Finite Group Databasebeta

← #13355331

Subgroups — #13355331

Order 1792.

1862 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6495C211yesmaximal, characteristic
2#11742799C211yesmaximal, characteristic, complemented
3C7×#2309C211yesmaximal, characteristic, complemented
4C2×#89987C211yesmaximal, characteristic, complemented
5C2×#89777C211yesmaximal, characteristic, complemented
6#11729341C211yesmaximal, characteristic, complemented
7#11729332C211yesmaximal, characteristic, complemented
8#11727417C218yesmaximal, complemented
9C23×C7×D4C2211yescharacteristic
10C22×#1157C2211yescharacteristic
11C2×C7×#117C2211yescharacteristic
12C2×C7×#117C2211yescharacteristic
13C22×#1142—21no—
14C2×#6495—24no—
15C2×#6495—22no—
16C2×#6495C418yescomplemented
17C2×#6495C2214yes—
18C22×#1141—21no—
19C2×#6489C2211yescharacteristic
20C22×#1136C2211yescharacteristic
21C22×#1136C2211yescharacteristic
22#90214—22no—
23#90213—22no—
24#90212—21no—
25D7×#117—22no—
26#90207—21no—
27#90195C2214yescomplemented
28#90177—22no—
29C4×#1157—22no—
30C7×#390—22no—
31C7×#390C2214yescomplemented
32C7×#379—22no—
33#89987—22no—
34#89987C2214yescomplemented
35C2×#6406—22no—
36#89777—24no—
37C2×#6379C2214yescomplemented
38#89325C2214yescomplemented
39#89318C2214yescomplemented
40#20945—71nomaximal, Sylow
41C25×C7—21no—
42C22×C7×D4—21no—
43C22×C7×D4—21no—
44C22×C7×D4—22no—
45C22×C7×D4—21no—
46C22×C7×D4—21no—
47C22×C7×D4—21no—
48C22×C7×D4—22no—
49C22×C7×D4—22no—
50C22×C7×D4C2×C414yescomplemented
51C22×C7×D4C2312yes—
52C23×C4×C7C2311yescharacteristic
53C22×C22⋊D7—21no—
54C2×#1157—21no—
55C2×#1157—21no—
56C2×#1157—22no—
57C2×#1157—22no—
58C2×#1157C2×C414yes—
59C2×#1157D414yes—
60C2×#1157D412yes—
61C2×#1157C2312yes—
62C22×D28D411yescharacteristic
63C22×C4×D7—21no—
64C22×Dic14D411yescharacteristic
65C7×#117—21no—
66C7×#117—21no—
67C7×#117—21no—
68C7×#117—21no—
69C7×#117—22no—
70C7×#117—22no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4C2312yes—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—28no—
75C2×C7×C22⋊C4D412yes—
76C2×C42×C7D412yes—
77C2×C42×C7C2312yes—
78C2×C22⋊Dic7—21no—
79C2×#1142—28no—
80C2×#1142—24no—
81#6495—216no—
82#6495—28no—
83#6495C2×C4116yescomplemented
84C2×#1141—28no—
85C2×#1141—24no—
86C4×C22⋊D7—28no—
87#6489—21no—
88#6489—21no—
89#6489D412yes—
90#6489C2312yes—
91C2×C4⋊Dic7D412yes—
92C2×C4×Dic7—21no—
93C2×#1136—22no—
94C2×#1136—21no—
95C2×#1136—21no—
96C2×#1136—22no—
97C2×#1136—22no—
98C2×#1136C2×C414yescomplemented
99C2×#1136C2312yes—
100C2×#1136C2312yes—
101C22×C7⋊C8—21no—
102C22×C7⋊C8—21no—
103#6456—24no—
104#6455—24no—
105D7×C4⋊C4—24no—
106#6450—24no—
107(C22⋊D7)⋊C4—28no—
108D7×C22⋊C4—24no—
109#6440—24no—
110C4×D28—24no—
111#6435—24no—
112C42×D7—24no—
113C4×Dic14—24no—
114#6406—28no—
115#6379—28no—
116C22×AGL(1,ℤ/8)—71no—
117#3428—71no—
118C2×#421—71no—
119C2×#418—71no—
120#2309D711yescharacteristic, complemented
121#1883—78no—
122#1742—71no—
123#1733—71no—
124C24×C7—22no—
125C24×C7—24no—
126C24×C7—41no—
127C24×C7—41no—
128C24×C7—42no—
129C24×C7—84no—
130C24×C7C2×D411yescharacteristic
131C23×D7—21no—
132C2×C7×D4—24no—
133C2×C7×D4—24no—
134C2×C7×D4—24no—
135C2×C7×D4—24no—
136C2×C7×D4—24no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4—22no—
142C2×C7×D4—41no—
143C2×C7×D4—41no—
144C2×C7×D4—42no—
145C2×C7×D4—42no—
146C2×C7×D4—42no—
147C2×C7×D4—42no—
148C2×C7×D4—42no—
149C2×C7×D4C22×C414yes—
150C22×C4×C7—22no—
151C22×C4×C7C22×C412yes—
152C22×C4×C7C2×D414yes—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D412yes—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7#4112yes—
160C22×C4×C7#4112yes—
161C22×C4×C7C2411yescharacteristic
162C2×C22⋊D7—24no—
163C2×C22⋊D7—22no—
164C2×C22⋊D7—24no—
165C2×C22⋊D7—22no—
166C22×Dic7—21no—
167C22×Dic7—22no—
168#1157—24no—
169#1157—24no—
170#1157—24no—
171#1157—24no—
172#1157—24no—
173#1157—22no—
174#1157—22no—
175#1157C22⋊C418yescomplemented
176#1157C22×C414yes—
177#1157C2×D414yes—
178C2×D28—21no—
179C2×D28—22no—
180C2×D28C22⋊C414yescomplemented
181C2×D28C2×D412yes—
182C2×C4×D7—24no—
183C2×C4×D7—22no—
184C2×C4×D7—24no—
185C2×C4×D7—22no—
186C2×C4×D7—28no—
187C2×Dic14—21no—
188C2×Dic14—22no—
189C2×Dic14C22⋊C414yescomplemented
190C2×Dic14C2×D412yes—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—42no—
194C7×C22⋊C4—22no—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—42no—
197C7×C22⋊C4—88no—
198C42×C7—22no—
199C42×C7—22no—
200C42×C7—22no—
201C42×C7—22no—
202C22⋊Dic7—22no—
203C22⋊Dic7—22no—
204#1142—216no—
205#1141—216no—
206(C2×D7)⋊C4—28no—
207C4⋊Dic7—22no—
208C4⋊Dic7C2×D414yes—
209Dic7⋊C4—28no—
210C4×Dic7—28no—
211C4×Dic7—22no—
212C4×Dic7—22no—
213#1136—24no—
214#1136—22no—
215#1136—22no—
216#1136—22no—
217#1136—42no—
218#1136C22×C414yes—
219C2×C7⋊C8—24no—
220C2×C7⋊C8—22no—
221C2×C7⋊C8—22no—
222C2×C7⋊C8—42no—
223C22×#41—71no—
224C23×D4C2×D711yescharacteristic
225C2×AGL(1,ℤ/8)—78no—
226C2×AGL(1,ℤ/8)—74no—
227C2×AGL(1,ℤ/8)—142no—
228C2×AGL(1,ℤ/8)—144no—
229C22×SD16—141no—
230C22×D8—141no—
231C22×M16—71no—
232C22×M16—71no—
233#520—74no—
234#520—142no—
235#519—141no—
236#518—141no—
237#518—142no—
238C4×#41—142no—
239C4×#41—142no—
240C2×#117—71no—
241C2×#117—142no—
242C2×#117C2×D711yescharacteristic
243C2×#117C2×D711yescharacteristic
244#421—144no—
245C2×#104—74no—
246#418—74no—
247#418—142no—
248C2×#102—142no—
249#390—22no—
250#390C2×D714yescomplemented
251#379—22no—
252#343—74no—
253#339—74no—
254C23×C7—24no—
255C23×C7—42no—
256C23×C7—48no—
257C23×C7—42no—
258C23×C7—42no—
259C23×C7—44no—
260C23×C7—44no—
261C23×C7—84no—
262C23×C7—84no—
263C23×C7—88no—
264C23×C7—88no—
265C23×C7C4×D412yes—
266C23×C7#12011yescharacteristic
267C23×C7#12011yescharacteristic
268C23×C7#12112yes—
269C23×C7C22×D411yescharacteristic
270C22×D7—22no—
271C22×D7—24no—
272C22×D7—82no—
273C7×D4—24no—
274C7×D4—24no—
275C7×D4—42no—
276C7×D4—42no—
277C7×D4—44no—
278C7×D4—44no—
279C2×C4×C7—24no—
280C2×C4×C7—44no—
281C2×C4×C7—44no—
282C2×C4×C7—42no—
283C2×C4×C7C2×C22⋊C412yes—
284C2×C4×C7C2×C22⋊C412yes—
285C2×C4×C7C4×D412yes—
286C2×C4×C7C4×D412yes—
287C2×C4×C7C4×D412yes—
288C2×C4×C7#12011yescharacteristic
289C2×C4×C7#12011yescharacteristic
290C2×C4×C7#12112yes—
291C2×C4×C7#12112yes—
292C2×C4×C7#12112yes—
293C2×C4×C7#12312yes—
294C2×C4×C7#12312yes—
295C2×C4×C7C23×C411yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C22×D411yescharacteristic
298C2×C4×C7C22×D411yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C22⋊D7—24no—
302C22⋊D7—24no—
303C22⋊D7—28no—
304C2×Dic7—22no—
305C2×Dic7—24no—
306C2×Dic7—24no—
307C2×Dic7—82no—
308D28—22no—
309D28—24no—
310D28C2×C22⋊C414yes—
311C4×D7—24no—
312C4×D7—24no—
313C4×D7—28no—
314C4×D7—88no—
315Dic14—22no—
316Dic14—24no—
317Dic14C2×C22⋊C414yes—
318C7⋊C8—24no—
319C7⋊C8—42no—
320C25—21no—
321C2×#41—74no—
322C2×#41—72no—
323C2×#41—74no—
324C2×#41—72no—
325C2×#41—142no—
326C2×#41—141no—
327C2×#41—142no—
328C2×#41—141no—
329C22×Q8—71no—
330C22×D4—22no—
331C22×D4—21no—
332C22×D4—21no—
333C22×D4—22no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—22no—
337C22×D4—21no—
338C22×D4—71no—
339C22×D4—141no—
340C22×D4C4×D714yescomplemented
341C22×D4C22×D712yes—
342C23×C4—141no—
343C23×C4C22×D711yescharacteristic
344AGL(1,ℤ/8)—716no—
345AGL(1,ℤ/8)—1416no—
346AGL(1,ℤ/8)—148no—
347C2×SD16—148no—
348C2×SD16—144no—
349C2×D8—148no—
350C2×D8—144no—
351C2×M16—74no—
352C2×M16—72no—
353C2×M16—72no—
354C2×M16—142no—
355C2×M16—141no—
356C2×M16—141no—
357C2×M16—142no—
358C2×M16—142no—
359C22×C8—141no—
360C22×C8—141no—
361C4×Q8—144no—
362C4×Q8—144no—
363C4×D4—144no—
364C4×D4—148no—
365C4×D4—144no—
366C4×D4—148no—
367#117—21no—
368#117—21no—
369#117—21no—
370#117—21no—
371#117—22no—
372#117—22no—
373#117—72no—
374#117—72no—
375#117—144no—
376#117—144no—
377#117—144no—
378#117—141no—
379#117—141no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—72no—
382C2×C4⋊C4—144no—
383C2×C4⋊C4C22×D712yes—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—28no—
386C2×C22⋊C4—144no—
387C2×C22⋊C4—141no—
388C2×C22⋊C4C22⋊D712yes—
389C2×C42—144no—
390C2×C42—141no—
391C2×C42C22⋊D712yes—
392C2×C42C22×D712yes—
393#104—148no—
394#102—148no—
395C22×C7—22no—
396C22×C7—22no—
397C22×C7—22no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—48no—
401C22×C7—44no—
402C22×C7—84no—
403C22×C7—88no—
404C22×C7—88no—
405C22×C7—84no—
406C22×C7#38612yes—
407C22×C7#38612yes—
408C22×C7C2×C4×D411yescharacteristic
409C22×C7C2×#12011yescharacteristic
410C22×C7C2×#12111yescharacteristic
411C2×D7—24no—
412C2×D7—44no—
413C2×D7—84no—
414C4×C7—44no—
415C4×C7—82no—
416C4×C7#38612yes—
417C4×C7#38612yes—
418C4×C7C22×C22⋊C411yescharacteristic
419C4×C7C2×C4×D411yescharacteristic
420C4×C7C2×#12111yescharacteristic
421C4×C7C2×#12311yescharacteristic
422Dic7—24no—
423Dic7—44no—
424C24—22no—
425C24—24no—
426C24—41no—
427C24—41no—
428C24—42no—
429C24—84no—
430C24—141no—
431C24C2×C22⋊D711yescharacteristic
432#41—78no—
433#41—74no—
434#41—74no—
435#41—144no—
436#41—144no—
437#41—144no—
438#41—144no—
439#41—144no—
440#41—142no—
441#41—142no—
442C2×Q8—74no—
443C2×Q8—72no—
444C2×Q8—142no—
445C2×Q8—141no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—42no—
463C2×D4—74no—
464C2×D4—72no—
465C2×D4—144no—
466C2×D4—142no—
467C2×D4—144no—
468C2×D4—142no—
469C2×D4—142no—
470C2×D4—141no—
471C2×D4C2×C4×D714yes—
472C22×C4—22no—
473C22×C4—148no—
474C22×C4—141no—
475C22×C4—144no—
476C22×C4—144no—
477C22×C4—142no—
478C22×C4—142no—
479C22×C4—142no—
480C22×C4C2×C4×D712yes—
481C22×C4#115712yes—
482C22×C4D4×D711yescharacteristic
483C22×C4D4×D711yescharacteristic
484C22×C4D4×D712yes—
485C22×C4D4×D711yescharacteristic
486C22×C4D4×D711yescharacteristic
487C22×C4#115912yes—
488C22×C4C2×C22⋊D714yes—
489C22×C4C2×C22⋊D711yescharacteristic
490C22×C4C23×D711yescharacteristic
491SD16—1416no—
492D8—1416no—
493M16—74no—
494M16—144no—
495M16—142no—
496M16—142no—
497M16—142no—
498M16—282no—
499C2×C8—144no—
500C2×C8—142no—
501C2×C8—142no—
502C2×C8—282no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—42no—
506C4⋊C4—74no—
507C4⋊C4—148no—
508C4⋊C4—142no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—42no—
512C22⋊C4—88no—
513C22⋊C4—148no—
514C22⋊C4—142no—
515C22⋊C4—142no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—148no—
521C42—142no—
522C42—142no—
523C2×C7—21no—
524C2×C7—22no—
525C2×C7—21no—
526C2×C7—22no—
527C2×C7—84no—
528C2×C7C2×#38611yescharacteristic
529C2×C7#302112yes—
530D7—44no—
531C23—24no—
532C23—42no—
533C23—48no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—84no—
539C23—84no—
540C23—88no—
541C23—88no—
542C23—142no—
543C23—144no—
544C23—562no—
545C23C4×C22⋊D712yes—
546C23#650011yescharacteristic
547C23#650011yescharacteristic
548C23#650112yes—
549C23C22×C22⋊D711yescharacteristic
550Q8—74no—
551Q8—142no—
552Q8—144no—
553D4—24no—
554D4—24no—
555D4—42no—
556D4—42no—
557D4—44no—
558D4—44no—
559D4—74no—
560D4—144no—
561D4—144no—
562D4—148no—
563D4—142no—
564D4—144no—
565C2×C4—24no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—142no—
570C2×C4—144no—
571C2×C4—144no—
572C2×C4—144no—
573C2×C4—148no—
574C2×C4—144no—
575C2×C4—568no—
576C2×C4—562no—
577C2×C4D7×C22⋊C412yes—
578C2×C4D7×C22⋊C412yes—
579C2×C4(C22⋊D7)⋊C412yes—
580C2×C4(C22⋊D7)⋊C412yes—
581C2×C4#644612yes—
582C2×C4#644612yes—
583C2×C4#644712yes—
584C2×C4#644712yes—
585C2×C4C4×C22⋊D712yes—
586C2×C4#650011yescharacteristic
587C2×C4#650011yescharacteristic
588C2×C4#650112yes—
589C2×C4C22×C4×D711yescharacteristic
590C2×C4C2×#115711yescharacteristic
591C2×C4C2×D4×D711yescharacteristic
592C2×C4C2×D4×D711yescharacteristic
593C2×C4C2×D4×D711yescharacteristic
594C2×C4C2×#115911yescharacteristic
595C8—144no—
596C8—282no—
597C7#2094511yescharacteristic, complemented, Sylow
598C22—22no—
599C22—22no—
600C22—22no—
601C22—42no—
602C22—42no—
603C22—48no—
604C22—44no—
605C22—84no—
606C22—88no—
607C22—88no—
608C22—84no—
609C22—144no—
610C22—284no—
611C22—564no—
612C22#8972812yes—
613C22#8972812yes—
614C22C2×C4×C22⋊D711yescharacteristic
615C22C2×#650011yescharacteristic
616C22C2×#650111yescharacteristic
617C4—44no—
618C4—82no—
619C4—144no—
620C4—284no—
621C4#8972812yes—
622C4#8972812yes—
623C4C2×D7×C22⋊C411yescharacteristic
624C4C2×(C22⋊D7)⋊C411yescharacteristic
625C4C2×#644611yescharacteristic
626C4C2×#644711yescharacteristic
627C2—21no—
628C2—21no—
629C2—22no—
630C2—22no—
631C2—84no—
632C2—284no—
633C2C2×#8972811yescharacteristic
634C2#1174084012yes—
subgroupG/H
centerC22C2×#6500
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2309C2
socleC22×C7C2×#120
prime radicalC22×#6495C2
cosocleC22×C4×C7C24
hypercenter#2309D7
hypocenterC7#20945
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. #11742799C2semidirect
3. C7×#2309C2semidirect
4. C2×#89987C2semidirect
5. C2×#89777C2semidirect
6. #11729341C2semidirect
7. #11729332C2semidirect
8. #11727417C2semidirect
16. C2×#6495C4semidirect
27. #90195C22semidirect
31. C7×#390C22semidirect
34. #89987C22semidirect
37. C2×#6379C22semidirect
38. #89325C22semidirect
39. #89318C22semidirect
50. C22×C7×D4C2×C4semidirect
83. #6495C2×C4semidirect
98. C2×#1136C2×C4semidirect
120. #2309D7semidirect
175. #1157C22⋊C4semidirect
180. C2×D28C22⋊C4semidirect
189. C2×Dic14C22⋊C4semidirect
250. #390C2×D7semidirect
340. C22×D4C4×D7semidirect
597. C7#20945semidirect

Maximal subgroups

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

subgroupindex
C22×#64952
#117427992
C7×#23092
C2×#899872
C2×#897772
#117293412
#117293322
#117274172
#209457