Finite Group Databasebeta

← #11752873

Subgroups — #11752873

Order 960.

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

normal subgroups only

This group has 668 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
1C22×#1203C211yesmaximal, characteristic, complemented
2C2×#6782C211yesmaximal, characteristic, complemented
3#91771C211yesmaximal, characteristic, complemented
4C5×#882C211yesmaximal, characteristic, complemented
5C3×#4023C211yesmaximal, characteristic, complemented
6#91573C211yesmaximal, characteristic, complemented
7#91570C211yesmaximal, characteristic, complemented
8#91451C212yesmaximal, complemented
9#91420C212yesmaximal, complemented
10#91416C212yesmaximal, complemented
11#91350C212yesmaximal, complemented
12#65926—31nomaximal, Hall
13C23×D15C2211yescharacteristic
14C23×C5×D3C2211yescharacteristic
15C2×#1220C2211yescharacteristic
16C2×D60C2212yescomplemented
17C2×C5×#67C2211yescharacteristic
18C2×C5×D12C2212yescomplemented
19C22×C3×Dic5C2211yescharacteristic
20C2×#1204C2211yescharacteristic, complemented
21C2×#1203—24no—
22C2×#1203C2212yes—
23C2×#1203C2212yes—
24C2×#1203C2211yescharacteristic, complemented
25C2×#1203C2212yescomplemented
26C2×#1203C2212yescomplemented
27C2×#1200C2211yescharacteristic, complemented
28#6782—24no—
29C2×D3×Dic5C2211yescharacteristic, complemented
30C3×C5×C22⋊C4C2211yescharacteristic
31#6714C2212yescomplemented
32C5×(C2×D3)⋊C4C2212yescomplemented
33C3×C22⋊Dic5C2211yescharacteristic
34C3×#683C2212yescomplemented
35C3×C4×Dic5C2212yescomplemented
36#6667C2212yescomplemented
37(C2×D3)⋊Dic5C2212yescomplemented
38#5525—51nomaximal, Hall
39C22×C22⋊D5—31no—
40C2×#711—31no—
41C5×#120—31no—
42#4099—31no—
43#4086—32no—
44#4085—32no—
45#4083—31no—
46#4031—32no—
47#4030—32no—
48#4028—31no—
49#4023D311yescharacteristic, complemented
50C23×C3×C5C2311yescharacteristic
51C22×D15—22no—
52C22×D15—42no—
53C22×D15C2312yes—
54C22×D15C2311yescharacteristic
55C22×C5×D3—22no—
56C22×C5×D3—42no—
57C22×C5×D3C2312yes—
58C22×C5×D3C2311yescharacteristic
59C2×C3×C4×C5C2312yes—
60#1220—22no—
61C2×Dic15C2311yescharacteristic
62D60—42no—
63C5×#67—22no—
64C2×C5×Dic3C2311yescharacteristic
65C5×D12—42no—
66C2×C3×Dic5D414yescomplemented
67C2×C3×Dic5C2312yes—
68C2×C3×Dic5C2311yescharacteristic
69C2×C3×Dic5C2311yescharacteristic
70#1204—22no—
71#1203—22no—
72#1203—24no—
73#1203—24no—
74#1203—42no—
75#1203—42no—
76#1200—22no—
77D3×Dic5—22no—
78C2×#235—51no—
79C22×D12—51no—
80C3×#117—51no—
81#895—52no—
82#895—52no—
83#882—51no—
84#882—51no—
85#882—51no—
86#882D511yescharacteristic, complemented
87#875—52no—
88#874—52no—
89C24×C5—31no—
90C23×D5—31no—
91C2×C5×D4—31no—
92C2×C5×D4—32no—
93C2×C22⋊D5—31no—
94C2×C22⋊D5—31no—
95C2×C22⋊D5—32no—
96C2×C22⋊D5—32no—
97C2×C22⋊D5—32no—
98C2×C22⋊D5—32no—
99C2×C22⋊D5—64no—
100C22×Dic5—31no—
101C22×Dic5C2×D311yescharacteristic
102#711—64no—
103C2×D20—32no—
104C2×C4×D5—31no—
105C2×Dic10—31no—
106C5×C22⋊C4—32no—
107C5×C22⋊C4C2×D311yescharacteristic
108C22⋊Dic5—32no—
109C22⋊Dic5C2×D311yescharacteristic
110#685—32no—
111#685—32no—
112#683C2×D312yescomplemented
113C4×Dic5C2×D312yescomplemented
114#535—151noSylow
115C22×C3×C5—22no—
116C22×C3×C5C2×D412yes—
117C22×C3×C5C2411yescharacteristic
118C2×D15—21no—
119C2×D15—22no—
120C2×D15—22no—
121C2×D15—22no—
122C2×D15—22no—
123C2×D15—44no—
124C2×C5×D3—21no—
125C2×C5×D3—22no—
126C2×C5×D3—22no—
127C2×C5×D3—22no—
128C2×C5×D3—22no—
129C2×C5×D3—44no—
130C3×C4×C5—22no—
131Dic15—21no—
132C3×Dic5—22no—
133C3×Dic5C2×D412yes—
134C3×Dic5C2×D412yes—
135C5×Dic3—21no—
136C23×D3—51no—
137C23×D3C2×D511yescharacteristic
138C22×C3×C4—51no—
139C2×#67—51no—
140C2×#67C2×D511yescharacteristic
141#235—104no—
142C2×D12—51no—
143C2×D12—52no—
144C2×D12—52no—
145C2×D12—52no—
146C2×D12—52no—
147C2×D12—52no—
148C2×D12—104no—
149C2×D12C2×D512yescomplemented
150C2×C4×D3—51no—
151C2×C4×D3—51no—
152C2×Dic6—51no—
153C3×C4⋊C4—52no—
154C3×C22⋊C4—51no—
155C3×C22⋊C4C2×D511yescharacteristic
156C3×C42—52no—
157(C2×D3)⋊C4—52no—
158(C2×D3)⋊C4—52no—
159(C2×D3)⋊C4—52no—
160(C2×D3)⋊C4C2×D512yescomplemented
161C23×C5—32no—
162C23×C5—31no—
163C23×C5—62no—
164C23×C5—122no—
165C23×C5C22×D311yescharacteristic
166C22×D5—32no—
167C22×D5—31no—
168C22×D5—62no—
169C22×D5—122no—
170C5×D4—62no—
171C5×D4—122no—
172C2×C4×C5—31no—
173C2×C4×C5C22×D312yes—
174C22⋊D5—62no—
175C22⋊D5—62no—
176C22⋊D5—64no—
177C22⋊D5—64no—
178C22⋊D5—122no—
179C22⋊D5—122no—
180C2×Dic5—31no—
181C2×Dic5—62no—
182C2×Dic5D1214yescomplemented
183C2×Dic5C22×D312yes—
184C2×Dic5C22×D311yescharacteristic
185C2×Dic5C22×D311yescharacteristic
186D20—122no—
187C4×D5—62no—
188Dic10—62no—
189C2×#41—151no—
190C22×D4—151no—
191#127—152no—
192#124—152no—
193#121—152no—
194#121—152no—
195#120—31no—
196#120—151no—
197#120—151no—
198#120—151no—
199#117—51no—
200C2×C3×C5—22no—
201C2×C3×C5C22×D411yescharacteristic
202C2×C3×C5H(5,𝔽2)12yes—
203D15—41no—
204D15—42no—
205C5×D3—41no—
206C5×D3—42no—
207C23×C3C22×D511yescharacteristic
208C22×D3—22no—
209C22×D3—42no—
210C22×D3—52no—
211C22×D3—51no—
212C22×D3—102no—
213C22×D3—202no—
214C22×D3C22×D512yes—
215C22×D3C22×D511yescharacteristic
216C2×C3×C4—54no—
217C2×C3×C4—51no—
218C2×C3×C4—51no—
219C2×C3×C4—52no—
220C2×C3×C4C22×D512yes—
221#67—22no—
222#67—102no—
223C2×Dic3—51no—
224C2×Dic3C22×D511yescharacteristic
225D12—42no—
226D12—104no—
227D12—104no—
228D12—102no—
229D12—202no—
230D12—202no—
231D12—202no—
232C4×D3—102no—
233C4×D3—102no—
234Dic6—102no—
235C22×C5—22no—
236C22×C5—61no—
237C22×C5—62no—
238C22×C5—62no—
239C22×C5—62no—
240C22×C5—62no—
241C22×C5—124no—
242C22×C5C2×D1212yes—
243C22×C5C23×D311yescharacteristic
244C2×D5—61no—
245C2×D5—62no—
246C2×D5—62no—
247C2×D5—62no—
248C2×D5—62no—
249C2×D5—124no—
250C4×C5—22no—
251C4×C5—61no—
252Dic5—22no—
253Dic5—61no—
254Dic5C2×D1212yes—
255Dic5C2×D1212yes—
256C24—31no—
257C24—151no—
258#41—304no—
259C2×Q8—151no—
260C2×D4—31no—
261C2×D4—32no—
262C2×D4—152no—
263C2×D4—152no—
264C2×D4—151no—
265C2×D4—152no—
266C2×D4—152no—
267C2×D4—151no—
268C2×D4—152no—
269C2×D4—304no—
270C22×C4—51no—
271C22×C4—151no—
272C22×C4—151no—
273C4⋊C4—52no—
274C22⋊C4—32no—
275C22⋊C4—51no—
276C22⋊C4—152no—
277C22⋊C4—152no—
278C22⋊C4—152no—
279C22⋊C4D3×D511yescharacteristic
280C42—52no—
281C3×C5#53511yescharacteristic, complemented, Hall
282C22×C3—22no—
283C22×C3D4×D512yes—
284C22×C3C23×D511yescharacteristic
285C2×D3—21no—
286C2×D3—22no—
287C2×D3—22no—
288C2×D3—22no—
289C2×D3—22no—
290C2×D3—44no—
291C2×D3—101no—
292C2×D3—102no—
293C2×D3—102no—
294C2×D3—102no—
295C2×D3—102no—
296C2×D3—204no—
297C3×C4—22no—
298C3×C4—52no—
299C3×C4—52no—
300C3×C4—102no—
301Dic3—21no—
302Dic3—101no—
303C2×C5—22no—
304C2×C5—122no—
305C2×C5—121no—
306C2×C5C22×D1211yescharacteristic
307C2×C5#100912yes—
308D5—122no—
309D5—121no—
310C23—32no—
311C23—31no—
312C23—62no—
313C23—122no—
314C23—152no—
315C23—151no—
316C23—302no—
317C23—602no—
318C23C2×D3×D511yescharacteristic
319Q8—302no—
320D4—62no—
321D4—122no—
322D4—304no—
323D4—304no—
324D4—302no—
325D4—302no—
326D4—602no—
327D4—602no—
328D4—602no—
329C2×C4—31no—
330C2×C4—54no—
331C2×C4—51no—
332C2×C4—51no—
333C2×C4—52no—
334C2×C4—151no—
335C2×C4—302no—
336C2×C4—302no—
337C2×C4C2×D3×D512yes—
338C2×C3—22no—
339C2×C3C2×D4×D511yescharacteristic
340C2×C3#414412yes—
341D3—42no—
342D3—41no—
343D3—202no—
344D3—201no—
345C5#552511yescharacteristic, complemented, Sylow
346C22—22no—
347C22—61no—
348C22—62no—
349C22—62no—
350C22—62no—
351C22—62no—
352C22—124no—
353C22—301no—
354C22—302no—
355C22—302no—
356C22—302no—
357C22—302no—
358C22—604no—
359C22D5×D1212yes—
360C22C22×D3×D511yescharacteristic
361C4—22no—
362C4—52no—
363C4—52no—
364C4—61no—
365C4—102no—
366C4—301no—
367C3#6592611yescharacteristic, complemented, Sylow
368C2—22no—
369C2—122no—
370C2—121no—
371C2—602no—
372C2—601no—
373C2C2×D5×D1211yescharacteristic
374C2#9202712yes—
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C22⋊C4C22
hypercenterC22⋊C4D3×D5
hypocenterC3×C5#535
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×#1203C2semidirect
2. C2×#6782C2semidirect
3. #91771C2semidirect
4. C5×#882C2semidirect
5. C3×#4023C2semidirect
6. #91573C2semidirect
7. #91570C2semidirect
8. #91451C2semidirect
9. #91420C2semidirect
10. #91416C2semidirect
11. #91350C2semidirect
16. C2×D60C22semidirect
18. C2×C5×D12C22semidirect
20. C2×#1204C22semidirect
24. C2×#1203C22semidirect
25. C2×#1203C22semidirect
26. C2×#1203C22semidirect
27. C2×#1200C22semidirect
29. C2×D3×Dic5C22semidirect
31. #6714C22semidirect
32. C5×(C2×D3)⋊C4C22semidirect
34. C3×#683C22semidirect
35. C3×C4×Dic5C22semidirect
36. #6667C22semidirect
37. (C2×D3)⋊Dic5C22semidirect
49. #4023D3semidirect
66. C2×C3×Dic5D4semidirect
86. #882D5semidirect
112. #683C2×D3semidirect
113. C4×Dic5C2×D3semidirect
149. C2×D12C2×D5semidirect
160. (C2×D3)⋊C4C2×D5semidirect
182. C2×Dic5D12semidirect
281. C3×C5#535semidirect
345. C5#5525semidirect
367. C3#65926semidirect

Maximal subgroups

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

subgroupindex
C22×#12032
C2×#67822
#917712
C5×#8822
C3×#40232
#915732
#915702
#914512
#914202
#914162
#913502
#659263
#55255