Finite Group Databasebeta

← #13373237

Subgroups — #13373237

Order 1792.

866 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89482C211yesmaximal, characteristic
2#11731083C211yesmaximal, characteristic
3#11731082C211yesmaximal, characteristic, complemented
4C7×#1771C211yesmaximal, characteristic, complemented
5#11728239C211yesmaximal, characteristic, complemented
6#11728234C211yesmaximal, characteristic
7#11727572C211yesmaximal, characteristic
8#11727569C211yesmaximal, characteristic, complemented
9#11727567C211yesmaximal, characteristic, complemented
10#11727522C211yesmaximal, characteristic
11C4×#6390C211yesmaximal, characteristic, complemented
12#11726440C211yesmaximal, characteristic
13#11726440C211yesmaximal, characteristic
14#11726378C211yesmaximal, characteristic, complemented
15#11726375C211yesmaximal, characteristic, complemented
16C22×#1132C2211yescharacteristic
17C2×#6455—21no—
18C2×C4×D28C2211yescharacteristic
19C2×C4×Dic14C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C2×C7×C4⋊C8C2211yescharacteristic
23C2×C7×#97C2211yescharacteristic
24C2×C7×#97C2211yescharacteristic
25C2×C4×C7×C8C2211yescharacteristic
26C4×C7×C4⋊C4C2211yescharacteristic
27C2×#6395C2211yescharacteristic
28C2×#6395C2211yescharacteristic
29C2×#6395C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6390C2211yescharacteristic
32C2×#6390C2211yescharacteristic
33C2×#6390C2211yescharacteristic
34#89759—21no—
35#89744—21no—
36#89744—21no—
37#89711C2211yescharacteristic
38#89711C2211yescharacteristic
39#89708C2211yescharacteristic
40#89707C2211yescharacteristic
41C4×C4⋊Dic7C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45#89482C418yescomplemented
46#89442—21no—
47#89442—21no—
48#89422—21no—
49#89421—21no—
50C7×#336C2211yescharacteristic
51C7×#336C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#89346C2211yescharacteristic
60#22781—71nomaximal, Sylow
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C22×Dic14C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C2×C7×C4⋊C4D411yescharacteristic
69C2×C7×C4⋊C4D411yescharacteristic
70C2×C42×C7C2311yescharacteristic
71C2×C42×C7C2311yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96C2×#1132—22no—
97C2×#1132—22no—
98C2×#1132C2×C414yes—
99#6455—44no—
100C4×D28—22no—
101C4×D28C2×C414yes—
102C4×Dic14—22no—
103C4×Dic14—22no—
104C4×Dic14—22no—
105C4×Dic14—41no—
106C4×Dic14—41no—
107C4×Dic14C2×C414yes—
108C7×C4⋊C8—22no—
109C7×#97—41no—
110C7×#97C2×C414yes—
111C4×C7×C8—41no—
112C7×#95D411yescharacteristic
113C7×#95D411yescharacteristic
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120#6395—41no—
121#6395—41no—
122#6395C2×C414yes—
123#6392C2×C414yes—
124#6390—22no—
125#6390—22no—
126#6390C2×C414yes—
127C2×#440—71no—
128#2354—71no—
129#2351—71no—
130#1936—71no—
131#1934—71no—
132#1921—71no—
133#1918—71no—
134#1880—71no—
135#1880—71no—
136#1836—71no—
137#1804—71no—
138#1803—71no—
139#1771D711yescharacteristic, complemented
140#1763—71no—
141C4×#103—71no—
142C23×D7—21no—
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2411yescharacteristic
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C2×D28—21no—
158C2×D28—21no—
159C2×D28—21no—
160C2×D28—21no—
161C2×D28—21no—
162C2×D28C22×C412yes—
163C2×C4×D7—24no—
164C2×C4×D7—42no—
165C2×C4×D7—42no—
166C2×Dic14—21no—
167C2×Dic14—21no—
168C2×Dic14—21no—
169C2×Dic14C22×C412yes—
170C2×Dic14#4112yes—
171C2×Dic14#4112yes—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8C22×C412yes—
183C2×C7×C8C22×C412yes—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—41no—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7C22×C412yes—
191C42×C7#4112yes—
192(C2×D7)⋊C4—24no—
193(C2×D7)⋊C4—42no—
194(C2×D7)⋊C4—42no—
195(C2×D7)⋊C4—81no—
196(C2×D7)⋊C4—81no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—42no—
208C4⋊Dic7—81no—
209C4⋊Dic7C22×C412yes—
210C4⋊Dic7C22×C412yes—
211Dic7⋊C4—24no—
212Dic7⋊C4—42no—
213Dic7⋊C4—42no—
214Dic7⋊C4—42no—
215Dic7⋊C4—42no—
216C4×Dic7—24no—
217C4×Dic7—42no—
218C4×Dic7—42no—
219C4×Dic7—42no—
220C4×Dic7—42no—
221#1132—28no—
222C22×SD16—71no—
223C2×C4×Q8—71no—
224C2×C4×Q8—71no—
225C2×C4×Q8—71no—
226C2×C4×D4—71no—
227C2×#117—141no—
228#440—78no—
229C2×#107—71no—
230C2×C4⋊C8C2×D711yescharacteristic
231C2×#103—71no—
232C2×#103—71no—
233C2×#103—71no—
234C2×#102—71no—
235C2×#102—71no—
236C2×#102—71no—
237C2×#97C2×D711yescharacteristic
238C2×#97C2×D711yescharacteristic
239C2×C4×C8C2×D711yescharacteristic
240#390—71no—
241#390—71no—
242#389—71no—
243#389—71no—
244#389—141no—
245#389—141no—
246#388—141no—
247#388—141no—
248#384—71no—
249#382—71no—
250#382—141no—
251#380—141no—
252C4×C4⋊C4—71no—
253C4×C4⋊C4—71no—
254C4×C4⋊C4—141no—
255C4×C4⋊C4C2×D711yescharacteristic
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#336C2×D711yescharacteristic
265#336C2×D711yescharacteristic
266C23×C7C22×D411yescharacteristic
267C22×D7—22no—
268C22×D7—41no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—41no—
272C22×D7—41no—
273C22×D7—41no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7C4×D412yes—
290C2×C4×C7C4×D412yes—
291C2×C4×C7AGL(1,ℤ/8)12yes—
292C2×C4×C7C23×C411yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7H(5,𝔽2)11yescharacteristic
297C2×C4×C72−1+411yescharacteristic
298C2×Dic7—22no—
299C2×Dic7—22no—
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319D28—22no—
320D28—22no—
321D28—41no—
322D28—41no—
323C4×D7—48no—
324Dic14—22no—
325Dic14—22no—
326Dic14—22no—
327Dic14#11714yescomplemented
328C7×C8—22no—
329C7×C8—22no—
330C7×C8—41no—
331C7×C8—41no—
332C22×Q8—71no—
333C22×D4—71no—
334C23×C4—141no—
335C2×SD16—74no—
336C2×SD16—142no—
337C2×SD16—142no—
338C22×C8C22×D711yescharacteristic
339C22×C8C22×D711yescharacteristic
340C22×C8C22×D711yescharacteristic
341C22×C8C22×D711yescharacteristic
342C4×Q8—74no—
343C4×Q8—142no—
344C4×Q8—142no—
345C4×Q8—142no—
346C4×Q8—281no—
347C4×Q8—281no—
348C4×D4—74no—
349C4×D4—142no—
350#117—284no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4D2811yescharacteristic
365C2×C4⋊C4D2811yescharacteristic
366C2×C22⋊C4—141no—
367C2×C22⋊C4—141no—
368C2×C22⋊C4—141no—
369C2×C22⋊C4—141no—
370C2×C22⋊C4—141no—
371C2×C42—141no—
372C2×C42—141no—
373C2×C42—141no—
374C2×C42—141no—
375C2×C42—141no—
376C2×C42C22×D711yescharacteristic
377C2×C42C22×D711yescharacteristic
378C2×C42C22×D711yescharacteristic
379#107—74no—
380C4⋊C8—22no—
381#103—74no—
382#103—142no—
383#103—142no—
384#102—74no—
385#102—281no—
386#102—281no—
387#97—41no—
388#97C4×D714yes—
389C4×C8—41no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95D2811yescharacteristic
397#95D2811yescharacteristic
398C22×C7C2×C4×D411yescharacteristic
399C22×C7#54611yescharacteristic
400C22×C7#54811yescharacteristic
401C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
402C22×C7#57511yescharacteristic
403C22×C7#57711yescharacteristic
404C22×C7#57811yescharacteristic
405C2×D7—42no—
406C2×D7—42no—
407C2×D7—42no—
408C2×D7—81no—
409C2×D7—81no—
410C2×D7—81no—
411C2×D7—81no—
412C4×C7—22no—
413C4×C7—22no—
414C4×C7—22no—
415C4×C7—41no—
416C4×C7—41no—
417C4×C7—41no—
418C4×C7C2×#11711yescharacteristic
419C4×C7#52011yescharacteristic
420C4×C7#55211yescharacteristic
421C4×C7#55311yescharacteristic
422Dic7—24no—
423Dic7—42no—
424Dic7—42no—
425Dic7—81no—
426Dic7—81no—
427Dic7—81no—
428Dic7—81no—
429C24—141no—
430C2×Q8—72no—
431C2×Q8—72no—
432C2×Q8—72no—
433C2×Q8—141no—
434C2×Q8—141no—
435C2×Q8—141no—
436C2×D4—72no—
437C2×D4—141no—
438C2×D4—141no—
439C2×D4—141no—
440C2×D4—141no—
441C2×D4—141no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—144no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—282no—
451C22×C4—282no—
452C22×C4C2×D2811yescharacteristic
453C22×C4C2×D2811yescharacteristic
454C22×C4C2×D2811yescharacteristic
455C22×C4C2×D2811yescharacteristic
456C22×C4C2×D2811yescharacteristic
457C22×C4C2×D2811yescharacteristic
458C22×C4C23×D711yescharacteristic
459SD16—148no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8C2×C4×D712yes—
471C2×C8C2×C4×D712yes—
472C4⋊C4—22no—
473C4⋊C4—41no—
474C4⋊C4—72no—
475C4⋊C4—72no—
476C4⋊C4—144no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—282no—
488C4⋊C4—282no—
489C4⋊C4—282no—
490C4⋊C4—282no—
491C4⋊C4—282no—
492C4⋊C4—561no—
493C22⋊C4—144no—
494C22⋊C4—282no—
495C22⋊C4—282no—
496C22⋊C4—561no—
497C22⋊C4—561no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—144no—
503C42—282no—
504C42—282no—
505C42—282no—
506C42—282no—
507C42C2×C4×D712yes—
508C42#115712yes—
509C2×C7#235211yescharacteristic
510C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
511C2×C7#294711yescharacteristic
512C2×C7#330811yescharacteristic
513C2×C7#332111yescharacteristic
514C2×C7#336811yescharacteristic
515C2×C7#337711yescharacteristic
516D7—82no—
517C23—142no—
518C23—281no—
519C23—281no—
520C23—281no—
521C23—281no—
522C23—281no—
523C23—281no—
524C23C22×D2811yescharacteristic
525Q8—74no—
526Q8—142no—
527Q8—142no—
528Q8—142no—
529D4—142no—
530D4—142no—
531D4—281no—
532D4—281no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—142no—
552C2×C4—142no—
553C2×C4—288no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4C4×D2812yes—
571C2×C4C4×D2812yes—
572C2×C4#647112yes—
573C2×C4C22×C4×D711yescharacteristic
574C2×C4C2×#115711yescharacteristic
575C2×C4#654811yescharacteristic
576C2×C4#655111yescharacteristic
577C2×C4D7×#4111yescharacteristic
578C2×C4D7×#4111yescharacteristic
579C8—22no—
580C8—22no—
581C8—41no—
582C8—41no—
583C7#2278111yescharacteristic, complemented, Sylow
584C22—282no—
585C22—282no—
586C22—282no—
587C22—561no—
588C22—561no—
589C22—561no—
590C22—561no—
591C22C2×C4×D2811yescharacteristic
592C22#9024511yescharacteristic
593C22#9026711yescharacteristic
594C22C2×#647111yescharacteristic
595C22#9043911yescharacteristic
596C22#9044211yescharacteristic
597C22#9044411yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—144no—
605C4—282no—
606C4—282no—
607C4—561no—
608C4—561no—
609C4—561no—
610C4—561no—
611C4#9019511yescharacteristic
612C4D7×#11711yescharacteristic
613C4#9024911yescharacteristic
614C4#9027211yescharacteristic
615C2—562no—
616C2#1173121611yescharacteristic
617C2C4×#647111yescharacteristic
618C2#1173601611yescharacteristic
619C2#1173695411yescharacteristic
620C2#1173696111yescharacteristic
621C2#1173742511yescharacteristic
622C2#1173743011yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1771C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1771D7
hypocenterC7#22781
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
3. #11731082C2semidirect
4. C7×#1771C2semidirect
5. #11728239C2semidirect
8. #11727569C2semidirect
9. #11727567C2semidirect
11. C4×#6390C2semidirect
14. #11726378C2semidirect
15. #11726375C2semidirect
45. #89482C4semidirect
139. #1771D7semidirect
327. Dic14#117semidirect
583. C7#22781semidirect

Maximal subgroups

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

subgroupindex
C2×#894822
#117310832
#117310822
C7×#17712
#117282392
#117282342
#117275722
#117275692
#117275672
#117275222
C4×#63902
#117264402
#117263782
#117263752
#227817