Finite Group Databasebeta

← #11754753

Subgroups — #11754753

Order 960.

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

normal subgroups only

This group has 432 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×#936C211yesmaximal, characteristic, complemented
3C3×#4084C211yesmaximal, characteristic
4#91576C212yesmaximal, complemented
5#91537C212yesmaximal, complemented
6#91534C212yesmaximal, complemented
7#91530C212yesmaximal, complemented
8#91469C211yesmaximal, characteristic, complemented
9#91359C211yesmaximal, characteristic, complemented
10#91357C211yesmaximal, characteristic, complemented
11#91344C211yesmaximal, characteristic, complemented
12#66016—31nomaximal, Hall
13C2×C3×C5×D4C2211yescharacteristic
14C22×Dic15C2212yes—
15C2×C5×Dic6C2211yescharacteristic
16C2×C3×C22⋊D5C2212yes—
17C2×C3×C4×D5C2211yescharacteristic
18C2×#1204C2212yes—
19#6719C2212yes—
20C4⋊Dic15C2211yescharacteristic
21C4×Dic15C2211yescharacteristic
22C5×#217C2212yes—
23C5×#217C2212yes—
24C4×C5×Dic3C2211yescharacteristic
25C3×C22⋊Dic5C2212yes—
26C3×C4⋊Dic5C2211yescharacteristic
27Dic3⋊Dic5C2212yes—
28Dic3⋊Dic5C2212yes—
29#6669C2212yes—
30#6668C2211yescharacteristic
31#6668C2211yescharacteristic
32#6665C2211yescharacteristic
33#6665C2212yes—
34#6665C2211yescharacteristic
35Dic3×Dic5C2212yes—
36#5603—51nomaximal, Hall
37C5×#124—31no—
38#4092—31no—
39Q8×Dic5—31no—
40#4084D311yescharacteristic
41D4×Dic5—31no—
42#4032—32no—
43#4031—32no—
44#4025—32no—
45#4024—32no—
46#4018—31no—
47#4015—31no—
48C23×C3×C5C2312yes—
49C22×C3×D5C2311yescharacteristic
50C3×C5×D4—22no—
51C2×C3×C4×C5C2311yescharacteristic
52C2×Dic15—42no—
53C2×Dic15C2311yescharacteristic
54C2×Dic15C2311yescharacteristic
55C2×Dic15C2312yes—
56C2×C5×Dic3C2312yes—
57C2×C5×Dic3C2311yescharacteristic
58C2×C5×Dic3C2311yescharacteristic
59C5×Dic6—22no—
60C3×C22⋊D5—42no—
61C2×C3×Dic5C2312yes—
62C2×C3×Dic5C2311yescharacteristic
63C3×C4×D5—22no—
64#1204—42no—
65C3×#121—51no—
66#936—52no—
67#936D511yescharacteristic, complemented
68#935—52no—
69D4×Dic3—51no—
70#926—51no—
71#924—51no—
72#888—51no—
73#887—51no—
74#879—52no—
75#878—52no—
76C2×C5×Q8—31no—
77C2×C5×D4C2×D311yescharacteristic
78C2×C22⋊D5C2×D312yes—
79C22×Dic5—32no—
80C2×C4×D5C2×D311yescharacteristic
81C2×Dic10—32no—
82C5×C22⋊C4—32no—
83C5×C22⋊C4—32no—
84C42×C5—31no—
85C22⋊Dic5—32no—
86C22⋊Dic5C2×D312yes—
87#685—31no—
88#685—31no—
89#685—32no—
90C4⋊Dic5—31no—
91C4⋊Dic5—32no—
92C4⋊Dic5—32no—
93C4⋊Dic5C2×D311yescharacteristic
94#683—31no—
95#683—31no—
96#683—32no—
97C4×Dic5—31no—
98C4×Dic5—32no—
99#542—151noSylow
100C22×C3×C5—22no—
101C22×C3×C5—22no—
102C22×C3×C5—22no—
103C22×C3×C5C2411yescharacteristic
104C2×C3×D5—21no—
105C2×C3×D5—21no—
106C2×C3×D5—21no—
107C3×C4×C5#4112yes—
108Dic15—22no—
109Dic15—21no—
110Dic15—21no—
111C3×Dic5—21no—
112C3×Dic5—22no—
113C5×Dic3—22no—
114C5×Dic3—21no—
115C5×Dic3—21no—
116C2×C3×D4—52no—
117C2×C3×D4C2×D511yescharacteristic
118C22×C3×C4—51no—
119C22×Dic3—52no—
120C2×Dic6—52no—
121C2×Dic6C2×D511yescharacteristic
122C3×C4⋊C4—51no—
123C3×C22⋊C4—52no—
124#217—52no—
125#217—52no—
126#217—51no—
127#217—51no—
128#217C2×D512yes—
129#217C2×D512yes—
130C4⋊Dic3—51no—
131C4⋊Dic3—51no—
132C4⋊Dic3—51no—
133Dic3⋊C4—52no—
134Dic3⋊C4—52no—
135Dic3⋊C4—52no—
136C4×Dic3—51no—
137C4×Dic3—52no—
138C4×Dic3C2×D511yescharacteristic
139C23×C5C22×D312yes—
140C22×D5C22×D311yescharacteristic
141C5×Q8—62no—
142C5×D4—22no—
143C2×C4×C5—32no—
144C2×C4×C5—31no—
145C2×C4×C5—31no—
146C2×C4×C5C22×D311yescharacteristic
147C22⋊D5—42no—
148C2×Dic5—31no—
149C2×Dic5—31no—
150C2×Dic5—32no—
151C2×Dic5—122no—
152C2×Dic5C22×D312yes—
153C2×Dic5C22×D311yescharacteristic
154C4×D5—22no—
155Dic10—122no—
156C4⋊Q8—151no—
157#126—152no—
158#124—31no—
159#124—152no—
160#123—152no—
161C22⋊Q8—152no—
162C22⋊Q8—151no—
163#121—51no—
164C4×Q8—151no—
165C4×D4—151no—
166#117—151no—
167C2×C3×C5—42no—
168C2×C3×C5C2×#4111yescharacteristic
169C2×C3×C5H(5,𝔽2)11yescharacteristic
170C2×C3×C52−1+411yescharacteristic
171C3×D5—41no—
172C23×C3—51no—
173C23×C3C22×D512yes—
174C3×D4—22no—
175C3×D4—202no—
176C2×C3×C4—52no—
177C2×C3×C4—51no—
178C2×C3×C4—102no—
179C2×C3×C4C22×D511yescharacteristic
180C2×Dic3—51no—
181C2×Dic3—51no—
182C2×Dic3—52no—
183C2×Dic3—202no—
184C2×Dic3C22×D512yes—
185C2×Dic3C22×D511yescharacteristic
186C2×Dic3C22×D511yescharacteristic
187Dic6—22no—
188Dic6—202no—
189C22×C5—22no—
190C22×C5—22no—
191C22×C5—22no—
192C22×C5C23×D311yescharacteristic
193C2×D5—21no—
194C2×D5—21no—
195C2×D5—21no—
196C4×C5—62no—
197C4×C5—61no—
198C4×C5—61no—
199C4×C5#23712yes—
200Dic5—21no—
201Dic5—22no—
202Dic5—62no—
203Dic5—61no—
204Dic5—61no—
205C2×Q8—31no—
206C2×Q8—152no—
207C2×D4—52no—
208C2×D4D3×D511yescharacteristic
209C22×C4—51no—
210C22×C4—152no—
211C4⋊C4—51no—
212C4⋊C4—151no—
213C4⋊C4—151no—
214C4⋊C4—152no—
215C4⋊C4—152no—
216C4⋊C4—152no—
217C4⋊C4—151no—
218C22⋊C4—32no—
219C22⋊C4—32no—
220C22⋊C4—52no—
221C22⋊C4—151no—
222C22⋊C4—151no—
223C22⋊C4—152no—
224C22⋊C4—152no—
225C42—31no—
226C42—152no—
227C42—151no—
228C3×C5#54211yescharacteristic, complemented, Hall
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—101no—
233C22×C3—101no—
234C22×C3—101no—
235C22×C3C23×D511yescharacteristic
236C3×C4—101no—
237C3×C4—102no—
238C3×C4#71112yes—
239Dic3—22no—
240Dic3—21no—
241Dic3—21no—
242Dic3—102no—
243Dic3—101no—
244Dic3—101no—
245C2×C5—42no—
246C2×C5C2×#23711yescharacteristic
247C2×C5#100411yescharacteristic
248C2×C5#101011yescharacteristic
249D5—41no—
250C23—51no—
251C23C2×D3×D512yes—
252Q8—62no—
253Q8—602no—
254D4—22no—
255D4—202no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—32no—
259C2×C4—52no—
260C2×C4—51no—
261C2×C4—102no—
262C2×C4—151no—
263C2×C4—151no—
264C2×C4—152no—
265C2×C4—602no—
266C2×C4C2×D3×D511yescharacteristic
267C2×C3—42no—
268C2×C3—201no—
269C2×C3C2×#71111yescharacteristic
270C2×C3#414911yescharacteristic
271C2×C3#415011yescharacteristic
272C5#560311yescharacteristic, complemented, Sylow
273C22—22no—
274C22—22no—
275C22—22no—
276C22—101no—
277C22—101no—
278C22—101no—
279C22C22×D3×D511yescharacteristic
280C4—61no—
281C4—61no—
282C4—62no—
283C4—102no—
284C4—101no—
285C4—301no—
286C4—301no—
287C4—302no—
288C4#678012yes—
289C3#6601611yescharacteristic, complemented, Sylow
290C2—42no—
291C2—201no—
292C2#9202011yescharacteristic
293C2#9202611yescharacteristic
294C2C2×#678011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
prime radicalC3×#4084C2
hypercenterC2×D4D3×D5
hypocenterC3×C5#542
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×#936C2semidirect
4. #91576C2semidirect
5. #91537C2semidirect
6. #91534C2semidirect
7. #91530C2semidirect
8. #91469C2semidirect
9. #91359C2semidirect
10. #91357C2semidirect
11. #91344C2semidirect
67. #936D5semidirect
228. C3×C5#542semidirect
272. C5#5603semidirect
289. C3#66016semidirect

Maximal subgroups

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

subgroupindex
D4×Dic152
C5×#9362
C3×#40842
#915762
#915372
#915342
#915302
#914692
#913592
#913572
#913442
#660163
#56035