Finite Group Databasebeta

← #13284483

Subgroups — #13284483

Order 1792.

1390 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90003C211yesmaximal, characteristic
2C2×#90002C211yesmaximal, characteristic, complemented
3C2×#89441C211yesmaximal, characteristic
4C7×#1446C211yesmaximal, characteristic, complemented
5#11727183C212yesmaximal
6#11727182C212yesmaximal
7#11727181C211yesmaximal, characteristic
8#11727174C211yesmaximal, characteristic, complemented
9#11727168C212yesmaximal, complemented
10#11727167C212yesmaximal, complemented
11#11727155C211yesmaximal, characteristic, complemented
12C23×C7×Q8C2211yescharacteristic
13C22×(C2×D7)⋊C4C2211yescharacteristic
14C22×(C2×D7)⋊C4C2211yescharacteristic
15C22×(C2×D7)⋊C4C2211yescharacteristic
16C22×(C2×D7)⋊C4C2212yes—
17C22×(C2×D7)⋊C4C2212yes—
18C22×C4⋊Dic7C2211yescharacteristic
19C22×Dic7⋊C4C2211yescharacteristic
20C22×Dic7⋊C4C2212yes—
21C22×C4×Dic7C2211yescharacteristic
22C2×C7×#95C2211yescharacteristic
23C2×C7×#95C2211yescharacteristic
24C2×C7×#95C2212yes—
25C2×C7×#95C2212yes—
26#90003C418yescomplemented
27#90002—24no—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2211yescharacteristic
36C2×C7⋊#95C2212yes—
37#89441—24no—
38#14625—71nomaximal, Sylow
39C24×D7C2311yescharacteristic
40C22×C7×Q8—22no—
41C22×C7×Q8—22no—
42C22×C7×Q8—42no—
43C22×C7×Q8—42no—
44C22×C7×Q8C2×C414yes—
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2312yes—
48C23×C4×C7C2312yes—
49C23×C4×C7C2311yescharacteristic
50C23×Dic7C2312yes—
51C23×Dic7C2311yescharacteristic
52C23×Dic7C2312yes—
53C23×Dic7C2311yescharacteristic
54C23×Dic7C2311yescharacteristic
55C2×(C2×D7)⋊C4—22no—
56C2×(C2×D7)⋊C4—22no—
57C2×(C2×D7)⋊C4—22no—
58C2×(C2×D7)⋊C4—22no—
59C2×(C2×D7)⋊C4—42no—
60C2×(C2×D7)⋊C4—42no—
61C2×(C2×D7)⋊C4—42no—
62C2×(C2×D7)⋊C4—42no—
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—42no—
70C2×Dic7⋊C4—22no—
71C2×Dic7⋊C4—42no—
72C2×Dic7⋊C4—42no—
73C2×Dic7⋊C4—42no—
74C2×Dic7⋊C4—42no—
75C2×C4×Dic7—42no—
76C2×C4×Dic7D414yes—
77C7×#95—22no—
78C7×#95—22no—
79C7×#95—42no—
80C7×#95—42no—
81C7⋊#95—22no—
82C7⋊#95—24no—
83C7⋊#95—24no—
84C7⋊#95—42no—
85C7⋊#95—42no—
86C7⋊#95—42no—
87C7⋊#95—42no—
88C7⋊#95C2×C418yes—
89C7⋊#95C2×C414yes—
90C2×#393—71no—
91C2×#391—71no—
92C2×#388—71no—
93#1446D711yescharacteristic, complemented
94#1444—71no—
95#1442—72no—
96#1442—72no—
97#1440—72no—
98#1439—71no—
99#1439—72no—
100#1434—71no—
101C24×C7C2411yescharacteristic
102C23×D7—21no—
103C23×D7—21no—
104C23×D7—22no—
105C23×D7—22no—
106C23×D7—22no—
107C23×D7—21no—
108C23×D7—21no—
109C23×D7—21no—
110C23×D7—21no—
111C23×D7—22no—
112C23×D7C22×C412yes—
113C2×C7×Q8—28no—
114C2×C7×Q8—48no—
115C2×C7×Q8—44no—
116C2×C7×Q8—44no—
117C2×C7×Q8—48no—
118C22×C4×C7—22no—
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—21no—
127C22×C4×C7—21no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—21no—
137C22×C4×C7—22no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7C22×C414yes—
144C22×C4×C7C22×C412yes—
145C22×C4×C7C2×D412yes—
146C22×C4×C7#4112yes—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—22no—
164C22×Dic7—21no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—22no—
172C22×Dic7C22×C412yes—
173C22×Dic7C22×C414yes—
174C22×Dic7C2×D412yes—
175C22×Dic7C2×D412yes—
176(C2×D7)⋊C4—28no—
177(C2×D7)⋊C4—48no—
178(C2×D7)⋊C4—44no—
179(C2×D7)⋊C4—44no—
180(C2×D7)⋊C4—48no—
181C4⋊Dic7—44no—
182Dic7⋊C4—44no—
183Dic7⋊C4—48no—
184C4×Dic7—44no—
185C23×Q8C2×D711yescharacteristic
186C22×C4⋊C4—71no—
187C22×C4⋊C4—71no—
188C22×C4⋊C4—72no—
189C22×C22⋊C4—71no—
190C22×C22⋊C4—72no—
191C22×C22⋊C4—71no—
192C22×C22⋊C4—71no—
193C22×C22⋊C4—72no—
194C22×C42—71no—
195#393—78no—
196#391—144no—
197#388—144no—
198C2×#95—71no—
199C2×#95—72no—
200C2×#95—72no—
201C2×#95—72no—
202C2×#95—71no—
203C2×#95—72no—
204C2×#95—72no—
205C2×#95—72no—
206C2×#95—72no—
207C2×#95C2×D711yescharacteristic
208C2×#95C2×D711yescharacteristic
209C2×#95C2×D712yes—
210C2×#95C2×D712yes—
211C23×C7C23×C411yescharacteristic
212C23×C7C22×D411yescharacteristic
213C23×C7C2×#4111yescharacteristic
214C23×C7H(5,𝔽2)12yes—
215C23×C7H(5,𝔽2)12yes—
216C23×C7H(5,𝔽2)12yes—
217C23×C7H(5,𝔽2)11yescharacteristic
218C23×C72−1+411yescharacteristic
219C23×C72−1+411yescharacteristic
220C23×C72−1+412yes—
221C23×C72−1+411yescharacteristic
222C22×D7—22no—
223C22×D7—24no—
224C22×D7—22no—
225C22×D7—22no—
226C22×D7—24no—
227C22×D7—41no—
228C22×D7—41no—
229C22×D7—41no—
230C22×D7—41no—
231C22×D7—41no—
232C22×D7—41no—
233C22×D7—42no—
234C22×D7—41no—
235C22×D7—41no—
236C22×D7—42no—
237C22×D7—42no—
238C22×D7—42no—
239C22×D7—42no—
240C22×D7—42no—
241C22×D7—42no—
242C22×D7—42no—
243C22×D7—42no—
244C22×D7—42no—
245C7×Q8—416no—
246C2×C4×C7—24no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
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—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7C4×D414yes—
275C2×Dic7—22no—
276C2×Dic7—24no—
277C2×Dic7—24no—
278C2×Dic7—24no—
279C2×Dic7—24no—
280C2×Dic7—22no—
281C2×Dic7—22no—
282C2×Dic7—22no—
283C2×Dic7—22no—
284C2×Dic7—22no—
285C2×Dic7—22no—
286C2×Dic7—41no—
287C2×Dic7—41no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7—42no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7—42no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302C2×Dic7—42no—
303C2×Dic7—42no—
304C25—71no—
305C22×Q8—22no—
306C22×Q8—22no—
307C22×Q8—42no—
308C22×Q8—42no—
309C22×Q8C4×D714yes—
310C23×C4—71no—
311C23×C4—72no—
312C23×C4—71no—
313C23×C4—71no—
314C23×C4—72no—
315C23×C4C22×D711yescharacteristic
316C23×C4C22×D711yescharacteristic
317C23×C4C22×D712yes—
318C23×C4C22×D712yes—
319C23×C4C22×D711yescharacteristic
320C2×C4⋊C4—142no—
321C2×C4⋊C4—142no—
322C2×C4⋊C4—282no—
323C2×C4⋊C4—282no—
324C2×C4⋊C4—282no—
325C2×C4⋊C4—282no—
326C2×C4⋊C4—282no—
327C2×C22⋊C4—74no—
328C2×C22⋊C4—78no—
329C2×C22⋊C4—142no—
330C2×C22⋊C4—142no—
331C2×C22⋊C4—142no—
332C2×C22⋊C4—142no—
333C2×C22⋊C4—282no—
334C2×C22⋊C4—282no—
335C2×C22⋊C4—282no—
336C2×C22⋊C4—282no—
337C2×C22⋊C4—282no—
338C2×C22⋊C4—282no—
339C2×C22⋊C4—282no—
340C2×C42—74no—
341C2×C42—282no—
342#95—22no—
343#95—22no—
344#95—42no—
345#95—42no—
346#95—78no—
347#95—74no—
348#95—142no—
349#95—144no—
350#95—144no—
351#95—282no—
352#95—282no—
353#95—282no—
354#95—282no—
355C22×C7C2×C4×D411yescharacteristic
356C22×C7#51812yes—
357C22×C7#51911yescharacteristic
358C22×C7#52011yescharacteristic
359C22×C7#52012yes—
360C22×C7#53411yescharacteristic
361C22×C7#53511yescharacteristic
362C22×C7#53511yescharacteristic
363C22×C7#53611yescharacteristic
364C22×C7#53611yescharacteristic
365C22×C7#53711yescharacteristic
366C22×C7#53812yes—
367C22×C7#54111yescharacteristic
368C22×C7#54211yescharacteristic
369C22×C7#54212yes—
370C22×C7H(3,𝔽4)12yes—
371C22×C7#56212yes—
372C22×C7#56212yes—
373C22×C7#56212yes—
374C22×C7#56212yes—
375C22×C7#56312yes—
376C22×C7#56312yes—
377C22×C7#56312yes—
378C2×D7—42no—
379C2×D7—42no—
380C2×D7—42no—
381C2×D7—44no—
382C2×D7—44no—
383C2×D7—81no—
384C2×D7—81no—
385C2×D7—81no—
386C2×D7—82no—
387C2×D7—82no—
388C2×D7—81no—
389C4×C7—24no—
390C4×C7—44no—
391C4×C7—81no—
392C4×C7—81no—
393C4×C7—82no—
394Dic7—42no—
395Dic7—42no—
396Dic7—44no—
397Dic7—42no—
398Dic7—82no—
399C24—72no—
400C24—141no—
401C24—141no—
402C24—142no—
403C24—142no—
404C24—141no—
405C24—142no—
406C24—141no—
407C24—141no—
408C24—142no—
409C24—141no—
410C24C23×D711yescharacteristic
411C2×Q8—28no—
412C2×Q8—48no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—48no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—72no—
442C22×C4—74no—
443C22×C4—72no—
444C22×C4—72no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—142no—
448C22×C4—142no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—141no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4—142no—
459C22×C4—142no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4—141no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—142no—
466C22×C4—142no—
467C22×C4—142no—
468C22×C4—142no—
469C22×C4—142no—
470C22×C4C2×C4×D714yes—
471C22×C4C2×C4×D712yes—
472C22×C4D4×D712yes—
473C22×C4#115912yes—
474C4⋊C4—284no—
475C4⋊C4—284no—
476C4⋊C4—288no—
477C22⋊C4—148no—
478C22⋊C4—288no—
479C22⋊C4—284no—
480C22⋊C4—284no—
481C22⋊C4—288no—
482C42—284no—
483C2×C7#232111yescharacteristic
484C2×C7#232311yescharacteristic
485C2×C7#232611yescharacteristic
486C2×C7#237812yes—
487C2×C7#237912yes—
488C2×C7#281411yescharacteristic
489C2×C7#281611yescharacteristic
490C2×C7#281911yescharacteristic
491C2×C7#282111yescharacteristic
492C2×C7#283112yes—
493C2×C7#283212yes—
494D7—82no—
495C23—142no—
496C23—144no—
497C23—142no—
498C23—142no—
499C23—144no—
500C23—281no—
501C23—282no—
502C23—282no—
503C23—281no—
504C23—281no—
505C23—282no—
506C23—282no—
507C23—281no—
508C23—281no—
509C23—282no—
510C23—281no—
511C23—281no—
512C23—282no—
513C23—282no—
514C23—282no—
515C23—282no—
516C23—282no—
517C23—281no—
518C23C22×C4×D711yescharacteristic
519C23C2×D4×D711yescharacteristic
520C23C2×#115911yescharacteristic
521C23#654812yes—
522C23#654812yes—
523C23#655112yes—
524C23#655111yescharacteristic
525C23#655111yescharacteristic
526C23#655311yescharacteristic
527C23#655312yes—
528C23#655411yescharacteristic
529Q8—416no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—144no—
561C2×C4—144no—
562C2×C4—144no—
563C2×C4—144no—
564C2×C4—142no—
565C2×C4—142no—
566C2×C4—142no—
567C2×C4—142no—
568C2×C4—142no—
569C2×C4—282no—
570C2×C4—282no—
571C2×C4—281no—
572C2×C4—282no—
573C2×C4—282no—
574C2×C4—282no—
575C2×C4—282no—
576C2×C4—282no—
577C2×C4—282no—
578C2×C4—282no—
579C2×C4—281no—
580C2×C4—282no—
581C2×C4—282no—
582C2×C4—282no—
583C2×C4—282no—
584C2×C4—282no—
585C2×C4—282no—
586C2×C4—282no—
587C2×C4(C22⋊D7)⋊C414yes—
588C7#1462511yescharacteristic, complemented, Sylow
589C22—282no—
590C22—284no—
591C22—282no—
592C22—282no—
593C22—284no—
594C22—561no—
595C22—562no—
596C22—561no—
597C22—561no—
598C22—561no—
599C22—562no—
600C22C2×(C22⋊D7)⋊C411yescharacteristic
601C22#9017712yes—
602C22#9026311yescharacteristic
603C22#9026511yescharacteristic
604C22#9026512yes—
605C22#9028011yescharacteristic
606C22#9028311yescharacteristic
607C22#9028412yes—
608C22#9031911yescharacteristic
609C22#9032011yescharacteristic
610C22#9033412yes—
611C22#9033712yes—
612C22#9033812yes—
613C22#9033812yes—
614C22#9033812yes—
615C22#9036511yescharacteristic
616C22#9037311yescharacteristic
617C22#9037412yes—
618C22#9037512yes—
619C22#9041511yescharacteristic
620C22#9042211yescharacteristic
621C22#9042612yes—
622C22#9042612yes—
623C4—24no—
624C4—44no—
625C4—81no—
626C4—82no—
627C4—81no—
628C4—282no—
629C4—282no—
630C4—284no—
631C4—282no—
632C4—562no—
633C2—562no—
634C2#1173126611yescharacteristic
635C2#1173392911yescharacteristic
636C2#1173394112yes—
637C2#1173405611yescharacteristic
638C2#1173406212yes—
639C2#1173429811yescharacteristic
640C2#1173430212yes—
641C2#1173433011yescharacteristic
642C2#1173471911yescharacteristic
643C2#1173472111yescharacteristic
644C2#1173472312yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1446C2
socle, cosocleC24×C7C24
prime radicalC24×D7C23
hypercenter#1446D7
hypocenterC7#14625
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
2. C2×#90002C2semidirect
4. C7×#1446C2semidirect
8. #11727174C2semidirect
9. #11727168C2semidirect
10. #11727167C2semidirect
11. #11727155C2semidirect
26. #90003C4semidirect
93. #1446D7semidirect
588. C7#14625semidirect

Maximal subgroups

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

subgroupindex
C2×#900032
C2×#900022
C2×#894412
C7×#14462
#117271832
#117271822
#117271812
#117271742
#117271682
#117271672
#117271552
#146257