Finite Group Databasebeta

← #12968941

Subgroups — #12968941

Order 1280.

1770 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10617626C212yesmaximal
2C5×#2911C211yesmaximal, characteristic
3#10615483C211yesmaximal, characteristic
4C2×#65411C211yesmaximal, characteristic
5C2×#65287C212yesmaximal
6#10610937C218yesmaximal, complemented
7#10610936C218yesmaximal, complemented
8#10610902C214yesmaximal, complemented
9#10610901C214yesmaximal, complemented
10C22×C5×M16C2211yescharacteristic
11C2×C5×#117C2212yes—
12C2×#4072C2212yes—
13C2×#4072C2211yescharacteristic
14C23×#168C2211yescharacteristic
15#65853—24no—
16#65853C2218yes—
17#65851—24no—
18#65851C2218yes—
19#65834C2218yes—
20#65817—24no—
21#65804—24no—
22C5×#428C2218yes—
23C5×#427C2214yes—
24#65411C2214yes—
25#65411C2214yes—
26#65410C2218yes—
27#65395C2214yes—
28C2×#3947C2212yes—
29C2×C8×Dic5C2212yes—
30#65287C2214yes—
31#65287C2214yes—
32#65287C2214yes—
33#65287C2214yes—
34C2×#3940C2214yes—
35C2×#3939C2214yes—
36#65167C2218yescomplemented
37#65165C22116yescomplemented
38#65163C2218yescomplemented
39#65147C2218yescomplemented
40#65145C22116yescomplemented
41#65144C2218yescomplemented
42#60660—51nomaximal, Sylow
43C23×C4×C5C2311yescharacteristic
44C2×C5×M16Q818yescomplemented
45C2×C5×M16C2314yes—
46C22×C5×C8C2312yes—
47C5×#117—22no—
48C5×#117—22no—
49C5×#117C2314yes—
50C5×#117C2314yes—
51C2×C5×C4⋊C4C2314yes—
52C2×C5×C22⋊C4—22no—
53C2×C42×C5—22no—
54C2×C22⋊Dic5—22no—
55C2×C22⋊Dic5D412yes—
56#4072—22no—
57#4072—22no—
58#4072D414yes—
59#4072C2314yes—
60#4072C2314yes—
61#4072C2314yes—
62C2×C4⋊Dic5D412yes—
63C2×C4⋊Dic5C2314yes—
64#4070—216no—
65C2×C4×Dic5—22no—
66C2×C4×Dic5C2312yes—
67C22×#168Q818yescomplemented
68C22×#168C2311yescharacteristic
69C22×#168C2311yescharacteristic
70C22×#168C2312yes—
71C22×#168C2312yes—
72#4036—28no—
73#4036C23116yes—
74#4034—28no—
75#4034C23116yes—
76#4024—216no—
77#4024—216no—
78#4016—28no—
79#4015—28no—
80C4×Dic10—216no—
81C5×#107C2318yes—
82C5×#106C2318yes—
83#3950C2318yes—
84#3949C2318yes—
85#3947C2318yes—
86C8×Dic5C2314yes—
87C8×Dic5C2314yes—
88#3940C2318yes—
89#3940C2318yes—
90#3939C2318yes—
91#3939C2318yes—
92#3458—52no—
93#3164—58no—
94#3163—58no—
95#3162—54no—
96#3160—54no—
97#2911—51no—
98#2911D511yescharacteristic
99C2×#427—52no—
100C2×#405—51no—
101C24×C5C2×D411yescharacteristic
102C22×C4×C5—24no—
103C22×C4×C5C2×D412yes—
104C22×C4×C5C2×D411yescharacteristic
105C22×C4×C5C2×Q814yes—
106C22×C4×C5C2×Q814yes—
107C22×C4×C5C2412yes—
108C22×C4×C5C2411yescharacteristic
109C22×Dic5—24no—
110C22×Dic5C2×D414yes—
111C2×Dic10—216no—
112C5×M16C2×Q8116yes—
113C2×C5×C8C2×Q818yes—
114C2×C5×C8C2414yes—
115C5×C4⋊C4—24no—
116C5×C4⋊C4C2418yes—
117C5×C22⋊C4—24no—
118C5×C22⋊C4—24no—
119C42×C5—24no—
120C42×C5—24no—
121C22⋊Dic5—24no—
122C22⋊Dic5—24no—
123C22⋊Dic5C2×D418yes—
124C4⋊Dic5—24no—
125C4⋊Dic5—216no—
126C4⋊Dic5C2×D418yes—
127C4⋊Dic5C2418yes—
128#683—216no—
129#683—216no—
130C4×Dic5—24no—
131C4×Dic5—24no—
132C4×Dic5C2×D414yes—
133C4×Dic5C2414yes—
134C2×#168C2×Q814yes—
135C2×#168C2×Q814yes—
136C2×#168C2×Q818yes—
137C2×#168C2×Q818yes—
138C2×#168C2411yescharacteristic
139C2×#168C2412yes—
140C2×#168C2411yescharacteristic
141C22×M16C2×D511yescharacteristic
142C23×C8—51no—
143#544—58no—
144#544—58no—
145#544—104no—
146#543—104no—
147#533—58no—
148#533—104no—
149#533—104no—
150C2×#117—51no—
151C2×#117—52no—
152C2×#117C2×D512yes—
153#501—516no—
154#501—516no—
155#500—58no—
156#499—58no—
157#499—58no—
158#498—58no—
159#428—58no—
160#428C2×D518yes—
161#427—54no—
162#427—54no—
163#427—54no—
164#427—54no—
165#427—54no—
166#427C2×D514yes—
167C2×#107—54no—
168C2×#106—54no—
169#405—54no—
170#405—54no—
171C2×#97—52no—
172C2×C4×C8—52no—
173C23×C5—42no—
174C23×C5C4⋊Q814yes—
175C23×C5C22×D411yescharacteristic
176C23×C5C22×D412yes—
177C2×C4×C5—28no—
178C2×C4×C5—84no—
179C2×C4×C5C4⋊Q818yes—
180C2×C4×C5C4⋊Q814yes—
181C2×C4×C5C22×D412yes—
182C2×C4×C5C22×D411yescharacteristic
183C2×C4×C5C22×Q812yes—
184C2×C4×C5C22×Q814yes—
185C2×C4×C5C22×Q814yes—
186C2×C4×C5C22×Q812yes—
187C2×C4×C5C2511yescharacteristic
188C2×Dic5—28no—
189C2×Dic5—44no—
190C2×Dic5—84no—
191C2×Dic5C22×D418yes—
192Dic10—816no—
193C5×C8C22×Q818yes—
194#168C22×Q812yes—
195#168C22×Q814yes—
196#168C22×Q812yes—
197C23×C4C22×D511yescharacteristic
198C2×M16Dic1018yescomplemented
199C2×M16C22×D514yes—
200C22×C8—58no—
201C22×C8—51no—
202C22×C8—51no—
203C22×C8—52no—
204C22×C8—52no—
205C22×C8C22×D512yes—
206C4⋊Q8—516no—
207C4⋊Q8—108no—
208C4⋊Q8—108no—
209#125—516no—
210#125—108no—
211#125—108no—
212C22⋊Q8—1016no—
213C22⋊Q8—1016no—
214C22⋊Q8—1016no—
215C4×Q8—1016no—
216#117—22no—
217#117—22no—
218#117—54no—
219#117—54no—
220#117—54no—
221#117—54no—
222#117—102no—
223#117—102no—
224#117C22×D514yes—
225#117C22×D514yes—
226C2×C4⋊C4—52no—
227C2×C4⋊C4—54no—
228C2×C4⋊C4C22×D514yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—52no—
231C2×C22⋊C4—102no—
232C2×C42—22no—
233C2×C42—52no—
234C2×C42—102no—
235#107—58no—
236#107—58no—
237#107—58no—
238#107C22×D518yes—
239#106—58no—
240#106—58no—
241#106—58no—
242#106C22×D518yes—
243#97—58no—
244C4×C8—54no—
245C4×C8—54no—
246C22×C5—22no—
247C22×C5—24no—
248C22×C5—44no—
249C22×C5C2×C4⋊Q812yes—
250C22×C5C2×C4⋊Q814yes—
251C22×C5C23×D411yescharacteristic
252C4×C5—48no—
253C4×C5C2×C4⋊Q814yes—
254C4×C5C2×C4⋊Q812yes—
255C4×C5C23×Q811yescharacteristic
256C4×C5C23×Q811yescharacteristic
257Dic5—28no—
258Dic5—48no—
259C24D4×D511yescharacteristic
260C2×Q8—1016no—
261C22×C4—24no—
262C22×C4—54no—
263C22×C4—104no—
264C22×C4C2×Dic1014yes—
265C22×C4D4×D511yescharacteristic
266C22×C4D4×D512yes—
267C22×C4Q8×D514yes—
268C22×C4C23×D512yes—
269C22×C4C23×D511yescharacteristic
270M16C2×Dic10116yes—
271C2×C8—54no—
272C2×C8—54no—
273C2×C8—58no—
274C2×C8—58no—
275C2×C8—51no—
276C2×C8—51no—
277C2×C8—52no—
278C2×C8C2×Dic1018yes—
279C2×C8C23×D514yes—
280C4⋊C4—24no—
281C4⋊C4—58no—
282C4⋊C4—58no—
283C4⋊C4—1016no—
284C4⋊C4—1016no—
285C4⋊C4—1016no—
286C4⋊C4—104no—
287C4⋊C4C23×D518yes—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—58no—
291C22⋊C4—104no—
292C22⋊C4—104no—
293C42—24no—
294C42—24no—
295C42—54no—
296C42—54no—
297C42—104no—
298C42—104no—
299C2×C5—22no—
300C2×C5—24no—
301C2×C5C22×C4⋊Q811yescharacteristic
302C2×C5#358412yes—
303C23—42no—
304C23#403414yes—
305C23C2×D4×D511yescharacteristic
306C23C2×D4×D512yes—
307Q8—4016no—
308C2×C4—28no—
309C2×C4—58no—
310C2×C4—84no—
311C2×C4—108no—
312C2×C4—204no—
313C2×C4—404no—
314C2×C4#403414yes—
315C2×C4#403418yes—
316C2×C4C22×Dic1012yes—
317C2×C4C22×Dic1014yes—
318C2×C4C2×D4×D512yes—
319C2×C4C2×D4×D511yescharacteristic
320C2×C4C2×Q8×D514yes—
321C2×C4C2×Q8×D512yes—
322C2×C4C24×D511yescharacteristic
323C8—52no—
324C8—52no—
325C8—54no—
326C8C22×Dic1018yes—
327C5#6066011yescharacteristic, complemented, Sylow
328C22—24no—
329C22—22no—
330C22—44no—
331C22C2×#403412yes—
332C22C2×#403414yes—
333C22C22×D4×D511yescharacteristic
334C4—48no—
335C4—108no—
336C4—208no—
337C4C2×#403414yes—
338C4C2×#403412yes—
339C4C23×Dic1011yescharacteristic
340C4C22×Q8×D511yescharacteristic
341C2—24no—
342C2—22no—
343C2C22×#403411yescharacteristic
344C2#1061910012yes—
subgroupG/H
centerC2×C4C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC5×#2911C2
socleC22×C5C23×D4
prime radicalC24×C5C2×D4
hypercenter#2911D5
hypocenterC5#60660
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2911

Complements

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

NH ≅ G/Ntype
6. #10610937C2semidirect
7. #10610936C2semidirect
8. #10610902C2semidirect
9. #10610901C2semidirect
36. #65167C22semidirect
37. #65165C22semidirect
38. #65163C22semidirect
39. #65147C22semidirect
40. #65145C22semidirect
41. #65144C22semidirect
44. C2×C5×M16Q8semidirect
67. C22×#168Q8semidirect
198. C2×M16Dic10semidirect
327. C5#60660semidirect

Maximal subgroups

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

subgroupindex
#106176262
C5×#29112
#106154832
C2×#654112
C2×#652872
#106109372
#106109362
#106109022
#106109012
#606605