Finite Group Databasebeta

← #11754776

Subgroups — #11754776

Order 960.

588 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6782C211yesmaximal, characteristic, complemented
2C2×#6779C211yesmaximal, characteristic, complemented
3#91827C211yesmaximal, characteristic, complemented
4C5×#939C211yesmaximal, characteristic, complemented
5C3×#4085C211yesmaximal, characteristic, complemented
6#91575C211yesmaximal, characteristic, complemented
7#91574C211yesmaximal, characteristic, complemented
8#91565C211yesmaximal, characteristic, complemented
9#91557C211yesmaximal, characteristic, complemented
10#91549C211yesmaximal, characteristic, complemented
11#91533C211yesmaximal, characteristic, complemented
12#91461C211yesmaximal, characteristic, complemented
13#91422C211yesmaximal, characteristic, complemented
14#91417C211yesmaximal, characteristic, complemented
15#91404C211yesmaximal, characteristic, complemented
16#65942—31nomaximal, Hall
17C2×C3×C5×D4C2211yescharacteristic
18C2×#1220C2211yescharacteristic
19C2×#1220C2211yescharacteristic
20C2×C4×D15C2211yescharacteristic
21C2×C5×#67C2211yescharacteristic
22C2×C5×#67C2211yescharacteristic
23C22×C5×Dic3C2211yescharacteristic
24C2×C3×C22⋊D5C2211yescharacteristic
25C2×C3×C22⋊D5C2211yescharacteristic
26C22×C3×Dic5C2211yescharacteristic
27C2×#1204C2211yescharacteristic
28C2×#1204C2211yescharacteristic
29C2×#1203C2211yescharacteristic
30C2×#1202C2211yescharacteristic
31C2×#1201C2211yescharacteristic
32C2×#1200C2211yescharacteristic
33C2×#1200C2211yescharacteristic
34#6782—24no—
35C2×D3×Dic5C2211yescharacteristic
36#6779—24no—
37C2×D5×Dic3C2211yescharacteristic
38#6719C2211yescharacteristic
39#6719C2211yescharacteristic
40C4⋊Dic15C2211yescharacteristic
41C5×#217C2211yescharacteristic
42C5×(C2×D3)⋊C4C2211yescharacteristic
43C5×Dic3⋊C4C2211yescharacteristic
44C3×C22⋊Dic5C2211yescharacteristic
45C3×#685C2211yescharacteristic
46C3×#683C2211yescharacteristic
47Dic3⋊Dic5C2211yescharacteristic
48Dic3⋊Dic5C2211yescharacteristic
49#6669C2211yescharacteristic
50#6668C2211yescharacteristic
51#6668C2211yescharacteristic
52(C2×D3)⋊Dic5C2211yescharacteristic
53#6665C2211yescharacteristic
54#5608—51nomaximal, Hall
55C2×#711—31no—
56C2×#709—31no—
57C5×#121—31no—
58#4085—31no—
59#4085D311yescharacteristic, complemented
60#4084—31no—
61#4084—31no—
62#4084—31no—
63#4070—31no—
64#4043—31no—
65D5×C4⋊C4—31no—
66#4030—31no—
67#4030—31no—
68#4024—31no—
69#4024—31no—
70C23×C3×C5C2311yescharacteristic
71C23×C3×C5C2311yescharacteristic
72C22×D15C2311yescharacteristic
73C22×C5×D3C2311yescharacteristic
74C22×C3×D5C2311yescharacteristic
75C3×C5×D4—41no—
76C2×C3×C4×C5C2311yescharacteristic
77#1220—22no—
78#1220—22no—
79C2×Dic15C2311yescharacteristic
80C2×Dic15C2311yescharacteristic
81C2×Dic15C2311yescharacteristic
82C4×D15—41no—
83C5×#67—22no—
84C5×#67—41no—
85C2×C5×Dic3—22no—
86C2×C5×Dic3C2311yescharacteristic
87C2×C5×Dic3C2311yescharacteristic
88C2×C5×Dic3C2311yescharacteristic
89C3×C22⋊D5—22no—
90C3×C22⋊D5—41no—
91C2×C3×Dic5—22no—
92C2×C3×Dic5C2311yescharacteristic
93C2×C3×Dic5C2311yescharacteristic
94C2×C3×Dic5C2311yescharacteristic
95#1204—22no—
96#1204—22no—
97#1203—22no—
98#1202—22no—
99#1201—41no—
100#1200—41no—
101#1200D414yescomplemented
102D3×Dic5—22no—
103D5×Dic3—22no—
104C2×#237—51no—
105C2×#235—51no—
106C3×#121—51no—
107#939—51no—
108#939D511yescharacteristic, complemented
109#938—51no—
110#938—51no—
111#938—51no—
112#924—51no—
113#897—51no—
114D3×C4⋊C4—51no—
115#884—51no—
116#884—51no—
117#878—51no—
118#878—51no—
119C2×C5×D4—31no—
120C2×C5×D4—31no—
121C2×C5×D4C2×D311yescharacteristic
122C22×C4×C5—31no—
123C2×C22⋊D5—31no—
124C2×C22⋊D5—31no—
125C2×C22⋊D5—31no—
126C2×C22⋊D5—31no—
127C2×C22⋊D5C2×D311yescharacteristic
128C2×C22⋊D5C2×D311yescharacteristic
129C22×Dic5—31no—
130C22×Dic5C2×D311yescharacteristic
131#711—64no—
132#709—64no—
133C2×D20—31no—
134C2×C4×D5—31no—
135C2×C4×D5—31no—
136C2×C4×D5—31no—
137C2×C4×D5—31no—
138C2×Dic10—31no—
139C2×Dic10—31no—
140C5×C4⋊C4—31no—
141C5×C22⋊C4—31no—
142C5×C22⋊C4—31no—
143C22⋊Dic5—31no—
144C22⋊Dic5—31no—
145C22⋊Dic5—31no—
146C22⋊Dic5C2×D311yescharacteristic
147#685—31no—
148#685C2×D311yescharacteristic
149C4⋊Dic5—31no—
150C4⋊Dic5—31no—
151C4⋊Dic5—31no—
152#683—31no—
153#683—31no—
154#683—31no—
155#683C2×D311yescharacteristic
156#537—151noSylow
157C22×C3×C5—21no—
158C22×C3×C5—21no—
159C22×C3×C5—21no—
160C22×C3×C5—21no—
161C22×C3×C5—21no—
162C22×C3×C5—21no—
163C22×C3×C5C2411yescharacteristic
164C2×D15—21no—
165C2×D15—21no—
166C2×D15C2×D412yes—
167C2×C5×D3—21no—
168C2×C5×D3—21no—
169C2×C5×D3—21no—
170C2×C3×D5—21no—
171C2×C3×D5—21no—
172C2×C3×D5—21no—
173C3×C4×C5—21no—
174Dic15—21no—
175Dic15—21no—
176Dic15—21no—
177C3×Dic5—21no—
178C3×Dic5—21no—
179C3×Dic5C2×D412yes—
180C5×Dic3—21no—
181C5×Dic3—21no—
182C5×Dic3C2×D412yes—
183C2×C3×D4—51no—
184C2×C3×D4—51no—
185C2×C3×D4C2×D511yescharacteristic
186C22×C3×C4—51no—
187C2×#67—51no—
188C2×#67—51no—
189C2×#67—51no—
190C2×#67—51no—
191C2×#67C2×D511yescharacteristic
192C2×#67C2×D511yescharacteristic
193C22×Dic3—51no—
194C22×Dic3C2×D511yescharacteristic
195#237—104no—
196#235—104no—
197C2×D12—51no—
198C2×C4×D3—51no—
199C2×C4×D3—51no—
200C2×C4×D3—51no—
201C2×C4×D3—51no—
202C2×Dic6—51no—
203C2×Dic6—51no—
204C3×C4⋊C4—51no—
205C3×C22⋊C4—51no—
206C3×C22⋊C4—51no—
207#217—51no—
208#217—51no—
209#217—51no—
210#217C2×D511yescharacteristic
211(C2×D3)⋊C4—51no—
212(C2×D3)⋊C4C2×D511yescharacteristic
213C4⋊Dic3—51no—
214C4⋊Dic3—51no—
215C4⋊Dic3—51no—
216Dic3⋊C4—51no—
217Dic3⋊C4—51no—
218Dic3⋊C4—51no—
219Dic3⋊C4C2×D511yescharacteristic
220C23×C5—31no—
221C23×C5C22×D311yescharacteristic
222C23×C5C22×D311yescharacteristic
223C22×D5—31no—
224C22×D5C22×D311yescharacteristic
225C5×D4—41no—
226C5×D4—62no—
227C5×D4—121no—
228C2×C4×C5—31no—
229C2×C4×C5—31no—
230C2×C4×C5—31no—
231C2×C4×C5—62no—
232C2×C4×C5C22×D311yescharacteristic
233C22⋊D5—22no—
234C22⋊D5—41no—
235C22⋊D5—62no—
236C22⋊D5—62no—
237C22⋊D5—62no—
238C22⋊D5—121no—
239C2×Dic5—22no—
240C2×Dic5—31no—
241C2×Dic5—31no—
242C2×Dic5—31no—
243C2×Dic5—62no—
244C2×Dic5C22×D311yescharacteristic
245C2×Dic5C22×D311yescharacteristic
246C2×Dic5C22×D311yescharacteristic
247D20—62no—
248C4×D5—34no—
249C4×D5—62no—
250C4×D5—121no—
251C4×D5—121no—
252Dic10—62no—
253Dic10—62no—
254C2×#41—151no—
255C2×#41—151no—
256C22⋊Q8—151no—
257C22⋊Q8—151no—
258C22⋊Q8—151no—
259C22⋊Q8—151no—
260#121—31no—
261#121—51no—
262#121—151no—
263#121—151no—
264#121—151no—
265#121—151no—
266#121—151no—
267#121—151no—
268C2×C4⋊C4—151no—
269C2×C3×C5—41no—
270C2×C3×C5—41no—
271C2×C3×C5C22×D411yescharacteristic
272C2×C3×C5H(5,𝔽2)11yescharacteristic
273C2×C3×C52−1+411yescharacteristic
274D15—22no—
275C3×D5—41no—
276C5×D3—41no—
277C23×C3—51no—
278C23×C3C22×D511yescharacteristic
279C23×C3C22×D511yescharacteristic
280C22×D3—51no—
281C22×D3C22×D511yescharacteristic
282C3×D4—41no—
283C3×D4—102no—
284C3×D4—201no—
285C2×C3×C4—51no—
286C2×C3×C4—51no—
287C2×C3×C4—51no—
288C2×C3×C4—102no—
289C2×C3×C4C22×D511yescharacteristic
290#67—22no—
291#67—41no—
292#67—102no—
293#67—102no—
294#67—102no—
295#67—201no—
296C2×Dic3—22no—
297C2×Dic3—51no—
298C2×Dic3—51no—
299C2×Dic3—51no—
300C2×Dic3—102no—
301C2×Dic3C22×D511yescharacteristic
302C2×Dic3C22×D511yescharacteristic
303C2×Dic3C22×D511yescharacteristic
304D12—102no—
305C4×D3—54no—
306C4×D3—102no—
307C4×D3—201no—
308C4×D3—201no—
309Dic6—102no—
310Dic6—102no—
311C22×C5—21no—
312C22×C5—21no—
313C22×C5—21no—
314C22×C5—21no—
315C22×C5—21no—
316C22×C5—21no—
317C22×C5—61no—
318C22×C5—61no—
319C22×C5—61no—
320C22×C5C23×D311yescharacteristic
321C2×D5—21no—
322C2×D5—21no—
323C2×D5—21no—
324C2×D5—32no—
325C2×D5—61no—
326C2×D5—61no—
327C4×C5—21no—
328C4×C5—32no—
329C4×C5—61no—
330C4×C5—61no—
331Dic5—21no—
332Dic5—21no—
333Dic5—61no—
334Dic5—61no—
335Dic5—61no—
336Dic5D3×D412yes—
337#41—304no—
338#41—304no—
339C2×Q8—151no—
340C2×Q8—151no—
341C2×D4—31no—
342C2×D4—31no—
343C2×D4—51no—
344C2×D4—51no—
345C2×D4—151no—
346C2×D4—151no—
347C2×D4—151no—
348C2×D4—151no—
349C2×D4—151no—
350C2×D4D3×D511yescharacteristic
351C22×C4—31no—
352C22×C4—51no—
353C22×C4—151no—
354C22×C4—151no—
355C22×C4—151no—
356C22×C4—151no—
357C22×C4—151no—
358C4⋊C4—31no—
359C4⋊C4—51no—
360C4⋊C4—151no—
361C4⋊C4—151no—
362C4⋊C4—151no—
363C4⋊C4—151no—
364C4⋊C4—151no—
365C4⋊C4—151no—
366C22⋊C4—31no—
367C22⋊C4—31no—
368C22⋊C4—51no—
369C22⋊C4—51no—
370C22⋊C4—151no—
371C22⋊C4—151no—
372C22⋊C4—151no—
373C22⋊C4—151no—
374C3×C5#53711yescharacteristic, complemented, Hall
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—101no—
382C22×C3—101no—
383C22×C3—101no—
384C22×C3C23×D511yescharacteristic
385C2×D3—21no—
386C2×D3—21no—
387C2×D3—21no—
388C2×D3—52no—
389C2×D3—101no—
390C2×D3—101no—
391C3×C4—21no—
392C3×C4—52no—
393C3×C4—101no—
394C3×C4—101no—
395Dic3—21no—
396Dic3—21no—
397Dic3—101no—
398Dic3—101no—
399Dic3—101no—
400Dic3D4×D512yes—
401C2×C5—41no—
402C2×C5—41no—
403C2×C5—121no—
404C2×C5C2×D3×D411yescharacteristic
405C2×C5#100411yescharacteristic
406C2×C5#101011yescharacteristic
407D5—41no—
408D5—62no—
409C23—31no—
410C23—51no—
411C23—151no—
412C23C2×D3×D511yescharacteristic
413C23C2×D3×D511yescharacteristic
414Q8—302no—
415Q8—302no—
416D4—41no—
417D4—62no—
418D4—102no—
419D4—121no—
420D4—201no—
421D4—302no—
422D4—302no—
423D4—302no—
424D4—302no—
425D4—601no—
426C2×C4—31no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—51no—
430C2×C4—51no—
431C2×C4—51no—
432C2×C4—62no—
433C2×C4—102no—
434C2×C4—154no—
435C2×C4—151no—
436C2×C4—151no—
437C2×C4—151no—
438C2×C4—302no—
439C2×C4—302no—
440C2×C4—601no—
441C2×C4—601no—
442C2×C4C2×D3×D511yescharacteristic
443C2×C3—41no—
444C2×C3—41no—
445C2×C3—201no—
446C2×C3C2×D4×D511yescharacteristic
447C2×C3#414411yescharacteristic
448C2×C3#415011yescharacteristic
449D3—41no—
450D3—102no—
451C5#560811yescharacteristic, complemented, Sylow
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—61no—
459C22—61no—
460C22—61no—
461C22—101no—
462C22—101no—
463C22—101no—
464C22—152no—
465C22—301no—
466C22—301no—
467C22C22×D3×D511yescharacteristic
468C4—21no—
469C4—32no—
470C4—52no—
471C4—61no—
472C4—61no—
473C4—101no—
474C4—101no—
475C4—301no—
476C4—301no—
477C4—301no—
478C3#6594211yescharacteristic, complemented, Sylow
479C2—41no—
480C2—41no—
481C2—121no—
482C2—201no—
483C2—302no—
484C2#9202011yescharacteristic
485C2D3×D4×D511yescharacteristic
486C2#9204911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#537
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#6782C2semidirect
2. C2×#6779C2semidirect
3. #91827C2semidirect
4. C5×#939C2semidirect
5. C3×#4085C2semidirect
6. #91575C2semidirect
7. #91574C2semidirect
8. #91565C2semidirect
9. #91557C2semidirect
10. #91549C2semidirect
11. #91533C2semidirect
12. #91461C2semidirect
13. #91422C2semidirect
14. #91417C2semidirect
15. #91404C2semidirect
59. #4085D3semidirect
101. #1200D4semidirect
108. #939D5semidirect
374. C3×C5#537semidirect
451. C5#5608semidirect
478. C3#65942semidirect

Maximal subgroups

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

subgroupindex
C2×#67822
C2×#67792
#918272
C5×#9392
C3×#40852
#915752
#915742
#915652
#915572
#915492
#915332
#914612
#914222
#914172
#914042
#659423
#56085