Finite Group Databasebeta

← #13280834

Subgroups — #13280834

Order 1792.

1390 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89765C211yesmaximal, characteristic
2C2×#89757C211yesmaximal, characteristic, complemented
3C2×#89441C211yesmaximal, characteristic
4C7×#1445C211yesmaximal, characteristic, complemented
5#11727181C211yesmaximal, characteristic
6#11727181C212yesmaximal
7#11727169C212yesmaximal, complemented
8#11727169C211yesmaximal, characteristic, complemented
9#11727162C212yesmaximal
10#11727160C212yesmaximal, complemented
11Dic7×#95C211yesmaximal, characteristic, complemented
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×(C2×D7)⋊C4C2211yescharacteristic
14C22×(C2×D7)⋊C4C2211yescharacteristic
15C22×(C2×D7)⋊C4C2212yes—
16C22×(C2×D7)⋊C4C2211yescharacteristic
17C22×(C2×D7)⋊C4C2212yes—
18C22×C4⋊Dic7C2211yescharacteristic
19C22×C4⋊Dic7C2212yes—
20C22×Dic7⋊C4C2211yescharacteristic
21C22×C4×Dic7C2212yes—
22C22×C4×Dic7C2211yescharacteristic
23C2×C7×#95C2211yescharacteristic
24C2×C7×#95C2211yescharacteristic
25C2×C7×#95C2212yes—
26C2×C7×#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2211yescharacteristic
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2211yescharacteristic
34C2×C7⋊#95C2212yes—
35#89765C418yescomplemented
36#89757—24no—
37#89441—24no—
38#14324—71nomaximal, Sylow
39C24×D7C2311yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2312yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×Dic7C2312yes—
46C23×Dic7C2312yes—
47C23×Dic7C2311yescharacteristic
48C23×Dic7C2311yescharacteristic
49C23×Dic7C2311yescharacteristic
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4—22no—
52C2×C7×C4⋊C4C2×C414yes—
53C2×(C2×D7)⋊C4—22no—
54C2×(C2×D7)⋊C4—22no—
55C2×(C2×D7)⋊C4—24no—
56C2×(C2×D7)⋊C4—22no—
57C2×(C2×D7)⋊C4—22no—
58C2×(C2×D7)⋊C4—24no—
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)⋊C4C2×C414yes—
65C2×(C2×D7)⋊C4C2×C418yes—
66C2×C4⋊Dic7—22no—
67C2×C4⋊Dic7—24no—
68C2×C4⋊Dic7—42no—
69C2×C4⋊Dic7—42no—
70C2×C4⋊Dic7—42no—
71C2×Dic7⋊C4—22no—
72C2×Dic7⋊C4—24no—
73C2×C4×Dic7—24no—
74C2×C4×Dic7—42no—
75C2×C4×Dic7—42no—
76C2×C4×Dic7—42no—
77C2×C4×Dic7D414yes—
78C7×#95—22no—
79C7×#95—22no—
80C7×#95—42no—
81C7×#95—42no—
82C7⋊#95—24no—
83C7⋊#95—24no—
84C7⋊#95—22no—
85C7⋊#95—24no—
86C7⋊#95—42no—
87C7⋊#95—42no—
88C7⋊#95C2×C418yes—
89C7⋊#95C2×C414yes—
90C2×#397—71no—
91C2×#389—71no—
92C2×#388—71no—
93#1445D711yescharacteristic, complemented
94#1442—71no—
95#1442—72no—
96#1439—71no—
97#1439—72no—
98#1437—72no—
99#1436—72no—
100C4×#95—71no—
101C24×C7C2411yescharacteristic
102C23×D7—21no—
103C23×D7—21no—
104C23×D7—22no—
105C23×D7—22no—
106C23×D7—21no—
107C23×D7—21no—
108C23×D7—22no—
109C23×D7—21no—
110C23×D7—22no—
111C23×D7—21no—
112C23×D7C22×C412yes—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
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—21no—
128C22×C4×C7—21no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—22no—
136C22×C4×C7—21no—
137C22×C4×C7C22×C414yes—
138C22×C4×C7C22×C412yes—
139C22×C4×C7C2×D412yes—
140C22×C4×C7#4114yes—
141C22×C4×C7#4112yes—
142C22×Dic7—22no—
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—21no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7C22×C412yes—
167C22×Dic7C22×C414yes—
168C22×Dic7C2×D412yes—
169C22×Dic7C2×D412yes—
170C22×Dic7#4114yes—
171C7×C4⋊C4—28no—
172(C2×D7)⋊C4—28no—
173(C2×D7)⋊C4—48no—
174(C2×D7)⋊C4—44no—
175(C2×D7)⋊C4—48no—
176(C2×D7)⋊C4—44no—
177C4⋊Dic7—48no—
178C4⋊Dic7—44no—
179Dic7⋊C4—44no—
180C4×Dic7—44no—
181C4×Dic7—48no—
182C22×C4⋊C4—71no—
183C22×C4⋊C4—72no—
184C22×C4⋊C4—71no—
185C22×C4⋊C4C2×D711yescharacteristic
186C22×C22⋊C4—71no—
187C22×C22⋊C4—72no—
188C22×C22⋊C4—71no—
189C22×C22⋊C4—72no—
190C22×C22⋊C4—71no—
191C22×C42—71no—
192C22×C42—72no—
193#397—78no—
194#389—144no—
195#388—144no—
196C2×#95—71no—
197C2×#95—72no—
198C2×#95—72no—
199C2×#95—72no—
200C2×#95—72no—
201C2×#95—71no—
202C2×#95—72no—
203C2×#95—72no—
204C2×#95C2×D711yescharacteristic
205C2×#95C2×D711yescharacteristic
206C2×#95C2×D712yes—
207C2×#95C2×D712yes—
208C23×C7C23×C411yescharacteristic
209C23×C7C22×D411yescharacteristic
210C23×C7C2×#4111yescharacteristic
211C23×C7C2×#4112yes—
212C23×C7C2×#4112yes—
213C23×C7H(5,𝔽2)11yescharacteristic
214C23×C7H(5,𝔽2)11yescharacteristic
215C23×C7H(5,𝔽2)11yescharacteristic
216C23×C7H(5,𝔽2)12yes—
217C23×C72−1+412yes—
218C23×C72−1+411yescharacteristic
219C22×D7—22no—
220C22×D7—24no—
221C22×D7—22no—
222C22×D7—24no—
223C22×D7—22no—
224C22×D7—41no—
225C22×D7—42no—
226C22×D7—41no—
227C22×D7—41no—
228C22×D7—41no—
229C22×D7—41no—
230C22×D7—41no—
231C22×D7—42no—
232C22×D7—42no—
233C22×D7—42no—
234C22×D7—42no—
235C22×D7—42no—
236C22×D7—42no—
237C22×D7—41no—
238C22×D7—42no—
239C22×D7—41no—
240C22×D7—42no—
241C22×D7—42no—
242C2×C4×C7—24no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—24no—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—42no—
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—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7C4×D414yes—
271C2×Dic7—22no—
272C2×Dic7—22no—
273C2×Dic7—24no—
274C2×Dic7—24no—
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—41no—
284C2×Dic7—42no—
285C2×Dic7—41no—
286C2×Dic7—42no—
287C2×Dic7—42no—
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#12414yes—
300C25—71no—
301C23×C4—71no—
302C23×C4—71no—
303C23×C4—71no—
304C23×C4—72no—
305C23×C4—72no—
306C23×C4C22×D711yescharacteristic
307C23×C4C22×D711yescharacteristic
308C23×C4C22×D712yes—
309C23×C4C22×D712yes—
310C23×C4C22×D711yescharacteristic
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—144no—
314C2×C4⋊C4—142no—
315C2×C4⋊C4—144no—
316C2×C4⋊C4—142no—
317C2×C4⋊C4—282no—
318C2×C4⋊C4—282no—
319C2×C4⋊C4—282no—
320C2×C4⋊C4C4×D714yes—
321C2×C22⋊C4—74no—
322C2×C22⋊C4—78no—
323C2×C22⋊C4—142no—
324C2×C22⋊C4—142no—
325C2×C22⋊C4—144no—
326C2×C22⋊C4—142no—
327C2×C22⋊C4—142no—
328C2×C22⋊C4—144no—
329C2×C22⋊C4—282no—
330C2×C22⋊C4—282no—
331C2×C22⋊C4—282no—
332C2×C22⋊C4—282no—
333C2×C22⋊C4—282no—
334C2×C42—74no—
335C2×C42—144no—
336C2×C42—282no—
337C2×C42—282no—
338C2×C42—282no—
339#95—22no—
340#95—22no—
341#95—42no—
342#95—42no—
343#95—78no—
344#95—74no—
345#95—142no—
346#95—144no—
347#95—144no—
348#95—144no—
349#95—282no—
350#95—282no—
351C22×C7C2×C4×D411yescharacteristic
352C22×C7C4×#4112yes—
353C22×C7#51811yescharacteristic
354C22×C7#52012yes—
355C22×C7#52011yescharacteristic
356C22×C7C2×#12411yescharacteristic
357C22×C7#52912yes—
358C22×C7#53211yescharacteristic
359C22×C7#53411yescharacteristic
360C22×C7#53511yescharacteristic
361C22×C7#53611yescharacteristic
362C22×C7#53711yescharacteristic
363C22×C7#53812yes—
364C22×C7#53812yes—
365C22×C7#53912yes—
366C22×C7#53912yes—
367C22×C7#53912yes—
368C22×C7#54012yes—
369C22×C7#54011yescharacteristic
370C22×C7#54112yes—
371C22×C7#54211yescharacteristic
372C22×C7#54212yes—
373C22×C7#54212yes—
374C2×D7—42no—
375C2×D7—42no—
376C2×D7—44no—
377C2×D7—44no—
378C2×D7—42no—
379C2×D7—81no—
380C2×D7—81no—
381C2×D7—82no—
382C2×D7—82no—
383C2×D7—81no—
384C2×D7—81no—
385C4×C7—24no—
386C4×C7—44no—
387C4×C7—44no—
388C4×C7—81no—
389C4×C7—81no—
390Dic7—24no—
391Dic7—44no—
392Dic7—42no—
393Dic7—42no—
394Dic7—82no—
395C24—72no—
396C24—141no—
397C24—141no—
398C24—141no—
399C24—141no—
400C24—142no—
401C24—142no—
402C24—142no—
403C24—141no—
404C24—142no—
405C24—141no—
406C24C23×D711yescharacteristic
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—72no—
432C22×C4—74no—
433C22×C4—72no—
434C22×C4—74no—
435C22×C4—72no—
436C22×C4—141no—
437C22×C4—142no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—141no—
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—142no—
451C22×C4—141no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—142no—
459C22×C4—142no—
460C22×C4C2×C4×D714yes—
461C22×C4C2×C4×D712yes—
462C22×C4D4×D712yes—
463C22×C4#115912yes—
464C22×C4#116114yes—
465C4⋊C4—28no—
466C4⋊C4—284no—
467C4⋊C4—284no—
468C4⋊C4—288no—
469C22⋊C4—148no—
470C22⋊C4—288no—
471C22⋊C4—288no—
472C22⋊C4—284no—
473C22⋊C4—284no—
474C42—284no—
475C42—288no—
476C2×C7C4×#12411yescharacteristic
477C2×C7#232211yescharacteristic
478C2×C7#232411yescharacteristic
479C2×C7#233412yes—
480C2×C7#233512yes—
481C2×C7#263911yescharacteristic
482C2×C7#264011yescharacteristic
483C2×C7#264912yes—
484C2×C7#265412yes—
485C2×C7#266111yescharacteristic
486C2×C7#266411yescharacteristic
487D7—82no—
488C23—142no—
489C23—144no—
490C23—142no—
491C23—142no—
492C23—144no—
493C23—281no—
494C23—282no—
495C23—282no—
496C23—282no—
497C23—282no—
498C23—281no—
499C23—281no—
500C23—281no—
501C23—281no—
502C23—282no—
503C23—282no—
504C23—282no—
505C23—281no—
506C23—281no—
507C23—282no—
508C23—282no—
509C23—282no—
510C23—281no—
511C23C22×C4×D711yescharacteristic
512C23C2×D4×D711yescharacteristic
513C23C2×#115911yescharacteristic
514C23#654811yescharacteristic
515C23#654811yescharacteristic
516C23C2×#116112yes—
517C23D7×#4112yes—
518C23#655311yescharacteristic
519C23#655312yes—
520C23#655411yescharacteristic
521C23#655412yes—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—74no—
551C2×C4—144no—
552C2×C4—142no—
553C2×C4—142no—
554C2×C4—144no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—144no—
558C2×C4—144no—
559C2×C4—144no—
560C2×C4—144no—
561C2×C4—142no—
562C2×C4—142no—
563C2×C4—282no—
564C2×C4—282no—
565C2×C4—282no—
566C2×C4—281no—
567C2×C4—282no—
568C2×C4—282no—
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(C22⋊D7)⋊C414yes—
580C7#1432411yescharacteristic, complemented, Sylow
581C22—282no—
582C22—282no—
583C22—282no—
584C22—284no—
585C22—284no—
586C22—561no—
587C22—562no—
588C22—562no—
589C22—561no—
590C22—561no—
591C22—561no—
592C22C2×(C22⋊D7)⋊C411yescharacteristic
593C22#9021412yes—
594C22#9023611yescharacteristic
595C22#9023711yescharacteristic
596C22C4×#116112yes—
597C22#9029811yescharacteristic
598C22#9029911yescharacteristic
599C22#9031512yes—
600C22#9033512yes—
601C22#9033612yes—
602C22#9033911yescharacteristic
603C22#9034411yescharacteristic
604C22#9034812yes—
605C22#9035312yes—
606C22#9035612yes—
607C22D7×#12411yescharacteristic
608C22#9037311yescharacteristic
609C22#9038812yes—
610C22#9038912yes—
611C22#9039812yes—
612C22#9040112yes—
613C22#9041011yescharacteristic
614C22#9041611yescharacteristic
615C4—24no—
616C4—44no—
617C4—44no—
618C4—81no—
619C4—81no—
620C4—144no—
621C4—282no—
622C4—284no—
623C4—282no—
624C4—562no—
625C2—562no—
626C2#1173150111yescharacteristic
627C2#1173240012yes—
628C2#1173240112yes—
629C2#1173387511yescharacteristic
630C2#1173392511yescharacteristic
631C2#1173499211yescharacteristic
632C2#1173513211yescharacteristic
633C2#1173528311yescharacteristic
634C2#1173528511yescharacteristic
635C2#1173549912yes—
636C2#1173567212yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1445C2
socle, cosocleC24×C7C24
prime radicalC24×D7C23
hypercenter#1445D7
hypocenterC7#14324
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
2. C2×#89757C2semidirect
4. C7×#1445C2semidirect
7. #11727169C2semidirect
8. #11727169C2semidirect
10. #11727160C2semidirect
11. Dic7×#95C2semidirect
35. #89765C4semidirect
93. #1445D7semidirect
580. C7#14324semidirect

Maximal subgroups

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

subgroupindex
C2×#897652
C2×#897572
C2×#894412
C7×#14452
#117271812
#117271692
#117271622
#117271602
Dic7×#952
#143247