Finite Group Databasebeta

← #13371100

Subgroups — #13371100

Order 1792.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#417C211yesmaximal, characteristic
2#11741534C211yesmaximal, characteristic
3#11739604C211yesmaximal, characteristic, complemented
4#11729889C211yesmaximal, characteristic
5#11729742C211yesmaximal, characteristic, complemented
6#11729737C211yesmaximal, characteristic, complemented
7#11729308C211yesmaximal, characteristic, complemented
8#11727992C211yesmaximal, characteristic, complemented
9#11727990C211yesmaximal, characteristic, complemented
10#11727974C211yesmaximal, characteristic, complemented
11Dic7×#103C211yesmaximal, characteristic, complemented
12#11727941C211yesmaximal, characteristic, complemented
13#11727939C211yesmaximal, characteristic, complemented
14#11727910C211yesmaximal, characteristic, complemented
15#11727909C211yesmaximal, characteristic, complemented
16C22×C7×#41C2211yescharacteristic
17C22×C7×M16C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C2×Q8×Dic7C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C2×#6489—21no—
22C7×#417C418yescomplemented
23C2×C7×#103C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C2×C7×#102C2211yescharacteristic
26C2×C7×#102C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29#89993—21no—
30#89993—21no—
31#89993C2211yescharacteristic
32#89993C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35#89984—21no—
36#89984C2211yescharacteristic
37#89984C2211yescharacteristic
38#89983—21no—
39C4×C22⋊Dic7—21no—
40C4×C22⋊Dic7C2211yescharacteristic
41C2×C22⋊(C7⋊C8)C2211yescharacteristic
42C2×#6392C2211yescharacteristic
43C2×#6391C2211yescharacteristic
44C2×#6389C2211yescharacteristic
45C2×C8×Dic7C2211yescharacteristic
46#89757C2211yescharacteristic
47#89757C2211yescharacteristic
48#89753—21no—
49#89753—21no—
50C4⋊C4×Dic7C2211yescharacteristic
51C4⋊C4×Dic7C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22583—71nomaximal, Sylow
61C2×C7×#41—22no—
62C2×C7×#41—22no—
63C2×C7×#41—22no—
64C2×C7×#41—41no—
65C2×C7×#41—41no—
66C2×C7×#41C2×C414yes—
67C22×C7×Q8C2311yescharacteristic
68C22×C7×D4—21no—
69C22×C7×D4C2311yescharacteristic
70C23×C4×C7—21no—
71C23×C4×C7—21no—
72C23×C4×C7C2311yescharacteristic
73C2×C7×M16—41no—
74C2×C7×M16—41no—
75C2×C7×M16C2×C414yes—
76C22×C7×C8C2311yescharacteristic
77C22×C7×C8C2311yescharacteristic
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—22no—
80C2×C7×C4⋊C4C2×C414yes—
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C22⋊Dic7—21no—
84C2×C22⋊Dic7—21no—
85C2×C22⋊Dic7—21no—
86C2×C22⋊Dic7—21no—
87C2×C22⋊Dic7—21no—
88C2×C22⋊Dic7—21no—
89C2×C22⋊Dic7D411yescharacteristic
90C2×C22⋊Dic7D411yescharacteristic
91Q8×Dic7—22no—
92Q8×Dic7—22no—
93Q8×Dic7—41no—
94Q8×Dic7—41no—
95#6489—44no—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7—21no—
98C2×C4⋊Dic7—21no—
99C2×C4⋊Dic7—21no—
100C2×C4⋊Dic7C2311yescharacteristic
101C2×C4⋊Dic7C2311yescharacteristic
102C2×C4⋊Dic7C2311yescharacteristic
103C2×C4⋊Dic7C2311yescharacteristic
104C2×C4×Dic7—21no—
105C2×C4×Dic7—21no—
106C2×C4×Dic7—21no—
107C2×C4×Dic7—21no—
108C2×C4×Dic7C2311yescharacteristic
109C2×C4×Dic7C2311yescharacteristic
110C22×C7⋊C8C2311yescharacteristic
111C22×C7⋊C8C2311yescharacteristic
112C7×#103C2×C414yes—
113C7×#103C2×C414yes—
114C7×#102C2×C414yes—
115C7×#102C2×C414yes—
116#6409—22no—
117#6409—22no—
118#6406—41no—
119#6406—41no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95D411yescharacteristic
127C7⋊#95D411yescharacteristic
128C22⋊(C7⋊C8)—22no—
129#6392—41no—
130#6391—41no—
131#6389—41no—
132C8×Dic7—41no—
133C2×#417D711yescharacteristic
134#2344—71no—
135#2341—71no—
136#1936—71no—
137#1934—71no—
138#1926—71no—
139#1925—71no—
140#1865—71no—
141#1863—71no—
142#1824—71no—
143#1809—71no—
144#1788—71no—
145#1763—71no—
146C4×#103—71no—
147#1753—71no—
148C24×C7—21no—
149C24×C7C2×D411yescharacteristic
150C7×#41—28no—
151C7×#41—44no—
152C7×#41—44no—
153C7×#41—44no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8C22×C412yes—
158C2×C7×Q8#4112yes—
159C2×C7×Q8#4112yes—
160C2×C7×D4—24no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4—21no—
165C2×C7×D4—21no—
166C2×C7×D4—42no—
167C2×C7×D4—42no—
168C2×C7×D4C22×C412yes—
169C22×C4×C7—21no—
170C22×C4×C7—21no—
171C22×C4×C7—21no—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—24no—
175C22×C4×C7—21no—
176C22×C4×C7—21no—
177C22×C4×C7—24no—
178C22×C4×C7—42no—
179C22×C4×C7—42no—
180C22×C4×C7—42no—
181C22×C4×C7—42no—
182C22×C4×C7C22×C412yes—
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7#4112yes—
185C22×C4×C7C2411yescharacteristic
186C22×Dic7—21no—
187C22×Dic7—21no—
188C22×Dic7—21no—
189C22×Dic7—21no—
190C22×Dic7C2×D411yescharacteristic
191C22×Dic7C2×D411yescharacteristic
192C22×Dic7C2×D411yescharacteristic
193C22×Dic7C2×D411yescharacteristic
194C7×M16—44no—
195C2×C7×C8—21no—
196C2×C7×C8—21no—
197C2×C7×C8—21no—
198C2×C7×C8—21no—
199C2×C7×C8C22×C412yes—
200C2×C7×C8C22×C412yes—
201C7×C4⋊C4—24no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4C22×C412yes—
205C7×C4⋊C4C22×C412yes—
206C22⋊Dic7—22no—
207C22⋊Dic7—41no—
208C22⋊Dic7—42no—
209C22⋊Dic7—42no—
210C22⋊Dic7—42no—
211C22⋊Dic7—81no—
212C22⋊Dic7—81no—
213C22⋊Dic7—81no—
214C4⋊Dic7—21no—
215C4⋊Dic7—21no—
216C4⋊Dic7—21no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7—21no—
220C4⋊Dic7—21no—
221C4⋊Dic7—21no—
222C4⋊Dic7—42no—
223C4⋊Dic7—42no—
224C4⋊Dic7—42no—
225C4⋊Dic7—42no—
226C4×Dic7—21no—
227C4×Dic7—21no—
228C4×Dic7—21no—
229C4×Dic7—21no—
230C4×Dic7—42no—
231C4×Dic7—42no—
232C4×Dic7—42no—
233C4×Dic7—42no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—21no—
237C2×C7⋊C8—21no—
238C2×C7⋊C8—21no—
239C2×C7⋊C8—21no—
240C22×#41C2×D711yescharacteristic
241C22×M16C2×D711yescharacteristic
242C2×C4×Q8—71no—
243C2×C4×Q8—71no—
244C2×#117—141no—
245C22×C4⋊C4C2×D711yescharacteristic
246C2×#107—71no—
247C2×#106—71no—
248#417Dic718yescomplemented
249C2×#103—71no—
250C2×#103—71no—
251C2×#103C2×D711yescharacteristic
252C2×#103C2×D711yescharacteristic
253C2×#102—71no—
254C2×#102—71no—
255C2×#102C2×D711yescharacteristic
256C2×#102C2×D711yescharacteristic
257C2×C22⋊C8—71no—
258C2×#97—71no—
259C2×C4×C8—71no—
260#390—71no—
261#390—71no—
262#390—141no—
263#390—141no—
264#389—71no—
265#389—71no—
266#387—141no—
267#387—141no—
268#381—141no—
269#380—71no—
270#380—71no—
271#380—141no—
272C4×C4⋊C4—71no—
273C4×C4⋊C4—71no—
274C4×C22⋊C4—71no—
275C4×C22⋊C4—141no—
276#340—71no—
277#340—71no—
278#340—71no—
279#340—71no—
280#340—71no—
281#340—71no—
282#336—71no—
283#336—71no—
284C23×C7—21no—
285C23×C7—21no—
286C23×C7—21no—
287C23×C7—21no—
288C23×C7—21no—
289C23×C7—21no—
290C23×C7—22no—
291C23×C7—41no—
292C23×C7—41no—
293C23×C7—41no—
294C23×C7—41no—
295C23×C7—41no—
296C23×C7—41no—
297C23×C7C4×D412yes—
298C23×C7C22×D411yescharacteristic
299C7×Q8—22no—
300C7×Q8—22no—
301C7×Q8—22no—
302C7×Q8#11714yescomplemented
303C7×D4—22no—
304C7×D4—22no—
305C7×D4—48no—
306C7×D4—41no—
307C7×D4—41no—
308C2×C4×C7—22no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—22no—
314C2×C4×C7—22no—
315C2×C4×C7—22no—
316C2×C4×C7—22no—
317C2×C4×C7—22no—
318C2×C4×C7—22no—
319C2×C4×C7—48no—
320C2×C4×C7—48no—
321C2×C4×C7—41no—
322C2×C4×C7—41no—
323C2×C4×C7—41no—
324C2×C4×C7—41no—
325C2×C4×C7C4×D412yes—
326C2×C4×C7C23×C411yescharacteristic
327C2×C4×C7C2×#4111yescharacteristic
328C2×C4×C7C2×#4111yescharacteristic
329C2×C4×C7C2×#4111yescharacteristic
330C2×C4×C7H(5,𝔽2)11yescharacteristic
331C2×C4×C7H(5,𝔽2)11yescharacteristic
332C2×Dic7—21no—
333C2×Dic7—21no—
334C2×Dic7—21no—
335C2×Dic7—21no—
336C2×Dic7—21no—
337C2×Dic7—21no—
338C2×Dic7—21no—
339C2×Dic7—21no—
340C2×Dic7—21no—
341C2×Dic7—21no—
342C2×Dic7—21no—
343C2×Dic7—41no—
344C2×Dic7—41no—
345C2×Dic7—41no—
346C2×Dic7—41no—
347C2×Dic7—41no—
348C2×Dic7—41no—
349C2×Dic7—41no—
350C2×Dic7—41no—
351C2×Dic7—41no—
352C2×Dic7—41no—
353C2×Dic7—41no—
354C2×Dic7—41no—
355C2×Dic7AGL(1,ℤ/8)12yes—
356C7×C8—22no—
357C7×C8—22no—
358C7⋊C8—41no—
359C7⋊C8—41no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41—22no—
363C2×#41—41no—
364C2×#41—41no—
365C2×#41C2×Dic714yes—
366C22×Q8C22×D711yescharacteristic
367C22×D4—21no—
368C22×D4C22×D711yescharacteristic
369C23×C4—21no—
370C23×C4—21no—
371C23×C4C22×D711yescharacteristic
372C2×M16—41no—
373C2×M16—41no—
374C2×M16C2×Dic714yes—
375C22×C8—71no—
376C22×C8—71no—
377C22×C8C22×D711yescharacteristic
378C22×C8C22×D711yescharacteristic
379C4×Q8—142no—
380C4×Q8—142no—
381C4×Q8—281no—
382C4×Q8—281no—
383#117—284no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—71no—
387C2×C4⋊C4—71no—
388C2×C4⋊C4—71no—
389C2×C4⋊C4—71no—
390C2×C4⋊C4—141no—
391C2×C4⋊C4—141no—
392C2×C4⋊C4—141no—
393C2×C4⋊C4—141no—
394C2×C4⋊C4C2×Dic714yes—
395C2×C4⋊C4C22×D711yescharacteristic
396C2×C4⋊C4C22×D711yescharacteristic
397C2×C22⋊C4—71no—
398C2×C22⋊C4—71no—
399C2×C22⋊C4—141no—
400C2×C22⋊C4—141no—
401C2×C22⋊C4—141no—
402C2×C22⋊C4—141no—
403C2×C22⋊C4—141no—
404C2×C22⋊C4—141no—
405C2×C42—71no—
406C2×C42—71no—
407C2×C42—141no—
408C2×C42—141no—
409C2×C42—141no—
410C2×C42—141no—
411#107—281no—
412#106—281no—
413#103—142no—
414#103—142no—
415#103C2×Dic714yes—
416#103C2×Dic714yes—
417#102—281no—
418#102—281no—
419#102C2×Dic714yes—
420#102C2×Dic714yes—
421C22⋊C8—142no—
422#97—281no—
423C4×C8—281no—
424#95—71no—
425#95—71no—
426#95—141no—
427#95—141no—
428#95—141no—
429#95—141no—
430#95—141no—
431#95—141no—
432C22×C7—22no—
433C22×C7—22no—
434C22×C7—22no—
435C22×C7—41no—
436C22×C7—41no—
437C22×C7—41no—
438C22×C7—41no—
439C22×C7—42no—
440C22×C7—42no—
441C22×C7—42no—
442C22×C7—81no—
443C22×C7—81no—
444C22×C7—81no—
445C22×C7—81no—
446C22×C7C2×C4×D411yescharacteristic
447C22×C7#54611yescharacteristic
448C22×C7#54611yescharacteristic
449C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
450C22×C7#57511yescharacteristic
451C22×C7#57711yescharacteristic
452C22×C7#57811yescharacteristic
453C4×C7—22no—
454C4×C7—24no—
455C4×C7—42no—
456C4×C7—42no—
457C4×C7C2×#11711yescharacteristic
458C4×C7#51811yescharacteristic
459C4×C7#55111yescharacteristic
460C4×C7#55111yescharacteristic
461Dic7—22no—
462Dic7—41no—
463Dic7—41no—
464Dic7—41no—
465Dic7—81no—
466Dic7—81no—
467Dic7—81no—
468Dic7—81no—
469C24—21no—
470C24D4×D711yescharacteristic
471#41—28no—
472#41—44no—
473#41—44no—
474#41—44no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8#115712yes—
479C2×Q8#115712yes—
480C2×Q8C22×Dic712yes—
481C2×D4—24no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4C22×Dic712yes—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—24no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—24no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—71no—
504C22×C4—71no—
505C22×C4—71no—
506C22×C4—71no—
507C22×C4—141no—
508C22×C4—141no—
509C22×C4—141no—
510C22×C4—141no—
511C22×C4D4×D711yescharacteristic
512C22×C4#115912yes—
513C22×C4C22×Dic712yes—
514C22×C4C23×D711yescharacteristic
515M16—44no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—141no—
521C2×C8—141no—
522C2×C8—141no—
523C2×C8—141no—
524C2×C8—141no—
525C2×C8—141no—
526C2×C8C22×Dic712yes—
527C2×C8C22×Dic712yes—
528C4⋊C4—24no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—141no—
532C4⋊C4—141no—
533C4⋊C4—141no—
534C4⋊C4—141no—
535C4⋊C4—141no—
536C4⋊C4—141no—
537C4⋊C4—141no—
538C4⋊C4—141no—
539C4⋊C4—282no—
540C4⋊C4—282no—
541C4⋊C4—282no—
542C4⋊C4—282no—
543C4⋊C4C22×Dic712yes—
544C4⋊C4C22×Dic712yes—
545C22⋊C4—142no—
546C22⋊C4—282no—
547C22⋊C4—281no—
548C22⋊C4—282no—
549C22⋊C4—282no—
550C22⋊C4—561no—
551C22⋊C4—561no—
552C22⋊C4—561no—
553C42—141no—
554C42—141no—
555C42—141no—
556C42—141no—
557C42—282no—
558C42—282no—
559C42—282no—
560C42—282no—
561C2×C7—42no—
562C2×C7—82no—
563C2×C7#234211yescharacteristic
564C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
565C2×C7#294711yescharacteristic
566C2×C7#330811yescharacteristic
567C2×C7#331711yescharacteristic
568C2×C7#332711yescharacteristic
569C2×C7#333211yescharacteristic
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23D4×Dic712yes—
584C23C2×D4×D711yescharacteristic
585Q8—22no—
586Q8—22no—
587Q8—22no—
588Q8#648914yescomplemented
589D4—22no—
590D4—22no—
591D4—48no—
592D4—41no—
593D4—41no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—72no—
612C2×C4—141no—
613C2×C4—141no—
614C2×C4—141no—
615C2×C4—141no—
616C2×C4—141no—
617C2×C4—141no—
618C2×C4—141no—
619C2×C4—141no—
620C2×C4—141no—
621C2×C4—141no—
622C2×C4—141no—
623C2×C4—281no—
624C2×C4—281no—
625C2×C4—281no—
626C2×C4—281no—
627C2×C4—281no—
628C2×C4—281no—
629C2×C4—281no—
630C2×C4—281no—
631C2×C4—281no—
632C2×C4—281no—
633C2×C4—281no—
634C2×C4—281no—
635C2×C4D4×Dic712yes—
636C2×C4C2×#115711yescharacteristic
637C2×C4C2×#115711yescharacteristic
638C2×C4C2×#115911yescharacteristic
639C2×C4#654811yescharacteristic
640C2×C4#654811yescharacteristic
641C2×C4C23×Dic711yescharacteristic
642C8—22no—
643C8—22no—
644C8—281no—
645C8—281no—
646C7#2258311yescharacteristic, complemented, Sylow
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22#9018111yescharacteristic
662C22#9018111yescharacteristic
663C22D7×AGL(1,ℤ/8)11yescharacteristic
664C22#9046411yescharacteristic
665C22#9046911yescharacteristic
666C22#9047011yescharacteristic
667C22C2×D4×Dic711yescharacteristic
668C4—22no—
669C4—24no—
670C4—42no—
671C4—42no—
672C4—142no—
673C4—281no—
674C4—281no—
675C4—281no—
676C4—561no—
677C4—561no—
678C4—561no—
679C4—561no—
680C4#9018611yescharacteristic
681C4#9018611yescharacteristic
682C4C2×#648911yescharacteristic
683C4#9048911yescharacteristic
684C2—42no—
685C2—82no—
686C2#1173634111yescharacteristic
687C2#1173634611yescharacteristic
688C2#1173635011yescharacteristic
689C2#1173635311yescharacteristic
690C2#1173954211yescharacteristic
691C2Dic7×AGL(1,ℤ/8)11yescharacteristic
692C2#1174141011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#417C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenterC2×#417D7
hypocenterC7#22583
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
3. #11739604C2semidirect
5. #11729742C2semidirect
6. #11729737C2semidirect
7. #11729308C2semidirect
8. #11727992C2semidirect
9. #11727990C2semidirect
10. #11727974C2semidirect
11. Dic7×#103C2semidirect
12. #11727941C2semidirect
13. #11727939C2semidirect
14. #11727910C2semidirect
15. #11727909C2semidirect
22. C7×#417C4semidirect
248. #417Dic7semidirect
302. C7×Q8#117semidirect
588. Q8#6489semidirect
646. C7#22583semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4172
#117415342
#117396042
#117298892
#117297422
#117297372
#117293082
#117279922
#117279902
#117279742
Dic7×#1032
#117279412
#117279392
#117279102
#117279092
#225837