Finite Group Databasebeta

← #13276621

Subgroups — #13276621

Order 1792.

1918 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7⋊#95C211yesmaximal, characteristic
2C2×#89722C211yesmaximal, characteristic
3C2×#89715C211yesmaximal, characteristic
4C2×#89715C211yesmaximal, characteristic
5C2×C22⋊C4×Dic7C211yesmaximal, characteristic
6C7×C22⋊#95C211yesmaximal, characteristic
7#11729881C211yesmaximal, characteristic
8#11727172C212yesmaximal, complemented
9#11727169C212yesmaximal, complemented
10#11727161C212yesmaximal, complemented
11#11727156C212yesmaximal, complemented
12C24×C4×C7C2211yescharacteristic
13C24×Dic7C2211yescharacteristic
14C24×Dic7C2211yescharacteristic
15C22×C7×C22⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C22⋊Dic7C2211yescharacteristic
18C22×C22⋊Dic7C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20C22×C4⋊Dic7C2211yescharacteristic
21C22×C4×Dic7C2212yes—
22C2×C7×#95C2212yes—
23C2×C7×#95C2212yes—
24C2×C7⋊#95—24no—
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⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36#89722—24no—
37#89715—24no—
38#89715C418yescomplemented
39C22⋊C4×Dic7—24no—
40#13972—71nomaximal, Sylow
41C25×C7C2311yescharacteristic
42C23×C4×C7—22no—
43C23×C4×C7—24no—
44C23×C4×C7—22no—
45C23×C4×C7—22no—
46C23×C4×C7—42no—
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2312yes—
50C23×C4×C7C2312yes—
51C23×Dic7—22no—
52C23×Dic7—22no—
53C23×Dic7—22no—
54C23×Dic7—22no—
55C23×Dic7—22no—
56C23×Dic7—22no—
57C23×Dic7—22no—
58C23×Dic7—22no—
59C23×Dic7—24no—
60C23×Dic7—42no—
61C23×Dic7C2×C414yes—
62C23×Dic7D414yes—
63C23×Dic7C2311yescharacteristic
64C23×Dic7C2311yescharacteristic
65C23×Dic7C2312yes—
66C23×Dic7C2312yes—
67C23×Dic7C2311yescharacteristic
68C23×Dic7C2311yescharacteristic
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4C2×C414yes—
74C2×C7×C22⋊C4D414yes—
75C2×C22⋊Dic7—22no—
76C2×C22⋊Dic7—22no—
77C2×C22⋊Dic7—22no—
78C2×C22⋊Dic7—22no—
79C2×C22⋊Dic7—22no—
80C2×C22⋊Dic7C2×C414yes—
81C2×C4⋊Dic7—22no—
82C2×C4⋊Dic7—22no—
83C2×C4⋊Dic7—24no—
84C2×C4⋊Dic7—42no—
85C2×C4×Dic7—24no—
86C2×C4×Dic7—24no—
87C2×C4×Dic7—42no—
88C7×#95—24no—
89C7×#95—42no—
90C7⋊#95—24no—
91C7⋊#95—24no—
92C7⋊#95—24no—
93C7⋊#95—24no—
94C7⋊#95—22no—
95C7⋊#95—22no—
96C7⋊#95—24no—
97C7⋊#95—24no—
98C7⋊#95—42no—
99C7⋊#95C2×C418yescomplemented
100C7⋊#95C2×C418yescomplemented
101C2×#385—71no—
102C2×#381—71no—
103C2×#381—71no—
104C2×C4×C22⋊C4—71no—
105C22×#95—71no—
106#1443—72no—
107#1442—72no—
108#1436—72no—
109#1434—72no—
110C22⋊#95—71no—
111C22⋊#95D711yescharacteristic
112C24×C7—22no—
113C24×C7—22no—
114C24×C7—22no—
115C24×C7—22no—
116C24×C7C22×C412yes—
117C24×C7C2×D412yes—
118C24×C7C2×D412yes—
119C24×C7#4112yes—
120C24×C7#4112yes—
121C24×C7#4112yes—
122C24×C7#4112yes—
123C24×C7C2411yescharacteristic
124C22×C4×C7—28no—
125C22×C4×C7—22no—
126C22×C4×C7—21no—
127C22×C4×C7—21no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—48no—
146C22×C4×C7—48no—
147C22×C4×C7—44no—
148C22×C4×C7—44no—
149C22×C4×C7C22×C412yes—
150C22×C4×C7C22×C412yes—
151C22×C4×C7C2×D414yes—
152C22×C4×C7#4114yes—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—22no—
156C22×Dic7—28no—
157C22×Dic7—28no—
158C22×Dic7—28no—
159C22×Dic7—28no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—21no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—21no—
172C22×Dic7—22no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—28no—
177C22×Dic7—21no—
178C22×Dic7—21no—
179C22×Dic7—21no—
180C22×Dic7—21no—
181C22×Dic7—22no—
182C22×Dic7—22no—
183C22×Dic7—21no—
184C22×Dic7—21no—
185C22×Dic7—22no—
186C22×Dic7—22no—
187C22×Dic7—44no—
188C22×Dic7—48no—
189C22×Dic7—48no—
190C22×Dic7—44no—
191C22×Dic7—44no—
192C22×Dic7—44no—
193C22×Dic7—44no—
194C22×Dic7C22×C414yes—
195C22×Dic7C22×C414yes—
196C22×Dic7C2×D412yes—
197C22×Dic7C2×D412yes—
198C22×Dic7#4114yes—
199C7×C22⋊C4—28no—
200C7×C22⋊C4—28no—
201C22⋊Dic7—28no—
202C22⋊Dic7—44no—
203C4⋊Dic7—44no—
204C4⋊Dic7—44no—
205C4×Dic7—48no—
206C24×C4—71no—
207C24×C4—71no—
208C24×C4C2×D711yescharacteristic
209C22×C4⋊C4—71no—
210C22×C4⋊C4—71no—
211C22×C22⋊C4—71no—
212C22×C22⋊C4—71no—
213C22×C22⋊C4C2×D711yescharacteristic
214C22×C22⋊C4C2×D711yescharacteristic
215C22×C42—72no—
216#385—144no—
217#381—78no—
218#381—144no—
219C4×C22⋊C4—144no—
220C2×#95—72no—
221C2×#95—72no—
222C2×#95—72no—
223C2×#95—71no—
224C2×#95—72no—
225C2×#95—72no—
226C2×#95—72no—
227C2×#95—72no—
228C2×#95—72no—
229C2×#95—71no—
230C2×#95—72no—
231C2×#95—144no—
232C2×#95C2×D712yes—
233C2×#95C2×D712yes—
234C23×C7—24no—
235C23×C7—24no—
236C23×C7—22no—
237C23×C7—22no—
238C23×C7—24no—
239C23×C7—22no—
240C23×C7—22no—
241C23×C7—24no—
242C23×C7—24no—
243C23×C7—24no—
244C23×C7—24no—
245C23×C7—24no—
246C23×C7—24no—
247C23×C7—24no—
248C23×C7—24no—
249C23×C7—24no—
250C23×C7#11714yes—
251C23×C7C4×D414yes—
252C23×C7C4×D414yes—
253C23×C7#12114yes—
254C23×C7#12314yes—
255C23×C7#12414yes—
256C23×C7#12614yes—
257C23×C7C23×C411yescharacteristic
258C23×C7C22×D411yescharacteristic
259C23×C7C22×D411yescharacteristic
260C23×C7C2×#4111yescharacteristic
261C23×C7C2×#4111yescharacteristic
262C23×C7C2×#4111yescharacteristic
263C23×C7C2×#4112yes—
264C23×C7C2×#4111yescharacteristic
265C23×C7C2×#4112yes—
266C23×C7H(5,𝔽2)12yes—
267C23×C72−1+412yes—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—24no—
277C2×C4×C7—24no—
278C2×C4×C7—416no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—41no—
285C2×C4×C7—42no—
286C2×C4×C7—41no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—24no—
296C2×Dic7—24no—
297C2×Dic7—24no—
298C2×Dic7—24no—
299C2×Dic7—24no—
300C2×Dic7—24no—
301C2×Dic7—24no—
302C2×Dic7—24no—
303C2×Dic7—22no—
304C2×Dic7—22no—
305C2×Dic7—22no—
306C2×Dic7—416no—
307C2×Dic7—42no—
308C2×Dic7—42no—
309C2×Dic7—42no—
310C2×Dic7—42no—
311C2×Dic7—42no—
312C2×Dic7—42no—
313C2×Dic7—42no—
314C2×Dic7—42no—
315C2×Dic7—42no—
316C2×Dic7—42no—
317C2×Dic7—416no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—42no—
325C2×Dic7—42no—
326C2×Dic7—42no—
327C2×Dic7—42no—
328C2×Dic7#12314yes—
329C25C22×D711yescharacteristic
330C23×C4—22no—
331C23×C4—24no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—42no—
335C23×C4—71no—
336C23×C4—74no—
337C23×C4—72no—
338C23×C4—71no—
339C23×C4—72no—
340C23×C4—74no—
341C23×C4—71no—
342C23×C4—71no—
343C23×C4—144no—
344C23×C4—142no—
345C23×C4—142no—
346C23×C4—142no—
347C23×C4—142no—
348C23×C4—142no—
349C23×C4—142no—
350C23×C4—142no—
351C23×C4—142no—
352C23×C4—282no—
353C23×C4C22×D711yescharacteristic
354C23×C4C22×D711yescharacteristic
355C23×C4C22×D712yes—
356C23×C4C22×D712yes—
357C2×C4⋊C4—144no—
358C2×C4⋊C4—142no—
359C2×C4⋊C4—142no—
360C2×C4⋊C4—282no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—74no—
366C2×C22⋊C4—142no—
367C2×C22⋊C4—142no—
368C2×C22⋊C4—142no—
369C2×C22⋊C4—142no—
370C2×C22⋊C4—142no—
371C2×C22⋊C4C4×D714yes—
372C2×C22⋊C4D2814yes—
373C2×C42—144no—
374C2×C42—144no—
375C2×C42—282no—
376#95—24no—
377#95—42no—
378#95—78no—
379#95—78no—
380#95—144no—
381#95—144no—
382#95—142no—
383#95—142no—
384#95—144no—
385#95—144no—
386#95—144no—
387#95—144no—
388#95—282no—
389C22×C7—24no—
390C22×C7—24no—
391C22×C7—28no—
392C22×C7—28no—
393C22×C7—24no—
394C22×C7—24no—
395C22×C7—28no—
396C22×C7—28no—
397C22×C7—24no—
398C22×C7—24no—
399C22×C7#38818yes—
400C22×C7C2×#11711yescharacteristic
401C22×C7C2×C4×D411yescharacteristic
402C22×C7C2×C4×D411yescharacteristic
403C22×C7C4×#4112yes—
404C22×C7#52012yes—
405C22×C7C2×#12111yescharacteristic
406C22×C7C2×#12311yescharacteristic
407C22×C7C2×#12311yescharacteristic
408C22×C7#52511yescharacteristic
409C22×C7C2×#12411yescharacteristic
410C22×C7C2×#12611yescharacteristic
411C22×C7#52912yes—
412C22×C7#53411yescharacteristic
413C22×C7#53611yescharacteristic
414C22×C7#53912yes—
415C22×C7#54612yes—
416C22×C7#54712yes—
417C22×C7#55112yes—
418C22×C7#55112yes—
419C22×C7#55112yes—
420C22×C7#55212yes—
421C22×C7#55212yes—
422C22×C7#55212yes—
423C4×C7—44no—
424C4×C7—42no—
425C4×C7—44no—
426C4×C7—42no—
427Dic7—24no—
428Dic7—44no—
429Dic7—42no—
430Dic7—44no—
431Dic7—81no—
432Dic7—81no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24C2×C4×D712yes—
438C24C2×D2812yes—
439C24#115712yes—
440C24D4×D712yes—
441C24#115912yes—
442C24#115912yes—
443C24#116112yes—
444C24C23×D711yescharacteristic
445C22×C4—28no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—72no—
471C22×C4—74no—
472C22×C4—74no—
473C22×C4—74no—
474C22×C4—72no—
475C22×C4—148no—
476C22×C4—141no—
477C22×C4—142no—
478C22×C4—141no—
479C22×C4—148no—
480C22×C4—148no—
481C22×C4—148no—
482C22×C4—148no—
483C22×C4—142no—
484C22×C4—141no—
485C22×C4—142no—
486C22×C4—142no—
487C22×C4—142no—
488C22×C4—142no—
489C22×C4—142no—
490C22×C4—142no—
491C22×C4—142no—
492C22×C4—142no—
493C22×C4—142no—
494C22×C4—142no—
495C22×C4—141no—
496C22×C4—142no—
497C22×C4—142no—
498C22×C4—142no—
499C22×C4—141no—
500C22×C4—141no—
501C22×C4—141no—
502C22×C4—141no—
503C22×C4—142no—
504C22×C4—142no—
505C22×C4—141no—
506C22×C4—141no—
507C22×C4—142no—
508C22×C4—142no—
509C22×C4—284no—
510C22×C4—288no—
511C22×C4—288no—
512C22×C4—284no—
513C22×C4—284no—
514C22×C4—284no—
515C22×C4—284no—
516C22×C4C2×C4×D712yes—
517C22×C4C2×C4×D712yes—
518C22×C4C2×D2814yes—
519C22×C4#115914yes—
520C4⋊C4—284no—
521C4⋊C4—284no—
522C22⋊C4—28no—
523C22⋊C4—28no—
524C22⋊C4—148no—
525C22⋊C4—284no—
526C42—288no—
527C2×C7—28no—
528C2×C7C2×#38811yescharacteristic
529C2×C7C4×#12311yescharacteristic
530C2×C7#232211yescharacteristic
531C2×C7#234312yes—
532C2×C7#235912yes—
533C2×C7#241211yescharacteristic
534C2×C7#241311yescharacteristic
535C2×C7#241912yes—
536C2×C7#249312yes—
537C2×C7#250311yescharacteristic
538C2×C7#256111yescharacteristic
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—22no—
546C23—24no—
547C23—22no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—22no—
553C23—24no—
554C23—22no—
555C23C4×D2814yes—
556C23(C22⋊D7)⋊C414yes—
557C23#644814yes—
558C23#644914yes—
559C23#645514yes—
560C23#645814yes—
561C23#646114yes—
562C23C22×C4×D711yescharacteristic
563C23C22×D2811yescharacteristic
564C23C2×#115711yescharacteristic
565C23C2×D4×D711yescharacteristic
566C23C2×#115911yescharacteristic
567C23C2×#115911yescharacteristic
568C23C2×#115912yes—
569C23#654812yes—
570C23C2×#116111yescharacteristic
571C23D7×#4112yes—
572C23#655412yes—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—416no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—74no—
599C2×C4—142no—
600C2×C4—144no—
601C2×C4—142no—
602C2×C4—144no—
603C2×C4—144no—
604C2×C4—144no—
605C2×C4—144no—
606C2×C4—144no—
607C2×C4—142no—
608C2×C4—144no—
609C2×C4—144no—
610C2×C4—142no—
611C2×C4—142no—
612C2×C4—2816no—
613C2×C4—282no—
614C2×C4—282no—
615C2×C4—282no—
616C2×C4—282no—
617C2×C4—282no—
618C2×C4—2816no—
619C2×C4—282no—
620C2×C4—282no—
621C2×C4—282no—
622C2×C4—282no—
623C2×C4—282no—
624C2×C4—281no—
625C2×C4—281no—
626C2×C4—282no—
627C2×C4—282no—
628C2×C4—281no—
629C2×C4—281no—
630C2×C4—282no—
631C2×C4—282no—
632C2×C4—281no—
633C2×C4—281no—
634C7#1397211yescharacteristic, complemented, Sylow
635C22—24no—
636C22—24no—
637C22—28no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—28no—
645C22#8944218yes—
646C22C2×C4×D2811yescharacteristic
647C22C2×(C22⋊D7)⋊C411yescharacteristic
648C22C2×#644811yescharacteristic
649C22C2×#644911yescharacteristic
650C22#9018612yes—
651C22C2×#645511yescharacteristic
652C22C2×#645811yescharacteristic
653C22C2×#646111yescharacteristic
654C22#9022212yes—
655C22C4×#115912yes—
656C22#9023712yes—
657C22#9024512yes—
658C22#9024612yes—
659C22#9027711yescharacteristic
660C22#9027812yes—
661C22#9028011yescharacteristic
662C22#9029112yes—
663C22#9029312yes—
664C22#9032812yes—
665C22#9033312yes—
666C22#9033911yescharacteristic
667C22#9034112yes—
668C22D7×#12311yescharacteristic
669C4—44no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—144no—
674C4—284no—
675C4—282no—
676C4—284no—
677C4—561no—
678C4—561no—
679C2—28no—
680C2C2×#8944211yescharacteristic
681C2#1173090912yes—
682C2#1173128011yescharacteristic
683C2#1173136212yes—
684C2#1173152411yescharacteristic
685C2#1173183712yes—
686C2#1173185111yescharacteristic
687C2#1173185211yescharacteristic
688C2#1173210611yescharacteristic
689C2#1173210911yescharacteristic
690C2#1173213912yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×C22⋊#95C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC22⋊#95D7
hypocenterC7#13972
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
8. #11727172C2semidirect
9. #11727169C2semidirect
10. #11727161C2semidirect
11. #11727156C2semidirect
38. #89715C4semidirect
99. C7⋊#95C2×C4semidirect
100. C7⋊#95C2×C4semidirect
634. C7#13972semidirect

Maximal subgroups

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

subgroupindex
C22×C7⋊#952
C2×#897222
C2×#897152
C2×C22⋊C4×Dic72
C7×C22⋊#952
#117298812
#117271722
#117271692
#117271612
#117271562
#139727