Finite Group Databasebeta

← #14301704

Subgroups — #14301704

Order 1920.

684 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756587C211yesmaximal, characteristic
2#11756578C211yesmaximal, characteristic, complemented
3C3×C5×#413C211yesmaximal, characteristic, complemented
4#11751227C212yesmaximal, complemented
5#11746380C212yesmaximal, complemented
6#10600929—31nomaximal, Hall
7C2×C3×C5×#41C2211yescharacteristic
8C22×C4×D15C2211yescharacteristic
9C2×C3×C5×M16C2212yes—
10#91837—24no—
11#91835—21no—
12#91827—21no—
13#91826C412yes—
14#91824C412yes—
15C4×#1220—22no—
16#91815C2211yescharacteristic
17C2×#6710—22no—
18C2×#6710C2212yes—
19C3×C5×#101—21no—
20C3×C5×#100—21no—
21#91104—24no—
22#70136—51nomaximal, Hall
23#66162—31no—
24#66153—31no—
25C5×#413D311yescharacteristic, complemented
26#65415—32no—
27#64779—32no—
28C23×D15C2×C411yescharacteristic
29C3×C5×#41—22no—
30C3×C5×#41—22no—
31C2×C3×C5×Q8D411yescharacteristic
32C2×C3×C5×D4C2×C412yes—
33C2×C3×C5×D4D411yescharacteristic
34C22×C3×C4×C5D412yes—
35C22×C3×C4×C5C2311yescharacteristic
36C2×#1220—22no—
37C22×Dic15C2×C411yescharacteristic
38C2×C4×D15—21no—
39C2×C4×D15—21no—
40C2×C4×D15—22no—
41C2×C4×D15—22no—
42C3×C5×M16—22no—
43C3×C5×M16—22no—
44C2×C3×C5×C8—22no—
45#6719—22no—
46#6719C2×C412yes—
47#6714—22no—
48C4⋊Dic15—21no—
49#6712—22no—
50C4×Dic15—21no—
51#6710—24no—
52#6710—22no—
53#6710—22no—
54C2×C3⋊#168—24no—
55C2×C3⋊#168D414yescomplemented
56#5828—51no—
57#5819—51no—
58C3×#413D511yescharacteristic, complemented
59#5126—52no—
60#4559—52no—
61C2×C5×#41C2×D311yescharacteristic
62C22×C4×D5—31no—
63C2×C5×M16C2×D312yes—
64#4094—64no—
65#4092—61no—
66#4084—61no—
67#4083—32no—
68#4081—32no—
69C4×C22⋊D5—62no—
70#4072—31no—
71C2×#681—32no—
72C2×#681—62no—
73C5×#101—21no—
74C5×#100—21no—
75#3952—64no—
76#1891—151noSylow
77C23×C3×C5C22⋊C412yes—
78C23×C3×C5C22×C411yescharacteristic
79C22×D15—21no—
80C22×D15—22no—
81C22×D15—42no—
82C3×C5×Q8—22no—
83C3×C5×D4—24no—
84C3×C5×D4—22no—
85C2×C3×C4×C5—22no—
86C2×C3×C4×C5—22no—
87C2×C3×C4×C5C22⋊C412yes—
88C2×C3×C4×C5C2×D412yes—
89C2×C3×C4×C5C2×D411yescharacteristic
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5#4112yes—
92#1220—44no—
93C2×Dic15—22no—
94C2×Dic15—21no—
95C2×Dic15—22no—
96C4×D15—41no—
97C4×D15—42no—
98C4×D15—41no—
99C3×C5×C8—42no—
100C3⋊#168—24no—
101C2×C3×#41C2×D511yescharacteristic
102C22×C4×D3—51no—
103C2×C3×M16C2×D512yes—
104#948—104no—
105#946—101no—
106#938—101no—
107#937—52no—
108#935—52no—
109C4×#67—102no—
110#926—51no—
111C2×#208—52no—
112C2×#208—102no—
113C3×#101—21no—
114C3×#100—21no—
115(C2×D3)⋊C8—104no—
116C23×D5—31no—
117C5×#41—22no—
118C5×#41—22no—
119C2×C5×Q8#6711yescharacteristic
120C2×C5×D4C4×D312yes—
121C2×C5×D4#6711yescharacteristic
122C22×C4×C5D1212yes—
123C22×C4×C5C22×D311yescharacteristic
124C2×C22⋊D5—62no—
125C22×Dic5—31no—
126C2×C4×D5—61no—
127C2×C4×D5—62no—
128C2×C4×D5—62no—
129C2×C4×D5—61no—
130C5×M16—22no—
131C5×M16—22no—
132C2×C5×C8—22no—
133C22⋊Dic5—32no—
134C22⋊Dic5—62no—
135#685—62no—
136C4⋊Dic5—61no—
137#683—62no—
138C4×Dic5—61no—
139#681—64no—
140#681—62no—
141#681—62no—
142C2×#168—34no—
143C2×#168—64no—
144#568—151no—
145#525—151no—
146#413D1511yescharacteristic, complemented
147#407—152no—
148#344—152no—
149C22×C3×C5—22no—
150C22×C3×C5—41no—
151C22×C3×C5—42no—
152C22×C3×C5C2×C22⋊C411yescharacteristic
153C22×C3×C5C4×D412yes—
154C2×D15—22no—
155C2×D15—24no—
156C2×D15—41no—
157C2×D15—42no—
158C2×D15—41no—
159C3×C4×C5—22no—
160C3×C4×C5#12011yescharacteristic
161C3×C4×C5#12112yes—
162C3×C4×C5#12311yescharacteristic
163Dic15—42no—
164Dic15—41no—
165C23×D3—51no—
166C3×#41—22no—
167C3×#41—22no—
168C2×C3×Q8C22⋊D511yescharacteristic
169C2×C3×D4C4×D512yes—
170C2×C3×D4C22⋊D511yescharacteristic
171C22×C3×C4D2012yes—
172C22×C3×C4C22×D511yescharacteristic
173C2×#67—102no—
174C22×Dic3—51no—
175C2×C4×D3—101no—
176C2×C4×D3—102no—
177C2×C4×D3—102no—
178C2×C4×D3—101no—
179C3×M16—22no—
180C3×M16—22no—
181C2×C3×C8—22no—
182#217—52no—
183#217—102no—
184(C2×D3)⋊C4—102no—
185C4⋊Dic3—101no—
186Dic3⋊C4—102no—
187C4×Dic3—101no—
188#208—104no—
189#208—102no—
190#208—102no—
191C2×C3⋊C8—54no—
192C2×C3⋊C8—104no—
193C23×C5(C2×D3)⋊C412yes—
194C23×C5C2×C4×D311yescharacteristic
195C22×D5—62no—
196C22×D5—61no—
197C22×D5—122no—
198C5×Q8—22no—
199C5×D4—24no—
200C5×D4—22no—
201C2×C4×C5—22no—
202C2×C4×C5—22no—
203C2×C4×C5(C2×D3)⋊C412yes—
204C2×C4×C5C2×D1211yescharacteristic
205C2×C4×C5D3×D412yes—
206C2×C4×C5#23712yes—
207C2×C4×C5C2×#6711yescharacteristic
208C22⋊D5—124no—
209C2×Dic5—62no—
210C2×Dic5—62no—
211C2×Dic5—61no—
212C4×D5—121no—
213C4×D5—121no—
214C4×D5—122no—
215C5×C8—42no—
216#168—64no—
217C2×#41C2×D1511yescharacteristic
218C23×C4—151no—
219#131—304no—
220C2×M16—152no—
221C2×M16—302no—
222C2×M16C2×D1512yes—
223#123—152no—
224C22⋊Q8—301no—
225#121—301no—
226#120—152no—
227C4×D4—302no—
228#117—151no—
229#101—21no—
230#100—21no—
231C22⋊C8—304no—
232C2×C3×C5—21no—
233C2×C3×C5—22no—
234C2×C3×C5—42no—
235C2×C3×C5#38611yescharacteristic
236D15—42no—
237C23×C3#68512yes—
238C23×C3C2×C4×D511yescharacteristic
239C22×D3—102no—
240C22×D3—101no—
241C22×D3—202no—
242C3×Q8—22no—
243C3×D4—24no—
244C3×D4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4#68512yes—
248C2×C3×C4C2×D2011yescharacteristic
249C2×C3×C4D4×D512yes—
250C2×C3×C4#71112yes—
251C2×C3×C4C2×C22⋊D511yescharacteristic
252#67—204no—
253C2×Dic3—102no—
254C2×Dic3—102no—
255C2×Dic3—101no—
256C4×D3—201no—
257C4×D3—201no—
258C4×D3—202no—
259C3×C8—42no—
260C3⋊C8—104no—
261C22×C5—22no—
262C22×C5—41no—
263C22×C5—42no—
264C22×C5#67⋊C412yes—
265C22×C5C2×(C2×D3)⋊C411yescharacteristic
266C2×D5—62no—
267C2×D5—64no—
268C2×D5—121no—
269C2×D5—122no—
270C2×D5—121no—
271C4×C5—22no—
272C4×C5#88211yescharacteristic
273C4×C5#88611yescharacteristic
274C4×C5#93912yes—
275Dic5—122no—
276Dic5—121no—
277C24—151no—
278#41—22no—
279#41—22no—
280C2×Q8#122011yescharacteristic
281C2×D4—302no—
282C2×D4C4×D1512yes—
283C2×D4#122011yescharacteristic
284C22×C4—151no—
285C22×C4—301no—
286C22×C4—302no—
287C22×C4—302no—
288C22×C4—301no—
289C22×C4D6012yes—
290C22×C4C22×D1511yescharacteristic
291M16—22no—
292M16—22no—
293M16—304no—
294M16—302no—
295M16—302no—
296C2×C8—22no—
297C2×C8—154no—
298C2×C8—304no—
299C4⋊C4—302no—
300C4⋊C4—301no—
301C22⋊C4—152no—
302C22⋊C4—302no—
303C22⋊C4—302no—
304C42—301no—
305C3×C5#189111yescharacteristic, complemented, Hall
306C22×C3—22no—
307C22×C3—41no—
308C22×C3—42no—
309C22×C3#402712yes—
310C22×C3C2×#68511yescharacteristic
311C2×D3—102no—
312C2×D3—104no—
313C2×D3—201no—
314C2×D3—202no—
315C2×D3—201no—
316C3×C4—22no—
317C3×C4#402811yescharacteristic
318C3×C4#403211yescharacteristic
319C3×C4#408512yes—
320Dic3—202no—
321Dic3—201no—
322C2×C5—21no—
323C2×C5—22no—
324C2×C5—42no—
325C2×C5#495811yescharacteristic
326D5—122no—
327C23—302no—
328C23—301no—
329C23—602no—
330C23#671412yes—
331C23C2×C4×D1511yescharacteristic
332Q8—22no—
333D4—24no—
334D4—22no—
335D4—604no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—302no—
339C2×C4—302no—
340C2×C4—301no—
341C2×C4—601no—
342C2×C4—601no—
343C2×C4—602no—
344C2×C4#671412yes—
345C2×C4C2×D6011yescharacteristic
346C2×C4D4×D1512yes—
347C2×C4#681912yes—
348C2×C4C2×#122011yescharacteristic
349C8—42no—
350C8—304no—
351C2×C3—21no—
352C2×C3—22no—
353C2×C3—42no—
354C2×C3#6524711yescharacteristic
355D3—202no—
356C5#7013611yescharacteristic, complemented, Sylow
357C22—22no—
358C22—41no—
359C22—42no—
360C22—302no—
361C22—304no—
362C22—601no—
363C22—602no—
364C22—601no—
365C22#9177012yes—
366C22C2×#671411yescharacteristic
367C4—22no—
368C4—602no—
369C4—601no—
370C4#9177111yescharacteristic
371C4#9177511yescharacteristic
372C4#9182812yes—
373C3#1060092911yescharacteristic, complemented, Sylow
374C2—21no—
375C2—22no—
376C2—42no—
377C2—602no—
378C2#1175105911yescharacteristic
subgroupG/H
centerC4#91771
commutatorC23×C3×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#413C2
socleC2×C3×C5#386
prime radical#11756587C2
cosocleC22×C3×C4×C5C23
hypercenter#413D15
hypocenterC3×C5#1891
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #11756578C2semidirect
3. C3×C5×#413C2semidirect
4. #11751227C2semidirect
5. #11746380C2semidirect
25. C5×#413D3semidirect
55. C2×C3⋊#168D4semidirect
58. C3×#413D5semidirect
146. #413D15semidirect
305. C3×C5#1891semidirect
356. C5#70136semidirect
373. C3#10600929semidirect

Maximal subgroups

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

subgroupindex
#117565872
#117565782
C3×C5×#4132
#117512272
#117463802
#106009293
#701365