Finite Group Databasebeta

← #13045225

Subgroups — #13045225

Order 1280.

1702 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#711C211yesmaximal, characteristic
2#10617523C211yesmaximal, characteristic, complemented
3#10617521C211yesmaximal, characteristic, complemented
4Q8×#704C211yesmaximal, characteristic, complemented
5C2×Q8×#170C211yesmaximal, characteristic, complemented
6#10617383C213yesmaximal, complemented
7#10617381C213yesmaximal, complemented
8#10617275C213yesmaximal, complemented
9#10617273C213yesmaximal, complemented
10#10617263C213yesmaximal, complemented
11#10617261C213yesmaximal, complemented
12#10617202C211yesmaximal, characteristic, complemented
13#10617199C211yesmaximal, characteristic, complemented
14#10617143C213yesmaximal
15#10617139C213yesmaximal
16C5×D4×Q8C411yescharacteristic
17Q8×C22⋊D5C411yescharacteristic
18Q8×C22⋊D5C411yescharacteristic
19C2×Q8×Dic5C2211yescharacteristic
20C2×Q8×Dic5C2211yescharacteristic
21#66062C2213yes—
22#66055C413yes—
23#65969C413yes—
24#65969C413yes—
25#65956C2213yes—
26#65956C2213yes—
27C4×Q8×D5C2211yescharacteristic
28Q8×Dic10C411yescharacteristic
29#65877C413yes—
30C4×#711C2213yes—
31#65788C2213yes—
32#65788C2213yes—
33#65788C2213yes—
34#65788C2213yes—
35Q8×#170C2213yes—
36Q8×#170C2213yes—
37Q8×#170C2213yes—
38Q8×#170C2213yes—
39Q8×#170C2211yescharacteristic
40Q8×#170C2211yescharacteristic
41Q8×#170C2211yescharacteristic
42Q8×#170C2211yescharacteristic
43#65775C2211yescharacteristic
44#65775C2213yes—
45#65759C2213yes—
46#65758C2213yes—
47#65754C2213yes—
48C4×#704C2213yes—
49#65748C2213yes—
50#65748C2213yes—
51C2×C4⋊#170C2213yes—
52C2×C4×#170C2213yes—
53#65709C2216yes—
54#65709C2213yes—
55#65709C2213yes—
56#65707C2216yes—
57#65707C2216yes—
58#65706C2213yes—
59#65706C2216yes—
60#65706C2213yes—
61#65703C2216yes—
62#65703C2213yes—
63#65703C2213yes—
64#65697C2213yes—
65#65697C2213yes—
66#65697C2216yes—
67#65696C2213yes—
68#65696C2213yes—
69#65691C2216yes—
70#65683C2213yes—
71#65680C2213yes—
72#65680C2213yes—
73#65678C2213yes—
74#60253—51nomaximal, Sylow
75C22×C5×Q8C2×C411yescharacteristic
76C22×C5×Q8C2×C411yescharacteristic
77C2×Q8×D5C2×C411yescharacteristic
78C2×#711C2311yescharacteristic
79C2×#704C2313yes—
80C2×#704C2311yescharacteristic
81C22×#170C2313yes—
82C22×#170C2311yescharacteristic
83C5×C4⋊Q8C2×C413yes—
84C5×C22⋊Q8C2×C413yes—
85C5×C22⋊Q8C2×C413yes—
86C4×C5×Q8C2×C411yescharacteristic
87C4×C5×D4C2×C413yes—
88#4092C2×C413yes—
89#4092C2×C413yes—
90Q8×Dic5C2×C413yes—
91Q8×Dic5C2×C413yes—
92Q8×Dic5C2×C411yescharacteristic
93Q8×Dic5C2×C411yescharacteristic
94Q8×Dic5C2313yes—
95Q8×Dic5C2313yes—
96Q8×Dic5C2311yescharacteristic
97#4090C2×C413yes—
98#4090C2×C413yes—
99D4×Dic5C2×C413yes—
100C4×C22⋊D5C2×C413yes—
101C4×C22⋊D5C2×C413yes—
102C2×C4⋊Dic5C2×C413yes—
103C2×C4⋊Dic5C2×C413yes—
104#4070C2×C413yes—
105#4070C2×C413yes—
106C2×C4×Dic5C2313yes—
107C2×C4×Dic5C2313yes—
108#4043C2×C416yes—
109#4038C2313yes—
110D5×C4⋊C4C2×C413yes—
111#4036C2×C416yes—
112#4035C2×C416yes—
113#4035C2×C416yes—
114#4034C2×C413yes—
115#4034C2×C413yes—
116#4033C2×C413yes—
117#4033C2316yes—
118#4033C2313yes—
119#4027C2×C413yes—
120#4027C2×C413yes—
121#4024C2×C416yes—
122#4024C2×C416yes—
123#4023C2313yes—
124#4023C2313yes—
125C42×D5C2313yes—
126#4015C2×C413yes—
127C4×Dic10C2×C413yes—
128C4×Dic10C2×C413yes—
129C22⋊#170C2313yes—
130C22⋊#170C2311yescharacteristic
131C22⋊#170C2311yescharacteristic
132C22⋊#170C2313yes—
133#4004C2313yes—
134#4004C2313yes—
135#4004C2313yes—
136#4004C2313yes—
137#4004C2316yes—
138#4003C2311yescharacteristic
139#4003C2313yes—
140#4003C2311yescharacteristic
141#4003C2313yes—
142#4002C2316yes—
143#4002C2316yes—
144#4002C2313yes—
145#4002C2313yes—
146C4⋊#170C2313yes—
147C4⋊#170C2313yes—
148C4⋊#170C2313yes—
149C4⋊#170C2313yes—
150C4⋊#170C2313yes—
151C4⋊#170C2313yes—
152C4⋊#170C2313yes—
153C4⋊#170C2313yes—
154C4⋊#170C2316yes—
155C4×#170C2313yes—
156C4×#170C2313yes—
157C4×#170C2316yes—
158C4×#170C2313yes—
159C4×#170C2313yes—
160C4×#170C2316yes—
161C4×#170C2313yes—
162C4×#170C2313yes—
163Q8×#41—51no—
164#2986—53no—
165#2986—53no—
166#2985—53no—
167#2985—53no—
168#2984—51no—
169#2984—51no—
170#2982—51no—
171#2982—51no—
172Q8×M16—51no—
173#2962—53no—
174#2960—53no—
175C2×C8×Q8—51no—
176#2925—53no—
177#2922—53no—
178C2×C5×Q8—24no—
179C2×C5×Q8—24no—
180C2×C5×Q8C22×C413yes—
181C2×C5×Q8C22×C413yes—
182C2×C5×Q8C22×C411yescharacteristic
183C2×C5×D4C22×C411yescharacteristic
184C22×C4×C5C22×C413yes—
185C22×C4×C5C22×C413yes—
186C2×C22⋊D5C22×C411yescharacteristic
187C2×C22⋊D5C22×C411yescharacteristic
188C22×Dic5C22×C413yes—
189C22×Dic5C22×C413yes—
190C22×Dic5C2411yescharacteristic
191C22×Dic5C2411yescharacteristic
192Q8×D5—24no—
193#711—22no—
194#711—22no—
195C2×C4×D5C22×C413yes—
196C2×C4×D5C22×C413yes—
197C2×C4×D5C2411yescharacteristic
198C2×Dic10C22×C411yescharacteristic
199C2×Dic10C22×C413yes—
200C2×Dic10C22×C416yes—
201#704—22no—
202#704—26no—
203C2×#170—26no—
204C2×#170—22no—
205C2×#170C2413yes—
206C2×#170C2411yescharacteristic
207C2×#170C2411yescharacteristic
208C2×#170C2413yes—
209C2×#170C2411yescharacteristic
210C2×#170C2413yes—
211C2×#170C2413yes—
212C2×#170C2411yescharacteristic
213C5×C4⋊C4C22×C413yes—
214C5×C4⋊C4C22×C413yes—
215C5×C4⋊C4C22×C416yes—
216C5×C22⋊C4C22×C413yes—
217C5×C22⋊C4C22×C413yes—
218C42×C5C22×C413yes—
219C22⋊Dic5C22×C413yes—
220C22⋊Dic5C22×C413yes—
221#685C22×C413yes—
222#685C22×C413yes—
223C4⋊Dic5C22×C413yes—
224C4⋊Dic5C22×C413yes—
225C4⋊Dic5C22×C413yes—
226C4⋊Dic5C22×C413yes—
227C4⋊Dic5C22×C416yes—
228C4⋊Dic5C22×C416yes—
229#683C22×C413yes—
230#683C22×C413yes—
231#683C22×C416yes—
232#683C22×C413yes—
233#683C22×C413yes—
234#683C22×C416yes—
235#683C22×C416yes—
236C4×Dic5C22×C413yes—
237C4×Dic5C22×C413yes—
238C4×Dic5C22×C413yes—
239C4×Dic5C22×C413yes—
240C4×Dic5C22×C413yes—
241C4×Dic5C2413yes—
242C4×Dic5C2413yes—
243C4×Dic5C2413yes—
244C4×Dic5C2413yes—
245Q82—51no—
246#557—53no—
247#554—53no—
248#554—53no—
249#554—53no—
250D4×Q8—51no—
251D4×Q8—51no—
252D4×Q8AGL(1,𝔽5)11yescharacteristic
253#533—53no—
254#533—53no—
255#533—53no—
256C4×#41—53no—
257C2×C4×Q8—51no—
258C2×C4×Q8—51no—
259C2×C4×Q8—51no—
260#446—53no—
261#446—53no—
262#446—56no—
263#446—53no—
264#446—53no—
265#446—56no—
266#446—56no—
267#446—53no—
268#446—53no—
269C8×Q8—51no—
270C8×Q8—51no—
271C8×Q8—53no—
272C8×Q8—53no—
273C8×Q8—53no—
274C8×Q8—53no—
275C8×Q8—56no—
276C8×Q8—53no—
277C8×Q8—53no—
278C8×Q8—51no—
279C8×Q8—51no—
280#433—53no—
281#433—53no—
282#433—53no—
283#433—53no—
284#433—56no—
285#433—56no—
286#432—53no—
287#432—53no—
288#431—53no—
289#431—53no—
290#431—53no—
291#431—53no—
292#431—53no—
293#431—53no—
294#424—53no—
295#424—53no—
296#423—53no—
297C2×C4⋊C8—53no—
298#408—51no—
299#408—53no—
300#405—56no—
301C4×M16—53no—
302C2×C4×C8—53no—
303C23×C5C23×C411yescharacteristic
304C23×C5C23×C411yescharacteristic
305C22×D5C23×C411yescharacteristic
306C5×Q8—26no—
307C5×Q8—26no—
308C5×Q8#13114yes—
309C5×D4—22no—
310C2×C4×C5—26no—
311C2×C4×C5—26no—
312C2×C4×C5C23×C411yescharacteristic
313C2×C4×C5C23×C413yes—
314C2×C4×C5C23×C413yes—
315C2×C4×C5C23×C413yes—
316C2×C4×C5C23×C413yes—
317C22⋊D5—22no—
318C22⋊D5—22no—
319C2×Dic5—21no—
320C2×Dic5—21no—
321C2×Dic5—43no—
322C2×Dic5—43no—
323C2×Dic5C23×C411yescharacteristic
324C2×Dic5C23×C413yes—
325C2×Dic5C23×C413yes—
326C2×Dic5C23×C413yes—
327C2×Dic5C23×C413yes—
328C2×Dic5C23×C411yescharacteristic
329C2×Dic52−1+412yes—
330C2×Dic52−1+412yes—
331C2×Dic5C2511yescharacteristic
332C4×D5—26no—
333C4×D5—21no—
334C4×D5—21no—
335C4×D5—43no—
336Dic10—26no—
337Dic10—22no—
338Dic10—46no—
339#170—23no—
340#170—21no—
341#170—21no—
342#170—23no—
343#170—21no—
344#170—23no—
345#170—23no—
346#170—21no—
347C2×#41—51no—
348C22×Q8—51no—
349C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
350C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
351C2×M16—51no—
352C2×M16—53no—
353C22×C8—51no—
354C22×C8—53no—
355C4⋊Q8—53no—
356C4⋊Q8—53no—
357C4⋊Q8—53no—
358C4⋊Q8—53no—
359C4⋊Q8—53no—
360C4⋊Q8C2×AGL(1,𝔽5)13yes—
361#125—56no—
362#125—56no—
363#125—56no—
364C22⋊Q8—53no—
365C22⋊Q8—53no—
366C22⋊Q8—53no—
367C22⋊Q8—53no—
368C22⋊Q8—56no—
369C22⋊Q8—56no—
370C22⋊Q8—56no—
371C22⋊Q8C2×AGL(1,𝔽5)13yes—
372C22⋊Q8C2×AGL(1,𝔽5)13yes—
373C4×Q8—51no—
374C4×Q8—53no—
375C4×Q8—53no—
376C4×Q8—51no—
377C4×Q8—51no—
378C4×Q8—53no—
379C4×Q8—53no—
380C4×Q8—53no—
381C4×Q8—53no—
382C4×Q8—56no—
383C4×Q8—53no—
384C4×Q8—53no—
385C4×Q8C2×AGL(1,𝔽5)11yescharacteristic
386C4×D4—53no—
387C4×D4—53no—
388C4×D4—53no—
389C4×D4—53no—
390C4×D4—53no—
391C4×D4C2×AGL(1,𝔽5)13yes—
392#117—53no—
393#117—53no—
394#117—53no—
395C2×C4⋊C4—53no—
396C2×C4⋊C4—53no—
397C2×C4⋊C4—53no—
398C2×C42—53no—
399C2×C42—53no—
400C2×C42—53no—
401C4⋊C8—53no—
402C4⋊C8—53no—
403C4⋊C8—56no—
404C4⋊C8—53no—
405C4⋊C8—53no—
406C4⋊C8—53no—
407C4⋊C8—53no—
408C4⋊C8—53no—
409C4⋊C8—53no—
410C4⋊C8—56no—
411C4⋊C8—53no—
412C4⋊C8—53no—
413C4⋊C8—53no—
414C4⋊C8—53no—
415C22⋊C8—51no—
416C22⋊C8—51no—
417C22⋊C8—53no—
418C22⋊C8—53no—
419C22⋊C8—51no—
420C22⋊C8—51no—
421C22⋊C8—53no—
422C22⋊C8—53no—
423#97—53no—
424#97—53no—
425#97—56no—
426#97—56no—
427C4×C8—53no—
428C4×C8—53no—
429C4×C8—53no—
430C4×C8—53no—
431C4×C8—53no—
432C4×C8—53no—
433C4×C8—56no—
434C4×C8—56no—
435C22×C5—21no—
436C22×C5—21no—
437C22×C5—21no—
438C22×C5—21no—
439C22×C5—21no—
440C22×C5#51912yes—
441C22×C5C24×C411yescharacteristic
442C2×D5—21no—
443C2×D5—21no—
444C2×D5—21no—
445C4×C5—21no—
446C4×C5—23no—
447C4×C5—23no—
448C4×C5—23no—
449C4×C5C2×#13116yes—
450Dic5—21no—
451Dic5—23no—
452Dic5—23no—
453Dic5—23no—
454Dic5—23no—
455Dic5#51912yes—
456Dic5C2×2−1+411yescharacteristic
457Dic5C2×2−1+411yescharacteristic
458#41—102no—
459#41—102no—
460C2×Q8—24no—
461C2×Q8—24no—
462C2×Q8—51no—
463C2×Q8—53no—
464C2×Q8—56no—
465C2×Q8—104no—
466C2×Q8C22×AGL(1,𝔽5)13yes—
467C2×Q8C22×AGL(1,𝔽5)13yes—
468C2×Q8C22×AGL(1,𝔽5)11yescharacteristic
469C2×D4—51no—
470C2×D4—51no—
471C2×D4C22×AGL(1,𝔽5)11yescharacteristic
472C22×C4—51no—
473C22×C4—51no—
474C22×C4—53no—
475C22×C4—53no—
476C22×C4—51no—
477C22×C4—53no—
478C22×C4—53no—
479C22×C4C22×AGL(1,𝔽5)13yes—
480C22×C4C22×AGL(1,𝔽5)13yes—
481M16—102no—
482M16—106no—
483C2×C8—51no—
484C2×C8—53no—
485C2×C8—51no—
486C2×C8—51no—
487C2×C8—51no—
488C2×C8—53no—
489C2×C8—53no—
490C2×C8—53no—
491C2×C8—106no—
492C2×C8—102no—
493C4⋊C4—53no—
494C4⋊C4—56no—
495C4⋊C4—56no—
496C4⋊C4—56no—
497C4⋊C4—53no—
498C4⋊C4—53no—
499C4⋊C4—53no—
500C4⋊C4—53no—
501C4⋊C4—53no—
502C4⋊C4—53no—
503C4⋊C4—53no—
504C4⋊C4—56no—
505C4⋊C4—56no—
506C4⋊C4C22×AGL(1,𝔽5)13yes—
507C4⋊C4C22×AGL(1,𝔽5)13yes—
508C4⋊C4C22×AGL(1,𝔽5)16yes—
509C22⋊C4—53no—
510C22⋊C4—53no—
511C22⋊C4—53no—
512C22⋊C4—53no—
513C22⋊C4C22×AGL(1,𝔽5)13yes—
514C22⋊C4C22×AGL(1,𝔽5)13yes—
515C42—53no—
516C42—53no—
517C42—53no—
518C42—53no—
519C42—53no—
520C42—53no—
521C42—53no—
522C42—53no—
523C42—53no—
524C42C22×AGL(1,𝔽5)13yes—
525C2×C5—22no—
526C2×C5—41no—
527C2×C5C2×#51911yescharacteristic
528C2×C5C22×#13111yescharacteristic
529C2×C5#357311yescharacteristic
530D5—41no—
531C23—51no—
532C23C23×AGL(1,𝔽5)11yescharacteristic
533C23C23×AGL(1,𝔽5)11yescharacteristic
534Q8—26no—
535Q8—26no—
536Q8—102no—
537Q8—106no—
538Q8—206no—
539Q8#413114yes—
540D4—22no—
541D4—102no—
542D4—102no—
543C2×C4—26no—
544C2×C4—26no—
545C2×C4—52no—
546C2×C4—52no—
547C2×C4—51no—
548C2×C4—51no—
549C2×C4—51no—
550C2×C4—53no—
551C2×C4—53no—
552C2×C4—53no—
553C2×C4—53no—
554C2×C4—101no—
555C2×C4—101no—
556C2×C4—101no—
557C2×C4—101no—
558C2×C4—106no—
559C2×C4—203no—
560C2×C4—203no—
561C2×C4—203no—
562C2×C4C23×AGL(1,𝔽5)11yescharacteristic
563C2×C4C23×AGL(1,𝔽5)13yes—
564C2×C4C23×AGL(1,𝔽5)13yes—
565C2×C4C23×AGL(1,𝔽5)13yes—
566C2×C4C23×AGL(1,𝔽5)13yes—
567C8—103no—
568C8—101no—
569C8—101no—
570C8—103no—
571C8—101no—
572C8—101no—
573C8—103no—
574C8—103no—
575C5#6025311yescharacteristic, complemented, Sylow
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—101no—
582C22—101no—
583C22—101no—
584C22#6626212yes—
585C22C24×AGL(1,𝔽5)11yescharacteristic
586C4—23no—
587C4—23no—
588C4—21no—
589C4—23no—
590C4—51no—
591C4—51no—
592C4—52no—
593C4—103no—
594C4—101no—
595C4—103no—
596C4—103no—
597C4—103no—
598C4C2×#413116yes—
599C2—22no—
600C2—41no—
601C2—201no—
602C2C22×#413111yescharacteristic
603C2C2×#6626211yescharacteristic
604C2#1061941511yescharacteristic
subgroupG/H
center, FrattiniC22C24×AGL(1,𝔽5)
commutator, socleC22×C5C24×C4
FittingC5×D4×Q8C4
prime radicalC2×#711C23
cosocleC2×Dic5C25
hypercenterD4×Q8AGL(1,𝔽5)
hypocenterC5#60253
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
2. #10617523C2semidirect
3. #10617521C2semidirect
4. Q8×#704C2semidirect
5. C2×Q8×#170C2semidirect
6. #10617383C2semidirect
7. #10617381C2semidirect
8. #10617275C2semidirect
9. #10617273C2semidirect
10. #10617263C2semidirect
11. #10617261C2semidirect
12. #10617202C2semidirect
13. #10617199C2semidirect
575. C5#60253semidirect

Maximal subgroups

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

subgroupindex
Q8×#7112
#106175232
#106175212
Q8×#7042
C2×Q8×#1702
#106173832
#106173812
#106172752
#106172732
#106172632
#106172612
#106172022
#106171992
#106171432
#106171392
#602535