Finite Group Databasebeta

← #13381275

Subgroups — #13381275

Order 1792.

1610 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90519C211yesmaximal, characteristic
2C2×C7×#416C211yesmaximal, characteristic, complemented
3C2×#90005C211yesmaximal, characteristic, complemented
4C22×(C2×D7)⋊C8C211yesmaximal, characteristic, complemented
5#11739712C211yesmaximal, characteristic, complemented
6C2×#89770C211yesmaximal, characteristic, complemented
7#11729723C211yesmaximal, characteristic, complemented
8#11728029C212yesmaximal, complemented
9#11728028C212yesmaximal, complemented
10#11727961C212yesmaximal, complemented
11#11727960C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C23×C4×D7C2211yescharacteristic
14C23×C7×C8C2211yescharacteristic
15C2×C7×#117C2211yescharacteristic
16#90519—24no—
17#90519—24no—
18C2×#6509C2212yes—
19C2×#6501C2212yes—
20C2×#6500—21no—
21C2×#6498—21no—
22C2×C4×C22⋊D7—22no—
23C2×#6489C2211yescharacteristic
24C2×#6489C2211yescharacteristic
25C23×C7⋊C8C2211yescharacteristic
26C7×#416—24no—
27C2×C7×#103C2212yescomplemented
28C2×C7×#102C2212yescomplemented
29#90005—24no—
30C2×#6409C2211yescharacteristic, complemented
31C2×#6409C2211yescharacteristic, complemented
32C2×#6406C2211yescharacteristic, complemented
33C2×#6406C2211yescharacteristic, complemented
34C2×(C2×D7)⋊C8—24no—
35C2×(C2×D7)⋊C8C2212yes—
36C2×(C2×D7)⋊C8C2212yes—
37#89770—24no—
38#89759C2212yes—
39#89759C2212yes—
40C2×#6382C2212yescomplemented
41C2×#6381C2212yescomplemented
42#89727—21no—
43#89715—21no—
44C4×(C2×D7)⋊C4—22no—
45#89346C2212yescomplemented
46#89346C2212yescomplemented
47#23479—71nomaximal, Sylow
48C24×D7D411yescharacteristic
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41—22no—
53C2×C7×#41—22no—
54C2×C7×#41D414yes—
55C22×C7×Q8C2311yescharacteristic
56C22×C7×D4—21no—
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C23×C4×C7C2311yescharacteristic
60C22×C22⋊D7—22no—
61C23×Dic7D411yescharacteristic
62C22×C4×D7—22no—
63C22×C4×D7—22no—
64C22×C4×D7—22no—
65C22×C4×D7—22no—
66C22×C4×D7—22no—
67C22×C4×D7—22no—
68C22×C4×D7D412yes—
69C22×C4×D7C2312yes—
70C22×C7×C8—22no—
71C22×C7×C8—22no—
72C22×C7×C8—22no—
73C22×C7×C8C2312yes—
74C7×#117—22no—
75C7×#117—22no—
76C2×C7×C4⋊C4C2312yes—
77C2×C7×C22⋊C4—21no—
78C2×C42×C7—21no—
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7—22no—
81C2×C22⋊Dic7—21no—
82#6509—24no—
83#6509—24no—
84#6501—24no—
85#6501—24no—
86#6500—28no—
87#6498—28no—
88C4×C22⋊D7—216no—
89C2×(C2×D7)⋊C4—22no—
90C2×(C2×D7)⋊C4—22no—
91C2×(C2×D7)⋊C4—22no—
92#6489—22no—
93#6489—22no—
94#6489—22no—
95#6489D414yes—
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2312yes—
99C2×Dic7⋊C4—22no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C22×C7⋊C8—22no—
103C22×C7⋊C8—22no—
104C22×C7⋊C8D414yescomplemented
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C7×#103—24no—
108C7×#102—24no—
109#6409—22no—
110#6409—22no—
111#6406—22no—
112#6406—22no—
113C7⋊#95—22no—
114C7⋊#95—22no—
115(C2×D7)⋊C8—24no—
116(C2×D7)⋊C8—24no—
117#6382—24no—
118#6381—24no—
119C2×#525—71no—
120C2×#427—71no—
121C2×#416—71no—
122C2×#416D711yescharacteristic, complemented
123C22×C22⋊C8—71no—
124#2280—71no—
125#2036—72no—
126#2035—72no—
127#2034—72no—
128#2033—72no—
129#1735—71no—
130C24×C7—21no—
131C24×C7C2×D411yescharacteristic
132C23×D7—22no—
133C23×D7—24no—
134C23×D7—22no—
135C23×D7—22no—
136C23×D7—42no—
137C23×D7C2×D412yes—
138C7×#41—28no—
139C7×#41—28no—
140C7×#41—44no—
141C7×#41—44no—
142C2×C7×Q8—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8C2×D412yes—
148C2×C7×D4—24no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4—42no—
155C2×C7×D4—42no—
156C2×C7×D4C2×D412yes—
157C22×C4×C7—21no—
158C22×C4×C7—24no—
159C22×C4×C7—22no—
160C22×C4×C7—42no—
161C22×C4×C7—42no—
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D412yes—
165C22×C4×C7C2×D412yes—
166C22×C4×C7#4112yes—
167C22×C4×C7#4112yes—
168C22×C4×C7#4112yes—
169C22×C4×C7C2411yescharacteristic
170C2×C22⋊D7—28no—
171C2×C22⋊D7—44no—
172C2×C22⋊D7—44no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—22no—
177C22×Dic7—21no—
178C22×Dic7—21no—
179C22×Dic7C2×D412yes—
180C2×C4×D7—24no—
181C2×C4×D7—24no—
182C2×C4×D7—24no—
183C2×C4×D7—22no—
184C2×C4×D7—22no—
185C2×C4×D7—22no—
186C2×C4×D7—22no—
187C2×C4×D7—22no—
188C2×C4×D7—22no—
189C2×C4×D7—44no—
190C2×C4×D7—44no—
191C2×C4×D7—44no—
192C2×C4×D7—44no—
193C2×C7×C8—22no—
194C2×C7×C8—22no—
195C2×C7×C8—22no—
196C2×C7×C8—44no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
199C7×C22⋊C4—42no—
200C42×C7—42no—
201C22⋊Dic7—28no—
202C22⋊Dic7—24no—
203C22⋊Dic7—42no—
204(C2×D7)⋊C4—28no—
205(C2×D7)⋊C4—44no—
206(C2×D7)⋊C4—44no—
207C4⋊Dic7—22no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7C2×D414yes—
213Dic7⋊C4—28no—
214C4×Dic7—24no—
215C4×Dic7—42no—
216C2×C7⋊C8—28no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8C2×D412yes—
222C2×C7⋊C8C2×D412yes—
223C22×#41C2×D711yescharacteristic
224C24×C4—71no—
225C23×C8—71no—
226C23×C8C2×D711yescharacteristic
227#525—144no—
228#525—144no—
229C2×#123—141no—
230C2×C22⋊Q8—72no—
231C2×#121—72no—
232C2×#120—141no—
233C2×C4×D4—142no—
234C2×#117—71no—
235C2×#117—71no—
236C2×#117C2×D711yescharacteristic
237#427—144no—
238C2×#107—72no—
239C2×#106—72no—
240#416—24no—
241#416—144no—
242C2×#103—71no—
243C2×#103—71no—
244C2×#103C2×D712yescomplemented
245C2×#102—71no—
246C2×#102—71no—
247C2×#102C2×D712yescomplemented
248C2×C22⋊C8—72no—
249C2×C22⋊C8—72no—
250C2×C22⋊C8—144no—
251#381—141no—
252#380—72no—
253#380—72no—
254#379—141no—
255C4×C22⋊C4—142no—
256#340—72no—
257#340—72no—
258C23×C7—22no—
259C23×C7—22no—
260C23×C7—22no—
261C23×C7—41no—
262C23×C7—41no—
263C23×C7—42no—
264C23×C7—42no—
265C23×C7#12012yes—
266C23×C7#12112yes—
267C23×C7#12312yes—
268C23×C7C22×D411yescharacteristic
269C22×D7—28no—
270C22×D7—22no—
271C22×D7—22no—
272C22×D7—22no—
273C22×D7—22no—
274C22×D7—22no—
275C22×D7—22no—
276C22×D7—48no—
277C22×D7—48no—
278C22×D7—44no—
279C22×D7—44no—
280C22×D7#13514yes—
281C7×Q8—22no—
282C7×Q8—22no—
283C7×Q8—41no—
284C7×Q8—41no—
285C7×D4—22no—
286C7×D4—22no—
287C7×D4—48no—
288C7×D4—41no—
289C7×D4—41no—
290C2×C4×C7—22no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—48no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7#12012yes—
298C2×C4×C7#12114yes—
299C2×C4×C7#12112yes—
300C2×C4×C7#12114yes—
301C2×C4×C7#12312yes—
302C2×C4×C7#12314yes—
303C2×C4×C7#12414yes—
304C2×C4×C7C22×D411yescharacteristic
305C2×C4×C7C22×D411yescharacteristic
306C2×C4×C7C22×D411yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C22⋊D7—416no—
311C2×Dic7—24no—
312C2×Dic7—22no—
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—42no—
316C2×Dic7—42no—
317C2×Dic7—44no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—42no—
321C2×Dic7—42no—
322C4×D7—42no—
323C4×D7—42no—
324C4×D7—42no—
325C4×D7—42no—
326C4×D7—48no—
327C7×C8—42no—
328C7⋊C8—22no—
329C7⋊C8—22no—
330C25—71no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41C22⋊D714yes—
337C22×Q8C22×D711yescharacteristic
338C22×D4—21no—
339C22×D4—142no—
340C22×D4C22×D711yescharacteristic
341C23×C4—21no—
342C23×C4—71no—
343C23×C4—72no—
344C23×C4—72no—
345C23×C4—142no—
346C23×C4—142no—
347C23×C4—142no—
348C23×C4—142no—
349C23×C4—142no—
350C23×C4—142no—
351C23×C4C22×D711yescharacteristic
352C22×C8—22no—
353C22×C8—22no—
354C22×C8—22no—
355C22×C8—74no—
356C22×C8—71no—
357C22×C8—71no—
358C22×C8—142no—
359C22×C8—142no—
360C22×C8C22×D712yes—
361#123—148no—
362C22⋊Q8—144no—
363C22⋊Q8—144no—
364#121—144no—
365#121—144no—
366#120—148no—
367C4×D4—1416no—
368#117—22no—
369#117—22no—
370#117—74no—
371#117—142no—
372#117—142no—
373#117—142no—
374C2×C4⋊C4—72no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—142no—
378C2×C4⋊C4C22×D712yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—142no—
381C2×C22⋊C4—142no—
382C2×C22⋊C4—141no—
383C2×C22⋊C4—142no—
384C2×C22⋊C4—142no—
385C2×C22⋊C4—141no—
386C2×C42—21no—
387C2×C42—141no—
388C2×C42—141no—
389#107—144no—
390#106—144no—
391#103—24no—
392#103—142no—
393#103—142no—
394#102—24no—
395#102—142no—
396#102—142no—
397C22⋊C8—144no—
398C22⋊C8—144no—
399#95—142no—
400#95—142no—
401C22×C7—24no—
402C22×C7—24no—
403C22×C7—22no—
404C22×C7—22no—
405C22×C7—42no—
406C22×C7—44no—
407C22×C7—82no—
408C22×C7—81no—
409C22×C7—81no—
410C22×C7#39414yes—
411C22×C7C2×#12011yescharacteristic
412C22×C7C2×#12111yescharacteristic
413C22×C7C2×#12311yescharacteristic
414C22×C7C2×#13512yes—
415C22×C7#57512yes—
416C2×D7—24no—
417C2×D7—24no—
418C2×D7—24no—
419C2×D7—416no—
420C2×D7—42no—
421C2×D7—42no—
422C2×D7—42no—
423C2×D7—42no—
424C4×C7—42no—
425C4×C7—82no—
426C4×C7#39414yes—
427C4×C7C2×#12111yescharacteristic
428C4×C7C2×#12111yescharacteristic
429C4×C7C2×#12311yescharacteristic
430C4×C7C2×#12411yescharacteristic
431Dic7—42no—
432Dic7—44no—
433Dic7—81no—
434Dic7—81no—
435C24—21no—
436C24—72no—
437C24—144no—
438C24—142no—
439C24—142no—
440C24—142no—
441C24—282no—
442C24D4×D711yescharacteristic
443#41—28no—
444#41—28no—
445#41—44no—
446#41—44no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8C2×C22⋊D712yes—
453C2×D4—24no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—148no—
462C2×D4—284no—
463C2×D4—284no—
464C2×D4C2×C22⋊D712yes—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—72no—
471C22×C4—142no—
472C22×C4—142no—
473C22×C4—142no—
474C22×C4—142no—
475C22×C4—144no—
476C22×C4—144no—
477C22×C4—144no—
478C22×C4—142no—
479C22×C4—142no—
480C22×C4—142no—
481C22×C4—142no—
482C22×C4—142no—
483C22×C4—142no—
484C22×C4—141no—
485C22×C4—141no—
486C22×C4—284no—
487C22×C4—284no—
488C22×C4—284no—
489C22×C4—284no—
490C22×C4#115712yes—
491C22×C4#115712yes—
492C22×C4D4×D712yes—
493C22×C4D4×D712yes—
494C22×C4D4×D711yescharacteristic
495C22×C4#115912yes—
496C22×C4C2×C22⋊D712yes—
497C22×C4C23×D711yescharacteristic
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—44no—
502C2×C8—72no—
503C2×C8—72no—
504C2×C8—148no—
505C2×C8—141no—
506C2×C8—141no—
507C2×C8—141no—
508C2×C8—141no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—74no—
512C4⋊C4—148no—
513C4⋊C4—142no—
514C4⋊C4—141no—
515C4⋊C4—141no—
516C4⋊C4—141no—
517C4⋊C4—141no—
518C22⋊C4—42no—
519C22⋊C4—148no—
520C22⋊C4—148no—
521C22⋊C4—144no—
522C22⋊C4—284no—
523C22⋊C4—284no—
524C22⋊C4—282no—
525C42—42no—
526C42—144no—
527C42—282no—
528C2×C7—24no—
529C2×C7—82no—
530C2×C7C2×#39411yescharacteristic
531C2×C7#300212yes—
532C2×C7#305011yescharacteristic
533C2×C7#305411yescharacteristic
534C2×C7#309112yes—
535D7—44no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—74no—
544C23—148no—
545C23—142no—
546C23—142no—
547C23—142no—
548C23—142no—
549C23—142no—
550C23—142no—
551C23—288no—
552C23—288no—
553C23—284no—
554C23—284no—
555C23#644612yes—
556C23#644712yes—
557C23#650012yes—
558C23C2×D4×D711yescharacteristic
559Q8—22no—
560Q8—22no—
561Q8—41no—
562Q8—41no—
563D4—22no—
564D4—22no—
565D4—48no—
566D4—41no—
567D4—41no—
568D4—2816no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—142no—
577C2×C4—142no—
578C2×C4—144no—
579C2×C4—142no—
580C2×C4—282no—
581C2×C4—282no—
582C2×C4—284no—
583C2×C4—282no—
584C2×C4—282no—
585C2×C4—282no—
586C2×C4—282no—
587C2×C4—288no—
588C2×C4—281no—
589C2×C4—281no—
590C2×C4—282no—
591C2×C4—282no—
592C2×C4#644612yes—
593C2×C4#644714yes—
594C2×C4#644712yes—
595C2×C4#644814yes—
596C2×C4#649214yes—
597C2×C4#650012yes—
598C2×C4#650214yes—
599C2×C4C2×#115711yescharacteristic
600C2×C4C2×#115711yescharacteristic
601C2×C4C2×D4×D711yescharacteristic
602C2×C4C2×D4×D711yescharacteristic
603C2×C4C2×#115911yescharacteristic
604C2×C4C22×C22⋊D711yescharacteristic
605C8—42no—
606C8—142no—
607C8—142no—
608C7#2347911yescharacteristic, complemented, Sylow
609C22—24no—
610C22—22no—
611C22—22no—
612C22—24no—
613C22—44no—
614C22—42no—
615C22—81no—
616C22—81no—
617C22—82no—
618C22—144no—
619C22—144no—
620C22—144no—
621C22—2816no—
622C22—282no—
623C22—282no—
624C22—282no—
625C22—282no—
626C22#8973114yes—
627C22C2×#644611yescharacteristic
628C22C2×#644711yescharacteristic
629C22D7×#13512yes—
630C22#9045912yes—
631C22C2×#650011yescharacteristic
632C4—42no—
633C4—82no—
634C4—284no—
635C4—282no—
636C4—561no—
637C4—561no—
638C4#8973114yes—
639C4C2×#644711yescharacteristic
640C4C2×#644811yescharacteristic
641C4C2×#649211yescharacteristic
642C4C2×#650211yescharacteristic
643C2—24no—
644C2—82no—
645C2—284no—
646C2#1173628311yescharacteristic
647C2#1173628411yescharacteristic
648C2#1173628712yes—
649C2C2×#8973111yescharacteristic
650C2#1174135612yes—
subgroupG/H
centerC22×C4D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#416C2
socleC23×C7C22×D4
prime radicalC2×#90519C2
hypercenterC2×#416D7
hypocenterC7#23479
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C2×C7×#416C2semidirect
3. C2×#90005C2semidirect
4. C22×(C2×D7)⋊C8C2semidirect
5. #11739712C2semidirect
6. C2×#89770C2semidirect
7. #11729723C2semidirect
8. #11728029C2semidirect
9. #11728028C2semidirect
10. #11727961C2semidirect
11. #11727960C2semidirect
27. C2×C7×#103C22semidirect
28. C2×C7×#102C22semidirect
30. C2×#6409C22semidirect
31. C2×#6409C22semidirect
32. C2×#6406C22semidirect
33. C2×#6406C22semidirect
40. C2×#6382C22semidirect
41. C2×#6381C22semidirect
45. #89346C22semidirect
46. #89346C22semidirect
104. C22×C7⋊C8D4semidirect
122. C2×#416D7semidirect
244. C2×#103C2×D7semidirect
247. C2×#102C2×D7semidirect
608. C7#23479semidirect

Maximal subgroups

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

subgroupindex
C2×#905192
C2×C7×#4162
C2×#900052
C22×(C2×D7)⋊C82
#117397122
C2×#897702
#117297232
#117280292
#117280282
#117279612
#117279602
#234797