Finite Group Databasebeta

← #13371130

Subgroups — #13371130

Order 1792.

934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#419C211yesmaximal, characteristic
2Q8×C22⋊Dic7C211yesmaximal, characteristic
3#11739684C211yesmaximal, characteristic
4#11729886C211yesmaximal, characteristic
5#11729742C211yesmaximal, characteristic
6#11729737C211yesmaximal, characteristic
7#11729329C211yesmaximal, characteristic
8#11728009C211yesmaximal, characteristic, complemented
9#11728007C211yesmaximal, characteristic, complemented
10#11727992C211yesmaximal, characteristic, complemented
11#11727990C211yesmaximal, characteristic, complemented
12#11727978C211yesmaximal, characteristic, complemented
13#11727977C211yesmaximal, characteristic, complemented
14#11727974C211yesmaximal, characteristic, complemented
15Dic7×#103C211yesmaximal, characteristic, complemented
16C23×C7×Q8C2211yescharacteristic
17C22×C7×M16C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C2×Q8×Dic7C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C7×#419C418yescomplemented
22C2×C7×#103C2211yescharacteristic
23C2×C7×#103C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C2×C7×#103C2211yescharacteristic
26#90002—21no—
27#90002—21no—
28#90002C2211yescharacteristic
29#90002C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34#89984—21no—
35#89984C2211yescharacteristic
36#89984C2211yescharacteristic
37C4×C22⋊Dic7—21no—
38C4×C22⋊Dic7C2211yescharacteristic
39C2×C22⋊(C7⋊C8)C2211yescharacteristic
40C2×#6392C2211yescharacteristic
41C2×#6391C2211yescharacteristic
42C2×#6389C2211yescharacteristic
43C2×C8×Dic7C2211yescharacteristic
44#89757C2211yescharacteristic
45#89757C2211yescharacteristic
46C4⋊C4×Dic7C2211yescharacteristic
47C4⋊C4×Dic7C2211yescharacteristic
48#89723—21no—
49#89723—21no—
50#89708—21no—
51#89704—21no—
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22585—71nomaximal, Sylow
61C22×C7×Q8—21no—
62C22×C7×Q8—22no—
63C22×C7×Q8—22no—
64C22×C7×Q8—22no—
65C22×C7×Q8—41no—
66C22×C7×Q8—41no—
67C22×C7×Q8C2×C414yes—
68C22×C7×Q8C2311yescharacteristic
69C22×C7×Q8C2311yescharacteristic
70C23×C4×C7—21no—
71C23×C4×C7C2311yescharacteristic
72C2×C7×M16—41no—
73C2×C7×M16—41no—
74C2×C7×M16C2×C414yes—
75C22×C7×C8C2311yescharacteristic
76C22×C7×C8C2311yescharacteristic
77C7×#117—22no—
78C7×#117C2×C414yes—
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C22⋊C4—21no—
82C2×C42×C7—21no—
83C2×C22⋊Dic7—21no—
84C2×C22⋊Dic7—21no—
85C2×C22⋊Dic7D411yescharacteristic
86C2×C22⋊Dic7D411yescharacteristic
87Q8×Dic7—22no—
88Q8×Dic7—22no—
89Q8×Dic7—41no—
90Q8×Dic7—41no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7C2311yescharacteristic
104C2×C4×Dic7C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C7×#103C2×C414yes—
108C7×#103C2×C414yes—
109C7×#103C2×C414yes—
110C7×#103C2×C414yes—
111#6409—22no—
112#6409—22no—
113#6409—41no—
114#6409—41no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95D411yescharacteristic
126C7⋊#95D411yescharacteristic
127C22⋊(C7⋊C8)—22no—
128#6392—41no—
129#6391—41no—
130#6389—41no—
131C8×Dic7—41no—
132C2×#419D711yescharacteristic
133#2346—71no—
134Q8×C22⋊C4—71no—
135#1935—71no—
136#1933—71no—
137#1928—71no—
138#1927—71no—
139#1865—71no—
140#1863—71no—
141#1825—71no—
142#1809—71no—
143#1787—71no—
144#1763—71no—
145C4×#103—71no—
146#1753—71no—
147C24×C7C2×D411yescharacteristic
148C2×C7×Q8—24no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—28no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—42no—
159C2×C7×Q8—42no—
160C2×C7×Q8—44no—
161C2×C7×Q8—44no—
162C2×C7×Q8—44no—
163C2×C7×Q8C22×C412yes—
164C2×C7×Q8C22×C412yes—
165C2×C7×Q8#4112yes—
166C2×C7×Q8#4112yes—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—21no—
171C22×C4×C7—21no—
172C22×C4×C7—24no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—42no—
177C22×C4×C7—42no—
178C22×C4×C7C22×C412yes—
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7#4112yes—
181C22×C4×C7C2411yescharacteristic
182C22×Dic7—21no—
183C22×Dic7—21no—
184C22×Dic7—21no—
185C22×Dic7—21no—
186C22×Dic7C2×D411yescharacteristic
187C22×Dic7C2×D411yescharacteristic
188C22×Dic7C2×D411yescharacteristic
189C22×Dic7C2×D411yescharacteristic
190C7×M16—44no—
191C2×C7×C8—21no—
192C2×C7×C8—21no—
193C2×C7×C8—21no—
194C2×C7×C8—21no—
195C2×C7×C8C22×C412yes—
196C2×C7×C8C22×C412yes—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4C22×C412yes—
200C7×C4⋊C4C22×C412yes—
201C7×C22⋊C4—24no—
202C42×C7—24no—
203C22⋊Dic7—22no—
204C22⋊Dic7—41no—
205C22⋊Dic7—42no—
206C22⋊Dic7—81no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7—42no—
216C4⋊Dic7—42no—
217C4⋊Dic7—81no—
218C4⋊Dic7—81no—
219C4×Dic7—21no—
220C4×Dic7—21no—
221C4×Dic7—21no—
222C4×Dic7—21no—
223C4×Dic7—42no—
224C4×Dic7—42no—
225C4×Dic7—42no—
226C4×Dic7—42no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—21no—
233C23×Q8C2×D711yescharacteristic
234C22×M16C2×D711yescharacteristic
235C2×C4×Q8—71no—
236C2×C4×Q8—71no—
237C2×#117C2×D711yescharacteristic
238C2×#107—71no—
239C2×#106—71no—
240#419Dic718yescomplemented
241C2×#103—71no—
242C2×#103—71no—
243C2×#103—71no—
244C2×#103—71no—
245C2×#103C2×D711yescharacteristic
246C2×#103C2×D711yescharacteristic
247C2×#103C2×D711yescharacteristic
248C2×#103C2×D711yescharacteristic
249C2×C22⋊C8—71no—
250C2×#97—71no—
251C2×C4×C8—71no—
252#391—71no—
253#391—71no—
254#391—141no—
255#391—141no—
256#389—71no—
257#389—71no—
258#388—141no—
259#388—141no—
260#382—141no—
261#380—71no—
262#380—71no—
263#380—141no—
264C4×C4⋊C4—71no—
265C4×C4⋊C4—71no—
266C4×C22⋊C4—71no—
267C4×C22⋊C4—141no—
268#376—141no—
269#340—71no—
270#340—71no—
271#340—71no—
272#340—71no—
273#340—71no—
274#340—71no—
275#336—71no—
276#336—71no—
277C23×C7—21no—
278C23×C7—21no—
279C23×C7—21no—
280C23×C7—21no—
281C23×C7—21no—
282C23×C7—21no—
283C23×C7C4×D412yes—
284C23×C7C22×D411yescharacteristic
285C7×Q8—22no—
286C7×Q8—22no—
287C7×Q8—22no—
288C7×Q8—22no—
289C7×Q8—22no—
290C7×Q8—48no—
291C7×Q8—41no—
292C7×Q8—41no—
293C7×Q8#11714yes—
294C2×C4×C7—22no—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—22no—
305C2×C4×C7—22no—
306C2×C4×C7—48no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7C4×D412yes—
314C2×C4×C7C23×C411yescharacteristic
315C2×C4×C7C2×#4111yescharacteristic
316C2×C4×C7C2×#4111yescharacteristic
317C2×C4×C7C2×#4111yescharacteristic
318C2×C4×C7H(5,𝔽2)11yescharacteristic
319C2×C4×C7H(5,𝔽2)11yescharacteristic
320C2×Dic7—21no—
321C2×Dic7—21no—
322C2×Dic7—21no—
323C2×Dic7—21no—
324C2×Dic7—21no—
325C2×Dic7—21no—
326C2×Dic7—21no—
327C2×Dic7—21no—
328C2×Dic7—21no—
329C2×Dic7—21no—
330C2×Dic7—21no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—41no—
342C2×Dic7—41no—
343C2×Dic7#13712yes—
344C7×C8—22no—
345C7×C8—22no—
346C7⋊C8—41no—
347C7⋊C8—41no—
348C22×Q8—21no—
349C22×Q8—22no—
350C22×Q8—22no—
351C22×Q8—22no—
352C22×Q8—41no—
353C22×Q8—41no—
354C22×Q8C2×Dic714yes—
355C22×Q8C22×D711yescharacteristic
356C22×Q8C22×D711yescharacteristic
357C23×C4—21no—
358C23×C4C22×D711yescharacteristic
359C2×M16—41no—
360C2×M16—41no—
361C2×M16C2×Dic714yes—
362C22×C8—71no—
363C22×C8—71no—
364C22×C8C22×D711yescharacteristic
365C22×C8C22×D711yescharacteristic
366C4×Q8—142no—
367C4×Q8—142no—
368C4×Q8—281no—
369C4×Q8—281no—
370#117—22no—
371#117C2×Dic714yes—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—71no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4C22×D711yescharacteristic
381C2×C4⋊C4C22×D711yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—71no—
384C2×C22⋊C4—71no—
385C2×C22⋊C4—141no—
386C2×C22⋊C4—141no—
387C2×C42—21no—
388C2×C42—71no—
389C2×C42—71no—
390C2×C42—141no—
391C2×C42—141no—
392C2×C42—141no—
393C2×C42—141no—
394#107—281no—
395#106—281no—
396#103—142no—
397#103—142no—
398#103—281no—
399#103—281no—
400#103C2×Dic714yes—
401#103C2×Dic714yes—
402#103C2×Dic714yes—
403#103C2×Dic714yes—
404C22⋊C8—142no—
405#97—281no—
406C4×C8—281no—
407#95—71no—
408#95—71no—
409#95—141no—
410#95—141no—
411#95—141no—
412#95—141no—
413#95—141no—
414#95—141no—
415#95—141no—
416#95—141no—
417#95—141no—
418#95—141no—
419C22×C7—22no—
420C22×C7—22no—
421C22×C7—22no—
422C22×C7—41no—
423C22×C7—41no—
424C22×C7—41no—
425C22×C7—41no—
426C22×C7C2×C4×D411yescharacteristic
427C22×C7#54611yescharacteristic
428C22×C7#54611yescharacteristic
429C22×C7C2×#13711yescharacteristic
430C22×C7#57511yescharacteristic
431C22×C7#57711yescharacteristic
432C22×C7#57811yescharacteristic
433C4×C7—22no—
434C4×C7—24no—
435C4×C7—42no—
436C4×C7—42no—
437C4×C7—42no—
438C4×C7C2×#11711yescharacteristic
439C4×C7#51811yescharacteristic
440C4×C7#55111yescharacteristic
441C4×C7#55111yescharacteristic
442Dic7—22no—
443Dic7—41no—
444Dic7—41no—
445Dic7—41no—
446Dic7—81no—
447Dic7—81no—
448Dic7—81no—
449Dic7—81no—
450C24D4×D711yescharacteristic
451C2×Q8—24no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—28no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—44no—
464C2×Q8—44no—
465C2×Q8—44no—
466C2×Q8#115712yes—
467C2×Q8#115712yes—
468C2×Q8C22×Dic712yes—
469C2×Q8C22×Dic712yes—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—71no—
482C22×C4—71no—
483C22×C4—71no—
484C22×C4—71no—
485C22×C4—141no—
486C22×C4—141no—
487C22×C4—141no—
488C22×C4—141no—
489C22×C4D4×D711yescharacteristic
490C22×C4#115912yes—
491C22×C4C22×Dic712yes—
492C22×C4C23×D711yescharacteristic
493M16—44no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—141no—
499C2×C8—141no—
500C2×C8—141no—
501C2×C8—141no—
502C2×C8—141no—
503C2×C8—141no—
504C2×C8C22×Dic712yes—
505C2×C8C22×Dic712yes—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—141no—
509C4⋊C4—141no—
510C4⋊C4—141no—
511C4⋊C4—141no—
512C4⋊C4—141no—
513C4⋊C4—141no—
514C4⋊C4—141no—
515C4⋊C4—141no—
516C4⋊C4—282no—
517C4⋊C4—282no—
518C4⋊C4—561no—
519C4⋊C4—561no—
520C4⋊C4C22×Dic712yes—
521C4⋊C4C22×Dic712yes—
522C22⋊C4—24no—
523C22⋊C4—142no—
524C22⋊C4—281no—
525C22⋊C4—282no—
526C22⋊C4—561no—
527C42—24no—
528C42—141no—
529C42—141no—
530C42—141no—
531C42—141no—
532C42—282no—
533C42—282no—
534C42—282no—
535C42—282no—
536C2×C7—42no—
537C2×C7#234211yescharacteristic
538C2×C7C4×#13711yescharacteristic
539C2×C7#294711yescharacteristic
540C2×C7#331211yescharacteristic
541C2×C7#332111yescharacteristic
542C2×C7#332711yescharacteristic
543C2×C7#332911yescharacteristic
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23D4×Dic712yes—
551C23C2×D4×D711yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—48no—
558Q8—41no—
559Q8—41no—
560Q8#648914yes—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—48no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—72no—
581C2×C4—141no—
582C2×C4—141no—
583C2×C4—141no—
584C2×C4—141no—
585C2×C4—141no—
586C2×C4—141no—
587C2×C4—141no—
588C2×C4—141no—
589C2×C4—141no—
590C2×C4—141no—
591C2×C4—141no—
592C2×C4—281no—
593C2×C4—281no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4D4×Dic712yes—
605C2×C4C2×#115711yescharacteristic
606C2×C4C2×#115711yescharacteristic
607C2×C4C2×#115911yescharacteristic
608C2×C4#654811yescharacteristic
609C2×C4#654811yescharacteristic
610C2×C4C23×Dic711yescharacteristic
611C8—22no—
612C8—22no—
613C8—281no—
614C8—281no—
615C7#2258511yescharacteristic, complemented, Sylow
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22#9018111yescharacteristic
624C22#9018111yescharacteristic
625C22D7×#13711yescharacteristic
626C22#9046811yescharacteristic
627C22#9046911yescharacteristic
628C22#9047011yescharacteristic
629C22C2×D4×Dic711yescharacteristic
630C4—22no—
631C4—24no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—142no—
636C4—281no—
637C4—281no—
638C4—281no—
639C4—561no—
640C4—561no—
641C4—561no—
642C4—561no—
643C4#9018611yescharacteristic
644C4#9018611yescharacteristic
645C4C2×#648911yescharacteristic
646C4#9048911yescharacteristic
647C2—42no—
648C2#1173641111yescharacteristic
649C2#1173641411yescharacteristic
650C2#1173641611yescharacteristic
651C2#1173641811yescharacteristic
652C2#1173954211yescharacteristic
653C2Dic7×#13711yescharacteristic
654C2#1174141011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#419C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenterC2×#419D7
hypocenterC7#22585
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
8. #11728009C2semidirect
9. #11728007C2semidirect
10. #11727992C2semidirect
11. #11727990C2semidirect
12. #11727978C2semidirect
13. #11727977C2semidirect
14. #11727974C2semidirect
15. Dic7×#103C2semidirect
21. C7×#419C4semidirect
240. #419Dic7semidirect
615. C7#22585semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4192
Q8×C22⋊Dic72
#117396842
#117298862
#117297422
#117297372
#117293292
#117280092
#117280072
#117279922
#117279902
#117279782
#117279772
#117279742
Dic7×#1032
#225857