Finite Group Databasebeta

← #11754110

Subgroups — #11754110

Order 960.

496 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 126 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#92030C212yesmaximal, complemented
2Q16×D15C211yesmaximal, characteristic, complemented
3C5×#919C211yesmaximal, characteristic, complemented
4C3×D5×Q16C211yesmaximal, characteristic, complemented
5#91524C212yesmaximal, complemented
6#91513C212yesmaximal, complemented
7#91510C212yesmaximal, complemented
8#91279C211yesmaximal, characteristic, complemented
9#91270C211yesmaximal, characteristic, complemented
10#91265C211yesmaximal, characteristic, complemented
11D3×Dic20C211yesmaximal, characteristic, complemented
12#66105—31nomaximal, Hall
13Q8×D15C2212yes—
14C5×#239C2212yes—
15C3×Q8×D5C2212yes—
16#6772—22no—
17#6772C2212yes—
18#6771—22no—
19#6771C2211yescharacteristic
20#6768—22no—
21#6768C2212yes—
22D3×Dic10—22no—
23D3×Dic10C2212yes—
24C3×C5×Q16C2211yescharacteristic
25#6718C2212yes—
26Dic60C2211yescharacteristic
27C8×D15C2211yescharacteristic
28C5×#215C2212yes—
29C5×D24C2211yescharacteristic
30C5×D3×C8C2211yescharacteristic
31C3×#689C2212yes—
32C3×Dic20C2211yescharacteristic
33C3×C8×D5C2211yescharacteristic
34#6662C2212yes—
35#6659C2212yes—
36#6656C2212yes—
37#6651C2211yescharacteristic
38#6650C2211yescharacteristic
39#6649C2211yescharacteristic
40#5766—51nomaximal, Hall
41#4150—32no—
42C5×#135—31no—
43#4097—32no—
44D5×Q16—31no—
45D5×Q16D311yescharacteristic, complemented
46#4061—32no—
47#4061—32no—
48#4058—31no—
49#4058—31no—
50C2×Dic20—31no—
51#4047—31no—
52C3×C5×Q8C2312yes—
53C4×D15—22no—
54C4×D15C2311yescharacteristic
55Dic30—22no—
56Dic30C2312yes—
57C5×D12—22no—
58C5×D12C2312yes—
59C4×C5×D3—22no—
60C4×C5×D3C2311yescharacteristic
61C3×C4×D5—22no—
62C3×C4×D5C2311yescharacteristic
63C3×Dic10—22no—
64C3×Dic10C2312yes—
65#1204—22no—
66#1204D411yescharacteristic
67#1203—22no—
68#1203D411yescharacteristic
69#1202D411yescharacteristic
70#1201—22no—
71#1201D411yescharacteristic
72D3×Dic5—22no—
73D3×Dic5—22no—
74D3×Dic5—22no—
75C3×C5×C8C2311yescharacteristic
76C3⋊#168C2311yescharacteristic
77C3×#168C2311yescharacteristic
78C5×C3⋊C8C2311yescharacteristic
79#1007—52no—
80C2×C3×Q16—51no—
81#942—52no—
82#919—51no—
83#919D511yescharacteristic, complemented
84#918—52no—
85#918—52no—
86D3×Q16—51no—
87D3×Q16—51no—
88#904—51no—
89#901—51no—
90C5×#41—32no—
91Q8×D5—32no—
92Q8×D5C2×D312yes—
93#711—32no—
94#711—32no—
95#711—62no—
96#711—62no—
97#709—31no—
98#709—62no—
99C2×Dic10—32no—
100C2×Dic10—62no—
101C5×Q16C2×D311yescharacteristic
102C5×SD16—32no—
103C5×D8—31no—
104C2×C5×C8—31no—
105#689—32no—
106#689C2×D312yes—
107#687—32no—
108#687—32no—
109#681—31no—
110Dic20—32no—
111Dic20—31no—
112Dic20C2×D311yescharacteristic
113#676—31no—
114#676—31no—
115C8×D5—31no—
116C8×D5C2×D311yescharacteristic
117#578—151noSylow
118C2×D15C2×D411yescharacteristic
119C2×C5×D3—22no—
120C2×C5×D3C2×D411yescharacteristic
121C2×C3×D5C2×D411yescharacteristic
122C3×C4×C5—22no—
123C3×C4×C5C2411yescharacteristic
124Dic15—22no—
125Dic15C2×D411yescharacteristic
126C3×Dic5—22no—
127C3×Dic5C2×D411yescharacteristic
128C5×Dic3C2×D411yescharacteristic
129C2×C3×Q8—52no—
130#239—52no—
131#239—102no—
132#239C2×D512yes—
133D3×Q8—52no—
134D3×Q8—52no—
135D3×Q8—102no—
136#235—51no—
137#235—52no—
138#235—102no—
139#235—102no—
140C3×Q16—52no—
141C3×Q16—51no—
142C3×Q16C2×D511yescharacteristic
143C2×C3×C8—51no—
144#216—52no—
145#216—52no—
146#215—52no—
147#215C2×D512yes—
148#208—51no—
149Dic12—51no—
150D24C2×D511yescharacteristic
151#204—52no—
152ΓL(1,𝔽25)—51no—
153ΓL(1,𝔽25)—51no—
154D3×C8—51no—
155D3×C8C2×D511yescharacteristic
156C5×Q8C22×D312yes—
157C5×D4—32no—
158C5×D4—62no—
159C2×C4×C5—31no—
160C2×C4×C5—62no—
161C22⋊D5—31no—
162C22⋊D5—31no—
163C22⋊D5—62no—
164C22⋊D5—62no—
165C2×Dic5—62no—
166C2×Dic5—62no—
167C2×Dic5—62no—
168D20—31no—
169C4×D5—22no—
170C4×D5—31no—
171C4×D5—62no—
172C4×D5C22×D311yescharacteristic
173Dic10—22no—
174Dic10—31no—
175Dic10—32no—
176Dic10—62no—
177Dic10—62no—
178Dic10C22×D312yes—
179C5×C8—31no—
180C5×C8C22×D311yescharacteristic
181#168—31no—
182#168C22×D311yescharacteristic
1832−1+4—152no—
184#137—152no—
185#137—152no—
186#137—152no—
187#135—31no—
188#135—151no—
189#135—151no—
190C2×Q16—51no—
191C2×Q16—151no—
192C2×Q16—151no—
193#131—151no—
194C2×C3×C5C22×D411yescharacteristic
195D15—21no—
196C3×D5—21no—
197C5×D3—21no—
198C5×D3—42no—
199C3×Q8—52no—
200C3×Q8—102no—
201C3×Q8C22×D512yes—
202C2×C3×C4—51no—
203C2×C3×C4—102no—
204#67—51no—
205#67—51no—
206#67—102no—
207D12—22no—
208D12—51no—
209D12—102no—
210D12C22×D512yes—
211C4×D3—22no—
212C4×D3—51no—
213C4×D3—102no—
214C4×D3—102no—
215C4×D3—102no—
216C4×D3—102no—
217C4×D3C22×D511yescharacteristic
218Dic6—51no—
219Dic6—52no—
220Dic6—102no—
221Dic6—102no—
222C3×C8—51no—
223C3×C8C22×D511yescharacteristic
224C3⋊C8—51no—
225C3⋊C8C22×D511yescharacteristic
226C22×C5—31no—
227C22×C5—62no—
228C2×D5—31no—
229C2×D5D3×D411yescharacteristic
230C4×C5—22no—
231C4×C5—31no—
232C4×C5C23×D311yescharacteristic
233Dic5—22no—
234Dic5—31no—
235Dic5—62no—
236Dic5D3×D411yescharacteristic
237#41—32no—
238#41—151no—
239#41—152no—
240#41—152no—
241#41—302no—
242#41—302no—
243#41—302no—
244C2×Q8—52no—
245C2×Q8—152no—
246C2×Q8—152no—
247C2×Q8—302no—
248Q16—52no—
249Q16—51no—
250Q16—152no—
251Q16—152no—
252Q16—151no—
253Q16D3×D511yescharacteristic
254SD16—32no—
255SD16—152no—
256SD16—152no—
257D8—31no—
258M16—151no—
259M16—151no—
260M16—151no—
261C2×C8—31no—
262C2×C8—51no—
263C2×C8—151no—
264C3×C5#57811yescharacteristic, complemented, Hall
265C22×C3—51no—
266C2×D3—22no—
267C2×D3—51no—
268C2×D3D4×D511yescharacteristic
269C3×C4—22no—
270C3×C4—51no—
271C3×C4—102no—
272C3×C4C23×D511yescharacteristic
273Dic3—51no—
274Dic3—102no—
275Dic3D4×D511yescharacteristic
276C2×C5—61no—
277C2×C5—122no—
278C2×C5C2×D3×D411yescharacteristic
279D5—21no—
280D5—61no—
281Q8—52no—
282Q8—102no—
283Q8—151no—
284Q8—152no—
285Q8—302no—
286Q8—302no—
287Q8C2×D3×D512yes—
288D4—32no—
289D4—62no—
290D4—151no—
291D4—151no—
292D4—151no—
293D4—302no—
294D4—302no—
295C2×C4—31no—
296C2×C4—51no—
297C2×C4—62no—
298C2×C4—102no—
299C2×C4—151no—
300C2×C4—302no—
301C2×C4—302no—
302C2×C4—302no—
303C2×C4—302no—
304C8—31no—
305C8—51no—
306C8—151no—
307C8C2×D3×D511yescharacteristic
308C2×C3—101no—
309C2×C3C2×D4×D511yescharacteristic
310D3—21no—
311D3—42no—
312D3—101no—
313C5#576611yescharacteristic, complemented, Sylow
314C22—31no—
315C22—51no—
316C22—62no—
317C22—151no—
318C4—22no—
319C4—31no—
320C4—51no—
321C4—102no—
322C4—151no—
323C4—302no—
324C4C22×D3×D511yescharacteristic
325C3#6610511yescharacteristic, complemented, Sylow
326C2—61no—
327C2—101no—
328C2—122no—
329C2—301no—
330C2D3×D4×D511yescharacteristic
subgroupG/H
centerC2D3×D4×D5
commutator, cosocleC3×C4×C5C24
FrattiniC4C22×D3×D5
FittingC3×C5×Q16C22
socleC2×C3×C5C22×D4
hypercenterQ16D3×D5
hypocenterC3×C5#578
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C4 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. #92030C2semidirect
2. Q16×D15C2semidirect
3. C5×#919C2semidirect
4. C3×D5×Q16C2semidirect
5. #91524C2semidirect
6. #91513C2semidirect
7. #91510C2semidirect
8. #91279C2semidirect
9. #91270C2semidirect
10. #91265C2semidirect
11. D3×Dic20C2semidirect
45. D5×Q16D3semidirect
83. #919D5semidirect
264. C3×C5#578semidirect
313. C5#5766semidirect
325. C3#66105semidirect

Maximal subgroups

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

subgroupindex
#920302
Q16×D152
C5×#9192
C3×D5×Q162
#915242
#915132
#915102
#912792
#912702
#912652
D3×Dic202
#661053
#57665