Finite Group Databasebeta

← #13282337

Subgroups — #13282337

Order 1792.

1450 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89984C211yesmaximal, characteristic
2C7×C22⋊#95C211yesmaximal, characteristic
3#11729881C211yesmaximal, characteristic
4#11729301C212yesmaximal
5#11729299C212yesmaximal
6#11727171C212yesmaximal, complemented
7#11727165C212yesmaximal, complemented
8#11727161C212yesmaximal, complemented
9#11727160C212yesmaximal, complemented
10C24×C4×C7C2211yescharacteristic
11C22×C7×C22⋊C4C2211yescharacteristic
12C22×C7×C22⋊C4C2211yescharacteristic
13C22×C22⋊Dic7C2212yes—
14C22×C22⋊Dic7C2212yes—
15C22×C4⋊Dic7C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×Dic7⋊C4C2212yes—
18C22×C4×Dic7C2212yes—
19C2×C7×#95C2212yes—
20C2×C7×#95C2212yes—
21#89984C418yescomplemented
22C2×C7⋊#95C2212yes—
23C2×C7⋊#95C2212yes—
24C2×C7⋊#95C2211yescharacteristic
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2211yescharacteristic
31C2×C7⋊#95C2212yes—
32#14453—71nomaximal, Sylow
33C25×C7C2311yescharacteristic
34C23×C4×C7—24no—
35C23×C4×C7—41no—
36C23×C4×C7—41no—
37C23×C4×C7—42no—
38C23×C4×C7C2×C414yes—
39C23×C4×C7C2312yes—
40C23×C4×C7C2311yescharacteristic
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×Dic7C2312yes—
44C23×Dic7C2312yes—
45C23×Dic7C2312yes—
46C23×Dic7C2312yes—
47C2×C7×C22⋊C4—22no—
48C2×C7×C22⋊C4—41no—
49C2×C7×C22⋊C4—41no—
50C2×C7×C22⋊C4—41no—
51C2×C7×C22⋊C4—41no—
52C2×C7×C22⋊C4Q814yes—
53C2×C22⋊Dic7—24no—
54C2×C22⋊Dic7—24no—
55C2×C22⋊Dic7—42no—
56C2×C22⋊Dic7—42no—
57C2×C22⋊Dic7—42no—
58C2×C22⋊Dic7C2×C418yes—
59C2×C4⋊Dic7—41no—
60C2×C4⋊Dic7—42no—
61C2×C4⋊Dic7—41no—
62C2×C4⋊Dic7C2×C414yes—
63C2×C4⋊Dic7C2×C414yes—
64C2×Dic7⋊C4—24no—
65C2×Dic7⋊C4—42no—
66C2×Dic7⋊C4—42no—
67C2×C4×Dic7—42no—
68C2×C4×Dic7—42no—
69C2×C4×Dic7—42no—
70C7×#95—24no—
71C7×#95—42no—
72C7⋊#95—24no—
73C7⋊#95—24no—
74C7⋊#95—42no—
75C7⋊#95—41no—
76C7⋊#95—42no—
77C7⋊#95—42no—
78C7⋊#95—42no—
79C7⋊#95—42no—
80C7⋊#95—41no—
81C7⋊#95C2×C418yes—
82C2×#380—71no—
83#1443—72no—
84#1441—72no—
85#1439—72no—
86#1437—72no—
87#1436—72no—
88#1436—72no—
89C22⋊#95—71no—
90C22⋊#95D711yescharacteristic
91C24×C7—22no—
92C24×C7—22no—
93C24×C7—22no—
94C24×C7—22no—
95C24×C7—21no—
96C24×C7—21no—
97C24×C7—21no—
98C24×C7—21no—
99C24×C7C22×C412yes—
100C24×C7C2×Q812yes—
101C24×C7#4112yes—
102C24×C7C2411yescharacteristic
103C22×C4×C7—21no—
104C22×C4×C7—21no—
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—21no—
112C22×C4×C7—21no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—28no—
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×C7—44no—
125C22×C4×C7—44no—
126C22×C4×C7—48no—
127C22×C4×C7—48no—
128C22×C4×C7C22×C412yes—
129C22×C4×C7C22×C412yes—
130C22×C4×C7C2×Q814yes—
131C22×C4×C7#4114yes—
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—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7C22×C414yes—
158C22×Dic7C22×C414yes—
159C22×Dic7#4114yes—
160C7×C22⋊C4—44no—
161C7×C22⋊C4—44no—
162C22⋊Dic7—48no—
163C22⋊Dic7—48no—
164C4⋊Dic7—44no—
165C4⋊Dic7—44no—
166Dic7⋊C4—48no—
167C4×Dic7—48no—
168C24×C4C2×D711yescharacteristic
169C22×C4⋊C4—71no—
170C22×C4⋊C4—71no—
171C22×C4⋊C4—72no—
172C22×C22⋊C4—72no—
173C22×C22⋊C4—72no—
174C22×C22⋊C4C2×D711yescharacteristic
175C22×C22⋊C4C2×D711yescharacteristic
176C22×C42—72no—
177#380—78no—
178C2×#95—72no—
179C2×#95—72no—
180C2×#95—72no—
181C2×#95—71no—
182C2×#95—71no—
183C2×#95—72no—
184C2×#95—72no—
185C2×#95—72no—
186C2×#95—72no—
187C2×#95—72no—
188C2×#95C2×D712yes—
189C2×#95C2×D712yes—
190C23×C7—22no—
191C23×C7—22no—
192C23×C7—22no—
193C23×C7—22no—
194C23×C7—22no—
195C23×C7—22no—
196C23×C7—22no—
197C23×C7—22no—
198C23×C7—22no—
199C23×C7—22no—
200C23×C7—24no—
201C23×C7—24no—
202C23×C7—24no—
203C23×C7—24no—
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×C7C4×Q814yes—
213C23×C7C23×C411yescharacteristic
214C23×C7C22×Q811yescharacteristic
215C23×C7C2×#4111yescharacteristic
216C23×C7C2×#4112yes—
217C23×C7C2×#4112yes—
218C23×C7H(5,𝔽2)11yescharacteristic
219C23×C7H(5,𝔽2)11yescharacteristic
220C23×C7H(5,𝔽2)12yes—
221C23×C7H(5,𝔽2)11yescharacteristic
222C23×C7H(5,𝔽2)12yes—
223C23×C72−1+411yescharacteristic
224C2×C4×C7—22no—
225C2×C4×C7—22no—
226C2×C4×C7—24no—
227C2×C4×C7—24no—
228C2×C4×C7—22no—
229C2×C4×C7—22no—
230C2×C4×C7—22no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—416no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7#11714yes—
249C2×Dic7—24no—
250C2×Dic7—24no—
251C2×Dic7—24no—
252C2×Dic7—24no—
253C2×Dic7—24no—
254C2×Dic7—24no—
255C2×Dic7—24no—
256C2×Dic7—24no—
257C2×Dic7—24no—
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×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C25C22×D711yescharacteristic
278C23×C4—24no—
279C23×C4—42no—
280C23×C4—41no—
281C23×C4—41no—
282C23×C4—72no—
283C23×C4—72no—
284C23×C4—72no—
285C23×C4—72no—
286C23×C4C4×D714yes—
287C23×C4C22×D712yes—
288C23×C4C22×D712yes—
289C23×C4C22×D711yescharacteristic
290C23×C4C22×D711yescharacteristic
291C2×C4⋊C4—74no—
292C2×C4⋊C4—74no—
293C2×C4⋊C4—144no—
294C2×C4⋊C4—282no—
295C2×C4⋊C4—282no—
296C2×C4⋊C4—281no—
297C2×C4⋊C4—282no—
298C2×C4⋊C4—281no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—41no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—41no—
304C2×C22⋊C4—78no—
305C2×C22⋊C4—144no—
306C2×C22⋊C4—144no—
307C2×C22⋊C4—282no—
308C2×C22⋊C4—282no—
309C2×C22⋊C4—282no—
310C2×C22⋊C4Dic1414yes—
311C2×C42—282no—
312C2×C42—282no—
313C2×C42—282no—
314#95—24no—
315#95—42no—
316#95—78no—
317#95—144no—
318#95—144no—
319#95—282no—
320#95—281no—
321#95—281no—
322#95—282no—
323#95—282no—
324#95—282no—
325#95—282no—
326C22×C7—24no—
327C22×C7—24no—
328C22×C7—24no—
329C22×C7—42no—
330C22×C7—42no—
331C22×C7—42no—
332C22×C7—42no—
333C22×C7—44no—
334C22×C7—44no—
335C22×C7—44no—
336C22×C7—44no—
337C22×C7C2×#11711yescharacteristic
338C22×C7C2×C4×Q811yescharacteristic
339C22×C7C4×#4111yescharacteristic
340C22×C7#51811yescharacteristic
341C22×C7#51811yescharacteristic
342C22×C7#51811yescharacteristic
343C22×C7#52011yescharacteristic
344C22×C7#53811yescharacteristic
345C22×C7#53812yes—
346C22×C7#53812yes—
347C22×C7#53812yes—
348C22×C7#53911yescharacteristic
349C22×C7#54012yes—
350C22×C7#54311yescharacteristic
351C22×C7#54411yescharacteristic
352C22×C7#55112yes—
353C22×C7#55312yes—
354C22×C7#55312yes—
355C22×C7#55412yes—
356C22×C7#55412yes—
357C22×C7#55612yes—
358C22×C7#56012yes—
359C22×C7#56212yes—
360C4×C7—24no—
361C4×C7—42no—
362C4×C7—44no—
363C4×C7—82no—
364Dic7—44no—
365Dic7—44no—
366Dic7—44no—
367Dic7—82no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24—22no—
373C24—21no—
374C24—22no—
375C24—22no—
376C24C2×Dic1412yes—
377C24C2×C4×D712yes—
378C24#115712yes—
379C24C23×D711yescharacteristic
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—74no—
406C22×C4—74no—
407C22×C4—74no—
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×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×C4C2×Dic1414yes—
434C22×C4C2×C4×D712yes—
435C22×C4C2×C4×D712yes—
436C22×C4#115914yes—
437C4⋊C4—284no—
438C4⋊C4—284no—
439C4⋊C4—288no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—288no—
443C22⋊C4—288no—
444C42—288no—
445C2×C7—44no—
446C2×C7#231211yescharacteristic
447C2×C7#232911yescharacteristic
448C2×C7#233311yescharacteristic
449C2×C7#234511yescharacteristic
450C2×C7#234711yescharacteristic
451C2×C7#236811yescharacteristic
452C2×C7#237711yescharacteristic
453C2×C7#277812yes—
454C2×C7#278012yes—
455C2×C7#278612yes—
456C2×C7#278812yes—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23C4×Dic1414yes—
480C23C22×Dic1411yescharacteristic
481C23C22×C4×D711yescharacteristic
482C23C2×#115711yescharacteristic
483C23C2×#115912yes—
484C23#654811yescharacteristic
485C23#654811yescharacteristic
486C23#654811yescharacteristic
487C23#654812yes—
488C23#655111yescharacteristic
489C23D7×#4112yes—
490C23#655312yes—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—416no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—144no—
516C2×C4—144no—
517C2×C4—144no—
518C2×C4—144no—
519C2×C4—144no—
520C2×C4—144no—
521C2×C4—144no—
522C2×C4—144no—
523C2×C4—144no—
524C2×C4—282no—
525C2×C4—282no—
526C2×C4—282no—
527C2×C4—282no—
528C2×C4—282no—
529C2×C4—282no—
530C2×C4—282no—
531C2×C4—282no—
532C2×C4—282no—
533C2×C4—282no—
534C2×C4—282no—
535C2×C4—282no—
536C2×C4—282no—
537C2×C4—282no—
538C2×C4—282no—
539C2×C4—282no—
540C2×C4—282no—
541C2×C4—282no—
542C2×C4—282no—
543C2×C4#644014yes—
544C7#1445311yescharacteristic, complemented, Sylow
545C22—24no—
546C22—24no—
547C22—24no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—42no—
553C22—44no—
554C22—42no—
555C22—44no—
556C22C2×C4×Dic1411yescharacteristic
557C22C2×#644011yescharacteristic
558C22#9017411yescharacteristic
559C22#9017711yescharacteristic
560C22#9017711yescharacteristic
561C22#9018511yescharacteristic
562C22#9019111yescharacteristic
563C22#9019511yescharacteristic
564C22#9020611yescharacteristic
565C22#9021211yescharacteristic
566C22#9021311yescharacteristic
567C22#9023012yes—
568C22#9023412yes—
569C22#9024712yes—
570C22#9025612yes—
571C22#9028412yes—
572C22#9028412yes—
573C22#9030512yes—
574C22#9031112yes—
575C22#9033212yes—
576C22#9033812yes—
577C22#9034812yes—
578C22#9035412yes—
579C4—24no—
580C4—42no—
581C4—44no—
582C4—82no—
583C4—284no—
584C4—284no—
585C4—284no—
586C4—562no—
587C2—44no—
588C2#1173077911yescharacteristic
589C2#1173090211yescharacteristic
590C2#1173091811yescharacteristic
591C2#1173126511yescharacteristic
592C2#1173132711yescharacteristic
593C2#1173145911yescharacteristic
594C2#1173150311yescharacteristic
595C2#1173201512yes—
596C2#1173201912yes—
597C2#1173230212yes—
598C2#1173237712yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×C22⋊#95C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC22⋊#95D7
hypocenterC7#14453
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
6. #11727171C2semidirect
7. #11727165C2semidirect
8. #11727161C2semidirect
9. #11727160C2semidirect
21. #89984C4semidirect
544. C7#14453semidirect

Maximal subgroups

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

subgroupindex
C2×#899842
C7×C22⋊#952
#117298812
#117293012
#117292992
#117271712
#117271652
#117271612
#117271602
#144537