Finite Group Databasebeta

← #13280841

Subgroups — #13280841

Order 1792.

1650 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#394C211yesmaximal, characteristic
2C2×#89993C211yesmaximal, characteristic
3C2×#89753C211yesmaximal, characteristic, complemented
4#11729885C211yesmaximal, characteristic
5#11729319C211yesmaximal, characteristic, complemented
6#11729302C212yesmaximal, complemented
7#11729300C212yesmaximal, complemented
8#11729299C212yesmaximal, complemented
9#11729299C212yesmaximal, complemented
10#11727166C211yesmaximal, characteristic, complemented
11Dic7×#95C211yesmaximal, characteristic, complemented
12C23×C7×D4C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2212yes—
15C22×C7×C22⋊C4C2212yes—
16C22×C22⋊Dic7C2212yes—
17C22×C22⋊Dic7C2212yes—
18C22×C22⋊Dic7C2212yes—
19C22×C22⋊Dic7C2212yes—
20C22×C4⋊Dic7C2211yescharacteristic
21C22×C4⋊Dic7C2211yescharacteristic
22C22×C4×Dic7C2212yes—
23C22×C4×Dic7C2211yescharacteristic
24C7×#394C418yescomplemented
25C2×C7×#95C2211yescharacteristic
26#89993—24no—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2211yescharacteristic
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2211yescharacteristic
31C2×C7⋊#95C2211yescharacteristic
32C2×C7⋊#95C2211yescharacteristic
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37#89753—24no—
38#14325—71nomaximal, Sylow
39C25×C7C2312yes—
40C22×C7×D4—22no—
41C22×C7×D4—22no—
42C22×C7×D4—42no—
43C22×C7×D4—42no—
44C22×C7×D4C2×C414yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2312yes—
51C23×Dic7C2311yescharacteristic
52C23×Dic7C2311yescharacteristic
53C23×Dic7C2312yes—
54C23×Dic7C2311yescharacteristic
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4C2×C414yes—
58C2×C7×C22⋊C4—42no—
59C2×C7×C22⋊C4—42no—
60C2×C7×C22⋊C4—42no—
61C2×C7×C22⋊C4—42no—
62C2×C7×C22⋊C4C2×C418yes—
63C2×C7×C22⋊C4C2×C418yes—
64C2×C22⋊Dic7—24no—
65C2×C22⋊Dic7—24no—
66C2×C22⋊Dic7—24no—
67C2×C22⋊Dic7—24no—
68C2×C22⋊Dic7—24no—
69C2×C22⋊Dic7—24no—
70C2×C22⋊Dic7—42no—
71C2×C22⋊Dic7—42no—
72C2×C22⋊Dic7—42no—
73C2×C22⋊Dic7—42no—
74C2×C22⋊Dic7—42no—
75C2×C22⋊Dic7—42no—
76C2×C4⋊Dic7—22no—
77C2×C4⋊Dic7—22no—
78C2×C4⋊Dic7—42no—
79C2×C4⋊Dic7—42no—
80C2×C4×Dic7—24no—
81C2×C4×Dic7—42no—
82C2×C4×Dic7—42no—
83C2×C4×Dic7—42no—
84C2×C4×Dic7D414yes—
85C7×#95C2×C414yes—
86C7⋊#95—22no—
87C7⋊#95—22no—
88C7⋊#95—22no—
89C7⋊#95—22no—
90C7⋊#95—22no—
91C7⋊#95—24no—
92C7⋊#95—42no—
93C7⋊#95—42no—
94C7⋊#95—42no—
95C7⋊#95—42no—
96C2×#394D711yescharacteristic
97C2×#390—71no—
98C2×#387—71no—
99#1442—71no—
100#1441—71no—
101#1440—72no—
102#1439—72no—
103#1438—71no—
104#1437—72no—
105#1437—72no—
106C4×#95—71no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7—22no—
110C24×C7—22no—
111C24×C7—22no—
112C24×C7—22no—
113C24×C7—22no—
114C24×C7—22no—
115C24×C7—22no—
116C24×C7—22no—
117C24×C7—22no—
118C24×C7—22no—
119C24×C7—22no—
120C24×C7—22no—
121C24×C7C22×C414yes—
122C24×C7C2411yescharacteristic
123C2×C7×D4—28no—
124C2×C7×D4—48no—
125C2×C7×D4—44no—
126C2×C7×D4—44no—
127C2×C7×D4—48no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7C22×C414yes—
147C22×C4×C7C22×C412yes—
148C22×C4×C7C22×C412yes—
149C22×C4×C7C22×C412yes—
150C22×C4×C7C2×D412yes—
151C22×C4×C7#4112yes—
152C22×Dic7—22no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—22no—
172C22×Dic7—21no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—22no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—21no—
180C22×Dic7—21no—
181C22×Dic7—21no—
182C22×Dic7C2×D412yes—
183C22×Dic7C2×D412yes—
184C22×Dic7#4114yes—
185C22×Dic7#4114yes—
186C7×C4⋊C4—28no—
187C7×C22⋊C4—48no—
188C7×C22⋊C4—48no—
189C22⋊Dic7—48no—
190C22⋊Dic7—48no—
191C22⋊Dic7—48no—
192C22⋊Dic7—48no—
193C4⋊Dic7—44no—
194C4⋊Dic7—44no—
195C4×Dic7—44no—
196C4×Dic7—48no—
197C23×D4C2×D711yescharacteristic
198C22×C4⋊C4—71no—
199C22×C4⋊C4—71no—
200C22×C4⋊C4C2×D711yescharacteristic
201C22×C22⋊C4—72no—
202C22×C22⋊C4—72no—
203C22×C22⋊C4—72no—
204C22×C22⋊C4—72no—
205C22×C22⋊C4C2×D712yes—
206C22×C22⋊C4C2×D712yes—
207C22×C42—71no—
208C22×C42—72no—
209#394Dic718yescomplemented
210#390—144no—
211#387—144no—
212C2×#95—71no—
213C2×#95—72no—
214C2×#95—71no—
215C2×#95—71no—
216C2×#95—71no—
217C2×#95—71no—
218C2×#95—72no—
219C2×#95—72no—
220C2×#95—72no—
221C2×#95—72no—
222C2×#95C2×D711yescharacteristic
223C23×C7—24no—
224C23×C7—24no—
225C23×C7—24no—
226C23×C7—24no—
227C23×C7—24no—
228C23×C7—24no—
229C23×C7—24no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7—42no—
249C23×C7—42no—
250C23×C7—42no—
251C23×C7—42no—
252C23×C7—42no—
253C23×C7—42no—
254C23×C7—42no—
255C23×C7—42no—
256C23×C7—42no—
257C23×C7—42no—
258C23×C7C23×C411yescharacteristic
259C23×C7C22×D411yescharacteristic
260C23×C7C2×#4111yescharacteristic
261C23×C7C2×#4112yes—
262C23×C7C2×#4112yes—
263C23×C7H(5,𝔽2)12yes—
264C23×C7H(5,𝔽2)12yes—
265C23×C7H(5,𝔽2)11yescharacteristic
266C23×C7H(5,𝔽2)11yescharacteristic
267C23×C7H(5,𝔽2)11yescharacteristic
268C23×C72−1+411yescharacteristic
269C7×D4—416no—
270C2×C4×C7—24no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—24no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—24no—
277C2×C4×C7—22no—
278C2×C4×C7—24no—
279C2×C4×C7—24no—
280C2×C4×C7—24no—
281C2×C4×C7—22no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7C4×D414yes—
294C2×Dic7—22no—
295C2×Dic7—22no—
296C2×Dic7—24no—
297C2×Dic7—24no—
298C2×Dic7—24no—
299C2×Dic7—22no—
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—24no—
303C2×Dic7—24no—
304C2×Dic7—42no—
305C2×Dic7—42no—
306C2×Dic7—42no—
307C2×Dic7—42no—
308C2×Dic7—42no—
309C2×Dic7—42no—
310C2×Dic7—42no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—42no—
315C2×Dic7—42no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
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#12414yes—
328C25C22×D712yes—
329C22×D4—22no—
330C22×D4—22no—
331C22×D4—42no—
332C22×D4—42no—
333C22×D4C2×Dic714yes—
334C23×C4—71no—
335C23×C4—71no—
336C23×C4—72no—
337C23×C4—71no—
338C23×C4—72no—
339C23×C4—71no—
340C23×C4C22×D711yescharacteristic
341C23×C4C22×D711yescharacteristic
342C23×C4C22×D712yes—
343C23×C4C22×D711yescharacteristic
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4—282no—
349C2×C4⋊C4—282no—
350C2×C4⋊C4C2×Dic714yes—
351C2×C22⋊C4—42no—
352C2×C22⋊C4—42no—
353C2×C22⋊C4—42no—
354C2×C22⋊C4—42no—
355C2×C22⋊C4—144no—
356C2×C22⋊C4—144no—
357C2×C22⋊C4—144no—
358C2×C22⋊C4—144no—
359C2×C22⋊C4—144no—
360C2×C22⋊C4—144no—
361C2×C22⋊C4—282no—
362C2×C22⋊C4—282no—
363C2×C22⋊C4—282no—
364C2×C22⋊C4—282no—
365C2×C22⋊C4—282no—
366C2×C22⋊C4—282no—
367C2×C22⋊C4C2×Dic718yes—
368C2×C22⋊C4C2×Dic718yes—
369C2×C42—74no—
370C2×C42—144no—
371C2×C42—282no—
372C2×C42—282no—
373C2×C42—282no—
374#95—144no—
375#95—142no—
376#95—142no—
377#95—142no—
378#95—142no—
379#95—142no—
380#95—282no—
381#95—282no—
382#95—282no—
383#95—282no—
384#95C2×Dic714yes—
385C22×C7—44no—
386C22×C7—44no—
387C22×C7—44no—
388C22×C7—44no—
389C22×C7—44no—
390C22×C7—44no—
391C22×C7—44no—
392C22×C7—82no—
393C22×C7—82no—
394C22×C7—82no—
395C22×C7—82no—
396C22×C7—82no—
397C22×C7—82no—
398C22×C7—82no—
399C22×C7—82no—
400C22×C7C2×C4×D411yescharacteristic
401C22×C7C4×#4112yes—
402C22×C7#51812yes—
403C22×C7#51811yescharacteristic
404C22×C7#52011yescharacteristic
405C22×C7C2×#12411yescharacteristic
406C22×C7#52912yes—
407C22×C7#53211yescharacteristic
408C22×C7#53411yescharacteristic
409C22×C7#53511yescharacteristic
410C22×C7#53511yescharacteristic
411C22×C7#53711yescharacteristic
412C22×C7#53812yes—
413C22×C7#53812yes—
414C22×C7#53812yes—
415C22×C7#53812yes—
416C22×C7#53912yes—
417C22×C7#54012yes—
418C22×C7#54012yes—
419C22×C7#54011yescharacteristic
420C22×C7#54012yes—
421C22×C7#54211yescharacteristic
422C22×C7#54212yes—
423C4×C7—24no—
424C4×C7—42no—
425C4×C7—42no—
426C4×C7—44no—
427Dic7—24no—
428Dic7—44no—
429Dic7—42no—
430Dic7—82no—
431Dic7—81no—
432Dic7—81no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24C22×Dic714yes—
448C24C23×D711yescharacteristic
449C2×D4—28no—
450C2×D4—48no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—48no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—72no—
473C22×C4—74no—
474C22×C4—72no—
475C22×C4—74no—
476C22×C4—142no—
477C22×C4—141no—
478C22×C4—141no—
479C22×C4—142no—
480C22×C4—142no—
481C22×C4—142no—
482C22×C4—142no—
483C22×C4—142no—
484C22×C4—141no—
485C22×C4—142no—
486C22×C4—142no—
487C22×C4—142no—
488C22×C4—141no—
489C22×C4—141no—
490C22×C4—141no—
491C22×C4—142no—
492C22×C4—141no—
493C22×C4—142no—
494C22×C4—142no—
495C22×C4—142no—
496C22×C4—142no—
497C22×C4—141no—
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×C4D4×D712yes—
507C22×C4#115912yes—
508C22×C4C22×Dic714yes—
509C22×C4C22×Dic712yes—
510C22×C4C22×Dic712yes—
511C22×C4C22×Dic712yes—
512C4⋊C4—28no—
513C4⋊C4—284no—
514C4⋊C4—284no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—288no—
518C22⋊C4—288no—
519C22⋊C4—288no—
520C22⋊C4—288no—
521C42—284no—
522C42—288no—
523C2×C7—84no—
524C2×C7C4×#12411yescharacteristic
525C2×C7#232111yescharacteristic
526C2×C7#232411yescharacteristic
527C2×C7#233312yes—
528C2×C7#233512yes—
529C2×C7#263911yescharacteristic
530C2×C7#264111yescharacteristic
531C2×C7#264812yes—
532C2×C7#265112yes—
533C2×C7#266111yescharacteristic
534C2×C7#266211yescharacteristic
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23C2×D4×D711yescharacteristic
571C23C2×#115911yescharacteristic
572C23#654812yes—
573C23#654812yes—
574C23#654811yescharacteristic
575C23#654811yescharacteristic
576C23D7×#4112yes—
577C23D7×#4112yes—
578C23#655311yescharacteristic
579C23#655411yescharacteristic
580C23C23×Dic711yescharacteristic
581D4—416no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—74no—
606C2×C4—142no—
607C2×C4—142no—
608C2×C4—144no—
609C2×C4—144no—
610C2×C4—144no—
611C2×C4—142no—
612C2×C4—142no—
613C2×C4—144no—
614C2×C4—144no—
615C2×C4—142no—
616C2×C4—282no—
617C2×C4—282no—
618C2×C4—282no—
619C2×C4—282no—
620C2×C4—282no—
621C2×C4—282no—
622C2×C4—281no—
623C2×C4—282no—
624C2×C4—281no—
625C2×C4—282no—
626C2×C4—281no—
627C2×C4—281no—
628C2×C4—282no—
629C2×C4—282no—
630C2×C4—282no—
631C2×C4—282no—
632C2×C4—282no—
633C2×C4—282no—
634C2×C4—282no—
635C2×C4—282no—
636C2×C4—281no—
637C2×C4—281no—
638C2×C4—282no—
639C2×C4D4×Dic714yes—
640C7#1432511yescharacteristic, complemented, Sylow
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22#9028011yescharacteristic
657C22#9028311yescharacteristic
658C22#9028412yes—
659C22#9029411yescharacteristic
660C22#9029811yescharacteristic
661C22#9029911yescharacteristic
662C22#9030412yes—
663C22#9030412yes—
664C22#9030612yes—
665C22#9030612yes—
666C22#9031011yescharacteristic
667C22#9034812yes—
668C22#9035112yes—
669C22#9035312yes—
670C22#9035412yes—
671C22#9035911yescharacteristic
672C22D7×#12411yescharacteristic
673C22#9037112yes—
674C22C2×D4×Dic711yescharacteristic
675C22#9048912yes—
676C22#9048911yescharacteristic
677C22#41×Dic712yes—
678C22#9051811yescharacteristic
679C4—24no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—144no—
684C4—284no—
685C4—282no—
686C4—562no—
687C4—561no—
688C4—561no—
689C2—84no—
690C2#1173442911yescharacteristic
691C2#1173443112yes—
692C2#1173443612yes—
693C2#1173444011yescharacteristic
694C2#1173444211yescharacteristic
695C2#1173457111yescharacteristic
696C2#1174002311yescharacteristic
697C2#1174007811yescharacteristic
698C2#1174008012yes—
699C2#1174028212yes—
700C2Dic7×#12411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#394C2
socle, cosocleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC2×#394D7
hypocenterC7#14325
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
3. C2×#89753C2semidirect
5. #11729319C2semidirect
6. #11729302C2semidirect
7. #11729300C2semidirect
8. #11729299C2semidirect
9. #11729299C2semidirect
10. #11727166C2semidirect
11. Dic7×#95C2semidirect
24. C7×#394C4semidirect
209. #394Dic7semidirect
640. C7#14325semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3942
C2×#899932
C2×#897532
#117298852
#117293192
#117293022
#117293002
#117292992
#117271662
Dic7×#952
#143257