Finite Group Databasebeta

← #13353627

Subgroups — #13353627

Order 1792.

1470 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D7×#117C211yesmaximal, characteristic
2C2×#89898C211yesmaximal, characteristic
3#11739712C211yesmaximal, characteristic
4C7×#1733C211yesmaximal, characteristic, complemented
5#11729736C211yesmaximal, characteristic, complemented
6#11729332C211yesmaximal, characteristic, complemented
7#11729328C211yesmaximal, characteristic, complemented
8#11726508C218yesmaximal, complemented
9C23×C4×D7C2211yescharacteristic
10C22×C7×M16C2211yescharacteristic
11C2×C7×#117C2211yescharacteristic
12C2×C7×#117C2211yescharacteristic
13C2×#6489C2211yescharacteristic
14C2×#6489C2211yescharacteristic
15C22×#1136C2211yescharacteristic
16D7×#117—22no—
17D7×#117—22no—
18D7×#117C414yescomplemented
19D7×#117C2214yes—
20C2×#6455C412yescomplemented
21C2×D7×C4⋊C4C412yescomplemented
22C2×D7×C22⋊C4—21no—
23C2×#6440—21no—
24C2×#6435—21no—
25C2×C42×D7—21no—
26#89898—21no—
27#89898—21no—
28#89898C2214yes—
29C2×(C2×D7)⋊C8—22no—
30#89759—22no—
31#89727—21no—
32#89715—21no—
33C4×(C2×D7)⋊C4C2214yes—
34C7×#339C2214yescomplemented
35#89350C2214yescomplemented
36#89327C2214yescomplemented
37#89318C2214yescomplemented
38#20705—71nomaximal, Sylow
39C24×D7D411yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×Dic7D411yescharacteristic
42C22×C4×D7—22no—
43C22×C4×D7—22no—
44C22×C4×D7—22no—
45C22×C4×D7—22no—
46C22×C4×D7—22no—
47C22×C4×D7C2×C411yescharacteristic
48C22×C4×D7C2×C411yescharacteristic
49C22×C4×D7C2×C412yes—
50C22×C4×D7D412yes—
51C22×C4×D7D412yes—
52C22×C4×D7D411yescharacteristic
53C22×C4×D7D411yescharacteristic
54C22×C4×D7C2312yes—
55C2×C7×M16—21no—
56C2×C7×M16—22no—
57C2×C7×M16—22no—
58C2×C7×M16C2312yes—
59C22×C7×C8—21no—
60C7×#117—21no—
61C7×#117—21no—
62C7×#117—22no—
63C7×#117—22no—
64C7×#117C2×C412yes—
65C7×#117C2312yes—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2×C412yes—
68C2×C7×C22⋊C4—21no—
69C2×C7×C22⋊C4—21no—
70C2×C42×C7—21no—
71C2×C42×C7C2312yes—
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—21no—
74C2×(C2×D7)⋊C4—22no—
75C2×(C2×D7)⋊C4—24no—
76#6489—21no—
77#6489—21no—
78#6489—22no—
79#6489—22no—
80#6489C2×C412yes—
81#6489C2312yes—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7C2×C412yes—
84C2×Dic7⋊C4—22no—
85C2×C4×Dic7—21no—
86C2×C4×Dic7—22no—
87C2×C4×Dic7C2312yes—
88C2×#1136—21no—
89C2×#1136—22no—
90C2×#1136—22no—
91C2×#1136C2312yes—
92C22×C7⋊C8—21no—
93#6455—24no—
94#6455C2×C418yes—
95D7×C4⋊C4—22no—
96D7×C4⋊C4—22no—
97D7×C4⋊C4C2×C414yes—
98D7×C4⋊C4C2×C414yes—
99D7×C22⋊C4—24no—
100D7×C22⋊C4—24no—
101#6440—24no—
102#6440—24no—
103#6435—24no—
104#6435—24no—
105C42×D7—24no—
106C42×D7—24no—
107C7⋊#95—24no—
108(C2×D7)⋊C8—24no—
109(C2×D7)⋊C8—24no—
110C22×#117—71no—
111C2×#407—71no—
112#2280—71no—
113#1826—78no—
114#1733—71no—
115#1733—71no—
116#1733—71no—
117#1733D711yescharacteristic, complemented
118C24×C7C2×D411yescharacteristic
119C23×D7—22no—
120C23×D7—22no—
121C23×D7—44no—
122C23×D7C22⋊C411yescharacteristic
123C23×D7C22⋊C412yes—
124C23×D7C22⋊C411yescharacteristic
125C23×D7C2×D412yes—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7C22×C411yescharacteristic
129C22×C4×C7C22×C411yescharacteristic
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7#4112yes—
134C22×C4×C7#4112yes—
135C22×C4×C7#4112yes—
136C22×C4×C7#4112yes—
137C22×C4×C7C2411yescharacteristic
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7C22⋊C412yes—
143C22×Dic7C22⋊C411yescharacteristic
144C22×Dic7C22⋊C411yescharacteristic
145C22×Dic7C2×D412yes—
146C2×C4×D7—22no—
147C2×C4×D7—22no—
148C2×C4×D7—24no—
149C2×C4×D7—22no—
150C2×C4×D7—22no—
151C2×C4×D7—24no—
152C2×C4×D7—24no—
153C2×C4×D7—22no—
154C2×C4×D7—22no—
155C2×C4×D7—24no—
156C2×C4×D7—22no—
157C2×C4×D7—22no—
158C2×C4×D7—24no—
159C2×C4×D7—24no—
160C2×C4×D7—22no—
161C2×C4×D7—22no—
162C2×C4×D7—22no—
163C2×C4×D7—22no—
164C2×C4×D7—24no—
165C2×C4×D7—24no—
166C2×C4×D7—48no—
167C2×C4×D7C22⋊C412yes—
168C2×C4×D7C22⋊C412yes—
169C2×C4×D7C22⋊C412yes—
170C2×C4×D7C22⋊C412yes—
171C2×C4×D7C22×C412yes—
172C2×C4×D7C22×C412yes—
173C2×C4×D7C2×D412yes—
174C2×C4×D7C2×D412yes—
175C7×M16—22no—
176C7×M16—22no—
177C7×M16—42no—
178C2×C7×C8—22no—
179C2×C7×C8—42no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—42no—
182C7×C4⋊C4C22×C414yes—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—42no—
186C42×C7—22no—
187C42×C7—22no—
188C42×C7—22no—
189C42×C7—22no—
190C22⋊Dic7—22no—
191C22⋊Dic7—22no—
192C22⋊Dic7—42no—
193(C2×D7)⋊C4—28no—
194(C2×D7)⋊C4—48no—
195C4⋊Dic7—22no—
196C4⋊Dic7—42no—
197C4⋊Dic7C22×C414yes—
198Dic7⋊C4—24no—
199Dic7⋊C4—24no—
200C4×Dic7—22no—
201C4×Dic7—22no—
202C4×Dic7—28no—
203C4×Dic7—22no—
204C4×Dic7—22no—
205#1136—22no—
206#1136—22no—
207#1136—42no—
208C2×C7⋊C8—22no—
209C2×C7⋊C8—42no—
210C24×C4—71no—
211C22×M16—71no—
212C22×M16C2×D711yescharacteristic
213C2×#117—71no—
214C2×#117—71no—
215C2×#117—74no—
216C2×#117—74no—
217C2×#117—72no—
218C2×#117—141no—
219C2×#117—141no—
220C2×#117—142no—
221C2×#117—142no—
222C2×#117C2×D711yescharacteristic
223C2×#117C2×D711yescharacteristic
224C22×C4⋊C4—72no—
225C22×C22⋊C4—141no—
226C22×C42—141no—
227#407—74no—
228#407—141no—
229#407—141no—
230C2×C22⋊C8—142no—
231#381—141no—
232#380—142no—
233#379—141no—
234C4×C22⋊C4—74no—
235#339—74no—
236#339—74no—
237#339—74no—
238#339C2×D714yescomplemented
239C23×C7—42no—
240C23×C7C2×C22⋊C411yescharacteristic
241C23×C7C2×C22⋊C411yescharacteristic
242C23×C7#12312yes—
243C23×C7#12312yes—
244C23×C7C22×D411yescharacteristic
245C22×D7—22no—
246C22×D7—24no—
247C22×D7—22no—
248C22×D7—22no—
249C22×D7—24no—
250C22×D7—22no—
251C22×D7—24no—
252C22×D7—24no—
253C22×D7—42no—
254C22×D7—42no—
255C22×D7—42no—
256C22×D7—48no—
257C22×D7—48no—
258C22×D7C2×C22⋊C412yes—
259C22×D7C2×C22⋊C412yes—
260C2×C4×C7—24no—
261C2×C4×C7—44no—
262C2×C4×C7—44no—
263C2×C4×C7—42no—
264C2×C4×C7C2×C22⋊C411yescharacteristic
265C2×C4×C7C2×C22⋊C411yescharacteristic
266C2×C4×C7#11712yes—
267C2×C4×C7#11712yes—
268C2×C4×C7#11712yes—
269C2×C4×C7#11712yes—
270C2×C4×C7#12312yes—
271C2×C4×C7#12312yes—
272C2×C4×C7#12312yes—
273C2×C4×C7#12312yes—
274C2×C4×C7#12312yes—
275C2×C4×C7#12312yes—
276C2×C4×C7C23×C411yescharacteristic
277C2×C4×C7C22×D411yescharacteristic
278C2×C4×C7C2×#4111yescharacteristic
279C2×C4×C7C2×#4111yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×Dic7—22no—
283C2×Dic7—22no—
284C2×Dic7—24no—
285C2×Dic7—22no—
286C2×Dic7—22no—
287C2×Dic7—24no—
288C2×Dic7—44no—
289C2×Dic7—44no—
290C2×Dic7—42no—
291C2×Dic7C2×C22⋊C412yes—
292C2×Dic7C2×C22⋊C412yes—
293C4×D7—22no—
294C4×D7—22no—
295C4×D7—22no—
296C4×D7—22no—
297C4×D7—22no—
298C4×D7—22no—
299C4×D7—22no—
300C4×D7—22no—
301C4×D7—24no—
302C4×D7—24no—
303C4×D7—24no—
304C4×D7—24no—
305C4×D7—48no—
306C4×D7—48no—
307C7×C8—42no—
308C7⋊C8—42no—
309C25—71no—
310C23×C4—71no—
311C23×C4—71no—
312C23×C4—71no—
313C23×C4—72no—
314C23×C4—72no—
315C23×C4—72no—
316C23×C4—71no—
317C23×C4—71no—
318C23×C4—72no—
319C23×C4—142no—
320C23×C4—142no—
321C23×C4—142no—
322C23×C4—142no—
323C23×C4—142no—
324C23×C4C22×D711yescharacteristic
325C2×M16—21no—
326C2×M16—22no—
327C2×M16—22no—
328C2×M16—72no—
329C2×M16—141no—
330C2×M16—142no—
331C2×M16—142no—
332C2×M16C22×D712yes—
333C22×C8—21no—
334C22×C8—141no—
335#117—21no—
336#117—21no—
337#117—22no—
338#117—22no—
339#117—72no—
340#117—72no—
341#117—78no—
342#117—144no—
343#117—144no—
344#117—144no—
345#117—144no—
346#117—144no—
347#117—141no—
348#117—141no—
349#117—142no—
350#117—142no—
351#117C4×D712yes—
352#117C22×D712yes—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—72no—
355C2×C4⋊C4—74no—
356C2×C4⋊C4—74no—
357C2×C4⋊C4—142no—
358C2×C4⋊C4—142no—
359C2×C4⋊C4—142no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4C4×D712yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—144no—
365C2×C22⋊C4—144no—
366C2×C22⋊C4—142no—
367C2×C22⋊C4—141no—
368C2×C22⋊C4—144no—
369C2×C22⋊C4—141no—
370C2×C42—21no—
371C2×C42—72no—
372C2×C42—144no—
373C2×C42—144no—
374C2×C42—142no—
375C2×C42—141no—
376C2×C42C22×D712yes—
377C22⋊C8—144no—
378C22⋊C8—144no—
379#95—144no—
380C22×C7—22no—
381C22×C7—22no—
382C22×C7—22no—
383C22×C7—42no—
384C22×C7—42no—
385C22×C7#38112yes—
386C22×C7#38112yes—
387C22×C7C22×C22⋊C411yescharacteristic
388C22×C7C2×#12311yescharacteristic
389C22×C7C2×#12311yescharacteristic
390C2×D7—28no—
391C2×D7—22no—
392C2×D7—22no—
393C2×D7—22no—
394C2×D7—22no—
395C2×D7—22no—
396C2×D7—44no—
397C2×D7—44no—
398C2×D7—48no—
399C2×D7—44no—
400C2×D7#42114yes—
401C4×C7—44no—
402C4×C7—82no—
403C4×C7#38112yes—
404C4×C7#38112yes—
405C4×C7C2×#11711yescharacteristic
406C4×C7C2×#11711yescharacteristic
407C4×C7C2×#12311yescharacteristic
408C4×C7C2×#12311yescharacteristic
409Dic7—22no—
410Dic7—22no—
411Dic7—44no—
412Dic7—82no—
413C24—71no—
414C24—72no—
415C24—72no—
416C24—71no—
417C24—142no—
418C24—142no—
419C24—284no—
420C24D4×D711yescharacteristic
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—71no—
424C22×C4—71no—
425C22×C4—72no—
426C22×C4—72no—
427C22×C4—72no—
428C22×C4—72no—
429C22×C4—72no—
430C22×C4—72no—
431C22×C4—72no—
432C22×C4—72no—
433C22×C4—72no—
434C22×C4—72no—
435C22×C4—144no—
436C22×C4—144no—
437C22×C4—142no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—144no—
442C22×C4—144no—
443C22×C4—144no—
444C22×C4—142no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—144no—
449C22×C4—144no—
450C22×C4—144no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—142no—
459C22×C4—288no—
460C22×C4C2×C4×D711yescharacteristic
461C22×C4C2×C4×D711yescharacteristic
462C22×C4#115712yes—
463C22×C4#115712yes—
464C22×C4D4×D711yescharacteristic
465C22×C4D4×D711yescharacteristic
466C22×C4D4×D711yescharacteristic
467C22×C4#115912yes—
468C22×C4#116112yes—
469C22×C4C23×D711yescharacteristic
470M16—22no—
471M16—22no—
472M16—42no—
473M16—142no—
474M16—142no—
475M16—282no—
476C2×C8—22no—
477C2×C8—42no—
478C2×C8—142no—
479C2×C8—282no—
480C4⋊C4—22no—
481C4⋊C4—42no—
482C4⋊C4—74no—
483C4⋊C4—144no—
484C4⋊C4—144no—
485C4⋊C4—142no—
486C4⋊C4—282no—
487C4⋊C4C2×C4×D714yes—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—42no—
491C22⋊C4—148no—
492C22⋊C4—142no—
493C22⋊C4—142no—
494C22⋊C4—288no—
495C22⋊C4—282no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—148no—
501C42—142no—
502C42—142no—
503C42—142no—
504C42—142no—
505C2×C7—21no—
506C2×C7—21no—
507C2×C7—22no—
508C2×C7—22no—
509C2×C7C2×#38111yescharacteristic
510C2×C7C2×#42112yes—
511D7—24no—
512D7—42no—
513C23—42no—
514C23—72no—
515C23—72no—
516C23—144no—
517C23—144no—
518C23—144no—
519C23—142no—
520C23—142no—
521C23—142no—
522C23—144no—
523C23—142no—
524C23—282no—
525C23—282no—
526C23—282no—
527C23—288no—
528C23—288no—
529C23D7×C22⋊C411yescharacteristic
530C23D7×C22⋊C411yescharacteristic
531C23#644612yes—
532C23#645712yes—
533C23C2×D4×D711yescharacteristic
534C2×C4—24no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—72no—
539C2×C4—72no—
540C2×C4—142no—
541C2×C4—142no—
542C2×C4—142no—
543C2×C4—142no—
544C2×C4—144no—
545C2×C4—142no—
546C2×C4—142no—
547C2×C4—142no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—142no—
552C2×C4—142no—
553C2×C4—144no—
554C2×C4—144no—
555C2×C4—144no—
556C2×C4—144no—
557C2×C4—144no—
558C2×C4—288no—
559C2×C4—288no—
560C2×C4—284no—
561C2×C4—284no—
562C2×C4—282no—
563C2×C4#643512yes—
564C2×C4#643512yes—
565C2×C4D7×C22⋊C411yescharacteristic
566C2×C4D7×C22⋊C411yescharacteristic
567C2×C4#644612yes—
568C2×C4#644612yes—
569C2×C4#644612yes—
570C2×C4#645512yes—
571C2×C4#645512yes—
572C2×C4#645712yes—
573C2×C4#645712yes—
574C2×C4#645712yes—
575C2×C4C22×C4×D711yescharacteristic
576C2×C4C2×#115711yescharacteristic
577C2×C4C2×#115711yescharacteristic
578C2×C4C2×D4×D711yescharacteristic
579C2×C4C2×#115911yescharacteristic
580C2×C4C2×#116111yescharacteristic
581C8—42no—
582C8—282no—
583C7#2070511yescharacteristic, complemented, Sylow
584C22—22no—
585C22—22no—
586C22—22no—
587C22—42no—
588C22—42no—
589C22—74no—
590C22—148no—
591C22—142no—
592C22—142no—
593C22—142no—
594C22—142no—
595C22—142no—
596C22—284no—
597C22—284no—
598C22—288no—
599C22—284no—
600C22#8943612yes—
601C22#8943612yes—
602C22C2×D7×C22⋊C411yescharacteristic
603C22C2×#644611yescharacteristic
604C22C2×#645711yescharacteristic
605C4—44no—
606C4—82no—
607C4—142no—
608C4—142no—
609C4—284no—
610C4—562no—
611C4#8943612yes—
612C4#8943612yes—
613C4C2×#643511yescharacteristic
614C4C2×#644611yescharacteristic
615C4C2×#645511yescharacteristic
616C4C2×#645711yescharacteristic
617C2—21no—
618C2—21no—
619C2—22no—
620C2—22no—
621C2—144no—
622C2—282no—
623C2C2×#8943611yescharacteristic
624C2D7×#42112yes—
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1733C2
socleC22×C7C22×C22⋊C4
prime radicalC24×D7D4
cosocleC22×C4×C7C24
hypercenter#1733D7
hypocenterC7#20705
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
4. C7×#1733C2semidirect
5. #11729736C2semidirect
6. #11729332C2semidirect
7. #11729328C2semidirect
8. #11726508C2semidirect
18. D7×#117C4semidirect
20. C2×#6455C4semidirect
21. C2×D7×C4⋊C4C4semidirect
34. C7×#339C22semidirect
35. #89350C22semidirect
36. #89327C22semidirect
37. #89318C22semidirect
117. #1733D7semidirect
238. #339C2×D7semidirect
583. C7#20705semidirect

Maximal subgroups

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

subgroupindex
C2×D7×#1172
C2×#898982
#117397122
C7×#17332
#117297362
#117293322
#117293282
#117265082
#207057