Finite Group Databasebeta

← #13276142

Subgroups — #13276142

Order 1792.

1586 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×C4×(C2×D7)⋊C4C211yesmaximal, characteristic
2C2×#89436C211yesmaximal, characteristic
3C2×#89422C211yesmaximal, characteristic, complemented
4C7×#1434C211yesmaximal, characteristic, complemented
5C4×C7⋊#95C211yesmaximal, characteristic, complemented
6#11727182C212yesmaximal
7(C2×D7)⋊#95C211yesmaximal, characteristic
8#11727171C212yesmaximal, complemented
9#11727162C212yesmaximal
10#11727160C212yesmaximal, complemented
11#11727156C211yesmaximal, characteristic, complemented
12C23×C4×D7C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×(C2×D7)⋊C4C2212yes—
15C22×(C2×D7)⋊C4C2212yes—
16C22×(C2×D7)⋊C4C2211yescharacteristic
17C22×(C2×D7)⋊C4C2211yescharacteristic
18C22×Dic7⋊C4C2212yes—
19C22×C4×Dic7C2211yescharacteristic
20C22×C4×Dic7C2212yes—
21C2×C7×#95C2211yescharacteristic
22C2×C7×#95C2212yes—
23C2×C7×#95C2211yescharacteristic
24C2×C7×#95C2212yes—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2211yescharacteristic
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2211yescharacteristic
33C2×C7⋊#95C2211yescharacteristic
34C2×C7⋊#95C2211yescharacteristic
35C4×(C2×D7)⋊C4C418yescomplemented
36#89436—24no—
37#89422—24no—
38#13917—71nomaximal, Sylow
39C24×D7C2311yescharacteristic
40C23×C4×C7C2312yes—
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×Dic7C2311yescharacteristic
46C23×Dic7C2311yescharacteristic
47C23×Dic7C2312yes—
48C23×Dic7C2312yes—
49C23×Dic7C2311yescharacteristic
50C22×C4×D7—24no—
51C22×C4×D7—22no—
52C22×C4×D7—22no—
53C22×C4×D7—42no—
54C22×C4×D7C2×C414yes—
55C2×C42×C7—42no—
56C2×C42×C7C2×C414yes—
57C2×(C2×D7)⋊C4—24no—
58C2×(C2×D7)⋊C4—22no—
59C2×(C2×D7)⋊C4—22no—
60C2×(C2×D7)⋊C4—24no—
61C2×(C2×D7)⋊C4—24no—
62C2×(C2×D7)⋊C4—24no—
63C2×(C2×D7)⋊C4—24no—
64C2×(C2×D7)⋊C4—42no—
65C2×(C2×D7)⋊C4—42no—
66C2×(C2×D7)⋊C4C2×C414yes—
67C2×(C2×D7)⋊C4C2×C414yes—
68C2×Dic7⋊C4—24no—
69C2×Dic7⋊C4—24no—
70C2×Dic7⋊C4—42no—
71C2×C4×Dic7—22no—
72C2×C4×Dic7—22no—
73C2×C4×Dic7—24no—
74C2×C4×Dic7—42no—
75C2×C4×Dic7—42no—
76C2×C4×Dic7C2×C414yes—
77C7×#95—22no—
78C7×#95—22no—
79C7×#95—24no—
80C7×#95—42no—
81C7⋊#95—22no—
82C7⋊#95—24no—
83C7⋊#95—24no—
84C7⋊#95—24no—
85C7⋊#95—24no—
86C7⋊#95—24no—
87C7⋊#95—42no—
88C7⋊#95C2×C414yes—
89C7⋊#95C2×C414yes—
90C7⋊#95D414yes—
91C2×#382—71no—
92C2×#381—71no—
93C2×C4×C22⋊C4—71no—
94#1443—72no—
95#1439—72no—
96#1437—72no—
97#1436—72no—
98#1434—71no—
99#1434D711yescharacteristic, complemented
100C22⋊#95—71no—
101C4×#95—71no—
102C24×C7C2411yescharacteristic
103C23×D7—21no—
104C23×D7—22no—
105C23×D7—21no—
106C23×D7—21no—
107C23×D7—22no—
108C23×D7—21no—
109C23×D7—21no—
110C23×D7—22no—
111C23×D7—21no—
112C23×D7C22×C412yes—
113C23×D7#4114yes—
114C22×C4×C7—22no—
115C22×C4×C7—21no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
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×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7C22×C412yes—
138C22×C4×C7C22×C412yes—
139C22×C4×C7C22×C412yes—
140C22×C4×C7C2×D412yes—
141C22×C4×C7#4114yes—
142C22×C4×C7#4112yes—
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—21no—
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—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7C22×C412yes—
168C22×Dic7C22×C412yes—
169C22×Dic7C22×C412yes—
170C22×Dic7C2×D414yes—
171C22×Dic7#4114yes—
172C22×Dic7#4112yes—
173C22×Dic7#4112yes—
174C2×C4×D7—28no—
175C2×C4×D7—28no—
176C2×C4×D7—44no—
177C2×C4×D7—48no—
178C2×C4×D7—48no—
179C42×C7—44no—
180(C2×D7)⋊C4—28no—
181(C2×D7)⋊C4—28no—
182(C2×D7)⋊C4—48no—
183(C2×D7)⋊C4—48no—
184Dic7⋊C4—48no—
185C4×Dic7—28no—
186C4×Dic7—48no—
187C24×C4—71no—
188C22×C4⋊C4—72no—
189C22×C22⋊C4—71no—
190C22×C22⋊C4—71no—
191C22×C22⋊C4—72no—
192C22×C22⋊C4—72no—
193C22×C42—71no—
194C22×C42—72no—
195C22×C42C2×D711yescharacteristic
196#382—144no—
197#381—144no—
198C4×C22⋊C4—78no—
199C2×#95—71no—
200C2×#95—71no—
201C2×#95—71no—
202C2×#95—71no—
203C2×#95—72no—
204C2×#95—72no—
205C2×#95—72no—
206C2×#95—72no—
207C2×#95—72no—
208C2×#95—72no—
209C2×#95C2×D711yescharacteristic
210C2×#95C2×D711yescharacteristic
211C2×#95C2×D712yes—
212C2×#95C2×D712yes—
213C23×C7C23×C411yescharacteristic
214C23×C7C22×D411yescharacteristic
215C23×C7C2×#4112yes—
216C23×C7C2×#4111yescharacteristic
217C23×C7C2×#4112yes—
218C23×C7C2×#4111yescharacteristic
219C23×C7C2×#4111yescharacteristic
220C23×C7C2×#4112yes—
221C23×C7H(5,𝔽2)12yes—
222C23×C7H(5,𝔽2)11yescharacteristic
223C23×C72−1+411yescharacteristic
224C22×D7—22no—
225C22×D7—22no—
226C22×D7—22no—
227C22×D7—22no—
228C22×D7—22no—
229C22×D7—22no—
230C22×D7—24no—
231C22×D7—24no—
232C22×D7—24no—
233C22×D7—24no—
234C22×D7—24no—
235C22×D7—24no—
236C22×D7—42no—
237C22×D7—42no—
238C22×D7—42no—
239C22×D7—42no—
240C22×D7—41no—
241C22×D7—41no—
242C22×D7—41no—
243C22×D7—41no—
244C22×D7—42no—
245C22×D7—42no—
246C22×D7#11714yes—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
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×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#11714yes—
275C2×C4×C7C4×D414yes—
276C2×Dic7—22no—
277C2×Dic7—22no—
278C2×Dic7—22no—
279C2×Dic7—22no—
280C2×Dic7—22no—
281C2×Dic7—22no—
282C2×Dic7—22no—
283C2×Dic7—22no—
284C2×Dic7—22no—
285C2×Dic7—22no—
286C2×Dic7—22no—
287C2×Dic7—22no—
288C2×Dic7—24no—
289C2×Dic7—24no—
290C2×Dic7—24no—
291C2×Dic7—24no—
292C2×Dic7—24no—
293C2×Dic7—24no—
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×Dic7#11714yes—
307C4×D7—416no—
308C25—71no—
309C23×C4—71no—
310C23×C4—72no—
311C23×C4—71no—
312C23×C4—72no—
313C23×C4—71no—
314C23×C4—74no—
315C23×C4—144no—
316C23×C4—142no—
317C23×C4—142no—
318C23×C4—282no—
319C23×C4C22×D712yes—
320C23×C4C22×D712yes—
321C23×C4C22×D711yescharacteristic
322C23×C4C22×D711yescharacteristic
323C23×C4C22×D711yescharacteristic
324C2×C4⋊C4—144no—
325C2×C4⋊C4—144no—
326C2×C4⋊C4—282no—
327C2×C22⋊C4—74no—
328C2×C22⋊C4—74no—
329C2×C22⋊C4—142no—
330C2×C22⋊C4—142no—
331C2×C22⋊C4—144no—
332C2×C22⋊C4—144no—
333C2×C22⋊C4—144no—
334C2×C22⋊C4—144no—
335C2×C22⋊C4—144no—
336C2×C22⋊C4—282no—
337C2×C22⋊C4—282no—
338C2×C42—42no—
339C2×C42—74no—
340C2×C42—142no—
341C2×C42—142no—
342C2×C42—144no—
343C2×C42—282no—
344C2×C42—282no—
345C2×C42C4×D714yes—
346#95—24no—
347#95—22no—
348#95—22no—
349#95—42no—
350#95—74no—
351#95—74no—
352#95—74no—
353#95—144no—
354#95—144no—
355#95—144no—
356#95—144no—
357#95—144no—
358#95—142no—
359#95—282no—
360C22×C7C2×#11711yescharacteristic
361C22×C7C2×#11711yescharacteristic
362C22×C7C2×#11711yescharacteristic
363C22×C7C2×C4×D411yescharacteristic
364C22×C7C4×#4111yescharacteristic
365C22×C7#51811yescharacteristic
366C22×C7#52011yescharacteristic
367C22×C7#52912yes—
368C22×C7#52912yes—
369C22×C7#52912yes—
370C22×C7#53212yes—
371C22×C7#53812yes—
372C22×C7#53912yes—
373C22×C7#54612yes—
374C22×C7#54611yescharacteristic
375C22×C7#54811yescharacteristic
376C22×C7#55112yes—
377C22×C7#55112yes—
378C22×C7#55211yescharacteristic
379C22×C7#55212yes—
380C22×C7#55311yescharacteristic
381C22×C7#55312yes—
382C22×C7#55912yes—
383C2×D7—24no—
384C2×D7—24no—
385C2×D7—24no—
386C2×D7—42no—
387C2×D7—44no—
388C2×D7—44no—
389C2×D7—42no—
390C2×D7—42no—
391C2×D7—42no—
392C2×D7—44no—
393C2×D7—44no—
394C4×C7—24no—
395C4×C7—24no—
396C4×C7—42no—
397C4×C7—82no—
398C4×C7—82no—
399Dic7—24no—
400Dic7—44no—
401Dic7—42no—
402Dic7—42no—
403Dic7—44no—
404C24—72no—
405C24—74no—
406C24—142no—
407C24—141no—
408C24—142no—
409C24—141no—
410C24—142no—
411C24—141no—
412C24—141no—
413C24—141no—
414C24—141no—
415C24C23×D711yescharacteristic
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
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—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—72no—
440C22×C4—72no—
441C22×C4—72no—
442C22×C4—72no—
443C22×C4—72no—
444C22×C4—74no—
445C22×C4—74no—
446C22×C4—148no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—142no—
459C22×C4—142no—
460C22×C4—141no—
461C22×C4—141no—
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—148no—
472C22×C4—284no—
473C22×C4—288no—
474C22×C4—288no—
475C22×C4C2×C4×D712yes—
476C22×C4C2×C4×D712yes—
477C22×C4C2×C4×D712yes—
478C22×C4#115714yes—
479C22×C4D4×D712yes—
480C22×C4#116112yes—
481C4⋊C4—288no—
482C22⋊C4—148no—
483C22⋊C4—148no—
484C22⋊C4—288no—
485C22⋊C4—288no—
486C42—44no—
487C42—148no—
488C42—288no—
489C2×C7#228311yescharacteristic
490C2×C7#229311yescharacteristic
491C2×C7#231811yescharacteristic
492C2×C7#234211yescharacteristic
493C2×C7#234311yescharacteristic
494C2×C7#235211yescharacteristic
495C2×C7#235711yescharacteristic
496C2×C7#255712yes—
497C2×C7#255812yes—
498C2×C7#257512yes—
499C2×C7#257612yes—
500D7—44no—
501C23—74no—
502C23—144no—
503C23—144no—
504C23—142no—
505C23—142no—
506C23—142no—
507C23—142no—
508C23—142no—
509C23—142no—
510C23—144no—
511C23—144no—
512C23—144no—
513C23—144no—
514C23—281no—
515C23—281no—
516C23—281no—
517C23—282no—
518C23—282no—
519C23—282no—
520C23—282no—
521C23—282no—
522C23—282no—
523C23—281no—
524C23C22×C4×D711yescharacteristic
525C23C2×#115712yes—
526C23C2×D4×D711yescharacteristic
527C23#654812yes—
528C23C2×#116111yescharacteristic
529C23D7×#4111yescharacteristic
530C23D7×#4111yescharacteristic
531C23D7×#4112yes—
532C23D7×#4112yes—
533C23#655311yescharacteristic
534C23#655411yescharacteristic
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—74no—
563C2×C4—142no—
564C2×C4—142no—
565C2×C4—142no—
566C2×C4—142no—
567C2×C4—142no—
568C2×C4—142no—
569C2×C4—144no—
570C2×C4—142no—
571C2×C4—142no—
572C2×C4—142no—
573C2×C4—142no—
574C2×C4—142no—
575C2×C4—142no—
576C2×C4—144no—
577C2×C4—144no—
578C2×C4—144no—
579C2×C4—144no—
580C2×C4—144no—
581C2×C4—2816no—
582C2×C4—282no—
583C2×C4—282no—
584C2×C4—282no—
585C2×C4—282no—
586C2×C4—282no—
587C2×C4—282no—
588C2×C4—282no—
589C2×C4—282no—
590C2×C4—282no—
591C2×C4—282no—
592C2×C4—282no—
593C2×C4—282no—
594C2×C4#643514yes—
595C2×C4#645614yes—
596C7#1391711yescharacteristic, complemented, Sylow
597C22—144no—
598C22—144no—
599C22—144no—
600C22—282no—
601C22—282no—
602C22—284no—
603C22—284no—
604C22—284no—
605C22—284no—
606C22—282no—
607C22—282no—
608C22C2×#643511yescharacteristic
609C22#9017711yescharacteristic
610C22C2×#645611yescharacteristic
611C22D7×#11711yescharacteristic
612C22D7×#11711yescharacteristic
613C22#9021311yescharacteristic
614C22#9021411yescharacteristic
615C22#9021912yes—
616C22#9021912yes—
617C22#9023912yes—
618C22#9025112yes—
619C22#9025512yes—
620C22#9027112yes—
621C22#9030112yes—
622C22#9032812yes—
623C22#9035212yes—
624C22#9035512yes—
625C22#9036711yescharacteristic
626C22#9036811yescharacteristic
627C22#9038511yescharacteristic
628C22#9038611yescharacteristic
629C22#9039612yes—
630C22#9039912yes—
631C4—24no—
632C4—24no—
633C4—42no—
634C4—82no—
635C4—82no—
636C4—144no—
637C4—284no—
638C4—284no—
639C4—282no—
640C4—282no—
641C2—284no—
642C2#1173085811yescharacteristic
643C2#1173101311yescharacteristic
644C2#1173106211yescharacteristic
645C2#1173145411yescharacteristic
646C2#1173150611yescharacteristic
647C2#1173167411yescharacteristic
648C2#1173174111yescharacteristic
649C2#1173280912yes—
650C2#1173281112yes—
651C2#1173332912yes—
652C2#1173333012yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1434C2
socle, cosocleC24×C7C24
prime radicalC24×D7C23
hypercenter#1434D7
hypocenterC7#13917
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
3. C2×#89422C2semidirect
4. C7×#1434C2semidirect
5. C4×C7⋊#95C2semidirect
8. #11727171C2semidirect
10. #11727160C2semidirect
11. #11727156C2semidirect
35. C4×(C2×D7)⋊C4C4semidirect
99. #1434D7semidirect
596. C7#13917semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C2×D7)⋊C42
C2×#894362
C2×#894222
C7×#14342
C4×C7⋊#952
#117271822
(C2×D7)⋊#952
#117271712
#117271622
#117271602
#117271562
#139177