Finite Group Databasebeta

← #11754371

Subgroups — #11754371

Order 960.

660 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 140 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
1C22×C4×D15C211yesmaximal, characteristic, complemented
2C2×#6780C211yesmaximal, characteristic, complemented
3C5×#929C211yesmaximal, characteristic, complemented
4C3×#4075C211yesmaximal, characteristic, complemented
5#91573C211yesmaximal, characteristic, complemented
6#91572C211yesmaximal, characteristic, complemented
7#91532C211yesmaximal, characteristic, complemented
8#91461C212yesmaximal, complemented
9#91406C212yesmaximal, complemented
10#91396C212yesmaximal, complemented
11#91377C212yesmaximal, complemented
12#65990—31nomaximal, Hall
13C23×D15C2211yescharacteristic
14C22×C3×C4×C5C2211yescharacteristic
15C22×Dic15C2211yescharacteristic
16C2×C4×D15—24no—
17C2×C4×D15C2212yes—
18C2×C4×D15C2212yes—
19C2×C5×#67C2211yescharacteristic
20C2×C3×C22⋊D5C2211yescharacteristic
21C2×#1204C2211yescharacteristic, complemented
22C2×#1203C2212yes—
23C2×#1202C2212yes—
24C2×#1201C2211yescharacteristic, complemented
25C2×D3×Dic5C2211yescharacteristic, complemented
26#6780—24no—
27C2×D5×Dic3C2211yescharacteristic, complemented
28C5×#217C2211yescharacteristic
29C5×(C2×D3)⋊C4C2212yescomplemented
30C5×Dic3⋊C4C2212yescomplemented
31C3×C22⋊Dic5C2211yescharacteristic
32C3×#685C2212yescomplemented
33C3×#683C2212yescomplemented
34#6669C2212yescomplemented
35#6667C2212yes—
36#6667C2212yes—
37Dic3×Dic5C2212yescomplemented
38#5656—51nomaximal, Hall
39C2×#711—31no—
40C22×C4×D5—31no—
41C5×#123—31no—
42#4083—31no—
43#4075D311yescharacteristic, complemented
44#4042—32no—
45#4039—32no—
46#4030—32no—
47#4028—31no—
48#4027—32no—
49#4023—31no—
50C23×C3×C5C2311yescharacteristic
51C22×D15—22no—
52C22×D15—24no—
53C22×D15C2312yes—
54C22×C5×D3C2311yescharacteristic
55C22×C3×D5C2311yescharacteristic
56C2×C3×C4×C5—22no—
57C2×C3×C4×C5C2312yes—
58C2×Dic15D414yescomplemented
59C2×Dic15C2311yescharacteristic
60C2×Dic15C2311yescharacteristic
61C4×D15—24no—
62C4×D15—24no—
63C5×#67—22no—
64C2×C5×Dic3C2312yes—
65C2×C5×Dic3C2311yescharacteristic
66C3×C22⋊D5—22no—
67C2×C3×Dic5C2312yes—
68C2×C3×Dic5C2311yescharacteristic
69#1204—22no—
70#1203—24no—
71#1202—24no—
72#1201—22no—
73D3×Dic5—22no—
74D5×Dic3—22no—
75C2×#237—51no—
76C22×C4×D3—51no—
77C3×#123—51no—
78#937—51no—
79#929D511yescharacteristic, complemented
80#896—52no—
81#893—52no—
82#884—52no—
83#882—51no—
84#67⋊C4—52no—
85#877—51no—
86C23×D5—31no—
87C2×C5×D4—31no—
88C22×C4×C5C2×D311yescharacteristic
89C2×C22⋊D5—31no—
90C2×C22⋊D5—32no—
91C2×C22⋊D5C2×D311yescharacteristic
92C22×Dic5—31no—
93C22×Dic5—31no—
94#711—64no—
95C2×D20—32no—
96C2×C4×D5—31no—
97C2×C4×D5—32no—
98C2×C4×D5—32no—
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×D312yescomplemented
108#683—32no—
109#683C2×D312yescomplemented
110C4×Dic5—32no—
111#525—151noSylow
112C22×C3×C5—22no—
113C22×C3×C5C2×D412yes—
114C22×C3×C5C2411yescharacteristic
115C2×D15—28no—
116C2×D15—22no—
117C2×D15—22no—
118C2×D15#4114yescomplemented
119C2×C5×D3—21no—
120C2×C5×D3—22no—
121C2×C3×D5—21no—
122C2×C3×D5—22no—
123C3×C4×C5—22no—
124Dic15C2×D412yes—
125Dic15C2×D412yes—
126C3×Dic5—21no—
127C3×Dic5—22no—
128C5×Dic3—21no—
129C5×Dic3—22no—
130C23×D3—51no—
131C2×C3×D4—51no—
132C22×C3×C4C2×D511yescharacteristic
133C2×#67—51no—
134C2×#67—52no—
135C2×#67C2×D511yescharacteristic
136C22×Dic3—51no—
137C22×Dic3—51no—
138#237—104no—
139C2×D12—52no—
140C2×C4×D3—51no—
141C2×C4×D3—52no—
142C2×C4×D3—52no—
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×D512yescomplemented
152Dic3⋊C4—52no—
153Dic3⋊C4C2×D512yescomplemented
154C4×Dic3—52no—
155C23×C5—31no—
156C23×C5C22×D311yescharacteristic
157C22×D5—32no—
158C22×D5—64no—
159C22×D5—62no—
160C22×D5C22×D311yescharacteristic
161C5×D4—62no—
162C2×C4×C5—22no—
163C2×C4×C5—32no—
164C2×C4×C5—31no—
165C2×C4×C5C22×D312yes—
166C22⋊D5—22no—
167C22⋊D5—64no—
168C22⋊D5—62no—
169C2×Dic5—34no—
170C2×Dic5—31no—
171C2×Dic5—31no—
172C2×Dic5—62no—
173C2×Dic5C22×D312yes—
174C2×Dic5C22×D311yescharacteristic
175D20—64no—
176C4×D5—64no—
177C4×D5—64no—
178C4×D5—62no—
179Dic10—62no—
180C2×#41—151no—
181C23×C4—151no—
182#123—31no—
183#123—51no—
184C22⋊Q8—152no—
185#121—152no—
186#120—151no—
187#120—151no—
188C4×D4—152no—
189C4×D4—152no—
190#117—151no—
191C2×C3×C5—22no—
192C2×C3×C5C22×D411yescharacteristic
193C2×C3×C5C2×#4112yes—
194D15—24no—
195C3×D5—41no—
196C5×D3—41no—
197C23×C3—51no—
198C23×C3C22×D511yescharacteristic
199C22×D3—52no—
200C22×D3—104no—
201C22×D3—102no—
202C22×D3C22×D511yescharacteristic
203C3×D4—102no—
204C2×C3×C4—22no—
205C2×C3×C4—52no—
206C2×C3×C4—51no—
207C2×C3×C4C22×D512yes—
208#67—22no—
209#67—104no—
210#67—102no—
211C2×Dic3—54no—
212C2×Dic3—51no—
213C2×Dic3—51no—
214C2×Dic3—102no—
215C2×Dic3C22×D512yes—
216C2×Dic3C22×D511yescharacteristic
217D12—104no—
218C4×D3—104no—
219C4×D3—104no—
220C4×D3—102no—
221Dic6—102no—
222C22×C5—22no—
223C22×C5—61no—
224C22×C5—62no—
225C22×C5D3×D412yes—
226C22×C5C23×D311yescharacteristic
227C2×D5—21no—
228C2×D5—22no—
229C2×D5—34no—
230C2×D5—68no—
231C2×D5—62no—
232C2×D5—62no—
233C4×C5—22no—
234C4×C5—61no—
235C4×C5—62no—
236Dic5—21no—
237Dic5—22no—
238Dic5—32no—
239Dic5—32no—
240C24—151no—
241#41—304no—
242C2×Q8—151no—
243C2×D4—31no—
244C2×D4—51no—
245C2×D4—152no—
246C2×D4—152no—
247C2×D4—151no—
248C22×C4—151no—
249C22×C4—151no—
250C22×C4—151no—
251C22×C4—152no—
252C22×C4—152no—
253C22×C4—304no—
254C22×C4D3×D511yescharacteristic
255C4⋊C4—32no—
256C4⋊C4—52no—
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—101no—
268C22×C3—102no—
269C22×C3D4×D512yes—
270C22×C3C23×D511yescharacteristic
271C2×D3—21no—
272C2×D3—22no—
273C2×D3—54no—
274C2×D3—108no—
275C2×D3—102no—
276C2×D3—102no—
277C3×C4—22no—
278C3×C4—101no—
279C3×C4—102no—
280Dic3—21no—
281Dic3—22no—
282Dic3—52no—
283Dic3—52no—
284C2×C5—22no—
285C2×C5—121no—
286C2×C5C2×D3×D411yescharacteristic
287C2×C5D3×#4112yes—
288D5—41no—
289D5—64no—
290C23—31no—
291C23—51no—
292C23—152no—
293C23—304no—
294C23—302no—
295C23C2×D3×D511yescharacteristic
296Q8—302no—
297D4—62no—
298D4—102no—
299D4—304no—
300D4—304no—
301D4—302no—
302C2×C4—22no—
303C2×C4—32no—
304C2×C4—31no—
305C2×C4—52no—
306C2×C4—51no—
307C2×C4—154no—
308C2×C4—151no—
309C2×C4—151no—
310C2×C4—304no—
311C2×C4—304no—
312C2×C4—302no—
313C2×C4—302no—
314C2×C4C2×D3×D512yes—
315C2×C3—22no—
316C2×C3—201no—
317C2×C3C2×D4×D511yescharacteristic
318C2×C3D5×#4112yes—
319D3—41no—
320D3—104no—
321C5#565611yescharacteristic, complemented, Sylow
322C22—22no—
323C22—62no—
324C22—61no—
325C22—102no—
326C22—101no—
327C22—154no—
328C22—308no—
329C22—302no—
330C22—302no—
331C22#679012yes—
332C22C22×D3×D511yescharacteristic
333C4—22no—
334C4—62no—
335C4—61no—
336C4—102no—
337C4—101no—
338C4—152no—
339C4—152no—
340C3#6599011yescharacteristic, complemented, Sylow
341C2—22no—
342C2—121no—
343C2—201no—
344C2—304no—
345C2#9201612yes—
346C2C2×#679011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#525
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×D15C2semidirect
2. C2×#6780C2semidirect
3. C5×#929C2semidirect
4. C3×#4075C2semidirect
5. #91573C2semidirect
6. #91572C2semidirect
7. #91532C2semidirect
8. #91461C2semidirect
9. #91406C2semidirect
10. #91396C2semidirect
11. #91377C2semidirect
21. C2×#1204C22semidirect
24. C2×#1201C22semidirect
25. C2×D3×Dic5C22semidirect
27. C2×D5×Dic3C22semidirect
29. C5×(C2×D3)⋊C4C22semidirect
30. C5×Dic3⋊C4C22semidirect
32. C3×#685C22semidirect
33. C3×#683C22semidirect
34. #6669C22semidirect
37. Dic3×Dic5C22semidirect
43. #4075D3semidirect
58. C2×Dic15D4semidirect
79. #929D5semidirect
107. #685C2×D3semidirect
109. #683C2×D3semidirect
118. C2×D15#41semidirect
151. (C2×D3)⋊C4C2×D5semidirect
153. Dic3⋊C4C2×D5semidirect
265. C3×C5#525semidirect
321. C5#5656semidirect
340. C3#65990semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D152
C2×#67802
C5×#9292
C3×#40752
#915732
#915722
#915322
#914612
#914062
#913962
#913772
#659903
#56565