Finite Group Databasebeta

← #14296972

Subgroups — #14296972

Order 1920.

1168 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C22×#6771C211yesmaximal, characteristic
2#11752414C211yesmaximal, characteristic, complemented
3C3×#65755C211yesmaximal, characteristic, complemented
4C2×#91175C212yesmaximal, complemented
5C2×#91174C212yesmaximal, complemented
6#10604906—31nomaximal, Hall
7C22×C4×D15C411yescharacteristic
8C22×C4×C5×D3C411yescharacteristic
9C22×C3×C4×D5C2211yescharacteristic
10C22×#1204C2211yescharacteristic
11C22×#1203C2211yescharacteristic
12C22×#1202C411yescharacteristic
13C22×#1201C411yescharacteristic
14C2×#6771—22no—
15C2×#6771—22no—
16C2×#6771—24no—
17C2×#6771—24no—
18C2×#6762C2212yes—
19C2×C3×#704C2212yes—
20#91232—22no—
21C3×#4003—22no—
22#91175—24no—
23#91174—24no—
24#69989—51nomaximal, Hall
25C22×#709—31no—
26C2×#4013—32no—
27C2×#4005—32no—
28#65755—31no—
29#65755D311yescharacteristic, complemented
30C23×D15C2×C411yescharacteristic
31C23×C5×D3C2×C411yescharacteristic
32C23×C3×D5C2×C411yescharacteristic
33C22×C3×C4×C5C2×C411yescharacteristic
34C22×Dic15C2×C411yescharacteristic
35C2×C4×D15—22no—
36C2×C4×D15—22no—
37C2×C4×D15—22no—
38C22×C5×Dic3C2×C411yescharacteristic
39C2×C4×C5×D3—22no—
40C2×C4×C5×D3—22no—
41C2×C4×C5×D3—22no—
42C22×C3×Dic5C2311yescharacteristic
43C2×C3×C4×D5—22no—
44C2×C3×C4×D5—22no—
45C2×C3×C4×D5D412yes—
46C2×C3×C4×D5D412yes—
47C2×#1204—21no—
48C2×#1204—22no—
49C2×#1204—21no—
50C2×#1204D414yes—
51C2×#1203—21no—
52C2×#1203—22no—
53C2×#1203—21no—
54C2×#1203D414yes—
55C2×#1202—22no—
56C2×#1202—22no—
57C2×#1202—22no—
58C2×#1201—22no—
59C2×#1201—22no—
60C2×#1201—22no—
61#6771—216no—
62#6771—216no—
63#6762—22no—
64#6762—22no—
65C2×C3⋊#170—22no—
66C3×#704—22no—
67C3×#704—22no—
68C2×C3×#170—22no—
69C22×#235—51no—
70C3×#407—51no—
71#5207—51no—
72C2×#824—52no—
73C2×#823—52no—
74C23×C4×C5—31no—
75C22×C22⋊D5—31no—
76C22×C22⋊D5—31no—
77C2×#709—64no—
78C2×#709—64no—
79C2×#709—62no—
80C2×#709—62no—
81C22×D20—31no—
82C22×C4×D5—31no—
83C22×C4×D5C2×D311yescharacteristic
84C22×Dic10—31no—
85C2×#704—32no—
86C2×#704C2×D312yes—
87#4013—64no—
88#4005—64no—
89#4003—22no—
90#4003—62no—
91#1785—151noSylow
92C23×C3×C5C22×C411yescharacteristic
93C22×D15—21no—
94C22×D15—21no—
95C22×D15—42no—
96C22×D15C22⋊C412yes—
97C22×C5×D3—21no—
98C22×C5×D3—21no—
99C22×C5×D3—42no—
100C22×C5×D3C22⋊C412yes—
101C22×C3×D5—21no—
102C22×C3×D5—21no—
103C22×C3×D5—42no—
104C22×C3×D5C22⋊C412yes—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5C22⋊C412yes—
108C2×Dic15—21no—
109C2×Dic15—21no—
110C2×Dic15C22⋊C412yes—
111C4×D15—28no—
112C2×C5×Dic3—21no—
113C2×C5×Dic3—21no—
114C2×C5×Dic3C22⋊C412yes—
115C4×C5×D3—28no—
116C2×C3×Dic5C2×D411yescharacteristic
117C2×C3×Dic5C2×D412yes—
118C2×C3×Dic5C2×D412yes—
119C2×C3×Dic5C2×D411yescharacteristic
120C3×C4×D5—24no—
121C3×C4×D5—24no—
122#1204—24no—
123#1204—24no—
124#1203—24no—
125#1203—24no—
126#1202—28no—
127#1201—28no—
128C3⋊#170—42no—
129C3×#170—42no—
130C23×C3×C4—51no—
131C22×#67—51no—
132C22×#67—51no—
133C2×#235—104no—
134C2×#235—104no—
135C2×#235—102no—
136C2×#235—102no—
137C22×D12—51no—
138C22×C4×D3—51no—
139C22×C4×D3AGL(1,𝔽5)11yescharacteristic
140C22×Dic6—51no—
141C2×C3×M16—52no—
142C2×#208—52no—
143C3×C22⋊C8—102no—
144#830—102no—
145#824—104no—
146#823—104no—
147C24×C5—31no—
148C23×D5—31no—
149C23×D5C4×D311yescharacteristic
150C22×C4×C5—31no—
151C22×C4×C5—62no—
152C22×C4×C5—62no—
153C22×C4×C5—62no—
154C22×C4×C5C4×D311yescharacteristic
155C2×C22⋊D5—34no—
156C2×C22⋊D5—62no—
157C2×C22⋊D5—62no—
158C2×C22⋊D5—62no—
159C2×C22⋊D5—61no—
160C2×C22⋊D5—62no—
161C2×C22⋊D5—61no—
162C22×Dic5—31no—
163C22×Dic5C22×D311yescharacteristic
164#709—616no—
165#709—616no—
166C2×D20—62no—
167C2×D20—62no—
168C2×D20—62no—
169C2×C4×D5—22no—
170C2×C4×D5—22no—
171C2×C4×D5—62no—
172C2×C4×D5—62no—
173C2×C4×D5—62no—
174C2×C4×D5D1212yes—
175C2×C4×D5#6712yes—
176C2×Dic10—34no—
177C2×Dic10—61no—
178C2×Dic10—62no—
179C2×Dic10—61no—
180#704—22no—
181#704—22no—
182#704—62no—
183#704—62no—
184C2×#170—22no—
185C2×#170—62no—
186C22×#41—151no—
187C2×#101—152no—
188C2×#100—152no—
189#407—51no—
190#407—151no—
191C22×C3×C5—22no—
192C22×C3×C5C2×C22⋊C411yescharacteristic
193C22×C3×C5C2×C22⋊C411yescharacteristic
194C22×C3×C5C2×C22⋊C411yescharacteristic
195C2×D15—22no—
196C2×D15—42no—
197C2×D15—42no—
198C2×D15—42no—
199C2×C5×D3—22no—
200C2×C5×D3—42no—
201C2×C5×D3—42no—
202C2×C5×D3—42no—
203C2×C3×D5—22no—
204C2×C3×D5—24no—
205C2×C3×D5—42no—
206C2×C3×D5—42no—
207C3×C4×C5—22no—
208Dic15—22no—
209C3×Dic5#12012yes—
210C3×Dic5#12012yes—
211C5×Dic3—22no—
212C24×C3—51no—
213C23×D3—51no—
214C23×D3C2×AGL(1,𝔽5)11yescharacteristic
215C22×C3×C4—51no—
216C22×C3×C4—52no—
217C22×C3×C4—52no—
218C22×C3×C4—102no—
219C22×C3×C4—102no—
220C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
221C2×#67—102no—
222C2×#67—102no—
223C2×#67—102no—
224C2×#67—102no—
225C2×#67—102no—
226C2×#67—102no—
227C22×Dic3—51no—
228C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
229#235—1016no—
230#235—1016no—
231C2×D12—54no—
232C2×D12—101no—
233C2×D12—102no—
234C2×D12—101no—
235C2×C4×D3—22no—
236C2×C4×D3—22no—
237C2×C4×D3—22no—
238C2×C4×D3—102no—
239C2×C4×D3—102no—
240C2×C4×D3—102no—
241C2×Dic6—54no—
242C2×Dic6—101no—
243C2×Dic6—102no—
244C2×Dic6—101no—
245C3×M16—102no—
246C3×M16—102no—
247C2×C3×C8—102no—
248#208—102no—
249#208—102no—
250C2×C3⋊C8—102no—
251C23×C5—32no—
252C23×C5—61no—
253C23×C5—61no—
254C23×C5—122no—
255C23×C5C2×C4×D311yescharacteristic
256C22×D5—21no—
257C22×D5—21no—
258C22×D5—32no—
259C22×D5—42no—
260C22×D5—61no—
261C22×D5—61no—
262C22×D5—122no—
263C22×D5(C2×D3)⋊C412yes—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—32no—
267C2×C4×C5—68no—
268C2×C4×C5—61no—
269C2×C4×C5—61no—
270C2×C4×C5(C2×D3)⋊C412yes—
271C22⋊D5—68no—
272C22⋊D5—64no—
273C22⋊D5—64no—
274C2×Dic5—32no—
275C2×Dic5—61no—
276C2×Dic5—61no—
277C2×Dic5C2×D1211yescharacteristic
278C2×Dic5D3×D412yes—
279C2×Dic5D3×D412yes—
280C2×Dic5C2×#6711yescharacteristic
281D20—68no—
282C4×D5—24no—
283C4×D5—24no—
284C4×D5—68no—
285Dic10—64no—
286Dic10—64no—
287#170—42no—
288#170—122no—
289C2×#41—302no—
290C2×#41—302no—
291C2×#41—304no—
292C2×#41—304no—
293C22×Q8—151no—
294C22×D4—151no—
295C22×D4—151no—
296C22×D4—151no—
297C23×C4—31no—
298C23×C4—51no—
299C23×C4—151no—
300C2×M16—52no—
301C2×M16—152no—
302#101—304no—
303#100—304no—
304C22⋊C8—102no—
305C22⋊C8—302no—
306C2×C3×C5—21no—
307C2×C3×C5—21no—
308C2×C3×C5#37911yescharacteristic
309C2×C3×C5#41312yes—
310D15—42no—
311C3×D5—42no—
312C5×D3—42no—
313C23×C3—52no—
314C23×C3—101no—
315C23×C3—101no—
316C23×C3—202no—
317C23×C3C22×AGL(1,𝔽5)11yescharacteristic
318C22×D3—21no—
319C22×D3—21no—
320C22×D3—42no—
321C22×D3—52no—
322C22×D3—101no—
323C22×D3—101no—
324C22×D3—202no—
325C22×D3C22⋊AGL(1,𝔽5)12yes—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—51no—
329C2×C3×C4—52no—
330C2×C3×C4—52no—
331C2×C3×C4—51no—
332C2×C3×C4—104no—
333C2×C3×C4—104no—
334C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
335#67—108no—
336#67—108no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—52no—
340C2×Dic3—101no—
341C2×Dic3—101no—
342C2×Dic3C22⋊AGL(1,𝔽5)12yes—
343D12—104no—
344D12—104no—
345C4×D3—28no—
346C4×D3—108no—
347Dic6—104no—
348Dic6—104no—
349C3×C8—202no—
350C3⋊C8—202no—
351C22×C5—22no—
352C22×C5—62no—
353C22×C5—122no—
354C22×C5—122no—
355C22×C5—122no—
356C22×C5D3×C22⋊C411yescharacteristic
357C22×C5D3×C22⋊C411yescharacteristic
358C22×C5C2×(C2×D3)⋊C411yescharacteristic
359C2×D5—24no—
360C2×D5—22no—
361C2×D5—42no—
362C2×D5—42no—
363C2×D5—62no—
364C2×D5—122no—
365C2×D5—122no—
366C2×D5—122no—
367C4×C5—22no—
368C4×C5—62no—
369Dic5—62no—
370Dic5#88212yes—
371Dic5#93712yes—
372C24—31no—
373C24—51no—
374C24—151no—
375#41—3016no—
376#41—3016no—
377C2×Q8—154no—
378C2×Q8—301no—
379C2×Q8—301no—
380C2×Q8—302no—
381C2×D4—154no—
382C2×D4—302no—
383C2×D4—301no—
384C2×D4—301no—
385C2×D4—302no—
386C2×D4—302no—
387C2×D4—302no—
388C2×D4—302no—
389C2×D4—302no—
390C2×D4—302no—
391C22×C4—31no—
392C22×C4—51no—
393C22×C4—52no—
394C22×C4—52no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—102no—
399C22×C4—102no—
400C22×C4—151no—
401C22×C4—302no—
402C22×C4—302no—
403C22×C4—302no—
404C22×C4D3×AGL(1,𝔽5)11yescharacteristic
405M16—102no—
406M16—102no—
407M16—302no—
408M16—302no—
409C2×C8—102no—
410C2×C8—302no—
411C3×C5#178511yescharacteristic, complemented, Hall
412C22×C3—22no—
413C22×C3—102no—
414C22×C3—104no—
415C22×C3—202no—
416C22×C3—202no—
417C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
418C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
419C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
420C2×D3—22no—
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—102no—
425C2×D3—202no—
426C2×D3—202no—
427C2×D3—202no—
428C3×C4—22no—
429C3×C4—52no—
430C3×C4—52no—
431Dic3—22no—
432Dic3—102no—
433C2×C5—21no—
434C2×C5—21no—
435C2×C5—122no—
436C2×C5#495511yescharacteristic
437C2×C5#513212yes—
438D5—42no—
439D5—122no—
440C23—32no—
441C23—52no—
442C23—61no—
443C23—61no—
444C23—101no—
445C23—101no—
446C23—122no—
447C23—152no—
448C23—202no—
449C23—301no—
450C23—301no—
451C23—602no—
452C23C2×D3×AGL(1,𝔽5)11yescharacteristic
453Q8—304no—
454Q8—304no—
455D4—304no—
456D4—304no—
457D4—308no—
458D4—308no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—32no—
462C2×C4—51no—
463C2×C4—51no—
464C2×C4—52no—
465C2×C4—52no—
466C2×C4—68no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—104no—
470C2×C4—104no—
471C2×C4—152no—
472C2×C4—308no—
473C2×C4—301no—
474C2×C4—301no—
475C2×C4(C2×D3)⋊AGL(1,𝔽5)12yes—
476C8—202no—
477C8—602no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—202no—
481C2×C3#6576112yes—
482C2×C3#6580011yescharacteristic
483D3—42no—
484D3—202no—
485C5#6998911yescharacteristic, complemented, Sylow
486C22—22no—
487C22—62no—
488C22—102no—
489C22—104no—
490C22—122no—
491C22—122no—
492C22—122no—
493C22—202no—
494C22—202no—
495C22—302no—
496C22—602no—
497C22—602no—
498C22—602no—
499C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
500C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
501C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
502C4—22no—
503C4—52no—
504C4—52no—
505C4—62no—
506C4—302no—
507C3#1060490611yescharacteristic, complemented, Sylow
508C2—21no—
509C2—21no—
510C2—122no—
511C2—202no—
512C2—602no—
513C2#1175181812yes—
514C2#1175202511yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC23×C3×C5C22×C4
FrattiniC23C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×#6771C2
cosocleC22×C3×Dic5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1785
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11752414C2semidirect
3. C3×#65755C2semidirect
4. C2×#91175C2semidirect
5. C2×#91174C2semidirect
29. #65755D3semidirect
411. C3×C5#1785semidirect
485. C5#69989semidirect
507. C3#10604906semidirect

Maximal subgroups

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

subgroupindex
C22×#67712
#117524142
C3×#657552
C2×#911752
C2×#911742
#106049063
#699895