Finite Group Databasebeta

← #13284166

Subgroups — #13284166

Order 1792.

1506 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89719C211yesmaximal, characteristic
2C7×#1439C211yesmaximal, characteristic
3#11729318C212yesmaximal, complemented
4#11729299C212yesmaximal
5#11729298C212yesmaximal
6#11729298C211yesmaximal, characteristic
7#11727178C212yesmaximal, complemented
8#11727172C212yesmaximal, complemented
9#11727156C212yesmaximal, complemented
10C24×Dic7C2211yescharacteristic
11C23×Dic14C2211yescharacteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C22⋊C4C2212yes—
14C22×C7×C22⋊C4C2211yescharacteristic
15C22×C22⋊Dic7C2212yes—
16C22×C22⋊Dic7C2211yescharacteristic
17C22×C4×Dic7C2212yes—
18C2×C7×#95C2212yes—
19C2×C7×#95C2211yescharacteristic
20C2×C7⋊#95C2212yes—
21C2×C7⋊#95C2212yes—
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#89719C418yescomplemented
32#14596—71nomaximal, Sylow
33C25×C7C2311yescharacteristic
34C23×C4×C7C2312yes—
35C23×C4×C7C2311yescharacteristic
36C23×C4×C7C2311yescharacteristic
37C23×C4×C7C2312yes—
38C23×Dic7—22no—
39C23×Dic7—24no—
40C23×Dic7—41no—
41C23×Dic7—42no—
42C23×Dic7—41no—
43C23×Dic7C2312yes—
44C23×Dic7C2312yes—
45C23×Dic7C2312yes—
46C23×Dic7C2312yes—
47C22×Dic14—41no—
48C22×Dic14—41no—
49C22×Dic14—42no—
50C22×Dic14—42no—
51C22×Dic14C2×C414yescomplemented
52C2×C7×C4⋊C4—24no—
53C2×C7×C4⋊C4—41no—
54C2×C7×C22⋊C4—24no—
55C2×C7×C22⋊C4—24no—
56C2×C7×C22⋊C4—41no—
57C2×C7×C22⋊C4—41no—
58C2×C7×C22⋊C4—42no—
59C2×C7×C22⋊C4C2×C414yes—
60C2×C22⋊Dic7—42no—
61C2×C22⋊Dic7—42no—
62C2×C22⋊Dic7—41no—
63C2×C22⋊Dic7—41no—
64C2×C22⋊Dic7C2×C418yes—
65C2×C22⋊Dic7Q814yescomplemented
66C2×C4×Dic7—24no—
67C2×C4×Dic7—42no—
68C2×C4×Dic7—42no—
69C7×#95—42no—
70C7×#95—41no—
71C7⋊#95—24no—
72C7⋊#95—24no—
73C7⋊#95—42no—
74C7⋊#95—42no—
75C7⋊#95—42no—
76C7⋊#95—42no—
77C7⋊#95—42no—
78C7⋊#95—42no—
79C7⋊#95—42no—
80C7⋊#95C2×C414yescomplemented
81C7⋊#95C2×C418yescomplemented
82C2×#393—71no—
83#1446—72no—
84#1443—72no—
85#1439D711yescharacteristic
86#1437—72no—
87#1436—72no—
88#1434—72no—
89C22⋊#95—72no—
90C22⋊#95—71no—
91C24×C7—22no—
92C24×C7—22no—
93C24×C7—22no—
94C24×C7—21no—
95C24×C7—22no—
96C24×C7—21no—
97C24×C7—21no—
98C24×C7—21no—
99C24×C7C22×C412yes—
100C24×C7C2×Q812yes—
101C24×C7#4112yes—
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—21no—
115C22×C4×C7—21no—
116C22×C4×C7—21no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7C22×C412yes—
125C22×C4×C7C22×C412yes—
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×Dic7—22no—
151C22×Dic7—44no—
152C22×Dic7—48no—
153C22×Dic7—48no—
154C22×Dic7—44no—
155C22×Dic7—44no—
156C22×Dic7C22×C414yes—
157C22×Dic7C22×C414yes—
158C22×Dic7C2×Q814yes—
159C22×Dic7#4114yes—
160C2×Dic14—48no—
161C2×Dic14—44no—
162C2×Dic14—44no—
163C2×Dic14—48no—
164C2×Dic14—44no—
165C7×C4⋊C4—44no—
166C7×C22⋊C4—44no—
167C7×C22⋊C4—48no—
168C22⋊Dic7—48no—
169C22⋊Dic7—44no—
170C4×Dic7—48no—
171C23×Q8—71no—
172C24×C4—71no—
173C22×C4⋊C4C2×D711yescharacteristic
174C22×C22⋊C4—72no—
175C22×C22⋊C4—71no—
176C22×C22⋊C4C2×D712yes—
177C22×C22⋊C4C2×D711yescharacteristic
178C22×C42—72no—
179#393—78no—
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×#95—72no—
189C2×#95—72no—
190C2×#95—72no—
191C2×#95C2×D712yes—
192C2×#95C2×D711yescharacteristic
193C23×C7—24no—
194C23×C7—24no—
195C23×C7—22no—
196C23×C7—22no—
197C23×C7—22no—
198C23×C7—22no—
199C23×C7—22no—
200C23×C7—22no—
201C23×C7—24no—
202C23×C7—24no—
203C23×C7—24no—
204C23×C7—24no—
205C23×C7—42no—
206C23×C7—42no—
207C23×C7—42no—
208C23×C7—42no—
209C23×C7—41no—
210C23×C7—41no—
211C23×C7—42no—
212C23×C7—41no—
213C23×C7—42no—
214C23×C7—41no—
215C23×C7C4×Q814yes—
216C23×C7C23×C411yescharacteristic
217C23×C7C22×Q811yescharacteristic
218C23×C7C2×#4111yescharacteristic
219C23×C7C2×#4112yes—
220C23×C7C2×#4112yes—
221C23×C7H(5,𝔽2)12yes—
222C23×C7H(5,𝔽2)11yescharacteristic
223C23×C7H(5,𝔽2)11yescharacteristic
224C23×C7H(5,𝔽2)12yes—
225C23×C7H(5,𝔽2)11yescharacteristic
226C23×C72−1+411yescharacteristic
227C2×C4×C7—24no—
228C2×C4×C7—22no—
229C2×C4×C7—22no—
230C2×C4×C7—24no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—41no—
237C2×C4×C7—42no—
238C2×C4×C7—41no—
239C2×C4×C7—42no—
240C2×C4×C7—41no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—41no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×Dic7—24no—
252C2×Dic7—24no—
253C2×Dic7—24no—
254C2×Dic7—24no—
255C2×Dic7—24no—
256C2×Dic7—24no—
257C2×Dic7—24no—
258C2×Dic7—24no—
259C2×Dic7—24no—
260C2×Dic7—24no—
261C2×Dic7—24no—
262C2×Dic7—24no—
263C2×Dic7—416no—
264C2×Dic7—42no—
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—
280Dic14—416no—
281C25C22×D711yescharacteristic
282C22×Q8—74no—
283C22×Q8—282no—
284C22×Q8—281no—
285C22×Q8—281no—
286C22×Q8—282no—
287C23×C4—72no—
288C23×C4—72no—
289C23×C4—72no—
290C23×C4—72no—
291C23×C4—144no—
292C23×C4—142no—
293C23×C4—281no—
294C23×C4—282no—
295C23×C4—281no—
296C23×C4C22×D712yes—
297C23×C4C22×D711yescharacteristic
298C23×C4C22×D711yescharacteristic
299C23×C4C22×D712yes—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—41no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—41no—
305C2×C22⋊C4—41no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—78no—
308C2×C22⋊C4—74no—
309C2×C22⋊C4—281no—
310C2×C22⋊C4—282no—
311C2×C22⋊C4—282no—
312C2×C22⋊C4—281no—
313C2×C22⋊C4C4×D714yes—
314C2×C42—144no—
315C2×C42—282no—
316C2×C42—282no—
317#95—42no—
318#95—41no—
319#95—74no—
320#95—78no—
321#95—144no—
322#95—144no—
323#95—282no—
324#95—282no—
325#95—282no—
326#95—282no—
327#95—282no—
328#95—282no—
329#95—282no—
330C22×C7—24no—
331C22×C7—28no—
332C22×C7—44no—
333C22×C7—42no—
334C22×C7—42no—
335C22×C7—42no—
336C22×C7—44no—
337C22×C7—42no—
338C22×C7—42no—
339C22×C7—42no—
340C22×C7—44no—
341C22×C7C2×C4×Q811yescharacteristic
342C22×C7C4×#4112yes—
343C22×C7#51811yescharacteristic
344C22×C7#51812yes—
345C22×C7#52011yescharacteristic
346C22×C7#53812yes—
347C22×C7#53811yescharacteristic
348C22×C7#53812yes—
349C22×C7#53911yescharacteristic
350C22×C7#54212yes—
351C22×C7#54212yes—
352C22×C7#54311yescharacteristic
353C22×C7#54411yescharacteristic
354C22×C7#55112yes—
355C22×C7#55111yescharacteristic
356C22×C7#55112yes—
357C22×C7#55111yescharacteristic
358C22×C7#55412yes—
359C22×C7#55412yes—
360C22×C7#55512yes—
361C22×C7#56011yescharacteristic
362C22×C7#56012yes—
363C22×C7H(3,𝔽4)11yescharacteristic
364C4×C7—42no—
365C4×C7—42no—
366C4×C7—44no—
367C4×C7—82no—
368Dic7—44no—
369Dic7—44no—
370Dic7—44no—
371Dic7—44no—
372C24—22no—
373C24—21no—
374C24—22no—
375C24—21no—
376C24—22no—
377C24—21no—
378C24—22no—
379C24—21no—
380C24C2×C4×D712yes—
381C24#115912yes—
382C24Q8×D712yes—
383C24C23×D711yescharacteristic
384C2×Q8—284no—
385C2×Q8—288no—
386C2×Q8—284no—
387C2×Q8—284no—
388C2×Q8—288no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—74no—
411C22×C4—74no—
412C22×C4—74no—
413C22×C4—74no—
414C22×C4—142no—
415C22×C4—142no—
416C22×C4—142no—
417C22×C4—142no—
418C22×C4—142no—
419C22×C4—142no—
420C22×C4—142no—
421C22×C4—142no—
422C22×C4—142no—
423C22×C4—142no—
424C22×C4—142no—
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—284no—
439C22×C4—288no—
440C22×C4—288no—
441C22×C4—284no—
442C22×C4—284no—
443C22×C4C2×C4×D712yes—
444C22×C4C2×C4×D712yes—
445C22×C4#115914yes—
446C4⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—48no—
449C22⋊C4—288no—
450C22⋊C4—284no—
451C42—288no—
452C2×C7—44no—
453C2×C7#232911yescharacteristic
454C2×C7#233712yes—
455C2×C7#236812yes—
456C2×C7#237211yescharacteristic
457C2×C7#237511yescharacteristic
458C2×C7#273611yescharacteristic
459C2×C7#273712yes—
460C2×C7#274211yescharacteristic
461C2×C7#276011yescharacteristic
462C2×C7#276311yescharacteristic
463C2×C7#278412yes—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—42no—
477C23—41no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—42no—
486C23#645014yes—
487C23C22×C4×D711yescharacteristic
488C23C2×#115912yes—
489C23C2×#115911yescharacteristic
490C23#654811yescharacteristic
491C23#654812yes—
492C23#654811yescharacteristic
493C23C2×Q8×D711yescharacteristic
494C23D7×#4112yes—
495C23#655312yes—
496C23#655311yescharacteristic
497C23#655411yescharacteristic
498Q8—2816no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—144no—
524C2×C4—144no—
525C2×C4—144no—
526C2×C4—144no—
527C2×C4—144no—
528C2×C4—144no—
529C2×C4—144no—
530C2×C4—144no—
531C2×C4—144no—
532C2×C4—144no—
533C2×C4—144no—
534C2×C4—144no—
535C2×C4—2816no—
536C2×C4—282no—
537C2×C4—282no—
538C2×C4—282no—
539C2×C4—282no—
540C2×C4—282no—
541C2×C4—282no—
542C2×C4—282no—
543C2×C4—282no—
544C2×C4—282no—
545C2×C4—282no—
546C2×C4—282no—
547C2×C4—282no—
548C2×C4—282no—
549C2×C4—282no—
550C2×C4—282no—
551C2×C4—282no—
552C7#1459611yescharacteristic, complemented, Sylow
553C22—24no—
554C22—28no—
555C22—42no—
556C22—44no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—42no—
564C22#9017711yescharacteristic
565C22C2×#645011yescharacteristic
566C22#9021411yescharacteristic
567C22C4×#115912yes—
568C22#9023612yes—
569C22#9027811yescharacteristic
570C22#9027812yes—
571C22#9029311yescharacteristic
572C22#9030712yes—
573C22#9030812yes—
574C22#9030811yescharacteristic
575C22#9031112yes—
576C22#9031411yescharacteristic
577C22#9032511yescharacteristic
578C22#9032612yes—
579C22#9032712yes—
580C22#9034912yes—
581C22#9035011yescharacteristic
582C22#9035312yes—
583C22#9037312yes—
584C22#9037411yescharacteristic
585C22#9039512yes—
586C22#9041111yescharacteristic
587C4—42no—
588C4—42no—
589C4—44no—
590C4—82no—
591C4—284no—
592C4—284no—
593C4—284no—
594C4—284no—
595C2—44no—
596C2#1173131911yescharacteristic
597C2#1173184412yes—
598C2#1173212511yescharacteristic
599C2#1173291712yes—
600C2#1173387911yescharacteristic
601C2#1173418211yescharacteristic
602C2#1173428811yescharacteristic
603C2#1173449312yes—
604C2#1173489112yes—
605C2#1173515111yescharacteristic
606C2#1173552111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1439C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenter#1439D7
hypocenterC7#14596
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
3. #11729318C2semidirect
7. #11727178C2semidirect
8. #11727172C2semidirect
9. #11727156C2semidirect
31. #89719C4semidirect
51. C22×Dic14C2×C4semidirect
65. C2×C22⋊Dic7Q8semidirect
80. C7⋊#95C2×C4semidirect
81. C7⋊#95C2×C4semidirect
552. C7#14596semidirect

Maximal subgroups

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

subgroupindex
C2×#897192
C7×#14392
#117293182
#117292992
#117292982
#117271782
#117271722
#117271562
#145967