Finite Group Databasebeta

← #13284639

Subgroups — #13284639

Order 1792.

1486 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#394C211yesmaximal, characteristic
2#11729885C212yesmaximal
3#11729323C211yesmaximal, characteristic, complemented
4#11729318C211yesmaximal, characteristic, complemented
5#11729302C212yesmaximal, complemented
6#11729300C212yesmaximal, complemented
7#11729300C212yesmaximal, complemented
8#11729299C212yesmaximal, complemented
9#11727166C212yesmaximal, complemented
10C23×C7×D4C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C7×C22⋊C4C2212yes—
13C22×C7×C22⋊C4C2212yes—
14C22×C22⋊Dic7C2212yes—
15C22×C22⋊Dic7C2212yes—
16C22×C22⋊Dic7C2212yes—
17C22×C22⋊Dic7C2212yes—
18C22×C4⋊Dic7C2212yes—
19C22×C4×Dic7C2211yescharacteristic
20C7×#394C418yescomplemented
21C2×C7×#95C2211yescharacteristic
22C2×C7⋊#95C2212yes—
23C2×C7⋊#95C2212yes—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2211yescharacteristic
31C2×C7⋊#95C2211yescharacteristic
32#14641—71nomaximal, Sylow
33C25×C7C2312yes—
34C22×C7×D4—42no—
35C22×C7×D4—42no—
36C22×C7×D4—41no—
37C22×C7×D4—41no—
38C22×C7×D4C2×C414yes—
39C23×C4×C7C2311yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×Dic7C2312yes—
44C23×Dic7C2312yes—
45C23×Dic7C2312yes—
46C23×Dic7C2312yes—
47C2×C7×C4⋊C4—41no—
48C2×C7×C4⋊C4—41no—
49C2×C7×C4⋊C4C2×C414yes—
50C2×C7×C22⋊C4—42no—
51C2×C7×C22⋊C4—42no—
52C2×C7×C22⋊C4—42no—
53C2×C7×C22⋊C4—42no—
54C2×C7×C22⋊C4C2×C418yes—
55C2×C7×C22⋊C4C2×C418yes—
56C2×C22⋊Dic7—24no—
57C2×C22⋊Dic7—24no—
58C2×C22⋊Dic7—42no—
59C2×C22⋊Dic7—42no—
60C2×C22⋊Dic7—42no—
61C2×C22⋊Dic7—42no—
62C2×C22⋊Dic7—42no—
63C2×C22⋊Dic7—42no—
64C2×C22⋊Dic7—42no—
65C2×C22⋊Dic7—42no—
66C2×C22⋊Dic7—42no—
67C2×C22⋊Dic7—42no—
68C2×C4⋊Dic7—42no—
69C2×C4⋊Dic7—42no—
70C2×C4⋊Dic7—42no—
71C2×C4×Dic7—42no—
72C2×C4×Dic7—41no—
73C7×#95C2×C414yes—
74C7⋊#95—24no—
75C7⋊#95—42no—
76C7⋊#95—42no—
77C7⋊#95—42no—
78C7⋊#95—42no—
79C7⋊#95—42no—
80C7⋊#95—42no—
81C7⋊#95—41no—
82C7⋊#95—41no—
83C7⋊#95—41no—
84C2×#394D711yescharacteristic
85#1445—71no—
86#1442—72no—
87#1440—72no—
88#1439—72no—
89#1439—72no—
90#1438—72no—
91#1437—72no—
92#1436—71no—
93C24×C7—22no—
94C24×C7—22no—
95C24×C7—22no—
96C24×C7—22no—
97C24×C7—22no—
98C24×C7—22no—
99C24×C7—22no—
100C24×C7—22no—
101C24×C7—22no—
102C24×C7—22no—
103C24×C7—22no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7—22no—
107C24×C7C22×C414yes—
108C24×C7C2411yescharacteristic
109C2×C7×D4—44no—
110C2×C7×D4—48no—
111C2×C7×D4—48no—
112C2×C7×D4—44no—
113C2×C7×D4—44no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—21no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—21no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7C22×C414yes—
133C22×C4×C7C22×C412yes—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C22×C412yes—
136C22×Dic7—22no—
137C22×Dic7—22no—
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7#4114yes—
164C7×C4⋊C4—44no—
165C7×C22⋊C4—48no—
166C7×C22⋊C4—48no—
167C22⋊Dic7—48no—
168C22⋊Dic7—48no—
169C22⋊Dic7—48no—
170C22⋊Dic7—48no—
171C4⋊Dic7—48no—
172C4×Dic7—44no—
173C23×D4C2×D711yescharacteristic
174C22×C4⋊C4—72no—
175C22×C4⋊C4C2×D711yescharacteristic
176C22×C22⋊C4—72no—
177C22×C22⋊C4—72no—
178C22×C22⋊C4—72no—
179C22×C22⋊C4—72no—
180C22×C22⋊C4C2×D712yes—
181C22×C22⋊C4C2×D712yes—
182C22×C42—71no—
183#394Dic718yescomplemented
184C2×#95—72no—
185C2×#95—72no—
186C2×#95—71no—
187C2×#95—72no—
188C2×#95—72no—
189C2×#95—72no—
190C2×#95—72no—
191C2×#95—71no—
192C2×#95—72no—
193C2×#95—71no—
194C2×#95C2×D711yescharacteristic
195C23×C7—24no—
196C23×C7—24no—
197C23×C7—24no—
198C23×C7—24no—
199C23×C7—24no—
200C23×C7—24no—
201C23×C7—24no—
202C23×C7—42no—
203C23×C7—42no—
204C23×C7—42no—
205C23×C7—42no—
206C23×C7—42no—
207C23×C7—42no—
208C23×C7—42no—
209C23×C7—42no—
210C23×C7—42no—
211C23×C7—42no—
212C23×C7—42no—
213C23×C7—42no—
214C23×C7—42no—
215C23×C7—42no—
216C23×C7—42no—
217C23×C7—42no—
218C23×C7—42no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7—42no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7C23×C411yescharacteristic
231C23×C7C2×#4112yes—
232C23×C7H(5,𝔽2)11yescharacteristic
233C23×C7H(5,𝔽2)12yes—
234C23×C7H(5,𝔽2)12yes—
235C23×C7H(5,𝔽2)11yescharacteristic
236C23×C7H(5,𝔽2)12yes—
237C23×C7H(5,𝔽2)12yes—
238C23×C72−1+412yes—
239C7×D4—416no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—24no—
243C2×C4×C7—24no—
244C2×C4×C7—24no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—41no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×Dic7—24no—
263C2×Dic7—24no—
264C2×Dic7—24no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C25C22×D712yes—
291C22×D4—42no—
292C22×D4—42no—
293C22×D4—41no—
294C22×D4—41no—
295C22×D4C2×Dic714yes—
296C23×C4—72no—
297C23×C4—72no—
298C23×C4—72no—
299C23×C4—72no—
300C23×C4C22×D712yes—
301C23×C4C22×D711yescharacteristic
302C23×C4C22×D711yescharacteristic
303C23×C4C22×D711yescharacteristic
304C2×C4⋊C4—41no—
305C2×C4⋊C4—41no—
306C2×C4⋊C4—282no—
307C2×C4⋊C4—282no—
308C2×C4⋊C4—282no—
309C2×C4⋊C4C2×Dic714yes—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—144no—
315C2×C22⋊C4—144no—
316C2×C22⋊C4—282no—
317C2×C22⋊C4—282no—
318C2×C22⋊C4—282no—
319C2×C22⋊C4—282no—
320C2×C22⋊C4—282no—
321C2×C22⋊C4—282no—
322C2×C22⋊C4—282no—
323C2×C22⋊C4—282no—
324C2×C22⋊C4—282no—
325C2×C22⋊C4—282no—
326C2×C22⋊C4C2×Dic718yes—
327C2×C22⋊C4C2×Dic718yes—
328C2×C42—282no—
329C2×C42—281no—
330#95—144no—
331#95—282no—
332#95—282no—
333#95—282no—
334#95—281no—
335#95—281no—
336#95—281no—
337#95—282no—
338#95—282no—
339#95—282no—
340#95C2×Dic714yes—
341C22×C7—44no—
342C22×C7—44no—
343C22×C7—44no—
344C22×C7—44no—
345C22×C7—44no—
346C22×C7—44no—
347C22×C7—44no—
348C22×C7—82no—
349C22×C7—82no—
350C22×C7—82no—
351C22×C7—82no—
352C22×C7—82no—
353C22×C7—82no—
354C22×C7—82no—
355C22×C7—82no—
356C22×C7C4×#4111yescharacteristic
357C22×C7#51811yescharacteristic
358C22×C7#51811yescharacteristic
359C22×C7#51812yes—
360C22×C7#52012yes—
361C22×C7#53812yes—
362C22×C7#53812yes—
363C22×C7#53912yes—
364C22×C7#54012yes—
365C22×C7#54012yes—
366C22×C7#54212yes—
367C22×C7#56012yes—
368C22×C7#56011yescharacteristic
369C22×C7#56012yes—
370C22×C7#56012yes—
371C22×C7H(3,𝔽4)11yescharacteristic
372C22×C7H(3,𝔽4)11yescharacteristic
373C22×C7#56212yes—
374C22×C7#56212yes—
375C22×C7#56212yes—
376C22×C7#56311yescharacteristic
377C4×C7—42no—
378C4×C7—42no—
379C4×C7—42no—
380C4×C7—44no—
381Dic7—44no—
382Dic7—82no—
383Dic7—82no—
384Dic7—82no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24C22×Dic714yes—
400C24C23×D711yescharacteristic
401C2×D4—44no—
402C2×D4—48no—
403C2×D4—44no—
404C2×D4—48no—
405C2×D4—44no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—74no—
425C22×C4—142no—
426C22×C4—142no—
427C22×C4—142no—
428C22×C4—142no—
429C22×C4—142no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4—142no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—142no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—142no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—142no—
452C22×C4C22×Dic714yes—
453C22×C4C22×Dic712yes—
454C22×C4C22×Dic712yes—
455C22×C4C22×Dic712yes—
456C4⋊C4—44no—
457C4⋊C4—288no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—288no—
461C22⋊C4—288no—
462C22⋊C4—288no—
463C22⋊C4—288no—
464C42—284no—
465C2×C7—84no—
466C2×C7#233311yescharacteristic
467C2×C7#233512yes—
468C2×C7#237511yescharacteristic
469C2×C7#237712yes—
470C2×C7#237911yescharacteristic
471C2×C7#282412yes—
472C2×C7#282812yes—
473C2×C7#282912yes—
474C2×C7#283712yes—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23#654811yescharacteristic
511C23#654811yescharacteristic
512C23#654812yes—
513C23#654812yes—
514C23#654812yes—
515C23D7×#4112yes—
516C23#655312yes—
517C23#655412yes—
518C23C23×Dic711yescharacteristic
519D4—416no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—144no—
543C2×C4—144no—
544C2×C4—144no—
545C2×C4—282no—
546C2×C4—282no—
547C2×C4—282no—
548C2×C4—282no—
549C2×C4—282no—
550C2×C4—282no—
551C2×C4—282no—
552C2×C4—282no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4—282no—
556C2×C4—282no—
557C2×C4—282no—
558C2×C4—282no—
559C2×C4—282no—
560C2×C4—282no—
561C2×C4—282no—
562C2×C4—282no—
563C2×C4—282no—
564C2×C4—282no—
565C2×C4—282no—
566C2×C4—282no—
567C2×C4—282no—
568C2×C4—282no—
569C2×C4—282no—
570C7#1464111yescharacteristic, complemented, Sylow
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22#9028511yescharacteristic
587C22#9028612yes—
588C22#9028611yescharacteristic
589C22#9030412yes—
590C22#9030612yes—
591C22#9031112yes—
592C22#9031112yes—
593C22#9031212yes—
594C22#9031212yes—
595C22#9034812yes—
596C22#9035112yes—
597C22#9035312yes—
598C22#9035412yes—
599C22#9036311yescharacteristic
600C22#9037411yescharacteristic
601C22#9037512yes—
602C22#9048912yes—
603C22#9048911yescharacteristic
604C22#9048911yescharacteristic
605C22#41×Dic711yescharacteristic
606C22#9051812yes—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—44no—
611C4—284no—
612C4—562no—
613C4—562no—
614C4—562no—
615C2—84no—
616C2#1173453812yes—
617C2#1173454112yes—
618C2#1173454812yes—
619C2#1173460912yes—
620C2#1174002511yescharacteristic
621C2#1174008011yescharacteristic
622C2#1174008412yes—
623C2#1174028212yes—
624C2#1174040211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#394C2
socle, cosocleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC2×#394D7
hypocenterC7#14641
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
3. #11729323C2semidirect
4. #11729318C2semidirect
5. #11729302C2semidirect
6. #11729300C2semidirect
7. #11729300C2semidirect
8. #11729299C2semidirect
9. #11727166C2semidirect
20. C7×#394C4semidirect
183. #394Dic7semidirect
570. C7#14641semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3942
#117298852
#117293232
#117293182
#117293022
#117293002
#117292992
#117271662
#146417