Finite Group Databasebeta

← #13280519

Subgroups — #13280519

Order 1792.

1566 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×#89759C211yesmaximal, characteristic
2C2×#89436C211yesmaximal, characteristic
3C2×#89438C211yesmaximal, characteristic
4C7×#1442C211yesmaximal, characteristic, complemented
5#11729322C211yesmaximal, characteristic, complemented
6#11727182C212yesmaximal
7#11727181C212yesmaximal
8#11727173C211yesmaximal, characteristic, complemented
9#11727169C211yesmaximal, characteristic, complemented
10#11727161C212yesmaximal, complemented
11#11727160C212yesmaximal, complemented
12C23×C4×D7C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×(C2×D7)⋊C4C2211yescharacteristic
15C22×(C2×D7)⋊C4C2211yescharacteristic
16C22×(C2×D7)⋊C4C2212yes—
17C22×(C2×D7)⋊C4C2212yes—
18C22×C4⋊Dic7C2212yes—
19C22×C4⋊Dic7C2211yescharacteristic
20C22×Dic7⋊C4C2212yes—
21C22×C4×Dic7C2211yescharacteristic
22C22×C4×Dic7C2211yescharacteristic
23C2×C7×#95C2212yes—
24C2×C7×#95C2211yescharacteristic
25C2×C7×#95C2212yes—
26C2×C7×#95C2211yescharacteristic
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2211yescharacteristic
35#89759—24no—
36#89438—24no—
37#89436C418yescomplemented
38#14295—71nomaximal, Sylow
39C24×D7C2311yescharacteristic
40C23×C4×C7C2312yes—
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×Dic7C2312yes—
46C23×Dic7C2312yes—
47C23×Dic7C2311yescharacteristic
48C23×Dic7C2311yescharacteristic
49C23×Dic7C2311yescharacteristic
50C22×C4×D7—42no—
51C22×C4×D7—42no—
52C22×C4×D7C2×C414yes—
53C22×C4×D7D414yes—
54C22×C4×D7Q814yes—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—42no—
57C2×(C2×D7)⋊C4—22no—
58C2×(C2×D7)⋊C4—22no—
59C2×(C2×D7)⋊C4—22no—
60C2×(C2×D7)⋊C4—22no—
61C2×(C2×D7)⋊C4—22no—
62C2×(C2×D7)⋊C4—22no—
63C2×(C2×D7)⋊C4—24no—
64C2×(C2×D7)⋊C4—42no—
65C2×(C2×D7)⋊C4—42no—
66C2×(C2×D7)⋊C4—42no—
67C2×(C2×D7)⋊C4—42no—
68C2×(C2×D7)⋊C4C2×C418yes—
69C2×C4⋊Dic7—22no—
70C2×C4⋊Dic7—24no—
71C2×C4⋊Dic7—42no—
72C2×C4⋊Dic7—42no—
73C2×C4⋊Dic7—42no—
74C2×Dic7⋊C4—24no—
75C2×Dic7⋊C4—42no—
76C2×Dic7⋊C4—42no—
77C2×C4×Dic7—22no—
78C2×C4×Dic7—22no—
79C2×C4×Dic7—42no—
80C2×C4×Dic7—42no—
81C7×#95—22no—
82C7×#95—24no—
83C7×#95—42no—
84C7×#95C2×C414yes—
85C7⋊#95—22no—
86C7⋊#95—24no—
87C7⋊#95—24no—
88C7⋊#95—42no—
89C7⋊#95—42no—
90C7⋊#95—42no—
91C7⋊#95C2×C414yes—
92C7⋊#95C2×C418yes—
93C2×#381—71no—
94C2×#380—71no—
95C2×C4×C22⋊C4—71no—
96#1444—71no—
97#1444—71no—
98#1442—71no—
99#1442D711yescharacteristic, complemented
100#1439—72no—
101#1439—72no—
102#1436—72no—
103#1436—72no—
104C24×C7C2411yescharacteristic
105C23×D7—21no—
106C23×D7—21no—
107C23×D7—21no—
108C23×D7—21no—
109C23×D7—22no—
110C23×D7—22no—
111C23×D7—22no—
112C23×D7—22no—
113C23×D7C22×C412yes—
114C23×D7C2×D412yes—
115C23×D7C2×Q812yes—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—21no—
119C22×C4×C7—21no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—21no—
123C22×C4×C7—21no—
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—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7C22×C414yes—
140C22×C4×C7C22×C412yes—
141C22×C4×C7C2×D412yes—
142C22×C4×C7C2×Q812yes—
143C22×C4×C7#4112yes—
144C22×C4×C7#4112yes—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—21no—
152C22×Dic7—21no—
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—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7C22×C414yes—
169C22×Dic7C22×C412yes—
170C22×Dic7C2×D412yes—
171C22×Dic7C2×Q812yes—
172C22×Dic7#4112yes—
173C22×Dic7#4112yes—
174C2×C4×D7—48no—
175C2×C4×D7—44no—
176C2×C4×D7—44no—
177C2×C4×D7—48no—
178C2×C4×D7C4⋊C4116yescomplemented
179C7×C4⋊C4—44no—
180(C2×D7)⋊C4—28no—
181(C2×D7)⋊C4—28no—
182(C2×D7)⋊C4—48no—
183(C2×D7)⋊C4—48no—
184C4⋊Dic7—48no—
185C4⋊Dic7—44no—
186Dic7⋊C4—48no—
187C4×Dic7—44no—
188C4×Dic7—44no—
189C24×C4—71no—
190C22×C4⋊C4—72no—
191C22×C4⋊C4—72no—
192C22×C4⋊C4—71no—
193C22×C4⋊C4C2×D711yescharacteristic
194C22×C22⋊C4—72no—
195C22×C22⋊C4—71no—
196C22×C22⋊C4—71no—
197C22×C22⋊C4—72no—
198C22×C42—71no—
199C22×C42—71no—
200#381—78no—
201#380—144no—
202C4×C22⋊C4—144no—
203C2×#95—72no—
204C2×#95—71no—
205C2×#95—72no—
206C2×#95—72no—
207C2×#95—71no—
208C2×#95—72no—
209C2×#95—72no—
210C2×#95—72no—
211C2×#95C2×D712yes—
212C2×#95C2×D711yescharacteristic
213C2×#95C2×D712yes—
214C2×#95C2×D711yescharacteristic
215C23×C7C23×C411yescharacteristic
216C23×C7C22×D411yescharacteristic
217C23×C7C22×Q811yescharacteristic
218C23×C7C2×#4111yescharacteristic
219C23×C7C2×#4111yescharacteristic
220C23×C7C2×#4111yescharacteristic
221C23×C7C2×#4111yescharacteristic
222C23×C7H(5,𝔽2)12yes—
223C23×C7H(5,𝔽2)12yes—
224C23×C7H(5,𝔽2)12yes—
225C23×C72−1+412yes—
226C22×D7—22no—
227C22×D7—22no—
228C22×D7—24no—
229C22×D7—24no—
230C22×D7—22no—
231C22×D7—22no—
232C22×D7—22no—
233C22×D7—22no—
234C22×D7—22no—
235C22×D7—22no—
236C22×D7—22no—
237C22×D7—22no—
238C22×D7—24no—
239C22×D7—24no—
240C22×D7—42no—
241C22×D7—42no—
242C22×D7—42no—
243C22×D7—42no—
244C22×D7—42no—
245C22×D7—42no—
246C22×D7—42no—
247C22×D7—42no—
248C22×D7C2×C4⋊C414yes—
249C2×C4×C7—24no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
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×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7C2×C4⋊C414yes—
278C2×Dic7—24no—
279C2×Dic7—22no—
280C2×Dic7—22no—
281C2×Dic7—24no—
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—22no—
286C2×Dic7—22no—
287C2×Dic7—24no—
288C2×Dic7—24no—
289C2×Dic7—41no—
290C2×Dic7—41no—
291C2×Dic7—41no—
292C2×Dic7—41no—
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—
304C2×Dic7—42no—
305C2×Dic7—42no—
306C2×Dic7C2×C4⋊C414yes—
307C4×D7—416no—
308C25—71no—
309C23×C4—71no—
310C23×C4—74no—
311C23×C4—74no—
312C23×C4—74no—
313C23×C4—71no—
314C23×C4—71no—
315C23×C4—72no—
316C23×C4—72no—
317C23×C4—282no—
318C23×C4—282no—
319C23×C4C22×D712yes—
320C23×C4C22×D712yes—
321C23×C4C22×D711yescharacteristic
322C23×C4C22×D711yescharacteristic
323C23×C4C22×D711yescharacteristic
324C2×C4⋊C4—22no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—144no—
327C2×C4⋊C4—144no—
328C2×C4⋊C4—142no—
329C2×C4⋊C4—282no—
330C2×C4⋊C4—282no—
331C2×C4⋊C4—282no—
332C2×C4⋊C4—282no—
333C2×C4⋊C4—282no—
334C2×C22⋊C4—78no—
335C2×C22⋊C4—142no—
336C2×C22⋊C4—142no—
337C2×C22⋊C4—142no—
338C2×C22⋊C4—142no—
339C2×C22⋊C4—142no—
340C2×C22⋊C4—142no—
341C2×C22⋊C4—144no—
342C2×C22⋊C4—282no—
343C2×C22⋊C4—282no—
344C2×C22⋊C4—282no—
345C2×C22⋊C4—282no—
346C2×C42—142no—
347C2×C42—142no—
348C2×C42—282no—
349C2×C42—282no—
350#95—22no—
351#95—24no—
352#95—42no—
353#95—78no—
354#95—74no—
355#95—144no—
356#95—142no—
357#95—144no—
358#95—282no—
359#95—282no—
360#95—282no—
361#95C4×D714yes—
362C22×C7C22×C4⋊C411yescharacteristic
363C22×C7C4×#4111yescharacteristic
364C22×C7C4×#4111yescharacteristic
365C22×C7#51812yes—
366C22×C7#52012yes—
367C22×C7#52511yescharacteristic
368C22×C7#52511yescharacteristic
369C22×C7#53311yescharacteristic
370C22×C7#53311yescharacteristic
371C22×C7#53411yescharacteristic
372C22×C7#53411yescharacteristic
373C22×C7#53511yescharacteristic
374C22×C7#53611yescharacteristic
375C22×C7#53812yes—
376C22×C7#53812yes—
377C22×C7#53912yes—
378C22×C7#53912yes—
379C22×C7#54012yes—
380C22×C7#54012yes—
381C22×C7#54212yes—
382C22×C7#54212yes—
383C22×C7#54312yes—
384C22×C7#54412yes—
385C2×D7—24no—
386C2×D7—24no—
387C2×D7—24no—
388C2×D7—44no—
389C2×D7—44no—
390C2×D7—42no—
391C2×D7—42no—
392C2×D7—42no—
393C2×D7—42no—
394C2×D7—44no—
395C2×D7—44no—
396C4×C7—24no—
397C4×C7—44no—
398C4×C7—42no—
399C4×C7—42no—
400C4×C7—82no—
401Dic7—24no—
402Dic7—44no—
403Dic7—42no—
404Dic7—42no—
405Dic7—82no—
406C24—72no—
407C24—72no—
408C24—72no—
409C24—141no—
410C24—142no—
411C24—141no—
412C24—141no—
413C24—141no—
414C24—142no—
415C24—142no—
416C24—142no—
417C24C23×D711yescharacteristic
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
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—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—72no—
442C22×C4—72no—
443C22×C4—72no—
444C22×C4—716no—
445C22×C4—74no—
446C22×C4—72no—
447C22×C4—72no—
448C22×C4—142no—
449C22×C4—142no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—141no—
457C22×C4—142no—
458C22×C4—142no—
459C22×C4—142no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—142no—
466C22×C4—142no—
467C22×C4—142no—
468C22×C4—142no—
469C22×C4—142no—
470C22×C4—142no—
471C22×C4—288no—
472C22×C4—284no—
473C22×C4—288no—
474C22×C4—284no—
475C22×C4C2×C4×D714yes—
476C22×C4C2×C4×D712yes—
477C22×C4D4×D712yes—
478C22×C4#115912yes—
479C22×C4Q8×D712yes—
480C22×C4#116112yes—
481C4⋊C4—44no—
482C4⋊C4—284no—
483C4⋊C4—288no—
484C4⋊C4—288no—
485C22⋊C4—148no—
486C22⋊C4—148no—
487C22⋊C4—288no—
488C22⋊C4—288no—
489C42—284no—
490C42—284no—
491C2×C7#229611yescharacteristic
492C2×C7#231511yescharacteristic
493C2×C7#231711yescharacteristic
494C2×C7#233512yes—
495C2×C7#233512yes—
496C2×C7#263511yescharacteristic
497C2×C7#263511yescharacteristic
498C2×C7#263811yescharacteristic
499C2×C7#263811yescharacteristic
500C2×C7#266512yes—
501C2×C7#266712yes—
502D7—44no—
503C23—74no—
504C23—142no—
505C23—144no—
506C23—142no—
507C23—144no—
508C23—142no—
509C23—144no—
510C23—142no—
511C23—142no—
512C23—142no—
513C23—142no—
514C23—144no—
515C23—142no—
516C23—142no—
517C23—142no—
518C23—282no—
519C23—282no—
520C23—282no—
521C23—282no—
522C23—282no—
523C23—282no—
524C23—282no—
525C23—282no—
526C23C22×C4×D711yescharacteristic
527C23C2×D4×D711yescharacteristic
528C23C2×#115911yescharacteristic
529C23#654812yes—
530C23C2×Q8×D711yescharacteristic
531C23C2×#116111yescharacteristic
532C23D7×#4111yescharacteristic
533C23D7×#4111yescharacteristic
534C23#655312yes—
535C23#655312yes—
536C23#655412yes—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—74no—
566C2×C4—144no—
567C2×C4—142no—
568C2×C4—144no—
569C2×C4—142no—
570C2×C4—144no—
571C2×C4—144no—
572C2×C4—144no—
573C2×C4—144no—
574C2×C4—142no—
575C2×C4—142no—
576C2×C4—144no—
577C2×C4—2816no—
578C2×C4—282no—
579C2×C4—282no—
580C2×C4—282no—
581C2×C4—281no—
582C2×C4—282no—
583C2×C4—282no—
584C2×C4—281no—
585C2×C4—282no—
586C2×C4—281no—
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×C4D7×C4⋊C414yes—
596C7#1429511yescharacteristic, complemented, Sylow
597C22—144no—
598C22—144no—
599C22—144no—
600C22—282no—
601C22—284no—
602C22—282no—
603C22—284no—
604C22—284no—
605C22—282no—
606C22—282no—
607C22—284no—
608C22C2×D7×C4⋊C411yescharacteristic
609C22#9021412yes—
610C22C4×#115911yescharacteristic
611C22#9023612yes—
612C22C4×#116111yescharacteristic
613C22#9028011yescharacteristic
614C22#9029711yescharacteristic
615C22#9031012yes—
616C22#9031811yescharacteristic
617C22#9032512yes—
618C22#9033612yes—
619C22#9034411yescharacteristic
620C22#9034812yes—
621C22#9035612yes—
622C22#9036012yes—
623C22#9036511yescharacteristic
624C22#9036611yescharacteristic
625C22#9037911yescharacteristic
626C22#9038012yes—
627C22#9038912yes—
628C22#9039512yes—
629C22#9040112yes—
630C22#9041911yescharacteristic
631C4—24no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—144no—
637C4—282no—
638C4—282no—
639C4—284no—
640C4—562no—
641C2—284no—
642C2#1173167011yescharacteristic
643C2#1173203611yescharacteristic
644C2#1173240112yes—
645C2#1173291512yes—
646C2#1173340811yescharacteristic
647C2#1173425911yescharacteristic
648C2#1173442511yescharacteristic
649C2#1173499011yescharacteristic
650C2#1173529012yes—
651C2#1173565711yescharacteristic
652C2#1173567912yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1442C2
socle, cosocleC24×C7C24
prime radicalC24×D7C23
hypercenter#1442D7
hypocenterC7#14295
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
4. C7×#1442C2semidirect
5. #11729322C2semidirect
8. #11727173C2semidirect
9. #11727169C2semidirect
10. #11727161C2semidirect
11. #11727160C2semidirect
37. #89436C4semidirect
99. #1442D7semidirect
178. C2×C4×D7C4⋊C4semidirect
596. C7#14295semidirect

Maximal subgroups

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

subgroupindex
C2×#897592
C2×#894362
C2×#894382
C7×#14422
#117293222
#117271822
#117271812
#117271732
#117271692
#117271612
#117271602
#142957