Finite Group Databasebeta

← #14295342

Subgroups — #14295342

Order 1920.

696 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6739C211yesmaximal, characteristic
2C2×C4×C3⋊#170C211yesmaximal, characteristic, complemented
3C2×(C2×D3)⋊#170C211yesmaximal, characteristic
4C2×#91169C211yesmaximal, characteristic, complemented
5Dic5×(C2×D3)⋊C4C211yesmaximal, characteristic
6C3×#64918C211yesmaximal, characteristic, complemented
7#11746570C211yesmaximal, characteristic, complemented
8#10604227—31nomaximal, Hall
9C22×D3×Dic5C2211yescharacteristic
10C22×C3⋊#170C2211yescharacteristic
11C22×C3⋊#170C2211yescharacteristic
12C22×C3×#170C2211yescharacteristic
13C22×C3×#170C2211yescharacteristic
14C2×#6739—22no—
15C2×#6739—22no—
16C2×#6739C414yescomplemented
17C2×C5×(C2×D3)⋊C4C411yescharacteristic
18C2×C3×C4×Dic5C2211yescharacteristic
19C2×(C2×D3)⋊Dic5C411yescharacteristic
20C2×Dic3×Dic5C2211yescharacteristic
21C4×C3⋊#170—22no—
22(C2×D3)⋊#170C414yescomplemented
23#91169—22no—
24#91115C411yescharacteristic
25#90994C411yescharacteristic
26#68649—51nomaximal, Hall
27C22×#704—31no—
28C2×C22⋊#170—31no—
29C2×#4002—31no—
30C2×C4×#170—31no—
31C22⋊C4×Dic5—31no—
32#64918—31no—
33#64918D311yescharacteristic, complemented
34C23×C5×D3C2×C411yescharacteristic
35C22×C3×C4×C5C2×C411yescharacteristic
36C22×Dic15C2×C411yescharacteristic
37C22×Dic15C2×C411yescharacteristic
38C22×C5×Dic3C2×C411yescharacteristic
39C22×C3×Dic5C2×C411yescharacteristic
40C22×C3×Dic5C2311yescharacteristic
41C2×D3×Dic5—21no—
42C2×D3×Dic5—21no—
43C2×D3×Dic5—21no—
44C2×D3×Dic5—21no—
45C2×D3×Dic5C2×C412yes—
46C2×D3×Dic5C2×C412yes—
47C2×C3⋊#170—21no—
48C2×C3⋊#170—21no—
49C2×C3⋊#170C2×C412yes—
50C2×C3⋊#170C2×C412yes—
51C2×C3⋊#170D412yes—
52C2×C3⋊#170D412yes—
53C2×C3×#170—21no—
54C2×C3×#170—21no—
55C2×C3×#170—21no—
56C2×C3×#170—21no—
57C2×C3×#170C2×C412yes—
58C2×C3×#170C2×C412yes—
59#6739—28no—
60C5×(C2×D3)⋊C4—22no—
61C3×C4×Dic5—21no—
62C3×C4×Dic5—21no—
63(C2×D3)⋊Dic5—22no—
64Dic3×Dic5—21no—
65Dic3×Dic5—21no—
66C22×ΓL(1,𝔽25)—51no—
67C2×(C2×D3)⋊C8—51no—
68C2×#815—51no—
69C4×(C2×D3)⋊C4—51no—
70C2×C4×C3⋊C8—51no—
71C3×#336—51no—
72#4550—51no—
73C23×Dic5—31no—
74C2×#704—34no—
75C2×#704—62no—
76C2×#704—62no—
77C22×#170—31no—
78C22×#170—31no—
79C22×#170C2×D311yescharacteristic
80C22×#170C2×D311yescharacteristic
81C2×C5×C22⋊C4—31no—
82C2×C22⋊Dic5—31no—
83C2×C4×Dic5—31no—
84C2×C4×Dic5C2×D311yescharacteristic
85C22⋊#170—34no—
86#4002—62no—
87C4×#170—62no—
88#3963—31no—
89#3963—31no—
90#1753—151noSylow
91C23×C3×C5C22×C411yescharacteristic
92C22×C5×D3—21no—
93C22×C5×D3—21no—
94C22×C5×D3—21no—
95C22×C5×D3—21no—
96C22×C5×D3—21no—
97C22×C5×D3—21no—
98C22×C5×D3C4212yes—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5M1612yescomplemented
102C2×Dic15—21no—
103C2×Dic15—21no—
104C2×Dic15—21no—
105C2×Dic15—21no—
106C2×Dic15—21no—
107C2×Dic15C4212yes—
108C2×C5×Dic3—21no—
109C2×C5×Dic3—21no—
110C2×C5×Dic3—21no—
111C2×C3×Dic5—21no—
112C2×C3×Dic5—21no—
113C2×C3×Dic5M1612yescomplemented
114C2×C3×Dic5C22×C411yescharacteristic
115C2×C3×Dic5C22×C411yescharacteristic
116C2×C3×Dic5C2×D411yescharacteristic
117C2×C3×Dic5C2×D411yescharacteristic
118C2×C3×Dic5#4111yescharacteristic
119C2×C3×Dic5#4111yescharacteristic
120D3×Dic5—22no—
121D3×Dic5—22no—
122D3×Dic5—22no—
123D3×Dic5—22no—
124C3⋊#170—22no—
125C3⋊#170C22⋊C414yescomplemented
126C3×#170—22no—
127C3×#170—22no—
128C22×C4×D3—51no—
129C22×C3×C8—51no—
130C22×C3×C8—51no—
131C2×C3×C42—51no—
132C2×(C2×D3)⋊C4—51no—
133C2×(C2×D3)⋊C4AGL(1,𝔽5)11yescharacteristic
134C2×C4×Dic3—51no—
135C22×C3⋊C8—51no—
136C22×C3⋊C8—51no—
137C2×ΓL(1,𝔽25)—54no—
138C2×ΓL(1,𝔽25)—102no—
139C2×ΓL(1,𝔽25)—102no—
140C3⋊#95—51no—
141C3⋊#95—51no—
142(C2×D3)⋊C8—54no—
143#815—102no—
144C4×C3⋊C8—102no—
145C24×C5—31no—
146C22×C4×C5—31no—
147C22×C4×C5C4×D311yescharacteristic
148C22×Dic5—31no—
149C22×Dic5—32no—
150C22×Dic5—32no—
151C22×Dic5—31no—
152C22×Dic5—61no—
153C22×Dic5—61no—
154C22×Dic5—61no—
155C22×Dic5—61no—
156C22×Dic5C4×D311yescharacteristic
157C22×Dic5C22×D311yescharacteristic
158#704—68no—
159C2×#170—21no—
160C2×#170—21no—
161C2×#170—21no—
162C2×#170—21no—
163C2×#170—32no—
164C2×#170—32no—
165C2×#170—32no—
166C2×#170—32no—
167C2×#170—61no—
168C2×#170—61no—
169C2×#170C4×D312yes—
170C2×#170C4×D312yes—
171C5×C22⋊C4—62no—
172C22⋊Dic5—62no—
173C4×Dic5—21no—
174C4×Dic5—21no—
175C4×Dic5—61no—
176C4×Dic5—61no—
177C22×M16—151no—
178C2×C22⋊C8—151no—
179C2×#97—151no—
180C2×C4×C8—151no—
181C4×C22⋊C4—151no—
182#336—51no—
183#336—151no—
184C22×C3×C5C2×C4211yescharacteristic
185C22×C3×C5C4×D411yescharacteristic
186C22×C3×C5C4×D411yescharacteristic
187C22×C3×C5C2×M1611yescharacteristic
188C22×C3×C5C2×M1611yescharacteristic
189C22×C3×C5#13111yescharacteristic
190C22×C3×C5#13111yescharacteristic
191C2×C5×D3—22no—
192C2×C5×D3—22no—
193C2×C5×D3—22no—
194C2×C5×D3—41no—
195C2×C5×D3—41no—
196C2×C5×D3—41no—
197C2×C5×D3—41no—
198C3×C4×C5—22no—
199Dic15—22no—
200Dic15—41no—
201C3×Dic5—22no—
202C3×Dic5C2×C22⋊C411yescharacteristic
203C3×Dic5#11711yescharacteristic
204C3×Dic5C4×D411yescharacteristic
205C3×Dic5C4×D411yescharacteristic
206C5×Dic3—41no—
207C23×D3C2×AGL(1,𝔽5)11yescharacteristic
208C22×C3×C4—51no—
209C22×C3×C4—51no—
210C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
211C22×Dic3—51no—
212C22×Dic3—51no—
213C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
214C2×C4×D3—52no—
215C2×C4×D3—52no—
216C2×C4×D3—101no—
217C2×C4×D3—101no—
218C2×C4×D3—101no—
219C2×C4×D3—101no—
220C2×C3×C8—52no—
221C2×C3×C8—52no—
222C2×C3×C8—101no—
223C2×C3×C8—101no—
224C2×C3×C8—101no—
225C2×C3×C8—101no—
226C3×C42—101no—
227C3×C42—101no—
228(C2×D3)⋊C4—22no—
229(C2×D3)⋊C4—102no—
230C4×Dic3—101no—
231C4×Dic3—101no—
232C2×C3⋊C8—52no—
233C2×C3⋊C8—52no—
234C2×C3⋊C8—52no—
235C2×C3⋊C8—52no—
236C2×C3⋊C8—101no—
237C2×C3⋊C8—101no—
238ΓL(1,𝔽25)—108no—
239C23×C5—32no—
240C23×C5—61no—
241C23×C5—61no—
242C23×C5—61no—
243C23×C5—61no—
244C23×C5—61no—
245C23×C5—61no—
246C23×C5C2×C4×D311yescharacteristic
247C2×C4×C5—21no—
248C2×C4×C5—21no—
249C2×C4×C5—61no—
250C2×C4×C5—61no—
251C2×C4×C5—61no—
252C2×C4×C5ΓL(1,𝔽25)12yescomplemented
253C2×Dic5—21no—
254C2×Dic5—21no—
255C2×Dic5—32no—
256C2×Dic5—61no—
257C2×Dic5—61no—
258C2×Dic5—62no—
259C2×Dic5—62no—
260C2×Dic5—62no—
261C2×Dic5—62no—
262C2×Dic5—61no—
263C2×Dic5—61no—
264C2×Dic5—61no—
265C2×Dic5ΓL(1,𝔽25)12yescomplemented
266C2×Dic5C2×C4×D311yescharacteristic
267C2×Dic5C2×C4×D311yescharacteristic
268C2×Dic5D3×D411yescharacteristic
269C2×Dic5D3×D411yescharacteristic
270C2×Dic5#23711yescharacteristic
271C2×Dic5#23911yescharacteristic
272#170—22no—
273#170—22no—
274#170—34no—
275#170—62no—
276C23×C4—151no—
277C2×M16—154no—
278C2×M16—302no—
279C2×M16—302no—
280C22×C8—51no—
281C22×C8—51no—
282C22×C8—151no—
283C22×C8—151no—
284C2×C22⋊C4—31no—
285C2×C22⋊C4—151no—
286C2×C42—51no—
287C2×C42—151no—
288C22⋊C8—154no—
289#97—302no—
290C4×C8—302no—
291#95—151no—
292#95—151no—
293C2×C3×C5C4×C22⋊C411yescharacteristic
294C2×C3×C5C4×M1611yescharacteristic
295C2×C3×C5#40511yescharacteristic
296C2×C3×C5#43511yescharacteristic
297C2×C3×C5#43511yescharacteristic
298C2×C3×C5#43611yescharacteristic
299C2×C3×C5#43611yescharacteristic
300C5×D3—42no—
301C23×C3C22×AGL(1,𝔽5)11yescharacteristic
302C22×D3—21no—
303C22×D3—21no—
304C22×D3—21no—
305C22×D3—21no—
306C22×D3—21no—
307C22×D3—21no—
308C22×D3C4×AGL(1,𝔽5)12yes—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—51no—
312C2×C3×C4—51no—
313C2×C3×C4—51no—
314C2×C3×C4—51no—
315C2×C3×C4—51no—
316C2×C3×C4—51no—
317C2×C3×C4—52no—
318C2×C3×C4—101no—
319C2×C3×C4—101no—
320C2×C3×C4#70412yescomplemented
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—52no—
325C2×Dic3—101no—
326C2×Dic3—101no—
327C2×Dic3—101no—
328C2×Dic3—101no—
329C2×Dic3—101no—
330C4×D3—102no—
331C4×D3—102no—
332C4×D3—102no—
333C4×D3—102no—
334C3×C8—102no—
335C3×C8—102no—
336C3⋊C8—54no—
337C3⋊C8—102no—
338C22×C5—62no—
339C22×C5—62no—
340C22×C5—62no—
341C22×C5—121no—
342C22×C5—121no—
343C22×C5—121no—
344C22×C5—121no—
345C22×C5C42×D311yescharacteristic
346C22×C5#67⋊C411yescharacteristic
347C22×C5#89311yescharacteristic
348C22×C5C2×ΓL(1,𝔽25)11yescharacteristic
349C22×C5C2×ΓL(1,𝔽25)11yescharacteristic
350C22×C5#90711yescharacteristic
351C22×C5#90711yescharacteristic
352C4×C5—22no—
353C4×C5—121no—
354Dic5—22no—
355Dic5—62no—
356Dic5—121no—
357Dic5D3×C22⋊C411yescharacteristic
358Dic5#67⋊C411yescharacteristic
359Dic5#89211yescharacteristic
360Dic5#89311yescharacteristic
361C24—31no—
362C22×C4—31no—
363C22×C4—51no—
364C22×C4—51no—
365C22×C4—152no—
366C22×C4—152no—
367C22×C4—151no—
368C22×C4—151no—
369C22×C4—301no—
370C22×C4—301no—
371C22×C4—301no—
372C22×C4—301no—
373C22×C4D3×AGL(1,𝔽5)11yescharacteristic
374M16—308no—
375C2×C8—52no—
376C2×C8—52no—
377C2×C8—101no—
378C2×C8—101no—
379C2×C8—101no—
380C2×C8—101no—
381C2×C8—152no—
382C2×C8—152no—
383C2×C8—152no—
384C2×C8—152no—
385C2×C8—301no—
386C2×C8—301no—
387C22⋊C4—62no—
388C22⋊C4—302no—
389C42—101no—
390C42—101no—
391C42—301no—
392C42—301no—
393C3×C5#175311yescharacteristic, complemented, Hall
394C22×C3D5⋊M1611yescharacteristic
395C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
396C22×C3#413111yescharacteristic
397C22×C3#413111yescharacteristic
398C22×C3D4×AGL(1,𝔽5)11yescharacteristic
399C22×C3D4×AGL(1,𝔽5)11yescharacteristic
400C22×C3C2×#70411yescharacteristic
401C2×D3—22no—
402C2×D3—22no—
403C2×D3—22no—
404C2×D3—41no—
405C2×D3—41no—
406C2×D3—41no—
407C2×D3—41no—
408C3×C4—22no—
409C3×C4—51no—
410C3×C4—51no—
411C3×C4—51no—
412C3×C4—51no—
413C3×C4—102no—
414Dic3—41no—
415Dic3—102no—
416Dic3—201no—
417C2×C5—122no—
418C2×C5#466611yescharacteristic
419C2×C5C4×ΓL(1,𝔽25)11yescharacteristic
420C2×C5#470811yescharacteristic
421C2×C5#473011yescharacteristic
422C2×C5#473011yescharacteristic
423C2×C5#483811yescharacteristic
424C2×C5#483811yescharacteristic
425C23—32no—
426C23—61no—
427C23—61no—
428C23—61no—
429C23—61no—
430C23—61no—
431C23—61no—
432C23C2×D3×AGL(1,𝔽5)11yescharacteristic
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—51no—
436C2×C4—51no—
437C2×C4—51no—
438C2×C4—51no—
439C2×C4—51no—
440C2×C4—51no—
441C2×C4—52no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—101no—
446C2×C4—101no—
447C2×C4—152no—
448C2×C4—302no—
449C2×C4—302no—
450C2×C4—302no—
451C2×C4—302no—
452C2×C4—301no—
453C2×C4—301no—
454C2×C4—301no—
455C2×C4—301no—
456C2×C4—301no—
457C2×C4#673912yescomplemented
458C8—102no—
459C8—102no—
460C8—154no—
461C8—302no—
462C2×C3#6569111yescharacteristic
463C2×C3#6569411yescharacteristic
464C2×C3#6569511yescharacteristic
465C2×C3C22⋊C4×AGL(1,𝔽5)11yescharacteristic
466C2×C3C4×#70411yescharacteristic
467C2×C3#6577311yescharacteristic
468C2×C3#6577411yescharacteristic
469D3—42no—
470C5#6864911yescharacteristic, complemented, Sylow
471C22—62no—
472C22—62no—
473C22—62no—
474C22—121no—
475C22—121no—
476C22—121no—
477C22—121no—
478C22#9191311yescharacteristic
479C22D5⋊ΓL(1,𝔽25)11yescharacteristic
480C22C4×D3×AGL(1,𝔽5)11yescharacteristic
481C22#9192111yescharacteristic
482C22#9192911yescharacteristic
483C22C2×#673911yescharacteristic
484C22#67⋊AGL(1,𝔽5)11yescharacteristic
485C4—22no—
486C4—51no—
487C4—51no—
488C4—51no—
489C4—51no—
490C4—102no—
491C4—121no—
492C4—302no—
493C4—601no—
494C3#1060422711yescharacteristic, complemented, Sylow
495C2—122no—
496C2#1175187011yescharacteristic
497C2C4×#673911yescharacteristic
498C2#1175192411yescharacteristic
499C2#1175192611yescharacteristic
500C2#1175192711yescharacteristic
501C2#1175193111yescharacteristic
502C2#1175205311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×AGL(1,𝔽5)
commutatorC22×C3×C5C2×C42
FittingC22×C3×C4×C5C2×C4
socleC23×C3×C5C22×C4
prime radicalC23×C5×D3C2×C4
cosocleC22×C3×Dic5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1753
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊#170C2semidirect
4. C2×#91169C2semidirect
6. C3×#64918C2semidirect
7. #11746570C2semidirect
16. C2×#6739C4semidirect
22. (C2×D3)⋊#170C4semidirect
33. #64918D3semidirect
101. C2×C3×C4×C5M16semidirect
113. C2×C3×Dic5M16semidirect
125. C3⋊#170C22⋊C4semidirect
252. C2×C4×C5ΓL(1,𝔽25)semidirect
265. C2×Dic5ΓL(1,𝔽25)semidirect
320. C2×C3×C4#704semidirect
393. C3×C5#1753semidirect
457. C2×C4#6739semidirect
470. C5#68649semidirect
494. C3#10604227semidirect

Maximal subgroups

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

subgroupindex
C22×#67392
C2×C4×C3⋊#1702
C2×(C2×D3)⋊#1702
C2×#911692
Dic5×(C2×D3)⋊C42
C3×#649182
#117465702
#106042273
#686495