Finite Group Databasebeta

← #14301440

Subgroups — #14301440

Order 1920.

720 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C3×C5×#518C211yesmaximal, characteristic
2C2×#91825C211yesmaximal, characteristic
3C2×#91117C211yesmaximal, characteristic
4#11751105C212yesmaximal, complemented
5#11746353C212yesmaximal, complemented
6#10602442—31nomaximal, Hall
7C22×C3×C5×D4C2211yescharacteristic
8C22×Dic30C2211yescharacteristic
9C3×C4×C5×D4—24no—
10C3×C5×#117C2212yes—
11C2×C3×C5×C22⋊C4—22no—
12C2×#6719—22no—
13#91825—22no—
14#91825C414yescomplemented
15C2×C4×Dic15C2211yescharacteristic
16C2×#6710C2212yes—
17#91117—22no—
18#91094—24no—
19#71649—51nomaximal, Hall
20C5×#518D311yescharacteristic
21C2×#4082—31no—
22C2×#3965—31no—
23#65293—32no—
24#64752—32no—
25C24×C3×C5D412yes—
26C2×C3×C5×D4—22no—
27C2×C3×C5×D4—22no—
28C2×C3×C5×D4C2×C412yes—
29C2×C3×C5×D4D412yes—
30C22×C3×C4×C5—24no—
31C22×C3×C4×C5C2311yescharacteristic
32C22×Dic15—21no—
33C22×Dic15—21no—
34C2×Dic30—22no—
35C2×Dic30—22no—
36C2×Dic30—21no—
37C2×Dic30C2×C412yescomplemented
38C3×C5×C4⋊C4D414yes—
39C3×C5×C22⋊C4—24no—
40C3×C5×C22⋊C4—22no—
41C3×C42×C5—22no—
42#6719—28no—
43C4×Dic15—21no—
44C4×Dic15C2×C412yescomplemented
45#6710—22no—
46#6710—22no—
47C2×C3⋊#168—22no—
48C3×#518D511yescharacteristic
49C2×#936—51no—
50C2×#833—51no—
51#5004—52no—
52#4532—52no—
53C22×C5×D4C2×D311yescharacteristic
54C22×Dic10—31no—
55C4×C5×D4—24no—
56C5×#117C2×D312yes—
57C2×C5×C22⋊C4—22no—
58C2×C22⋊Dic5—62no—
59#4082—34no—
60#4082—62no—
61C2×C4×Dic5—31no—
62C2×#681—32no—
63#3965—62no—
64#3942—64no—
65#1885—151noSylow
66C23×C3×C5—22no—
67C23×C3×C5—22no—
68C23×C3×C5—44no—
69C23×C3×C5C22⋊C414yes—
70C23×C3×C5C2×D411yescharacteristic
71C3×C5×D4—24no—
72C3×C5×D4—24no—
73C2×C3×C4×C5—24no—
74C2×C3×C4×C5—84no—
75C2×C3×C4×C5C22×C411yescharacteristic
76C2×C3×C4×C5C2×D412yes—
77C2×C3×C4×C5C2×D411yescharacteristic
78C2×C3×C4×C5#4112yes—
79C2×Dic15—22no—
80C2×Dic15—22no—
81C2×Dic15—42no—
82C2×Dic15—41no—
83C2×Dic15—41no—
84Dic30—24no—
85Dic30—42no—
86C3⋊#168—42no—
87C22×C3×D4C2×D511yescharacteristic
88C22×Dic6—51no—
89C3×C4×D4—24no—
90C3×#117C2×D512yes—
91C2×C3×C22⋊C4—22no—
92C2×#217—102no—
93#936—54no—
94#936—102no—
95C2×C4×Dic3—51no—
96C2×#208—52no—
97#833—102no—
98#810—104no—
99C24×C5#6712yes—
100C2×C5×D4—22no—
101C2×C5×D4—22no—
102C2×C5×D4C4×D312yes—
103C2×C5×D4D1212yes—
104C22×C4×C5—24no—
105C22×C4×C5C22×D311yescharacteristic
106C22×Dic5—61no—
107C22×Dic5—61no—
108C2×Dic10—32no—
109C2×Dic10—62no—
110C2×Dic10—62no—
111C2×Dic10—61no—
112C5×C4⋊C4#6714yes—
113C5×C22⋊C4—24no—
114C5×C22⋊C4—22no—
115C42×C5—22no—
116C22⋊Dic5—68no—
117C4×Dic5—32no—
118C4×Dic5—61no—
119#681—62no—
120#681—62no—
121C2×#168—62no—
122C2×#124—151no—
123#518D1511yescharacteristic
124#419—152no—
125C2×#100—151no—
126#339—152no—
127C22×C3×C5—22no—
128C22×C3×C5—24no—
129C22×C3×C5—44no—
130C22×C3×C5—44no—
131C22×C3×C5—82no—
132C22×C3×C5C2×C22⋊C411yescharacteristic
133C22×C3×C5#12011yescharacteristic
134C22×C3×C5#12311yescharacteristic
135C3×C4×C5—44no—
136C3×C4×C5C4×D412yes—
137C3×C4×C5#12112yes—
138Dic15—42no—
139Dic15—42no—
140C24×C3C22⋊D512yes—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4C4×D512yes—
144C2×C3×D4D2012yes—
145C22×C3×C4—24no—
146C22×C3×C4C22×D511yescharacteristic
147C22×Dic3—101no—
148C22×Dic3—101no—
149C2×Dic6—52no—
150C2×Dic6—102no—
151C2×Dic6—102no—
152C2×Dic6—101no—
153C3×C4⋊C4C22⋊D514yes—
154C3×C22⋊C4—24no—
155C3×C22⋊C4—22no—
156C3×C42—22no—
157#217—108no—
158C4×Dic3—52no—
159C4×Dic3—101no—
160#208—102no—
161#208—102no—
162C2×C3⋊C8—102no—
163C23×C5—22no—
164C23×C5—22no—
165C23×C5—44no—
166C23×C5(C2×D3)⋊C414yes—
167C23×C5C2×#6711yescharacteristic
168C5×D4—24no—
169C5×D4—24no—
170C2×C4×C5—24no—
171C2×C4×C5—84no—
172C2×C4×C5C2×C4×D311yescharacteristic
173C2×C4×C5C2×D1211yescharacteristic
174C2×C4×C5#23512yes—
175C2×C4×C5C2×#6712yes—
176C2×Dic5—62no—
177C2×Dic5—62no—
178C2×Dic5—122no—
179C2×Dic5—121no—
180C2×Dic5—121no—
181Dic10—64no—
182Dic10—122no—
183#168—122no—
184C22×Q8—151no—
185C22×D4C2×D1511yescharacteristic
186C2×M16—152no—
187#124—154no—
188#124—302no—
189C4×D4—24no—
190#117C2×D1512yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—302no—
193C2×C42—151no—
194#103—304no—
195#100—302no—
196C2×C3×C5—21no—
197C2×C3×C5—21no—
198C2×C3×C5—44no—
199C2×C3×C5#38611yescharacteristic
200C2×C3×C5#45512yes—
201C23×C3—22no—
202C23×C3—22no—
203C23×C3—44no—
204C23×C3#68514yes—
205C23×C3C2×C22⋊D511yescharacteristic
206C3×D4—24no—
207C3×D4—24no—
208C2×C3×C4—24no—
209C2×C3×C4—84no—
210C2×C3×C4C2×C4×D511yescharacteristic
211C2×C3×C4C2×D2011yescharacteristic
212C2×C3×C4#70912yes—
213C2×C3×C4C2×C22⋊D512yes—
214C2×Dic3—102no—
215C2×Dic3—102no—
216C2×Dic3—202no—
217C2×Dic3—201no—
218C2×Dic3—201no—
219Dic6—104no—
220Dic6—202no—
221C3⋊C8—202no—
222C22×C5—22no—
223C22×C5—24no—
224C22×C5—44no—
225C22×C5—44no—
226C22×C5—82no—
227C22×C5C2×(C2×D3)⋊C411yescharacteristic
228C22×C5#92911yescharacteristic
229C22×C5#95311yescharacteristic
230C4×C5—44no—
231C4×C5C4×#6712yes—
232C4×C5#93012yes—
233Dic5—122no—
234Dic5—122no—
235C24#122012yes—
236C2×Q8—152no—
237C2×Q8—301no—
238C2×Q8—302no—
239C2×Q8—302no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4C4×D1512yes—
243C2×D4D6012yes—
244C22×C4—24no—
245C22×C4—301no—
246C22×C4—301no—
247C22×C4C22×D1511yescharacteristic
248M16—302no—
249M16—302no—
250C2×C8—302no—
251C4⋊C4#122014yes—
252C22⋊C4—24no—
253C22⋊C4—22no—
254C22⋊C4—308no—
255C42—22no—
256C42—152no—
257C42—301no—
258C3×C5#188511yescharacteristic, complemented, Hall
259C22×C3—22no—
260C22×C3—24no—
261C22×C3—44no—
262C22×C3—44no—
263C22×C3—82no—
264C22×C3C2×#68511yescharacteristic
265C22×C3#407511yescharacteristic
266C22×C3#409911yescharacteristic
267C3×C4—44no—
268C3×C4C4×C22⋊D512yes—
269C3×C4#407612yes—
270Dic3—202no—
271Dic3—202no—
272C2×C5—21no—
273C2×C5—21no—
274C2×C5—44no—
275C2×C5#521311yescharacteristic
276C2×C5#524212yes—
277C23—22no—
278C23—22no—
279C23—44no—
280C23#671414yes—
281C23C2×#122011yescharacteristic
282Q8—304no—
283Q8—602no—
284D4—24no—
285D4—24no—
286C2×C4—24no—
287C2×C4—84no—
288C2×C4—302no—
289C2×C4—302no—
290C2×C4—602no—
291C2×C4—601no—
292C2×C4—601no—
293C2×C4C2×C4×D1511yescharacteristic
294C2×C4C2×D6011yescharacteristic
295C2×C4#681712yes—
296C2×C4C2×#122012yes—
297C8—602no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—44no—
301C2×C3#6550211yescharacteristic
302C2×C3#6553112yes—
303C5#7164911yescharacteristic, complemented, Sylow
304C22—22no—
305C22—24no—
306C22—44no—
307C22—44no—
308C22—82no—
309C22C2×#671411yescharacteristic
310C22#9181811yescharacteristic
311C22#9184211yescharacteristic
312C4—44no—
313C4—602no—
314C4—602no—
315C4C4×#122012yes—
316C4#9181912yes—
317C3#1060244211yescharacteristic, complemented, Sylow
318C2—21no—
319C2—21no—
320C2—44no—
321C2#1175131411yescharacteristic
322C2#1175134312yes—
subgroupG/H
centerC22#91842
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#518C2
socleC22×C3×C5#120
prime radicalC22×C3×C5×D4C22
cosocleC22×C3×C4×C5C23
hypercenter#518D15
hypocenterC3×C5#1885
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
4. #11751105C2semidirect
5. #11746353C2semidirect
14. #91825C4semidirect
37. C2×Dic30C2×C4semidirect
44. C4×Dic15C2×C4semidirect
258. C3×C5#1885semidirect
303. C5#71649semidirect
317. C3#10602442semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5182
C2×#918252
C2×#911172
#117511052
#117463532
#106024423
#716495