Finite Group Databasebeta

← #14244232

Subgroups — #14244232

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×#3550C211yesmaximal, characteristic
2#11743326C211yesmaximal, characteristic, complemented
3#11741164C212yesmaximal, complemented
4#11741162C212yesmaximal, complemented
5#11741154C212yesmaximal, complemented
6Q8×#1144C212yesmaximal, complemented
7#11741123C214yesmaximal, complemented
8#11741003C214yesmaximal, complemented
9#11740912C214yesmaximal, complemented
10#11740899C214yesmaximal, complemented
11#11740813C211yesmaximal, characteristic, complemented
12#11740776C211yesmaximal, characteristic, complemented
13#11740772C211yesmaximal, characteristic, complemented
14#11740761C211yesmaximal, characteristic
15#11740717C211yesmaximal, characteristic
16C7×#563—22no—
17C7×#559C2212yes—
18C7×Q82C2212yes—
19C7×#556—21no—
20C7×#556C2212yes—
21C7×D4×Q8—21no—
22C7×D4×Q8C2212yes—
23C7×#542—22no—
24C7×#542C2214yes—
25C7×#536—22no—
26C7×#533C2211yescharacteristic
27C7×#533C2211yescharacteristic
28C7×#532C2211yescharacteristic
29#90413—21no—
30#90412—21no—
31#90376—22no—
32#90314—22no—
33#90313—22no—
34#90258C2214yes—
35#90258C2212yes—
36Q8×Dic14C2212yes—
37#90210C2212yes—
38#90207C2212yes—
39#90161C2211yescharacteristic
40C2×C4×Dic14C2211yescharacteristic
41#89866C2214yes—
42#89863C2212yes—
43#89863C2214yes—
44#89863C2212yes—
45#89862C2212yes—
46#89854C2212yes—
47#89852C2212yes—
48#89852C2214yes—
49#89851C2211yescharacteristic
50#89850C2211yescharacteristic
51#89850C2214yes—
52#89847C2212yes—
53#89842C2211yescharacteristic
54#89842C2211yescharacteristic
55#89828C2214yes—
56#89827C2214yes—
57#89826C2214yes—
58#89826C2214yes—
59#89822C2212yes—
60#89817C2214yes—
61#89816C2212yes—
62#89812C2212yes—
63#89807C2214yes—
64#89805C2212yes—
65#89802C2214yes—
66C4×#1144C2212yes—
67C4×#1144C2212yes—
68C4×#1144C2214yes—
69#89796C2214yes—
70#89796C2212yes—
71Q8×C7⋊C8C2212yes—
72#89794C2214yes—
73#89793C2212yes—
74#89793C2214yes—
75#89792C2212yes—
76#89790C2214yes—
77#89790C2214yes—
78C4×#1142C2214yes—
79#89781C2214yes—
80#89781C2214yes—
81#89779C2214yes—
82#89772C2214yes—
83#89698C2211yescharacteristic
84#89697C2211yescharacteristic
85#61854—71nomaximal, Sylow
86C2×C7×#41—21no—
87C22×C7×Q8—22no—
88C22×Dic14C2311yescharacteristic
89C7×C4⋊Q8—24no—
90C7×C4⋊Q8—22no—
91C7×C4⋊Q8D411yescharacteristic
92C7×C4⋊Q8D411yescharacteristic
93C7×C4⋊Q8C2311yescharacteristic
94C7×C4⋊Q8C2311yescharacteristic
95C7×C4⋊Q8C2311yescharacteristic
96C7×C4⋊Q8C2314yes—
97C7×#127C2311yescharacteristic
98C7×#126—24no—
99C7×#126—24no—
100C7×#125—21no—
101C7×#125D412yes—
102C7×#124—22no—
103C7×#124—24no—
104C7×#124—21no—
105C7×#124C2314yes—
106C7×#123—24no—
107C7×#123—24no—
108C7×C22⋊Q8—22no—
109C7×C22⋊Q8—21no—
110C7×C22⋊Q8—22no—
111C7×C22⋊Q8—24no—
112C7×C22⋊Q8—24no—
113C7×C22⋊Q8D412yes—
114C7×C22⋊Q8D412yes—
115C7×C22⋊Q8C2312yes—
116C7×#121—21no—
117C7×#121C2312yes—
118C4×C7×Q8—22no—
119C4×C7×Q8—21no—
120C4×C7×Q8C2312yes—
121C4×C7×Q8C2314yes—
122C4×C7×Q8C2312yes—
123C4×C7×Q8C2312yes—
124C4×C7×D4—22no—
125C4×C7×D4—21no—
126C4×C7×D4C2314yes—
127C4×C7×D4C2312yes—
128C7×#117—21no—
129C7×#117—22no—
130C7×#117C2312yes—
131C2×C42×C7C2311yescharacteristic
132Q8×Dic7—22no—
133#6507—22no—
134#6507—24no—
135C2×#1144C2312yes—
136C2×#1144C2312yes—
137C2×#1142C2314yes—
138#6489C2312yes—
139C2×C4⋊Dic7C2311yescharacteristic
140#6487—21no—
141#6487—21no—
142C2×Dic7⋊C4—21no—
143C2×C4×Dic7—21no—
144#6453—22no—
145#6452—24no—
146#6452—24no—
147#6452—24no—
148#6451—22no—
149#6451—24no—
150#6451—22no—
151#6450—22no—
152#6450—22no—
153#6450—24no—
154#6450—22no—
155#6441—24no—
156#6440—24no—
157#6433C2312yes—
158#6433C2314yes—
159#6432C2311yescharacteristic
160#6432C2311yescharacteristic
161#6432C2314yes—
162C4×Dic14—21no—
163C4×Dic14—21no—
164C4×Dic14—21no—
165C4×Dic14—21no—
166C4×Dic14C2312yes—
167C4×Dic14C2312yes—
168C4×Dic14C2314yes—
169C4×Dic14C2314yes—
170#6409C2312yes—
171#6409C2312yes—
172#6409C2314yes—
173#6409C2314yes—
174#6406C2314yes—
175#6406C2314yes—
176#6406C2314yes—
177C22⋊(C7⋊C8)C2312yes—
178C22⋊(C7⋊C8)C2312yes—
179#6384C2312yes—
180#6384C2314yes—
181#6384C2312yes—
182#6382C2314yes—
183#6382C2314yes—
184#6381C2312yes—
185#6381C2314yes—
186#6381C2312yes—
187#6381C2314yes—
188#6381C2314yes—
189C4⋊(C7⋊C8)C2312yes—
190C4⋊(C7⋊C8)C2312yes—
191C4⋊(C7⋊C8)C2314yes—
192C4⋊(C7⋊C8)C2314yes—
193#6377C2312yes—
194#6377C2314yes—
195C4×C7⋊C8C2311yescharacteristic
196C4×C7⋊C8C2311yescharacteristic
197C4×C7⋊C8C2314yes—
198#3550D711yescharacteristic
199#3502—71no—
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—
213C7×#41—82no—
214C2×C7×Q8—21no—
215C2×C7×Q8—24no—
216C2×C7×Q8—21no—
217C2×C7×Q8—84no—
218C2×C7×Q8C2×D412yes—
219C2×C7×Q8C2×D412yes—
220C2×C7×Q8C2412yes—
221C2×C7×D4—22no—
222C2×C7×D4—21no—
223C2×C7×D4—21no—
224C2×C7×D4C2412yes—
225C22×C4×C7—22no—
226C22×C4×C7—24no—
227C22×C4×C7C2×D411yescharacteristic
228C22×C4×C7C2×D411yescharacteristic
229C22×C4×C7C2411yescharacteristic
230C22×Dic7—21no—
231C22×Dic7—21no—
232C2×Dic14—21no—
233C2×Dic14—24no—
234C2×Dic14—21no—
235C2×Dic14—21no—
236C2×Dic14—24no—
237C2×Dic14C2412yes—
238C7×C4⋊C4—21no—
239C7×C4⋊C4—21no—
240C7×C4⋊C4—21no—
241C7×C4⋊C4—21no—
242C7×C4⋊C4—22no—
243C7×C4⋊C4—24no—
244C7×C4⋊C4—24no—
245C7×C4⋊C4—22no—
246C7×C4⋊C4C2×D412yes—
247C7×C4⋊C4C2×D412yes—
248C7×C4⋊C4C2×D414yes—
249C7×C4⋊C4C2×D412yes—
250C7×C4⋊C4C2×D414yes—
251C7×C4⋊C4C2×D412yes—
252C7×C4⋊C4C2412yes—
253C7×C4⋊C4C2412yes—
254C7×C22⋊C4—22no—
255C7×C22⋊C4—24no—
256C7×C22⋊C4—22no—
257C7×C22⋊C4—24no—
258C7×C22⋊C4—22no—
259C7×C22⋊C4—24no—
260C7×C22⋊C4C2×D414yes—
261C42×C7—22no—
262C42×C7—24no—
263C42×C7C2×D411yescharacteristic
264C42×C7C2×D411yescharacteristic
265C42×C7C2411yescharacteristic
266C42×C7C2414yes—
267C42×C7C2411yescharacteristic
268C22⋊Dic7—22no—
269#1144—24no—
270#1144—24no—
271#1142—28no—
272C4⋊Dic7—21no—
273C4⋊Dic7—21no—
274C4⋊Dic7—21no—
275C4⋊Dic7—24no—
276C4⋊Dic7C2412yes—
277C4⋊Dic7C2412yes—
278C4⋊Dic7C2412yes—
279Dic7⋊C4—22no—
280Dic7⋊C4—22no—
281Dic7⋊C4—22no—
282Dic7⋊C4—22no—
283Dic7⋊C4—24no—
284Dic7⋊C4—24no—
285Dic7⋊C4—24no—
286Dic7⋊C4—24no—
287C4×Dic7—22no—
288C4×Dic7—22no—
289C4×Dic7—22no—
290C4×Dic7—24no—
291C4×Dic7—24no—
292C2×C7⋊C8C2412yes—
293C2×C7⋊C8C2412yes—
294C2×C7⋊C8C2414yes—
295#563—22no—
296#559C2×D712yes—
297Q82—72no—
298Q82—141no—
299Q82C2×D712yes—
300#557—72no—
301#557—74no—
302#557—141no—
303#557—142no—
304#556—21no—
305#556C2×D712yes—
306D4×Q8—21no—
307D4×Q8C2×D712yes—
308#544—72no—
309#544—142no—
310#542—22no—
311#542C2×D714yes—
312#536—22no—
313#533—71no—
314#533—142no—
315#533C2×D711yescharacteristic
316#533C2×D711yescharacteristic
317#532C2×D711yescharacteristic
318#519—72no—
319C2×C4×Q8—71no—
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—72no—
339#477—74no—
340#477—72no—
341#477—74no—
342#475—74no—
343#475—72no—
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×C7—22no—
365C23×C7C22×D411yescharacteristic
366C7×Q8—24no—
367C7×Q8—22no—
368C7×Q8—22no—
369C7×Q8—42no—
370C7×Q8—42no—
371C7×Q8—84no—
372C7×Q8—81no—
373C7×D4—24no—
374C7×D4—42no—
375C7×D4—81no—
376C7×D4—82no—
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×C7C22×D411yescharacteristic
386C2×C4×C7C22×D411yescharacteristic
387C2×C4×C7C22×D414yes—
388C2×C4×C7C22×D414yes—
389C2×C4×C7C22×D411yescharacteristic
390C2×C4×C7C22×D411yescharacteristic
391C2×C4×C7C22×D411yescharacteristic
392C2×C4×C7C2511yescharacteristic
393C2×Dic7—24no—
394C2×Dic7—22no—
395C2×Dic7—22no—
396C2×Dic7—22no—
397C2×Dic7—22no—
398C2×Dic7—81no—
399Dic14—24no—
400Dic14—22no—
401Dic14—42no—
402Dic14—81no—
403Dic14—84no—
404Dic142−1+414yes—
405C7⋊C8—24no—
406C7⋊C8—22no—
407C7⋊C8—22no—
408C2×#41—21no—
409C22×Q8—22no—
410C22×Q8—71no—
411C2×Q16—72no—
412C2×Q16—72no—
413C2×SD16—74no—
414C4⋊Q8—24no—
415C4⋊Q8—22no—
416C4⋊Q8—71no—
417C4⋊Q8—71no—
418C4⋊Q8—74no—
419C4⋊Q8—142no—
420C4⋊Q8—144no—
421C4⋊Q8—142no—
422C4⋊Q8—142no—
423C4⋊Q8—144no—
424C4⋊Q8C22⋊D711yescharacteristic
425C4⋊Q8C22⋊D711yescharacteristic
426C4⋊Q8C22×D711yescharacteristic
427C4⋊Q8C22×D711yescharacteristic
428C4⋊Q8C22×D711yescharacteristic
429C4⋊Q8C22×D714yes—
430#127C22×D711yescharacteristic
431#126—24no—
432#126—24no—
433#125—21no—
434#125—72no—
435#125—74no—
436#125—144no—
437#125—142no—
438#125—144no—
439#125—144no—
440#125C22⋊D712yes—
441#124—22no—
442#124—24no—
443#124—21no—
444#124C22×D714yes—
445#123—24no—
446#123—24no—
447C22⋊Q8—22no—
448C22⋊Q8—21no—
449C22⋊Q8—22no—
450C22⋊Q8—24no—
451C22⋊Q8—24no—
452C22⋊Q8—141no—
453C22⋊Q8—141no—
454C22⋊Q8—144no—
455C22⋊Q8C22⋊D712yes—
456C22⋊Q8C22⋊D712yes—
457C22⋊Q8C22×D712yes—
458#121—21no—
459#121C22×D712yes—
460C4×Q8—22no—
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—144no—
470C4×Q8—142no—
471C4×Q8—141no—
472C4×Q8—141no—
473C4×Q8—141no—
474C4×Q8—141no—
475C4×Q8C22×D712yes—
476C4×Q8C22×D714yes—
477C4×Q8C22×D712yes—
478C4×Q8C22×D712yes—
479C4×D4—22no—
480C4×D4—21no—
481C4×D4C22×D712yes—
482C4×D4C22×D714yes—
483#117—21no—
484#117—22no—
485#117—72no—
486#117—144no—
487#117C22×D712yes—
488C2×C4⋊C4—71no—
489C2×C4⋊C4—141no—
490C2×C42—141no—
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—74no—
507#103—72no—
508#103—72no—
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×C7—42no—
524C22×C7—42no—
525C22×C7—42no—
526C22×C7C23×D411yescharacteristic
527C4×C7—21no—
528C4×C7—24no—
529C4×C7—21no—
530C4×C7—21no—
531C4×C7—21no—
532C4×C7—24no—
533C4×C7—42no—
534C4×C7—42no—
535C4×C7—42no—
536C4×C7#57812yes—
537C4×C7C2×2−1+412yes—
538Dic7—24no—
539Dic7—42no—
540Dic7—42no—
541Dic7—42no—
542#41—82no—
543C2×Q8—21no—
544C2×Q8—24no—
545C2×Q8—21no—
546C2×Q8—72no—
547C2×Q8—84no—
548C2×Q8—144no—
549C2×Q8—141no—
550C2×Q8—141no—
551C2×Q8—141no—
552C2×Q8—144no—
553C2×Q8C2×C22⋊D712yes—
554C2×Q8C2×C22⋊D712yes—
555C2×Q8C23×D712yes—
556C2×D4—22no—
557C2×D4—21no—
558C2×D4—21no—
559C2×D4C23×D712yes—
560C22×C4—22no—
561C22×C4—24no—
562C22×C4—141no—
563C22×C4—141no—
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—21no—
574C4⋊C4—21no—
575C4⋊C4—21no—
576C4⋊C4—21no—
577C4⋊C4—22no—
578C4⋊C4—24no—
579C4⋊C4—24no—
580C4⋊C4—22no—
581C4⋊C4—72no—
582C4⋊C4—72no—
583C4⋊C4—72no—
584C4⋊C4—142no—
585C4⋊C4—142no—
586C4⋊C4—144no—
587C4⋊C4—144no—
588C4⋊C4—142no—
589C4⋊C4—144no—
590C4⋊C4—142no—
591C4⋊C4—144no—
592C4⋊C4—141no—
593C4⋊C4—141no—
594C4⋊C4—141no—
595C4⋊C4—144no—
596C4⋊C4C2×C22⋊D712yes—
597C4⋊C4C2×C22⋊D712yes—
598C4⋊C4C2×C22⋊D712yes—
599C4⋊C4C2×C22⋊D712yes—
600C4⋊C4C2×C22⋊D714yes—
601C4⋊C4C2×C22⋊D714yes—
602C4⋊C4C23×D712yes—
603C4⋊C4C23×D712yes—
604C22⋊C4—22no—
605C22⋊C4—24no—
606C22⋊C4—22no—
607C22⋊C4—24no—
608C22⋊C4—22no—
609C22⋊C4—24no—
610C22⋊C4—142no—
611C22⋊C4C2×C22⋊D714yes—
612C42—22no—
613C42—24no—
614C42—142no—
615C42—142no—
616C42—142no—
617C42—144no—
618C42—144no—
619C42C2×C22⋊D711yescharacteristic
620C42C2×C22⋊D711yescharacteristic
621C42C23×D711yescharacteristic
622C42C23×D714yes—
623C42C23×D711yescharacteristic
624C2×C7—41no—
625C2×C7—82no—
626C2×C7#348611yescharacteristic
627C2×C7C2×#57811yescharacteristic
628C2×C7#358511yescharacteristic
629C23—22no—
630C23C22×C22⋊D711yescharacteristic
631Q8—24no—
632Q8—22no—
633Q8—22no—
634Q8—42no—
635Q8—42no—
636Q8—74no—
637Q8—84no—
638Q8—81no—
639Q8—144no—
640Q8—142no—
641Q8—282no—
642Q8—561no—
643Q8—564no—
644D4—24no—
645D4—42no—
646D4—81no—
647D4—82no—
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—144no—
657C2×C4—142no—
658C2×C4—142no—
659C2×C4—142no—
660C2×C4—142no—
661C2×C4—561no—
662C2×C4C22×C22⋊D714yes—
663C2×C4C22×C22⋊D711yescharacteristic
664C2×C4C22×C22⋊D711yescharacteristic
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—42no—
678C22—42no—
679C22—42no—
680C22C23×C22⋊D711yescharacteristic
681C4—21no—
682C4—21no—
683C4—24no—
684C4—21no—
685C4—24no—
686C4—21no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—144no—
691C4—282no—
692C4—282no—
693C4—282no—
694C4#9052912yes—
695C4D7×2−1+412yes—
696C2—41no—
697C2—82no—
698C2#1174434511yescharacteristic
699C2C2×#9052911yescharacteristic
700C2#1174436211yescharacteristic
subgroupG/H
centerC22C23×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC7×#3550C2
socleC22×C7C23×D4
prime radicalC7×#532C22
hypercenter#3550D7
hypocenterC7#61854
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #533 ◁ #3550

Complements

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

NH ≅ G/Ntype
2. #11743326C2semidirect
3. #11741164C2semidirect
4. #11741162C2semidirect
5. #11741154C2semidirect
6. Q8×#1144C2semidirect
7. #11741123C2semidirect
8. #11741003C2semidirect
9. #11740912C2semidirect
10. #11740899C2semidirect
11. #11740813C2semidirect
12. #11740776C2semidirect
13. #11740772C2semidirect
673. C7#61854semidirect

Maximal subgroups

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

subgroupindex
C7×#35502
#117433262
#117411642
#117411622
#117411542
Q8×#11442
#117411232
#117410032
#117409122
#117408992
#117408132
#117407762
#117407722
#117407612
#117407172
#618547