Finite Group Databasebeta

← #13386779

Subgroups — #13386779

Order 1792.

1344 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#548C211yesmaximal, characteristic
2C22×#6409C211yesmaximal, characteristic, complemented
3C2×#89806C211yesmaximal, characteristic, complemented
4#11739819C211yesmaximal, characteristic, complemented
5#11729889C212yesmaximal, complemented
6#11729886C211yesmaximal, characteristic, complemented
7#11729476C212yesmaximal, complemented
8#11729448C212yesmaximal, complemented
9#11729435C211yesmaximal, characteristic, complemented
10#11729383C212yesmaximal, complemented
11#11729364C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C23×C7×Q8C2211yescharacteristic
14C7×#548—24no—
15C7×#548—24no—
16C2×C7×#124—21no—
17C2×C7×#124C2211yescharacteristic
18C2×C7×C22⋊Q8—21no—
19C2×C7×C22⋊Q8C2211yescharacteristic
20C2×C7×#121—21no—
21C2×C7×#121C2211yescharacteristic
22C2×C4×C7×Q8C2211yescharacteristic
23C2×C4×C7×D4—21no—
24C2×C4×C7×D4C2211yescharacteristic
25C22×C4⋊Dic7C2211yescharacteristic
26C23×C7⋊C8C2211yescharacteristic
27C2×#6409—24no—
28C2×#6409C2212yes—
29C2×#6409C2212yes—
30C2×#6409C2212yes—
31C2×#6409C2212yes—
32#89992—21no—
33C2×#6406C2212yes—
34C2×#6406C2212yes—
35#89984C2211yescharacteristic
36#89984C2212yes—
37C2×C22⋊(C7⋊C8)C2211yescharacteristic
38C2×C22⋊(C7⋊C8)C2212yes—
39#89806—24no—
40#89758—21no—
41C2×#6382C2212yes—
42#89721—21no—
43#89718—21no—
44#89700C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#24036—71nomaximal, Sylow
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—24no—
51C2×C7×#41—42no—
52C22×C7×Q8—21no—
53C22×C7×Q8—22no—
54C22×C7×Q8—24no—
55C22×C7×Q8—42no—
56C22×C7×Q8D414yes—
57C22×C7×Q8C2311yescharacteristic
58C22×C7×Q8C2311yescharacteristic
59C22×C7×D4—21no—
60C22×C7×D4—21no—
61C22×C7×D4D411yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C23×C4×C7—21no—
65C23×C4×C7C2311yescharacteristic
66C23×C4×C7C2311yescharacteristic
67C23×Dic7—21no—
68C7×#124—28no—
69C7×#124—22no—
70C7×#124—22no—
71C7×C22⋊Q8—28no—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8—22no—
74C7×#121—28no—
75C7×#121—22no—
76C7×#121D414yes—
77C4×C7×Q8—22no—
78C4×C7×Q8—22no—
79C4×C7×D4—28no—
80C4×C7×D4—22no—
81C4×C7×D4—22no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4D411yescharacteristic
85C2×C7×C4⋊C4C2311yescharacteristic
86C2×C7×C22⋊C4—21no—
87C2×C7×C22⋊C4—21no—
88C2×C7×C22⋊C4—21no—
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4D411yescharacteristic
91C2×C7×C22⋊C4D411yescharacteristic
92C2×C42×C7—21no—
93C2×C42×C7C2311yescharacteristic
94C2×C22⋊Dic7—21no—
95C2×C22⋊Dic7—22no—
96C2×C4⋊Dic7—22no—
97C2×C4⋊Dic7—22no—
98C2×C4⋊Dic7—21no—
99C2×C4⋊Dic7C2312yes—
100C2×C4⋊Dic7C2312yes—
101C2×Dic7⋊C4—22no—
102C22×C7⋊C8—22no—
103C22×C7⋊C8—42no—
104C22×C7⋊C8C2312yes—
105C22×C7⋊C8C2312yes—
106#6409—24no—
107#6409—24no—
108#6409—24no—
109#6409—42no—
110#6406—24no—
111#6406—42no—
112C7⋊#95—22no—
113C7⋊#95—22no—
114C7⋊#95—21no—
115C22⋊(C7⋊C8)—22no—
116C22⋊(C7⋊C8)—42no—
117#6382—24no—
118C2×#548D711yescharacteristic
119C2×#481—71no—
120C22×#103—71no—
121#2487—71no—
122#2042—72no—
123#2035—72no—
124#1893—71no—
125#1869—71no—
126#1865—72no—
127#1788—72no—
128#1787—71no—
129C24×C7—21no—
130C24×C7C2×D411yescharacteristic
131C24×C7C2×D411yescharacteristic
132C7×#41—48no—
133C7×#41—44no—
134C7×#41—44no—
135C2×C7×Q8—24no—
136C2×C7×Q8—21no—
137C2×C7×Q8—21no—
138C2×C7×Q8—22no—
139C2×C7×Q8—28no—
140C2×C7×Q8—22no—
141C2×C7×Q8—22no—
142C2×C7×Q8—44no—
143C2×C7×Q8—44no—
144C2×C7×Q8—48no—
145C2×C7×Q8C2×D412yes—
146C2×C7×Q8C2×D412yes—
147C2×C7×Q8#4114yes—
148C2×C7×D4—24no—
149C2×C7×D4—22no—
150C2×C7×D4—24no—
151C2×C7×D4—22no—
152C2×C7×D4—22no—
153C2×C7×D4—21no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—44no—
157C2×C7×D4—42no—
158C2×C7×D4—42no—
159C2×C7×D4C2×D412yes—
160C22×C4×C7—21no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—24no—
164C22×C4×C7—24no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—22no—
170C22×C4×C7—22no—
171C22×C4×C7—44no—
172C22×C4×C7—44no—
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D412yes—
178C22×C4×C7C2×D412yes—
179C22×C4×C7C2411yescharacteristic
180C22×Dic7—22no—
181C22×Dic7—22no—
182C22×Dic7—42no—
183C22×Dic7—82no—
184C7×C4⋊C4—24no—
185C7×C4⋊C4—24no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4C2×D412yes—
189C7×C22⋊C4—24no—
190C7×C22⋊C4—24no—
191C7×C22⋊C4—24no—
192C7×C22⋊C4—22no—
193C7×C22⋊C4—24no—
194C7×C22⋊C4—22no—
195C42×C7—24no—
196C42×C7—21no—
197C42×C7—21no—
198C22⋊Dic7—42no—
199C22⋊Dic7—82no—
200C4⋊Dic7—22no—
201C4⋊Dic7—22no—
202C4⋊Dic7—22no—
203C4⋊Dic7—22no—
204C4⋊Dic7—42no—
205Dic7⋊C4—82no—
206C2×C7⋊C8—22no—
207C2×C7⋊C8—22no—
208C2×C7⋊C8—22no—
209C2×C7⋊C8—22no—
210C2×C7⋊C8—22no—
211C2×C7⋊C8—22no—
212C2×C7⋊C8—44no—
213C22×#41C2×D711yescharacteristic
214C23×Q8C2×D711yescharacteristic
215C23×C8—71no—
216#548—24no—
217#548—24no—
218C2×#124—21no—
219C2×#124C2×D711yescharacteristic
220C2×C22⋊Q8—21no—
221C2×C22⋊Q8C2×D711yescharacteristic
222C2×#121—21no—
223C2×#121C2×D711yescharacteristic
224C2×C4×Q8C2×D711yescharacteristic
225C2×C4×D4—21no—
226C2×C4×D4C2×D711yescharacteristic
227C22×C4⋊C4—71no—
228#481—144no—
229C2×#106—72no—
230C2×#103—72no—
231C2×#103—72no—
232C2×#103—72no—
233C2×#103—72no—
234C2×#103—144no—
235C2×#102—72no—
236C2×#102—72no—
237C2×C22⋊C8—71no—
238C2×C22⋊C8—72no—
239#400—141no—
240#399—141no—
241#389—72no—
242#386—141no—
243#385—141no—
244#380—71no—
245#380—72no—
246#340—72no—
247#336—72no—
248C23×C7—22no—
249C23×C7—22no—
250C23×C7—21no—
251C23×C7—22no—
252C23×C7—22no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—22no—
256C23×C7—42no—
257C23×C7—41no—
258C23×C7—41no—
259C23×C7—42no—
260C23×C7#12012yes—
261C23×C7#13512yes—
262C23×C7#13712yes—
263C23×C7C22×D411yescharacteristic
264C7×Q8—22no—
265C7×Q8—24no—
266C7×Q8—22no—
267C7×Q8—22no—
268C7×Q8—48no—
269C7×Q8—42no—
270C7×Q8#12314yes—
271C7×D4—24no—
272C7×D4—48no—
273C7×D4—44no—
274C7×D4—48no—
275C7×D4—41no—
276C7×D4—41no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—24no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—24no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—21no—
288C2×C4×C7—22no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—22no—
292C2×C4×C7—48no—
293C2×C4×C7—48no—
294C2×C4×C7—42no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—42no—
300C2×C4×C7#12012yes—
301C2×C4×C7C22×D411yescharacteristic
302C2×C4×C7C22×D411yescharacteristic
303C2×C4×C7C2×#4112yes—
304C2×C4×C7H(5,𝔽2)12yes—
305C2×Dic7—42no—
306C2×Dic7—42no—
307C2×Dic7—42no—
308C2×Dic7—42no—
309C2×Dic7—42no—
310C2×Dic7—42no—
311C2×Dic7—84no—
312C7⋊C8—42no—
313C7⋊C8—42no—
314C2×#41—22no—
315C2×#41—22no—
316C2×#41—24no—
317C2×#41—42no—
318C22×Q8—21no—
319C22×Q8—22no—
320C22×Q8—24no—
321C22×Q8—42no—
322C22×Q8C22⋊D714yes—
323C22×Q8C22×D711yescharacteristic
324C22×Q8C22×D711yescharacteristic
325C22×D4—21no—
326C22×D4—21no—
327C22×D4C22⋊D711yescharacteristic
328C22×D4C22×D711yescharacteristic
329C23×C4—21no—
330C23×C4—21no—
331C23×C4—141no—
332C23×C4C22×D711yescharacteristic
333C23×C4C22×D711yescharacteristic
334C22×C8—72no—
335C22×C8—72no—
336C22×C8—142no—
337C22×C8—282no—
338#124—28no—
339#124—22no—
340#124—22no—
341C22⋊Q8—28no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344#121—28no—
345#121—22no—
346#121C22⋊D714yes—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×D4—28no—
350C4×D4—22no—
351C4×D4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—72no—
355C2×C4⋊C4—72no—
356C2×C4⋊C4—142no—
357C2×C4⋊C4—142no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—142no—
360C2×C4⋊C4C22⋊D711yescharacteristic
361C2×C4⋊C4C22×D711yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—141no—
367C2×C22⋊C4—142no—
368C2×C22⋊C4C22⋊D711yescharacteristic
369C2×C22⋊C4C22⋊D711yescharacteristic
370C2×C42—21no—
371C2×C42C22×D711yescharacteristic
372#106—144no—
373#103—144no—
374#103—144no—
375#103—144no—
376#103—282no—
377#102—144no—
378#102—282no—
379C22⋊C8—142no—
380C22⋊C8—282no—
381#95—142no—
382#95—141no—
383#95—142no—
384C22×C7—22no—
385C22×C7—22no—
386C22×C7—24no—
387C22×C7—24no—
388C22×C7—42no—
389C22×C7—41no—
390C22×C7—41no—
391C22×C7—42no—
392C22×C7—41no—
393C22×C7—42no—
394C22×C7—42no—
395C22×C7—42no—
396C22×C7—82no—
397C22×C7—82no—
398C22×C7#45214yes—
399C22×C7C2×#12011yescharacteristic
400C22×C7#54612yes—
401C22×C7C2×#13511yescharacteristic
402C22×C7C2×#13711yescharacteristic
403C22×C7#57712yes—
404C4×C7—22no—
405C4×C7—24no—
406C4×C7—41no—
407C4×C7—42no—
408C4×C7—41no—
409C4×C7—42no—
410C4×C7—41no—
411C4×C7C2×#12311yescharacteristic
412C4×C7#53411yescharacteristic
413C4×C7#54612yes—
414Dic7—82no—
415Dic7—82no—
416C24—21no—
417C24C2×C22⋊D711yescharacteristic
418C24C2×C22⋊D711yescharacteristic
419#41—48no—
420#41—44no—
421#41—44no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—28no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—48no—
432C2×Q8#115914yes—
433C2×Q8C2×C22⋊D712yes—
434C2×Q8C2×C22⋊D712yes—
435C2×D4—24no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—24no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—44no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4C2×C22⋊D712yes—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4—282no—
463C22×C4—562no—
464C22×C4C2×C22⋊D711yescharacteristic
465C22×C4C2×C22⋊D711yescharacteristic
466C22×C4C2×C22⋊D711yescharacteristic
467C22×C4C2×C22⋊D711yescharacteristic
468C22×C4C2×C22⋊D712yes—
469C22×C4C2×C22⋊D712yes—
470C22×C4C23×D711yescharacteristic
471C2×C8—142no—
472C2×C8—142no—
473C2×C8—142no—
474C2×C8—142no—
475C2×C8—142no—
476C2×C8—142no—
477C2×C8—284no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—142no—
483C4⋊C4—142no—
484C4⋊C4—142no—
485C4⋊C4—142no—
486C4⋊C4—282no—
487C4⋊C4—562no—
488C4⋊C4C2×C22⋊D712yes—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—22no—
493C22⋊C4—24no—
494C22⋊C4—22no—
495C22⋊C4—282no—
496C22⋊C4—562no—
497C42—24no—
498C42—21no—
499C42—21no—
500C2×C7—24no—
501C2×C7—81no—
502C2×C7—82no—
503C2×C7#240511yescharacteristic
504C2×C7C2×#45211yescharacteristic
505C2×C7#301011yescharacteristic
506C2×C7#330112yes—
507C2×C7#331612yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—21no—
512C23—21no—
513C23—22no—
514C23—21no—
515C23—22no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23#651312yes—
521C23#651412yes—
522C23#651612yes—
523C23C22×C22⋊D711yescharacteristic
524Q8—22no—
525Q8—24no—
526Q8—22no—
527Q8—22no—
528Q8—48no—
529Q8—42no—
530Q8#649814yes—
531D4—24no—
532D4—48no—
533D4—44no—
534D4—48no—
535D4—41no—
536D4—41no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—282no—
561C2×C4—282no—
562C2×C4—282no—
563C2×C4—282no—
564C2×C4—282no—
565C2×C4—282no—
566C2×C4—564no—
567C2×C4#651612yes—
568C2×C4C2×#115912yes—
569C2×C4#654812yes—
570C2×C4C22×C22⋊D711yescharacteristic
571C2×C4C22×C22⋊D711yescharacteristic
572C8—282no—
573C8—282no—
574C7#2403611yescharacteristic, complemented, Sylow
575C22—24no—
576C22—22no—
577C22—24no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—41no—
582C22—42no—
583C22—41no—
584C22—42no—
585C22—41no—
586C22—42no—
587C22—82no—
588C22—82no—
589C22#9001114yes—
590C22#9049712yes—
591C22C2×#651311yescharacteristic
592C22C2×#651411yescharacteristic
593C22#9052812yes—
594C22C2×#651611yescharacteristic
595C4—22no—
596C4—24no—
597C4—41no—
598C4—42no—
599C4—41no—
600C4—42no—
601C4—41no—
602C4—562no—
603C4—562no—
604C4C2×#649811yescharacteristic
605C4#9049511yescharacteristic
606C4#9049712yes—
607C2—24no—
608C2—81no—
609C2—82no—
610C2#1174098012yes—
611C2#1174098312yes—
612C2#1174149211yescharacteristic
613C2C2×#9001111yescharacteristic
614C2#1174155811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#548C2
socleC23×C7C22×D4
hypercenterC2×#548D7
hypocenterC7#24036
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
2. C22×#6409C2semidirect
3. C2×#89806C2semidirect
4. #11739819C2semidirect
5. #11729889C2semidirect
6. #11729886C2semidirect
7. #11729476C2semidirect
8. #11729448C2semidirect
9. #11729435C2semidirect
10. #11729383C2semidirect
11. #11729364C2semidirect
574. C7#24036semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5482
C22×#64092
C2×#898062
#117398192
#117298892
#117298862
#117294762
#117294482
#117294352
#117293832
#117293642
#240367