Finite Group Databasebeta

← #11754815

Subgroups — #11754815

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×D5×Dic3C211yesmaximal, characteristic, complemented
3C5×#935C211yesmaximal, characteristic, complemented
4C3×#4083C211yesmaximal, characteristic, complemented
5#91564C211yesmaximal, characteristic, complemented
6#91559C212yesmaximal, complemented
7#91550C212yesmaximal, complemented
8#91542C211yesmaximal, characteristic, complemented
9#91532C211yesmaximal, characteristic, complemented
10#91431C212yesmaximal, complemented
11#91396C212yesmaximal, complemented
12#65990—31nomaximal, Hall
13C23×C3×D5C2211yescharacteristic
14C2×C3×C5×D4C2211yescharacteristic
15C2×#1220C2211yescharacteristic
16C2×#1220C2211yescharacteristic, complemented
17C22×Dic15C2211yescharacteristic
18C22×Dic15C2211yescharacteristic, complemented
19#6819—24no—
20C2×C4×D15C2211yescharacteristic, complemented
21C2×Dic30C2211yescharacteristic, complemented
22C22×C5×Dic3C2211yescharacteristic
23C2×C3×C22⋊D5C2212yes—
24C2×#1202C2212yes—
25C2×D5×Dic3—24no—
26C2×D5×Dic3C2212yes—
27C2×D5×Dic3C2212yes—
28C5×#217C2211yescharacteristic
29C5×#217C2212yescomplemented
30C5×Dic3⋊C4C2212yescomplemented
31C3×C22⋊Dic5C2211yescharacteristic
32C3×#685C2212yes—
33#6669C2212yescomplemented
34#6668C2212yescomplemented
35#6667C2212yescomplemented
36#6665C2212yes—
37Dic3×Dic5C2212yescomplemented
38#5586—51nomaximal, Hall
39C2×#711—31no—
40C22×C4×D5—31no—
41C5×#123—31no—
42#4083D311yescharacteristic, complemented
43#4075—31no—
44#4042—32no—
45#4039—32no—
46#4030—32no—
47#4028—31no—
48#4027—32no—
49#4023—31no—
50C23×C3×C5C2311yescharacteristic
51C23×C3×C5C2311yescharacteristic
52C22×D15C2311yescharacteristic
53C22×C3×D5—22no—
54C22×C3×D5—24no—
55C22×C3×D5C2312yes—
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—
67C2×C5×Dic3—22no—
68C2×C5×Dic3C2312yes—
69C2×C5×Dic3C2312yes—
70C3×C22⋊D5—24no—
71C2×C3×Dic5C2312yes—
72#1202—24no—
73D5×Dic3—24no—
74D5×Dic3—24no—
75C23×Dic3—51no—
76C2×#237—51no—
77C3×#120—51no—
78#953—51no—
79#939—52no—
80#935D511yescharacteristic, complemented
81D4×Dic3—52no—
82#886—51no—
83#67⋊C4—52no—
84#878—52no—
85#877—51no—
86C23×D5C2×D311yescharacteristic
87C2×C5×D4C2×D311yescharacteristic
88C22×C4×C5—31no—
89C2×C22⋊D5—31no—
90C2×C22⋊D5—31no—
91C2×C22⋊D5C2×D312yes—
92C22×Dic5—31no—
93C22×Dic5—31no—
94#711—64no—
95C2×D20—32no—
96C2×C4×D5—32no—
97C2×C4×D5—32no—
98C2×C4×D5—31no—
99C2×C4×D5—64no—
100C2×Dic10—31no—
101C5×C4⋊C4—32no—
102C5×C22⋊C4—31no—
103C5×C22⋊C4—32no—
104C22⋊Dic5C2×D311yescharacteristic
105#685—32no—
106#685—32no—
107#685C2×D312yes—
108#683—32no—
109#683—32no—
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×C3×D5—28no—
120C2×C3×D5—22no—
121C2×C3×D5—22no—
122C2×C3×D5#4114yescomplemented
123C3×C4×C5—21no—
124Dic15—21no—
125Dic15C2×D412yes—
126Dic15C2×D412yes—
127C3×Dic5—22no—
128C5×Dic3—22no—
129C5×Dic3—22no—
130C24×C3—51no—
131C2×C3×D4—52no—
132C2×C3×D4C2×D511yescharacteristic
133C2×#67—51no—
134C2×#67—51no—
135C2×#67—52no—
136C22×Dic3—51no—
137C22×Dic3—51no—
138C22×Dic3—52no—
139C22×Dic3—52no—
140C22×Dic3—104no—
141C22×Dic3C2×D511yescharacteristic
142#237—104no—
143C2×C4×D3—51no—
144C2×Dic6—51no—
145C3×C22⋊C4—51no—
146C3×C22⋊C4—52no—
147#217—52no—
148#217C2×D511yescharacteristic
149#217C2×D512yescomplemented
150(C2×D3)⋊C4—52no—
151C4⋊Dic3—52no—
152Dic3⋊C4—52no—
153Dic3⋊C4C2×D512yescomplemented
154C4×Dic3—52no—
155C23×C5C22×D311yescharacteristic
156C23×C5C22×D311yescharacteristic
157C22×D5—22no—
158C22×D5—24no—
159C22×D5—31no—
160C22×D5C22×D312yes—
161C5×D4—22no—
162C2×C4×C5—32no—
163C2×C4×C5—32no—
164C2×C4×C5—62no—
165C2×C4×C5C22×D311yescharacteristic
166C22⋊D5—24no—
167C22⋊D5—62no—
168C22⋊D5—62no—
169C2×Dic5—34no—
170C2×Dic5—31no—
171C2×Dic5—31no—
172C2×Dic5—31no—
173C2×Dic5—62no—
174C2×Dic5C22×D312yes—
175D20—64no—
176C4×D5—62no—
177C4×D5—64no—
178C4×D5—64no—
179Dic10—62no—
180C2×#41—151no—
181C23×C4—151no—
182#123—31no—
183#123—151no—
184C22⋊Q8—152no—
185#121—152no—
186#120—51no—
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—
196C3×D5—24no—
197C23×C3—52no—
198C23×C3—104no—
199C23×C3—102no—
200C23×C3C22×D511yescharacteristic
201C23×C3C22×D511yescharacteristic
202C22×D3—51no—
203C3×D4—22no—
204C3×D4—104no—
205C2×C3×C4—52no—
206C2×C3×C4C22×D511yescharacteristic
207#67—104no—
208#67—102no—
209#67—102no—
210C2×Dic3—22no—
211C2×Dic3—54no—
212C2×Dic3—51no—
213C2×Dic3—51no—
214C2×Dic3—51no—
215C2×Dic3—104no—
216C2×Dic3—104no—
217C2×Dic3—102no—
218C2×Dic3C22×D512yes—
219C2×Dic3C22×D512yes—
220C4×D3—102no—
221Dic6—102no—
222C22×C5—22no—
223C22×C5—21no—
224C22×C5—22no—
225C22×C5D3×D412yes—
226C22×C5C23×D311yescharacteristic
227C2×D5—28no—
228C2×D5—22no—
229C2×D5—22no—
230C2×D5—61no—
231C2×D5—62no—
232C2×D5#23714yescomplemented
233C4×C5—21no—
234C4×C5—62no—
235C4×C5—62no—
236Dic5—22no—
237Dic5—32no—
238Dic5—32no—
239Dic5—61no—
240C24—51no—
241#41—304no—
242C2×Q8—151no—
243C2×D4—52no—
244C2×D4—152no—
245C2×D4—151no—
246C2×D4—151no—
247C2×D4D3×D511yescharacteristic
248C22×C4—31no—
249C22×C4—152no—
250C22×C4—152no—
251C22×C4—151no—
252C22×C4—151no—
253C22×C4—151no—
254C22×C4—304no—
255C4⋊C4—32no—
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×C3—54no—
270C22×C3—108no—
271C22×C3—102no—
272C22×C3—102no—
273C22×C3D4×D512yes—
274C22×C3C23×D511yescharacteristic
275C2×D3—101no—
276C2×D3—102no—
277C3×C4—21no—
278C3×C4—102no—
279Dic3—22no—
280Dic3—22no—
281Dic3—52no—
282Dic3—52no—
283Dic3—101no—
284C2×C5—22no—
285C2×C5—41no—
286C2×C5C2×D3×D411yescharacteristic
287C2×C5C2×#23712yes—
288D5—24no—
289D5—121no—
290C23—52no—
291C23—104no—
292C23—102no—
293C23—151no—
294C23C2×D3×D511yescharacteristic
295C23C2×D3×D511yescharacteristic
296Q8—302no—
297D4—22no—
298D4—104no—
299D4—304no—
300D4—302no—
301D4—302no—
302C2×C4—32no—
303C2×C4—32no—
304C2×C4—52no—
305C2×C4—62no—
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×C3—104no—
318C2×C3C2×D4×D511yescharacteristic
319C2×C3D5×#4112yes—
320D3—201no—
321C5#558611yescharacteristic, complemented, Sylow
322C22—22no—
323C22—22no—
324C22—21no—
325C22—54no—
326C22—108no—
327C22—102no—
328C22—102no—
329C22—302no—
330C22—301no—
331C22#679012yes—
332C22C22×D3×D511yescharacteristic
333C4—21no—
334C4—62no—
335C4—62no—
336C4—102no—
337C4—152no—
338C4—152no—
339C4—301no—
340C3#6599011yescharacteristic, complemented, Sylow
341C2—22no—
342C2—41no—
343C2—104no—
344C2—601no—
345C2D5×#23712yes—
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×D5×Dic3C2semidirect
3. C5×#935C2semidirect
4. C3×#4083C2semidirect
5. #91564C2semidirect
6. #91559C2semidirect
7. #91550C2semidirect
8. #91542C2semidirect
9. #91532C2semidirect
10. #91431C2semidirect
11. #91396C2semidirect
16. C2×#1220C22semidirect
18. C22×Dic15C22semidirect
20. C2×C4×D15C22semidirect
21. C2×Dic30C22semidirect
29. C5×#217C22semidirect
30. C5×Dic3⋊C4C22semidirect
33. #6669C22semidirect
34. #6668C22semidirect
35. #6667C22semidirect
37. Dic3×Dic5C22semidirect
42. #4083D3semidirect
61. C2×Dic15D4semidirect
80. #935D5semidirect
122. C2×C3×D5#41semidirect
149. #217C2×D5semidirect
153. Dic3⋊C4C2×D5semidirect
232. C2×D5#237semidirect
265. C3×C5#525semidirect
321. C5#5586semidirect
340. C3#65990semidirect

Maximal subgroups

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

subgroupindex
C2×#68192
C22×D5×Dic32
C5×#9352
C3×#40832
#915642
#915592
#915502
#915422
#915322
#914312
#913962
#659903
#55865