Finite Group Databasebeta

← #13360507

Subgroups — #13360507

Order 1792.

1002 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×Dic7C211yesmaximal, characteristic
2Dic7×C22⋊Q8C211yesmaximal, characteristic
3C2×#89824C211yesmaximal, characteristic
4C2×#89561C211yesmaximal, characteristic, complemented
5#11731311C211yesmaximal, characteristic
6C7×#1894C211yesmaximal, characteristic
7#11729743C211yesmaximal, characteristic
8#11728012C211yesmaximal, characteristic, complemented
9#11728008C211yesmaximal, characteristic, complemented
10#11727974C211yesmaximal, characteristic, complemented
11#11727267C211yesmaximal, characteristic
12#11726612C211yesmaximal, characteristic, complemented
13#11726605C211yesmaximal, characteristic, complemented
14#11726540C211yesmaximal, characteristic, complemented
15#11726537C211yesmaximal, characteristic, complemented
16C22×C7×M16C2211yescharacteristic
17C2×C7×C22⋊Q8C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C22×#1144C2211yescharacteristic
20C2×#6487C2211yescharacteristic
21C22×C4×Dic7C2211yescharacteristic
22C22×#1136C2211yescharacteristic
23C2×#6450C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C2×C7×#103C2211yescharacteristic
26C7×#380C2211yescharacteristic
27#90002C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29#89984C2211yescharacteristic
30M16×Dic7—24no—
31C2×#6390C2211yescharacteristic
32C2×#6390C2211yescharacteristic
33C2×#6389C2211yescharacteristic
34C2×#6389C2211yescharacteristic
35C2×C8×Dic7C2211yescharacteristic
36C2×C8×Dic7C2211yescharacteristic
37#89824C418yescomplemented
38#89757C2211yescharacteristic
39#89753—21no—
40C4⋊C4×Dic7—21no—
41C4⋊C4×Dic7C2211yescharacteristic
42C4⋊C4×Dic7C2211yescharacteristic
43#89746C2211yescharacteristic
44#89745C2211yescharacteristic
45C2×#6384C2211yescharacteristic
46C2×#6382C2211yescharacteristic
47C2×#6382C2211yescharacteristic
48C2×#6381C2211yescharacteristic
49#89722—21no—
50#89718—21no—
51C22⋊C4×Dic7—21no—
52C22⋊C4×Dic7—21no—
53#89561—24no—
54#89424—21no—
55#89421—21no—
56C7×#340C2211yescharacteristic
57C7×#340C2211yescharacteristic
58#89346C2211yescharacteristic
59#89346C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#21539—71nomaximal, Sylow
63C22×C7×Q8C2311yescharacteristic
64C23×C4×C7C2311yescharacteristic
65C23×Dic7D411yescharacteristic
66C23×Dic7D411yescharacteristic
67C22×Dic14C2311yescharacteristic
68C2×C7×M16—22no—
69C2×C7×M16—22no—
70C2×C7×M16—22no—
71C22×C7×C8C2311yescharacteristic
72C22×C7×C8C2311yescharacteristic
73C7×C22⋊Q8—22no—
74C7×C22⋊Q8C2×C414yes—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4—21no—
81C2×C22⋊Dic7—21no—
82C2×C22⋊Dic7—21no—
83Q8×Dic7—22no—
84Q8×Dic7—22no—
85C2×#1144—22no—
86C2×#1144—22no—
87C2×#1144C2×C414yescomplemented
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93#6487—22no—
94#6487C2×C414yes—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×C4×Dic7—22no—
99C2×C4×Dic7—22no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7—21no—
104C2×C4×Dic7D411yescharacteristic
105C2×C4×Dic7D411yescharacteristic
106C2×C4×Dic7C2311yescharacteristic
107C2×C4×Dic7C2311yescharacteristic
108C2×#1136—22no—
109C2×#1136—22no—
110C2×#1136C2×C414yes—
111C22×C7⋊C8C2311yescharacteristic
112C22×C7⋊C8C2311yescharacteristic
113#6450—22no—
114#6450—22no—
115C7×#103—22no—
116C7×#103—22no—
117C7×#95—21no—
118#6409C2×C414yescomplemented
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95—21no—
128#6390—22no—
129#6390—22no—
130#6389—22no—
131#6389D414yes—
132C8×Dic7—22no—
133C8×Dic7—22no—
134#6384C2×C414yescomplemented
135#6382—22no—
136#6382C2×C414yescomplemented
137#6381—22no—
138C2×#470—71no—
139C2×#441—71no—
140C2×C4×M16—71no—
141C4×C22⋊Q8—71no—
142C4×C22⋊Q8—71no—
143#1969—71no—
144#1941—71no—
145#1933—71no—
146#1933—71no—
147#1894—71no—
148#1894D711yescharacteristic
149#1809—71no—
150#1776—71no—
151#1763—71no—
152#1763—71no—
153C24×C7C2×D411yescharacteristic
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—21no—
159C2×C7×Q8C22×C412yes—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7C22×C412yes—
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7#4112yes—
171C22×C4×C7C2411yescharacteristic
172C22×Dic7—21no—
173C22×Dic7—21no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—22no—
177C22×Dic7—22no—
178C22×Dic7—21no—
179C22×Dic7—21no—
180C22×Dic7—41no—
181C22×Dic7—41no—
182C22×Dic7C2×D411yescharacteristic
183C22×Dic7C2×D411yescharacteristic
184C22×Dic7C2×D411yescharacteristic
185C22×Dic7C2×D411yescharacteristic
186C2×Dic14—21no—
187C2×Dic14—21no—
188C2×Dic14—21no—
189C2×Dic14—21no—
190C2×Dic14—21no—
191C2×Dic14C22×C412yes—
192C7×M16—28no—
193C2×C7×C8—21no—
194C2×C7×C8—21no—
195C2×C7×C8—21no—
196C2×C7×C8—21no—
197C2×C7×C8C2×D412yes—
198C2×C7×C8#4112yes—
199C7×C4⋊C4—24no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4C22×C412yes—
206C7×C22⋊C4—24no—
207C7×C22⋊C4—42no—
208C22⋊Dic7—24no—
209C22⋊Dic7—42no—
210#1144—28no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7—21no—
216C4⋊Dic7—42no—
217C4⋊Dic7—42no—
218C4⋊Dic7C22×C412yes—
219Dic7⋊C4—24no—
220Dic7⋊C4—42no—
221Dic7⋊C4—42no—
222C4×Dic7—21no—
223C4×Dic7—21no—
224C4×Dic7—22no—
225C4×Dic7—22no—
226C4×Dic7—42no—
227C4×Dic7—42no—
228C4×Dic7—42no—
229C4×Dic7—42no—
230C4×Dic7C2×D412yes—
231C4×Dic7#4112yes—
232#1136—28no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8C22×C412yes—
236C2×C7⋊C8C22×C412yes—
237C2×C7⋊C8C2×D412yes—
238C2×C7⋊C8#4112yes—
239C22×Q16—71no—
240C22×M16—71no—
241C22×M16C2×D711yescharacteristic
242C2×C22⋊Q8—71no—
243C2×C22⋊Q8C2×D711yescharacteristic
244C2×C4×Q8—71no—
245C2×C4×Q8—71no—
246C22×C42—71no—
247#470—78no—
248#441—144no—
249C2×#107—71no—
250C2×#106—71no—
251C2×#106—71no—
252C2×#103—71no—
253C2×#103—71no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#103C2×D711yescharacteristic
257C2×#103C2×D711yescharacteristic
258C4×M16—144no—
259C2×#97—71no—
260C2×#97—71no—
261C2×C4×C8—71no—
262C2×C4×C8—71no—
263#391—71no—
264#391—71no—
265#389—71no—
266#389—71no—
267#387—141no—
268#387—141no—
269#385—141no—
270#385—141no—
271#380—71no—
272#380C2×D711yescharacteristic
273C4×C4⋊C4—71no—
274C4×C4⋊C4—71no—
275C4×C4⋊C4—141no—
276C4×C4⋊C4—141no—
277C4×C22⋊C4—141no—
278C4×C22⋊C4—141no—
279#340—71no—
280#340—71no—
281#340—71no—
282#340—71no—
283#340C2×D711yescharacteristic
284#340C2×D711yescharacteristic
285C23×C7—21no—
286C23×C7—21no—
287C23×C7—21no—
288C23×C7—21no—
289C23×C7—21no—
290C23×C7—21no—
291C23×C7C4×D412yes—
292C23×C7C22×D411yescharacteristic
293C7×Q8—22no—
294C7×Q8—22no—
295C7×Q8—41no—
296C7×Q8—41no—
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—41no—
306C2×C4×C7—41no—
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×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7C4×D412yes—
316C2×C4×C7C23×C411yescharacteristic
317C2×C4×C7C22×D411yescharacteristic
318C2×C4×C7C2×#4111yescharacteristic
319C2×C4×C7C2×#4111yescharacteristic
320C2×C4×C7C2×#4111yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
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—22no—
331C2×Dic7—22no—
332C2×Dic7—44no—
333C2×Dic7—44no—
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—41no—
344C2×Dic7#12112yes—
345C2×Dic7#12112yes—
346C2×Dic7#13712yes—
347C2×Dic7#13712yes—
348Dic14—22no—
349Dic14—22no—
350Dic14—41no—
351Dic14—41no—
352C7×C8—22no—
353C7×C8—22no—
354C7⋊C8—22no—
355C7⋊C8C4×D414yes—
356C22×Q8—71no—
357C22×Q8C22×D711yescharacteristic
358C23×C4—71no—
359C23×C4—71no—
360C23×C4C22×D711yescharacteristic
361C2×Q16—74no—
362C2×Q16—142no—
363C2×Q16—142no—
364C2×M16—22no—
365C2×M16—22no—
366C2×M16—22no—
367C2×M16—74no—
368C2×M16—142no—
369C2×M16—142no—
370C22×C8—71no—
371C22×C8—71no—
372C22×C8C22×D711yescharacteristic
373C22×C8C22×D711yescharacteristic
374C22⋊Q8—22no—
375C22⋊Q8—74no—
376C22⋊Q8—142no—
377C22⋊Q8C4×D714yes—
378C4×Q8—142no—
379C4×Q8—142no—
380C4×Q8—142no—
381C4×Q8—142no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—71no—
384C2×C4⋊C4—71no—
385C2×C4⋊C4—71no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4—141no—
388C2×C4⋊C4—141no—
389C2×C4⋊C4—141no—
390C2×C4⋊C4—141no—
391C2×C4⋊C4C22×D711yescharacteristic
392C2×C4⋊C4C22×D711yescharacteristic
393C2×C4⋊C4C22×D711yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—141no—
397C2×C22⋊C4—141no—
398C2×C42—71no—
399C2×C42—71no—
400C2×C42—71no—
401C2×C42—71no—
402C2×C42—142no—
403C2×C42—142no—
404C2×C42—141no—
405C2×C42—141no—
406C2×C42—141no—
407C2×C42—141no—
408#107—142no—
409#106—74no—
410#106—142no—
411#103—22no—
412#103—22no—
413#103—74no—
414#103—74no—
415#103—142no—
416#103—142no—
417#97—74no—
418#97—142no—
419C4×C8—142no—
420C4×C8—142no—
421#95—21no—
422#95—141no—
423#95—141no—
424#95—141no—
425#95—141no—
426#95—141no—
427#95—141no—
428#95—141no—
429#95—141no—
430#95—141no—
431C22×C7—22no—
432C22×C7—22no—
433C22×C7—22no—
434C22×C7—41no—
435C22×C7—41no—
436C22×C7—41no—
437C22×C7—41no—
438C22×C7C2×C4×D411yescharacteristic
439C22×C7C2×#12111yescharacteristic
440C22×C7#53211yescharacteristic
441C22×C7C2×#13711yescharacteristic
442C22×C7C2×#13711yescharacteristic
443C22×C7#57511yescharacteristic
444C22×C7#57511yescharacteristic
445C4×C7—22no—
446C4×C7—42no—
447C4×C7—42no—
448C4×C7—81no—
449C4×C7—81no—
450C4×C7C2×C4×D411yescharacteristic
451C4×C7C4×#4111yescharacteristic
452C4×C7#52511yescharacteristic
453C4×C7#52911yescharacteristic
454Dic7—22no—
455Dic7—22no—
456Dic7—22no—
457Dic7—22no—
458Dic7—42no—
459Dic7—42no—
460Dic7—81no—
461Dic7—81no—
462C24D4×D711yescharacteristic
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—72no—
469C2×Q8—141no—
470C2×Q8—141no—
471C2×Q8—141no—
472C2×Q8—141no—
473C2×Q8—141no—
474C2×Q8C2×C4×D712yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—71no—
484C22×C4—71no—
485C22×C4—71no—
486C22×C4—71no—
487C22×C4—142no—
488C22×C4—142no—
489C22×C4—142no—
490C22×C4—142no—
491C22×C4—141no—
492C22×C4—141no—
493C22×C4—141no—
494C22×C4—141no—
495C22×C4—281no—
496C22×C4—281no—
497C22×C4C2×C4×D712yes—
498C22×C4D4×D711yescharacteristic
499C22×C4#115912yes—
500C22×C4C23×D711yescharacteristic
501Q16—148no—
502M16—28no—
503M16—148no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—72no—
509C2×C8—72no—
510C2×C8—72no—
511C2×C8—72no—
512C2×C8—141no—
513C2×C8—141no—
514C2×C8D4×D712yes—
515C2×C8#115912yes—
516C4⋊C4—24no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—72no—
523C4⋊C4—144no—
524C4⋊C4—141no—
525C4⋊C4—141no—
526C4⋊C4—141no—
527C4⋊C4—141no—
528C4⋊C4—141no—
529C4⋊C4—282no—
530C4⋊C4—282no—
531C4⋊C4—282no—
532C4⋊C4—282no—
533C4⋊C4C2×C4×D712yes—
534C22⋊C4—24no—
535C22⋊C4—42no—
536C22⋊C4—144no—
537C22⋊C4—282no—
538C42—72no—
539C42—72no—
540C42—142no—
541C42—142no—
542C42—141no—
543C42—141no—
544C42—282no—
545C42—282no—
546C42—282no—
547C42—282no—
548C2×C7—42no—
549C2×C7C4×#12111yescharacteristic
550C2×C7C4×#13711yescharacteristic
551C2×C7C4×#13711yescharacteristic
552C2×C7#305611yescharacteristic
553C2×C7#305611yescharacteristic
554C2×C7#318311yescharacteristic
555C2×C7#318511yescharacteristic
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23(C22⋊D7)⋊C412yes—
563C23C2×D4×D711yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—142no—
569Q8—142no—
570Q8—281no—
571Q8—281no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—72no—
591C2×C4—72no—
592C2×C4—72no—
593C2×C4—72no—
594C2×C4—141no—
595C2×C4—141no—
596C2×C4—141no—
597C2×C4—141no—
598C2×C4—141no—
599C2×C4—141no—
600C2×C4—141no—
601C2×C4—141no—
602C2×C4—142no—
603C2×C4—142no—
604C2×C4—284no—
605C2×C4—284no—
606C2×C4—281no—
607C2×C4—281no—
608C2×C4—281no—
609C2×C4—281no—
610C2×C4—281no—
611C2×C4—281no—
612C2×C4—281no—
613C2×C4—281no—
614C2×C4—281no—
615C2×C4—281no—
616C2×C4(C22⋊D7)⋊C412yes—
617C2×C4C22×C4×D711yescharacteristic
618C2×C4C2×D4×D711yescharacteristic
619C2×C4C2×#115911yescharacteristic
620C2×C4C2×#115911yescharacteristic
621C2×C4D7×#4111yescharacteristic
622C2×C4D7×#4111yescharacteristic
623C8—22no—
624C8—22no—
625C8—74no—
626C8—142no—
627C7#2153911yescharacteristic, complemented, Sylow
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×(C22⋊D7)⋊C411yescharacteristic
636C22#9028711yescharacteristic
637C22D7×#12111yescharacteristic
638C22#9046411yescharacteristic
639C22D7×#13711yescharacteristic
640C22D7×#13711yescharacteristic
641C22#9046811yescharacteristic
642C4—22no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—81no—
647C4—142no—
648C4—142no—
649C4—142no—
650C4—142no—
651C4—282no—
652C4—282no—
653C4—561no—
654C4—561no—
655C4C4×#115911yescharacteristic
656C4C4×D4×D711yescharacteristic
657C4#9027711yescharacteristic
658C4#9034011yescharacteristic
659C2—42no—
660C2#1173210511yescharacteristic
661C2#1173631211yescharacteristic
662C2#1173639111yescharacteristic
663C2#1173823611yescharacteristic
664C2#1173825111yescharacteristic
665C2#1173833811yescharacteristic
666C2#1173834911yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1894C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1894D7
hypocenterC7#21539
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
4. C2×#89561C2semidirect
8. #11728012C2semidirect
9. #11728008C2semidirect
10. #11727974C2semidirect
12. #11726612C2semidirect
13. #11726605C2semidirect
14. #11726540C2semidirect
15. #11726537C2semidirect
37. #89824C4semidirect
87. C2×#1144C2×C4semidirect
118. #6409C2×C4semidirect
134. #6384C2×C4semidirect
136. #6382C2×C4semidirect
627. C7#21539semidirect

Maximal subgroups

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

subgroupindex
C2×M16×Dic72
Dic7×C22⋊Q82
C2×#898242
C2×#895612
#117313112
C7×#18942
#117297432
#117280122
#117280082
#117279742
#117272672
#117266122
#117266052
#117265402
#117265372
#215397