Finite Group Databasebeta

← #13526550

Subgroups — #13526550

Order 1792.

2214 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2315C211yesmaximal, characteristic, complemented
2#11739566C211yesmaximal, characteristic
3#11739566C211yesmaximal, characteristic
4#11739539C211yesmaximal, characteristic, complemented
5C2×#89730C211yesmaximal, characteristic
6C2×#89718C211yesmaximal, characteristic, complemented
7C2×C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
8#11732070C214yesmaximal, complemented
9#11732032C214yesmaximal, complemented
10#11731904C214yesmaximal, complemented
11#11731898C214yesmaximal, complemented
12#11731373C214yesmaximal, complemented
13#11731355C214yesmaximal, complemented
14C23×C22⋊D7C2211yescharacteristic
15C24×Dic7C2211yescharacteristic
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C7×C22⋊C4C2211yescharacteristic
18C22×C7×C22⋊C4C2211yescharacteristic
19C22×C22⋊Dic7C2211yescharacteristic
20C22×C22⋊Dic7C2211yescharacteristic
21C22×(C2×D7)⋊C4C2212yes—
22C22×Dic7⋊C4C2212yescomplemented
23C22×C4×Dic7C2212yescomplemented
24C2×#6461C2214yescomplemented
25C7×#385C2214yes—
26C7×#385C2214yescomplemented
27C7×#380C2212yescomplemented
28C7×#380C2212yescomplemented
29#89985C2214yes—
30#89985C2212yes—
31#89985C2212yes—
32#89984C2214yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yescomplemented
35C2×C7⋊#95C2212yescomplemented
36#89766C2214yescomplemented
37#89765C2214yescomplemented
38#89762C2214yescomplemented
39#89758C2214yescomplemented
40#89754C2214yescomplemented
41#89753C2214yescomplemented
42#89750C2214yescomplemented
43#89730C2212yes—
44#89730C2214yes—
45#89730C2212yes—
46#89730C2214yes—
47#89730C2214yes—
48#89727C2212yes—
49#89727C2212yes—
50#89722C2214yescomplemented
51#89722C2214yescomplemented
52#89718C2214yes—
53#89718C2212yescomplemented
54#89718C2212yescomplemented
55#89715C2212yes—
56#89715C2212yes—
57C22⋊C4×Dic7C2214yes—
58C22⋊C4×Dic7C2214yescomplemented
59#89447C2214yescomplemented
60#89446C2214yescomplemented
61#89442C2214yescomplemented
62#89441C2214yescomplemented
63C7⋊#401C2214yescomplemented
64#89432C2214yescomplemented
65#89426C2214yescomplemented
66#89423C2214yescomplemented
67#34995—71nomaximal, Sylow
68C25×C7C2311yescharacteristic
69C24×D7C2311yescharacteristic
70C23×C4×C7C2312yes—
71C23×C4×C7C2311yescharacteristic
72C23×C4×C7C2312yes—
73C23×C4×C7C2311yescharacteristic
74C22×C22⋊D7—24no—
75C22×C22⋊D7—24no—
76C22×C22⋊D7—24no—
77C22×C22⋊D7C2312yes—
78C22×C22⋊D7C2312yes—
79C23×Dic7—24no—
80C23×Dic7—44no—
81C23×Dic7C2311yescharacteristic
82C23×Dic7C2311yescharacteristic
83C23×Dic7C2312yes—
84C23×Dic7C2312yes—
85C23×Dic7C2312yes—
86C23×Dic7C2312yes—
87C23×Dic7C2311yescharacteristic
88C2×C7×C4⋊C4C2314yes—
89C2×C7×C4⋊C4C2314yes—
90C2×C7×C22⋊C4—24no—
91C2×C7×C22⋊C4—24no—
92C2×C7×C22⋊C4—24no—
93C2×C7×C22⋊C4C2312yes—
94C2×C7×C22⋊C4C2312yes—
95C2×C7×C22⋊C4C2312yes—
96C2×C7×C22⋊C4C2312yes—
97C2×C7×C22⋊C4C2312yes—
98C2×C7×C22⋊C4C2312yes—
99C2×C22⋊Dic7—24no—
100C2×C22⋊Dic7—24no—
101C2×C22⋊Dic7C2312yes—
102C2×C22⋊Dic7C2312yes—
103C2×C22⋊Dic7C2312yes—
104C2×C22⋊Dic7C2312yes—
105C2×(C2×D7)⋊C4—44no—
106C2×(C2×D7)⋊C4C2314yes—
107C2×(C2×D7)⋊C4C2314yes—
108C2×(C2×D7)⋊C4C2314yes—
109C2×(C2×D7)⋊C4C2314yes—
110C2×(C2×D7)⋊C4C2314yes—
111C2×(C2×D7)⋊C4C2314yes—
112C2×C4⋊Dic7C2314yes—
113C2×C4⋊Dic7C2314yes—
114C2×Dic7⋊C4—44no—
115C2×Dic7⋊C4C2314yes—
116C2×Dic7⋊C4C2314yes—
117C2×C4×Dic7—44no—
118C2×C4×Dic7C2314yes—
119C2×C4×Dic7C2314yes—
120#6461—48no—
121C7×#95C2314yes—
122C7×#95C2314yes—
123C7⋊#95C2314yes—
124C7⋊#95C2314yes—
125C7⋊#95C2314yes—
126C7⋊#95C2314yes—
127C7⋊#95C2314yes—
128C7⋊#95C2314yes—
129C7⋊#95C2314yes—
130C7⋊#95C2314yes—
131C7⋊#95C2314yes—
132C7⋊#95C2314yes—
133C7⋊#95C2314yes—
134C7⋊#95C2314yes—
135#2852—74no—
136#2643—74no—
137#2510—74no—
138#2493—74no—
139#2481—74no—
140#2480—74no—
141#2315—71no—
142#2315D711yescharacteristic, complemented
143#2308—71no—
144#2308—71no—
145C2×#386—71no—
146C2×#385—71no—
147C2×C4×C22⋊C4—71no—
148C24×C7—42no—
149C24×C7—42no—
150C24×C7#4114yes—
151C24×C7#4114yes—
152C24×C7#4114yes—
153C24×C7C2411yescharacteristic
154C24×C7C2411yescharacteristic
155C24×C7C2411yescharacteristic
156C23×D7—22no—
157C23×D7—22no—
158C23×D7—22no—
159C23×D7—42no—
160C23×D7—42no—
161C23×D7C2412yes—
162C22×C4×C7—24no—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7—24no—
166C22×C4×C7—42no—
167C22×C4×C7—42no—
168C22×C4×C7—42no—
169C22×C4×C7—42no—
170C22×C4×C7—42no—
171C22×C4×C7—42no—
172C22×C4×C7C2412yes—
173C22×C4×C7C2414yes—
174C22×C4×C7C2414yes—
175C22×C4×C7C2412yes—
176C2×C22⋊D7—24no—
177C2×C22⋊D7—24no—
178C2×C22⋊D7—24no—
179C2×C22⋊D7—24no—
180C2×C22⋊D7—24no—
181C2×C22⋊D7—24no—
182C2×C22⋊D7—416no—
183C22×Dic7—22no—
184C22×Dic7—24no—
185C22×Dic7—24no—
186C22×Dic7—24no—
187C22×Dic7—24no—
188C22×Dic7—24no—
189C22×Dic7—22no—
190C22×Dic7—22no—
191C22×Dic7—22no—
192C22×Dic7—22no—
193C22×Dic7—24no—
194C22×Dic7—24no—
195C22×Dic7—42no—
196C22×Dic7—42no—
197C22×Dic7—42no—
198C22×Dic7—42no—
199C22×Dic7—42no—
200C22×Dic7—42no—
201C22×Dic7—42no—
202C22×Dic7—416no—
203C22×Dic7C2412yes—
204C22×Dic7C2412yes—
205C22×Dic7C2414yes—
206C22×Dic7C2414yes—
207C22×Dic7C2412yes—
208C7×C4⋊C4—44no—
209C7×C4⋊C4—44no—
210C7×C22⋊C4—24no—
211C7×C22⋊C4—24no—
212C7×C22⋊C4—24no—
213C7×C22⋊C4—24no—
214C7×C22⋊C4—24no—
215C7×C22⋊C4—24no—
216C22⋊Dic7—24no—
217C22⋊Dic7—24no—
218C22⋊Dic7—24no—
219C22⋊Dic7—24no—
220(C2×D7)⋊C4—28no—
221(C2×D7)⋊C4—44no—
222(C2×D7)⋊C4—44no—
223(C2×D7)⋊C4—44no—
224(C2×D7)⋊C4—44no—
225(C2×D7)⋊C4—44no—
226C4⋊Dic7—44no—
227C4⋊Dic7—44no—
228Dic7⋊C4—28no—
229Dic7⋊C4—44no—
230C4×Dic7—28no—
231C4×Dic7—44no—
232C23×D4—71no—
233C24×C4—71no—
234C2×#126—74no—
235C22×C4⋊C4—72no—
236C22×C22⋊C4—71no—
237C22×C22⋊C4—71no—
238C22×C22⋊C4—72no—
239C22×C22⋊C4C2×D711yescharacteristic
240C22×C22⋊C4C2×D711yescharacteristic
241C22×C22⋊C4C2×D711yescharacteristic
242C22×C42—72no—
243#401—74no—
244#400—74no—
245#400—74no—
246#400—74no—
247#397—74no—
248#397—74no—
249#397—74no—
250#396—74no—
251#388—74no—
252#388—74no—
253#388—74no—
254#387—74no—
255#387—74no—
256#387—74no—
257#386—74no—
258#386—72no—
259#386—72no—
260#386—74no—
261#386—74no—
262#386—72no—
263#386—72no—
264#386—74no—
265#385—74no—
266#385—74no—
267#385—72no—
268#385—72no—
269#385—74no—
270#385C2×D714yes—
271#385C2×D714yescomplemented
272#383—74no—
273#381—72no—
274#381—72no—
275#380—74no—
276#380C2×D712yescomplemented
277#380C2×D712yescomplemented
278#379—72no—
279#379—72no—
280C4×C22⋊C4—74no—
281C4×C22⋊C4—74no—
282C2×#95—72no—
283C2×#95—72no—
284C2×#95—72no—
285C23×C7—22no—
286C23×C7—22no—
287C23×C7—22no—
288C23×C7—22no—
289C23×C7—22no—
290C23×C7—22no—
291C23×C7—44no—
292C23×C7—44no—
293C23×C7—48no—
294C23×C7—48no—
295C23×C7#126116yescomplemented
296C23×C7C2×#4112yes—
297C23×C7C2×#4112yes—
298C23×C7C2×#4112yes—
299C23×C7C2×#4112yes—
300C23×C7C2×#4112yes—
301C23×C7C2×#4112yes—
302C23×C7C2×#4112yes—
303C23×C7C2×#4112yes—
304C23×C7C2×#4112yes—
305C23×C7C2511yescharacteristic
306C22×D7—22no—
307C22×D7—22no—
308C22×D7—22no—
309C22×D7—24no—
310C22×D7—24no—
311C22×D7—48no—
312C22×D7—44no—
313C22×D7—82no—
314C22×D7—82no—
315C22×D7—84no—
316C2×C4×C7—22no—
317C2×C4×C7—22no—
318C2×C4×C7—24no—
319C2×C4×C7—24no—
320C2×C4×C7—24no—
321C2×C4×C7—24no—
322C2×C4×C7—24no—
323C2×C4×C7—24no—
324C2×C4×C7—24no—
325C2×C4×C7—24no—
326C2×C4×C7—48no—
327C2×C4×C7—82no—
328C2×C4×C7—82no—
329C2×C4×C7—84no—
330C22⋊D7—48no—
331C22⋊D7—48no—
332C2×Dic7—28no—
333C2×Dic7—22no—
334C2×Dic7—22no—
335C2×Dic7—24no—
336C2×Dic7—24no—
337C2×Dic7—22no—
338C2×Dic7—22no—
339C2×Dic7—22no—
340C2×Dic7—22no—
341C2×Dic7—24no—
342C2×Dic7—24no—
343C2×Dic7—22no—
344C2×Dic7—22no—
345C2×Dic7—24no—
346C2×Dic7—24no—
347C2×Dic7—48no—
348C2×Dic7—48no—
349C2×Dic7—48no—
350C2×Dic7—48no—
351C2×Dic7—44no—
352C2×Dic7—44no—
353C2×Dic7H(5,𝔽2)14yes—
354C25—71no—
355C25C22×D711yescharacteristic
356C22×D4—72no—
357C22×D4—72no—
358C22×D4—144no—
359C22×D4—144no—
360C22×D4—144no—
361C23×C4—71no—
362C23×C4—71no—
363C23×C4—72no—
364C23×C4—72no—
365C23×C4—72no—
366C23×C4—72no—
367C23×C4—71no—
368C23×C4—144no—
369C23×C4—284no—
370C23×C4C22×D712yes—
371C23×C4C22×D711yescharacteristic
372C23×C4C22×D711yescharacteristic
373C23×C4C22×D712yes—
374#126—288no—
375C2×C4⋊C4—74no—
376C2×C4⋊C4—74no—
377C2×C4⋊C4—74no—
378C2×C4⋊C4—74no—
379C2×C4⋊C4—284no—
380C2×C4⋊C4C22×D714yes—
381C2×C4⋊C4C22×D714yes—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—72no—
386C2×C22⋊C4—72no—
387C2×C22⋊C4—72no—
388C2×C22⋊C4—72no—
389C2×C22⋊C4—74no—
390C2×C22⋊C4—74no—
391C2×C22⋊C4—74no—
392C2×C22⋊C4—74no—
393C2×C22⋊C4—74no—
394C2×C22⋊C4—74no—
395C2×C22⋊C4—144no—
396C2×C22⋊C4—144no—
397C2×C22⋊C4—284no—
398C2×C22⋊C4C22×D712yes—
399C2×C22⋊C4C22×D712yes—
400C2×C22⋊C4C22×D712yes—
401C2×C22⋊C4C22×D712yes—
402C2×C22⋊C4C22×D712yes—
403C2×C22⋊C4C22×D712yes—
404C2×C42—74no—
405C2×C42—74no—
406C2×C42—284no—
407#95—74no—
408#95—74no—
409#95—74no—
410#95—74no—
411#95—74no—
412#95—74no—
413#95—74no—
414#95—74no—
415#95—74no—
416#95—74no—
417#95—74no—
418#95—74no—
419#95C22×D714yes—
420#95C22×D714yes—
421C22×C7—24no—
422C22×C7—24no—
423C22×C7—24no—
424C22×C7—22no—
425C22×C7—22no—
426C22×C7—22no—
427C22×C7—22no—
428C22×C7—22no—
429C22×C7—22no—
430C22×C7—24no—
431C22×C7—24no—
432C22×C7—24no—
433C22×C7—416no—
434C22×C7C2×#12614yes—
435C22×C7C2×#12614yes—
436C22×C7C2×#12614yes—
437C22×C7C22×#4111yescharacteristic
438C22×C7C22×#4111yescharacteristic
439C22×C7C22×#4111yescharacteristic
440C22×C7C2×H(5,𝔽2)12yes—
441C22×C7#58512yes—
442C2×D7—42no—
443C2×D7—44no—
444C2×D7—42no—
445C2×D7—42no—
446C2×D7—44no—
447C2×D7—88no—
448C4×C7—42no—
449C4×C7—42no—
450C4×C7—44no—
451C4×C7—44no—
452Dic7—24no—
453Dic7—42no—
454Dic7—44no—
455Dic7—44no—
456Dic7—42no—
457C24—42no—
458C24—42no—
459C24—72no—
460C24—142no—
461C24—142no—
462C24—142no—
463C24—282no—
464C24—282no—
465C24#115714yes—
466C24#115914yes—
467C24#116114yes—
468C24C23×D711yescharacteristic
469C24C23×D711yescharacteristic
470C24C23×D711yescharacteristic
471C2×D4—144no—
472C2×D4—144no—
473C2×D4—144no—
474C2×D4—144no—
475C2×D4—144no—
476C2×D4—144no—
477C2×D4—2816no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—72no—
489C22×C4—72no—
490C22×C4—72no—
491C22×C4—74no—
492C22×C4—74no—
493C22×C4—144no—
494C22×C4—144no—
495C22×C4—144no—
496C22×C4—142no—
497C22×C4—144no—
498C22×C4—144no—
499C22×C4—142no—
500C22×C4—142no—
501C22×C4—142no—
502C22×C4—144no—
503C22×C4—144no—
504C22×C4—142no—
505C22×C4—282no—
506C22×C4—282no—
507C22×C4—282no—
508C22×C4—282no—
509C22×C4—282no—
510C22×C4—282no—
511C22×C4—282no—
512C22×C4—2816no—
513C22×C4C23×D712yes—
514C22×C4C23×D714yes—
515C22×C4C23×D714yes—
516C22×C4C23×D712yes—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—148no—
520C4⋊C4—284no—
521C4⋊C4—284no—
522C4⋊C4—284no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—148no—
530C22⋊C4—144no—
531C22⋊C4—144no—
532C22⋊C4—144no—
533C22⋊C4—144no—
534C22⋊C4—284no—
535C22⋊C4—284no—
536C22⋊C4—284no—
537C22⋊C4—284no—
538C22⋊C4—284no—
539C42—148no—
540C42—284no—
541C2×C7—24no—
542C2×C7—24no—
543C2×C7—24no—
544C2×C7C22×#12611yescharacteristic
545C2×C7#345912yes—
546C2×C7#349011yescharacteristic
547C2×C7#349111yescharacteristic
548C2×C7#350512yes—
549D7—82no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—44no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—142no—
561C23—142no—
562C23—142no—
563C23—144no—
564C23—144no—
565C23—288no—
566C23—284no—
567C23—564no—
568C23—562no—
569C23—562no—
570C23#6461116yescomplemented
571C23C2×#115712yes—
572C23C2×#115712yes—
573C23C2×#115712yes—
574C23C2×#115912yes—
575C23C2×#115912yes—
576C23C2×#115912yes—
577C23C2×#116112yes—
578C23C2×#116112yes—
579C23C2×#116112yes—
580C23C24×D711yescharacteristic
581D4—288no—
582D4—288no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—48no—
594C2×C4—74no—
595C2×C4—84no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—148no—
599C2×C4—142no—
600C2×C4—142no—
601C2×C4—142no—
602C2×C4—142no—
603C2×C4—142no—
604C2×C4—144no—
605C2×C4—144no—
606C2×C4—142no—
607C2×C4—144no—
608C2×C4—142no—
609C2×C4—144no—
610C2×C4—142no—
611C2×C4—144no—
612C2×C4—144no—
613C2×C4—288no—
614C2×C4—288no—
615C2×C4—288no—
616C2×C4—288no—
617C2×C4—284no—
618C2×C4—284no—
619C7#3499511yescharacteristic, complemented, Sylow
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—416no—
633C22—282no—
634C22—284no—
635C22—284no—
636C22—282no—
637C22—282no—
638C22—568no—
639C22C2×#646114yes—
640C22C2×#646114yes—
641C22C2×#646114yes—
642C22C22×#115711yescharacteristic
643C22C22×#115911yescharacteristic
644C22C22×#116111yescharacteristic
645C22D7×H(5,𝔽2)12yes—
646C22#9061812yes—
647C4—44no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—144no—
652C4—282no—
653C4—284no—
654C4—284no—
655C4—282no—
656C2—24no—
657C2—24no—
658C2—24no—
659C2—562no—
660C2C22×#646111yescharacteristic
661C2#1174292711yescharacteristic
662C2#1174293611yescharacteristic
663C2#1174336712yes—
664C2#1174364012yes—
subgroupG/H
center, FrattiniC23C24×D7
commutator, socle, cosocleC23×C7C25
FittingC7×#2315C2
prime radicalC23×C22⋊D7C22
hypercenter#2315D7
hypocenterC7#34995
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2315

Complements

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

NH ≅ G/Ntype
1. C7×#2315C2semidirect
4. #11739539C2semidirect
6. C2×#89718C2semidirect
7. C2×C22⋊C4×Dic7C2semidirect
8. #11732070C2semidirect
9. #11732032C2semidirect
10. #11731904C2semidirect
11. #11731898C2semidirect
12. #11731373C2semidirect
13. #11731355C2semidirect
22. C22×Dic7⋊C4C22semidirect
23. C22×C4×Dic7C22semidirect
24. C2×#6461C22semidirect
26. C7×#385C22semidirect
27. C7×#380C22semidirect
28. C7×#380C22semidirect
34. C2×C7⋊#95C22semidirect
35. C2×C7⋊#95C22semidirect
36. #89766C22semidirect
37. #89765C22semidirect
38. #89762C22semidirect
39. #89758C22semidirect
40. #89754C22semidirect
41. #89753C22semidirect
42. #89750C22semidirect
50. #89722C22semidirect
51. #89722C22semidirect
53. #89718C22semidirect
54. #89718C22semidirect
58. C22⋊C4×Dic7C22semidirect
59. #89447C22semidirect
60. #89446C22semidirect
61. #89442C22semidirect
62. #89441C22semidirect
63. C7⋊#401C22semidirect
64. #89432C22semidirect
65. #89426C22semidirect
66. #89423C22semidirect
142. #2315D7semidirect
271. #385C2×D7semidirect
276. #380C2×D7semidirect
277. #380C2×D7semidirect
295. C23×C7#126semidirect
570. C23#6461semidirect
619. C7#34995semidirect

Maximal subgroups

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

subgroupindex
C7×#23152
#117395662
#117395392
C2×#897302
C2×#897182
C2×C22⋊C4×Dic72
#117320702
#117320322
#117319042
#117318982
#117313732
#117313552
#349957