Finite Group Databasebeta

← #13276331

Subgroups — #13276331

Order 1792.

1862 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6455C211yesmaximal, characteristic
2C2×#89759C211yesmaximal, characteristic
3C2×C4×(C2×D7)⋊C4C211yesmaximal, characteristic
4C7×C4⋊#95C211yesmaximal, characteristic, complemented
5#11729317C211yesmaximal, characteristic, complemented
6#11729317C211yesmaximal, characteristic, complemented
7#11729289C211yesmaximal, characteristic, complemented
8#11727183C214yesmaximal
9#11727157C214yesmaximal, complemented
10C23×C4×D7C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×(C2×D7)⋊C4C2212yes—
15C22×(C2×D7)⋊C4C2212yes—
16C22×(C2×D7)⋊C4C2212yes—
17C22×C4⋊Dic7C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C22×C4×Dic7C2211yescharacteristic
20C22×C4×Dic7C2212yes—
21C2×#6455—24no—
22C2×#6455—24no—
23C2×#6455C418yescomplemented
24C2×C7×#95C2214yes—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2214yes—
28C2×C7⋊#95C2214yes—
29C2×C7⋊#95C2214yes—
30#89759—24no—
31C4×(C2×D7)⋊C4—24no—
32#13939—71nomaximal, Sylow
33C24×D7C2311yescharacteristic
34C23×C4×C7C2312yes—
35C23×C4×C7C2312yes—
36C23×C4×C7C2312yes—
37C23×C4×C7C2311yescharacteristic
38C23×Dic7C2312yes—
39C23×Dic7C2312yes—
40C23×Dic7C2312yes—
41C23×Dic7C2311yescharacteristic
42C22×C4×D7—22no—
43C22×C4×D7—22no—
44C22×C4×D7—22no—
45C22×C4×D7—22no—
46C22×C4×D7C2×C414yes—
47C22×C4×D7D414yes—
48C22×C4×D7D414yes—
49C2×C7×C4⋊C4—22no—
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4—22no—
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4C2×C414yes—
55C2×C42×C7—22no—
56C2×C42×C7—22no—
57C2×C42×C7—22no—
58C2×(C2×D7)⋊C4—24no—
59C2×(C2×D7)⋊C4—24no—
60C2×(C2×D7)⋊C4—24no—
61C2×(C2×D7)⋊C4—24no—
62C2×(C2×D7)⋊C4—24no—
63C2×(C2×D7)⋊C4—24no—
64C2×(C2×D7)⋊C4—44no—
65C2×(C2×D7)⋊C4C2×C418yes—
66C2×C4⋊Dic7—22no—
67C2×C4⋊Dic7—22no—
68C2×C4⋊Dic7—22no—
69C2×C4⋊Dic7—22no—
70C2×C4⋊Dic7—22no—
71C2×C4⋊Dic7C2×C414yes—
72C2×C4×Dic7—24no—
73C2×C4×Dic7—24no—
74C2×C4×Dic7—22no—
75C2×C4×Dic7—22no—
76C2×C4×Dic7—22no—
77C2×C4×Dic7C2×C418yes—
78#6455—232no—
79C7×#95—44no—
80C7⋊#95—28no—
81C7⋊#95—24no—
82C7⋊#95—24no—
83C7⋊#95—28no—
84C7⋊#95—44no—
85C22×#117—71no—
86C2×#380—71no—
87C2×C4×C22⋊C4—71no—
88#1440—74no—
89C4⋊#95—71no—
90C4⋊#95—71no—
91C4⋊#95—71no—
92C4⋊#95D711yescharacteristic, complemented
93#1434—74no—
94C24×C7C2411yescharacteristic
95C23×D7—22no—
96C23×D7—22no—
97C23×D7—21no—
98C23×D7—21no—
99C23×D7—21no—
100C23×D7—21no—
101C23×D7—22no—
102C23×D7—22no—
103C23×D7C22×C412yes—
104C23×D7C2×D412yes—
105C23×D7C2×D412yes—
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—24no—
113C22×C4×C7—22no—
114C22×C4×C7—24no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—24no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—24no—
121C22×C4×C7C22×C414yes—
122C22×C4×C7C22×C412yes—
123C22×C4×C7C2×D412yes—
124C22×C4×C7C2×D412yes—
125C22×C4×C7#4114yes—
126C22×C4×C7#4112yes—
127C22×C4×C7#4112yes—
128C22×C4×C7#4112yes—
129C22×C4×C7#4112yes—
130C22×Dic7—21no—
131C22×Dic7—21no—
132C22×Dic7—21no—
133C22×Dic7—21no—
134C22×Dic7—22no—
135C22×Dic7—22no—
136C22×Dic7—22no—
137C22×Dic7—24no—
138C22×Dic7—24no—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—24no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—24no—
149C22×Dic7C22×C414yes—
150C22×Dic7C22×C412yes—
151C22×Dic7C2×D412yes—
152C22×Dic7C2×D412yes—
153C22×Dic7#4114yes—
154C2×C4×D7—28no—
155C2×C4×D7—28no—
156C2×C4×D7—28no—
157C2×C4×D7—28no—
158C2×C4×D7—28no—
159C2×C4×D7—28no—
160C2×C4×D7C22⋊C4116yes—
161C7×C4⋊C4—28no—
162C7×C4⋊C4—44no—
163C42×C7—44no—
164(C2×D7)⋊C4—216no—
165(C2×D7)⋊C4—216no—
166(C2×D7)⋊C4—48no—
167C4⋊Dic7—28no—
168C4⋊Dic7—44no—
169C4×Dic7—216no—
170C4×Dic7—44no—
171C24×C4—71no—
172C2×#117—78no—
173C2×#117—144no—
174C2×#117—144no—
175C22×C4⋊C4—71no—
176C22×C4⋊C4—71no—
177C22×C4⋊C4C2×D711yescharacteristic
178C22×C4⋊C4C2×D711yescharacteristic
179C22×C22⋊C4—72no—
180C22×C22⋊C4—72no—
181C22×C22⋊C4—72no—
182C22×C42—72no—
183C22×C42—71no—
184C22×C42C2×D711yescharacteristic
185#380—144no—
186C4×C22⋊C4—144no—
187C2×#95—72no—
188C2×#95—72no—
189C2×#95—74no—
190C2×#95—74no—
191C2×#95—74no—
192C2×#95C2×D714yes—
193C23×C7C23×C411yescharacteristic
194C23×C7C22×D411yescharacteristic
195C23×C7C22×D411yescharacteristic
196C23×C7C2×#4111yescharacteristic
197C23×C7C2×#4111yescharacteristic
198C23×C7C2×#4111yescharacteristic
199C23×C7C2×#4111yescharacteristic
200C23×C7C2×#4112yes—
201C23×C7C2×#4112yes—
202C23×C7H(5,𝔽2)12yes—
203C23×C72−1+412yes—
204C22×D7—22no—
205C22×D7—22no—
206C22×D7—24no—
207C22×D7—24no—
208C22×D7—22no—
209C22×D7—22no—
210C22×D7—22no—
211C22×D7—22no—
212C22×D7—24no—
213C22×D7—22no—
214C22×D7—22no—
215C22×D7—22no—
216C22×D7—22no—
217C22×D7—24no—
218C22×D7—42no—
219C22×D7—42no—
220C22×D7—42no—
221C22×D7—42no—
222C22×D7—42no—
223C22×D7—42no—
224C22×D7—42no—
225C22×D7—42no—
226C22×D7C2×C22⋊C414yes—
227C2×C4×C7—24no—
228C2×C4×C7—28no—
229C2×C4×C7—24no—
230C2×C4×C7—24no—
231C2×C4×C7—24no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—44no—
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—44no—
243C2×C4×C7—44no—
244C2×C4×C7C2×C22⋊C414yes—
245C2×C4×C7#11714yes—
246C2×C4×C7#11714yes—
247C2×C4×C7#12314yes—
248C2×C4×C7#12314yes—
249C2×C4×C7#12314yes—
250C2×C4×C7#12314yes—
251C2×Dic7—22no—
252C2×Dic7—22no—
253C2×Dic7—22no—
254C2×Dic7—22no—
255C2×Dic7—22no—
256C2×Dic7—22no—
257C2×Dic7—24no—
258C2×Dic7—28no—
259C2×Dic7—24no—
260C2×Dic7—24no—
261C2×Dic7—24no—
262C2×Dic7—42no—
263C2×Dic7—44no—
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—44no—
273C2×Dic7—44no—
274C2×Dic7C2×C22⋊C414yes—
275C4×D7—232no—
276C25—71no—
277C23×C4—71no—
278C23×C4—74no—
279C23×C4—74no—
280C23×C4—74no—
281C23×C4—72no—
282C23×C4—72no—
283C23×C4—72no—
284C23×C4—142no—
285C23×C4—142no—
286C23×C4—142no—
287C23×C4—142no—
288C23×C4C22×D712yes—
289C23×C4C22×D712yes—
290C23×C4C22×D712yes—
291C23×C4C22×D711yescharacteristic
292#117—1432no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—74no—
299C2×C4⋊C4—142no—
300C2×C4⋊C4—142no—
301C2×C4⋊C4—142no—
302C2×C4⋊C4—142no—
303C2×C4⋊C4—142no—
304C2×C4⋊C4C4×D714yes—
305C2×C22⋊C4—78no—
306C2×C22⋊C4—144no—
307C2×C22⋊C4—144no—
308C2×C22⋊C4—144no—
309C2×C22⋊C4—144no—
310C2×C22⋊C4—144no—
311C2×C22⋊C4—144no—
312C2×C22⋊C4—284no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—78no—
317C2×C42—144no—
318C2×C42—144no—
319C2×C42—142no—
320C2×C42—142no—
321C2×C42—142no—
322#95—44no—
323#95—148no—
324#95—144no—
325#95—144no—
326#95—148no—
327#95—284no—
328C22×C7C22×C22⋊C411yescharacteristic
329C22×C7C2×#11711yescharacteristic
330C22×C7C2×#11711yescharacteristic
331C22×C7C4×#4112yes—
332C22×C7#52012yes—
333C22×C7C2×#12311yescharacteristic
334C22×C7C2×#12311yescharacteristic
335C22×C7C2×#12311yescharacteristic
336C22×C7C2×#12311yescharacteristic
337C22×C7#52512yes—
338C22×C7#53211yescharacteristic
339C22×C7#53211yescharacteristic
340C22×C7#53511yescharacteristic
341C22×C7#53611yescharacteristic
342C22×C7#53712yes—
343C22×C7#55212yes—
344C22×C7#55212yes—
345C22×C7#55312yes—
346C22×C7#55312yes—
347C22×C7#55512yes—
348C22×C7#55512yes—
349C22×C7#55612yes—
350C22×C7#55612yes—
351C2×D7—24no—
352C2×D7—24no—
353C2×D7—24no—
354C2×D7—44no—
355C2×D7—44no—
356C2×D7—44no—
357C2×D7—44no—
358C2×D7—42no—
359C2×D7—42no—
360C2×D7—42no—
361C2×D7—42no—
362C4×C7—44no—
363C4×C7—44no—
364C4×C7—82no—
365C4×C7#38118yes—
366Dic7—24no—
367Dic7—44no—
368Dic7—44no—
369Dic7—82no—
370C24—72no—
371C24—72no—
372C24—72no—
373C24—141no—
374C24—141no—
375C24—141no—
376C24—142no—
377C24—142no—
378C24—141no—
379C24—142no—
380C24—142no—
381C24C23×D711yescharacteristic
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—72no—
398C22×C4—72no—
399C22×C4—72no—
400C22×C4—716no—
401C22×C4—74no—
402C22×C4—74no—
403C22×C4—141no—
404C22×C4—141no—
405C22×C4—141no—
406C22×C4—141no—
407C22×C4—148no—
408C22×C4—148no—
409C22×C4—148no—
410C22×C4—148no—
411C22×C4—148no—
412C22×C4—148no—
413C22×C4—142no—
414C22×C4—142no—
415C22×C4—142no—
416C22×C4—142no—
417C22×C4—142no—
418C22×C4—144no—
419C22×C4—142no—
420C22×C4—142no—
421C22×C4—144no—
422C22×C4—142no—
423C22×C4—142no—
424C22×C4—142no—
425C22×C4—144no—
426C22×C4—144no—
427C22×C4—142no—
428C22×C4C2×C4×D714yes—
429C22×C4C2×C4×D712yes—
430C22×C4#115712yes—
431C22×C4#115712yes—
432C22×C4D4×D712yes—
433C22×C4D4×D712yes—
434C22×C4#115912yes—
435C22×C4#116114yes—
436C22×C4#116112yes—
437C4⋊C4—28no—
438C4⋊C4—44no—
439C4⋊C4—148no—
440C4⋊C4—284no—
441C22⋊C4—1416no—
442C22⋊C4—1416no—
443C22⋊C4—288no—
444C42—44no—
445C42—1416no—
446C42—284no—
447C2×C7C2×#38111yescharacteristic
448C2×C7#229711yescharacteristic
449C2×C7#231111yescharacteristic
450C2×C7#234612yes—
451C2×C7#234712yes—
452C2×C7#248512yes—
453C2×C7#248512yes—
454C2×C7#249911yescharacteristic
455C2×C7#249911yescharacteristic
456C2×C7#250011yescharacteristic
457C2×C7#250011yescharacteristic
458D7—44no—
459C23—74no—
460C23—142no—
461C23—142no—
462C23—142no—
463C23—142no—
464C23—142no—
465C23—142no—
466C23—142no—
467C23—142no—
468C23—144no—
469C23—144no—
470C23—144no—
471C23—144no—
472C23—142no—
473C23—142no—
474C23—282no—
475C23—282no—
476C23—282no—
477C23—282no—
478C23—282no—
479C23—282no—
480C23—282no—
481C23—282no—
482C23C22×C4×D711yescharacteristic
483C23C2×#115711yescharacteristic
484C23C2×#115711yescharacteristic
485C23C2×D4×D711yescharacteristic
486C23C2×D4×D711yescharacteristic
487C23C2×#115911yescharacteristic
488C23#654812yes—
489C23C2×#116111yescharacteristic
490C23C2×#116112yes—
491C23D7×#4112yes—
492C23#655412yes—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—74no—
511C2×C4—142no—
512C2×C4—142no—
513C2×C4—142no—
514C2×C4—142no—
515C2×C4—142no—
516C2×C4—142no—
517C2×C4—1432no—
518C2×C4—144no—
519C2×C4—144no—
520C2×C4—144no—
521C2×C4—148no—
522C2×C4—144no—
523C2×C4—282no—
524C2×C4—282no—
525C2×C4—282no—
526C2×C4—282no—
527C2×C4—282no—
528C2×C4—284no—
529C2×C4—282no—
530C2×C4—282no—
531C2×C4—282no—
532C2×C4—284no—
533C2×C4—282no—
534C2×C4—284no—
535C2×C4#643514yes—
536C2×C4D7×C22⋊C414yes—
537C2×C4#644614yes—
538C2×C4#644614yes—
539C2×C4#645514yes—
540C2×C4#645714yes—
541C2×C4#645714yes—
542C7#1393911yescharacteristic, complemented, Sylow
543C22—144no—
544C22—144no—
545C22—144no—
546C22—284no—
547C22—282no—
548C22—282no—
549C22—282no—
550C22—284no—
551C22—282no—
552C22—284no—
553C22—284no—
554C22C2×#643511yescharacteristic
555C22C2×D7×C22⋊C411yescharacteristic
556C22C2×#644611yescharacteristic
557C22C2×#644611yescharacteristic
558C22C2×#645511yescharacteristic
559C22C2×#645711yescharacteristic
560C22C2×#645711yescharacteristic
561C22#9020512yes—
562C22#9022412yes—
563C22#9023712yes—
564C22#9024912yes—
565C22C4×#116112yes—
566C22#9027512yes—
567C22#9029912yes—
568C22#9030712yes—
569C22#9031812yes—
570C22#9033312yes—
571C22#9035911yescharacteristic
572C22#9036611yescharacteristic
573C22#9038512yes—
574C22#9040611yescharacteristic
575C22#9041611yescharacteristic
576C22#9042512yes—
577C4—44no—
578C4—44no—
579C4—82no—
580C4—144no—
581C4—284no—
582C4—284no—
583C4—562no—
584C4#8943618yes—
585C2—284no—
586C2C2×#8943611yescharacteristic
587C2#1173107912yes—
588C2#1173147012yes—
589C2#1173161011yescharacteristic
590C2#1173181211yescharacteristic
591C2#1173342412yes—
592C2#1173367512yes—
593C2#1173373911yescharacteristic
594C2#1173377911yescharacteristic
595C2#1173392311yescharacteristic
596C2#1173407911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×C4⋊#95C2
socle, cosocleC24×C7C24
prime radicalC24×D7C23
hypercenterC4⋊#95D7
hypocenterC7#13939
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
4. C7×C4⋊#95C2semidirect
5. #11729317C2semidirect
6. #11729317C2semidirect
7. #11729289C2semidirect
9. #11727157C2semidirect
23. C2×#6455C4semidirect
92. C4⋊#95D7semidirect
542. C7#13939semidirect

Maximal subgroups

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

subgroupindex
C22×#64552
C2×#897592
C2×C4×(C2×D7)⋊C42
C7×C4⋊#952
#117293172
#117292892
#117271832
#117271572
#139397