4—302no—
311C4#678012yes—
312C3#6605311yescharacteristic, complemented, Sylow
313C2—42no—
314C2—122no—
315C2—201no—
316C2#9202511yescharacteristic
317C2#9202711yescharacteristic
318C2C2×#678011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#540
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. D4×Dic15C2semidirect
2. C5×#940C2semidirect
3. C3×#4084C2semidirect
4. #91571C2semidirect
5. #91565C2semidirect
6. #91555C2semidirect
7. #91543C2semidirect
8. #91449C2semidirect
9. #91435C2semidirect
10. #91381C2semidirect
11. #91357C2semidirect
41. #4084D3semidirect
69. #940D5semidirect
242. C3×C5#540semidirect
293. C5#5619semidirect
312. C3#66053semidirect

Maximal subgroups

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

subgroupindex
D4×Dic152
C5×#9402
C3×#40842
#915712
#915652
#915552
#915432
#914492
#914352
#913812
#913572
#660533
#56195