Finite Group Databasebeta

← #13280808

Subgroups — #13280808

Order 1792.

1590 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89764C211yesmaximal, characteristic
2C2×#89757C211yesmaximal, characteristic, complemented
3C2×#89419C212yesmaximal, complemented
4C2×#89442C212yesmaximal
5C2×#89441C211yesmaximal, characteristic
6C7×#1445C211yesmaximal, characteristic, complemented
7#11727182C212yesmaximal
8#11727181C211yesmaximal, characteristic
9#11727175C212yesmaximal, complemented
10#11727174C211yesmaximal, characteristic, complemented
11Dic7×#95C211yesmaximal, characteristic, complemented
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×(C2×D7)⋊C4C2211yescharacteristic
14C22×(C2×D7)⋊C4C2211yescharacteristic
15C22×(C2×D7)⋊C4C2212yes—
16C22×(C2×D7)⋊C4C2212yes—
17C22×(C2×D7)⋊C4C2211yescharacteristic
18C22×C4⋊Dic7C2212yes—
19C22×C4⋊Dic7C2211yescharacteristic
20C22×Dic7⋊C4C2211yescharacteristic
21C22×Dic7⋊C4C2212yes—
22C22×Dic7⋊C4C2212yes—
23C22×C4×Dic7C2211yescharacteristic
24C22×C4×Dic7C2212yes—
25C2×C7×#95C2211yescharacteristic
26C2×C7×#95C2211yescharacteristic
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2211yescharacteristic
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2211yescharacteristic
35#89764C418yescomplemented
36#89757—24no—
37#89442—28no—
38#89441—24no—
39#89419—28no—
40#14322—71nomaximal, Sylow
41C24×D7C2311yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2312yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2311yescharacteristic
47C23×Dic7C2312yes—
48C23×Dic7C2311yescharacteristic
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2312yes—
51C23×Dic7C2311yescharacteristic
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4C2×C414yes—
55C2×(C2×D7)⋊C4—22no—
56C2×(C2×D7)⋊C4—22no—
57C2×(C2×D7)⋊C4—24no—
58C2×(C2×D7)⋊C4—24no—
59C2×(C2×D7)⋊C4—24no—
60C2×(C2×D7)⋊C4—22no—
61C2×(C2×D7)⋊C4—22no—
62C2×(C2×D7)⋊C4—24no—
63C2×(C2×D7)⋊C4—42no—
64C2×(C2×D7)⋊C4—42no—
65C2×(C2×D7)⋊C4—42no—
66C2×(C2×D7)⋊C4C2×C414yes—
67C2×(C2×D7)⋊C4C2×C418yes—
68C2×C4⋊Dic7—22no—
69C2×C4⋊Dic7—24no—
70C2×C4⋊Dic7—24no—
71C2×C4⋊Dic7—24no—
72C2×C4⋊Dic7—42no—
73C2×Dic7⋊C4—24no—
74C2×Dic7⋊C4—24no—
75C2×Dic7⋊C4—24no—
76C2×Dic7⋊C4—24no—
77C2×Dic7⋊C4—22no—
78C2×Dic7⋊C4—42no—
79C2×Dic7⋊C4—42no—
80C2×Dic7⋊C4C2×C418yes—
81C2×C4×Dic7—42no—
82C2×C4×Dic7—42no—
83C2×C4×Dic7—42no—
84C2×C4×Dic7D418yes—
85C2×C4×Dic7D414yes—
86C7×#95—24no—
87C7×#95—22no—
88C7×#95—24no—
89C7×#95—22no—
90C7⋊#95—24no—
91C7⋊#95—22no—
92C7⋊#95—24no—
93C7⋊#95—24no—
94C7⋊#95—24no—
95C7⋊#95C2×C414yes—
96C2×#399—71no—
97C2×#389—72no—
98C2×#389—71no—
99C2×#388—71no—
100C2×#388—72no—
101#1445—72no—
102#1445D711yescharacteristic, complemented
103#1444—71no—
104#1439—71no—
105#1439—72no—
106C4×#95—71no—
107C24×C7C2411yescharacteristic
108C23×D7—21no—
109C23×D7—22no—
110C23×D7—21no—
111C23×D7—22no—
112C23×D7—22no—
113C23×D7—21no—
114C23×D7—21no—
115C23×D7—21no—
116C23×D7—21no—
117C23×D7—22no—
118C23×D7C22×C412yes—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
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—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—21no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—22no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—22no—
142C22×C4×C7—21no—
143C22×C4×C7C22×C414yes—
144C22×C4×C7C22×C412yes—
145C22×C4×C7C2×D414yes—
146C22×C4×C7C2×D412yes—
147C22×C4×C7#4112yes—
148C22×Dic7—21no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—21no—
155C22×Dic7—22no—
156C22×Dic7—21no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—21no—
160C22×Dic7—22no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7C22×C412yes—
171C22×Dic7C22×C414yes—
172C22×Dic7C2×D412yes—
173C22×Dic7C2×D414yes—
174C22×Dic7C2×D414yes—
175C22×Dic7C2×D412yes—
176C22×Dic7#4114yes—
177C7×C4⋊C4—28no—
178(C2×D7)⋊C4—28no—
179(C2×D7)⋊C4—216no—
180(C2×D7)⋊C4—44no—
181(C2×D7)⋊C4—48no—
182(C2×D7)⋊C4—44no—
183C4⋊Dic7—44no—
184C4⋊Dic7—48no—
185Dic7⋊C4—216no—
186Dic7⋊C4—48no—
187Dic7⋊C4—44no—
188C4×Dic7—48no—
189C4×Dic7—44no—
190C22×C4⋊C4—72no—
191C22×C4⋊C4—71no—
192C22×C4⋊C4—72no—
193C22×C4⋊C4—72no—
194C22×C4⋊C4—71no—
195C22×C4⋊C4C2×D711yescharacteristic
196C22×C22⋊C4—72no—
197C22×C22⋊C4—71no—
198C22×C22⋊C4—71no—
199C22×C22⋊C4—71no—
200C22×C22⋊C4—72no—
201C22×C42—72no—
202C22×C42—71no—
203#399—78no—
204#389—148no—
205#389—144no—
206#388—144no—
207#388—148no—
208C2×#95—71no—
209C2×#95—72no—
210C2×#95—72no—
211C2×#95—72no—
212C2×#95—72no—
213C2×#95—71no—
214C2×#95C2×D711yescharacteristic
215C2×#95C2×D712yes—
216C2×#95C2×D711yescharacteristic
217C2×#95C2×D712yes—
218C23×C7C23×C411yescharacteristic
219C23×C7C22×D411yescharacteristic
220C23×C7C22×D412yes—
221C23×C7C2×#4111yescharacteristic
222C23×C7C2×#4112yes—
223C23×C7H(5,𝔽2)11yescharacteristic
224C23×C7H(5,𝔽2)12yes—
225C23×C72−1+411yescharacteristic
226C23×C72−1+411yescharacteristic
227C23×C72−1+411yescharacteristic
228C23×C72−1+412yes—
229C22×D7—22no—
230C22×D7—24no—
231C22×D7—22no—
232C22×D7—24no—
233C22×D7—22no—
234C22×D7—41no—
235C22×D7—42no—
236C22×D7—41no—
237C22×D7—41no—
238C22×D7—42no—
239C22×D7—42no—
240C22×D7—41no—
241C22×D7—41no—
242C22×D7—42no—
243C22×D7—41no—
244C22×D7—42no—
245C22×D7—42no—
246C22×D7—42no—
247C22×D7—42no—
248C22×D7—42no—
249C22×D7—41no—
250C22×D7—41no—
251C22×D7—42no—
252C2×C4×C7—24no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—24no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—42no—
273C2×C4×C7—41no—
274C2×C4×C7—42no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—41no—
280C2×C4×C7C4×D414yes—
281C2×Dic7—24no—
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—22no—
286C2×Dic7—24no—
287C2×Dic7—22no—
288C2×Dic7—24no—
289C2×Dic7—24no—
290C2×Dic7—24no—
291C2×Dic7—24no—
292C2×Dic7—22no—
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—22no—
296C2×Dic7—41no—
297C2×Dic7—42no—
298C2×Dic7—41no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302C2×Dic7—42no—
303C2×Dic7—42no—
304C2×Dic7—42no—
305C2×Dic7—42no—
306C2×Dic7—42no—
307C2×Dic7—42no—
308C2×Dic7C4×D418yes—
309C2×Dic7#12714yes—
310C25—71no—
311C23×C4—71no—
312C23×C4—72no—
313C23×C4—71no—
314C23×C4—71no—
315C23×C4—72no—
316C23×C4C22×D711yescharacteristic
317C23×C4C22×D711yescharacteristic
318C23×C4C22×D712yes—
319C23×C4C22×D712yes—
320C23×C4C22×D711yescharacteristic
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—78no—
324C2×C4⋊C4—144no—
325C2×C4⋊C4—144no—
326C2×C4⋊C4—142no—
327C2×C4⋊C4—144no—
328C2×C4⋊C4—144no—
329C2×C4⋊C4—144no—
330C2×C4⋊C4—144no—
331C2×C4⋊C4—144no—
332C2×C4⋊C4—142no—
333C2×C4⋊C4—282no—
334C2×C4⋊C4—282no—
335C2×C4⋊C4—282no—
336C2×C4⋊C4C4×D714yes—
337C2×C22⋊C4—78no—
338C2×C22⋊C4—74no—
339C2×C22⋊C4—144no—
340C2×C22⋊C4—144no—
341C2×C22⋊C4—142no—
342C2×C22⋊C4—142no—
343C2×C22⋊C4—142no—
344C2×C22⋊C4—144no—
345C2×C22⋊C4—142no—
346C2×C22⋊C4—144no—
347C2×C22⋊C4—282no—
348C2×C22⋊C4—282no—
349C2×C22⋊C4—282no—
350C2×C42—78no—
351C2×C42—74no—
352C2×C42—282no—
353C2×C42—282no—
354C2×C42—282no—
355#95—24no—
356#95—22no—
357#95—24no—
358#95—22no—
359#95—74no—
360#95—144no—
361#95—144no—
362#95—142no—
363#95—144no—
364#95—144no—
365C22×C7C2×C4×D411yescharacteristic
366C22×C7C2×C4×D412yes—
367C22×C7#51911yescharacteristic
368C22×C7#52012yes—
369C22×C7#52011yescharacteristic
370C22×C7C2×#12711yescharacteristic
371C22×C7#53212yes—
372C22×C7#53211yescharacteristic
373C22×C7#53412yes—
374C22×C7#53411yescharacteristic
375C22×C7#53612yes—
376C22×C7#53611yescharacteristic
377C22×C7#53611yescharacteristic
378C22×C7#53612yes—
379C22×C7#53711yescharacteristic
380C22×C7#53712yes—
381C22×C7#53912yes—
382C22×C7#53912yes—
383C22×C7#53912yes—
384C22×C7#54112yes—
385C22×C7#54111yescharacteristic
386C22×C7#54211yescharacteristic
387C22×C7#54212yes—
388C2×D7—42no—
389C2×D7—44no—
390C2×D7—42no—
391C2×D7—42no—
392C2×D7—44no—
393C2×D7—81no—
394C2×D7—81no—
395C2×D7—81no—
396C2×D7—82no—
397C2×D7—81no—
398C2×D7—82no—
399C4×C7—24no—
400C4×C7—44no—
401C4×C7—44no—
402C4×C7—81no—
403C4×C7—81no—
404Dic7—28no—
405Dic7—24no—
406Dic7—42no—
407Dic7—42no—
408Dic7—82no—
409C24—72no—
410C24—141no—
411C24—142no—
412C24—141no—
413C24—141no—
414C24—141no—
415C24—142no—
416C24—142no—
417C24—141no—
418C24—142no—
419C24—141no—
420C24C23×D711yescharacteristic
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—74no—
446C22×C4—74no—
447C22×C4—72no—
448C22×C4—74no—
449C22×C4—74no—
450C22×C4—72no—
451C22×C4—72no—
452C22×C4—142no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—142no—
459C22×C4—141no—
460C22×C4—142no—
461C22×C4—141no—
462C22×C4—141no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—142no—
466C22×C4—141no—
467C22×C4—142no—
468C22×C4—142no—
469C22×C4—142no—
470C22×C4—142no—
471C22×C4—142no—
472C22×C4—142no—
473C22×C4—142no—
474C22×C4C2×C4×D714yes—
475C22×C4C2×C4×D712yes—
476C22×C4D4×D714yes—
477C22×C4D4×D712yes—
478C22×C4#115912yes—
479C4⋊C4—28no—
480C4⋊C4—1416no—
481C4⋊C4—284no—
482C4⋊C4—284no—
483C4⋊C4—288no—
484C4⋊C4—288no—
485C22⋊C4—1416no—
486C22⋊C4—148no—
487C22⋊C4—288no—
488C22⋊C4—284no—
489C22⋊C4—284no—
490C42—284no—
491C42—288no—
492C2×C7C4×#12711yescharacteristic
493C2×C7#232212yes—
494C2×C7#232211yescharacteristic
495C2×C7#232311yescharacteristic
496C2×C7#232312yes—
497C2×C7#265611yescharacteristic
498C2×C7#265811yescharacteristic
499C2×C7#266311yescharacteristic
500C2×C7#266312yes—
501C2×C7#266411yescharacteristic
502C2×C7#266412yes—
503D7—82no—
504C23—142no—
505C23—144no—
506C23—144no—
507C23—142no—
508C23—142no—
509C23—281no—
510C23—282no—
511C23—282no—
512C23—282no—
513C23—282no—
514C23—281no—
515C23—281no—
516C23—282no—
517C23—282no—
518C23—281no—
519C23—281no—
520C23—282no—
521C23—281no—
522C23—282no—
523C23—282no—
524C23—281no—
525C23—282no—
526C23—281no—
527C23C22×C4×D711yescharacteristic
528C23C2×D4×D711yescharacteristic
529C23C2×D4×D712yes—
530C23C2×#115911yescharacteristic
531C23#655111yescharacteristic
532C23#655111yescharacteristic
533C23D7×#4112yes—
534C23#655311yescharacteristic
535C23#655312yes—
536C23#655411yescharacteristic
537C23#655412yes—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—74no—
567C2×C4—78no—
568C2×C4—144no—
569C2×C4—144no—
570C2×C4—144no—
571C2×C4—144no—
572C2×C4—144no—
573C2×C4—142no—
574C2×C4—144no—
575C2×C4—144no—
576C2×C4—144no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—144no—
580C2×C4—142no—
581C2×C4—142no—
582C2×C4—142no—
583C2×C4—282no—
584C2×C4—282no—
585C2×C4—281no—
586C2×C4—282no—
587C2×C4—281no—
588C2×C4—282no—
589C2×C4—282no—
590C2×C4—282no—
591C2×C4—282no—
592C2×C4—282no—
593C2×C4—282no—
594C2×C4—282no—
595C2×C4(C22⋊D7)⋊C414yes—
596C7#1432211yescharacteristic, complemented, Sylow
597C22—284no—
598C22—282no—
599C22—282no—
600C22—284no—
601C22—282no—
602C22—561no—
603C22—561no—
604C22—561no—
605C22—561no—
606C22—562no—
607C22—562no—
608C22C2×(C22⋊D7)⋊C411yescharacteristic
609C22#9021412yes—
610C22C4×D4×D712yes—
611C22#9026311yescharacteristic
612C22#9026511yescharacteristic
613C22#9028712yes—
614C22#9031912yes—
615C22#9031911yescharacteristic
616C22#9032011yescharacteristic
617C22#9032012yes—
618C22#9033911yescharacteristic
619C22#9033912yes—
620C22#9034412yes—
621C22#9034411yescharacteristic
622C22#9035612yes—
623C22#9035911yescharacteristic
624C22#9037311yescharacteristic
625C22#9038312yes—
626C22#9038712yes—
627C22#9039812yes—
628C22#9040112yes—
629C22D7×#12711yescharacteristic
630C22#9042211yescharacteristic
631C4—24no—
632C4—44no—
633C4—44no—
634C4—81no—
635C4—81no—
636C4—144no—
637C4—148no—
638C4—282no—
639C4—282no—
640C4—562no—
641C2—562no—
642C2#1173150111yescharacteristic
643C2#1173239212yes—
644C2#1173274212yes—
645C2#1173373811yescharacteristic
646C2#1173405611yescharacteristic
647C2#1173501012yes—
648C2#1173528412yes—
649C2#1173550211yescharacteristic
650C2#1173555611yescharacteristic
651C2#1173567711yescharacteristic
652C2#1173567811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1445C2
socle, cosocleC24×C7C24
prime radicalC24×D7C23
hypercenter#1445D7
hypocenterC7#14322
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
2. C2×#89757C2semidirect
3. C2×#89419C2semidirect
6. C7×#1445C2semidirect
9. #11727175C2semidirect
10. #11727174C2semidirect
11. Dic7×#95C2semidirect
35. #89764C4semidirect
102. #1445D7semidirect
596. C7#14322semidirect

Maximal subgroups

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

subgroupindex
C2×#897642
C2×#897572
C2×#894192
C2×#894422
C2×#894412
C7×#14452
#117271822
#117271812
#117271752
#117271742
Dic7×#952
#143227