Finite Group Databasebeta

← #13283460

Subgroups — #13283460

Order 1792.

1490 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89721C211yesmaximal, characteristic
2C2×#89716C212yesmaximal
3C2×#89705C212yesmaximal, complemented
4C7×#1437C211yesmaximal, characteristic
5#11729302C211yesmaximal, characteristic
6#11729301C212yesmaximal
7#11727165C212yesmaximal, complemented
8#11727159C212yesmaximal, complemented
9Dic7⋊#95C212yesmaximal, complemented
10C22×C7×C22⋊C4C2212yes—
11C22×C7×C22⋊C4C2211yescharacteristic
12C22×C42×C7C2211yescharacteristic
13C22×C22⋊Dic7C2211yescharacteristic
14C22×C22⋊Dic7C2212yes—
15C22×C22⋊Dic7C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×Dic7⋊C4C2212yes—
18C22×Dic7⋊C4C2212yes—
19C22×Dic7⋊C4C2212yes—
20C22×C4×Dic7C2212yes—
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⋊#95C2212yes—
31#89721C418yescomplemented
32#89716—28no—
33#89705—28no—
34#14543—71nomaximal, Sylow
35C25×C7C2311yescharacteristic
36C23×C4×C7C2311yescharacteristic
37C23×C4×C7C2311yescharacteristic
38C23×C4×C7C2312yes—
39C23×C4×C7C2312yes—
40C23×Dic7C2312yes—
41C23×Dic7C2312yes—
42C23×Dic7C2312yes—
43C23×Dic7C2312yes—
44C2×C7×C22⋊C4—24no—
45C2×C7×C22⋊C4—24no—
46C2×C7×C22⋊C4—41no—
47C2×C7×C22⋊C4—41no—
48C2×C7×C22⋊C4—42no—
49C2×C7×C22⋊C4C2×C414yes—
50C2×C42×C7—24no—
51C2×C42×C7—41no—
52C2×C22⋊Dic7—24no—
53C2×C22⋊Dic7—24no—
54C2×C22⋊Dic7—24no—
55C2×C22⋊Dic7—41no—
56C2×C22⋊Dic7—41no—
57C2×C22⋊Dic7—42no—
58C2×C22⋊Dic7C2×C418yes—
59C2×C4⋊Dic7—41no—
60C2×C4⋊Dic7—41no—
61C2×C4⋊Dic7C2×C414yes—
62C2×Dic7⋊C4—24no—
63C2×Dic7⋊C4—24no—
64C2×Dic7⋊C4—24no—
65C2×Dic7⋊C4—24no—
66C2×Dic7⋊C4—24no—
67C2×Dic7⋊C4—24no—
68C2×Dic7⋊C4—42no—
69C2×Dic7⋊C4—42no—
70C2×Dic7⋊C4C2×C418yes—
71C2×C4×Dic7—24no—
72C2×C4×Dic7—24no—
73C2×C4×Dic7—42no—
74C7×#95—41no—
75C7×#95—42no—
76C7⋊#95—24no—
77C7⋊#95—24no—
78C7⋊#95—24no—
79C7⋊#95—24no—
80C7⋊#95—42no—
81C7⋊#95—42no—
82C7⋊#95C2×C414yes—
83C7⋊#95D418yes—
84C2×#399—71no—
85C2×#388—72no—
86C2×#387—72no—
87#1441—72no—
88#1440—71no—
89#1439—72no—
90#1437D711yescharacteristic
91#1436—72no—
92C4⋊#95—72no—
93C24×C7—22no—
94C24×C7—22no—
95C24×C7—22no—
96C24×C7—21no—
97C24×C7—21no—
98C24×C7—22no—
99C24×C7—22no—
100C24×C7—22no—
101C24×C7C22×C412yes—
102C24×C7C2411yescharacteristic
103C22×C4×C7—22no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—21no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—21no—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7C22×C412yes—
124C22×C4×C7C22×C412yes—
125C22×C4×C7C2×D414yes—
126C22×C4×C7#4114yes—
127C22×Dic7—22no—
128C22×Dic7—22no—
129C22×Dic7—22no—
130C22×Dic7—22no—
131C22×Dic7—22no—
132C22×Dic7—22no—
133C22×Dic7—22no—
134C22×Dic7—22no—
135C22×Dic7—22no—
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×Dic7C22×C414yes—
151C22×Dic7C22×C414yes—
152C22×Dic7C2×D414yes—
153C22×Dic7C2×D414yes—
154C22×Dic7#4114yes—
155C7×C22⋊C4—44no—
156C7×C22⋊C4—48no—
157C42×C7—44no—
158C22⋊Dic7—216no—
159C22⋊Dic7—44no—
160C22⋊Dic7—44no—
161C4⋊Dic7—44no—
162Dic7⋊C4—216no—
163Dic7⋊C4—48no—
164Dic7⋊C4—48no—
165C4×Dic7—48no—
166C22×C4⋊C4—72no—
167C22×C4⋊C4—71no—
168C22×C4⋊C4—72no—
169C22×C4⋊C4—72no—
170C22×C22⋊C4—72no—
171C22×C22⋊C4—71no—
172C22×C22⋊C4—71no—
173C22×C22⋊C4C2×D712yes—
174C22×C22⋊C4C2×D711yescharacteristic
175C22×C42—72no—
176C22×C42C2×D711yescharacteristic
177#399—78no—
178#388—148no—
179#387—148no—
180C2×#95—71no—
181C2×#95—72no—
182C2×#95—72no—
183C2×#95—72no—
184C2×#95—72no—
185C2×#95—72no—
186C2×#95—72no—
187C2×#95—72no—
188C2×#95C2×D711yescharacteristic
189C2×#95C2×D712yes—
190C23×C7—24no—
191C23×C7—22no—
192C23×C7—22no—
193C23×C7—22no—
194C23×C7—24no—
195C23×C7—42no—
196C23×C7—42no—
197C23×C7—42no—
198C23×C7—42no—
199C23×C7—42no—
200C23×C7—42no—
201C23×C7—41no—
202C23×C7—42no—
203C23×C7—42no—
204C23×C7—42no—
205C23×C7—42no—
206C23×C7—42no—
207C23×C7—41no—
208C23×C7—41no—
209C23×C7—42no—
210C23×C7—41no—
211C23×C7C23×C411yescharacteristic
212C23×C7C22×D412yes—
213C23×C7C2×#4112yes—
214C23×C7C2×#4112yes—
215C23×C7H(5,𝔽2)11yescharacteristic
216C23×C7H(5,𝔽2)11yescharacteristic
217C23×C7H(5,𝔽2)12yes—
218C23×C72−1+412yes—
219C23×C72−1+412yes—
220C2×C4×C7—24no—
221C2×C4×C7—22no—
222C2×C4×C7—22no—
223C2×C4×C7—24no—
224C2×C4×C7—24no—
225C2×C4×C7—24no—
226C2×C4×C7—24no—
227C2×C4×C7—24no—
228C2×C4×C7—24no—
229C2×C4×C7—24no—
230C2×C4×C7—42no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×C4×C7—41no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—41no—
244C2×Dic7—24no—
245C2×Dic7—24no—
246C2×Dic7—24no—
247C2×Dic7—24no—
248C2×Dic7—24no—
249C2×Dic7—24no—
250C2×Dic7—24no—
251C2×Dic7—24no—
252C2×Dic7—24no—
253C2×Dic7—24no—
254C2×Dic7—24no—
255C2×Dic7—24no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7—42no—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—42no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7C4×D418yes—
272C25C22×D711yescharacteristic
273C23×C4—72no—
274C23×C4—72no—
275C23×C4—72no—
276C23×C4—72no—
277C23×C4C22×D711yescharacteristic
278C23×C4C22×D711yescharacteristic
279C23×C4C22×D712yes—
280C23×C4C22×D712yes—
281C2×C4⋊C4—78no—
282C2×C4⋊C4—74no—
283C2×C4⋊C4—144no—
284C2×C4⋊C4—144no—
285C2×C4⋊C4—144no—
286C2×C4⋊C4—144no—
287C2×C4⋊C4—144no—
288C2×C4⋊C4—144no—
289C2×C4⋊C4—282no—
290C2×C4⋊C4—282no—
291C2×C4⋊C4—281no—
292C2×C4⋊C4—281no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—78no—
299C2×C22⋊C4—144no—
300C2×C22⋊C4—144no—
301C2×C22⋊C4—144no—
302C2×C22⋊C4—281no—
303C2×C22⋊C4—281no—
304C2×C22⋊C4—282no—
305C2×C22⋊C4C4×D714yes—
306C2×C42—24no—
307C2×C42—41no—
308C2×C42—144no—
309C2×C42—144no—
310C2×C42—282no—
311#95—41no—
312#95—42no—
313#95—74no—
314#95—78no—
315#95—144no—
316#95—144no—
317#95—144no—
318#95—144no—
319#95—282no—
320#95—282no—
321C22×C7—44no—
322C22×C7—42no—
323C22×C7—44no—
324C22×C7—42no—
325C22×C7—42no—
326C22×C7—82no—
327C22×C7—82no—
328C22×C7—82no—
329C22×C7—82no—
330C22×C7C2×C4×D412yes—
331C22×C7C4×#4111yescharacteristic
332C22×C7#51811yescharacteristic
333C22×C7#51911yescharacteristic
334C22×C7#52012yes—
335C22×C7#53612yes—
336C22×C7#53712yes—
337C22×C7#53912yes—
338C22×C7#54112yes—
339C22×C7#54212yes—
340C22×C7#54212yes—
341C22×C7D4211yescharacteristic
342C22×C7#54612yes—
343C22×C7#54712yes—
344C22×C7#55012yes—
345C22×C7#55112yes—
346C22×C7#55212yes—
347C22×C7#55911yescharacteristic
348C22×C7#56011yescharacteristic
349C22×C7#56212yes—
350C22×C7#56311yescharacteristic
351C4×C7—42no—
352C4×C7—42no—
353C4×C7—44no—
354C4×C7—44no—
355Dic7—28no—
356Dic7—44no—
357Dic7—44no—
358Dic7—82no—
359C24—22no—
360C24—22no—
361C24—21no—
362C24—21no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24C2×C4×D712yes—
368C24C23×D711yescharacteristic
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—74no—
390C22×C4—74no—
391C22×C4—74no—
392C22×C4—74no—
393C22×C4—74no—
394C22×C4—142no—
395C22×C4—142no—
396C22×C4—142no—
397C22×C4—142no—
398C22×C4—142no—
399C22×C4—142no—
400C22×C4—142no—
401C22×C4—142no—
402C22×C4—142no—
403C22×C4—142no—
404C22×C4—142no—
405C22×C4—142no—
406C22×C4—142no—
407C22×C4—142no—
408C22×C4—142no—
409C22×C4—142no—
410C22×C4—142no—
411C22×C4—142no—
412C22×C4—142no—
413C22×C4—142no—
414C22×C4—142no—
415C22×C4—142no—
416C22×C4—142no—
417C22×C4C2×C4×D712yes—
418C22×C4C2×C4×D712yes—
419C22×C4#115714yes—
420C22×C4D4×D714yes—
421C4⋊C4—1416no—
422C4⋊C4—284no—
423C4⋊C4—288no—
424C4⋊C4—288no—
425C22⋊C4—44no—
426C22⋊C4—48no—
427C22⋊C4—1416no—
428C22⋊C4—284no—
429C22⋊C4—284no—
430C42—44no—
431C42—288no—
432C2×C7—82no—
433C2×C7#232312yes—
434C2×C7#233211yescharacteristic
435C2×C7#235912yes—
436C2×C7#236611yescharacteristic
437C2×C7#237611yescharacteristic
438C2×C7#267212yes—
439C2×C7#270512yes—
440C2×C7#272112yes—
441C2×C7#275412yes—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—24no—
447C23—42no—
448C23—41no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23—42no—
462C23—41no—
463C23C22×C4×D711yescharacteristic
464C23C2×#115712yes—
465C23C2×D4×D712yes—
466C23#654811yescharacteristic
467C23#654812yes—
468C23#654811yescharacteristic
469C23D7×#4112yes—
470C23#655412yes—
471C23#655412yes—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—78no—
497C2×C4—144no—
498C2×C4—144no—
499C2×C4—144no—
500C2×C4—144no—
501C2×C4—144no—
502C2×C4—144no—
503C2×C4—144no—
504C2×C4—144no—
505C2×C4—144no—
506C2×C4—144no—
507C2×C4—144no—
508C2×C4—144no—
509C2×C4—282no—
510C2×C4—282no—
511C2×C4—282no—
512C2×C4—282no—
513C2×C4—282no—
514C2×C4—282no—
515C2×C4—282no—
516C2×C4—282no—
517C2×C4—282no—
518C2×C4—282no—
519C2×C4—282no—
520C2×C4—282no—
521C2×C4—282no—
522C2×C4—282no—
523C2×C4—282no—
524C7#1454311yescharacteristic, complemented, Sylow
525C22—44no—
526C22—42no—
527C22—44no—
528C22—42no—
529C22—42no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22C4×#115711yescharacteristic
535C22#9017711yescharacteristic
536C22#9018112yes—
537C22#9020711yescharacteristic
538C22#9022012yes—
539C22C4×D4×D712yes—
540C22#9023712yes—
541C22#9024212yes—
542C22#9024712yes—
543C22#9025212yes—
544C22#9025512yes—
545C22#9028111yescharacteristic
546C22#9029012yes—
547C22#9029912yes—
548C22#9031212yes—
549C22#9033912yes—
550C22#9035511yescharacteristic
551C22#9036012yes—
552C22#9036311yescharacteristic
553C22#9039212yes—
554C22#9041111yescharacteristic
555C4—44no—
556C4—44no—
557C4—42no—
558C4—42no—
559C4—148no—
560C4—284no—
561C4—284no—
562C4—562no—
563C2—82no—
564C2#1173089311yescharacteristic
565C2#1173102312yes—
566C2#1173212411yescharacteristic
567C2#1173235912yes—
568C2#1173290512yes—
569C2#1173314512yes—
570C2#1173318212yes—
571C2#1173375411yescharacteristic
572C2#1173383412yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1437C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenter#1437D7
hypocenterC7#14543
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
3. C2×#89705C2semidirect
7. #11727165C2semidirect
8. #11727159C2semidirect
9. Dic7⋊#95C2semidirect
31. #89721C4semidirect
524. C7#14543semidirect

Maximal subgroups

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

subgroupindex
C2×#897212
C2×#897162
C2×#897052
C7×#14372
#117293022
#117293012
#117271652
#117271592
Dic7⋊#952
#145437