Finite Group Databasebeta

← #13362261

Subgroups — #13362261

Order 1792.

1230 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
1C2×C7×#440C211yesmaximal, characteristic, complemented
2C2×#89947C211yesmaximal, characteristic
3C4×#6507C211yesmaximal, characteristic, complemented
4C2×#89798C211yesmaximal, characteristic, complemented
5C4×#6503C211yesmaximal, characteristic
6C2×#89790C211yesmaximal, characteristic, complemented
7C2×#89482C211yesmaximal, characteristic, complemented
8#11729804C211yesmaximal, characteristic, complemented
9#11728383C211yesmaximal, characteristic, complemented
10#11728360C211yesmaximal, characteristic, complemented
11#11728014C211yesmaximal, characteristic, complemented
12#11727979C211yesmaximal, characteristic, complemented
13#11727946C211yesmaximal, characteristic, complemented
14#11727911C211yesmaximal, characteristic
15Dic7×#97C211yesmaximal, characteristic, complemented
16C22×C7×SD16C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C2×#6507C2211yescharacteristic
20C22×#1143C2211yescharacteristic
21C2×#6503C2211yescharacteristic
22C22×#1142C2211yescharacteristic
23C2×C4×C22⋊D7—22no—
24C22×#1132C2211yescharacteristic
25C2×C4×D28C2211yescharacteristic
26C2×C4×Dic14C2211yescharacteristic
27C7×#440—24no—
28C2×C7×#107C2211yescharacteristic
29C2×C7×#103C2211yescharacteristic
30C2×C7×#102C2211yescharacteristic
31C2×C7×#97C2211yescharacteristic
32#90002C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34#89993C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36#89947C418yescomplemented
37C2×#6395C2211yescharacteristic
38C2×#6392C2211yescharacteristic
39C2×#6390C2211yescharacteristic
40C2×#6389C2211yescharacteristic
41C2×#6389C2211yescharacteristic
42C2×C8×Dic7C2211yescharacteristic
43C2×C8×Dic7C2211yescharacteristic
44#89798—24no—
45#89790—24no—
46#89760C2211yescharacteristic
47#89748C2211yescharacteristic
48#89748C2211yescharacteristic
49#89746C2211yescharacteristic
50#89744—22no—
51C2×#6384C2211yescharacteristic
52C2×#6383C2211yescharacteristic
53C2×#6381C2211yescharacteristic
54C2×#6381C2211yescharacteristic
55#89729—22no—
56C4×Dic7⋊C4—22no—
57C42×Dic7C2211yescharacteristic
58C2×#6377C2211yescharacteristic
59#89482—24no—
60#21692—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C22×C22⋊D7—22no—
65C22×D28C2311yescharacteristic
66C22×C4×D7—21no—
67C22×Dic14C2311yescharacteristic
68C2×C7×SD16—22no—
69C2×C7×SD16—22no—
70C2×C7×SD16C2×C414yescomplemented
71C22×C7×C8C2311yescharacteristic
72C22×C7×C8C2311yescharacteristic
73C4×C7×Q8—22no—
74C4×C7×Q8—22no—
75C4×C7×D4—22no—
76C4×C7×D4—22no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—21no—
81C2×C42×C7—21no—
82C2×C42×C7C2311yescharacteristic
83C2×C22⋊Dic7—22no—
84#6507—22no—
85#6507C2×C414yescomplemented
86C2×#1143—22no—
87C2×#1143—22no—
88C2×#1143C2×C414yescomplemented
89#6503—22no—
90#6503C2×C414yes—
91C2×#1142—22no—
92C2×#1142—22no—
93C2×#1142C2×C414yescomplemented
94C4×C22⋊D7—48no—
95C2×(C2×D7)⋊C4—21no—
96C2×(C2×D7)⋊C4—22no—
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×Dic7⋊C4—22no—
100C2×Dic7⋊C4—21no—
101C2×Dic7⋊C4—22no—
102C2×Dic7⋊C4—22no—
103C2×C4×Dic7—21no—
104C2×C4×Dic7D412yes—
105C2×C4×Dic7D412yes—
106C2×C4×Dic7C2311yescharacteristic
107C2×C4×Dic7C2311yescharacteristic
108C22×C7⋊C8C2311yescharacteristic
109C22×C7⋊C8C2311yescharacteristic
110C2×#1132—22no—
111C2×#1132—22no—
112C2×#1132C2×C414yescomplemented
113C4×D28—22no—
114C4×D28—22no—
115C4×Dic14—22no—
116C4×Dic14—22no—
117C7×#107—22no—
118C7×#103—22no—
119C7×#102—22no—
120C7×#97D414yescomplemented
121#6409—22no—
122#6406—22no—
123C7⋊#95—22no—
124C7⋊#95—22no—
125#6395—22no—
126#6392—22no—
127#6390—22no—
128#6389—22no—
129#6389—22no—
130C8×Dic7—22no—
131C8×Dic7C2×C414yescomplemented
132#6384—22no—
133#6383—22no—
134#6381—22no—
135#6381—22no—
136#6377D414yescomplemented
137C2×#492—71no—
138C2×#440—71no—
139C2×#440—71no—
140C2×#440—71no—
141C2×#440D711yescharacteristic, complemented
142C4×C4⋊Q8—71no—
143C4×#127—71no—
144#1973—71no—
145#1973—71no—
146#1973—71no—
147#1973—71no—
148#1957—71no—
149#1956—71no—
150#1845—71no—
151C4×#97—71no—
152C24×C7—21no—
153C23×D7—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—21no—
159C2×C7×Q8C22×C412yes—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4—21no—
165C2×C7×D4C22×C412yes—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—44no—
170C22×C4×C7—42no—
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2411yescharacteristic
174C2×C22⋊D7—28no—
175C2×C22⋊D7—44no—
176C2×C22⋊D7—44no—
177C22×Dic7—21no—
178C22×Dic7—21no—
179C22×Dic7—21no—
180C22×Dic7C2×D412yes—
181C22×Dic7C2×D412yes—
182C2×D28—21no—
183C2×D28—21no—
184C2×D28—21no—
185C2×D28—21no—
186C2×D28—21no—
187C2×D28C22×C412yes—
188C2×C4×D7—44no—
189C2×C4×D7—42no—
190C2×Dic14—21no—
191C2×Dic14—21no—
192C2×Dic14—21no—
193C2×Dic14—21no—
194C2×Dic14—21no—
195C2×Dic14C22×C412yes—
196C7×SD16—28no—
197C2×C7×C8—21no—
198C2×C7×C8—21no—
199C2×C7×C8C22×C412yes—
200C2×C7×C8C2×D412yes—
201C2×C7×C8C2×D412yes—
202C2×C7×C8#4112yes—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—42no—
208C7×C22⋊C4—42no—
209C42×C7—42no—
210C42×C7C2×D412yes—
211C42×C7C2×D412yes—
212C22⋊Dic7—44no—
213#1143—28no—
214#1142—28no—
215(C2×D7)⋊C4—44no—
216(C2×D7)⋊C4—42no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7—21no—
220C4⋊Dic7—21no—
221Dic7⋊C4—28no—
222Dic7⋊C4—42no—
223Dic7⋊C4—44no—
224Dic7⋊C4—44no—
225C4×Dic7—21no—
226C4×Dic7—21no—
227C4×Dic7—24no—
228C4×Dic7—24no—
229C4×Dic7—42no—
230C4×Dic7C22×C412yes—
231C4×Dic7#4112yes—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8C22×C412yes—
235C2×C7⋊C8C2×D412yes—
236C2×C7⋊C8C2×D412yes—
237C2×C7⋊C8#4112yes—
238#1132—28no—
239C22×SD16—71no—
240C22×SD16—71no—
241C22×SD16—71no—
242C22×SD16C2×D711yescharacteristic
243C2×C4⋊Q8—71no—
244C2×#127—71no—
245C2×C4×Q8—71no—
246C2×C4×Q8C2×D711yescharacteristic
247C2×C4×D4—71no—
248C2×C4×D4—142no—
249C2×C4×D4C2×D711yescharacteristic
250#492—78no—
251#440—24no—
252#440—144no—
253#440—144no—
254#440—144no—
255C2×#107—71no—
256C2×#107—71no—
257C2×#107—71no—
258C2×#107C2×D711yescharacteristic
259C2×#103—71no—
260C2×#103—71no—
261C2×#103—71no—
262C2×#103C2×D711yescharacteristic
263C2×#102—71no—
264C2×#102—71no—
265C2×#102—71no—
266C2×#102C2×D711yescharacteristic
267C2×#97—71no—
268C2×#97—71no—
269C2×#97—71no—
270C2×#97C2×D711yescharacteristic
271C2×C4×C8—71no—
272C2×C4×C8—71no—
273#391—71no—
274#391—71no—
275#390—71no—
276#390—71no—
277#390—142no—
278#389—142no—
279#384—71no—
280#384—71no—
281C4×C4⋊C4—142no—
282C43—71no—
283C23×C7—22no—
284C23×C7—41no—
285C23×C7—41no—
286C23×C7—42no—
287C23×C7—42no—
288C23×C7C22×D411yescharacteristic
289C22×D7—22no—
290C22×D7—41no—
291C22×D7—41no—
292C22×D7—42no—
293C22×D7—42no—
294C7×Q8—22no—
295C7×Q8—22no—
296C7×Q8—41no—
297C7×Q8—41no—
298C7×D4—22no—
299C7×D4—22no—
300C7×D4—41no—
301C7×D4—41no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—22no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—42no—
310C2×C4×C7—42no—
311C2×C4×C7—84no—
312C2×C4×C7#12712yes—
313C2×C4×C7#12712yes—
314C2×C4×C7C23×C411yescharacteristic
315C2×C4×C7C22×D411yescharacteristic
316C2×C4×C7C22×D411yescharacteristic
317C2×C4×C7C2×#4111yescharacteristic
318C2×C4×C7C2×#4111yescharacteristic
319C2×C4×C7C2×#4111yescharacteristic
320C22⋊D7—416no—
321C2×Dic7—22no—
322C2×Dic7—22no—
323C2×Dic7—22no—
324C2×Dic7—22no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—42no—
330C2×Dic7—42no—
331C2×Dic7C4×D414yes—
332C2×Dic7AGL(1,ℤ/8)14yes—
333C2×Dic7#13714yes—
334D28—22no—
335D28—22no—
336D28—41no—
337D28—41no—
338C4×D7—84no—
339Dic14—22no—
340Dic14—22no—
341Dic14—41no—
342Dic14—41no—
343C7×C8—22no—
344C7×C8C4×D414yescomplemented
345C7⋊C8—22no—
346C7⋊C8C4×D414yescomplemented
347C22×Q8—71no—
348C22×Q8C22×D711yescharacteristic
349C22×D4—71no—
350C22×D4—142no—
351C22×D4C22×D711yescharacteristic
352C23×C4—21no—
353C23×C4—141no—
354C2×SD16—22no—
355C2×SD16—22no—
356C2×SD16—74no—
357C2×SD16—74no—
358C2×SD16—74no—
359C2×SD16—142no—
360C2×SD16—142no—
361C2×SD16—142no—
362C2×SD16—142no—
363C2×SD16—142no—
364C2×SD16—142no—
365C2×SD16C4×D714yescomplemented
366C22×C8—71no—
367C22×C8—71no—
368C22×C8C22×D711yescharacteristic
369C22×C8C22×D711yescharacteristic
370C4⋊Q8—74no—
371C4⋊Q8—142no—
372#127—74no—
373#127—142no—
374C4×Q8—22no—
375C4×Q8—22no—
376C4×Q8—142no—
377C4×Q8—142no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—142no—
381C4×D4—142no—
382C4×D4—288no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—71no—
385C2×C4⋊C4—71no—
386C2×C4⋊C4—142no—
387C2×C4⋊C4—142no—
388C2×C4⋊C4—142no—
389C2×C4⋊C4—141no—
390C2×C4⋊C4C22×D711yescharacteristic
391C2×C4⋊C4C22×D711yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—142no—
394C2×C22⋊C4—142no—
395C2×C22⋊C4—141no—
396C2×C42—21no—
397C2×C42—71no—
398C2×C42—72no—
399C2×C42—72no—
400C2×C42—71no—
401C2×C42—141no—
402C2×C42C22×D711yescharacteristic
403#107—22no—
404#107—142no—
405#107—142no—
406#107—142no—
407#103—22no—
408#103—142no—
409#103—142no—
410#103—142no—
411#102—22no—
412#102—142no—
413#102—142no—
414#102—142no—
415#97—74no—
416#97—142no—
417#97—142no—
418#97C22⋊D714yescomplemented
419C4×C8—74no—
420C4×C8—142no—
421#95—142no—
422#95—142no—
423C22×C7—42no—
424C22×C7—44no—
425C22×C7—81no—
426C22×C7—81no—
427C22×C7—82no—
428C22×C7C2×C4×D411yescharacteristic
429C22×C7C2×#12711yescharacteristic
430C22×C7#53211yescharacteristic
431C22×C7C2×AGL(1,ℤ/8)12yes—
432C22×C7C2×#13712yes—
433C2×D7—42no—
434C2×D7—44no—
435C2×D7—81no—
436C2×D7—81no—
437C2×D7—82no—
438C4×C7—22no—
439C4×C7—22no—
440C4×C7—42no—
441C4×C7—81no—
442C4×C7—81no—
443C4×C7C2×C4×D411yescharacteristic
444C4×C7C2×C4×D411yescharacteristic
445C4×C7#53211yescharacteristic
446C4×C7#53211yescharacteristic
447Dic7—42no—
448Dic7—42no—
449Dic7—81no—
450Dic7—81no—
451Dic7#44018yescomplemented
452C24—21no—
453C24—141no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—72no—
460C2×Q8—141no—
461C2×Q8—141no—
462C2×Q8—141no—
463C2×Q8—141no—
464C2×Q8—141no—
465C2×Q8C2×C4×D712yes—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—72no—
472C2×D4—148no—
473C2×D4—141no—
474C2×D4—141no—
475C2×D4—141no—
476C2×D4—141no—
477C2×D4—141no—
478C2×D4—284no—
479C2×D4—284no—
480C2×D4C2×C4×D712yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—72no—
487C22×C4—72no—
488C22×C4—141no—
489C22×C4—141no—
490C22×C4—141no—
491C22×C4—284no—
492C22×C4—282no—
493C22×C4D4×D711yescharacteristic
494C22×C4D4×D711yescharacteristic
495C22×C4C23×D711yescharacteristic
496SD16—28no—
497SD16—148no—
498SD16—148no—
499SD16—148no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—72no—
503C2×C8—72no—
504C2×C8—72no—
505C2×C8—72no—
506C2×C8—141no—
507C2×C8—141no—
508C2×C8C2×C4×D712yes—
509C2×C8#115712yes—
510C2×C8C2×C22⋊D712yes—
511C2×C8C2×C22⋊D712yes—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—42no—
517C4⋊C4—148no—
518C4⋊C4—141no—
519C4⋊C4—141no—
520C4⋊C4—141no—
521C4⋊C4—141no—
522C4⋊C4—284no—
523C4⋊C4—284no—
524C4⋊C4—282no—
525C22⋊C4—42no—
526C22⋊C4—284no—
527C22⋊C4—284no—
528C22⋊C4—282no—
529C42—42no—
530C42—72no—
531C42—72no—
532C42—144no—
533C42—144no—
534C42—141no—
535C42—141no—
536C42—282no—
537C42D4×D712yes—
538C42C2×C22⋊D712yes—
539C2×C7—82no—
540C2×C7C4×#12711yescharacteristic
541C2×C7C2×#44012yes—
542C2×C7#315111yescharacteristic
543C2×C7#315211yescharacteristic
544C2×C7#318212yes—
545D7—82no—
546C23—22no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—41no—
551C23—142no—
552C23—282no—
553C23—282no—
554C23—281no—
555C23—281no—
556C23C2×D4×D711yescharacteristic
557Q8—22no—
558Q8—22no—
559Q8—41no—
560Q8—41no—
561Q8—142no—
562Q8—142no—
563Q8—281no—
564Q8—281no—
565D4—22no—
566D4—22no—
567D4—41no—
568D4—41no—
569D4—142no—
570D4—142no—
571D4—2816no—
572D4—281no—
573D4—281no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—74no—
584C2×C4—74no—
585C2×C4—74no—
586C2×C4—84no—
587C2×C4—142no—
588C2×C4—142no—
589C2×C4—142no—
590C2×C4—142no—
591C2×C4—281no—
592C2×C4—281no—
593C2×C4—282no—
594C2×C4—282no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—564no—
598C2×C4#650312yes—
599C2×C4#650312yes—
600C2×C4C22×C4×D711yescharacteristic
601C2×C4C2×#115711yescharacteristic
602C2×C4C2×D4×D711yescharacteristic
603C2×C4D7×#4111yescharacteristic
604C2×C4D7×#4111yescharacteristic
605C2×C4C22×C22⋊D711yescharacteristic
606C8—22no—
607C8—74no—
608C8—142no—
609C8C4×C22⋊D714yescomplemented
610C7#2169211yescharacteristic, complemented, Sylow
611C22—42no—
612C22—44no—
613C22—82no—
614C22—81no—
615C22—81no—
616C22—282no—
617C22—284no—
618C22—562no—
619C22—561no—
620C22—561no—
621C22C4×D4×D711yescharacteristic
622C22#9023911yescharacteristic
623C22D7×AGL(1,ℤ/8)12yes—
624C22D7×#13712yes—
625C22C2×#650311yescharacteristic
626C4—22no—
627C4—22no—
628C4—42no—
629C4—78no—
630C4—81no—
631C4—81no—
632C4—282no—
633C4—282no—
634C4—561no—
635C4—561no—
636C4C4×D4×D711yescharacteristic
637C4#9023911yescharacteristic
638C4C2×C4×C22⋊D711yescharacteristic
639C4#9052311yescharacteristic
640C2—82no—
641C2—562no—
642C2D7×#44012yes—
643C2#1173723712yes—
644C2C4×#650311yescharacteristic
645C2#1174138211yescharacteristic
646C2#1174141711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#440C2
socleC23×C7C22×D4
prime radicalC2×#6503C22
cosocleC22×C4×C7C24
hypercenterC2×#440D7
hypocenterC7#21692
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. C2×C7×#440C2semidirect
3. C4×#6507C2semidirect
4. C2×#89798C2semidirect
6. C2×#89790C2semidirect
7. C2×#89482C2semidirect
8. #11729804C2semidirect
9. #11728383C2semidirect
10. #11728360C2semidirect
11. #11728014C2semidirect
12. #11727979C2semidirect
13. #11727946C2semidirect
15. Dic7×#97C2semidirect
36. #89947C4semidirect
70. C2×C7×SD16C2×C4semidirect
85. #6507C2×C4semidirect
88. C2×#1143C2×C4semidirect
93. C2×#1142C2×C4semidirect
112. C2×#1132C2×C4semidirect
120. C7×#97D4semidirect
131. C8×Dic7C2×C4semidirect
136. #6377D4semidirect
141. C2×#440D7semidirect
344. C7×C8C4×D4semidirect
346. C7⋊C8C4×D4semidirect
365. C2×SD16C4×D7semidirect
418. #97C22⋊D7semidirect
451. Dic7#440semidirect
609. C8C4×C22⋊D7semidirect
610. C7#21692semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4402
C2×#899472
C4×#65072
C2×#897982
C4×#65032
C2×#897902
C2×#894822
#117298042
#117283832
#117283602
#117280142
#117279792
#117279462
#117279112
Dic7×#972
#216927