Finite Group Databasebeta

← #13416864

Subgroups — #13416864

Order 1792.

900 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2451C211yesmaximal, characteristic, complemented
2C2×#89856C211yesmaximal, characteristic
3C2×C4×#1143C211yesmaximal, characteristic
4C4⋊C4×D28C211yesmaximal, characteristic
5#11729537C211yesmaximal, characteristic, complemented
6#11729383C211yesmaximal, characteristic, complemented
7#11727441C211yesmaximal, characteristic
8#11727441C211yesmaximal, characteristic
9#11727439C211yesmaximal, characteristic, complemented
10#11727436C211yesmaximal, characteristic, complemented
11#11727380C211yesmaximal, characteristic
12#11727380C211yesmaximal, characteristic
13#11727372C211yesmaximal, characteristic, complemented
14#11727362C211yesmaximal, characteristic, complemented
15#11727236C211yesmaximal, characteristic, complemented
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C22×#1143C2211yescharacteristic
19C2×D7×C4⋊C4—21no—
20C2×C4×D28C2211yescharacteristic
21C2×C4×D28C2211yescharacteristic
22C2×C4×D28C2211yescharacteristic
23C7×#398—21no—
24C7×#395—21no—
25C7×#391—21no—
26C7×#391C2211yescharacteristic
27C7×#391C2211yescharacteristic
28C7×#389C2211yescharacteristic
29C7×#389C2211yescharacteristic
30C7×#387—21no—
31C4×C7×C4⋊C4C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35#89856—24no—
36C4×#1143—24no—
37#89759—21no—
38C2×#6383C2211yescharacteristic
39C2×#6383C2211yescharacteristic
40C2×#6383C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6382C2211yescharacteristic
45C2×#6382C2211yescharacteristic
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#26376—71nomaximal, Sylow
62C22×C7×Q8—21no—
63C22×C7×Q8C2311yescharacteristic
64C22×D28C2311yescharacteristic
65C22×C4×D7—21no—
66C22×C4×D7—21no—
67C22×C4×D7—21no—
68C7×C4⋊Q8—22no—
69C7×C4⋊Q8—41no—
70C4×C7×Q8—22no—
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⋊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×#1143Q814yescomplemented
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#6382—22no—
128#6382—22no—
129#6382—41no—
130C4⋊(C7⋊C8)—22no—
131C4×C7⋊C8—22no—
132C2×#476—71no—
133C2×C4×SD16—71no—
134#2451D711yescharacteristic, complemented
135D4×C4⋊C4—71no—
136#2096—71no—
137#2084—71no—
138#2066—71no—
139#2058—71no—
140#1941—71no—
141#1916—71no—
142#1877—71no—
143#1877—71no—
144#1865—71no—
145#1864—71no—
146#1864—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×Q8—42no—
154C2×C7×Q8—42no—
155C2×C7×Q8—81no—
156C2×C7×Q8C2×Q812yes—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2411yescharacteristic
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171C2×D28—21no—
172C2×D28—21no—
173C2×D28—21no—
174C2×D28C2×D412yes—
175C2×D28C2×Q812yes—
176C2×D28#4112yes—
177C2×C4×D7—24no—
178C2×C4×D7—42no—
179C2×C4×D7—42no—
180C2×C4×D7—42no—
181C2×C4×D7—42no—
182C2×C4×D7—42no—
183C2×C4×D7—42no—
184C2×C4×D7—81no—
185C2×C4×D7—81no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—41no—
192C7×C4⋊C4—42no—
193C7×C4⋊C4—42no—
194C7×C4⋊C4—81no—
195C7×C4⋊C4—81no—
196C7×C4⋊C4SD1614yescomplemented
197C7×C4⋊C4C2×D412yes—
198C42×C7—21no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—21no—
202C42×C7—21no—
203C42×C7—21no—
204C42×C7—42no—
205#1143—28no—
206(C2×D7)⋊C4—24no—
207(C2×D7)⋊C4—42no—
208(C2×D7)⋊C4—42no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—42no—
215C4⋊Dic7—81no—
216C4⋊Dic7#4112yes—
217Dic7⋊C4—42no—
218Dic7⋊C4—42no—
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⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8C2×D412yes—
230C2×C7⋊C8C2×Q812yes—
231C22×SD16—71no—
232C2×C4⋊Q8C2×D711yescharacteristic
233C2×C4×Q8C2×D711yescharacteristic
234C2×C4×D4—71no—
235C2×C4×D4—71no—
236C2×C4×D4—71no—
237C22×C4⋊C4—141no—
238#476—144no—
239C4×SD16—144no—
240C2×#106—71no—
241C2×#106—71no—
242C2×#106—71no—
243C2×#106—71no—
244C2×#106—71no—
245C2×C4⋊C8—71no—
246C2×#103—71no—
247C2×#103—71no—
248C2×#103—71no—
249C2×#102—71no—
250C2×#102—71no—
251C2×#102—71no—
252C2×C4×C8—71no—
253#398—21no—
254#395—21no—
255#391—21no—
256#391C2×D711yescharacteristic
257#391C2×D711yescharacteristic
258#389—71no—
259#389—71no—
260#389C2×D711yescharacteristic
261#389C2×D711yescharacteristic
262#387—21no—
263#385—141no—
264#385—141no—
265#384—71no—
266#380—141no—
267C4×C4⋊C4C2×D711yescharacteristic
268#340—71no—
269#340—71no—
270#340—71no—
271#340—71no—
272#336—71no—
273#336—71no—
274#336—71no—
275#336—71no—
276C23×C7C22×D411yescharacteristic
277C22×D7—22no—
278C22×D7—22no—
279C22×D7—22no—
280C22×D7—41no—
281C22×D7—41no—
282C22×D7—41no—
283C22×D7—41no—
284C7×Q8—22no—
285C7×Q8—22no—
286C7×Q8—41no—
287C7×Q8—41no—
288C7×Q8—84no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
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—41no—
315C2×C4×C7—41no—
316C2×C4×C7C2×SD1612yes—
317C2×C4×C7C2×SD1612yes—
318C2×C4×C7C2×SD1612yes—
319C2×C4×C7C22×D411yescharacteristic
320C2×C4×C7C22×Q811yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C7C2×#4111yescharacteristic
323C2×C4×C7H(5,𝔽2)11yescharacteristic
324C2×C4×C72−1+411yescharacteristic
325C2×Dic7—22no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334D28—22no—
335D28—22no—
336D28—22no—
337D28C22⋊Q814yes—
338C4×D7—48no—
339C4×D7—84no—
340C4×D7—84no—
341C7⋊C8—22no—
342C7⋊C8—22no—
343C7⋊C8—41no—
344C7⋊C8—41no—
345C22×Q8—21no—
346C22×Q8C22×D711yescharacteristic
347C22×D4—71no—
348C23×C4—141no—
349C23×C4—141no—
350C23×C4—141no—
351C2×SD16—74no—
352C2×SD16—142no—
353C2×SD16—142no—
354C22×C8—71no—
355C22×C8—71no—
356C22×C8—71no—
357C22×C8—71no—
358C4⋊Q8—22no—
359C4⋊Q8—41no—
360C4×Q8—22no—
361C4×Q8—41no—
362C4×D4—142no—
363C4×D4—142no—
364C4×D4—142no—
365C4×D4—142no—
366C4×D4—281no—
367C4×D4—281no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—71no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—284no—
380C2×C4⋊C4C22⋊D711yescharacteristic
381C2×C4⋊C4C22⋊D711yescharacteristic
382C2×C4⋊C4C22×D711yescharacteristic
383C2×C4⋊C4C22×D711yescharacteristic
384C2×C4⋊C4C22×D711yescharacteristic
385C2×C22⋊C4—141no—
386C2×C22⋊C4—141no—
387C2×C22⋊C4—141no—
388C2×C42—21no—
389C2×C42C22×D711yescharacteristic
390C2×C42C22×D711yescharacteristic
391C2×C42C22×D711yescharacteristic
392#106—142no—
393#106—142no—
394#106—142no—
395#106—142no—
396#106—281no—
397C4⋊C8—142no—
398#103—142no—
399#103—142no—
400#103—281no—
401#102—74no—
402#102—142no—
403#102—142no—
404C4×C8—142no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—141no—
412#95—141no—
413#95C22⋊D711yescharacteristic
414#95C22⋊D711yescharacteristic
415C22×C7#52511yescharacteristic
416C22×C7D4211yescharacteristic
417C22×C7D4×Q811yescharacteristic
418C22×C7C22×SD1611yescharacteristic
419C22×C7#57511yescharacteristic
420C22×C7#57711yescharacteristic
421C22×C7#57711yescharacteristic
422C2×D7—24no—
423C2×D7—42no—
424C2×D7—42no—
425C2×D7—42no—
426C2×D7—42no—
427C2×D7—42no—
428C2×D7—42no—
429C4×C7—22no—
430C4×C7—22no—
431C4×C7—22no—
432C4×C7—41no—
433C4×C7—42no—
434C4×C7—41no—
435C4×C7—41no—
436C4×C7—42no—
437C4×C7—81no—
438C4×C7—81no—
439C4×C7C2×C22⋊Q811yescharacteristic
440C4×C7#53911yescharacteristic
441C4×C7#54711yescharacteristic
442C4×C7#55411yescharacteristic
443Dic7—42no—
444Dic7—81no—
445Dic7—81no—
446C24—141no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—81no—
455C2×Q8Q8×D712yes—
456C2×D4—72no—
457C2×D4—72no—
458C2×D4—72no—
459C2×D4—141no—
460C2×D4—141no—
461C2×D4—141no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—144no—
467C22×C4—141no—
468C22×C4—141no—
469C22×C4—141no—
470C22×C4—282no—
471C22×C4—282no—
472C22×C4—282no—
473C22×C4—282no—
474C22×C4—282no—
475C22×C4—282no—
476C22×C4—561no—
477C22×C4—561no—
478C22×C4C2×C22⋊D711yescharacteristic
479C22×C4C2×C22⋊D711yescharacteristic
480C22×C4C2×C22⋊D711yescharacteristic
481C22×C4C2×C22⋊D711yescharacteristic
482C22×C4C2×C22⋊D711yescharacteristic
483C22×C4C2×C22⋊D711yescharacteristic
484C22×C4C23×D711yescharacteristic
485SD16—148no—
486C2×C8—72no—
487C2×C8—72no—
488C2×C8—141no—
489C2×C8—141no—
490C2×C8—141no—
491C2×C8—141no—
492C2×C8—141no—
493C2×C8—141no—
494C2×C8—141no—
495C2×C8—141no—
496C2×C8—141no—
497C2×C8—141no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—41no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—72no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4—141no—
510C4⋊C4—141no—
511C4⋊C4—141no—
512C4⋊C4—141no—
513C4⋊C4—141no—
514C4⋊C4—282no—
515C4⋊C4—282no—
516C4⋊C4—282no—
517C4⋊C4—561no—
518C4⋊C4#114314yescomplemented
519C4⋊C4D4×D712yes—
520C22⋊C4—144no—
521C22⋊C4—282no—
522C22⋊C4—282no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—42no—
530C2×C7#244911yescharacteristic
531C2×C7#310111yescharacteristic
532C2×C7#312511yescharacteristic
533C2×C7#326711yescharacteristic
534C2×C7D4×SD1611yescharacteristic
535C2×C7Q8×SD1611yescharacteristic
536C2×C7#338811yescharacteristic
537D7—44no—
538C23—142no—
539C23—142no—
540C23—142no—
541C23—281no—
542C23—281no—
543C23—281no—
544C23—281no—
545C23C22×C22⋊D711yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—84no—
551D4—74no—
552D4—142no—
553D4—142no—
554D4—142no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—142no—
583C2×C4—288no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—281no—
588C2×C4—281no—
589C2×C4—281no—
590C2×C4—281no—
591C2×C4—281no—
592C2×C4—564no—
593C2×C4—564no—
594C2×C4C2×#114312yes—
595C2×C4C2×#114312yes—
596C2×C4C2×#114312yes—
597C2×C4C2×D4×D711yescharacteristic
598C2×C4#654811yescharacteristic
599C2×C4C2×Q8×D711yescharacteristic
600C2×C4#655111yescharacteristic
601C2×C4D7×#4111yescharacteristic
602C2×C4D7×#4111yescharacteristic
603C8—142no—
604C8—142no—
605C8—281no—
606C8—281no—
607C7#2637611yescharacteristic, complemented, Sylow
608C22—144no—
609C22—282no—
610C22—282no—
611C22—282no—
612C22—282no—
613C22—282no—
614C22—282no—
615C22D4×C22⋊D711yescharacteristic
616C22C22×#114311yescharacteristic
617C22Q8×C22⋊D711yescharacteristic
618C22#9051311yescharacteristic
619C22#9051911yescharacteristic
620C22#9052711yescharacteristic
621C22#9052811yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—81no—
631C4—81no—
632C4—282no—
633C4—561no—
634C4—561no—
635C4D7×C22⋊Q811yescharacteristic
636C4#9032211yescharacteristic
637C4#9032711yescharacteristic
638C4#9032911yescharacteristic
639C2—284no—
640C2#1174026211yescharacteristic
641C2#1174079611yescharacteristic
642C2#1174090711yescharacteristic
643C2#1174094811yescharacteristic
644C2D4×#114311yescharacteristic
645C2#1174114011yescharacteristic
646C2Q8×#114311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2451C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2451D7
hypocenterC7#26376
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2451

Complements

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

NH ≅ G/Ntype
1. C7×#2451C2semidirect
5. #11729537C2semidirect
6. #11729383C2semidirect
9. #11727439C2semidirect
10. #11727436C2semidirect
13. #11727372C2semidirect
14. #11727362C2semidirect
15. #11727236C2semidirect
87. C2×#1143Q8semidirect
134. #2451D7semidirect
196. C7×C4⋊C4SD16semidirect
518. C4⋊C4#1143semidirect
607. C7#26376semidirect

Maximal subgroups

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

subgroupindex
C7×#24512
C2×#898562
C2×C4×#11432
C4⋊C4×D282
#117295372
#117293832
#117274412
#117274392
#117274362
#117273802
#117273722
#117273622
#117272362
#263767