Finite Group Databasebeta

← #14244149

Subgroups — #14244149

Order 1792.

1660 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3501C211yesmaximal, characteristic
2#11743338C211yesmaximal, characteristic, complemented
3#11741089C214yesmaximal, complemented
4#11741089C212yesmaximal, complemented
5#11741077C214yesmaximal, complemented
6#11741074C212yesmaximal, complemented
7#11741013C212yesmaximal, complemented
8#11741003C212yesmaximal, complemented
9#11740926C214yesmaximal, complemented
10#11740817C211yesmaximal, characteristic, complemented
11#11740816C211yesmaximal, characteristic, complemented
12#11740776C211yesmaximal, characteristic, complemented
13#11739990C214yesmaximal
14#11739760C211yesmaximal, characteristic
15#11739759C211yesmaximal, characteristic
16C7×#554—21no—
17C7×#554—22no—
18C7×#554C2212yes—
19C7×#554C2214yes—
20C7×D4×Q8—21no—
21C7×D4×Q8C2212yes—
22C7×#544C2212yes—
23C7×#543—22no—
24C7×#533C2211yescharacteristic
25C7×#533C2211yescharacteristic
26C2×C7×C22⋊Q8—22no—
27C7×#518C2212yes—
28C2×C4×C7×D4C2211yescharacteristic
29#90412—21no—
30#90412—21no—
31C7⋊#564—22no—
32#90316—22no—
33#90316—22no—
34#90261C2212yescomplemented
35#90261C2212yescomplemented
36#90258C2212yescomplemented
37#90258C2212yescomplemented
38#90209C2214yes—
39#90161C2211yescharacteristic
40#90161C2211yescharacteristic
41#89990C2212yes—
42#89990C2212yes—
43#89866C2212yescomplemented
44#89866C2214yescomplemented
45#89865C2211yescharacteristic, complemented
46#89865C2214yescomplemented
47#89865C2211yescharacteristic, complemented
48#89863C2214yescomplemented
49#89863C2212yescomplemented
50#89852C2214yescomplemented
51#89851C2211yescharacteristic, complemented
52#89850C2211yescharacteristic, complemented
53#89841C2212yescomplemented
54#89841C2214yescomplemented
55#89837C2212yescomplemented
56#89836C2214yescomplemented
57#89835C2214yescomplemented
58#89835C2214yescomplemented
59#89835C2214yescomplemented
60#89796C2212yescomplemented
61#89796C2214yescomplemented
62Q8×C7⋊C8C2212yescomplemented
63#89791C2214yes—
64#89791C2212yes—
65#89791C2214yes—
66#89791C2212yes—
67#89790C2214yes—
68#89790C2214yes—
69#89790C2214yes—
70C4×#1142C2212yes—
71C4×#1142C2214yes—
72C4×#1142C2212yes—
73C4×#1142C2214yes—
74#89781C2214yes—
75#89781C2214yes—
76#89780C2212yes—
77#89779C2212yes—
78#89772C2214yes—
79#89768C2214yes—
80#89768C2214yes—
81#89743C2212yes—
82#89742C2212yes—
83#89698C2211yescharacteristic
84#89697C2211yescharacteristic
85#61848—71nomaximal, Sylow
86C22×C7×Q8—22no—
87C22×C7×D4C2311yescharacteristic
88C23×C4×C7—21no—
89C7×C4⋊Q8—22no—
90C7×C4⋊Q8D411yescharacteristic
91C7×C4⋊Q8D411yescharacteristic
92C7×C4⋊Q8C2311yescharacteristic
93C7×C4⋊Q8C2314yes—
94C7×C4⋊Q8C2311yescharacteristic
95C7×#125—22no—
96C7×#125D412yes—
97C7×#125C2312yes—
98C7×#125C2314yes—
99C7×C22⋊Q8—22no—
100C7×C22⋊Q8—24no—
101C7×C22⋊Q8—21no—
102C7×C22⋊Q8—21no—
103C7×C22⋊Q8—22no—
104C7×C22⋊Q8—24no—
105C7×C22⋊Q8—24no—
106C7×C22⋊Q8—24no—
107C7×C22⋊Q8—24no—
108C7×C22⋊Q8—24no—
109C7×C22⋊Q8D412yes—
110C7×C22⋊Q8D412yes—
111C4×C7×Q8—21no—
112C4×C7×Q8—21no—
113C4×C7×Q8C2312yes—
114C4×C7×Q8C2312yes—
115C4×C7×D4—22no—
116C4×C7×D4—22no—
117C4×C7×D4—21no—
118C4×C7×D4—21no—
119C4×C7×D4C2312yes—
120C4×C7×D4C2312yes—
121C4×C7×D4C2314yes—
122C4×C7×D4C2314yes—
123C7×#117C2312yes—
124C7×#117C2312yes—
125C2×C7×C4⋊C4—24no—
126C2×C7×C4⋊C4—22no—
127C2×C7×C4⋊C4C2311yescharacteristic
128C2×C7×C22⋊C4—21no—
129C2×C7×C22⋊C4—24no—
130C2×C42×C7C2311yescharacteristic
131Q8×Dic7—22no—
132Q8×Dic7—22no—
133#6507—22no—
134#6507—22no—
135C2×#1142C2312yes—
136C2×#1142C2312yes—
137C2×#1142C2314yes—
138#6489—21no—
139#6489—21no—
140#6487—21no—
141#6487—21no—
142#6487C2312yes—
143#6487C2312yes—
144#6453—24no—
145#6453—24no—
146#6453—22no—
147#6453—22no—
148#6452—24no—
149#6452—24no—
150#6452—24no—
151#6452—24no—
152#6450—22no—
153#6450—22no—
154#6442—24no—
155#6442—24no—
156#6433—21no—
157#6433—21no—
158#6433C2314yes—
159#6432C2311yescharacteristic
160#6432C2314yes—
161#6432C2311yescharacteristic
162#6432C2311yescharacteristic
163#6432C2311yescharacteristic
164C4×Dic14—21no—
165C4×Dic14—21no—
166C4×Dic14C2312yes—
167C4×Dic14C2314yes—
168C4×Dic14C2314yes—
169C4×Dic14C2312yes—
170#6406C2314yes—
171#6406C2312yes—
172#6406C2312yes—
173C22⋊(C7⋊C8)C2312yes—
174C22⋊(C7⋊C8)C2312yes—
175#6384C2312yes—
176#6384C2314yes—
177#6384C2314yes—
178#6384C2314yes—
179#6384C2312yes—
180#6384C2314yes—
181#6384C2314yes—
182#6382C2312yes—
183#6382C2314yes—
184#6382C2312yes—
185#6382C2314yes—
186#6381C2314yes—
187#6381C2314yes—
188#6381C2314yes—
189C4⋊(C7⋊C8)C2312yes—
190C4⋊(C7⋊C8)C2314yes—
191C4⋊(C7⋊C8)C2314yes—
192C4⋊(C7⋊C8)C2312yes—
193#6377C2312yes—
194#6377C2314yes—
195C4×C7⋊C8C2311yescharacteristic
196C4×C7⋊C8C2311yescharacteristic
197C4×C7⋊C8C2314yes—
198#3556—71no—
199#3501D711yescharacteristic
200#3388—74no—
201#3386—72no—
202#3381—72no—
203#3372—72no—
204#3372—74no—
205#3359—72no—
206#3242—74no—
207#3237—71no—
208#3231—71no—
209#3123—74no—
210#3110—71no—
211#3101—71no—
212#2985—71no—
213C24×C7—21no—
214C2×C7×Q8—21no—
215C2×C7×Q8—21no—
216C2×C7×Q8—24no—
217C2×C7×Q8—84no—
218C2×C7×Q8C2×D412yes—
219C2×C7×Q8C2×D412yes—
220C2×C7×D4—24no—
221C2×C7×D4—21no—
222C2×C7×D4C2412yes—
223C22×C4×C7—21no—
224C22×C4×C7—22no—
225C22×C4×C7—24no—
226C22×C4×C7—24no—
227C22×C4×C7—22no—
228C22×C4×C7—82no—
229C22×C4×C7C2×D411yescharacteristic
230C22×C4×C7C2×D411yescharacteristic
231C22×C4×C7C2411yescharacteristic
232C2×Dic14—21no—
233C2×Dic14—21no—
234C2×Dic14C2412yes—
235C2×Dic14C2412yes—
236C7×C4⋊C4—22no—
237C7×C4⋊C4—21no—
238C7×C4⋊C4—24no—
239C7×C4⋊C4—24no—
240C7×C4⋊C4—24no—
241C7×C4⋊C4—21no—
242C7×C4⋊C4—22no—
243C7×C4⋊C4—21no—
244C7×C4⋊C4C2×D412yes—
245C7×C4⋊C4C2×D412yes—
246C7×C4⋊C4C2×D412yes—
247C7×C4⋊C4C2×D412yes—
248C7×C4⋊C4C2×D414yes—
249C7×C4⋊C4C2×D414yes—
250C7×C4⋊C4C2412yes—
251C7×C4⋊C4C2412yes—
252C7×C4⋊C4C2412yes—
253C7×C22⋊C4—22no—
254C7×C22⋊C4—24no—
255C7×C22⋊C4—24no—
256C7×C22⋊C4—22no—
257C7×C22⋊C4—22no—
258C7×C22⋊C4C2×D414yes—
259C42×C7—22no—
260C42×C7C2×D411yescharacteristic
261C42×C7C2×D411yescharacteristic
262C42×C7C2411yescharacteristic
263C42×C7C2411yescharacteristic
264C42×C7C2414yes—
265C22⋊Dic7—22no—
266C22⋊Dic7—22no—
267#1142—24no—
268#1142—24no—
269#1142—28no—
270C4⋊Dic7—21no—
271C4⋊Dic7—21no—
272C4⋊Dic7—21no—
273C4⋊Dic7—21no—
274C4⋊Dic7—21no—
275C4⋊Dic7—21no—
276C4⋊Dic7—24no—
277C4⋊Dic7—24no—
278C4⋊Dic7C2412yes—
279C4⋊Dic7C2412yes—
280Dic7⋊C4—22no—
281Dic7⋊C4—22no—
282Dic7⋊C4—22no—
283Dic7⋊C4—22no—
284Dic7⋊C4—24no—
285Dic7⋊C4—24no—
286Dic7⋊C4—24no—
287Dic7⋊C4—24no—
288C4×Dic7—22no—
289C4×Dic7—22no—
290C4×Dic7—24no—
291C4×Dic7—24no—
292C2×C7⋊C8C2412yes—
293C2×C7⋊C8C2412yes—
294C2×C7⋊C8C2414yes—
295#564—142no—
296#557—72no—
297#557—72no—
298#557—72no—
299#557—72no—
300#557—141no—
301#557—141no—
302#554—21no—
303#554—22no—
304#554C2×D712yes—
305#554C2×D714yes—
306D4×Q8—21no—
307D4×Q8C2×D712yes—
308#544C2×D712yes—
309#543—22no—
310#541—74no—
311#541—142no—
312#541—142no—
313#533—71no—
314#533—71no—
315#533C2×D711yescharacteristic
316#533C2×D711yescharacteristic
317C2×C22⋊Q8—22no—
318#518C2×D712yes—
319C2×C4×D4C2×D711yescharacteristic
320#501—74no—
321#501—72no—
322#500—71no—
323#499—74no—
324#498—71no—
325#491—72no—
326#491—74no—
327#487—71no—
328#487—74no—
329#487—71no—
330#485—74no—
331#485—74no—
332#484—72no—
333#483—72no—
334#479—74no—
335#479—74no—
336#479—74no—
337#478—74no—
338#478—74no—
339#477—72no—
340#477—74no—
341#476—72no—
342#475—72no—
343#475—74no—
344#474—72no—
345#461—72no—
346#461—72no—
347#461—74no—
348#461—74no—
349#450—72no—
350#450—72no—
351#446—72no—
352#446—74no—
353C8×Q8—72no—
354#440—74no—
355#440—74no—
356#440—74no—
357C4×SD16—72no—
358C4×SD16—72no—
359C4×SD16—74no—
360C4×SD16—74no—
361#433—74no—
362#432—71no—
363#431—71no—
364C23×C7—24no—
365C23×C7—22no—
366C23×C7—81no—
367C23×C7—81no—
368C23×C7C22×D411yescharacteristic
369C7×Q8—24no—
370C7×Q8—42no—
371C7×Q8—42no—
372C7×Q8—81no—
373C7×Q8—84no—
374C7×D4—24no—
375C7×D4—22no—
376C7×D42−1+414yes—
377C2×C4×C7—22no—
378C2×C4×C7—22no—
379C2×C4×C7—22no—
380C2×C4×C7—22no—
381C2×C4×C7—22no—
382C2×C4×C7—41no—
383C2×C4×C7—44no—
384C2×C4×C7—84no—
385C2×C4×C7—82no—
386C2×C4×C7—84no—
387C2×C4×C7—81no—
388C2×C4×C7C22×D411yescharacteristic
389C2×C4×C7C22×D414yes—
390C2×C4×C7C22×D414yes—
391C2×C4×C7C22×D411yescharacteristic
392C2×C4×C7C22×D411yescharacteristic
393C2×C4×C7C22×D411yescharacteristic
394C2×C4×C7C22×D411yescharacteristic
395C2×C4×C7C2511yescharacteristic
396C2×Dic7—22no—
397C2×Dic7—22no—
398C2×Dic7—22no—
399C2×Dic7—22no—
400Dic14—22no—
401Dic14—22no—
402Dic14—24no—
403Dic14—42no—
404Dic14—42no—
405C7⋊C8—22no—
406C7⋊C8—22no—
407C7⋊C8—24no—
408C22×Q8—22no—
409C22×D4C22×D711yescharacteristic
410C23×C4—21no—
411C2×SD16—72no—
412C2×SD16—72no—
413C2×SD16—74no—
414C4⋊Q8—22no—
415C4⋊Q8—71no—
416C4⋊Q8—71no—
417C4⋊Q8—71no—
418C4⋊Q8—71no—
419C4⋊Q8—74no—
420C4⋊Q8—142no—
421C4⋊Q8—142no—
422C4⋊Q8C22⋊D711yescharacteristic
423C4⋊Q8C22⋊D711yescharacteristic
424C4⋊Q8C22×D711yescharacteristic
425C4⋊Q8C22×D711yescharacteristic
426C4⋊Q8C22×D714yes—
427#126—144no—
428#126—144no—
429#125—22no—
430#125—74no—
431#125—144no—
432#125—144no—
433#125—142no—
434#125—142no—
435#125—144no—
436#125—144no—
437#125—144no—
438#125—144no—
439#125—141no—
440#125—141no—
441#125C22⋊D712yes—
442#125C22×D712yes—
443#125C22×D714yes—
444C22⋊Q8—22no—
445C22⋊Q8—24no—
446C22⋊Q8—21no—
447C22⋊Q8—21no—
448C22⋊Q8—22no—
449C22⋊Q8—24no—
450C22⋊Q8—24no—
451C22⋊Q8—24no—
452C22⋊Q8—24no—
453C22⋊Q8—24no—
454C22⋊Q8—72no—
455C22⋊Q8—72no—
456C22⋊Q8—141no—
457C22⋊Q8—141no—
458C22⋊Q8C22⋊D712yes—
459C22⋊Q8C22⋊D712yes—
460C4×Q8—21no—
461C4×Q8—21no—
462C4×Q8—72no—
463C4×Q8—72no—
464C4×Q8—74no—
465C4×Q8—74no—
466C4×Q8—142no—
467C4×Q8—142no—
468C4×Q8—142no—
469C4×Q8—142no—
470C4×Q8—141no—
471C4×Q8—141no—
472C4×Q8C22×D712yes—
473C4×Q8C22×D712yes—
474C4×D4—22no—
475C4×D4—22no—
476C4×D4—21no—
477C4×D4—21no—
478C4×D4C22×D712yes—
479C4×D4C22×D712yes—
480C4×D4C22×D714yes—
481C4×D4C22×D714yes—
482#117—141no—
483#117—141no—
484#117C22×D712yes—
485#117C22×D712yes—
486C2×C4⋊C4—24no—
487C2×C4⋊C4—22no—
488C2×C4⋊C4C22×D711yescharacteristic
489C2×C22⋊C4—21no—
490C2×C22⋊C4—24no—
491C2×C42C22×D711yescharacteristic
492#107—74no—
493#107—74no—
494#107—74no—
495#106—72no—
496#106—74no—
497#106—72no—
498#106—74no—
499C4⋊C8—72no—
500C4⋊C8—72no—
501C4⋊C8—74no—
502C4⋊C8—74no—
503#103—74no—
504#103—72no—
505#103—74no—
506#103—72no—
507#103—74no—
508#103—74no—
509#103—74no—
510#102—72no—
511#102—74no—
512#102—72no—
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—24no—
523C22×C7—21no—
524C22×C7—42no—
525C22×C7—42no—
526C22×C7—84no—
527C22×C7C23×D411yescharacteristic
528C4×C7—21no—
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#57712yes—
538C4×C7C2×2−1+412yes—
539Dic7—42no—
540Dic7—42no—
541Dic7—42no—
542Dic7—42no—
543C24—21no—
544C2×Q8—21no—
545C2×Q8—21no—
546C2×Q8—24no—
547C2×Q8—72no—
548C2×Q8—72no—
549C2×Q8—84no—
550C2×Q8—141no—
551C2×Q8—141no—
552C2×Q8C2×C22⋊D712yes—
553C2×Q8C2×C22⋊D712yes—
554C2×D4—24no—
555C2×D4—21no—
556C2×D4C23×D712yes—
557C22×C4—21no—
558C22×C4—22no—
559C22×C4—24no—
560C22×C4—24no—
561C22×C4—22no—
562C22×C4—82no—
563C22×C4C2×C22⋊D711yescharacteristic
564C22×C4C2×C22⋊D711yescharacteristic
565C22×C4C23×D711yescharacteristic
566SD16—144no—
567SD16—144no—
568SD16—148no—
569C2×C8—72no—
570C2×C8—72no—
571C2×C8—74no—
572C4⋊C4—22no—
573C4⋊C4—21no—
574C4⋊C4—24no—
575C4⋊C4—24no—
576C4⋊C4—24no—
577C4⋊C4—21no—
578C4⋊C4—22no—
579C4⋊C4—21no—
580C4⋊C4—72no—
581C4⋊C4—72no—
582C4⋊C4—142no—
583C4⋊C4—142no—
584C4⋊C4—144no—
585C4⋊C4—144no—
586C4⋊C4—142no—
587C4⋊C4—142no—
588C4⋊C4—144no—
589C4⋊C4—144no—
590C4⋊C4—141no—
591C4⋊C4—141no—
592C4⋊C4—141no—
593C4⋊C4—141no—
594C4⋊C4—141no—
595C4⋊C4—141no—
596C4⋊C4—144no—
597C4⋊C4—144no—
598C4⋊C4C2×C22⋊D712yes—
599C4⋊C4C2×C22⋊D714yes—
600C4⋊C4C2×C22⋊D712yes—
601C4⋊C4C2×C22⋊D712yes—
602C4⋊C4C2×C22⋊D712yes—
603C4⋊C4C2×C22⋊D714yes—
604C4⋊C4C23×D712yes—
605C4⋊C4C23×D712yes—
606C4⋊C4C23×D712yes—
607C22⋊C4—22no—
608C22⋊C4—24no—
609C22⋊C4—24no—
610C22⋊C4—22no—
611C22⋊C4—22no—
612C22⋊C4—142no—
613C22⋊C4—142no—
614C22⋊C4C2×C22⋊D714yes—
615C42—22no—
616C42—142no—
617C42—142no—
618C42—144no—
619C42—144no—
620C42C2×C22⋊D711yescharacteristic
621C42C2×C22⋊D711yescharacteristic
622C42C23×D711yescharacteristic
623C42C23×D714yes—
624C42C23×D711yescharacteristic
625C2×C7—41no—
626C2×C7—44no—
627C2×C7#348611yescharacteristic
628C2×C7C2×#57711yescharacteristic
629C2×C7#358611yescharacteristic
630C23—24no—
631C23—22no—
632C23—81no—
633C23—81no—
634C23C22×C22⋊D711yescharacteristic
635Q8—24no—
636Q8—42no—
637Q8—42no—
638Q8—81no—
639Q8—84no—
640Q8—142no—
641Q8—144no—
642Q8—142no—
643Q8—282no—
644Q8—282no—
645D4—24no—
646D4—22no—
647D4#655114yes—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—41no—
654C2×C4—44no—
655C2×C4—84no—
656C2×C4—82no—
657C2×C4—84no—
658C2×C4—81no—
659C2×C4—142no—
660C2×C4—142no—
661C2×C4—142no—
662C2×C4—142no—
663C2×C4C22×C22⋊D711yescharacteristic
664C2×C4C22×C22⋊D711yescharacteristic
665C2×C4C22×C22⋊D714yes—
666C2×C4C22×C22⋊D714yes—
667C2×C4C22×C22⋊D711yescharacteristic
668C2×C4C22×C22⋊D711yescharacteristic
669C2×C4C22×C22⋊D711yescharacteristic
670C2×C4C24×D711yescharacteristic
671C8—142no—
672C8—142no—
673C8—144no—
674C7#6184811yescharacteristic, complemented, Sylow
675C22—21no—
676C22—24no—
677C22—21no—
678C22—21no—
679C22—42no—
680C22—42no—
681C22—84no—
682C22C23×C22⋊D711yescharacteristic
683C4—21no—
684C4—21no—
685C4—24no—
686C4—21no—
687C4—21no—
688C4—24no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—282no—
693C4—282no—
694C4—282no—
695C4—282no—
696C4#9052712yes—
697C4C2×#655112yes—
698C2—41no—
699C2—44no—
700C2#1174431911yescharacteristic
701C2C2×#9052711yescharacteristic
702C2#1174436711yescharacteristic
subgroupG/H
centerC22C23×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC7×#3501C2
socleC22×C7C23×D4
prime radicalC22×C7×D4C23
hypercenter#3501D7
hypocenterC7#61848
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #533 ◁ #3501

Complements

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

NH ≅ G/Ntype
2. #11743338C2semidirect
3. #11741089C2semidirect
4. #11741089C2semidirect
5. #11741077C2semidirect
6. #11741074C2semidirect
7. #11741013C2semidirect
8. #11741003C2semidirect
9. #11740926C2semidirect
10. #11740817C2semidirect
11. #11740816C2semidirect
12. #11740776C2semidirect
34. #90261C22semidirect
35. #90261C22semidirect
36. #90258C22semidirect
37. #90258C22semidirect
43. #89866C22semidirect
44. #89866C22semidirect
45. #89865C22semidirect
46. #89865C22semidirect
47. #89865C22semidirect
48. #89863C22semidirect
49. #89863C22semidirect
50. #89852C22semidirect
51. #89851C22semidirect
52. #89850C22semidirect
53. #89841C22semidirect
54. #89841C22semidirect
55. #89837C22semidirect
56. #89836C22semidirect
57. #89835C22semidirect
58. #89835C22semidirect
59. #89835C22semidirect
60. #89796C22semidirect
61. #89796C22semidirect
62. Q8×C7⋊C8C22semidirect
674. C7#61848semidirect

Maximal subgroups

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

subgroupindex
C7×#35012
#117433382
#117410892
#117410772
#117410742
#117410132
#117410032
#117409262
#117408172
#117408162
#117407762
#117399902
#117397602
#117397592
#618487