Finite Group Databasebeta

← #11754380

Subgroups — #11754380

Order 960.

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

normal subgroups only

This group has 588 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×#6817C211yesmaximal, characteristic, complemented
2C2×#6768C211yesmaximal, characteristic, complemented
3C5×#930C211yesmaximal, characteristic, complemented
4C3×#4070C211yesmaximal, characteristic, complemented
5#91558C212yesmaximal, complemented
6#91557C212yesmaximal, complemented
7#91466C211yesmaximal, characteristic, complemented
8#91437C211yesmaximal, characteristic, complemented
9#91432C212yesmaximal, complemented
10#91420C212yesmaximal, complemented
11#91390C211yesmaximal, characteristic, complemented
12#65942—31nomaximal, Hall
13C22×C3×C4×C5C2211yescharacteristic
14C2×#1220C2211yescharacteristic
15C2×#1220C2211yescharacteristic
16#6817—24no—
17C2×D60C2211yescharacteristic
18C2×C4×D15C2211yescharacteristic
19C2×C4×D15C2211yescharacteristic
20C2×Dic30C2211yescharacteristic
21C2×C5×#67C2212yes—
22C2×C5×D12C2211yescharacteristic
23C2×C3×Dic10C2211yescharacteristic
24C2×#1203C2212yes—
25C2×#1203C2212yes—
26C2×#1200C2212yes—
27C2×D3×Dic5C2212yes—
28#6768—24no—
29C5×(C2×D3)⋊C4C2212yes—
30C5×C4⋊Dic3C2211yescharacteristic
31C3×C22⋊Dic5C2212yes—
32C3×C4⋊Dic5C2211yescharacteristic
33C3×#683C2212yes—
34Dic3⋊Dic5C2212yes—
35#6669C2212yes—
36#6667C2212yes—
37(C2×D3)⋊Dic5C2212yes—
38#5629—51nomaximal, Hall
39C2×#711—31no—
40C2×#709—31no—
41C5×#121—31no—
42#4085—32no—
43#4084—32no—
44#4084—31no—
45#4070D311yescharacteristic, complemented
46#4043—31no—
47D5×C4⋊C4—31no—
48#4030—32no—
49#4024—32no—
50C23×C3×C5C2311yescharacteristic
51C22×D15C2311yescharacteristic
52C22×D15C2311yescharacteristic
53C22×C5×D3C2312yes—
54C2×C3×C4×C5—22no—
55C2×C3×C4×C5C2311yescharacteristic
56C2×C3×C4×C5C2311yescharacteristic
57#1220—22no—
58#1220—22no—
59C2×Dic15C2311yescharacteristic
60C2×Dic15C2311yescharacteristic
61D60—22no—
62C4×D15—22no—
63C4×D15D414yescomplemented
64Dic30—22no—
65C5×#67—42no—
66C2×C5×Dic3C2312yes—
67C5×D12—22no—
68C2×C3×Dic5C2312yes—
69C2×C3×Dic5C2312yes—
70C3×Dic10—22no—
71#1203—24no—
72#1203—42no—
73#1200—42no—
74D3×Dic5—24no—
75C2×#239—51no—
76C2×#235—51no—
77C3×C22⋊Q8—51no—
78#946—51no—
79#930D511yescharacteristic, complemented
80#896—52no—
81#895—51no—
82#895—52no—
83D3×C4⋊C4—51no—
84#884—52no—
85#884—52no—
86C2×C5×D4—32no—
87C2×C5×D4—31no—
88C22×C4×C5C2×D311yescharacteristic
89C2×C22⋊D5—31no—
90C2×C22⋊D5—31no—
91C2×C22⋊D5—32no—
92C2×C22⋊D5—32no—
93C22×Dic5—32no—
94#711—64no—
95#709—64no—
96C2×D20—31no—
97C2×C4×D5—31no—
98C2×C4×D5—31no—
99C2×C4×D5—32no—
100C2×Dic10—31no—
101C2×Dic10C2×D311yescharacteristic
102C5×C4⋊C4—31no—
103C5×C22⋊C4—32no—
104C22⋊Dic5—32no—
105C22⋊Dic5C2×D312yes—
106#685—32no—
107C4⋊Dic5—32no—
108C4⋊Dic5C2×D311yescharacteristic
109#683—32no—
110#683C2×D312yes—
111#537—151noSylow
112C22×C3×C5—21no—
113C22×C3×C5—21no—
114C22×C3×C5—21no—
115C22×C3×C5C2411yescharacteristic
116C2×D15—21no—
117C2×D15—21no—
118C2×D15—21no—
119C2×D15—21no—
120C2×D15—21no—
121C2×D15C2×D412yes—
122C2×C5×D3—22no—
123C2×C5×D3—22no—
124C2×C5×D3—22no—
125C3×C4×C5—21no—
126C3×C4×C5C2×D412yes—
127Dic15—21no—
128Dic15C2×D412yes—
129C3×Dic5—22no—
130C3×Dic5—22no—
131C5×Dic3—22no—
132C2×C3×Q8—51no—
133C22×C3×C4C2×D511yescharacteristic
134C2×#67—51no—
135C2×#67—51no—
136C2×#67C2×D512yes—
137#239—104no—
138#235—104no—
139C2×D12—52no—
140C2×D12—51no—
141C2×D12—52no—
142C2×D12C2×D511yescharacteristic
143C2×C4×D3—51no—
144C2×C4×D3—52no—
145C2×C4×D3—51no—
146C2×C4×D3—52no—
147C2×Dic6—51no—
148C3×C4⋊C4—52no—
149C3×C4⋊C4—51no—
150C3×C22⋊C4—52no—
151(C2×D3)⋊C4—52no—
152(C2×D3)⋊C4—52no—
153(C2×D3)⋊C4C2×D512yes—
154C4⋊Dic3C2×D511yescharacteristic
155Dic3⋊C4—52no—
156Dic3⋊C4—52no—
157C23×C5—32no—
158C23×C5C22×D311yescharacteristic
159C22×D5—31no—
160C22×D5—31no—
161C5×D4—62no—
162C5×D4—122no—
163C2×C4×C5—22no—
164C2×C4×C5—32no—
165C2×C4×C5C22×D311yescharacteristic
166C2×C4×C5C22×D311yescharacteristic
167C22⋊D5—64no—
168C22⋊D5—62no—
169C22⋊D5—62no—
170C22⋊D5—122no—
171C2×Dic5—31no—
172C2×Dic5—31no—
173C2×Dic5—64no—
174C2×Dic5C22×D312yes—
175C2×Dic5C22×D312yes—
176D20—62no—
177C4×D5—34no—
178C4×D5—62no—
179C4×D5—122no—
180Dic10—22no—
181Dic10—62no—
182C2×#41—151no—
183C2×#41—151no—
184C22⋊Q8—51no—
185C22⋊Q8—152no—
186C22⋊Q8—151no—
187#121—31no—
188#121—152no—
189#121—152no—
190#121—152no—
191#121—151no—
192C2×C4⋊C4—151no—
193C2×C3×C5—41no—
194C2×C3×C5C22×D411yescharacteristic
195C2×C3×C5H(5,𝔽2)11yescharacteristic
196C2×C3×C52−1+411yescharacteristic
197D15—22no—
198D15—41no—
199C5×D3—42no—
200C23×C3C22×D511yescharacteristic
201C22×D3—51no—
202C22×D3—51no—
203C22×D3C22×D512yes—
204C3×Q8—102no—
205C2×C3×C4—22no—
206C2×C3×C4—52no—
207C2×C3×C4—52no—
208C2×C3×C4C22×D511yescharacteristic
209C2×C3×C4C22×D511yescharacteristic
210#67—42no—
211#67—102no—
212#67—102no—
213C2×Dic3—51no—
214C2×Dic3—51no—
215C2×Dic3C22×D512yes—
216D12—22no—
217D12—102no—
218D12—104no—
219D12—202no—
220C4×D3—54no—
221C4×D3—104no—
222C4×D3—102no—
223C4×D3—202no—
224Dic6—102no—
225C22×C5—21no—
226C22×C5—21no—
227C22×C5—21no—
228C22×C5—62no—
229C22×C5—62no—
230C22×C5—62no—
231C22×C5C23×D311yescharacteristic
232C2×D5—32no—
233C2×D5—61no—
234C2×D5—61no—
235C2×D5—61no—
236C2×D5—61no—
237C2×D5—61no—
238C4×C5—21no—
239C4×C5—62no—
240C4×C5D3×D412yes—
241Dic5—22no—
242Dic5—22no—
243Dic5—32no—
244Dic5—61no—
245#41—304no—
246#41—304no—
247C2×Q8—51no—
248C2×Q8—151no—
249C2×D4—32no—
250C2×D4—31no—
251C2×D4—152no—
252C2×D4—152no—
253C2×D4—151no—
254C2×D4—151no—
255C2×D4—151no—
256C22×C4—152no—
257C22×C4—152no—
258C22×C4—151no—
259C22×C4—151no—
260C22×C4D3×D511yescharacteristic
261C4⋊C4—31no—
262C4⋊C4—51no—
263C4⋊C4—52no—
264C4⋊C4—152no—
265C4⋊C4—152no—
266C22⋊C4—32no—
267C22⋊C4—52no—
268C22⋊C4—152no—
269C22⋊C4—152no—
270C3×C5#53711yescharacteristic, complemented, Hall
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3C23×D511yescharacteristic
275C2×D3—22no—
276C2×D3—22no—
277C2×D3—22no—
278C2×D3—52no—
279C2×D3—101no—
280C2×D3—101no—
281C2×D3—101no—
282C2×D3—101no—
283C2×D3—101no—
284C3×C4—21no—
285C3×C4—102no—
286C3×C4—102no—
287C3×C4D4×D512yes—
288Dic3—22no—
289Dic3—52no—
290Dic3—101no—
291C2×C5—41no—
292C2×C5—122no—
293C2×C5C2×D3×D411yescharacteristic
294C2×C5#100711yescharacteristic
295C2×C5#100911yescharacteristic
296D5—62no—
297D5—121no—
298C23—32no—
299C23—151no—
300C23—151no—
301C23C2×D3×D511yescharacteristic
302Q8—102no—
303Q8—302no—
304D4—62no—
305D4—122no—
306D4—302no—
307D4—304no—
308D4—302no—
309D4—302no—
310D4—602no—
311C2×C4—22no—
312C2×C4—32no—
313C2×C4—52no—
314C2×C4—52no—
315C2×C4—151no—
316C2×C4—154no—
317C2×C4—151no—
318C2×C4—304no—
319C2×C4—302no—
320C2×C4—602no—
321C2×C4C2×D3×D511yescharacteristic
322C2×C4C2×D3×D511yescharacteristic
323C2×C3—41no—
324C2×C3C2×D4×D511yescharacteristic
325C2×C3#414411yescharacteristic
326C2×C3#415011yescharacteristic
327D3—42no—
328D3—102no—
329D3—201no—
330C5#562911yescharacteristic, complemented, Sylow
331C22—21no—
332C22—21no—
333C22—21no—
334C22—62no—
335C22—62no—
336C22—62no—
337C22—152no—
338C22—301no—
339C22—301no—
340C22—301no—
341C22—301no—
342C22—301no—
343C22C22×D3×D511yescharacteristic
344C4—21no—
345C4—62no—
346C4—102no—
347C4—102no—
348C4—152no—
349C4—301no—
350C4#679012yes—
351C3#6594211yescharacteristic, complemented, Sylow
352C2—41no—
353C2—122no—
354C2—302no—
355C2—601no—
356C2#9200411yescharacteristic
357C2#9201811yescharacteristic
358C2C2×#679011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#537
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. C2×#6817C2semidirect
2. C2×#6768C2semidirect
3. C5×#930C2semidirect
4. C3×#4070C2semidirect
5. #91558C2semidirect
6. #91557C2semidirect
7. #91466C2semidirect
8. #91437C2semidirect
9. #91432C2semidirect
10. #91420C2semidirect
11. #91390C2semidirect
45. #4070D3semidirect
63. C4×D15D4semidirect
79. #930D5semidirect
270. C3×C5#537semidirect
330. C5#5629semidirect
351. C3#65942semidirect

Maximal subgroups

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

subgroupindex
C2×#68172
C2×#67682
C5×#9302
C3×#40702
#915582
#915572
#914662
#914372
#914322
#914202
#913902
#659423
#56295