Finite Group Databasebeta

← #14223473

Subgroups — #14223473

Order 1792.

1834 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11743349C211yesmaximal, characteristic
2#11743283C211yesmaximal, characteristic
3C7×#3165C211yesmaximal, characteristic, complemented
4#11740784C211yesmaximal, characteristic, complemented
5C2×#89851C211yesmaximal, characteristic, complemented
6#11739419C218yesmaximal, complemented
7#11739385C212yesmaximal, complemented
8#11739383C212yesmaximal, complemented
9#11739338C212yesmaximal, complemented
10D7×#498C212yesmaximal, complemented
11#11736654C214yesmaximal
12#11736650C214yesmaximal, complemented
13C4×#6470C211yesmaximal, characteristic
14#11735951C211yesmaximal, characteristic, complemented
15C7×#533C2211yescharacteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17Q8×C22⋊D7—22no—
18C2×#6470C2212yes—
19#90421—21no—
20#90420C2214yes—
21#90419C2212yes—
22#90419C2212yes—
23#90418—21no—
24D7×C4⋊Q8C2212yes—
25D7×C4⋊Q8C2212yes—
26#90413—21no—
27#90412—21no—
28#90380C2214yes—
29#90326—22no—
30#90323—22no—
31#90270—21no—
32Q8×D28—21no—
33#90258—21no—
34Q8×Dic14—21no—
35#90214C2212yes—
36#90210C2212yes—
37#90202—22no—
38#90195C2212yes—
39#90191C2212yes—
40C4×#1157C2211yescharacteristic
41#90161C2211yescharacteristic
42C2×#6432C2211yescharacteristic
43C7×#501C2214yes—
44C7×#500C2212yes—
45C7×#498C2212yes—
46C7×#428C2214yes—
47C4×C7×M16C2211yescharacteristic
48#89894C2212yes—
49#89893C2214yes—
50#89851C2214yes—
51#89851C2211yescharacteristic
52#89851C2211yescharacteristic
53#89851C2212yes—
54#89851C2212yes—
55#89850C2211yescharacteristic
56#89850C2211yescharacteristic
57#89836C2214yes—
58#89770C2214yes—
59C2×#6382C2214yes—
60C2×C4×C7⋊C8C2211yescharacteristic
61#89653C2218yes—
62#89652C2218yes—
63#89650C2218yes—
64#89648C2218yes—
65#89633C2218yes—
66D7×#106C2218yes—
67#89628C2218yes—
68#89627C2218yes—
69#89480C2214yes—
70#89474C2214yes—
71C4×#1131C2212yes—
72C4×C8×D7C2212yes—
73#89454C2212yes—
74#89452C2212yes—
75#60665—71nomaximal, Sylow
76C22×C7×Q8—21no—
77C2×Q8×D7—22no—
78C2×Q8×D7—22no—
79C2×#1157C2311yescharacteristic
80C22×Dic14—21no—
81C2×C7×M16C2312yes—
82C7×C4⋊Q8—21no—
83C7×C4⋊Q8—21no—
84C7×C4⋊Q8C2311yescharacteristic
85C7×C4⋊Q8C2312yes—
86C7×C4⋊Q8C2311yescharacteristic
87C7×C4⋊Q8C2312yes—
88C7×#125C2312yes—
89C7×C22⋊Q8—21no—
90C7×C22⋊Q8—21no—
91C4×C7×Q8—21no—
92C4×C7×Q8—21no—
93C7×#117C2312yes—
94C2×C7×C4⋊C4—21no—
95C2×C7×C4⋊C4C2312yes—
96C2×C42×C7C2311yescharacteristic
97#6509—22no—
98#6509—22no—
99#6509—24no—
100Q8×Dic7—22no—
101Q8×Dic7—22no—
102#6507—24no—
103#6507—22no—
104#6507—22no—
105C4×C22⋊D7—24no—
106C4×C22⋊D7D412yes—
107C4×C22⋊D7D412yes—
108#6489C2312yes—
109#6489C2311yescharacteristic
110C2×C4⋊Dic7—21no—
111C2×C4⋊Dic7C2312yes—
112#6487—21no—
113#6487—21no—
114#6487—24no—
115C22×C7⋊C8C2312yes—
116#6470Q8116yescomplemented
117C2×#1131C2314yes—
118C2×C8×D7C2314yes—
119#6460—22no—
120#6460—22no—
121#6460—24no—
122#6459—24no—
123#6459—24no—
124#6456—22no—
125#6455C2314yes—
126#6455C2314yes—
127D7×C4⋊C4—22no—
128D7×C4⋊C4—22no—
129D7×C4⋊C4C2314yes—
130D7×C4⋊C4C2314yes—
131#6453—22no—
132#6453C2314yes—
133#6453C2314yes—
134#6452—24no—
135#6452—24no—
136#6451—22no—
137#6451—24no—
138#6451—22no—
139#6451—22no—
140#6451C2314yes—
141#6451C2314yes—
142#6450—22no—
143#6450—22no—
144#6450—22no—
145(C22⋊D7)⋊C4—24no—
146#6441—24no—
147C4×D28—22no—
148C4×D28D411yescharacteristic
149C4×D28D411yescharacteristic
150#6435C2312yes—
151C42×D7C2312yes—
152#6433—22no—
153#6433C2312yes—
154#6432—21no—
155#6432—21no—
156#6432—22no—
157#6432C2312yes—
158#6432C2311yescharacteristic
159#6432C2312yes—
160#6432C2311yescharacteristic
161C4×Dic14—22no—
162C4×Dic14—21no—
163C4×Dic14—21no—
164C4×Dic14D411yescharacteristic
165C4×Dic14D411yescharacteristic
166C7×#107C2318yes—
167C7×#106C2318yes—
168C7×#97C2312yes—
169C4×C7×C8C2312yes—
170#6392C2318yes—
171#6391C2318yes—
172#6389C2314yes—
173C8×Dic7C2314yes—
174#6382C2314yes—
175#6382C2318yes—
176#6382C2318yes—
177#6382C2314yes—
178#6381C2314yes—
179#6381C2314yes—
180C4×C7⋊C8C2311yescharacteristic
181C4×C7⋊C8C2311yescharacteristic
182C4×C7⋊C8C2312yes—
183#3503—71no—
184#3503—71no—
185#3165—72no—
186#3165—72no—
187#3165—71no—
188#3165D711yescharacteristic, complemented
189#3163—78no—
190#3163—74no—
191#3160—72no—
192#3160—71no—
193C2×#498—72no—
194C2×#498—71no—
195#2912—74no—
196C4×#131—71no—
197C2×C7×Q8—21no—
198C2×C7×Q8—22no—
199C2×C7×Q8—21no—
200C2×C7×Q8—22no—
201C2×C7×Q8—82no—
202C22×C4×C7—22no—
203C22×C4×C7C2×D411yescharacteristic
204C22×C4×C7C2×D411yescharacteristic
205C22×C4×C7C2411yescharacteristic
206C2×C22⋊D7C2×D412yes—
207C22×Dic7—22no—
208Q8×D7—84no—
209Q8×D7—84no—
210#1157C2×Q818yes—
211C2×D28C2×D411yescharacteristic
212C2×C4×D7—24no—
213C2×C4×D7—24no—
214C2×C4×D7C2×D412yes—
215C2×C4×D7C2×D412yes—
216C2×C4×D7C2412yes—
217C2×Dic14—21no—
218C2×Dic14—21no—
219C2×Dic14—22no—
220C2×Dic14—22no—
221C2×Dic14—24no—
222C2×Dic14—24no—
223C2×Dic14—82no—
224C2×Dic14C2×D411yescharacteristic
225C7×M16C2×Q818yescomplemented
226C2×C7×C8C2414yes—
227C7×C4⋊C4—21no—
228C7×C4⋊C4—22no—
229C7×C4⋊C4—22no—
230C7×C4⋊C4—21no—
231C7×C4⋊C4—22no—
232C7×C4⋊C4—22no—
233C7×C4⋊C4—22no—
234C7×C4⋊C4C2412yes—
235C7×C4⋊C4C2414yes—
236C7×C4⋊C4C2412yes—
237C7×C22⋊C4—22no—
238C42×C7—22no—
239C42×C7C2×D411yescharacteristic
240C42×C7C2×D411yescharacteristic
241C42×C7C2411yescharacteristic
242C42×C7C2411yescharacteristic
243C22⋊Dic7—22no—
244C22⋊Dic7C2×D412yes—
245(C2×D7)⋊C4—24no—
246(C2×D7)⋊C4—24no—
247(C2×D7)⋊C4C2×D412yes—
248(C2×D7)⋊C4C2×D412yes—
249C4⋊Dic7—22no—
250C4⋊Dic7—21no—
251C4⋊Dic7—21no—
252C4⋊Dic7—22no—
253C4⋊Dic7—22no—
254C4⋊Dic7—24no—
255C4⋊Dic7—24no—
256C4⋊Dic7C2×D411yescharacteristic
257C4⋊Dic7C2×D411yescharacteristic
258C4⋊Dic7C2414yes—
259C4⋊Dic7C2412yes—
260C4⋊Dic7C2412yes—
261Dic7⋊C4—22no—
262Dic7⋊C4—22no—
263Dic7⋊C4—24no—
264Dic7⋊C4—24no—
265Dic7⋊C4—24no—
266Dic7⋊C4—24no—
267Dic7⋊C4—24no—
268Dic7⋊C4—24no—
269Dic7⋊C4C2×D412yes—
270Dic7⋊C4C2×D412yes—
271C4×Dic7—22no—
272C4×Dic7—24no—
273C4×Dic7—24no—
274C4×Dic7C2×D412yes—
275C4×Dic7C2×D412yes—
276C4×Dic7C2412yes—
277C2×C7⋊C8C2×Q818yescomplemented
278C2×C7⋊C8C2412yes—
279C2×C7⋊C8C2412yes—
280#1131C2×Q8116yescomplemented
281C8×D7C2×Q8116yescomplemented
282C2×#131—72no—
283Q82—141no—
284Q82—141no—
285#557—141no—
286#557—141no—
287#554—142no—
288#554—141no—
289#554—142no—
290#554—141no—
291D4×Q8—141no—
292D4×Q8—142no—
293D4×Q8—141no—
294D4×Q8—142no—
295#544—74no—
296#544—74no—
297#544—72no—
298#544—72no—
299#533—72no—
300#533—72no—
301#533—71no—
302#533C2×D711yescharacteristic
303C2×C4⋊Q8—72no—
304C2×C4⋊Q8—72no—
305C2×C4⋊Q8—71no—
306C2×C4⋊Q8C2×D711yescharacteristic
307#520—72no—
308#520—72no—
309C4×#41—71no—
310#501—78no—
311#501—78no—
312#501—74no—
313#501C2×D714yes—
314#500—71no—
315#500—71no—
316#500—72no—
317#500C2×D712yes—
318#499—74no—
319#499—78no—
320#498—71no—
321#498—72no—
322#498—74no—
323#498—72no—
324#498—71no—
325#498—78no—
326#498—72no—
327#498C2×D712yes—
328#428—74no—
329#428—78no—
330#428—78no—
331#428C2×D714yes—
332#427—74no—
333#427—78no—
334C2×#106—74no—
335C2×#106—78no—
336#405—72no—
337#405—74no—
338C4×M16—72no—
339C4×M16C2×D711yescharacteristic
340C2×C4×C8—71no—
341C2×C4×C8—72no—
342C23×C7C22×D411yescharacteristic
343C22×D7C22×D412yes—
344C7×Q8—42no—
345C7×Q8—44no—
346C7×Q8—42no—
347C7×Q8—82no—
348C2×C4×C7—22no—
349C2×C4×C7—22no—
350C2×C4×C7—22no—
351C2×C4×C7—24no—
352C2×C4×C7—22no—
353C2×C4×C7—82no—
354C2×C4×C7C22×D411yescharacteristic
355C2×C4×C7C22×D411yescharacteristic
356C2×C4×C7C22×D411yescharacteristic
357C2×C4×C7C22×D411yescharacteristic
358C2×C4×C7C22×D411yescharacteristic
359C2×C4×C7C22×Q814yes—
360C2×C4×C7C2511yescharacteristic
361C22⋊D7C4⋊Q818yes—
362C2×Dic7—22no—
363C2×Dic7—22no—
364C2×Dic7—24no—
365C2×Dic7—82no—
366C2×Dic7C22×D412yes—
367C2×Dic7C22×D412yes—
368C2×Dic7C22×D412yes—
369D28C4⋊Q814yes—
370C4×D7—24no—
371C4×D7—24no—
372C4×D7—84no—
373C4×D7—84no—
374C4×D7C22×Q818yes—
375Dic14—42no—
376Dic14—44no—
377Dic14—42no—
378Dic14—84no—
379Dic14—84no—
380Dic14—82no—
381Dic14C4⋊Q814yes—
382C7×C8C22×Q818yes—
383C7⋊C8C22×Q814yes—
384C7⋊C8C22×Q814yes—
385C2×#41—71no—
386C22×Q8—21no—
387C22×Q8—142no—
388C22×Q8—142no—
389C22×Q8—141no—
390#131—716no—
391C2×M16—74no—
392C2×M16C22×D712yes—
393C22×C8—72no—
394C22×C8—74no—
395C4⋊Q8—21no—
396C4⋊Q8—21no—
397C4⋊Q8—74no—
398C4⋊Q8—74no—
399C4⋊Q8—71no—
400C4⋊Q8—72no—
401C4⋊Q8—72no—
402C4⋊Q8—71no—
403C4⋊Q8—142no—
404C4⋊Q8—142no—
405C4⋊Q8—142no—
406C4⋊Q8—142no—
407C4⋊Q8—142no—
408C4⋊Q8—144no—
409C4⋊Q8—142no—
410C4⋊Q8—144no—
411C4⋊Q8—141no—
412C4⋊Q8—141no—
413C4⋊Q8C22×D711yescharacteristic
414C4⋊Q8C22×D712yes—
415C4⋊Q8C22×D712yes—
416C4⋊Q8C22×D711yescharacteristic
417#125—74no—
418#125—74no—
419#125—72no—
420#125—142no—
421#125—144no—
422#125—144no—
423#125—142no—
424#125C22×D712yes—
425C22⋊Q8—21no—
426C22⋊Q8—21no—
427C22⋊Q8—144no—
428C22⋊Q8—144no—
429C22⋊Q8—144no—
430C22⋊Q8—144no—
431C22⋊Q8—144no—
432C22⋊Q8—144no—
433C22⋊Q8—142no—
434C22⋊Q8—142no—
435C22⋊Q8—142no—
436C22⋊Q8—142no—
437C22⋊Q8—141no—
438C22⋊Q8—141no—
439C4×Q8—21no—
440C4×Q8—21no—
441C4×Q8—71no—
442C4×Q8—71no—
443C4×Q8—142no—
444C4×Q8—142no—
445C4×Q8—142no—
446C4×Q8—142no—
447C4×Q8—142no—
448C4×Q8—142no—
449C4×Q8—141no—
450C4×Q8—141no—
451C4×D4—72no—
452C4×D4—72no—
453C4×D4—71no—
454C4×D4—71no—
455C4×D4—142no—
456C4×D4—144no—
457C4×D4—142no—
458C4×D4—144no—
459#117—71no—
460#117—72no—
461#117—72no—
462#117—74no—
463#117—74no—
464#117C22×D712yes—
465C2×C4⋊C4—21no—
466C2×C4⋊C4—72no—
467C2×C4⋊C4—74no—
468C2×C4⋊C4—74no—
469C2×C4⋊C4—142no—
470C2×C4⋊C4—142no—
471C2×C4⋊C4—141no—
472C2×C4⋊C4C22×D712yes—
473C2×C42—72no—
474C2×C42C22×D711yescharacteristic
475#107—78no—
476#107—74no—
477#107—74no—
478#107C22×D718yes—
479#106—78no—
480#106—78no—
481#106—74no—
482#106—78no—
483#106—74no—
484#106C22×D718yes—
485#97—74no—
486#97C22×D712yes—
487C4×C8—74no—
488C4×C8—71no—
489C4×C8—71no—
490C4×C8—72no—
491C4×C8C22×D712yes—
492C22×C7—21no—
493C22×C7—21no—
494C22×C7C2×C4⋊Q812yes—
495C22×C7C23×D411yescharacteristic
496C2×D7—22no—
497C2×D7—22no—
498C2×D7C2×C4⋊Q814yes—
499C4×C7—21no—
500C4×C7—21no—
501C4×C7—42no—
502C4×C7—44no—
503C4×C7—42no—
504C4×C7C2×C4⋊Q812yes—
505C4×C7#57712yes—
506C4×C7#57712yes—
507C4×C7C23×Q812yes—
508Dic7—22no—
509Dic7—22no—
510Dic7—42no—
511Dic7—44no—
512Dic7—42no—
513Dic7C2×C4⋊Q814yes—
514#41—78no—
515C2×Q8—21no—
516C2×Q8—21no—
517C2×Q8—22no—
518C2×Q8—22no—
519C2×Q8—71no—
520C2×Q8—82no—
521C2×Q8—144no—
522C2×Q8—144no—
523C2×Q8—141no—
524C2×Q8—141no—
525C2×Q8—142no—
526C2×Q8—142no—
527C2×Q8—562no—
528C2×Q8—564no—
529C2×Q8—564no—
530C2×D4—72no—
531C2×D4—71no—
532C22×C4—22no—
533C22×C4—72no—
534C22×C4—72no—
535C22×C4—72no—
536C22×C4—144no—
537C22×C4—144no—
538C22×C4—142no—
539C22×C4D4×D711yescharacteristic
540C22×C4D4×D711yescharacteristic
541C22×C4C23×D711yescharacteristic
542M16—716no—
543M16Q8×D718yescomplemented
544C2×C8—78no—
545C2×C8—72no—
546C2×C8—72no—
547C2×C8—716no—
548C2×C8C23×D714yes—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—21no—
552C4⋊C4—22no—
553C4⋊C4—21no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—72no—
557C4⋊C4—72no—
558C4⋊C4—71no—
559C4⋊C4—71no—
560C4⋊C4—72no—
561C4⋊C4—74no—
562C4⋊C4—72no—
563C4⋊C4—144no—
564C4⋊C4—144no—
565C4⋊C4—144no—
566C4⋊C4—144no—
567C4⋊C4—144no—
568C4⋊C4—144no—
569C4⋊C4—144no—
570C4⋊C4—144no—
571C4⋊C4—142no—
572C4⋊C4—142no—
573C4⋊C4—142no—
574C4⋊C4—142no—
575C4⋊C4—141no—
576C4⋊C4—142no—
577C4⋊C4—141no—
578C4⋊C4C23×D714yes—
579C4⋊C4C23×D712yes—
580C4⋊C4C23×D712yes—
581C22⋊C4—22no—
582C22⋊C4—72no—
583C22⋊C4—72no—
584C22⋊C4—72no—
585C22⋊C4—144no—
586C22⋊C4—144no—
587C22⋊C4—142no—
588C42—22no—
589C42—72no—
590C42—72no—
591C42—72no—
592C42—144no—
593C42—144no—
594C42—142no—
595C42D4×D711yescharacteristic
596C42D4×D711yescharacteristic
597C42C23×D711yescharacteristic
598C42C23×D711yescharacteristic
599C2×C7—22no—
600C2×C7C22×C4⋊Q811yescharacteristic
601C2×C7C2×#57711yescharacteristic
602C2×C7C2×#57711yescharacteristic
603D7—24no—
604C23—72no—
605C23C2×D4×D711yescharacteristic
606Q8—42no—
607Q8—44no—
608Q8—42no—
609Q8—74no—
610Q8—82no—
611Q8—282no—
612Q8—282no—
613Q8—284no—
614Q8—562no—
615Q8—564no—
616Q8—564no—
617D4—78no—
618D4—74no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—72no—
625C2×C4—78no—
626C2×C4—72no—
627C2×C4—72no—
628C2×C4—82no—
629C2×C4—144no—
630C2×C4—144no—
631C2×C4—142no—
632C2×C4—142no—
633C2×C4—144no—
634C2×C4—564no—
635C2×C4—564no—
636C2×C4—562no—
637C2×C4C2×D4×D711yescharacteristic
638C2×C4C2×D4×D711yescharacteristic
639C2×C4C2×D4×D711yescharacteristic
640C2×C4C2×D4×D711yescharacteristic
641C2×C4C2×D4×D711yescharacteristic
642C2×C4C2×Q8×D714yes—
643C2×C4C24×D711yescharacteristic
644C8—74no—
645C8—74no—
646C8C2×Q8×D718yes—
647C7#6066511yescharacteristic, complemented, Sylow
648C22—21no—
649C22—21no—
650C22—74no—
651C22—142no—
652C22—142no—
653C22D7×C4⋊Q812yes—
654C22C22×D4×D711yescharacteristic
655C4—21no—
656C4—21no—
657C4—44no—
658C4—42no—
659C4—42no—
660C4—74no—
661C4—142no—
662C4—142no—
663C4—282no—
664C4—284no—
665C4—282no—
666C4D7×C4⋊Q812yes—
667C4#9046612yes—
668C4#9046912yes—
669C4C22×Q8×D712yes—
670C2—22no—
671C2—144no—
672C2C2×D7×C4⋊Q811yescharacteristic
673C2C2×#9046611yescharacteristic
674C2C2×#9046911yescharacteristic
subgroupG/H
centerC22C22×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC7×#3165C2
socleC22×C7C23×D4
prime radicalC2×#1157C23
hypercenter#3165D7
hypocenterC7#60665
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
3. C7×#3165C2semidirect
4. #11740784C2semidirect
5. C2×#89851C2semidirect
6. #11739419C2semidirect
7. #11739385C2semidirect
8. #11739383C2semidirect
9. #11739338C2semidirect
10. D7×#498C2semidirect
12. #11736650C2semidirect
14. #11735951C2semidirect
116. #6470Q8semidirect
188. #3165D7semidirect
225. C7×M16C2×Q8semidirect
277. C2×C7⋊C8C2×Q8semidirect
280. #1131C2×Q8semidirect
281. C8×D7C2×Q8semidirect
543. M16Q8×D7semidirect
647. C7#60665semidirect

Maximal subgroups

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

subgroupindex
#117433492
#117432832
C7×#31652
#117407842
C2×#898512
#117394192
#117393852
#117393832
#117393382
D7×#4982
#117366542
#117366502
C4×#64702
#117359512
#606657