Finite Group Databasebeta

← #14244239

Subgroups — #14244239

Order 1792.

1648 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3502C211yesmaximal, characteristic, complemented
2#11743335C211yesmaximal, characteristic
3#11741153C212yesmaximal, complemented
4Q8×#1144C212yesmaximal, complemented
5#11741145C214yesmaximal, complemented
6#11741142C212yesmaximal, complemented
7#11741139C214yesmaximal, complemented
8#11741128C212yesmaximal, complemented
9#11740923C214yesmaximal, complemented
10#11740817C211yesmaximal, characteristic, complemented
11#11740804C211yesmaximal, characteristic
12#11740776C211yesmaximal, characteristic, complemented
13#11739997C214yesmaximal, complemented
14#11739892C211yesmaximal, characteristic, complemented
15#11739889C211yesmaximal, characteristic
16C7×Q82—21no—
17C7×Q82C2212yes—
18C7×#557—21no—
19C7×#557—22no—
20C7×#557C2212yes—
21C7×#557C2214yes—
22C7×#544—22no—
23C7×#544C2212yes—
24C7×#533—22no—
25C7×#533C2211yescharacteristic
26C7×#533C2211yescharacteristic
27C7×#519C2212yes—
28C2×C4×C7×Q8C2211yescharacteristic
29#90423—21no—
30#90418—21no—
31#90390—22no—
32#90334—22no—
33#90319—22no—
34#90276C2212yes—
35#90275C2212yes—
36Q8×D28C2212yes—
37Q8×Dic14C2212yes—
38#90225C2214yes—
39#90168C2211yescharacteristic
40#90161C2211yescharacteristic
41#90000C2212yes—
42#89999C2212yes—
43#89866C2212yes—
44#89863C2212yes—
45#89863C2214yes—
46#89861C2214yes—
47#89860C2211yescharacteristic
48#89860C2211yescharacteristic
49#89859C2214yes—
50#89851C2211yescharacteristic
51#89850C2211yescharacteristic
52#89850C2214yes—
53#89841C2212yes—
54#89840C2214yes—
55#89838C2214yes—
56#89838C2214yes—
57#89837C2212yes—
58#89837C2214yes—
59#89835C2214yes—
60#89803C2212yes—
61#89802C2214yes—
62C4×#1144C2214yes—
63C4×#1144C2212yes—
64C4×#1144C2212yes—
65#89800C2214yes—
66#89800C2214yes—
67#89799C2212yes—
68#89798C2214yes—
69#89798C2214yes—
70C4×#1143C2214yes—
71#89796C2212yes—
72#89796C2214yes—
73Q8×C7⋊C8C2212yes—
74#89793C2214yes—
75#89793C2212yes—
76#89793C2212yes—
77#89792C2214yes—
78#89775C2214yes—
79#89772C2214yes—
80#89768C2214yes—
81#89742C2212yes—
82#89739C2212yes—
83#89698C2211yescharacteristic
84#89697C2211yescharacteristic
85#61854—71nomaximal, Sylow
86C22×C7×Q8C2311yescharacteristic
87C2×Q8×D7—22no—
88C2×#1157—21no—
89C7×C4⋊Q8—22no—
90C7×C4⋊Q8—24no—
91C7×C4⋊Q8—22no—
92C7×C4⋊Q8—24no—
93C7×C4⋊Q8—22no—
94C7×C4⋊Q8D411yescharacteristic
95C7×C4⋊Q8D411yescharacteristic
96C7×C4⋊Q8C2314yes—
97C7×C4⋊Q8C2311yescharacteristic
98C7×C4⋊Q8C2311yescharacteristic
99C7×#125—24no—
100C7×#125—24no—
101C7×#125—24no—
102C7×#125—22no—
103C7×#125D412yes—
104C7×#125C2312yes—
105C7×#125C2314yes—
106C7×C22⋊Q8—21no—
107C7×C22⋊Q8—21no—
108C7×C22⋊Q8—24no—
109C7×C22⋊Q8D412yes—
110C7×C22⋊Q8D412yes—
111C4×C7×Q8—22no—
112C4×C7×Q8—22no—
113C4×C7×Q8—22no—
114C4×C7×Q8—24no—
115C4×C7×Q8—22no—
116C4×C7×Q8—21no—
117C4×C7×Q8—21no—
118C4×C7×Q8—21no—
119C4×C7×Q8—21no—
120C4×C7×Q8C2312yes—
121C4×C7×Q8C2312yes—
122C4×C7×Q8C2314yes—
123C4×C7×Q8C2314yes—
124C4×C7×Q8C2312yes—
125C4×C7×Q8C2312yes—
126C7×#117—24no—
127C7×#117C2312yes—
128C7×#117C2312yes—
129C2×C7×C4⋊C4—21no—
130C2×C7×C4⋊C4C2311yescharacteristic
131C2×C42×C7—21no—
132C2×C42×C7C2311yescharacteristic
133#6510—22no—
134#6509—22no—
135Q8×Dic7—22no—
136C2×#1144C2312yes—
137C2×#1144C2312yes—
138C2×#1143C2314yes—
139#6493—21no—
140#6493C2312yes—
141#6489—21no—
142#6487—21no—
143#6487C2312yes—
144#6461—24no—
145#6461—24no—
146#6460—24no—
147#6460—22no—
148#6459—24no—
149#6457—24no—
150#6456—22no—
151#6455—22no—
152#6451—24no—
153#6451—22no—
154#6448—24no—
155#6446—24no—
156#6438—21no—
157#6438C2314yes—
158#6437C2311yescharacteristic
159C4×D28—21no—
160C4×D28C2314yes—
161C4×D28C2312yes—
162#6433—21no—
163#6432C2311yescharacteristic
164#6432C2311yescharacteristic
165#6432C2311yescharacteristic
166#6432C2314yes—
167C4×Dic14—21no—
168C4×Dic14C2312yes—
169C4×Dic14C2314yes—
170#6409C2314yes—
171#6409C2312yes—
172#6409C2312yes—
173C22⋊(C7⋊C8)C2312yes—
174C22⋊(C7⋊C8)C2312yes—
175#6384C2312yes—
176#6384C2314yes—
177#6384C2312yes—
178#6384C2314yes—
179#6383C2314yes—
180#6383C2314yes—
181#6383C2314yes—
182#6382C2314yes—
183#6382C2314yes—
184#6381C2312yes—
185#6381C2314yes—
186#6381C2314yes—
187#6381C2314yes—
188#6381C2312yes—
189C4⋊(C7⋊C8)C2314yes—
190C4⋊(C7⋊C8)C2312yes—
191C4⋊(C7⋊C8)C2314yes—
192C4⋊(C7⋊C8)C2312yes—
193#6377C2314yes—
194#6377C2312yes—
195C4×C7⋊C8C2311yescharacteristic
196C4×C7⋊C8C2311yescharacteristic
197C4×C7⋊C8C2314yes—
198#3550—71no—
199#3502D711yescharacteristic, complemented
200#3387—72no—
201#3386—74no—
202Q8×Q16—72no—
203#3377—74no—
204#3375—72no—
205#3365—72no—
206#3243—74no—
207#3237—71no—
208#3228—71no—
209#3121—74no—
210#3112—71no—
211#3104—71no—
212#2985—71no—
213C2×C7×Q8—24no—
214C2×C7×Q8—21no—
215C2×C7×Q8—21no—
216C2×C7×Q8—21no—
217C2×C7×Q8—24no—
218C2×C7×Q8C2×D412yes—
219C2×C7×Q8C2×D412yes—
220C2×C7×Q8C2412yes—
221C22×C4×C7—21no—
222C22×C4×C7—21no—
223C22×C4×C7C2×D411yescharacteristic
224C22×C4×C7C2×D411yescharacteristic
225C22×C4×C7C2411yescharacteristic
226C2×C22⋊D7—22no—
227Q8×D7—84no—
228#1157—82no—
229C2×D28—21no—
230C2×D28—21no—
231C2×D28C2412yes—
232C2×C4×D7—22no—
233C2×C4×D7—24no—
234C2×Dic14—21no—
235C2×Dic14—21no—
236C2×Dic14—24no—
237C2×Dic14C2412yes—
238C7×C4⋊C4—22no—
239C7×C4⋊C4—22no—
240C7×C4⋊C4—21no—
241C7×C4⋊C4—24no—
242C7×C4⋊C4—24no—
243C7×C4⋊C4—22no—
244C7×C4⋊C4—21no—
245C7×C4⋊C4—21no—
246C7×C4⋊C4—24no—
247C7×C4⋊C4—24no—
248C7×C4⋊C4—22no—
249C7×C4⋊C4—24no—
250C7×C4⋊C4C2×D412yes—
251C7×C4⋊C4C2×D414yes—
252C7×C4⋊C4C2×D412yes—
253C7×C4⋊C4C2×D412yes—
254C7×C4⋊C4C2×D414yes—
255C7×C4⋊C4C2×D412yes—
256C7×C4⋊C4C2412yes—
257C7×C4⋊C4C2412yes—
258C7×C4⋊C4C2412yes—
259C7×C22⋊C4—22no—
260C7×C22⋊C4C2×D414yes—
261C42×C7—22no—
262C42×C7—22no—
263C42×C7—24no—
264C42×C7—24no—
265C42×C7—22no—
266C42×C7C2×D411yescharacteristic
267C42×C7C2×D411yescharacteristic
268C42×C7C2411yescharacteristic
269C42×C7C2414yes—
270C42×C7C2411yescharacteristic
271C22⋊Dic7—22no—
272#1144—24no—
273#1144—24no—
274#1143—28no—
275(C2×D7)⋊C4—22no—
276(C2×D7)⋊C4—22no—
277(C2×D7)⋊C4—24no—
278(C2×D7)⋊C4—24no—
279(C2×D7)⋊C4—24no—
280C4⋊Dic7—21no—
281C4⋊Dic7—21no—
282C4⋊Dic7—21no—
283C4⋊Dic7—21no—
284C4⋊Dic7—24no—
285C4⋊Dic7C2412yes—
286C4⋊Dic7C2412yes—
287Dic7⋊C4—22no—
288Dic7⋊C4—22no—
289Dic7⋊C4—24no—
290C4×Dic7—22no—
291C4×Dic7—24no—
292C2×C7⋊C8C2414yes—
293C2×C7⋊C8C2412yes—
294C2×C7⋊C8C2412yes—
295#563—142no—
296#559—72no—
297Q82—21no—
298Q82—72no—
299Q82C2×D712yes—
300#557—21no—
301#557—22no—
302#557C2×D712yes—
303#557C2×D714yes—
304#556—72no—
305#556—141no—
306D4×Q8—72no—
307D4×Q8—141no—
308#544—22no—
309#544C2×D712yes—
310#542—74no—
311#542—142no—
312#536—142no—
313#533—22no—
314#533—71no—
315#533C2×D711yescharacteristic
316#533C2×D711yescharacteristic
317#532—71no—
318#519C2×D712yes—
319C2×C4×Q8C2×D711yescharacteristic
320#501—72no—
321#501—74no—
322#500—71no—
323#500—74no—
324#498—71no—
325#491—72no—
326#489—74no—
327#488—74no—
328#486—71no—
329#486—71no—
330#485—74no—
331#484—72no—
332#482—74no—
333#480—72no—
334#479—74no—
335#478—74no—
336#478—74no—
337#477—72no—
338#477—74no—
339#477—72no—
340#477—72no—
341#477—74no—
342#475—72no—
343#475—74no—
344#474—74no—
345#464—74no—
346#464—74no—
347#462—72no—
348#460—72no—
349#451—72no—
350#448—72no—
351#446—72no—
352#446—74no—
353C8×Q8—72no—
354#441—74no—
355#440—74no—
356#440—74no—
357C4×Q16—72no—
358C4×Q16—72no—
359C4×Q16—74no—
360C4×SD16—74no—
361#433—74no—
362#432—71no—
363#431—71no—
364C23×C7C22×D411yescharacteristic
365C22×D7—22no—
366C7×Q8—24no—
367C7×Q8—24no—
368C7×Q8—22no—
369C7×Q8—42no—
370C7×Q8—42no—
371C7×Q8—84no—
372C7×Q8—81no—
373C7×Q82−1+414yes—
374C2×C4×C7—24no—
375C2×C4×C7—22no—
376C2×C4×C7—22no—
377C2×C4×C7—22no—
378C2×C4×C7—22no—
379C2×C4×C7—22no—
380C2×C4×C7—22no—
381C2×C4×C7—41no—
382C2×C4×C7—81no—
383C2×C4×C7C22×D414yes—
384C2×C4×C7C22×D411yescharacteristic
385C2×C4×C7C22×D411yescharacteristic
386C2×C4×C7C22×D414yes—
387C2×C4×C7C22×D411yescharacteristic
388C2×C4×C7C22×D411yescharacteristic
389C2×C4×C7C22×D411yescharacteristic
390C2×C4×C7C2511yescharacteristic
391C22⋊D7—82no—
392C2×Dic7—22no—
393C2×Dic7—22no—
394C2×Dic7—22no—
395D28—24no—
396D28—42no—
397D28—81no—
398C4×D7—44no—
399C4×D7—84no—
400Dic14—22no—
401Dic14—22no—
402Dic14—42no—
403Dic14—81no—
404Dic14—84no—
405C7⋊C8—24no—
406C7⋊C8—22no—
407C7⋊C8—22no—
408C2×#41—141no—
409C22×Q8—142no—
410C22×Q8C22×D711yescharacteristic
411C2×Q16—72no—
412C2×Q16—72no—
413C2×SD16—74no—
414C4⋊Q8—22no—
415C4⋊Q8—24no—
416C4⋊Q8—22no—
417C4⋊Q8—24no—
418C4⋊Q8—22no—
419C4⋊Q8—71no—
420C4⋊Q8—71no—
421C4⋊Q8—74no—
422C4⋊Q8—71no—
423C4⋊Q8—144no—
424C4⋊Q8—142no—
425C4⋊Q8C22⋊D711yescharacteristic
426C4⋊Q8C22⋊D711yescharacteristic
427C4⋊Q8C22×D711yescharacteristic
428C4⋊Q8C22×D711yescharacteristic
429C4⋊Q8C22×D714yes—
430#127—71no—
431#126—144no—
432#126—144no—
433#125—24no—
434#125—24no—
435#125—24no—
436#125—22no—
437#125—141no—
438#125C22⋊D712yes—
439#125C22×D712yes—
440#125C22×D714yes—
441#124—74no—
442#124—142no—
443#124—144no—
444#124—141no—
445#123—144no—
446#123—144no—
447C22⋊Q8—21no—
448C22⋊Q8—21no—
449C22⋊Q8—24no—
450C22⋊Q8—72no—
451C22⋊Q8—142no—
452C22⋊Q8—141no—
453C22⋊Q8—142no—
454C22⋊Q8—144no—
455C22⋊Q8—144no—
456C22⋊Q8C22⋊D712yes—
457C22⋊Q8C22⋊D712yes—
458#121—72no—
459#121—141no—
460C4×Q8—22no—
461C4×Q8—22no—
462C4×Q8—22no—
463C4×Q8—24no—
464C4×Q8—22no—
465C4×Q8—21no—
466C4×Q8—21no—
467C4×Q8—21no—
468C4×Q8—21no—
469C4×Q8—72no—
470C4×Q8—74no—
471C4×Q8—142no—
472C4×Q8—141no—
473C4×Q8C22×D712yes—
474C4×Q8C22×D712yes—
475C4×Q8C22×D714yes—
476C4×Q8C22×D714yes—
477C4×Q8C22×D712yes—
478C4×Q8C22×D712yes—
479C4×D4—72no—
480C4×D4—74no—
481C4×D4—142no—
482C4×D4—141no—
483#117—24no—
484#117—141no—
485#117—142no—
486#117C22×D712yes—
487#117C22×D712yes—
488C2×C4⋊C4—21no—
489C2×C4⋊C4C22×D711yescharacteristic
490C2×C42—21no—
491C2×C42C22×D711yescharacteristic
492#107—72no—
493#107—74no—
494#107—72no—
495#107—74no—
496#107—74no—
497#106—74no—
498#106—74no—
499C4⋊C8—72no—
500C4⋊C8—72no—
501C4⋊C8—74no—
502C4⋊C8—74no—
503#103—72no—
504#103—72no—
505#103—74no—
506#103—72no—
507#103—72no—
508#103—74no—
509#103—74no—
510#102—74no—
511#102—74no—
512#102—74no—
513C22⋊C8—72no—
514C22⋊C8—72no—
515#97—72no—
516#97—74no—
517C4×C8—71no—
518C4×C8—71no—
519C4×C8—74no—
520C22×C7—21no—
521C22×C7—21no—
522C22×C7—21no—
523C22×C7C23×D411yescharacteristic
524C2×D7—42no—
525C2×D7—42no—
526C2×D7—42no—
527C4×C7—21no—
528C4×C7—24no—
529C4×C7—21no—
530C4×C7—24no—
531C4×C7—21no—
532C4×C7—21no—
533C4×C7—24no—
534C4×C7—42no—
535C4×C7—42no—
536C4×C7—42no—
537C4×C7#57812yes—
538C4×C7C2×2−1+412yes—
539Dic7—42no—
540Dic7—42no—
541Dic7—42no—
542#41—562no—
543C2×Q8—24no—
544C2×Q8—21no—
545C2×Q8—21no—
546C2×Q8—21no—
547C2×Q8—24no—
548C2×Q8—72no—
549C2×Q8—141no—
550C2×Q8—144no—
551C2×Q8—141no—
552C2×Q8—564no—
553C2×Q8C2×C22⋊D712yes—
554C2×Q8C2×C22⋊D712yes—
555C2×Q8C23×D712yes—
556C2×D4—72no—
557C2×D4—142no—
558C2×D4—141no—
559C2×D4—141no—
560C22×C4—21no—
561C22×C4—21no—
562C22×C4—142no—
563C22×C4—144no—
564C22×C4C2×C22⋊D711yescharacteristic
565C22×C4C2×C22⋊D711yescharacteristic
566C22×C4C23×D711yescharacteristic
567Q16—144no—
568Q16—144no—
569SD16—148no—
570C2×C8—72no—
571C2×C8—72no—
572C2×C8—74no—
573C4⋊C4—22no—
574C4⋊C4—22no—
575C4⋊C4—21no—
576C4⋊C4—24no—
577C4⋊C4—24no—
578C4⋊C4—22no—
579C4⋊C4—21no—
580C4⋊C4—21no—
581C4⋊C4—24no—
582C4⋊C4—24no—
583C4⋊C4—22no—
584C4⋊C4—24no—
585C4⋊C4—72no—
586C4⋊C4—72no—
587C4⋊C4—144no—
588C4⋊C4—142no—
589C4⋊C4—142no—
590C4⋊C4—141no—
591C4⋊C4—141no—
592C4⋊C4—141no—
593C4⋊C4—141no—
594C4⋊C4—144no—
595C4⋊C4C2×C22⋊D712yes—
596C4⋊C4C2×C22⋊D712yes—
597C4⋊C4C2×C22⋊D712yes—
598C4⋊C4C2×C22⋊D712yes—
599C4⋊C4C2×C22⋊D714yes—
600C4⋊C4C2×C22⋊D714yes—
601C4⋊C4C23×D712yes—
602C4⋊C4C23×D712yes—
603C4⋊C4C23×D712yes—
604C22⋊C4—22no—
605C22⋊C4—142no—
606C22⋊C4—144no—
607C22⋊C4—142no—
608C22⋊C4—144no—
609C22⋊C4—142no—
610C22⋊C4—144no—
611C22⋊C4C2×C22⋊D714yes—
612C42—22no—
613C42—22no—
614C42—24no—
615C42—24no—
616C42—22no—
617C42—142no—
618C42—144no—
619C42C2×C22⋊D711yescharacteristic
620C42C2×C22⋊D711yescharacteristic
621C42C23×D711yescharacteristic
622C42C23×D714yes—
623C42C23×D711yescharacteristic
624C2×C7—41no—
625C2×C7#348611yescharacteristic
626C2×C7C2×#57811yescharacteristic
627C2×C7#358511yescharacteristic
628D7—82no—
629C23—142no—
630C23C22×C22⋊D711yescharacteristic
631Q8—24no—
632Q8—24no—
633Q8—22no—
634Q8—42no—
635Q8—42no—
636Q8—81no—
637Q8—84no—
638Q8—142no—
639Q8—142no—
640Q8—282no—
641Q8—564no—
642Q8—561no—
643Q8#655114yes—
644D4—144no—
645D4—282no—
646D4—561no—
647D4—562no—
648C2×C4—24no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—41no—
656C2×C4—81no—
657C2×C4—142no—
658C2×C4—142no—
659C2×C4—142no—
660C2×C4—284no—
661C2×C4—564no—
662C2×C4C22×C22⋊D711yescharacteristic
663C2×C4C22×C22⋊D711yescharacteristic
664C2×C4C22×C22⋊D714yes—
665C2×C4C22×C22⋊D714yes—
666C2×C4C22×C22⋊D711yescharacteristic
667C2×C4C22×C22⋊D711yescharacteristic
668C2×C4C22×C22⋊D711yescharacteristic
669C2×C4C24×D711yescharacteristic
670C8—144no—
671C8—142no—
672C8—142no—
673C7#6185411yescharacteristic, complemented, Sylow
674C22—21no—
675C22—21no—
676C22—21no—
677C22—282no—
678C22—282no—
679C22—282no—
680C22C23×C22⋊D711yescharacteristic
681C4—24no—
682C4—21no—
683C4—21no—
684C4—24no—
685C4—21no—
686C4—21no—
687C4—24no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—282no—
692C4—282no—
693C4—282no—
694C4#9052912yes—
695C4C2×#655112yes—
696C2—41no—
697C2—562no—
698C2#1174431911yescharacteristic
699C2C2×#9052911yescharacteristic
700C2#1174436611yescharacteristic
subgroupG/H
centerC22C23×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC7×#3502C2
socleC22×C7C23×D4
prime radical#90168C22
hypercenter#3502D7
hypocenterC7#61854
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #533 ◁ #3502

Complements

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

NH ≅ G/Ntype
1. C7×#3502C2semidirect
3. #11741153C2semidirect
4. Q8×#1144C2semidirect
5. #11741145C2semidirect
6. #11741142C2semidirect
7. #11741139C2semidirect
8. #11741128C2semidirect
9. #11740923C2semidirect
10. #11740817C2semidirect
12. #11740776C2semidirect
13. #11739997C2semidirect
14. #11739892C2semidirect
199. #3502D7semidirect
673. C7#61854semidirect

Maximal subgroups

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

subgroupindex
C7×#35022
#117433352
#117411532
Q8×#11442
#117411452
#117411422
#117411392
#117411282
#117409232
#117408172
#117408042
#117407762
#117399972
#117398922
#117398892
#618547