Finite Group Databasebeta

← #11755417

Subgroups — #11755417

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×C3×C22⋊D5C211yesmaximal, characteristic, complemented
2C2×#6782C211yesmaximal, characteristic, complemented
3#91842C211yesmaximal, characteristic, complemented
4C5×#953C211yesmaximal, characteristic, complemented
5#91567C212yesmaximal, complemented
6#91565C212yesmaximal, complemented
7#91564C211yesmaximal, characteristic, complemented
8#91563C211yesmaximal, characteristic, complemented
9#91534C212yesmaximal, complemented
10#91533C212yesmaximal, complemented
11#91528C211yesmaximal, characteristic, complemented
12#65926—31nomaximal, Hall
13C24×C3×C5C2211yescharacteristic
14C23×C3×D5C2211yescharacteristic
15C2×#1220C2211yescharacteristic
16C2×#1220C2212yescomplemented
17C2×C5×#67C2211yescharacteristic
18C2×C5×#67C2212yescomplemented
19C2×C3×C22⋊D5—24no—
20C2×C3×C22⋊D5C2212yes—
21C2×C3×C22⋊D5C2212yes—
22C22×C3×Dic5C2211yescharacteristic
23C2×#1204C2211yescharacteristic, complemented
24C2×#1203C2211yescharacteristic, complemented
25C2×#1202C2212yescomplemented
26C2×#1201C2212yescomplemented
27C2×#1200C2211yescharacteristic, complemented
28#6782—24no—
29C2×D3×Dic5C2211yescharacteristic, complemented
30#6719C2211yescharacteristic
31#6719C2212yescomplemented
32C5×#217C2211yescharacteristic
33C5×#217C2212yescomplemented
34#6668C2212yescomplemented
35#6665C2212yescomplemented
36#6665C2212yescomplemented
37Dic3×Dic5C2212yescomplemented
38#5805—51nomaximal, Hall
39C22×C22⋊D5D311yescharacteristic, complemented
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#4023—31no—
50C23×C3×C5—22no—
51C23×C3×C5—42no—
52C23×C3×C5C2312yes—
53C23×C3×C5C2311yescharacteristic
54C22×D15C2311yescharacteristic
55C22×C5×D3C2311yescharacteristic
56C22×C3×D5—22no—
57C22×C3×D5—42no—
58C22×C3×D5C2312yes—
59#1220—22no—
60#1220—42no—
61C2×Dic15C2312yes—
62C2×Dic15C2311yescharacteristic
63C5×#67—22no—
64C5×#67—42no—
65C2×C5×Dic3C2312yes—
66C2×C5×Dic3C2311yescharacteristic
67C3×C22⋊D5—24no—
68C3×C22⋊D5—24no—
69C2×C3×Dic5D414yescomplemented
70C2×C3×Dic5C2311yescharacteristic
71C2×C3×Dic5C2311yescharacteristic
72#1204—22no—
73#1203—22no—
74#1202—42no—
75#1201—42no—
76#1200—22no—
77D3×Dic5—22no—
78C22×C3×D4—51no—
79C2×#235—51no—
80#953—51no—
81#953—51no—
82#953—51no—
83#953D511yescharacteristic, complemented
84#940—52no—
85#938—52no—
86#938—52no—
87#936—52no—
88#926—51no—
89C24×C5C2×D311yescharacteristic
90C23×D5C2×D311yescharacteristic
91C2×C5×D4—31no—
92C2×C5×D4—32no—
93C2×C22⋊D5—24no—
94C2×C22⋊D5—31no—
95C2×C22⋊D5—32no—
96C2×C22⋊D5—32no—
97C2×C22⋊D5—31no—
98C2×C22⋊D5C2×D312yes—
99C2×C22⋊D5C2×D312yes—
100C22×Dic5—31no—
101C22×Dic5C2×D311yescharacteristic
102#711—64no—
103C2×D20—32no—
104C2×C4×D5—31no—
105C2×Dic10—31no—
106C5×C22⋊C4—31no—
107C5×C22⋊C4—32no—
108C22⋊Dic5—31no—
109C22⋊Dic5—32no—
110#685—32no—
111#685—32no—
112#683—32no—
113C4×Dic5—32no—
114#535—151noSylow
115C22×C3×C5—22no—
116C22×C3×C5—22no—
117C22×C3×C5—22no—
118C22×C3×C5—22no—
119C22×C3×C5—44no—
120C22×C3×C5C2×D412yes—
121C22×C3×C5C2411yescharacteristic
122C2×D15—21no—
123C2×D15—22no—
124C2×C5×D3—21no—
125C2×C5×D3—22no—
126C2×C3×D5—22no—
127C2×C3×D5—22no—
128C2×C3×D5—22no—
129C2×C3×D5—44no—
130Dic15—21no—
131Dic15—22no—
132C3×Dic5C2×D412yes—
133C3×Dic5C2×D412yes—
134C5×Dic3—21no—
135C5×Dic3—22no—
136C24×C3—51no—
137C24×C3C2×D511yescharacteristic
138C2×C3×D4—52no—
139C2×C3×D4—52no—
140C2×C3×D4—104no—
141C22×C3×C4—51no—
142C2×#67—51no—
143C2×#67—52no—
144C2×#67—52no—
145C2×#67—52no—
146C2×#67C2×D511yescharacteristic
147C2×#67C2×D512yescomplemented
148#235—104no—
149C2×D12—51no—
150C2×C4×D3—51no—
151C2×C4×D3—51no—
152C2×Dic6—51no—
153#217—51no—
154#217—52no—
155#217—52no—
156#217—52no—
157#217C2×D511yescharacteristic
158#217C2×D512yescomplemented
159C4⋊Dic3—52no—
160C4×Dic3—52no—
161C23×C5—22no—
162C23×C5—31no—
163C23×C5—42no—
164C23×C5C22×D312yes—
165C23×C5C22×D311yescharacteristic
166C22×D5—22no—
167C22×D5—31no—
168C22×D5—42no—
169C22×D5C22×D312yes—
170C5×D4—62no—
171C5×D4—122no—
172C2×C4×C5—32no—
173C2×C4×C5—31no—
174C22⋊D5—24no—
175C22⋊D5—24no—
176C22⋊D5—62no—
177C22⋊D5—62no—
178C22⋊D5—122no—
179C22⋊D5—122no—
180C2×Dic5—32no—
181C2×Dic5—31no—
182C2×Dic5—62no—
183C2×Dic5#6714yescomplemented
184C2×Dic5C22×D311yescharacteristic
185C2×Dic5C22×D311yescharacteristic
186D20—122no—
187C4×D5—62no—
188Dic10—62no—
189C2×#41—151no—
190C22×D4—51no—
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—151no—
200C2×C3×C5—22no—
201C2×C3×C5—42no—
202C2×C3×C5C22×D411yescharacteristic
203C2×C3×C5H(5,𝔽2)12yes—
204D15—41no—
205C3×D5—42no—
206C5×D3—41no—
207C23×C3—22no—
208C23×C3—42no—
209C23×C3—52no—
210C23×C3—102no—
211C23×C3—202no—
212C23×C3C22×D512yes—
213C23×C3C22×D511yescharacteristic
214C22×D3—51no—
215C22×D3C22×D511yescharacteristic
216C3×D4—104no—
217C3×D4—104no—
218C2×C3×C4—54no—
219C2×C3×C4—51no—
220C2×C3×C4—51no—
221#67—22no—
222#67—42no—
223#67—102no—
224#67—202no—
225#67—202no—
226#67—202no—
227C2×Dic3—52no—
228C2×Dic3—51no—
229C2×Dic3C22×D512yes—
230C2×Dic3C22×D511yescharacteristic
231D12—102no—
232C4×D3—102no—
233C4×D3—102no—
234Dic6—102no—
235C22×C5—22no—
236C22×C5—22no—
237C22×C5—22no—
238C22×C5—22no—
239C22×C5—44no—
240C22×C5—62no—
241C22×C5—61no—
242C22×C5C2×#6712yes—
243C22×C5C23×D311yescharacteristic
244C2×D5—22no—
245C2×D5—22no—
246C2×D5—22no—
247C2×D5—44no—
248C2×D5—62no—
249C2×D5—61no—
250C4×C5—62no—
251C4×C5—61no—
252Dic5—62no—
253Dic5—61no—
254Dic5C2×#6712yes—
255Dic5C2×#6712yes—
256C24—51no—
257C24D3×D511yescharacteristic
258#41—304no—
259C2×Q8—151no—
260C2×D4—31no—
261C2×D4—32no—
262C2×D4—52no—
263C2×D4—52no—
264C2×D4—104no—
265C2×D4—152no—
266C2×D4—151no—
267C2×D4—151no—
268C2×D4—152no—
269C2×D4—152no—
270C22×C4—51no—
271C22×C4—151no—
272C22×C4—151no—
273C4⋊C4—152no—
274C22⋊C4—31no—
275C22⋊C4—32no—
276C22⋊C4—152no—
277C22⋊C4—152no—
278C22⋊C4—151no—
279C22⋊C4—152no—
280C42—152no—
281C3×C5#53511yescharacteristic, complemented, Hall
282C22×C3—22no—
283C22×C3—22no—
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—44no—
287C22×C3—102no—
288C22×C3—102no—
289C22×C3—102no—
290C22×C3—204no—
291C22×C3D4×D512yes—
292C22×C3C23×D511yescharacteristic
293C2×D3—21no—
294C2×D3—22no—
295C2×D3—101no—
296C2×D3—102no—
297C3×C4—52no—
298C3×C4—52no—
299Dic3—21no—
300Dic3—22no—
301Dic3—101no—
302Dic3—102no—
303C2×C5—22no—
304C2×C5—42no—
305C2×C5—121no—
306C2×C5#100412yes—
307C2×C5C22×#6711yescharacteristic
308D5—42no—
309D5—121no—
310C23—22no—
311C23—31no—
312C23—42no—
313C23—52no—
314C23—102no—
315C23—151no—
316C23—202no—
317C23C2×D3×D512yes—
318C23C2×D3×D511yescharacteristic
319Q8—302no—
320D4—62no—
321D4—104no—
322D4—104no—
323D4—122no—
324D4—302no—
325D4—302no—
326D4—602no—
327D4—602no—
328D4—602no—
329C2×C4—32no—
330C2×C4—31no—
331C2×C4—54no—
332C2×C4—51no—
333C2×C4—51no—
334C2×C4—152no—
335C2×C4—151no—
336C2×C4—302no—
337C2×C4—302no—
338C2×C3—22no—
339C2×C3—42no—
340C2×C3—202no—
341C2×C3C2×D4×D511yescharacteristic
342C2×C3#414412yes—
343D3—41no—
344D3—201no—
345C5#580511yescharacteristic, complemented, Sylow
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—44no—
351C22—61no—
352C22—62no—
353C22—102no—
354C22—102no—
355C22—102no—
356C22—204no—
357C22—301no—
358C22—302no—
359C22D5×#6712yes—
360C22C22×D3×D511yescharacteristic
361C4—52no—
362C4—52no—
363C4—61no—
364C4—62no—
365C4—301no—
366C4—302no—
367C3#6592611yescharacteristic, complemented, Sylow
368C2—22no—
369C2—42no—
370C2—121no—
371C2—202no—
372C2—601no—
373C2C2×D5×#6711yescharacteristic
374C2#9204912yes—
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC24×C3×C5C22
hypercenterC24D3×D5
hypocenterC3×C5#535
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C3×C22⋊D5C2semidirect
2. C2×#6782C2semidirect
3. #91842C2semidirect
4. C5×#953C2semidirect
5. #91567C2semidirect
6. #91565C2semidirect
7. #91564C2semidirect
8. #91563C2semidirect
9. #91534C2semidirect
10. #91533C2semidirect
11. #91528C2semidirect
16. C2×#1220C22semidirect
18. C2×C5×#67C22semidirect
23. C2×#1204C22semidirect
24. C2×#1203C22semidirect
25. C2×#1202C22semidirect
26. C2×#1201C22semidirect
27. C2×#1200C22semidirect
29. C2×D3×Dic5C22semidirect
31. #6719C22semidirect
33. C5×#217C22semidirect
34. #6668C22semidirect
35. #6665C22semidirect
36. #6665C22semidirect
37. Dic3×Dic5C22semidirect
39. C22×C22⋊D5D3semidirect
69. C2×C3×Dic5D4semidirect
83. #953D5semidirect
147. C2×#67C2×D5semidirect
158. #217C2×D5semidirect
183. C2×Dic5#67semidirect
281. C3×C5#535semidirect
345. C5#5805semidirect
367. C3#65926semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊D52
C2×#67822
#918422
C5×#9532
#915672
#915652
#915642
#915632
#915342
#915332
#915282
#659263
#58055