Finite Group Databasebeta

← #13285808

Subgroups — #13285808

Order 1792.

1938 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#95C211yesmaximal, characteristic
2C2×#89982C211yesmaximal, characteristic
3C2×#89984C211yesmaximal, characteristic
4C2×#89984C212yesmaximal
5C2×#89433C211yesmaximal, characteristic, complemented
6#11729881C212yesmaximal
7#11727177C211yesmaximal, characteristic, complemented
8#11727173C211yesmaximal, characteristic, complemented
9#11727169C212yesmaximal, complemented
10#11727165C212yesmaximal, complemented
11#11727164C211yesmaximal, characteristic, complemented
12C24×C4×C7C2211yescharacteristic
13C24×C4×C7C2212yes—
14C22×C22⋊Dic7C2212yes—
15C22×C22⋊Dic7C2211yescharacteristic
16C22×C22⋊Dic7C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C22×C4⋊Dic7C2212yes—
19C22×C4⋊Dic7C2212yes—
20C22×C4⋊Dic7C2211yescharacteristic
21C22×Dic7⋊C4C2211yescharacteristic
22C22×Dic7⋊C4C2211yescharacteristic
23C2×C7×#95—24no—
24C2×C7×#95C2211yescharacteristic
25C2×C7×#95C2212yes—
26C2×C7×#95C2211yescharacteristic
27#89984—24no—
28#89984—28no—
29#89982—24no—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2211yescharacteristic
34C2×C7⋊#95C2211yescharacteristic
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38C2×C7⋊#95C2212yes—
39#89433—24no—
40#14766—71nomaximal, Sylow
41C25×C7C2311yescharacteristic
42C23×C4×C7—24no—
43C23×C4×C7—24no—
44C23×C4×C7—24no—
45C23×C4×C7—22no—
46C23×C4×C7—24no—
47C23×C4×C7—22no—
48C23×C4×C7—24no—
49C23×C4×C7—42no—
50C23×C4×C7—42no—
51C23×C4×C7—42no—
52C23×C4×C7D414yes—
53C23×C4×C7Q818yes—
54C23×C4×C7C2311yescharacteristic
55C23×C4×C7C2312yes—
56C23×C4×C7C2311yescharacteristic
57C23×C4×C7C2312yes—
58C23×Dic7C2311yescharacteristic
59C23×Dic7C2312yes—
60C23×Dic7C2312yes—
61C23×Dic7C2311yescharacteristic
62C23×Dic7C2311yescharacteristic
63C23×Dic7C2311yescharacteristic
64C2×C22⋊Dic7—22no—
65C2×C22⋊Dic7—22no—
66C2×C22⋊Dic7—22no—
67C2×C22⋊Dic7—24no—
68C2×C22⋊Dic7—24no—
69C2×C22⋊Dic7—24no—
70C2×C22⋊Dic7—24no—
71C2×C22⋊Dic7D414yes—
72C2×C4⋊Dic7—24no—
73C2×C4⋊Dic7—22no—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7—22no—
76C2×C4⋊Dic7—24no—
77C2×C4⋊Dic7—22no—
78C2×C4⋊Dic7—24no—
79C2×C4⋊Dic7—24no—
80C2×C4⋊Dic7—42no—
81C2×C4⋊Dic7—42no—
82C2×C4⋊Dic7—42no—
83C2×Dic7⋊C4—22no—
84C2×Dic7⋊C4—22no—
85C2×Dic7⋊C4—22no—
86C2×Dic7⋊C4—22no—
87C2×Dic7⋊C4—42no—
88C7×#95—22no—
89C7×#95—24no—
90C7×#95D414yes—
91C7⋊#95—24no—
92C7⋊#95—22no—
93C7⋊#95—22no—
94C7⋊#95—24no—
95C7⋊#95—24no—
96C7⋊#95—24no—
97C7⋊#95—24no—
98C7⋊#95—42no—
99C7⋊#95—42no—
100C2×#398—71no—
101C2×#385—71no—
102C2×#380—72no—
103C2×#380—71no—
104C22×#95D711yescharacteristic
105#1445—71no—
106#1444—71no—
107#1442—72no—
108#1441—71no—
109#1441—72no—
110C22⋊#95—72no—
111C24×C7—21no—
112C24×C7—21no—
113C24×C7—22no—
114C24×C7—22no—
115C24×C7—21no—
116C24×C7—21no—
117C24×C7C2×D412yes—
118C24×C7C2×D412yes—
119C24×C7C2×Q814yes—
120C24×C7#4112yes—
121C24×C7#4114yes—
122C24×C7C2411yescharacteristic
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—216no—
130C22×C4×C7—28no—
131C22×C4×C7—22no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—28no—
142C22×C4×C7—216no—
143C22×C4×C7—44no—
144C22×C4×C7—48no—
145C22×C4×C7—48no—
146C22×C4×C7—48no—
147C22×C4×C7—48no—
148C22×C4×C7—48no—
149C22×C4×C7—48no—
150C22×C4×C7—48no—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D414yes—
155C22×C4×C7C2×Q814yes—
156C22×C4×C7C2×Q814yes—
157C22×C4×C7#4112yes—
158C22×C4×C7#4114yes—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—22no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—22no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—21no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—21no—
180C22×Dic7—22no—
181C22×Dic7—22no—
182C22×Dic7—22no—
183C22×Dic7—22no—
184C22×Dic7—22no—
185C22×Dic7—22no—
186C22×Dic7—21no—
187C22×Dic7—22no—
188C22×Dic7—22no—
189C22×Dic7—22no—
190C22×Dic7—22no—
191C22×Dic7—22no—
192C22×Dic7—21no—
193C22×Dic7C2×D412yes—
194C22×Dic7C2×D412yes—
195C22⋊Dic7—28no—
196C22⋊Dic7—48no—
197C22⋊Dic7—44no—
198C4⋊Dic7—48no—
199C4⋊Dic7—48no—
200C4⋊Dic7—44no—
201C4⋊Dic7—44no—
202Dic7⋊C4—44no—
203Dic7⋊C4—44no—
204C24×C4C2×D712yes—
205C24×C4C2×D711yescharacteristic
206C22×C4⋊C4—71no—
207C22×C4⋊C4—71no—
208C22×C4⋊C4—72no—
209C22×C4⋊C4—72no—
210C22×C4⋊C4—71no—
211C22×C4⋊C4—71no—
212C22×C22⋊C4—71no—
213C22×C22⋊C4—71no—
214C22×C22⋊C4—72no—
215#398—144no—
216#385—144no—
217#380—148no—
218#380—144no—
219C2×#95—24no—
220C2×#95—72no—
221C2×#95—72no—
222C2×#95—72no—
223C2×#95—72no—
224C2×#95—72no—
225C2×#95—71no—
226C2×#95—71no—
227C2×#95—72no—
228C2×#95—72no—
229C2×#95C2×D712yes—
230C2×#95C2×D711yescharacteristic
231C2×#95C2×D711yescharacteristic
232C23×C7—22no—
233C23×C7—24no—
234C23×C7—22no—
235C23×C7—24no—
236C23×C7—22no—
237C23×C7—22no—
238C23×C7—24no—
239C23×C7—24no—
240C23×C7—22no—
241C23×C7—24no—
242C23×C7—24no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—24no—
246C23×C7—24no—
247C23×C7—24no—
248C23×C7—22no—
249C23×C7—24no—
250C23×C7#12114yes—
251C23×C7C22⋊Q818yes—
252C23×C7#12314yes—
253C23×C7#12518yes—
254C23×C7C4⋊Q814yes—
255C23×C7C22×D411yescharacteristic
256C23×C7C22×D411yescharacteristic
257C23×C7C22×D411yescharacteristic
258C23×C7C22×Q812yes—
259C23×C7C2×#4111yescharacteristic
260C23×C7C2×#4112yes—
261C23×C7C2×#4112yes—
262C23×C7C2×#4111yescharacteristic
263C23×C7H(5,𝔽2)12yes—
264C23×C7H(5,𝔽2)11yescharacteristic
265C23×C72−1+411yescharacteristic
266C2×C4×C7—24no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—22no—
277C2×C4×C7—24no—
278C2×C4×C7—24no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—24no—
282C2×C4×C7—416no—
283C2×C4×C7—432no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7#12114yes—
291C2×C4×C7C22⋊Q818yes—
292C2×Dic7—22no—
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—24no—
296C2×Dic7—22no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—42no—
304C2×Dic7—41no—
305C2×Dic7—42no—
306C2×Dic7—42no—
307C2×Dic7—42no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—42no—
312C2×Dic7—41no—
313C2×Dic7—42no—
314C2×Dic7—42no—
315C2×Dic7—41no—
316C2×Dic7—42no—
317C2×Dic7—42no—
318C2×Dic7—42no—
319C2×Dic7—42no—
320C2×Dic7—42no—
321C2×Dic7—42no—
322C2×Dic7—42no—
323C2×Dic7—42no—
324C2×Dic7—42no—
325C2×Dic7—42no—
326C2×Dic7—42no—
327C2×Dic7—42no—
328C25C22×D711yescharacteristic
329C23×C4—24no—
330C23×C4—24no—
331C23×C4—24no—
332C23×C4—24no—
333C23×C4—22no—
334C23×C4—24no—
335C23×C4—22no—
336C23×C4—42no—
337C23×C4—42no—
338C23×C4—42no—
339C23×C4—71no—
340C23×C4—71no—
341C23×C4—72no—
342C23×C4—72no—
343C23×C4—71no—
344C23×C4—71no—
345C23×C4Dic1418yes—
346C23×C4D2814yes—
347C23×C4C22×D712yes—
348C23×C4C22×D711yescharacteristic
349C23×C4C22×D712yes—
350C23×C4C22×D711yescharacteristic
351C2×C4⋊C4—144no—
352C2×C4⋊C4—144no—
353C2×C4⋊C4—142no—
354C2×C4⋊C4—142no—
355C2×C4⋊C4—142no—
356C2×C4⋊C4—142no—
357C2×C4⋊C4—142no—
358C2×C4⋊C4—144no—
359C2×C4⋊C4—142no—
360C2×C4⋊C4—142no—
361C2×C4⋊C4—144no—
362C2×C4⋊C4—142no—
363C2×C4⋊C4—282no—
364C2×C4⋊C4—282no—
365C2×C4⋊C4—282no—
366C2×C4⋊C4—282no—
367C2×C22⋊C4—74no—
368C2×C22⋊C4—142no—
369C2×C22⋊C4—142no—
370C2×C22⋊C4—144no—
371C2×C22⋊C4—142no—
372C2×C22⋊C4—144no—
373C2×C22⋊C4—144no—
374C2×C22⋊C4—144no—
375#95—24no—
376#95—22no—
377#95—144no—
378#95—144no—
379#95—144no—
380#95—142no—
381#95—142no—
382#95—144no—
383#95—144no—
384#95—282no—
385#95—282no—
386#95C22⋊D714yes—
387C22×C7—28no—
388C22×C7—24no—
389C22×C7—24no—
390C22×C7—24no—
391C22×C7—24no—
392C22×C7—28no—
393C22×C7—28no—
394C22×C7—28no—
395C22×C7—24no—
396C22×C7—24no—
397C22×C7#39818yescomplemented
398C22×C7C2×#12111yescharacteristic
399C22×C7C2×#12111yescharacteristic
400C22×C7C2×C22⋊Q812yes—
401C22×C7C2×C22⋊Q812yes—
402C22×C7C2×#12311yescharacteristic
403C22×C7C2×#12512yes—
404C22×C7C2×C4⋊Q811yescharacteristic
405C22×C7#53211yescharacteristic
406C22×C7#53312yes—
407C22×C7#53411yescharacteristic
408C22×C7#53711yescharacteristic
409C22×C7#54312yes—
410C22×C7#54412yes—
411C22×C7D4211yescharacteristic
412C22×C7#54612yes—
413C22×C7#54612yes—
414C22×C7#54711yescharacteristic
415C22×C7#54811yescharacteristic
416C22×C7#55112yes—
417C22×C7#55112yes—
418C22×C7#55212yes—
419C22×C7#55311yescharacteristic
420C22×C7#55312yes—
421C4×C7—24no—
422C4×C7—28no—
423C4×C7—42no—
424C4×C7—44no—
425Dic7—42no—
426Dic7—42no—
427Dic7—81no—
428Dic7—82no—
429Dic7—81no—
430Dic7—82no—
431C24—21no—
432C24—22no—
433C24—21no—
434C24—22no—
435C24—21no—
436C24—21no—
437C24C2×Dic1414yes—
438C24C2×D2812yes—
439C24D4×D712yes—
440C24#115914yes—
441C24#116112yes—
442C24C23×D711yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—216no—
450C22×C4—28no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—216no—
462C22×C4—28no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—72no—
472C22×C4—72no—
473C22×C4—141no—
474C22×C4—141no—
475C22×C4—142no—
476C22×C4—141no—
477C22×C4—142no—
478C22×C4—141no—
479C22×C4—141no—
480C22×C4—141no—
481C22×C4—141no—
482C22×C4—141no—
483C22×C4—141no—
484C22×C4—142no—
485C22×C4—142no—
486C22×C4—141no—
487C22×C4—142no—
488C22×C4—142no—
489C22×C4—142no—
490C22×C4—141no—
491C22×C4—142no—
492C22×C4—142no—
493C22×C4—142no—
494C22×C4—142no—
495C22×C4—142no—
496C22×C4—141no—
497C22×C4—142no—
498C22×C4—142no—
499C22×C4—142no—
500C22×C4—142no—
501C22×C4—142no—
502C22×C4—142no—
503C22×C4—142no—
504C22×C4—141no—
505C22×C4—141no—
506C22×C4—142no—
507C22×C4C2×Dic1414yes—
508C22×C4C2×Dic1414yes—
509C22×C4C2×D2812yes—
510C22×C4C2×D2812yes—
511C22×C4#115714yes—
512C22×C4#115712yes—
513C22×C4C2×C22⋊D714yes—
514C22×C4C2×C22⋊D712yes—
515C4⋊C4—288no—
516C4⋊C4—288no—
517C4⋊C4—284no—
518C4⋊C4—284no—
519C4⋊C4—284no—
520C4⋊C4—284no—
521C22⋊C4—148no—
522C22⋊C4—284no—
523C22⋊C4—288no—
524C2×C7—28no—
525C2×C7C2×#39811yescharacteristic
526C2×C7#241211yescharacteristic
527C2×C7#241611yescharacteristic
528C2×C7#243712yes—
529C2×C7#243812yes—
530C2×C7#244212yes—
531C2×C7#249811yescharacteristic
532C2×C7#254311yescharacteristic
533C2×C7#256912yes—
534C2×C7#265911yescharacteristic
535C2×C7#285711yescharacteristic
536C23—24no—
537C23—22no—
538C23—24no—
539C23—24no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—22no—
548C23—24no—
549C23—24no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—22no—
554C23#643214yes—
555C23#644118yes—
556C23#644914yes—
557C23#645318yes—
558C23#645814yes—
559C23C22×Dic1412yes—
560C23C22×D2811yescharacteristic
561C23C2×#115711yescharacteristic
562C23C2×#115712yes—
563C23C2×D4×D711yescharacteristic
564C23C2×#115912yes—
565C23#654811yescharacteristic
566C23#654812yes—
567C23C2×#116111yescharacteristic
568C23#655111yescharacteristic
569C23C22×C22⋊D711yescharacteristic
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—416no—
587C2×C4—432no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—144no—
595C2×C4—142no—
596C2×C4—142no—
597C2×C4—142no—
598C2×C4—142no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—282no—
603C2×C4—281no—
604C2×C4—281no—
605C2×C4—281no—
606C2×C4—281no—
607C2×C4—282no—
608C2×C4—282no—
609C2×C4—282no—
610C2×C4—281no—
611C2×C4—281no—
612C2×C4—282no—
613C2×C4—281no—
614C2×C4—282no—
615C2×C4—282no—
616C2×C4—282no—
617C2×C4—282no—
618C2×C4—282no—
619C2×C4—281no—
620C2×C4—282no—
621C2×C4—282no—
622C2×C4—282no—
623C2×C4—282no—
624C2×C4—282no—
625C2×C4—281no—
626C2×C4—282no—
627C2×C4—282no—
628C2×C4—282no—
629C2×C4—282no—
630C2×C4#648718yes—
631C2×C4#649314yes—
632C7#1476611yescharacteristic, complemented, Sylow
633C22—28no—
634C22—24no—
635C22—28no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—28no—
641C22—28no—
642C22—24no—
643C22#8943318yescomplemented
644C22C2×#643211yescharacteristic
645C22#9016112yes—
646C22#9016811yescharacteristic
647C22C2×#644112yes—
648C22#9017412yes—
649C22#9018112yes—
650C22C2×#644911yescharacteristic
651C22#9018411yescharacteristic
652C22#9018612yes—
653C22#9018612yes—
654C22C2×#645312yes—
655C22#9019112yes—
656C22C2×#645811yescharacteristic
657C22#9019811yescharacteristic
658C22#9020111yescharacteristic
659C22#9020412yes—
660C22#9020511yescharacteristic
661C22#9020512yes—
662C22C2×#648712yes—
663C22C2×#649311yescharacteristic
664C22D4×C22⋊D711yescharacteristic
665C22#9049712yes—
666C22#9050811yescharacteristic
667C4—24no—
668C4—28no—
669C4—42no—
670C4—44no—
671C4—282no—
672C4—282no—
673C4—561no—
674C4—562no—
675C4—561no—
676C4—562no—
677C2—28no—
678C2C2×#8943311yescharacteristic
679C2#1173079512yes—
680C2#1173079911yescharacteristic
681C2#1173080412yes—
682C2#1173080611yescharacteristic
683C2#1173082311yescharacteristic
684C2#1173952111yescharacteristic
685C2#1173955311yescharacteristic
686C2#1173955512yes—
687C2#1173961712yes—
688C2#1173963811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutator, socle, cosocleC24×C7C24
FittingC22×C7×#95C2
prime radicalC25×C7C23
hypercenterC22×#95D7
hypocenterC7#14766
LCSG ▷ C24×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
5. C2×#89433C2semidirect
7. #11727177C2semidirect
8. #11727173C2semidirect
9. #11727169C2semidirect
10. #11727165C2semidirect
11. #11727164C2semidirect
397. C22×C7#398semidirect
632. C7#14766semidirect
643. C22#89433semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#952
C2×#899822
C2×#899842
C2×#894332
#117298812
#117271772
#117271732
#117271692
#117271652
#117271642
#147667