Finite Group Databasebeta

← #13284482

Subgroups — #13284482

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×#89746C211yesmaximal, characteristic, complemented
2C2×#89719C211yesmaximal, characteristic
3C2×#89723C211yesmaximal, characteristic
4C7×#1439C211yesmaximal, characteristic
5#11729322C211yesmaximal, characteristic, complemented
6#11729303C212yesmaximal
7#11729301C212yesmaximal
8#11727178C211yesmaximal, characteristic, complemented
9#11727169C212yesmaximal, complemented
10#11727168C212yesmaximal, complemented
11#11727155C211yesmaximal, characteristic, complemented
12C23×Dic14C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2211yescharacteristic
15C22×C7×C22⋊C4C2212yes—
16C22×C22⋊Dic7C2211yescharacteristic
17C22×C22⋊Dic7C2211yescharacteristic
18C22×C22⋊Dic7C2212yes—
19C22×C4⋊Dic7C2211yescharacteristic
20C22×Dic7⋊C4C2212yes—
21C22×C4×Dic7C2211yescharacteristic
22C2×C7×#95C2211yescharacteristic
23C2×C7×#95C2212yes—
24C2×C7⋊#95C2211yescharacteristic
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2211yescharacteristic
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2211yescharacteristic
34C2×C7⋊#95C2212yes—
35#89746—24no—
36#89723—24no—
37#89719C418yescomplemented
38#14625—71nomaximal, Sylow
39C25×C7C2311yescharacteristic
40C23×C4×C7C2312yes—
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×Dic7C2311yescharacteristic
45C23×Dic7C2311yescharacteristic
46C23×Dic7C2312yes—
47C23×Dic7C2312yes—
48C23×Dic7C2311yescharacteristic
49C23×Dic7C2311yescharacteristic
50C22×Dic14—22no—
51C22×Dic14—22no—
52C22×Dic14—42no—
53C22×Dic14—42no—
54C22×Dic14C2×C414yes—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—42no—
57C2×C7×C22⋊C4—22no—
58C2×C7×C22⋊C4—22no—
59C2×C7×C22⋊C4—42no—
60C2×C7×C22⋊C4—42no—
61C2×C7×C22⋊C4—42no—
62C2×C7×C22⋊C4C2×C414yes—
63C2×C22⋊Dic7—22no—
64C2×C22⋊Dic7—22no—
65C2×C22⋊Dic7—42no—
66C2×C22⋊Dic7—42no—
67C2×C22⋊Dic7—42no—
68C2×C22⋊Dic7—42no—
69C2×C22⋊Dic7C2×C418yes—
70C2×C4⋊Dic7—22no—
71C2×C4⋊Dic7—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—42no—
79C7⋊#95—22no—
80C7⋊#95—22no—
81C7⋊#95—24no—
82C7⋊#95—24no—
83C7⋊#95—42no—
84C7⋊#95—42no—
85C7⋊#95—42no—
86C7⋊#95—42no—
87C7⋊#95—42no—
88C7⋊#95C2×C414yes—
89C7⋊#95C2×C418yes—
90C2×#393—71no—
91C2×#391—71no—
92C2×#388—71no—
93#1446—71no—
94#1444—71no—
95#1442—72no—
96#1442—72no—
97#1440—72no—
98#1439—72no—
99#1439D711yescharacteristic
100#1434—71no—
101C24×C7—21no—
102C24×C7—21no—
103C24×C7—22no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7—21no—
107C24×C7—21no—
108C24×C7—21no—
109C24×C7—21no—
110C24×C7—22no—
111C24×C7C22×C412yes—
112C24×C7C2411yescharacteristic
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—21no—
116C22×C4×C7—21no—
117C22×C4×C7—22no—
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—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×C7C22×C412yes—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C2×D412yes—
136C22×C4×C7#4112yes—
137C22×Dic7—22no—
138C22×Dic7—22no—
139C22×Dic7—21no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—22no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—21no—
163C22×Dic7—22no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7C22×C414yes—
168C22×Dic7C22×C414yes—
169C22×Dic7C2×D412yes—
170C22×Dic7C2×D412yes—
171C2×Dic14—28no—
172C2×Dic14—48no—
173C2×Dic14—44no—
174C2×Dic14—44no—
175C2×Dic14—48no—
176C7×C4⋊C4—44no—
177C7×C22⋊C4—28no—
178C7×C22⋊C4—48no—
179C22⋊Dic7—48no—
180C22⋊Dic7—44no—
181C22⋊Dic7—44no—
182C4⋊Dic7—44no—
183Dic7⋊C4—48no—
184C4×Dic7—44no—
185C23×Q8—71no—
186C22×C4⋊C4—71no—
187C22×C4⋊C4—72no—
188C22×C4⋊C4C2×D711yescharacteristic
189C22×C22⋊C4—72no—
190C22×C22⋊C4—71no—
191C22×C22⋊C4—71no—
192C22×C22⋊C4C2×D711yescharacteristic
193C22×C22⋊C4C2×D712yes—
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—71no—
204C2×#95—72no—
205C2×#95—72no—
206C2×#95—72no—
207C2×#95—72no—
208C2×#95—72no—
209C2×#95C2×D711yescharacteristic
210C2×#95C2×D712yes—
211C23×C7—22no—
212C23×C7—24no—
213C23×C7—22no—
214C23×C7—22no—
215C23×C7—24no—
216C23×C7—41no—
217C23×C7—41no—
218C23×C7—41no—
219C23×C7—41no—
220C23×C7—41no—
221C23×C7—41no—
222C23×C7—42no—
223C23×C7—41no—
224C23×C7—41no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7C23×C411yescharacteristic
235C23×C7C22×D411yescharacteristic
236C23×C7C2×#4111yescharacteristic
237C23×C7H(5,𝔽2)12yes—
238C23×C7H(5,𝔽2)12yes—
239C23×C7H(5,𝔽2)11yescharacteristic
240C23×C7H(5,𝔽2)12yes—
241C23×C72−1+411yescharacteristic
242C23×C72−1+411yescharacteristic
243C23×C72−1+411yescharacteristic
244C23×C72−1+412yes—
245C2×C4×C7—24no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—24no—
249C2×C4×C7—22no—
250C2×C4×C7—24no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
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×C7C4×D414yes—
269C2×Dic7—24no—
270C2×Dic7—24no—
271C2×Dic7—24no—
272C2×Dic7—24no—
273C2×Dic7—24no—
274C2×Dic7—24no—
275C2×Dic7—22no—
276C2×Dic7—22no—
277C2×Dic7—22no—
278C2×Dic7—22no—
279C2×Dic7—22no—
280C2×Dic7—22no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7—42no—
292C2×Dic7—42no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302C2×Dic7—42no—
303Dic14—416no—
304C25C22×D711yescharacteristic
305C22×Q8—74no—
306C22×Q8—142no—
307C22×Q8—142no—
308C22×Q8—282no—
309C22×Q8—282no—
310C23×C4—72no—
311C23×C4—71no—
312C23×C4—72no—
313C23×C4—71no—
314C23×C4—71no—
315C23×C4—71no—
316C23×C4C22×D712yes—
317C23×C4C22×D712yes—
318C23×C4C22×D711yescharacteristic
319C23×C4C22×D711yescharacteristic
320C2×C4⋊C4—22no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—142no—
323C2×C4⋊C4—282no—
324C2×C4⋊C4—282no—
325C2×C4⋊C4—282no—
326C2×C4⋊C4—282no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—42no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—42no—
332C2×C22⋊C4—78no—
333C2×C22⋊C4—142no—
334C2×C22⋊C4—142no—
335C2×C22⋊C4—282no—
336C2×C22⋊C4—282no—
337C2×C22⋊C4—282no—
338C2×C22⋊C4—282no—
339C2×C22⋊C4C4×D714yes—
340C2×C42—74no—
341C2×C42—282no—
342#95—22no—
343#95—42no—
344#95—74no—
345#95—78no—
346#95—142no—
347#95—144no—
348#95—144no—
349#95—142no—
350#95—282no—
351#95—282no—
352#95—282no—
353#95—282no—
354#95—282no—
355C22×C7—42no—
356C22×C7—42no—
357C22×C7—42no—
358C22×C7—44no—
359C22×C7—44no—
360C22×C7—81no—
361C22×C7—81no—
362C22×C7—81no—
363C22×C7—82no—
364C22×C7—82no—
365C22×C7—81no—
366C22×C7C2×C4×D411yescharacteristic
367C22×C7#51812yes—
368C22×C7#51911yescharacteristic
369C22×C7#52011yescharacteristic
370C22×C7#52012yes—
371C22×C7#53411yescharacteristic
372C22×C7#53511yescharacteristic
373C22×C7#53511yescharacteristic
374C22×C7#53611yescharacteristic
375C22×C7#53611yescharacteristic
376C22×C7#53711yescharacteristic
377C22×C7#53812yes—
378C22×C7#54111yescharacteristic
379C22×C7#54212yes—
380C22×C7#54211yescharacteristic
381C22×C7H(3,𝔽4)12yes—
382C22×C7#56212yes—
383C22×C7#56212yes—
384C22×C7#56212yes—
385C22×C7#56212yes—
386C22×C7#56312yes—
387C22×C7#56312yes—
388C22×C7#56312yes—
389C4×C7—24no—
390C4×C7—42no—
391C4×C7—82no—
392C4×C7—82no—
393Dic7—42no—
394Dic7—44no—
395Dic7—44no—
396Dic7—42no—
397Dic7—81no—
398Dic7—81no—
399C24—21no—
400C24—21no—
401C24—21no—
402C24—22no—
403C24—22no—
404C24—21no—
405C24—22no—
406C24—21no—
407C24—21no—
408C24—22no—
409C24C2×C4×D712yes—
410C24C23×D711yescharacteristic
411C2×Q8—148no—
412C2×Q8—288no—
413C2×Q8—284no—
414C2×Q8—288no—
415C2×Q8—284no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
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—22no—
436C22×C4—74no—
437C22×C4—74no—
438C22×C4—72no—
439C22×C4—72no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—141no—
443C22×C4—142no—
444C22×C4—142no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—142no—
449C22×C4—142no—
450C22×C4—141no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—141no—
458C22×C4—141no—
459C22×C4—142no—
460C22×C4—141no—
461C22×C4—141no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—141no—
466C22×C4—141no—
467C22×C4—142no—
468C22×C4—141no—
469C22×C4—141no—
470C22×C4C2×C4×D712yes—
471C22×C4C2×C4×D712yes—
472C22×C4D4×D712yes—
473C22×C4#115912yes—
474C4⋊C4—44no—
475C4⋊C4—284no—
476C4⋊C4—288no—
477C22⋊C4—28no—
478C22⋊C4—48no—
479C22⋊C4—288no—
480C22⋊C4—284no—
481C22⋊C4—284no—
482C42—284no—
483C2×C7—82no—
484C2×C7#232111yescharacteristic
485C2×C7#232311yescharacteristic
486C2×C7#232611yescharacteristic
487C2×C7#237812yes—
488C2×C7#237912yes—
489C2×C7#281411yescharacteristic
490C2×C7#281611yescharacteristic
491C2×C7#281911yescharacteristic
492C2×C7#282111yescharacteristic
493C2×C7#283112yes—
494C2×C7#283212yes—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23C22×C4×D711yescharacteristic
519C23C2×D4×D711yescharacteristic
520C23C2×#115911yescharacteristic
521C23#654812yes—
522C23#654811yescharacteristic
523C23#654812yes—
524C23#655111yescharacteristic
525C23#655312yes—
526C23#655411yescharacteristic
527C23#655412yes—
528C23#655411yescharacteristic
529Q8—2816no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—142no—
554C2×C4—144no—
555C2×C4—144no—
556C2×C4—144no—
557C2×C4—144no—
558C2×C4—144no—
559C2×C4—144no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—142no—
565C2×C4—282no—
566C2×C4—282no—
567C2×C4—282no—
568C2×C4—282no—
569C2×C4—282no—
570C2×C4—282no—
571C2×C4—282no—
572C2×C4—282no—
573C2×C4—282no—
574C2×C4—282no—
575C2×C4—282no—
576C2×C4—282no—
577C2×C4—281no—
578C2×C4—281no—
579C2×C4—281no—
580C2×C4—282no—
581C2×C4—281no—
582C2×C4—282no—
583C2×C4—281no—
584C2×C4—282no—
585C2×C4—281no—
586C2×C4—282no—
587C2×C4(C22⋊D7)⋊C414yes—
588C7#1462511yescharacteristic, complemented, Sylow
589C22—42no—
590C22—42no—
591C22—44no—
592C22—44no—
593C22—42no—
594C22—82no—
595C22—81no—
596C22—82no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22C2×(C22⋊D7)⋊C411yescharacteristic
601C22#9019511yescharacteristic
602C22#9020711yescharacteristic
603C22#9023612yes—
604C22#9023712yes—
605C22#9028311yescharacteristic
606C22#9028512yes—
607C22#9029411yescharacteristic
608C22#9029412yes—
609C22#9029911yescharacteristic
610C22#9030812yes—
611C22#9031212yes—
612C22#9031611yescharacteristic
613C22#9031911yescharacteristic
614C22#9033712yes—
615C22#9033812yes—
616C22#9034411yescharacteristic
617C22#9035812yes—
618C22#9035812yes—
619C22#9036312yes—
620C22#9036611yescharacteristic
621C22#9040312yes—
622C22#9040611yescharacteristic
623C4—24no—
624C4—42no—
625C4—82no—
626C4—82no—
627C4—284no—
628C4—282no—
629C4—284no—
630C4—282no—
631C4—561no—
632C4—561no—
633C2—82no—
634C2#1173131611yescharacteristic
635C2#1173184712yes—
636C2#1173232011yescharacteristic
637C2#1173293612yes—
638C2#1173374211yescharacteristic
639C2#1173419011yescharacteristic
640C2#1173429911yescharacteristic
641C2#1173454512yes—
642C2#1173485012yes—
643C2#1173517311yescharacteristic
644C2#1173553511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1439C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenter#1439D7
hypocenterC7#14625
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C2×#89746C2semidirect
5. #11729322C2semidirect
8. #11727178C2semidirect
9. #11727169C2semidirect
10. #11727168C2semidirect
11. #11727155C2semidirect
37. #89719C4semidirect
588. C7#14625semidirect

Maximal subgroups

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

subgroupindex
C2×#897462
C2×#897192
C2×#897232
C7×#14392
#117293222
#117293032
#117293012
#117271782
#117271692
#117271682
#117271552
#146257