Finite Group Databasebeta

← #13373162

Subgroups — #13373162

Order 1792.

866 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89482C211yesmaximal, characteristic
2#11731215C211yesmaximal, characteristic
3#11731085C211yesmaximal, characteristic, complemented
4C7×#1771C211yesmaximal, characteristic, complemented
5#11728240C211yesmaximal, characteristic, complemented
6#11728233C211yesmaximal, characteristic
7#11727571C211yesmaximal, characteristic, complemented
8#11727569C211yesmaximal, characteristic, complemented
9#11727566C211yesmaximal, characteristic
10#11727521C211yesmaximal, characteristic, complemented
11C4×#6395C211yesmaximal, characteristic
12#11726426C211yesmaximal, characteristic
13#11726426C211yesmaximal, characteristic
14#11726378C211yesmaximal, characteristic, complemented
15#11726375C211yesmaximal, characteristic, complemented
16C22×#1132C2211yescharacteristic
17C2×C4×D28C2211yescharacteristic
18C2×C4×D28C2211yescharacteristic
19C2×C4×D28C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C2×C7×C4⋊C8C2211yescharacteristic
22C2×C7×#97C2211yescharacteristic
23C2×C7×#97C2211yescharacteristic
24C2×C4×C7×C8C2211yescharacteristic
25C4×C7×C4⋊C4C2211yescharacteristic
26C2×#6395C2211yescharacteristic
27C2×#6395C2211yescharacteristic
28C2×#6395C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×#6390C2211yescharacteristic
31C2×#6390C2211yescharacteristic
32C2×#6390C2211yescharacteristic
33#89761—21no—
34#89761—21no—
35#89749—21no—
36C4⋊C4×Dic7—21no—
37#89708C2211yescharacteristic
38#89707C2211yescharacteristic
39C4×C4⋊Dic7C2211yescharacteristic
40C4×C4⋊Dic7C2211yescharacteristic
41#89701C2211yescharacteristic
42#89701C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45#89482C418yescomplemented
46#89439—21no—
47D7×#95—21no—
48#89427—21no—
49#89427—21no—
50C7×#336C2211yescharacteristic
51C7×#336C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#89346C2211yescharacteristic
60#22776—71nomaximal, Sylow
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C22×C4×D7—21no—
64C22×C4×D7—21no—
65C22×Dic14C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×C42×C7C2311yescharacteristic
74C2×C42×C7C2311yescharacteristic
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×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×#1132—22no—
93C2×#1132—22no—
94C2×#1132C2×C414yes—
95C4×D28—22no—
96C4×D28—22no—
97C4×D28—22no—
98C4×D28—41no—
99C4×D28—41no—
100C4×D28C2×C414yes—
101C4×Dic14—22no—
102C4×Dic14C2×C414yescomplemented
103C7×C4⋊C8—22no—
104C7×#97—41no—
105C7×#97C2×C414yescomplemented
106C4×C7×C8—41no—
107C7×#95D411yescharacteristic
108C7×#95D411yescharacteristic
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119#6395—22no—
120#6395—22no—
121#6395C2×C414yes—
122#6392C2×C414yescomplemented
123#6390—41no—
124#6390—41no—
125#6390C2×C414yescomplemented
126C2×#440—71no—
127#2355—71no—
128#2349—71no—
129#1935—71no—
130#1933—71no—
131#1921—71no—
132#1918—71no—
133#1877—71no—
134#1877—71no—
135#1836—71no—
136#1805—71no—
137#1802—71no—
138#1771D711yescharacteristic, complemented
139#1762—71no—
140C4×#102—71no—
141C23×D7—21no—
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2411yescharacteristic
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C2×D28—21no—
157C2×D28—21no—
158C2×D28—21no—
159C2×D28C22×C412yes—
160C2×D28#4112yes—
161C2×D28#4112yes—
162C2×C4×D7—24no—
163C2×C4×D7—42no—
164C2×C4×D7—42no—
165C2×C4×D7—42no—
166C2×C4×D7—42no—
167C2×C4×D7—81no—
168C2×C4×D7—81no—
169C2×C4×D7—81no—
170C2×C4×D7—81no—
171C2×Dic14—21no—
172C2×Dic14—21no—
173C2×Dic14—21no—
174C2×Dic14—21no—
175C2×Dic14—21no—
176C2×Dic14C22×C412yes—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8—21no—
183C2×C7×C8—21no—
184C2×C7×C8—21no—
185C2×C7×C8—21no—
186C2×C7×C8—21no—
187C2×C7×C8C22×C412yes—
188C2×C7×C8C22×C412yes—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—41no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7—21no—
195C42×C7C22×C412yes—
196C42×C7#4112yes—
197(C2×D7)⋊C4—24no—
198(C2×D7)⋊C4—42no—
199(C2×D7)⋊C4—42no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—42no—
211C4⋊Dic7—81no—
212C4⋊Dic7C22×C412yes—
213C4⋊Dic7C22×C412yes—
214Dic7⋊C4—24no—
215Dic7⋊C4—81no—
216Dic7⋊C4—81no—
217C4×Dic7—24no—
218C4×Dic7—81no—
219C4×Dic7—81no—
220#1132—28no—
221C22×SD16—71no—
222C2×C4×Q8—71no—
223C2×C4×D4—71no—
224C2×C4×D4—71no—
225C2×C4×D4—71no—
226#440—78no—
227C2×#107—71no—
228C2×C4⋊C8C2×D711yescharacteristic
229C2×#103—71no—
230C2×#103—71no—
231C2×#103—71no—
232C2×#102—71no—
233C2×#102—71no—
234C2×#102—71no—
235C2×#97C2×D711yescharacteristic
236C2×#97C2×D711yescharacteristic
237C2×C4×C8C2×D711yescharacteristic
238#391—71no—
239#391—71no—
240#389—71no—
241#389—71no—
242#387—141no—
243#387—141no—
244#385—141no—
245#385—141no—
246#384—71no—
247#382—71no—
248#382—141no—
249#380—141no—
250C4×C4⋊C4—71no—
251C4×C4⋊C4—71no—
252C4×C4⋊C4—141no—
253C4×C4⋊C4C2×D711yescharacteristic
254C2×#95—141no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#336C2×D711yescharacteristic
264#336C2×D711yescharacteristic
265C23×C7C22×D411yescharacteristic
266C22×D7—22no—
267C22×D7—22no—
268C22×D7—22no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—41no—
272C22×D7—41no—
273C2×C4×C7—21no—
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×C7C4×D412yes—
289C2×C4×C7C4×D412yes—
290C2×C4×C7#13712yes—
291C2×C4×C7C23×C411yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7H(5,𝔽2)11yescharacteristic
296C2×C4×C72−1+411yescharacteristic
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—22no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
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—
318D28—22no—
319D28—22no—
320D28—22no—
321D28#11714yes—
322C4×D7—84no—
323C4×D7—84no—
324C4×D7—84no—
325Dic14—22no—
326Dic14—22no—
327Dic14—41no—
328Dic14—41no—
329C7×C8—22no—
330C7×C8—22no—
331C7×C8—41no—
332C7×C8—41no—
333C22×Q8—71no—
334C22×D4—71no—
335C23×C4—141no—
336C23×C4—141no—
337C23×C4—141no—
338C2×SD16—74no—
339C2×SD16—142no—
340C2×SD16—142no—
341C22×C8C22×D711yescharacteristic
342C22×C8C22×D711yescharacteristic
343C22×C8C22×D711yescharacteristic
344C22×C8C22×D711yescharacteristic
345C4×Q8—74no—
346C4×Q8—142no—
347C4×D4—74no—
348C4×D4—142no—
349C4×D4—142no—
350C4×D4—142no—
351C4×D4—281no—
352C4×D4—281no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—71no—
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×C42—141no—
370C2×C42—141no—
371C2×C42—141no—
372C2×C42C22×D711yescharacteristic
373C2×C42C22×D711yescharacteristic
374C2×C42C22×D711yescharacteristic
375#107—74no—
376C4⋊C8—22no—
377#103—74no—
378#103—281no—
379#103—281no—
380#102—74no—
381#102—142no—
382#102—142no—
383#97—41no—
384#97C4×D714yescomplemented
385C4×C8—41no—
386#95—141no—
387#95—141no—
388#95—141no—
389#95—141no—
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×#13711yescharacteristic
402C22×C7#57511yescharacteristic
403C22×C7#57611yescharacteristic
404C22×C7#57711yescharacteristic
405C2×D7—24no—
406C2×D7—42no—
407C2×D7—42no—
408C2×D7—42no—
409C2×D7—42no—
410C2×D7—42no—
411C2×D7—42no—
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—42no—
423Dic7—42no—
424Dic7—42no—
425Dic7—81no—
426Dic7—81no—
427Dic7—81no—
428Dic7—81no—
429C24—141no—
430C2×Q8—72no—
431C2×Q8—141no—
432C2×Q8—141no—
433C2×Q8—141no—
434C2×Q8—141no—
435C2×Q8—141no—
436C2×D4—72no—
437C2×D4—72no—
438C2×D4—72no—
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×C4—282no—
453C22×C4—282no—
454C22×C4—561no—
455C22×C4—561no—
456C22×C4—561no—
457C22×C4—561no—
458C22×C4C2×D2811yescharacteristic
459C22×C4C2×D2811yescharacteristic
460C22×C4C2×D2811yescharacteristic
461C22×C4C2×D2811yescharacteristic
462C22×C4C2×D2811yescharacteristic
463C22×C4C2×D2811yescharacteristic
464C22×C4C23×D711yescharacteristic
465SD16—148no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8C2×C4×D712yes—
477C2×C8C2×C4×D712yes—
478C4⋊C4—22no—
479C4⋊C4—41no—
480C4⋊C4—72no—
481C4⋊C4—72no—
482C4⋊C4—144no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—141no—
492C4⋊C4—141no—
493C4⋊C4—282no—
494C4⋊C4—561no—
495C4⋊C4—561no—
496C4⋊C4—561no—
497C22⋊C4—144no—
498C22⋊C4—282no—
499C22⋊C4—282no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—144no—
505C42—561no—
506C42—561no—
507C42C2×C4×D712yes—
508C42#115712yes—
509C2×C7#235211yescharacteristic
510C2×C7C4×#13711yescharacteristic
511C2×C7#294611yescharacteristic
512C2×C7#331111yescharacteristic
513C2×C7#331411yescharacteristic
514C2×C7#337011yescharacteristic
515C2×C7#337211yescharacteristic
516D7—44no—
517C23—142no—
518C23—142no—
519C23—142no—
520C23—281no—
521C23—281no—
522C23—281no—
523C23—281no—
524C23C22×D2811yescharacteristic
525Q8—142no—
526Q8—142no—
527Q8—281no—
528Q8—281no—
529D4—74no—
530D4—142no—
531D4—142no—
532D4—142no—
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—281no—
552C2×C4—281no—
553C2×C4—281no—
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—564no—
570C2×C4—564no—
571C2×C4—564no—
572C2×C4C4×D2812yes—
573C2×C4C4×D2812yes—
574C2×C4#647212yes—
575C2×C4C22×C4×D711yescharacteristic
576C2×C4C2×#115711yescharacteristic
577C2×C4#654811yescharacteristic
578C2×C4#655111yescharacteristic
579C2×C4D7×#4111yescharacteristic
580C2×C4D7×#4111yescharacteristic
581C8—22no—
582C8—22no—
583C8—41no—
584C8—41no—
585C7#2277611yescharacteristic, complemented, Sylow
586C22—144no—
587C22—282no—
588C22—282no—
589C22—282no—
590C22—282no—
591C22—282no—
592C22—282no—
593C22C2×C4×D2811yescharacteristic
594C22#9024511yescharacteristic
595C22#9026711yescharacteristic
596C22C2×#647211yescharacteristic
597C22#9043911yescharacteristic
598C22#9044211yescharacteristic
599C22#9044311yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—282no—
607C4—282no—
608C4—282no—
609C4—561no—
610C4—561no—
611C4—561no—
612C4—561no—
613C4#9019511yescharacteristic
614C4D7×#11711yescharacteristic
615C4#9024911yescharacteristic
616C4#9027211yescharacteristic
617C2—284no—
618C2#1173121611yescharacteristic
619C2C4×#647211yescharacteristic
620C2#1173600511yescharacteristic
621C2#1173694711yescharacteristic
622C2#1173694811yescharacteristic
623C2#1173741611yescharacteristic
624C2#1173741711yescharacteristic
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#22776
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. #11731085C2semidirect
4. C7×#1771C2semidirect
5. #11728240C2semidirect
7. #11727571C2semidirect
8. #11727569C2semidirect
10. #11727521C2semidirect
14. #11726378C2semidirect
15. #11726375C2semidirect
45. #89482C4semidirect
102. C4×Dic14C2×C4semidirect
105. C7×#97C2×C4semidirect
122. #6392C2×C4semidirect
125. #6390C2×C4semidirect
138. #1771D7semidirect
384. #97C4×D7semidirect
585. C7#22776semidirect

Maximal subgroups

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

subgroupindex
C2×#894822
#117312152
#117310852
C7×#17712
#117282402
#117282332
#117275712
#117275692
#117275662
#117275212
C4×#63952
#117264262
#117263782
#117263752
#227767