Finite Group Databasebeta

← #11754953

Subgroups — #11754953

Order 960.

668 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 136 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×#1201C211yesmaximal, characteristic, complemented
3C5×#937C211yesmaximal, characteristic, complemented
4C3×#4083C211yesmaximal, characteristic, complemented
5#91571C212yesmaximal, complemented
6#91570C211yesmaximal, characteristic, complemented
7#91567C212yesmaximal, complemented
8#91564C211yesmaximal, characteristic, complemented
9#91532C211yesmaximal, characteristic, complemented
10#91462C212yesmaximal, complemented
11#91461C212yesmaximal, complemented
12#65926—31nomaximal, Hall
13C23×C5×D3C2211yescharacteristic
14C23×C3×D5C2211yescharacteristic
15C2×C3×C5×D4C2211yescharacteristic
16C2×#1220C2211yescharacteristic
17C2×#1220C2211yescharacteristic, complemented
18C22×Dic15C2211yescharacteristic
19C22×Dic15C2211yescharacteristic, complemented
20#6819—24no—
21C2×C4×D15C2211yescharacteristic, complemented
22C2×Dic30C2211yescharacteristic, complemented
23C2×C5×#67C2212yescomplemented
24C2×C3×C22⋊D5C2212yescomplemented
25C2×#1203C2212yescomplemented
26C2×#1202C2212yescomplemented
27C2×#1201—24no—
28C2×#1201C2212yes—
29C2×#1201C2212yes—
30C5×#217C2211yescharacteristic
31C5×(C2×D3)⋊C4C2212yescomplemented
32C3×C22⋊Dic5C2211yescharacteristic
33C3×#685C2212yescomplemented
34#6669C2212yescomplemented
35(C2×D3)⋊Dic5C2212yescomplemented
36#6665C2212yescomplemented
37Dic3×Dic5C2212yescomplemented
38#5592—51nomaximal, Hall
39C22×C22⋊D5—31no—
40C2×#711—31no—
41C5×#120—31no—
42#4099—31no—
43#4086—32no—
44#4085—32no—
45#4083D311yescharacteristic, complemented
46#4031—32no—
47#4030—32no—
48#4028—31no—
49#4023—31no—
50C23×C3×C5C2311yescharacteristic
51C23×C3×C5C2311yescharacteristic
52C22×D15C2311yescharacteristic
53C22×C5×D3—22no—
54C22×C5×D3—42no—
55C22×C5×D3C2312yes—
56C22×C3×D5—22no—
57C22×C3×D5—42no—
58C22×C3×D5C2312yes—
59C3×C5×D4—22no—
60C2×C3×C4×C5C2311yescharacteristic
61#1220—22no—
62#1220—22no—
63C2×Dic15—22no—
64C2×Dic15D414yescomplemented
65C2×Dic15C2311yescharacteristic
66C2×Dic15C2311yescharacteristic
67C2×Dic15C2311yescharacteristic
68C4×D15—22no—
69Dic30—22no—
70C5×#67—42no—
71C2×C5×Dic3C2312yes—
72C3×C22⋊D5—42no—
73C2×C3×Dic5C2312yes—
74#1203—42no—
75#1202—42no—
76#1201—24no—
77#1201—24no—
78C22×#67—51no—
79C2×#237—51no—
80C3×#120—51no—
81#953—51no—
82#940—52no—
83#939—52no—
84#937D511yescharacteristic, complemented
85#885—52no—
86#884—52no—
87#882—51no—
88#877—51no—
89C24×C5—31no—
90C23×D5C2×D311yescharacteristic
91C2×C5×D4—32no—
92C2×C5×D4C2×D311yescharacteristic
93C2×C22⋊D5—31no—
94C2×C22⋊D5—31no—
95C2×C22⋊D5—32no—
96C2×C22⋊D5—32no—
97C2×C22⋊D5—32no—
98C2×C22⋊D5—64no—
99C2×C22⋊D5C2×D312yescomplemented
100C22×Dic5—31no—
101C22×Dic5—31no—
102#711—64no—
103C2×D20—32no—
104C2×C4×D5—31no—
105C2×Dic10—31no—
106C5×C22⋊C4—31no—
107C5×C22⋊C4—32no—
108C22⋊Dic5—32no—
109C22⋊Dic5C2×D311yescharacteristic
110#685—32no—
111#685C2×D312yescomplemented
112#683—32no—
113C4×Dic5—32no—
114#535—151noSylow
115C22×C3×C5—22no—
116C22×C3×C5—21no—
117C22×C3×C5—22no—
118C22×C3×C5C2×D412yes—
119C22×C3×C5C2411yescharacteristic
120C2×D15—21no—
121C2×D15—22no—
122C2×C5×D3—22no—
123C2×C5×D3—22no—
124C2×C5×D3—22no—
125C2×C5×D3—44no—
126C2×C3×D5—22no—
127C2×C3×D5—22no—
128C2×C3×D5—22no—
129C2×C3×D5—44no—
130C3×C4×C5—21no—
131Dic15—21no—
132Dic15C2×D412yes—
133Dic15C2×D412yes—
134C3×Dic5—22no—
135C5×Dic3—22no—
136C24×C3—51no—
137C23×D3C2×D511yescharacteristic
138C2×C3×D4—52no—
139C2×C3×D4C2×D511yescharacteristic
140C2×#67—51no—
141C2×#67—51no—
142C2×#67—52no—
143C2×#67—52no—
144C2×#67—52no—
145C2×#67—104no—
146C2×#67C2×D512yescomplemented
147C22×Dic3—51no—
148C22×Dic3—51no—
149#237—104no—
150C2×D12—52no—
151C2×C4×D3—51no—
152C2×Dic6—51no—
153C3×C22⋊C4—51no—
154C3×C22⋊C4—52no—
155#217—52no—
156#217C2×D511yescharacteristic
157(C2×D3)⋊C4—52no—
158(C2×D3)⋊C4C2×D512yescomplemented
159Dic3⋊C4—52no—
160C4×Dic3—52no—
161C23×C5—32no—
162C23×C5—62no—
163C23×C5—122no—
164C23×C5C22×D311yescharacteristic
165C23×C5C22×D311yescharacteristic
166C22×D5—22no—
167C22×D5—31no—
168C22×D5—42no—
169C22×D5C22×D312yes—
170C5×D4—22no—
171C5×D4—122no—
172C2×C4×C5—32no—
173C2×C4×C5C22×D311yescharacteristic
174C22⋊D5—42no—
175C22⋊D5—64no—
176C22⋊D5—64no—
177C22⋊D5—62no—
178C22⋊D5—62no—
179C22⋊D5—122no—
180C2×Dic5—34no—
181C2×Dic5—31no—
182C2×Dic5—31no—
183C2×Dic5—31no—
184C2×Dic5—62no—
185C2×Dic5C22×D312yes—
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—51no—
197#120—151no—
198#120—151no—
199#117—151no—
200C2×C3×C5—22no—
201C2×C3×C5—41no—
202C2×C3×C5C22×D411yescharacteristic
203C2×C3×C5H(5,𝔽2)12yes—
204D15—41no—
205C3×D5—42no—
206C5×D3—42no—
207C23×C3—52no—
208C23×C3—102no—
209C23×C3—202no—
210C23×C3C22×D511yescharacteristic
211C23×C3C22×D511yescharacteristic
212C22×D3—22no—
213C22×D3—42no—
214C22×D3—51no—
215C22×D3C22×D512yes—
216C3×D4—22no—
217C3×D4—202no—
218C2×C3×C4—52no—
219C2×C3×C4C22×D511yescharacteristic
220#67—42no—
221#67—104no—
222#67—104no—
223#67—102no—
224#67—102no—
225#67—202no—
226C2×Dic3—54no—
227C2×Dic3—51no—
228C2×Dic3—51no—
229C2×Dic3—51no—
230C2×Dic3—102no—
231C2×Dic3C22×D512yes—
232D12—202no—
233C4×D3—102no—
234Dic6—102no—
235C22×C5—22no—
236C22×C5—21no—
237C22×C5—22no—
238C22×C5—62no—
239C22×C5—62no—
240C22×C5—62no—
241C22×C5—124no—
242C22×C5D3×D412yes—
243C22×C5C23×D311yescharacteristic
244C2×D5—22no—
245C2×D5—22no—
246C2×D5—22no—
247C2×D5—44no—
248C2×D5—61no—
249C2×D5—62no—
250C4×C5—21no—
251C4×C5—62no—
252Dic5—22no—
253Dic5—32no—
254Dic5—32no—
255Dic5—61no—
256C24—31no—
257C24—51no—
258#41—304no—
259C2×Q8—151no—
260C2×D4—32no—
261C2×D4—52no—
262C2×D4—152no—
263C2×D4—152no—
264C2×D4—152no—
265C2×D4—152no—
266C2×D4—151no—
267C2×D4—151no—
268C2×D4—304no—
269C2×D4D3×D511yescharacteristic
270C22×C4—151no—
271C22×C4—151no—
272C22×C4—151no—
273C4⋊C4—152no—
274C22⋊C4—31no—
275C22⋊C4—32no—
276C22⋊C4—51no—
277C22⋊C4—52no—
278C22⋊C4—152no—
279C22⋊C4—152no—
280C42—152no—
281C3×C5#53511yescharacteristic, complemented, Hall
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—22no—
285C22×C3—102no—
286C22×C3—102no—
287C22×C3—102no—
288C22×C3—204no—
289C22×C3D4×D512yes—
290C22×C3C23×D511yescharacteristic
291C2×D3—22no—
292C2×D3—22no—
293C2×D3—22no—
294C2×D3—44no—
295C2×D3—101no—
296C2×D3—102no—
297C3×C4—21no—
298C3×C4—102no—
299Dic3—22no—
300Dic3—52no—
301Dic3—52no—
302Dic3—101no—
303C2×C5—22no—
304C2×C5—41no—
305C2×C5—122no—
306C2×C5C2×D3×D411yescharacteristic
307C2×C5#100412yes—
308D5—42no—
309D5—121no—
310C23—32no—
311C23—52no—
312C23—62no—
313C23—102no—
314C23—122no—
315C23—151no—
316C23—202no—
317C23C2×D3×D511yescharacteristic
318C23C2×D3×D511yescharacteristic
319Q8—302no—
320D4—22no—
321D4—122no—
322D4—202no—
323D4—302no—
324D4—304no—
325D4—304no—
326D4—302no—
327D4—602no—
328D4—602no—
329C2×C4—32no—
330C2×C4—52no—
331C2×C4—154no—
332C2×C4—151no—
333C2×C4—151no—
334C2×C4—151no—
335C2×C4—302no—
336C2×C4—302no—
337C2×C4C2×D3×D511yescharacteristic
338C2×C3—22no—
339C2×C3—41no—
340C2×C3—202no—
341C2×C3C2×D4×D511yescharacteristic
342C2×C3#414412yes—
343D3—42no—
344D3—201no—
345C5#559211yescharacteristic, complemented, Sylow
346C22—22no—
347C22—22no—
348C22—21no—
349C22—62no—
350C22—62no—
351C22—62no—
352C22—102no—
353C22—102no—
354C22—102no—
355C22—124no—
356C22—204no—
357C22—302no—
358C22—301no—
359C22#679012yes—
360C22C22×D3×D511yescharacteristic
361C4—21no—
362C4—62no—
363C4—102no—
364C4—152no—
365C4—152no—
366C4—301no—
367C3#6592611yescharacteristic, complemented, Sylow
368C2—22no—
369C2—41no—
370C2—122no—
371C2—202no—
372C2—601no—
373C2#9202512yes—
374C2C2×#679011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#535
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×#1201C2semidirect
3. C5×#937C2semidirect
4. C3×#4083C2semidirect
5. #91571C2semidirect
6. #91570C2semidirect
7. #91567C2semidirect
8. #91564C2semidirect
9. #91532C2semidirect
10. #91462C2semidirect
11. #91461C2semidirect
17. C2×#1220C22semidirect
19. C22×Dic15C22semidirect
21. C2×C4×D15C22semidirect
22. C2×Dic30C22semidirect
23. C2×C5×#67C22semidirect
24. C2×C3×C22⋊D5C22semidirect
25. C2×#1203C22semidirect
26. C2×#1202C22semidirect
31. C5×(C2×D3)⋊C4C22semidirect
33. C3×#685C22semidirect
34. #6669C22semidirect
35. (C2×D3)⋊Dic5C22semidirect
36. #6665C22semidirect
37. Dic3×Dic5C22semidirect
45. #4083D3semidirect
64. C2×Dic15D4semidirect
84. #937D5semidirect
99. C2×C22⋊D5C2×D3semidirect
111. #685C2×D3semidirect
146. C2×#67C2×D5semidirect
158. (C2×D3)⋊C4C2×D5semidirect
281. C3×C5#535semidirect
345. C5#5592semidirect
367. C3#65926semidirect

Maximal subgroups

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

subgroupindex
C2×#68192
C22×#12012
C5×#9372
C3×#40832
#915712
#915702
#915672
#915642
#915322
#914622
#914612
#659263
#55925