Finite Group Databasebeta

← #13356198

Subgroups — #13356198

Order 1792.

1218 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2297C211yesmaximal, characteristic
2C2×#90004C211yesmaximal, characteristic
3C2×#90004C211yesmaximal, characteristic
4C2×#89767C211yesmaximal, characteristic, complemented
5C2×C4×C4⋊Dic7C211yesmaximal, characteristic, complemented
6#11729335C211yesmaximal, characteristic, complemented
7#11729281C211yesmaximal, characteristic, complemented
8#11727421C212yesmaximal, complemented
9#11727420C212yesmaximal, complemented
10#11727391C212yesmaximal, complemented
11#11727390C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C22×C42×C7C2211yescharacteristic
15C22×C4⋊Dic7C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×C4×Dic7—21no—
18C22×#1136C2211yescharacteristic
19C22×#1136C2211yescharacteristic
20C7×#391C2211yescharacteristic
21C7×#391C2211yescharacteristic
22C7×#390C2211yescharacteristic
23C7×#390C2211yescharacteristic
24C7×#386—21no—
25C7×#386—21no—
26C4×C7×C22⋊C4—21no—
27C4×C7×C22⋊C4—21no—
28#90004—24no—
29#90004C418yescomplemented
30C2×#6409C2212yes—
31C2×#6409C2212yes—
32C2×#6406C2212yes—
33C2×#6406C2212yes—
34C2×C7⋊#95—21no—
35#89767—24no—
36C2×#6382C2212yes—
37C2×#6381C2212yes—
38C4×C4⋊Dic7—21no—
39C4×C4⋊Dic7—21no—
40C4×C4⋊Dic7C2212yes—
41C4×C4⋊Dic7C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89319C2212yes—
45#89319C2212yes—
46#21059—71nomaximal, Sylow
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41C2×C414yes—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7—21no—
57C23×C4×C7D411yescharacteristic
58C23×C4×C7D411yescharacteristic
59C23×C4×C7C2311yescharacteristic
60C23×Dic7—21no—
61C23×Dic7—21no—
62C7×#117—22no—
63C7×#117—22no—
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4—21no—
68C2×C7×C22⋊C4—21no—
69C2×C7×C22⋊C4—21no—
70C2×C7×C22⋊C4—21no—
71C2×C42×C7—22no—
72C2×C42×C7—22no—
73C2×C42×C7—21no—
74C2×C42×C7D411yescharacteristic
75C2×C42×C7D411yescharacteristic
76C2×C42×C7C2311yescharacteristic
77C2×C42×C7C2311yescharacteristic
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—22no—
81C2×C4⋊Dic7—22no—
82C2×C4⋊Dic7C2×C414yes—
83C2×C4⋊Dic7D414yes—
84C2×C4⋊Dic7C2312yes—
85C2×C4⋊Dic7C2312yes—
86C2×C4×Dic7—22no—
87C2×C4×Dic7—22no—
88C2×C4×Dic7—44no—
89C2×#1136—22no—
90C2×#1136—22no—
91C2×#1136—22no—
92C2×#1136—22no—
93C2×#1136—22no—
94C2×#1136C2×C414yes—
95C22×C7⋊C8C2312yes—
96C22×C7⋊C8C2312yes—
97C7×#95—21no—
98C7×#95—21no—
99C7×#95—21no—
100C7×#95—21no—
101#6409—24no—
102#6409C2×C418yescomplemented
103#6406—24no—
104#6406C2×C418yescomplemented
105C7⋊#95—22no—
106C7⋊#95—22no—
107#6382—24no—
108#6381—24no—
109C2×#428—71no—
110C2×#417—71no—
111C2×#417—71no—
112#2297D711yescharacteristic
113C2×C4×C4⋊C4—71no—
114#1928—72no—
115#1927—72no—
116#1926—72no—
117#1925—72no—
118#1736—71no—
119#1728—71no—
120C24×C7—21no—
121C24×C7C2×D411yescharacteristic
122C7×#41—28no—
123C7×#41—28no—
124C7×#41—44no—
125C7×#41—44no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8—21no—
129C2×C7×Q8—21no—
130C2×C7×Q8—21no—
131C2×C7×Q8C22×C412yes—
132C2×C7×D4—24no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—42no—
139C2×C7×D4—42no—
140C2×C7×D4C22×C412yes—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—21no—
146C22×C4×C7—24no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—41no—
150C22×C4×C7—41no—
151C22×C4×C7—42no—
152C22×C4×C7—42no—
153C22×C4×C7C22×C412yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D412yes—
160C22×C4×C7#4112yes—
161C22×C4×C7#4112yes—
162C22×C4×C7#4112yes—
163C22×C4×C7#4112yes—
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—24no—
166C22×Dic7—24no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—42no—
170C22×Dic7—42no—
171C22×Dic7—42no—
172C22×Dic7—42no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C22⋊C4—42no—
178C7×C22⋊C4—42no—
179C7×C22⋊C4—42no—
180C7×C22⋊C4—42no—
181C7×C22⋊C4—42no—
182C42×C7—22no—
183C42×C7—22no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—42no—
189C4⋊Dic7—24no—
190C4⋊Dic7—24no—
191C4⋊Dic7—22no—
192C4⋊Dic7—22no—
193C4⋊Dic7C22×C414yes—
194C4⋊Dic7C2×D414yes—
195C4×Dic7—44no—
196C4×Dic7—44no—
197#1136—28no—
198#1136—44no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8—22no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8C22×C414yes—
205C22×#41C2×D711yescharacteristic
206C22×M16—71no—
207C22×M16—71no—
208C2×#117C2×D711yescharacteristic
209C22×C4⋊C4—71no—
210C22×C4⋊C4—71no—
211C22×C42—141no—
212C22×C42C2×D711yescharacteristic
213#428—144no—
214C2×#107—72no—
215C2×#106—72no—
216#417—78no—
217#417—144no—
218C2×#103—72no—
219C2×#103—72no—
220C2×#102—72no—
221C2×#102—72no—
222#391C2×D711yescharacteristic
223#391C2×D711yescharacteristic
224#390C2×D711yescharacteristic
225#390C2×D711yescharacteristic
226#386—21no—
227#386—21no—
228C4×C4⋊C4—72no—
229C4×C4⋊C4—72no—
230C4×C4⋊C4—141no—
231C4×C4⋊C4—141no—
232C4×C22⋊C4—21no—
233C4×C22⋊C4—21no—
234C2×#95—141no—
235#340—72no—
236#340—72no—
237#336—72no—
238#336—72no—
239C23×C7—21no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7—21no—
243C23×C7—22no—
244C23×C7—41no—
245C23×C7—41no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—41no—
249C23×C7—41no—
250C23×C7C4×D412yes—
251C23×C7#12112yes—
252C23×C7#12312yes—
253C23×C7C22×D411yescharacteristic
254C7×Q8—22no—
255C7×Q8—22no—
256C7×Q8—41no—
257C7×Q8—41no—
258C7×D4—22no—
259C7×D4—22no—
260C7×D4—48no—
261C7×D4—41no—
262C7×D4—41no—
263C2×C4×C7—28no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—22no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—44no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—48no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7#11714yes—
286C2×C4×C7C4×D414yes—
287C2×C4×C7C4×D412yes—
288C2×C4×C7#12112yes—
289C2×C4×C7#12312yes—
290C2×C4×C7#12414yes—
291C2×C4×C7#12614yes—
292C2×C4×C7AGL(1,ℤ/8)12yes—
293C2×C4×C7#13712yes—
294C2×C4×C7C23×C411yescharacteristic
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×Dic7—24no—
301C2×Dic7—24no—
302C2×Dic7—48no—
303C2×Dic7—48no—
304C2×Dic7—42no—
305C2×Dic7—42no—
306C2×Dic7—42no—
307C2×Dic7—42no—
308C7⋊C8—24no—
309C7⋊C8—42no—
310C2×#41—22no—
311C2×#41—22no—
312C2×#41—22no—
313C2×#41—22no—
314C2×#41—22no—
315C2×#41C4×D714yes—
316C22×Q8C22×D711yescharacteristic
317C22×D4—21no—
318C22×D4C22×D711yescharacteristic
319C23×C4—21no—
320C23×C4—141no—
321C23×C4—141no—
322C23×C4C22⋊D711yescharacteristic
323C23×C4C22⋊D711yescharacteristic
324C23×C4C22×D711yescharacteristic
325C2×M16—74no—
326C2×M16—142no—
327C2×M16—142no—
328C2×M16—142no—
329C2×M16—142no—
330C2×M16—142no—
331C22×C8—72no—
332C22×C8—72no—
333#117—22no—
334#117—22no—
335C2×C4⋊C4—74no—
336C2×C4⋊C4—74no—
337C2×C4⋊C4—72no—
338C2×C4⋊C4—72no—
339C2×C4⋊C4—142no—
340C2×C4⋊C4—142no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—142no—
354C2×C42—142no—
355C2×C42—284no—
356C2×C42C22⋊D711yescharacteristic
357C2×C42C22⋊D711yescharacteristic
358C2×C42C22×D711yescharacteristic
359C2×C42C22×D711yescharacteristic
360#107—144no—
361#106—144no—
362#103—78no—
363#103—144no—
364#102—78no—
365#102—144no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—142no—
371#95—142no—
372C22×C7—22no—
373C22×C7—22no—
374C22×C7—22no—
375C22×C7—22no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7—81no—
382C22×C7—81no—
383C22×C7—81no—
384C22×C7—81no—
385C22×C7#38814yes—
386C22×C7C2×C4×D411yescharacteristic
387C22×C7C2×#12111yescharacteristic
388C22×C7C2×#12311yescharacteristic
389C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
390C22×C7C2×#13711yescharacteristic
391C22×C7#57511yescharacteristic
392C22×C7#57511yescharacteristic
393C4×C7—22no—
394C4×C7—22no—
395C4×C7—41no—
396C4×C7—41no—
397C4×C7—42no—
398C4×C7—81no—
399C4×C7—81no—
400C4×C7#38814yes—
401C4×C7C2×#11711yescharacteristic
402C4×C7C2×C4×D411yescharacteristic
403C4×C7C2×#12411yescharacteristic
404C4×C7C2×#12611yescharacteristic
405Dic7—44no—
406Dic7—44no—
407C24—21no—
408C24C2×C22⋊D711yescharacteristic
409#41—28no—
410#41—28no—
411#41—44no—
412#41—44no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8C2×C4×D712yes—
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C2×C4×D712yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—144no—
441C22×C4—144no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—282no—
445C22×C4—282no—
446C22×C4—282no—
447C22×C4—282no—
448C22×C4C2×C4×D712yes—
449C22×C4#115712yes—
450C22×C4D4×D712yes—
451C22×C4#115912yes—
452C22×C4#115912yes—
453C22×C4#115912yes—
454C22×C4C2×C22⋊D711yescharacteristic
455C22×C4C2×C22⋊D711yescharacteristic
456C22×C4C2×C22⋊D711yescharacteristic
457C22×C4C2×C22⋊D711yescharacteristic
458C22×C4C2×C22⋊D711yescharacteristic
459C22×C4C23×D711yescharacteristic
460M16—148no—
461M16—284no—
462C2×C8—74no—
463C2×C8—142no—
464C2×C8—142no—
465C2×C8—142no—
466C2×C8—142no—
467C2×C8—142no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—74no—
473C4⋊C4—74no—
474C4⋊C4—144no—
475C4⋊C4—144no—
476C4⋊C4—142no—
477C4⋊C4—142no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C42—22no—
484C42—22no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—42no—
490C42—284no—
491C42—284no—
492C2×C7—24no—
493C2×C7—82no—
494C2×C7C2×#38811yescharacteristic
495C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
496C2×C7C4×#13711yescharacteristic
497C2×C7#303611yescharacteristic
498C2×C7#303711yescharacteristic
499C2×C7#309211yescharacteristic
500C2×C7#309311yescharacteristic
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C4×C22⋊D712yes—
513C23#649812yes—
514C23#650112yes—
515C23C22×C22⋊D711yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520D4—22no—
521D4—22no—
522D4—48no—
523D4—41no—
524D4—41no—
525C2×C4—28no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—48no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—144no—
548C2×C4—144no—
549C2×C4—288no—
550C2×C4—288no—
551C2×C4—282no—
552C2×C4—282no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4#644014yes—
556C2×C4#644214yes—
557C2×C4(C22⋊D7)⋊C414yes—
558C2×C4#644814yes—
559C2×C4C4×C22⋊D712yes—
560C2×C4#649812yes—
561C2×C4#650112yes—
562C2×C4#651212yes—
563C2×C4#651412yes—
564C2×C4C22×C4×D711yescharacteristic
565C2×C4C2×#115711yescharacteristic
566C2×C4C2×D4×D711yescharacteristic
567C2×C4C2×#115911yescharacteristic
568C2×C4C2×#115911yescharacteristic
569C2×C4C2×#115911yescharacteristic
570C8—144no—
571C8—282no—
572C7#2105911yescharacteristic, complemented, Sylow
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22#8971614yes—
587C22C2×C4×C22⋊D711yescharacteristic
588C22C2×#649811yescharacteristic
589C22C2×#650111yescharacteristic
590C22C2×#651211yescharacteristic
591C22#9051311yescharacteristic
592C22#9051311yescharacteristic
593C22C2×#651411yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—284no—
602C4—284no—
603C4#8971614yes—
604C4C2×#644011yescharacteristic
605C4C2×#644211yescharacteristic
606C4C2×(C22⋊D7)⋊C411yescharacteristic
607C4C2×#644811yescharacteristic
608C2—24no—
609C2—82no—
610C2C2×#8971611yescharacteristic
611C2C4×#651211yescharacteristic
612C2C4×#651411yescharacteristic
613C2#1174035911yescharacteristic
614C2#1174036011yescharacteristic
615C2#1174074211yescharacteristic
616C2#1174075911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2297C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenter#2297D7
hypocenterC7#21059
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
4. C2×#89767C2semidirect
5. C2×C4×C4⋊Dic7C2semidirect
6. #11729335C2semidirect
7. #11729281C2semidirect
8. #11727421C2semidirect
9. #11727420C2semidirect
10. #11727391C2semidirect
11. #11727390C2semidirect
29. #90004C4semidirect
102. #6409C2×C4semidirect
104. #6406C2×C4semidirect
572. C7#21059semidirect

Maximal subgroups

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

subgroupindex
C7×#22972
C2×#900042
C2×#897672
C2×C4×C4⋊Dic72
#117293352
#117292812
#117274212
#117274202
#117273912
#117273902
#210597