Finite Group Databasebeta

← #14294770

Subgroups — #14294770

Order 1920.

864 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91931C212yesmaximal
2C2×#91404C211yesmaximal, characteristic
3#11746790C211yesmaximal, characteristic, complemented
4C3×#64919C211yesmaximal, characteristic, complemented
5#11746574C212yesmaximal, complemented
6#10604334—31nomaximal, Hall
7C22×C4×D15C411yescharacteristic
8C22×#1200C411yescharacteristic
9C22×#1200C2211yescharacteristic
10C2×#6762C2212yes—
11C2×C3×#704C2212yes—
12C2×#6740—22no—
13#91931—24no—
14#91931C418yescomplemented
15C2×#6738—22no—
16C2×C5×Dic3⋊C4C411yescharacteristic
17C2×C3×#683C2211yescharacteristic
18C2×Dic3⋊Dic5C411yescharacteristic
19C2×#6668C2211yescharacteristic
20#91404—22no—
21#91404—22no—
22D3×#343—51nomaximal, Hall
23C2×D5⋊M16—32no—
24C2×D5×C4⋊C4—31no—
25#64919—31no—
26#64919—32no—
27#64919D311yescharacteristic, complemented
28C23×D15C2×C411yescharacteristic
29C22×C3×C4×C5C2×C411yescharacteristic
30C22×Dic15C2×C411yescharacteristic
31C2×C4×D15—21no—
32C2×C4×D15—21no—
33C2×C4×D15—42no—
34C22×C5×Dic3C2×C411yescharacteristic
35C22×C5×Dic3C2×C411yescharacteristic
36C22×C3×Dic5C2×C411yescharacteristic
37C22×C3×Dic5C2311yescharacteristic
38C2×#1200—21no—
39C2×#1200—21no—
40C2×#1200—42no—
41C2×#1200C2×C414yes—
42C2×#1200D412yes—
43C2×#1200D412yes—
44C2×#1200D412yes—
45C2×#1200Q812yes—
46#6762—22no—
47#6762C2×C414yescomplemented
48C2×C3⋊#170—22no—
49C3×#704—22no—
50C3×#704C2×C414yescomplemented
51C2×C3×#170—22no—
52#6740—28no—
53#6738—28no—
54C5×Dic3⋊C4—22no—
55C3×#683—22no—
56Dic3⋊Dic5—22no—
57#6668—22no—
58C2×D3×M16—52no—
59C2×D3×C4⋊C4—51no—
60C3×#343—51no—
61#4554—52no—
62#4530—51no—
63C22×C4×D5—31no—
64C22×C4×D5—31no—
65C22×C4×D5—31no—
66C2×#704—32no—
67C2×#704C2×D312yes—
68D5⋊M16—38no—
69D5⋊M16—64no—
70C2×#700—62no—
71C2×D5⋊C8—62no—
72C2×C5×C4⋊C4—31no—
73C2×C4⋊Dic5—31no—
74C2×#683—31no—
75C2×#683C2×D311yescharacteristic
76D5×C4⋊C4—62no—
77D5×C4⋊C4—62no—
78C2×#343—151noSylow
79C23×C3×C5C22×C411yescharacteristic
80C22×D15—22no—
81C22×D15—22no—
82C22×D15C4212yes—
83C22×D15C22⋊C411yescharacteristic
84C22×D15C22⋊C411yescharacteristic
85C22×D15C4⋊C412yes—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—41no—
88C2×Dic15—21no—
89C2×Dic15—41no—
90C4×D15—42no—
91C4×D15—42no—
92C2×C5×Dic3—21no—
93C2×C5×Dic3—41no—
94C2×C5×Dic3C4212yes—
95C2×C5×Dic3C22⋊C412yes—
96C2×C5×Dic3C4⋊C412yes—
97C2×C3×Dic5—21no—
98C2×C3×Dic5—41no—
99C2×C3×Dic5C22×C412yes—
100C2×C3×Dic5C2×D411yescharacteristic
101C2×C3×Dic5C2×D412yes—
102C2×C3×Dic5C2×Q811yescharacteristic
103#1200—42no—
104#1200—42no—
105#1200C22⋊C414yes—
106#1200C22⋊C414yes—
107#1200C4⋊C414yes—
108#1200C4⋊C414yes—
109C3⋊#170—24no—
110C3×#170—24no—
111C22×C4×D3—51no—
112C22×C4×D3—51no—
113C22×C4×D3—51no—
114C2×C3×M16—52no—
115C2×C3×C4⋊C4—51no—
116C2×C4⋊Dic3—51no—
117C2×Dic3⋊C4—51no—
118C2×Dic3⋊C4AGL(1,𝔽5)11yescharacteristic
119C2×#208—52no—
120D3×M16—58no—
121D3×M16—104no—
122C2×ΓL(1,𝔽25)—102no—
123C2×D3×C8—102no—
124D3×C4⋊C4—102no—
125D3×C4⋊C4—102no—
126C23×D5—31no—
127C22×C4×C5—31no—
128C22×C4×C5—31no—
129C22×C4×C5C4×D311yescharacteristic
130C22×Dic5—31no—
131C22×Dic5C4×D311yescharacteristic
132C22×Dic5C22×D311yescharacteristic
133C2×C4×D5—34no—
134C2×C4×D5—32no—
135C2×C4×D5—32no—
136C2×C4×D5—32no—
137C2×C4×D5—32no—
138C2×C4×D5—61no—
139C2×C4×D5—61no—
140C2×C4×D5—61no—
141C2×C4×D5—61no—
142C2×C4×D5—122no—
143C2×C4×D5—122no—
144#704—22no—
145#704—34no—
146#704—62no—
147#704C4×D314yescomplemented
148C2×#170—22no—
149C2×#170—62no—
150#700—68no—
151D5⋊C8—68no—
152C5×C4⋊C4—62no—
153C4⋊Dic5—62no—
154#683—22no—
155#683—62no—
156C22×M16—152no—
157C22×C4⋊C4—151no—
158#343—51no—
159#343—151no—
160#343—152no—
161C22×C3×C5—21no—
162C22×C3×C5—21no—
163C22×C3×C5C2×C4211yescharacteristic
164C22×C3×C5C2×C22⋊C411yescharacteristic
165C22×C3×C5C2×C4⋊C411yescharacteristic
166C2×D15—21no—
167C2×D15—21no—
168C2×D15—24no—
169C2×D15—24no—
170C2×D15#9512yes—
171C2×D15#9512yes—
172C2×D15#10012yes—
173C2×D15#10112yes—
174C3×C4×C5—41no—
175Dic15—41no—
176C3×Dic5—41no—
177C3×Dic5C2×C22⋊C412yes—
178C3×Dic5C2×C4⋊C412yes—
179C5×Dic3—41no—
180C5×Dic3#9514yes—
181C23×D3—51no—
182C22×C3×C4—51no—
183C22×C3×C4—51no—
184C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
185C22×Dic3—51no—
186C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
187C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
188C2×C4×D3—54no—
189C2×C4×D3—52no—
190C2×C4×D3—52no—
191C2×C4×D3—52no—
192C2×C4×D3—52no—
193C2×C4×D3—101no—
194C2×C4×D3—101no—
195C2×C4×D3—101no—
196C2×C4×D3—101no—
197C2×C4×D3—202no—
198C2×C4×D3—202no—
199C3×M16—54no—
200C3×M16—102no—
201C2×C3×C8—102no—
202C3×C4⋊C4—102no—
203C4⋊Dic3—102no—
204Dic3⋊C4—22no—
205Dic3⋊C4—102no—
206#208—54no—
207#208—102no—
208C2×C3⋊C8—102no—
209ΓL(1,𝔽25)—108no—
210D3×C8—108no—
211C23×C5C2×C4×D311yescharacteristic
212C22×D5—31no—
213C22×D5—32no—
214C22×D5—31no—
215C22×D5—32no—
216C22×D5—62no—
217C22×D5—62no—
218C2×C4×C5—21no—
219C2×C4×C5—32no—
220C2×C4×C5—32no—
221C2×C4×C5—32no—
222C2×C4×C5—41no—
223C2×C4×C5—61no—
224C2×C4×C5—121no—
225C2×Dic5—21no—
226C2×Dic5—41no—
227C2×Dic5—61no—
228C2×Dic5—121no—
229C2×Dic5C2×C4×D312yes—
230C2×Dic5D3×D411yescharacteristic
231C2×Dic5D3×D412yes—
232C2×Dic5D3×Q811yescharacteristic
233C4×D5—34no—
234C4×D5—34no—
235C4×D5—34no—
236C4×D5—34no—
237C4×D5—122no—
238C4×D5—122no—
239C4×D5—122no—
240C4×D5—122no—
241#170—24no—
242#170—64no—
243C23×C4—151no—
244C23×C4—151no—
245C23×C4—151no—
246C2×M16—52no—
247C2×M16—158no—
248C2×M16—152no—
249C2×M16—302no—
250C2×M16—304no—
251C22×C8—302no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—51no—
254C2×C4⋊C4—151no—
255C2×C4⋊C4—151no—
256C2×C4⋊C4—302no—
257C2×C4⋊C4—302no—
258C2×C3×C5—21no—
259C2×C3×C5—21no—
260C2×C3×C5C2×#9511yescharacteristic
261C2×C3×C5C2×#10011yescharacteristic
262C2×C3×C5C2×#10111yescharacteristic
263D15—22no—
264D15#34314yescomplemented
265C23×C3C22×AGL(1,𝔽5)11yescharacteristic
266C22×D3—51no—
267C22×D3—52no—
268C22×D3—52no—
269C22×D3—51no—
270C22×D3—102no—
271C22×D3—102no—
272C2×C3×C4—21no—
273C2×C3×C4—41no—
274C2×C3×C4—52no—
275C2×C3×C4—51no—
276C2×C3×C4—51no—
277C2×C3×C4—52no—
278C2×C3×C4—101no—
279C2×C3×C4—201no—
280C2×Dic3—21no—
281C2×Dic3—41no—
282C2×Dic3—101no—
283C2×Dic3—201no—
284C2×Dic3C4×AGL(1,𝔽5)12yes—
285C2×Dic3C4⋊AGL(1,𝔽5)12yes—
286C2×Dic3C22⋊AGL(1,𝔽5)12yes—
287C4×D3—54no—
288C4×D3—54no—
289C4×D3—54no—
290C4×D3—54no—
291C4×D3—202no—
292C4×D3—202no—
293C4×D3—202no—
294C4×D3—202no—
295C3×C8—104no—
296C3⋊C8—104no—
297C22×C5—21no—
298C22×C5—21no—
299C22×C5C42×D311yescharacteristic
300C22×C5D3×C22⋊C411yescharacteristic
301C22×C5D3×C4⋊C411yescharacteristic
302C2×D5—32no—
303C2×D5—32no—
304C2×D5—32no—
305C2×D5—32no—
306C2×D5—64no—
307C2×D5—64no—
308C2×D5—61no—
309C2×D5—61no—
310C4×C5—34no—
311C4×C5—41no—
312C4×C5—121no—
313Dic5—41no—
314Dic5—121no—
315Dic5D3×C22⋊C412yes—
316Dic5D3×C4⋊C412yes—
317C24—151no—
318C22×C4—31no—
319C22×C4—31no—
320C22×C4—51no—
321C22×C4—51no—
322C22×C4—152no—
323C22×C4—152no—
324C22×C4—154no—
325C22×C4—152no—
326C22×C4—152no—
327C22×C4—151no—
328C22×C4—301no—
329C22×C4—301no—
330C22×C4—301no—
331C22×C4—301no—
332C22×C4—602no—
333C22×C4—602no—
334C22×C4D3×AGL(1,𝔽5)11yescharacteristic
335M16—54no—
336M16—102no—
337M16—154no—
338M16—308no—
339M16—302no—
340C2×C8—102no—
341C2×C8—308no—
342C2×C8—302no—
343C4⋊C4—62no—
344C4⋊C4—102no—
345C4⋊C4—302no—
346C4⋊C4—302no—
347C3×C5C2×#34311yescharacteristic, complemented, Hall
348C22×C3—21no—
349C22×C3—21no—
350C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
351C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
352C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
353C2×D3—52no—
354C2×D3—52no—
355C2×D3—52no—
356C2×D3—52no—
357C2×D3—104no—
358C2×D3—104no—
359C2×D3—101no—
360C2×D3—101no—
361C3×C4—41no—
362C3×C4—52no—
363C3×C4—52no—
364C3×C4—201no—
365Dic3—41no—
366Dic3—201no—
367Dic3(C2×AGL(1,𝔽5))⋊C414yes—
368C2×C5—21no—
369C2×C5—21no—
370C2×C5D3×#9511yescharacteristic
371C2×C5D3×#10011yescharacteristic
372C2×C5D3×#10111yescharacteristic
373D5—34no—
374D5—62no—
375C23—151no—
376C23—152no—
377C23—152no—
378C23—151no—
379C23—302no—
380C23—302no—
381C23C2×D3×AGL(1,𝔽5)11yescharacteristic
382C2×C4—21no—
383C2×C4—32no—
384C2×C4—32no—
385C2×C4—32no—
386C2×C4—41no—
387C2×C4—52no—
388C2×C4—51no—
389C2×C4—51no—
390C2×C4—52no—
391C2×C4—61no—
392C2×C4—101no—
393C2×C4—121no—
394C2×C4—154no—
395C2×C4—154no—
396C2×C4—154no—
397C2×C4—154no—
398C2×C4—201no—
399C2×C4—301no—
400C2×C4—602no—
401C2×C4—602no—
402C2×C4—602no—
403C2×C4—602no—
404C2×C4—601no—
405C8—104no—
406C8—304no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
410C2×C3D5⋊#10011yescharacteristic
411C2×C3D5⋊#10111yescharacteristic
412D3—54no—
413D3—102no—
414C5D3×#34311yescharacteristic, complemented, Sylow
415C22—21no—
416C22—21no—
417C22—152no—
418C22—152no—
419C22—152no—
420C22—152no—
421C22—304no—
422C22—304no—
423C22—301no—
424C22—301no—
425C22C4×D3×AGL(1,𝔽5)11yescharacteristic
426C22D3×C4⋊AGL(1,𝔽5)11yescharacteristic
427C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
428C4—34no—
429C4—41no—
430C4—52no—
431C4—52no—
432C4—121no—
433C4—201no—
434C4—601no—
435C3#1060433411yescharacteristic, complemented, Sylow
436C2—21no—
437C2—21no—
438C2—154no—
439C2—302no—
440C2#1175191811yescharacteristic
441C2D3×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
442C2#1175205011yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC22×C3×C5C2×C42
FrattiniC23C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×D15C2×C4
cosocleC22×C3×Dic5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5C2×#343
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11746790C2semidirect
4. C3×#64919C2semidirect
5. #11746574C2semidirect
14. #91931C4semidirect
27. #64919D3semidirect
47. #6762C2×C4semidirect
50. C3×#704C2×C4semidirect
147. #704C4×D3semidirect
264. D15#343semidirect
347. C3×C5C2×#343semidirect
414. C5D3×#343semidirect
435. C3#10604334semidirect

Maximal subgroups

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

subgroupindex
C2×#919312
C2×#914042
#117467902
C3×#649192
#117465742
#106043343
D3×#3435