Finite Group Databasebeta

← #13416882

Subgroups — #13416882

Order 1792.

856 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2351C211yesmaximal, characteristic, complemented
2C2×#89793C211yesmaximal, characteristic, complemented
3C2×C4×#1143C211yesmaximal, characteristic
4#11731120C211yesmaximal, characteristic
5#11729387C211yesmaximal, characteristic
6#11729387C211yesmaximal, characteristic
7#11729383C211yesmaximal, characteristic, complemented
8#11729382C211yesmaximal, characteristic, complemented
9#11727384C211yesmaximal, characteristic
10#11727379C211yesmaximal, characteristic
11#11727373C211yesmaximal, characteristic, complemented
12#11727359C211yesmaximal, characteristic, complemented
13#11727312C211yesmaximal, characteristic
14#11727295C211yesmaximal, characteristic, complemented
15#11727236C211yesmaximal, characteristic, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C22×#1143C2211yescharacteristic
20C2×#6459—21no—
21C2×C4×D28C2211yescharacteristic
22C2×#6432C2211yescharacteristic
23C7×#389—21no—
24C7×#389—21no—
25C7×#389C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C7×#382—21no—
28C7×#382C2211yescharacteristic
29C4×C7×C4⋊C4—21no—
30C4×C7×C4⋊C4C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C4×#1143—24no—
35#89793—24no—
36#89766—21no—
37C2×#6383C2211yescharacteristic
38C2×#6383C2211yescharacteristic
39C2×#6383C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45#89711C2211yescharacteristic
46#89711C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51#89445—21no—
52#89440—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26377—71nomaximal, Sylow
62C22×C7×Q8C2311yescharacteristic
63C22×D28C2311yescharacteristic
64C22×C4×D7—21no—
65C22×Dic14—21no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C4×C7×Q8—22no—
69C4×C7×Q8—22no—
70C4×C7×Q8—41no—
71C4×C7×Q8—41no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C42×C7—21no—
83C2×C42×C7—21no—
84C2×C42×C7—21no—
85C2×C42×C7C2311yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2311yescharacteristic
88C2×#1143—22no—
89C2×#1143—22no—
90C2×#1143Q814yescomplemented
91C2×(C2×D7)⋊C4—21no—
92C2×(C2×D7)⋊C4—21no—
93C2×(C2×D7)⋊C4—21no—
94C2×(C2×D7)⋊C4—21no—
95C2×(C2×D7)⋊C4—21no—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4⋊Dic7C2311yescharacteristic
100C2×Dic7⋊C4—21no—
101C2×Dic7⋊C4—21no—
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106#6459—44no—
107C4×D28—22no—
108C4×D28—41no—
109#6432—22no—
110#6432—41no—
111C7×#95—21no—
112C7×#95—21no—
113C7×#95—21no—
114C7×#95—21no—
115C7×#95D411yescharacteristic
116C7×#95D411yescharacteristic
117#6409—22no—
118#6409—22no—
119#6409D414yes—
120C7⋊#95—21no—
121C7⋊#95—21no—
122#6383—22no—
123#6383—22no—
124#6383—41no—
125#6382—22no—
126#6382—41no—
127#6381—22no—
128#6381—22no—
129#6381—22no—
130C4⋊(C7⋊C8)—22no—
131C4×C7⋊C8—22no—
132C2×#477—71no—
133C2×C4×SD16—71no—
134#2452—71no—
135#2351D711yescharacteristic, complemented
136#2092—71no—
137#2085—71no—
138#2067—71no—
139#2058—71no—
140#1942—71no—
141#1916—71no—
142#1880—71no—
143#1880—71no—
144#1865—71no—
145#1863—71no—
146#1862—71no—
147C23×D7—21no—
148C2×C7×Q8—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8C2×D412yes—
152C2×C7×Q8C2×Q812yes—
153C2×C7×Q8#4112yes—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C2×D28—21no—
169C2×D28—21no—
170C2×D28—21no—
171C2×D28—21no—
172C2×D28—21no—
173C2×D28C2×Q812yes—
174C2×C4×D7—42no—
175C2×C4×D7—42no—
176C2×C4×D7—81no—
177C2×Dic14—42no—
178C2×Dic14—42no—
179C2×Dic14—81no—
180C7×C4⋊C4—24no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—42no—
189C7×C4⋊C4—42no—
190C7×C4⋊C4—42no—
191C7×C4⋊C4—42no—
192C7×C4⋊C4#4112yes—
193C42×C7—24no—
194C42×C7—21no—
195C42×C7—21no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—21no—
199C42×C7—21no—
200C42×C7—42no—
201C42×C7—42no—
202#1143—28no—
203(C2×D7)⋊C4—42no—
204(C2×D7)⋊C4—42no—
205(C2×D7)⋊C4—42no—
206(C2×D7)⋊C4—42no—
207(C2×D7)⋊C4—81no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—42no—
214C4⋊Dic7C2×D412yes—
215Dic7⋊C4—42no—
216Dic7⋊C4—42no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8C2×D412yes—
228C2×C7⋊C8C2×Q812yes—
229C22×SD16—71no—
230C2×C4⋊Q8—71no—
231C2×C22⋊Q8—141no—
232C2×C4×Q8C2×D711yescharacteristic
233C2×C4×Q8C2×D711yescharacteristic
234C2×C4×Q8C2×D711yescharacteristic
235C2×C4×D4—71no—
236#477—144no—
237C4×SD16—144no—
238C2×#107—71no—
239C2×#107—71no—
240C2×#107—71no—
241C2×#106—71no—
242C2×#106—71no—
243C2×C4⋊C8—71no—
244C2×#103—71no—
245C2×#103—71no—
246C2×#103—71no—
247C2×#102—71no—
248C2×#102—71no—
249C2×#102—71no—
250C2×C4×C8—71no—
251#396—141no—
252#393—141no—
253#390—71no—
254#390—71no—
255#389—21no—
256#389—21no—
257#389—71no—
258#389—71no—
259#389C2×D711yescharacteristic
260#389C2×D711yescharacteristic
261#385—141no—
262#382—21no—
263#382C2×D711yescharacteristic
264C4×C4⋊C4—21no—
265C4×C4⋊C4C2×D711yescharacteristic
266#340—71no—
267#340—71no—
268#340—71no—
269#340—71no—
270#336—71no—
271#336—71no—
272#336—71no—
273#336—71no—
274C23×C7C22×D411yescharacteristic
275C22×D7—22no—
276C22×D7—41no—
277C22×D7—41no—
278C22×D7—41no—
279C22×D7—41no—
280C22×D7—41no—
281C22×D7—41no—
282C7×Q8—22no—
283C7×Q8—22no—
284C7×Q8—22no—
285C7×Q8C22⋊Q814yescomplemented
286C2×C4×C7—22no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—22no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7#13512yes—
315C2×C4×C7AGL(1,ℤ/8)12yes—
316C2×C4×C7C22×D411yescharacteristic
317C2×C4×C7C22×Q811yescharacteristic
318C2×C4×C7C2×#4111yescharacteristic
319C2×C4×C7C2×#4111yescharacteristic
320C2×C4×C7H(5,𝔽2)11yescharacteristic
321C2×C4×C72−1+411yescharacteristic
322C2×Dic7—22no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331D28—22no—
332D28—22no—
333D28—41no—
334D28—41no—
335C4×D7—84no—
336Dic14—84no—
337C7⋊C8—22no—
338C7⋊C8—22no—
339C7⋊C8—41no—
340C7⋊C8—41no—
341C22×Q8—141no—
342C22×Q8C22×D711yescharacteristic
343C22×D4—71no—
344C23×C4—141no—
345C2×SD16—74no—
346C2×SD16—142no—
347C2×SD16—142no—
348C22×C8—71no—
349C22×C8—71no—
350C22×C8—71no—
351C22×C8—71no—
352C4⋊Q8—142no—
353C4⋊Q8—281no—
354C22⋊Q8—284no—
355C4×Q8—22no—
356C4×Q8—22no—
357C4×Q8—22no—
358C4×Q8—22no—
359C4×Q8—41no—
360C4×Q8—41no—
361C4×D4—142no—
362C4×D4—281no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4C22⋊D711yescharacteristic
375C2×C4⋊C4C22⋊D711yescharacteristic
376C2×C4⋊C4C22×D711yescharacteristic
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C22⋊C4—141no—
380C2×C22⋊C4—141no—
381C2×C22⋊C4—141no—
382C2×C22⋊C4—141no—
383C2×C22⋊C4—141no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42C22×D711yescharacteristic
388C2×C42C22×D711yescharacteristic
389C2×C42C22×D711yescharacteristic
390#107—142no—
391#107—142no—
392#107—142no—
393#106—142no—
394#106—281no—
395C4⋊C8—142no—
396#103—74no—
397#103—142no—
398#103—142no—
399#102—142no—
400#102—142no—
401#102—281no—
402C4×C8—142no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—141no—
408#95—141no—
409#95C22⋊D711yescharacteristic
410#95C22⋊D711yescharacteristic
411C22×C7#52511yescharacteristic
412C22×C7D4211yescharacteristic
413C22×C7D4×Q811yescharacteristic
414C22×C7C2×#13511yescharacteristic
415C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
416C22×C7#57711yescharacteristic
417C22×C7#57811yescharacteristic
418C2×D7—42no—
419C2×D7—42no—
420C2×D7—42no—
421C2×D7—81no—
422C2×D7—81no—
423C2×D7—81no—
424C2×D7—81no—
425C4×C7—24no—
426C4×C7—22no—
427C4×C7—22no—
428C4×C7—41no—
429C4×C7—42no—
430C4×C7—41no—
431C4×C7—41no—
432C4×C7—41no—
433C4×C7—81no—
434C4×C7—81no—
435C4×C7C2×C22⋊Q811yescharacteristic
436C4×C7#53911yescharacteristic
437C4×C7#54711yescharacteristic
438C4×C7#55411yescharacteristic
439Dic7—42no—
440Dic7—81no—
441Dic7—81no—
442C24—141no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—282no—
447C2×Q8—282no—
448C2×Q8—561no—
449C2×Q8#115712yes—
450C2×Q8D4×D712yes—
451C2×Q8Q8×D712yes—
452C2×D4—72no—
453C2×D4—141no—
454C2×D4—141no—
455C2×D4—141no—
456C2×D4—141no—
457C2×D4—141no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—141no—
463C22×C4—141no—
464C22×C4—141no—
465C22×C4—282no—
466C22×C4—282no—
467C22×C4—561no—
468C22×C4C2×C22⋊D711yescharacteristic
469C22×C4C2×C22⋊D711yescharacteristic
470C22×C4C2×C22⋊D711yescharacteristic
471C22×C4C2×C22⋊D711yescharacteristic
472C22×C4C2×C22⋊D711yescharacteristic
473C22×C4C2×C22⋊D711yescharacteristic
474C22×C4C23×D711yescharacteristic
475SD16—148no—
476C2×C8—72no—
477C2×C8—72no—
478C2×C8—141no—
479C2×C8—141no—
480C2×C8—141no—
481C2×C8—141no—
482C2×C8—141no—
483C2×C8—141no—
484C2×C8—141no—
485C2×C8—141no—
486C2×C8—141no—
487C2×C8—141no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—72no—
501C4⋊C4—141no—
502C4⋊C4—141no—
503C4⋊C4—141no—
504C4⋊C4—141no—
505C4⋊C4—141no—
506C4⋊C4—282no—
507C4⋊C4—282no—
508C4⋊C4—282no—
509C4⋊C4#115712yes—
510C22⋊C4—282no—
511C22⋊C4—282no—
512C22⋊C4—282no—
513C22⋊C4—282no—
514C22⋊C4—561no—
515C42—24no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—42no—
523C42—42no—
524C2×C7#244911yescharacteristic
525C2×C7#310611yescharacteristic
526C2×C7#312111yescharacteristic
527C2×C7#326511yescharacteristic
528C2×C7#328511yescharacteristic
529C2×C7#338111yescharacteristic
530C2×C7#338611yescharacteristic
531D7—82no—
532C23—142no—
533C23—281no—
534C23—281no—
535C23—281no—
536C23—281no—
537C23—281no—
538C23—281no—
539C23C22×C22⋊D711yescharacteristic
540Q8—22no—
541Q8—22no—
542Q8—22no—
543Q8—564no—
544Q8#645914yescomplemented
545D4—142no—
546D4—142no—
547D4—281no—
548D4—281no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—142no—
578C2×C4—281no—
579C2×C4—281no—
580C2×C4—281no—
581C2×C4—281no—
582C2×C4—281no—
583C2×C4—281no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—564no—
587C2×C4#651212yes—
588C2×C4#651312yes—
589C2×C4C2×#115711yescharacteristic
590C2×C4C2×#115711yescharacteristic
591C2×C4C2×D4×D711yescharacteristic
592C2×C4#654811yescharacteristic
593C2×C4C2×Q8×D711yescharacteristic
594C2×C4#655111yescharacteristic
595C8—142no—
596C8—142no—
597C8—281no—
598C8—281no—
599C7#2637711yescharacteristic, complemented, Sylow
600C22—282no—
601C22—282no—
602C22—282no—
603C22—561no—
604C22—561no—
605C22—561no—
606C22—561no—
607C22#9048211yescharacteristic
608C22D4×C22⋊D711yescharacteristic
609C22Q8×C22⋊D711yescharacteristic
610C22C2×#651211yescharacteristic
611C22C2×#651311yescharacteristic
612C22#9052811yescharacteristic
613C22#9052911yescharacteristic
614C4—22no—
615C4—22no—
616C4—24no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—282no—
625C4—561no—
626C4—561no—
627C4#9019811yescharacteristic
628C4C2×#645911yescharacteristic
629C4#9020211yescharacteristic
630C4#9020611yescharacteristic
631C2—562no—
632C2#1173961811yescharacteristic
633C2#1173997411yescharacteristic
634C2#1173999711yescharacteristic
635C2#1174098711yescharacteristic
636C2#1174099411yescharacteristic
637C2#1174113811yescharacteristic
638C2#1174113911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2351C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2351D7
hypocenterC7#26377
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C7×#2351C2semidirect
2. C2×#89793C2semidirect
7. #11729383C2semidirect
8. #11729382C2semidirect
11. #11727373C2semidirect
12. #11727359C2semidirect
14. #11727295C2semidirect
15. #11727236C2semidirect
90. C2×#1143Q8semidirect
135. #2351D7semidirect
285. C7×Q8C22⋊Q8semidirect
544. Q8#6459semidirect
599. C7#26377semidirect

Maximal subgroups

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

subgroupindex
C7×#23512
C2×#897932
C2×C4×#11432
#117311202
#117293872
#117293832
#117293822
#117273842
#117273792
#117273732
#117273592
#117273122
#117272952
#117272362
#263777