Finite Group Databasebeta

← #14244187

Subgroups — #14244187

Order 1792.

1704 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3548C211yesmaximal, characteristic
2#11743326C211yesmaximal, characteristic, complemented
3#11741077C214yesmaximal, complemented
4#11741076C212yesmaximal, complemented
5#11741074C212yesmaximal, complemented
6#11741032C212yesmaximal, complemented
7#11741032C214yesmaximal, complemented
8#11741031C212yesmaximal, complemented
9#11740884C214yesmaximal, complemented
10#11740875C214yesmaximal, complemented
11#11740776C211yesmaximal, characteristic, complemented
12#11740761C211yesmaximal, characteristic, complemented
13#11740760C211yesmaximal, characteristic, complemented
14#11740719C211yesmaximal, characteristic, complemented
15#11740717C211yesmaximal, characteristic, complemented
16C7×#562—22no—
17C7×#554C2212yes—
18C7×#554C2212yes—
19C7×#550C2212yes—
20C7×#550C2212yes—
21C7×#547—21no—
22C7×#547—21no—
23C7×#540C2214yes—
24C7×#537—22no—
25C7×#537—22no—
26C7×#533C2211yescharacteristic
27C7×#532C2211yescharacteristic
28C7×#532C2211yescharacteristic
29#90413—21no—
30#90412—21no—
31#90376—22no—
32#90314—22no—
33#90313—22no—
34#90258C2212yes—
35#90258C2214yes—
36Q8×Dic14C2212yes—
37#90210C2212yes—
38#90207C2212yes—
39#90161C2211yescharacteristic
40C2×C4×Dic14C2211yescharacteristic
41#89866C2212yes—
42#89863C2212yes—
43#89852C2212yes—
44#89851C2211yescharacteristic
45#89850C2211yescharacteristic
46#89847C2212yes—
47#89847C2212yes—
48#89847C2214yes—
49#89847C2214yes—
50#89843C2212yes—
51#89843C2214yes—
52#89842C2211yescharacteristic
53#89842C2211yescharacteristic
54#89842C2214yes—
55#89837C2214yes—
56#89836C2214yes—
57#89835C2214yes—
58#89835C2214yes—
59#89812C2212yes—
60#89812C2212yes—
61#89807C2214yes—
62#89807C2214yes—
63#89806C2212yes—
64#89805C2212yes—
65#89796C2212yes—
66#89796C2214yes—
67Q8×C7⋊C8C2212yes—
68#89790C2214yes—
69#89790C2214yes—
70#89790C2214yes—
71C4×#1142C2212yes—
72C4×#1142C2212yes—
73C4×#1142C2214yes—
74C4×#1142C2214yes—
75#89781C2214yes—
76#89781C2214yes—
77#89781C2214yes—
78#89780C2212yes—
79#89780C2214yes—
80#89779C2212yes—
81#89779C2214yes—
82#89772C2214yes—
83#89698C2211yescharacteristic
84#89697C2211yescharacteristic
85#61851—71nomaximal, Sylow
86C2×C7×#41—22no—
87C2×C7×#41—22no—
88C2×C7×#41—21no—
89C2×C7×#41—21no—
90C22×Dic14C2311yescharacteristic
91C7×C4⋊Q8D411yescharacteristic
92C7×C4⋊Q8D411yescharacteristic
93C7×C4⋊Q8C2311yescharacteristic
94C7×C4⋊Q8C2311yescharacteristic
95C7×#127C2311yescharacteristic
96C7×#127C2311yescharacteristic
97C7×#127C2314yes—
98C7×#125D412yes—
99C7×#125C2314yes—
100C7×#124—21no—
101C7×#124—21no—
102C7×#123—24no—
103C7×#123—24no—
104C7×C22⋊Q8—22no—
105C7×C22⋊Q8—22no—
106C7×C22⋊Q8—24no—
107C7×C22⋊Q8—24no—
108C7×C22⋊Q8D412yes—
109C7×C22⋊Q8D412yes—
110C7×#121—21no—
111C7×#121—21no—
112C7×#121—22no—
113C7×#121—22no—
114C7×#121—24no—
115C7×#121—24no—
116C7×#121—24no—
117C7×#121—24no—
118C7×#121C2312yes—
119C7×#121C2312yes—
120C4×C7×Q8C2312yes—
121C4×C7×Q8C2312yes—
122C4×C7×D4—21no—
123C4×C7×D4—21no—
124C4×C7×D4C2312yes—
125C4×C7×D4C2314yes—
126C4×C7×D4C2314yes—
127C4×C7×D4C2312yes—
128C7×#117C2312yes—
129C2×C7×C4⋊C4—22no—
130C2×C7×C4⋊C4—22no—
131C2×C42×C7C2311yescharacteristic
132Q8×Dic7—22no—
133#6507—22no—
134#6507—24no—
135C2×#1142C2312yes—
136C2×#1142C2312yes—
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#6406C2312yes—
171#6406C2314yes—
172#6406C2314yes—
173#6406C2314yes—
174#6406C2314yes—
175#6406C2312yes—
176#6406C2314yes—
177C22⋊(C7⋊C8)C2312yes—
178C22⋊(C7⋊C8)C2312yes—
179#6384C2314yes—
180#6384C2312yes—
181#6384C2312yes—
182#6382C2312yes—
183#6382C2312yes—
184#6382C2314yes—
185#6382C2314yes—
186#6381C2314yes—
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#3548D711yescharacteristic
199#3502—71no—
200#3388—74no—
201#3385—72no—
202#3381—72no—
203#3368—72no—
204#3368—74no—
205#3362—72no—
206#3241—74no—
207#3235—71no—
208#3228—71no—
209#3126—74no—
210#3112—71no—
211#3103—71no—
212#2985—71no—
213C7×#41—82no—
214C7×#41—82no—
215C7×#41—84no—
216C7×#41—84no—
217C2×C7×Q8—21no—
218C2×C7×Q8—21no—
219C2×C7×Q8C2×D412yes—
220C2×C7×Q8C2×D412yes—
221C2×C7×D4—22no—
222C2×C7×D4—22no—
223C2×C7×D4—21no—
224C2×C7×D4—21no—
225C2×C7×D4—24no—
226C2×C7×D4—24no—
227C2×C7×D4—21no—
228C2×C7×D4—21no—
229C2×C7×D4C2412yes—
230C2×C7×D4C2412yes—
231C22×C4×C7—22no—
232C22×C4×C7—22no—
233C22×C4×C7—24no—
234C22×C4×C7—24no—
235C22×C4×C7C2×D411yescharacteristic
236C22×C4×C7C2×D411yescharacteristic
237C22×C4×C7C2411yescharacteristic
238C22×Dic7—21no—
239C22×Dic7—21no—
240C2×Dic14—21no—
241C2×Dic14—24no—
242C2×Dic14—21no—
243C2×Dic14—21no—
244C2×Dic14—24no—
245C2×Dic14C2412yes—
246C7×C4⋊C4—21no—
247C7×C4⋊C4—21no—
248C7×C4⋊C4—24no—
249C7×C4⋊C4—24no—
250C7×C4⋊C4C2×D412yes—
251C7×C4⋊C4C2×D412yes—
252C7×C4⋊C4C2×D414yes—
253C7×C4⋊C4C2×D412yes—
254C7×C4⋊C4C2×D414yes—
255C7×C4⋊C4C2×D412yes—
256C7×C4⋊C4C2412yes—
257C7×C4⋊C4C2412yes—
258C7×C22⋊C4—22no—
259C7×C22⋊C4—22no—
260C7×C22⋊C4—24no—
261C7×C22⋊C4—24no—
262C7×C22⋊C4—22no—
263C7×C22⋊C4—22no—
264C7×C22⋊C4C2×D414yes—
265C42×C7C2×D411yescharacteristic
266C42×C7C2×D411yescharacteristic
267C42×C7C2411yescharacteristic
268C42×C7C2414yes—
269C42×C7C2411yescharacteristic
270C22⋊Dic7—22no—
271#1142—24no—
272#1142—24no—
273#1142—28no—
274C4⋊Dic7—21no—
275C4⋊Dic7—21no—
276C4⋊Dic7—21no—
277C4⋊Dic7—24no—
278C4⋊Dic7C2412yes—
279C4⋊Dic7C2412yes—
280C4⋊Dic7C2412yes—
281Dic7⋊C4—22no—
282Dic7⋊C4—22no—
283Dic7⋊C4—22no—
284Dic7⋊C4—22no—
285Dic7⋊C4—24no—
286Dic7⋊C4—24no—
287Dic7⋊C4—24no—
288Dic7⋊C4—24no—
289C4×Dic7—22no—
290C4×Dic7—22no—
291C4×Dic7—22no—
292C4×Dic7—24no—
293C4×Dic7—24no—
294C2×C7⋊C8C2412yes—
295C2×C7⋊C8C2412yes—
296C2×C7⋊C8C2414yes—
297#562—22no—
298Q82—72no—
299Q82—141no—
300#557—72no—
301#557—74no—
302#557—141no—
303#557—142no—
304#554C2×D712yes—
305#554C2×D712yes—
306#550C2×D712yes—
307#550C2×D712yes—
308#547—21no—
309#547—21no—
310#544—72no—
311#544—142no—
312#540C2×D714yes—
313#537—22no—
314#537—22no—
315#533—71no—
316#533—142no—
317#533C2×D711yescharacteristic
318#532C2×D711yescharacteristic
319#532C2×D711yescharacteristic
320#519—72no—
321C2×C4×Q8—71no—
322#501—72no—
323#501—74no—
324#500—71no—
325#499—74no—
326#498—71no—
327#490—72no—
328#490—74no—
329#486—71no—
330#486—74no—
331#486—71no—
332#482—74no—
333#482—74no—
334#481—72no—
335#480—72no—
336#479—74no—
337#479—74no—
338#478—74no—
339#478—74no—
340#478—74no—
341#477—72no—
342#476—72no—
343#476—74no—
344#475—72no—
345#474—72no—
346#474—74no—
347#460—72no—
348#460—72no—
349#460—74no—
350#460—74no—
351#448—72no—
352#448—72no—
353#446—72no—
354#446—74no—
355C8×Q8—72no—
356#440—74no—
357#440—74no—
358#440—74no—
359C4×SD16—72no—
360C4×SD16—72no—
361C4×SD16—74no—
362C4×SD16—74no—
363#433—74no—
364#432—71no—
365#431—71no—
366C23×C7—22no—
367C23×C7—22no—
368C23×C7C22×D411yescharacteristic
369C7×Q8—24no—
370C7×Q8—42no—
371C7×Q8—81no—
372C7×Q8—81no—
373C7×D4—22no—
374C7×D4—22no—
375C7×D4—24no—
376C7×D4—42no—
377C7×D4—42no—
378C7×D4—81no—
379C7×D4—84no—
380C7×D4—81no—
381C7×D4—84no—
382C7×D4—82no—
383C7×D4—82no—
384C2×C4×C7—22no—
385C2×C4×C7—22no—
386C2×C4×C7—22no—
387C2×C4×C7—22no—
388C2×C4×C7—41no—
389C2×C4×C7—44no—
390C2×C4×C7—44no—
391C2×C4×C7—84no—
392C2×C4×C7—84no—
393C2×C4×C7C22×D411yescharacteristic
394C2×C4×C7C22×D411yescharacteristic
395C2×C4×C7C22×D414yes—
396C2×C4×C7C22×D414yes—
397C2×C4×C7C22×D411yescharacteristic
398C2×C4×C7C22×D411yescharacteristic
399C2×C4×C7C22×D411yescharacteristic
400C2×C4×C7C2511yescharacteristic
401C2×Dic7—24no—
402C2×Dic7—22no—
403C2×Dic7—22no—
404C2×Dic7—22no—
405C2×Dic7—22no—
406C2×Dic7—81no—
407Dic14—24no—
408Dic14—22no—
409Dic14—42no—
410Dic14—81no—
411Dic14—84no—
412Dic142−1+414yes—
413C7⋊C8—22no—
414C7⋊C8—22no—
415C7⋊C8—24no—
416C2×#41—22no—
417C2×#41—22no—
418C2×#41—21no—
419C2×#41—21no—
420C22×Q8—71no—
421C2×SD16—72no—
422C2×SD16—72no—
423C2×SD16—74no—
424C4⋊Q8—71no—
425C4⋊Q8—71no—
426C4⋊Q8—74no—
427C4⋊Q8—142no—
428C4⋊Q8—144no—
429C4⋊Q8—142no—
430C4⋊Q8—142no—
431C4⋊Q8—144no—
432C4⋊Q8C22⋊D711yescharacteristic
433C4⋊Q8C22⋊D711yescharacteristic
434C4⋊Q8C22×D711yescharacteristic
435C4⋊Q8C22×D711yescharacteristic
436#127C22×D711yescharacteristic
437#127C22×D711yescharacteristic
438#127C22×D714yes—
439#125—72no—
440#125—74no—
441#125—144no—
442#125—142no—
443#125—144no—
444#125—144no—
445#125C22⋊D712yes—
446#125C22×D714yes—
447#124—21no—
448#124—21no—
449#123—24no—
450#123—24no—
451C22⋊Q8—22no—
452C22⋊Q8—22no—
453C22⋊Q8—24no—
454C22⋊Q8—24no—
455C22⋊Q8—141no—
456C22⋊Q8—141no—
457C22⋊Q8—144no—
458C22⋊Q8C22⋊D712yes—
459C22⋊Q8C22⋊D712yes—
460#121—21no—
461#121—21no—
462#121—22no—
463#121—22no—
464#121—24no—
465#121—24no—
466#121—24no—
467#121—24no—
468#121C22×D712yes—
469#121C22×D712yes—
470C4×Q8—72no—
471C4×Q8—72no—
472C4×Q8—74no—
473C4×Q8—74no—
474C4×Q8—142no—
475C4×Q8—142no—
476C4×Q8—142no—
477C4×Q8—144no—
478C4×Q8—142no—
479C4×Q8—141no—
480C4×Q8—141no—
481C4×Q8—141no—
482C4×Q8—141no—
483C4×Q8C22×D712yes—
484C4×Q8C22×D712yes—
485C4×D4—21no—
486C4×D4—21no—
487C4×D4C22×D712yes—
488C4×D4C22×D712yes—
489C4×D4C22×D714yes—
490C4×D4C22×D714yes—
491#117—72no—
492#117—144no—
493#117C22×D712yes—
494C2×C4⋊C4—22no—
495C2×C4⋊C4—22no—
496C2×C4⋊C4—71no—
497C2×C4⋊C4—141no—
498C2×C42—141no—
499C2×C42C22×D711yescharacteristic
500#107—74no—
501#107—74no—
502#107—74no—
503#106—72no—
504#106—72no—
505#106—74no—
506#106—74no—
507C4⋊C8—72no—
508C4⋊C8—72no—
509C4⋊C8—74no—
510C4⋊C8—74no—
511#103—72no—
512#103—72no—
513#103—74no—
514#102—72no—
515#102—74no—
516#102—74no—
517#102—72no—
518#102—74no—
519#102—74no—
520#102—74no—
521C22⋊C8—72no—
522C22⋊C8—72no—
523#97—72no—
524#97—74no—
525C4×C8—71no—
526C4×C8—71no—
527C4×C8—74no—
528C22×C7—21no—
529C22×C7—21no—
530C22×C7—21no—
531C22×C7—42no—
532C22×C7—42no—
533C22×C7—42no—
534C22×C7—42no—
535C22×C7—42no—
536C22×C7—42no—
537C22×C7C23×D411yescharacteristic
538C4×C7—21no—
539C4×C7—24no—
540C4×C7—21no—
541C4×C7—21no—
542C4×C7—21no—
543C4×C7—24no—
544C4×C7—42no—
545C4×C7—42no—
546C4×C7#57712yes—
547C4×C7C2×2−1+412yes—
548Dic7—24no—
549Dic7—42no—
550Dic7—42no—
551Dic7—42no—
552#41—82no—
553#41—82no—
554#41—84no—
555#41—84no—
556C2×Q8—21no—
557C2×Q8—21no—
558C2×Q8—72no—
559C2×Q8—144no—
560C2×Q8—141no—
561C2×Q8—141no—
562C2×Q8—141no—
563C2×Q8—144no—
564C2×Q8C2×C22⋊D712yes—
565C2×Q8C2×C22⋊D712yes—
566C2×D4—22no—
567C2×D4—22no—
568C2×D4—21no—
569C2×D4—21no—
570C2×D4—24no—
571C2×D4—24no—
572C2×D4—21no—
573C2×D4—21no—
574C2×D4C23×D712yes—
575C2×D4C23×D712yes—
576C22×C4—22no—
577C22×C4—22no—
578C22×C4—24no—
579C22×C4—24no—
580C22×C4—141no—
581C22×C4—141no—
582C22×C4C2×C22⋊D711yescharacteristic
583C22×C4C2×C22⋊D711yescharacteristic
584C22×C4C23×D711yescharacteristic
585SD16—144no—
586SD16—144no—
587SD16—148no—
588C2×C8—72no—
589C2×C8—72no—
590C2×C8—74no—
591C4⋊C4—21no—
592C4⋊C4—21no—
593C4⋊C4—24no—
594C4⋊C4—24no—
595C4⋊C4—72no—
596C4⋊C4—72no—
597C4⋊C4—72no—
598C4⋊C4—142no—
599C4⋊C4—142no—
600C4⋊C4—144no—
601C4⋊C4—144no—
602C4⋊C4—142no—
603C4⋊C4—144no—
604C4⋊C4—142no—
605C4⋊C4—144no—
606C4⋊C4—141no—
607C4⋊C4—141no—
608C4⋊C4—141no—
609C4⋊C4—144no—
610C4⋊C4C2×C22⋊D712yes—
611C4⋊C4C2×C22⋊D712yes—
612C4⋊C4C2×C22⋊D712yes—
613C4⋊C4C2×C22⋊D712yes—
614C4⋊C4C2×C22⋊D714yes—
615C4⋊C4C2×C22⋊D714yes—
616C4⋊C4C23×D712yes—
617C4⋊C4C23×D712yes—
618C22⋊C4—22no—
619C22⋊C4—22no—
620C22⋊C4—24no—
621C22⋊C4—24no—
622C22⋊C4—22no—
623C22⋊C4—22no—
624C22⋊C4—142no—
625C22⋊C4C2×C22⋊D714yes—
626C42—142no—
627C42—142no—
628C42—142no—
629C42—144no—
630C42—144no—
631C42C2×C22⋊D711yescharacteristic
632C42C2×C22⋊D711yescharacteristic
633C42C23×D711yescharacteristic
634C42C23×D714yes—
635C42C23×D711yescharacteristic
636C2×C7—41no—
637C2×C7—82no—
638C2×C7—82no—
639C2×C7#348611yescharacteristic
640C2×C7C2×#57711yescharacteristic
641C2×C7#358511yescharacteristic
642C23—22no—
643C23—22no—
644C23C22×C22⋊D711yescharacteristic
645Q8—24no—
646Q8—42no—
647Q8—74no—
648Q8—81no—
649Q8—81no—
650Q8—144no—
651Q8—142no—
652Q8—282no—
653Q8—561no—
654Q8—564no—
655D4—22no—
656D4—22no—
657D4—24no—
658D4—42no—
659D4—42no—
660D4—81no—
661D4—84no—
662D4—81no—
663D4—84no—
664D4—82no—
665D4—82no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—41no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4—144no—
676C2×C4—142no—
677C2×C4—142no—
678C2×C4—142no—
679C2×C4—142no—
680C2×C4—561no—
681C2×C4C22×C22⋊D711yescharacteristic
682C2×C4C22×C22⋊D711yescharacteristic
683C2×C4C22×C22⋊D714yes—
684C2×C4C22×C22⋊D714yes—
685C2×C4C22×C22⋊D711yescharacteristic
686C2×C4C22×C22⋊D711yescharacteristic
687C2×C4C22×C22⋊D711yescharacteristic
688C2×C4C24×D711yescharacteristic
689C8—142no—
690C8—142no—
691C8—144no—
692C7#6185111yescharacteristic, complemented, Sylow
693C22—21no—
694C22—21no—
695C22—21no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22C23×C22⋊D711yescharacteristic
703C4—21no—
704C4—21no—
705C4—24no—
706C4—21no—
707C4—24no—
708C4—21no—
709C4—42no—
710C4—42no—
711C4—144no—
712C4—282no—
713C4—282no—
714C4—282no—
715C4#9052712yes—
716C4D7×2−1+412yes—
717C2—41no—
718C2—82no—
719C2—82no—
720C2#1174434511yescharacteristic
721C2C2×#9052711yescharacteristic
722C2#1174436211yescharacteristic
subgroupG/H
centerC22C23×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
Fitting, prime radicalC7×#3548C2
socleC22×C7C23×D4
hypercenter#3548D7
hypocenterC7#61851
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #533 ◁ #3548

Complements

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

NH ≅ G/Ntype
2. #11743326C2semidirect
3. #11741077C2semidirect
4. #11741076C2semidirect
5. #11741074C2semidirect
6. #11741032C2semidirect
7. #11741032C2semidirect
8. #11741031C2semidirect
9. #11740884C2semidirect
10. #11740875C2semidirect
11. #11740776C2semidirect
12. #11740761C2semidirect
13. #11740760C2semidirect
14. #11740719C2semidirect
15. #11740717C2semidirect
692. C7#61851semidirect

Maximal subgroups

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

subgroupindex
C7×#35482
#117433262
#117410772
#117410762
#117410742
#117410322
#117410312
#117408842
#117408752
#117407762
#117407612
#117407602
#117407192
#117407172
#618517