Finite Group Databasebeta

← #11754824

Subgroups — #11754824

Order 960.

660 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C2×#6819C211yesmaximal, characteristic, complemented
2C22×D3×Dic5C211yesmaximal, characteristic, complemented
3C5×#937C211yesmaximal, characteristic, complemented
4C3×#4081C211yesmaximal, characteristic, complemented
5#91570C211yesmaximal, characteristic, complemented
6#67⋊Dic5C212yesmaximal, complemented
7#91558C212yesmaximal, complemented
8#91546C211yesmaximal, characteristic, complemented
9#91532C211yesmaximal, characteristic, complemented
10#91432C212yesmaximal, complemented
11#91406C212yesmaximal, complemented
12#65920—31nomaximal, Hall
13C23×C5×D3C2211yescharacteristic
14C2×C3×C5×D4C2211yescharacteristic
15C2×#1220C2211yescharacteristic
16C2×#1220C2211yescharacteristic, complemented
17C22×Dic15C2211yescharacteristic
18C22×Dic15C2211yescharacteristic, complemented
19#6819—24no—
20C2×C4×D15C2211yescharacteristic, complemented
21C2×Dic30C2211yescharacteristic, complemented
22C2×C5×#67C2212yes—
23C22×C3×Dic5C2211yescharacteristic
24C2×#1203C2212yes—
25C2×D3×Dic5—24no—
26C2×D3×Dic5C2212yes—
27C2×D3×Dic5C2212yes—
28C5×#217C2211yescharacteristic
29C5×(C2×D3)⋊C4C2212yes—
30C3×C22⋊Dic5C2211yescharacteristic
31C3×C22⋊Dic5C2212yescomplemented
32C3×#683C2212yescomplemented
33Dic3⋊Dic5C2212yescomplemented
34#6669C2212yescomplemented
35#6667C2212yescomplemented
36(C2×D3)⋊Dic5C2212yes—
37Dic3×Dic5C2212yescomplemented
38#5656—51nomaximal, Hall
39C23×Dic5—31no—
40C2×#711—31no—
41C5×#120—31no—
42#4099—31no—
43#4085—32no—
44#4081D311yescharacteristic, complemented
45D4×Dic5—32no—
46#4032—31no—
47#4027—32no—
48#4024—32no—
49#4023—31no—
50C23×C3×C5C2311yescharacteristic
51C23×C3×C5C2311yescharacteristic
52C22×D15C2311yescharacteristic
53C22×C5×D3—22no—
54C22×C5×D3—24no—
55C22×C5×D3C2312yes—
56C3×C5×D4—22no—
57C2×C3×C4×C5C2311yescharacteristic
58#1220—22no—
59#1220—22no—
60C2×Dic15—22no—
61C2×Dic15D414yescomplemented
62C2×Dic15C2311yescharacteristic
63C2×Dic15C2311yescharacteristic
64C2×Dic15C2311yescharacteristic
65C4×D15—22no—
66Dic30—22no—
67C5×#67—24no—
68C2×C5×Dic3C2312yes—
69C2×C3×Dic5—22no—
70C2×C3×Dic5C2312yes—
71C2×C3×Dic5C2312yes—
72#1203—24no—
73D3×Dic5—24no—
74D3×Dic5—24no—
75C2×#237—51no—
76C22×C4×D3—51no—
77C3×#123—51no—
78#937D511yescharacteristic, complemented
79#929—51no—
80#896—52no—
81#893—52no—
82#884—52no—
83#882—51no—
84#67⋊C4—52no—
85#877—51no—
86C24×C5—31no—
87C2×C5×D4—32no—
88C2×C5×D4C2×D311yescharacteristic
89C2×C22⋊D5—31no—
90C2×C22⋊D5—31no—
91C2×C22⋊D5—32no—
92C22×Dic5—31no—
93C22×Dic5—31no—
94C22×Dic5—32no—
95C22×Dic5—32no—
96C22×Dic5—64no—
97C22×Dic5C2×D311yescharacteristic
98#711—64no—
99C2×C4×D5—31no—
100C2×Dic10—31no—
101C5×C22⋊C4—31no—
102C5×C22⋊C4—32no—
103C22⋊Dic5—32no—
104C22⋊Dic5C2×D311yescharacteristic
105C22⋊Dic5C2×D312yescomplemented
106#685—32no—
107C4⋊Dic5—32no—
108#683—32no—
109#683C2×D312yescomplemented
110C4×Dic5—32no—
111#525—151noSylow
112C22×C3×C5—22no—
113C22×C3×C5—21no—
114C22×C3×C5—22no—
115C22×C3×C5C2×D412yes—
116C22×C3×C5C2411yescharacteristic
117C2×D15—21no—
118C2×D15—22no—
119C2×C5×D3—28no—
120C2×C5×D3—22no—
121C2×C5×D3—22no—
122C2×C5×D3#4114yescomplemented
123C3×C4×C5—21no—
124Dic15—21no—
125Dic15C2×D412yes—
126Dic15C2×D412yes—
127C3×Dic5—22no—
128C3×Dic5—22no—
129C5×Dic3—22no—
130C23×D3C2×D511yescharacteristic
131C2×C3×D4C2×D511yescharacteristic
132C22×C3×C4—51no—
133C2×#67—51no—
134C2×#67—51no—
135C2×#67C2×D512yes—
136C22×Dic3—51no—
137C22×Dic3—51no—
138#237—104no—
139C2×D12—52no—
140C2×C4×D3—52no—
141C2×C4×D3—52no—
142C2×C4×D3—51no—
143C2×C4×D3—104no—
144C2×Dic6—51no—
145C3×C4⋊C4—52no—
146C3×C22⋊C4—51no—
147C3×C22⋊C4—52no—
148#217C2×D511yescharacteristic
149(C2×D3)⋊C4—52no—
150(C2×D3)⋊C4—52no—
151(C2×D3)⋊C4C2×D512yes—
152Dic3⋊C4—52no—
153Dic3⋊C4—52no—
154C4×Dic3—52no—
155C23×C5—32no—
156C23×C5—64no—
157C23×C5—62no—
158C23×C5C22×D311yescharacteristic
159C23×C5C22×D311yescharacteristic
160C22×D5—31no—
161C5×D4—22no—
162C5×D4—64no—
163C2×C4×C5—32no—
164C2×C4×C5C22×D311yescharacteristic
165C22⋊D5—64no—
166C22⋊D5—62no—
167C22⋊D5—62no—
168C2×Dic5—22no—
169C2×Dic5—34no—
170C2×Dic5—31no—
171C2×Dic5—31no—
172C2×Dic5—31no—
173C2×Dic5—64no—
174C2×Dic5—64no—
175C2×Dic5—62no—
176C2×Dic5C22×D312yes—
177C2×Dic5C22×D312yes—
178C4×D5—62no—
179Dic10—62no—
180C2×#41—151no—
181C23×C4—151no—
182#123—51no—
183#123—151no—
184C22⋊Q8—152no—
185#121—152no—
186#120—31no—
187#120—151no—
188C4×D4—152no—
189C4×D4—152no—
190#117—151no—
191C2×C3×C5—22no—
192C2×C3×C5—41no—
193C2×C3×C5C22×D411yescharacteristic
194C2×C3×C5C2×#4112yes—
195D15—41no—
196C5×D3—24no—
197C23×C3C22×D511yescharacteristic
198C23×C3C22×D511yescharacteristic
199C22×D3—22no—
200C22×D3—24no—
201C22×D3—51no—
202C22×D3C22×D512yes—
203C3×D4—22no—
204C2×C3×C4—52no—
205C2×C3×C4—52no—
206C2×C3×C4—102no—
207C2×C3×C4C22×D511yescharacteristic
208#67—24no—
209#67—102no—
210#67—102no—
211C2×Dic3—54no—
212C2×Dic3—51no—
213C2×Dic3—51no—
214C2×Dic3—51no—
215C2×Dic3—102no—
216C2×Dic3C22×D512yes—
217D12—104no—
218C4×D3—102no—
219C4×D3—104no—
220C4×D3—104no—
221Dic6—102no—
222C22×C5—22no—
223C22×C5—21no—
224C22×C5—22no—
225C22×C5—34no—
226C22×C5—68no—
227C22×C5—62no—
228C22×C5—62no—
229C22×C5D3×D412yes—
230C22×C5C23×D311yescharacteristic
231C2×D5—61no—
232C2×D5—62no—
233C4×C5—21no—
234C4×C5—62no—
235Dic5—22no—
236Dic5—22no—
237Dic5—32no—
238Dic5—32no—
239Dic5—61no—
240C24—31no—
241#41—304no—
242C2×Q8—151no—
243C2×D4—32no—
244C2×D4—152no—
245C2×D4—151no—
246C2×D4—151no—
247C2×D4D3×D511yescharacteristic
248C22×C4—51no—
249C22×C4—152no—
250C22×C4—152no—
251C22×C4—151no—
252C22×C4—151no—
253C22×C4—151no—
254C22×C4—304no—
255C4⋊C4—52no—
256C4⋊C4—152no—
257C4⋊C4—152no—
258C22⋊C4—31no—
259C22⋊C4—32no—
260C22⋊C4—51no—
261C22⋊C4—52no—
262C22⋊C4—152no—
263C22⋊C4—152no—
264C42—152no—
265C3×C5#52511yescharacteristic, complemented, Hall
266C22×C3—22no—
267C22×C3—21no—
268C22×C3—22no—
269C22×C3D4×D512yes—
270C22×C3C23×D511yescharacteristic
271C2×D3—28no—
272C2×D3—22no—
273C2×D3—22no—
274C2×D3—101no—
275C2×D3—102no—
276C2×D3#71114yescomplemented
277C3×C4—21no—
278C3×C4—102no—
279C3×C4—102no—
280Dic3—22no—
281Dic3—52no—
282Dic3—52no—
283Dic3—101no—
284C2×C5—22no—
285C2×C5—41no—
286C2×C5—64no—
287C2×C5C2×D3×D411yescharacteristic
288C2×C5D3×#4112yes—
289D5—121no—
290C23—32no—
291C23—64no—
292C23—62no—
293C23—151no—
294C23C2×D3×D511yescharacteristic
295C23C2×D3×D511yescharacteristic
296Q8—302no—
297D4—22no—
298D4—64no—
299D4—304no—
300D4—302no—
301D4—302no—
302C2×C4—32no—
303C2×C4—52no—
304C2×C4—52no—
305C2×C4—102no—
306C2×C4—154no—
307C2×C4—151no—
308C2×C4—151no—
309C2×C4—151no—
310C2×C4—302no—
311C2×C4—304no—
312C2×C4—304no—
313C2×C4—302no—
314C2×C4C2×D3×D511yescharacteristic
315C2×C3—22no—
316C2×C3—41no—
317C2×C3C2×D4×D511yescharacteristic
318C2×C3C2×#71112yes—
319D3—24no—
320D3—201no—
321C5#565611yescharacteristic, complemented, Sylow
322C22—22no—
323C22—22no—
324C22—21no—
325C22—34no—
326C22—68no—
327C22—62no—
328C22—62no—
329C22—302no—
330C22—301no—
331C22#679012yes—
332C22C22×D3×D511yescharacteristic
333C4—21no—
334C4—62no—
335C4—102no—
336C4—102no—
337C4—152no—
338C4—152no—
339C4—301no—
340C3#6592011yescharacteristic, complemented, Sylow
341C2—22no—
342C2—41no—
343C2—64no—
344C2—601no—
345C2D3×#71112yes—
346C2C2×#679011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#525
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. C2×#6819C2semidirect
2. C22×D3×Dic5C2semidirect
3. C5×#937C2semidirect
4. C3×#4081C2semidirect
5. #91570C2semidirect
6. #67⋊Dic5C2semidirect
7. #91558C2semidirect
8. #91546C2semidirect
9. #91532C2semidirect
10. #91432C2semidirect
11. #91406C2semidirect
16. C2×#1220C22semidirect
18. C22×Dic15C22semidirect
20. C2×C4×D15C22semidirect
21. C2×Dic30C22semidirect
31. C3×C22⋊Dic5C22semidirect
32. C3×#683C22semidirect
33. Dic3⋊Dic5C22semidirect
34. #6669C22semidirect
35. #6667C22semidirect
37. Dic3×Dic5C22semidirect
44. #4081D3semidirect
61. C2×Dic15D4semidirect
78. #937D5semidirect
105. C22⋊Dic5C2×D3semidirect
109. #683C2×D3semidirect
122. C2×C5×D3#41semidirect
265. C3×C5#525semidirect
276. C2×D3#711semidirect
321. C5#5656semidirect
340. C3#65920semidirect

Maximal subgroups

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

subgroupindex
C2×#68192
C22×D3×Dic52
C5×#9372
C3×#40812
#915702
#67⋊Dic52
#915582
#915462
#915322
#914322
#914062
#659203
#56565