Finite Group Databasebeta

← #13416834

Subgroups — #13416834

Order 1792.

884 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2453C211yesmaximal, characteristic, complemented
2C2×#89838C211yesmaximal, characteristic
3C2×C4×#1143C211yesmaximal, characteristic
4C4⋊C4×D28C211yesmaximal, characteristic
5#11729493C211yesmaximal, characteristic, complemented
6#11729383C211yesmaximal, characteristic, complemented
7#11727441C211yesmaximal, characteristic
8#11727441C211yesmaximal, characteristic
9#11727438C211yesmaximal, characteristic, complemented
10#11727437C211yesmaximal, characteristic, complemented
11#11727379C211yesmaximal, characteristic
12#11727379C211yesmaximal, characteristic
13#11727373C211yesmaximal, characteristic, complemented
14#11727362C211yesmaximal, characteristic, complemented
15#11727236C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C22×#1143C2211yescharacteristic
19C2×D7×C4⋊C4—21no—
20C2×C4×D28C2211yescharacteristic
21C2×C4×D28C2211yescharacteristic
22C2×C4×D28C2211yescharacteristic
23C7×#400—21no—
24C7×#398—21no—
25C7×#391C2211yescharacteristic
26C7×#391C2211yescharacteristic
27C7×#389—21no—
28C7×#389C2211yescharacteristic
29C7×#389C2211yescharacteristic
30C7×#387—21no—
31C4×C7×C4⋊C4C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35#89838—24no—
36C4×#1143—24no—
37#89759—21no—
38C2×#6383C2211yescharacteristic
39C2×#6383C2211yescharacteristic
40C2×#6383C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46#89707C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51#89440—21no—
52#89440—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26374—71nomaximal, Sylow
62C22×C7×Q8C2311yescharacteristic
63C22×D28C2311yescharacteristic
64C22×C4×D7—21no—
65C22×C4×D7—21no—
66C22×C4×D7—21no—
67C7×#125—22no—
68C7×#125—41no—
69C4×C7×Q8—22no—
70C4×C7×Q8—41no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C42×C7—21no—
82C2×C42×C7C2311yescharacteristic
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×#1143—22no—
86C2×#1143—22no—
87C2×#1143Q814yes—
88C2×(C2×D7)⋊C4—21no—
89C2×(C2×D7)⋊C4—21no—
90C2×(C2×D7)⋊C4—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102D7×C4⋊C4—44no—
103C4×D28—22no—
104C4×D28—22no—
105C4×D28—22no—
106C4×D28—22no—
107C4×D28—41no—
108C4×D28—41no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95—21no—
112C7×#95—21no—
113C7×#95—21no—
114C7×#95—21no—
115C7×#95D411yescharacteristic
116C7×#95D411yescharacteristic
117#6409—22no—
118#6409—22no—
119#6409—41no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122#6383—22no—
123#6383—22no—
124#6383D414yes—
125#6382—22no—
126#6382—22no—
127#6381—22no—
128#6381—22no—
129#6381—41no—
130C4⋊(C7⋊C8)—22no—
131C4×C7⋊C8—22no—
132C2×#478—71no—
133C2×C4×SD16—71no—
134#2453D711yescharacteristic, complemented
135D4×C4⋊C4—71no—
136#2095—71no—
137#2084—71no—
138#2068—71no—
139#2059—71no—
140#1942—71no—
141#1916—71no—
142#1877—71no—
143#1877—71no—
144#1865—71no—
145#1862—71no—
146#1862—71no—
147C23×D7—21no—
148C2×C7×Q8—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8C2×Q812yes—
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×D28C2×D412yes—
172C2×D28C2×Q812yes—
173C2×D28#4112yes—
174C2×C4×D7—24no—
175C2×C4×D7—42no—
176C2×C4×D7—42no—
177C2×C4×D7—42no—
178C2×C4×D7—42no—
179C2×C4×D7—42no—
180C2×C4×D7—42no—
181C2×C4×D7—81no—
182C2×C4×D7—81no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—42no—
191C7×C4⋊C4—42no—
192C7×C4⋊C4—42no—
193C7×C4⋊C4—42no—
194C7×C4⋊C4—42no—
195C7×C4⋊C4C2×D412yes—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—21no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—21no—
202C42×C7—42no—
203#1143—28no—
204(C2×D7)⋊C4—24no—
205(C2×D7)⋊C4—42no—
206(C2×D7)⋊C4—42no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—42no—
213C4⋊Dic7—81no—
214C4⋊Dic7#4112yes—
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×#125C2×D711yescharacteristic
231C2×C4×Q8C2×D711yescharacteristic
232C2×C4×D4—71no—
233C2×C4×D4—71no—
234C2×C4×D4—71no—
235C22×C4⋊C4—141no—
236#478—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#400—21no—
252#398—21no—
253#391C2×D711yescharacteristic
254#391C2×D711yescharacteristic
255#389—21no—
256#389—71no—
257#389—71no—
258#389C2×D711yescharacteristic
259#389C2×D711yescharacteristic
260#387—21no—
261#385—141no—
262#385—141no—
263#384—71no—
264#380—141no—
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—22no—
277C22×D7—22no—
278C22×D7—41no—
279C22×D7—41no—
280C22×D7—41no—
281C22×D7—41no—
282C7×Q8—22no—
283C7×Q8—22no—
284C7×Q8—41no—
285C7×Q8—41no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—22no—
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—41no—
305C2×C4×C7—41no—
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—22no—
334D28C22⋊Q814yes—
335C4×D7—48no—
336C4×D7—84no—
337C4×D7—84no—
338C7⋊C8—22no—
339C7⋊C8—22no—
340C7⋊C8—41no—
341C7⋊C8—41no—
342C22×Q8C22×D711yescharacteristic
343C22×D4—71no—
344C23×C4—141no—
345C23×C4—141no—
346C23×C4—141no—
347C2×SD16—74no—
348C2×SD16—142no—
349C2×SD16—142no—
350C22×C8—71no—
351C22×C8—71no—
352C22×C8—71no—
353C22×C8—71no—
354#125—22no—
355#125—41no—
356C4×Q8—22no—
357C4×Q8—41no—
358C4×D4—142no—
359C4×D4—142no—
360C4×D4—142no—
361C4×D4—142no—
362C4×D4—281no—
363C4×D4—281no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—71no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—284no—
377C2×C4⋊C4C22⋊D711yescharacteristic
378C2×C4⋊C4C22⋊D711yescharacteristic
379C2×C4⋊C4C22×D711yescharacteristic
380C2×C4⋊C4C22×D711yescharacteristic
381C2×C4⋊C4C22×D711yescharacteristic
382C2×C22⋊C4—141no—
383C2×C22⋊C4—141no—
384C2×C22⋊C4—141no—
385C2×C42—21no—
386C2×C42C22×D711yescharacteristic
387C2×C42C22×D711yescharacteristic
388C2×C42C22×D711yescharacteristic
389#107—142no—
390#107—142no—
391#107—281no—
392#106—142no—
393#106—142no—
394C4⋊C8—142no—
395#103—142no—
396#103—142no—
397#103—281no—
398#102—74no—
399#102—142no—
400#102—142no—
401C4×C8—142no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—141no—
409#95—141no—
410#95C22⋊D711yescharacteristic
411#95C22⋊D711yescharacteristic
412C22×C7#52511yescharacteristic
413C22×C7D4211yescharacteristic
414C22×C7D4×Q811yescharacteristic
415C22×C7C2×#13511yescharacteristic
416C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
417C22×C7#57711yescharacteristic
418C22×C7#57811yescharacteristic
419C2×D7—24no—
420C2×D7—42no—
421C2×D7—42no—
422C2×D7—42no—
423C2×D7—42no—
424C2×D7—42no—
425C2×D7—42no—
426C4×C7—22no—
427C4×C7—22no—
428C4×C7—41no—
429C4×C7—42no—
430C4×C7—41no—
431C4×C7—41no—
432C4×C7—41no—
433C4×C7—42no—
434C4×C7—81no—
435C4×C7—81no—
436C4×C7C2×C22⋊Q811yescharacteristic
437C4×C7#53911yescharacteristic
438C4×C7#54711yescharacteristic
439C4×C7#55411yescharacteristic
440Dic7—42no—
441Dic7—81no—
442Dic7—81no—
443C24—141no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8Q8×D712yes—
450C2×D4—72no—
451C2×D4—72no—
452C2×D4—72no—
453C2×D4—141no—
454C2×D4—141no—
455C2×D4—141no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—144no—
461C22×C4—141no—
462C22×C4—141no—
463C22×C4—141no—
464C22×C4—282no—
465C22×C4—282no—
466C22×C4—282no—
467C22×C4—282no—
468C22×C4—282no—
469C22×C4—282no—
470C22×C4—561no—
471C22×C4—561no—
472C22×C4C2×C22⋊D711yescharacteristic
473C22×C4C2×C22⋊D711yescharacteristic
474C22×C4C2×C22⋊D711yescharacteristic
475C22×C4C2×C22⋊D711yescharacteristic
476C22×C4C2×C22⋊D711yescharacteristic
477C22×C4C2×C22⋊D711yescharacteristic
478C22×C4C23×D711yescharacteristic
479SD16—148no—
480C2×C8—72no—
481C2×C8—72no—
482C2×C8—141no—
483C2×C8—141no—
484C2×C8—141no—
485C2×C8—141no—
486C2×C8—141no—
487C2×C8—141no—
488C2×C8—141no—
489C2×C8—141no—
490C2×C8—141no—
491C2×C8—141no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—72no—
505C4⋊C4—141no—
506C4⋊C4—141no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—141no—
510C4⋊C4—282no—
511C4⋊C4—282no—
512C4⋊C4—282no—
513C4⋊C4—561no—
514C4⋊C4D4×D712yes—
515C22⋊C4—144no—
516C22⋊C4—282no—
517C22⋊C4—282no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—42no—
525C2×C7#244911yescharacteristic
526C2×C7#310511yescharacteristic
527C2×C7#312011yescharacteristic
528C2×C7#326511yescharacteristic
529C2×C7#328311yescharacteristic
530C2×C7#338111yescharacteristic
531C2×C7#338611yescharacteristic
532D7—44no—
533C23—142no—
534C23—142no—
535C23—142no—
536C23—281no—
537C23—281no—
538C23—281no—
539C23—281no—
540C23C22×C22⋊D711yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—41no—
544Q8—41no—
545D4—74no—
546D4—142no—
547D4—142no—
548D4—142no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
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—41no—
568C2×C4—41no—
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—288no—
579C2×C4—281no—
580C2×C4—281no—
581C2×C4—281no—
582C2×C4—281no—
583C2×C4—281no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—564no—
588C2×C4—564no—
589C2×C4#651212yes—
590C2×C4#651312yes—
591C2×C4C2×D4×D711yescharacteristic
592C2×C4#654811yescharacteristic
593C2×C4C2×Q8×D711yescharacteristic
594C2×C4#655111yescharacteristic
595C2×C4D7×#4111yescharacteristic
596C2×C4D7×#4111yescharacteristic
597C8—142no—
598C8—142no—
599C8—281no—
600C8—281no—
601C7#2637411yescharacteristic, complemented, Sylow
602C22—144no—
603C22—282no—
604C22—282no—
605C22—282no—
606C22—282no—
607C22—282no—
608C22—282no—
609C22D4×C22⋊D711yescharacteristic
610C22Q8×C22⋊D711yescharacteristic
611C22C2×#651211yescharacteristic
612C22C2×#651311yescharacteristic
613C22#9051911yescharacteristic
614C22#9052811yescharacteristic
615C22#9052911yescharacteristic
616C4—22no—
617C4—22no—
618C4—41no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—42no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—282no—
627C4—561no—
628C4—561no—
629C4D7×C22⋊Q811yescharacteristic
630C4#9032211yescharacteristic
631C4#9032711yescharacteristic
632C4#9032911yescharacteristic
633C2—284no—
634C2#1174026211yescharacteristic
635C2#1174058611yescharacteristic
636C2#1174089111yescharacteristic
637C2#1174096911yescharacteristic
638C2#1174098711yescharacteristic
639C2#1174113811yescharacteristic
640C2#1174113911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2453C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2453D7
hypocenterC7#26374
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2453

Complements

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

NH ≅ G/Ntype
1. C7×#2453C2semidirect
5. #11729493C2semidirect
6. #11729383C2semidirect
9. #11727438C2semidirect
10. #11727437C2semidirect
13. #11727373C2semidirect
14. #11727362C2semidirect
15. #11727236C2semidirect
134. #2453D7semidirect
601. C7#26374semidirect

Maximal subgroups

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

subgroupindex
C7×#24532
C2×#898382
C2×C4×#11432
C4⋊C4×D282
#117294932
#117293832
#117274412
#117274382
#117274372
#117273792
#117273732
#117273622
#117272362
#263747