Finite Group Databasebeta

← #13375664

Subgroups — #13375664

Order 1792.

914 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic7⋊C4C211yesmaximal, characteristic
2C2×#89531C211yesmaximal, characteristic
3#11733431C211yesmaximal, characteristic, complemented
4C7×#1864C211yesmaximal, characteristic
5#11729363C211yesmaximal, characteristic
6#11728317C211yesmaximal, characteristic, complemented
7#11728315C211yesmaximal, characteristic, complemented
8#11727923C211yesmaximal, characteristic
9#11727906C211yesmaximal, characteristic
10Dic7×#102C211yesmaximal, characteristic
11#11727374C211yesmaximal, characteristic, complemented
12#11726557C211yesmaximal, characteristic, complemented
13#11726544C211yesmaximal, characteristic, complemented
14#11726406C211yesmaximal, characteristic, complemented
15#11726378C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×D4×Dic7C2211yescharacteristic
18C2×D4×Dic7C2211yescharacteristic
19C22×Dic7⋊C4—21no—
20C2×#6451C2211yescharacteristic
21C2×#6450—21no—
22C2×C7×#106C2211yescharacteristic
23C2×C7×#102C2211yescharacteristic
24C2×C7×#102C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C2×#6406C2211yescharacteristic
27C2×#6406C2211yescharacteristic
28#89982—21no—
29C2×#6392C2211yescharacteristic
30C2×#6389C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33#89750—21no—
34#89748C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37#89745C2211yescharacteristic
38#89745C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41#89718—21no—
42#89714—21no—
43C4×C4⋊Dic7C2211yescharacteristic
44C4×Dic7⋊C4C2211yescharacteristic
45#89700C2211yescharacteristic
46C2×#6377C2211yescharacteristic
47#89531C418yescomplemented
48#89424—21no—
49#89418—21no—
50C7×#340C2211yescharacteristic
51C7×#336C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22968—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C23×Dic7—21no—
64C23×Dic7—21no—
65C22×Dic14—21no—
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C4×C7×D4—22no—
69C4×C7×D4—41no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4—21no—
75C2×C42×C7C2311yescharacteristic
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—21no—
78D4×Dic7—22no—
79D4×Dic7—22no—
80D4×Dic7—41no—
81D4×Dic7C2×C414yes—
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×Dic7⋊C4—21no—
92C2×Dic7⋊C4—44no—
93C2×Dic7⋊C4D411yescharacteristic
94C2×Dic7⋊C4D411yescharacteristic
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7C2311yescharacteristic
99C2×C4×Dic7C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102#6451—22no—
103#6451C2×C414yescomplemented
104#6450—44no—
105C7×#106—41no—
106C7×#102—22no—
107C7×#102C2×C414yes—
108C7×#95—21no—
109#6406C2×C414yes—
110#6406Q814yescomplemented
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95D411yescharacteristic
118C7⋊#95D411yescharacteristic
119#6392C2×C414yescomplemented
120#6389—41no—
121Dic7⋊C8C2×C414yescomplemented
122C8×Dic7—22no—
123#6382—41no—
124#6381C2×C414yescomplemented
125#6377—41no—
126C2×#474—71no—
127#2365—71no—
128D4×C4⋊C4—71no—
129#1990—71no—
130#1948—71no—
131#1943—71no—
132#1943—71no—
133#1921—71no—
134#1921—71no—
135#1864—71no—
136#1864D711yescharacteristic
137#1770—71no—
138#1762—71no—
139#1762—71no—
140C4×#102—71no—
141C24×C7—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4C22×C412yes—
146C2×C7×D4C2×Q812yes—
147C2×C7×D4#4112yes—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—42no—
152C22×C4×C7—42no—
153C22×C4×C7—81no—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2411yescharacteristic
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—24no—
161C22×Dic7—21no—
162C22×Dic7—42no—
163C22×Dic7—42no—
164C22×Dic7—42no—
165C22×Dic7—42no—
166C22×Dic7—81no—
167C22×Dic7C2×D411yescharacteristic
168C22×Dic7C2×D411yescharacteristic
169C22×Dic7C2×D411yescharacteristic
170C22×Dic7C2×D411yescharacteristic
171C2×Dic14—24no—
172C2×Dic14—42no—
173C2×Dic14—42no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8C22×C412yes—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—42no—
186C7×C4⋊C4C22×C412yes—
187C7×C22⋊C4—42no—
188C42×C7—21no—
189C42×C7—21no—
190C22⋊Dic7—24no—
191C22⋊Dic7—42no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7C22×C412yes—
198C4⋊Dic7C22×C412yes—
199C4⋊Dic7C2×Q812yes—
200Dic7⋊C4—24no—
201Dic7⋊C4—22no—
202Dic7⋊C4—22no—
203Dic7⋊C4—42no—
204Dic7⋊C4—42no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207Dic7⋊C4—81no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—42no—
211C4×Dic7—42no—
212C4×Dic7—42no—
213C4×Dic7C22×C412yes—
214C4×Dic7#4112yes—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8C22×C412yes—
220C2×C7⋊C8C2×Q812yes—
221C2×C4⋊Q8—71no—
222C2×C4×Q8—141no—
223C2×C4×D4—71no—
224C2×C4×D4—71no—
225C2×C4×D4C2×D711yescharacteristic
226C22×C4⋊C4—141no—
227#474—78no—
228C2×#107—71no—
229C2×#107—71no—
230C2×#106—71no—
231C2×#106C2×D711yescharacteristic
232C2×C4⋊C8—71no—
233C2×#102—71no—
234C2×#102—71no—
235C2×#102C2×D711yescharacteristic
236C2×#102C2×D711yescharacteristic
237C2×#97—71no—
238C2×#97—71no—
239C2×C4×C8—71no—
240#391—141no—
241#389—71no—
242#389—71no—
243#389—71no—
244#389C2×D711yescharacteristic
245#387—141no—
246#387—141no—
247#385—141no—
248#385—141no—
249#384—71no—
250#380—141no—
251C4×C4⋊C4—71no—
252C4×C4⋊C4—71no—
253C4×C4⋊C4—71no—
254C4×C4⋊C4—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340C2×D711yescharacteristic
263#336—71no—
264#336C2×D711yescharacteristic
265C23×C7—22no—
266C23×C7—22no—
267C23×C7—22no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7C22×D411yescharacteristic
273C7×D4—22no—
274C7×D4—22no—
275C7×D4—22no—
276C7×D4C4×Q814yescomplemented
277C2×C4×C7—22no—
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—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—84no—
293C2×C4×C7C23×C411yescharacteristic
294C2×C4×C7C22×Q811yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7H(5,𝔽2)11yescharacteristic
298C2×C4×C72−1+411yescharacteristic
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—22no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—48no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—84no—
321C2×Dic7C4×D412yes—
322C2×Dic7C4×D412yes—
323C2×Dic7AGL(1,ℤ/8)12yes—
324C2×Dic7AGL(1,ℤ/8)12yes—
325Dic14—48no—
326C7×C8—22no—
327C7×C8—41no—
328C7⋊C8—22no—
329C7⋊C8—22no—
330C22×Q8—141no—
331C22×D4C22×D711yescharacteristic
332C23×C4—21no—
333C23×C4—141no—
334C23×C4—141no—
335C22×C8—71no—
336C22×C8—71no—
337C22×C8C22×D711yescharacteristic
338C22×C8C22×D711yescharacteristic
339C4⋊Q8—74no—
340C4⋊Q8—142no—
341C4×Q8—284no—
342C4×D4—22no—
343C4×D4—41no—
344C4×D4—74no—
345C4×D4—142no—
346C4×D4—142no—
347C4×D4—281no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—284no—
362C2×C4⋊C4C22×D711yescharacteristic
363C2×C4⋊C4C22×D711yescharacteristic
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—141no—
367C2×C22⋊C4—141no—
368C2×C42—71no—
369C2×C42—71no—
370C2×C42—141no—
371C2×C42—141no—
372C2×C42—141no—
373C2×C42C22×D711yescharacteristic
374#107—74no—
375#107—74no—
376#106—41no—
377#106—281no—
378C4⋊C8—74no—
379#102—22no—
380#102—74no—
381#102—74no—
382#102C4×D714yes—
383#97—281no—
384#97—281no—
385C4×C8—142no—
386#95—21no—
387#95—71no—
388#95—71no—
389#95—141no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395C22×C7—24no—
396C22×C7—42no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—42no—
401C22×C7—42no—
402C22×C7C2×C4×D411yescharacteristic
403C22×C7#54611yescharacteristic
404C22×C7D4×Q811yescharacteristic
405C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
406C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
407C22×C7#57711yescharacteristic
408C22×C7#57811yescharacteristic
409C4×C7—41no—
410C4×C7—42no—
411C4×C7—41no—
412C4×C7—81no—
413C4×C7—81no—
414C4×C7C2×C4×Q811yescharacteristic
415C4×C7#52011yescharacteristic
416C4×C7#55211yescharacteristic
417C4×C7#55411yescharacteristic
418Dic7—22no—
419Dic7—41no—
420Dic7—42no—
421Dic7—42no—
422Dic7—42no—
423Dic7—41no—
424Dic7—81no—
425Dic7#44214yes—
426C24—21no—
427C2×Q8—144no—
428C2×Q8—282no—
429C2×Q8—282no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4C2×C4×D712yes—
434C2×D4#115912yes—
435C2×D4Q8×D712yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—71no—
442C22×C4—71no—
443C22×C4—71no—
444C22×C4—71no—
445C22×C4—81no—
446C22×C4—144no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—282no—
452C22×C4—282no—
453C22×C4—282no—
454C22×C4—282no—
455C22×C4—561no—
456C22×C4D4×D711yescharacteristic
457C22×C4D4×D711yescharacteristic
458C22×C4C23×D711yescharacteristic
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—72no—
465C2×C8—72no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8C2×C4×D712yes—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—72no—
478C4⋊C4—72no—
479C4⋊C4—72no—
480C4⋊C4—142no—
481C4⋊C4—142no—
482C4⋊C4—144no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—282no—
489C4⋊C4—282no—
490C4⋊C4—282no—
491C4⋊C4—282no—
492C4⋊C4—561no—
493C4⋊C4C2×C4×D712yes—
494C22⋊C4—42no—
495C22⋊C4—144no—
496C22⋊C4—282no—
497C42—21no—
498C42—21no—
499C42—72no—
500C42—72no—
501C42—141no—
502C42—141no—
503C42—282no—
504C42—282no—
505C42—282no—
506C2×C7—44no—
507C2×C7#236011yescharacteristic
508C2×C7C2×#44211yescharacteristic
509C2×C7#294711yescharacteristic
510C2×C7#332411yescharacteristic
511C2×C7#333011yescharacteristic
512C2×C7#338411yescharacteristic
513C2×C7#338611yescharacteristic
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D711yescharacteristic
522Q8—288no—
523D4—22no—
524D4—22no—
525D4—22no—
526D4#645014yescomplemented
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—72no—
543C2×C4—72no—
544C2×C4—72no—
545C2×C4—72no—
546C2×C4—84no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—142no—
554C2×C4—142no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—142no—
558C2×C4—288no—
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—564no—
569C2×C4C22×C4×D711yescharacteristic
570C2×C4C2×#115911yescharacteristic
571C2×C4#654811yescharacteristic
572C2×C4C2×Q8×D711yescharacteristic
573C2×C4D7×#4111yescharacteristic
574C2×C4#655411yescharacteristic
575C8—22no—
576C8—41no—
577C8—142no—
578C8—142no—
579C7#2296811yescharacteristic, complemented, Sylow
580C22—24no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22C4×D4×D711yescharacteristic
588C22#9028211yescharacteristic
589C22#9032311yescharacteristic
590C22#9045311yescharacteristic
591C22#9046211yescharacteristic
592C22D7×AGL(1,ℤ/8)11yescharacteristic
593C22D7×AGL(1,ℤ/8)11yescharacteristic
594C4—41no—
595C4—41no—
596C4—42no—
597C4—74no—
598C4—81no—
599C4—81no—
600C4—142no—
601C4—281no—
602C4—282no—
603C4—281no—
604C4—282no—
605C4—282no—
606C4—561no—
607C4C2×#645011yescharacteristic
608C4#9023711yescharacteristic
609C4#9032711yescharacteristic
610C4#9034111yescharacteristic
611C2—44no—
612C2#1173184111yescharacteristic
613C2#1173709311yescharacteristic
614C2D7×#44211yescharacteristic
615C2#1173756111yescharacteristic
616C2#1173761011yescharacteristic
617C2#1173850011yescharacteristic
618C2#1173860411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1864C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#1864D7
hypocenterC7#22968
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
3. #11733431C2semidirect
6. #11728317C2semidirect
7. #11728315C2semidirect
11. #11727374C2semidirect
12. #11726557C2semidirect
13. #11726544C2semidirect
14. #11726406C2semidirect
15. #11726378C2semidirect
47. #89531C4semidirect
103. #6451C2×C4semidirect
110. #6406Q8semidirect
119. #6392C2×C4semidirect
121. Dic7⋊C8C2×C4semidirect
124. #6381C2×C4semidirect
276. C7×D4C4×Q8semidirect
526. D4#6450semidirect
579. C7#22968semidirect

Maximal subgroups

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

subgroupindex
D4×Dic7⋊C42
C2×#895312
#117334312
C7×#18642
#117293632
#117283172
#117283152
#117279232
#117279062
Dic7×#1022
#117273742
#117265572
#117265442
#117264062
#117263782
#229687