Finite Group Databasebeta

← #13515015

Subgroups — #13515015

Order 1792.

2570 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90174C211yesmaximal, characteristic
2C2×C7×#385C211yesmaximal, characteristic
3C2×#89982C211yesmaximal, characteristic
4C2×C4×C22⋊Dic7C211yesmaximal, characteristic
5#11739607C214yesmaximal, complemented
6#11739540C211yesmaximal, characteristic
7#11739528C211yesmaximal, characteristic
8#11739528C211yesmaximal, characteristic
9#11731844C214yesmaximal, complemented
10#11731841C214yesmaximal, complemented
11#11731832C214yesmaximal, complemented
12#11731826C214yesmaximal, complemented
13#11730784C214yesmaximal, complemented
14C24×C4×C7C2211yescharacteristic
15C22×C7×C4⋊C4C2212yes—
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C7×C22⋊C4C2211yescharacteristic
18C22×C22⋊Dic7C2211yescharacteristic
19C22×C22⋊Dic7C2211yescharacteristic
20C22×C22⋊Dic7C2211yescharacteristic
21C22×C22⋊Dic7C2211yescharacteristic
22C2×#6487C2214yes—
23C22×Dic7⋊C4C2212yes—
24C22×C4×Dic7C2212yes—
25C2×#6450C2214yescomplemented
26#90174C4116yescomplemented
27C2×#6441C2214yes—
28C2×#6441C2214yes—
29C7×#385C2212yes—
30C7×#385C2212yes—
31C7×#385C2214yes—
32C2×C7×#95C2212yes—
33#89982C2214yes—
34#89982C2212yes—
35#89982C2212yes—
36#89982C2212yes—
37#89982C2212yes—
38C4×C22⋊Dic7C2214yes—
39C4×C22⋊Dic7C2214yes—
40C4×C22⋊Dic7C2212yes—
41C4×C22⋊Dic7C2214yes—
42C4×C22⋊Dic7C2212yes—
43C2×C7⋊#95C2212yes—
44C2×C7⋊#95C2212yes—
45#89750C2214yescomplemented
46#89750C2214yescomplemented
47C4⋊C4×Dic7C2214yescomplemented
48#89746C2214yescomplemented
49#89745C2214yescomplemented
50#89744C2214yescomplemented
51#89744C2214yescomplemented
52#89718C2212yes—
53#89718C2212yes—
54#89714C2214yes—
55C22⋊C4×Dic7C2214yes—
56C22⋊C4×Dic7C2214yes—
57C22⋊C4×Dic7C2212yes—
58C22⋊C4×Dic7C2214yes—
59C22⋊C4×Dic7C2212yes—
60#89425C2214yescomplemented
61#89425C2214yescomplemented
62#89424C2214yescomplemented
63#89424C2214yescomplemented
64#89421C2214yescomplemented
65#89421C2214yescomplemented
66#89418C2214yescomplemented
67#89418C2214yescomplemented
68#34270—71nomaximal, Sylow
69C25×C7C2311yescharacteristic
70C23×C4×C7—24no—
71C23×C4×C7—44no—
72C23×C4×C7C2311yescharacteristic
73C23×C4×C7C2312yes—
74C23×C4×C7C2312yes—
75C23×C4×C7C2311yescharacteristic
76C23×C4×C7C2311yescharacteristic
77C23×C4×C7C2311yescharacteristic
78C23×C4×C7C2312yes—
79C23×Dic7C2311yescharacteristic
80C23×Dic7C2311yescharacteristic
81C23×Dic7C2312yes—
82C23×Dic7C2312yes—
83C23×Dic7C2312yes—
84C22×Dic14C2314yes—
85C2×C7×C4⋊C4—44no—
86C2×C7×C4⋊C4C2314yes—
87C2×C7×C4⋊C4C2314yes—
88C2×C7×C22⋊C4—24no—
89C2×C7×C22⋊C4C2×C418yes—
90C2×C7×C22⋊C4C2312yes—
91C2×C7×C22⋊C4C2312yes—
92C2×C7×C22⋊C4C2312yes—
93C2×C7×C22⋊C4C2312yes—
94C2×C22⋊Dic7—24no—
95C2×C22⋊Dic7C2×C418yes—
96C2×C22⋊Dic7C2×C418yes—
97C2×C22⋊Dic7Q818yescomplemented
98C2×C22⋊Dic7C2312yes—
99C2×C22⋊Dic7C2312yes—
100C2×C22⋊Dic7C2312yes—
101C2×C22⋊Dic7C2312yes—
102C2×C22⋊Dic7C2312yes—
103C2×C22⋊Dic7C2312yes—
104C2×C22⋊Dic7C2312yes—
105C2×C22⋊Dic7C2312yes—
106C2×C4⋊Dic7C2314yes—
107#6487C2×C4132yescomplemented
108C2×Dic7⋊C4—44no—
109C2×Dic7⋊C4C2314yes—
110C2×Dic7⋊C4C2314yes—
111C2×Dic7⋊C4C2314yes—
112C2×Dic7⋊C4C2314yes—
113C2×Dic7⋊C4C2314yes—
114C2×Dic7⋊C4C2314yes—
115C2×C4×Dic7—44no—
116C2×C4×Dic7C2314yes—
117C2×C4×Dic7C2314yes—
118C2×C4×Dic7C2314yes—
119C2×C4×Dic7C2314yes—
120C2×C4×Dic7C2314yes—
121C2×C4×Dic7C2314yes—
122#6450—48no—
123#6441C2×C4132yescomplemented
124#6441C2×C4132yescomplemented
125C7×#95C2314yes—
126C7×#95C2314yes—
127C7⋊#95C2314yes—
128C7⋊#95C2314yes—
129C7⋊#95C2314yes—
130C7⋊#95C2314yes—
131C7⋊#95C2314yes—
132C7⋊#95C2314yes—
133C7⋊#95C2314yes—
134C7⋊#95C2314yes—
135C7⋊#95C2314yes—
136C7⋊#95C2314yes—
137C2×#543—71no—
138#2368—74no—
139#2368—74no—
140#2360—74no—
141#2360—74no—
142#2329—74no—
143#2315—71no—
144C4×C22⋊Q8—74no—
145C2×#385—71no—
146C2×#385D711yescharacteristic
147#2273—71no—
148#2273—71no—
149C2×C4×C22⋊C4—71no—
150C24×C7—42no—
151C24×C7—42no—
152C24×C7C22×C414yes—
153C24×C7C2×Q814yes—
154C24×C7#4114yes—
155C24×C7C2411yescharacteristic
156C24×C7C2411yescharacteristic
157C24×C7C2411yescharacteristic
158C22×C4×C7—22no—
159C22×C4×C7—24no—
160C22×C4×C7—24no—
161C22×C4×C7—24no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7—24no—
166C22×C4×C7—24no—
167C22×C4×C7—42no—
168C22×C4×C7—42no—
169C22×C4×C7—42no—
170C22×C4×C7—42no—
171C22×C4×C7—42no—
172C22×C4×C7—42no—
173C22×C4×C7—42no—
174C22×C4×C7—416no—
175C22×C4×C7C22×C418yes—
176C22×C4×C7C2412yes—
177C22×C4×C7C2412yes—
178C22×C4×C7C2412yes—
179C22×C4×C7C2414yes—
180C22×C4×C7C2412yes—
181C22×Dic7—22no—
182C22×Dic7—22no—
183C22×Dic7—42no—
184C22×Dic7—42no—
185C22×Dic7—42no—
186C22×Dic7—42no—
187C22×Dic7—42no—
188C22×Dic7—42no—
189C22×Dic7—42no—
190C22×Dic7—42no—
191C22×Dic7C22×C418yes—
192C22×Dic7C22×C418yes—
193C22×Dic7C2×Q818yes—
194C22×Dic7C2412yes—
195C22×Dic7C2412yes—
196C22×Dic7C2414yes—
197C22×Dic7C2414yes—
198C22×Dic7C2414yes—
199C2×Dic14—44no—
200C2×Dic14—44no—
201C2×Dic14C22×C4116yescomplemented
202C7×C4⋊C4—28no—
203C7×C4⋊C4—44no—
204C7×C22⋊C4—24no—
205C7×C22⋊C4—24no—
206C7×C22⋊C4C22×C418yes—
207C7×C22⋊C4C22×C418yes—
208C22⋊Dic7—24no—
209C22⋊Dic7—24no—
210C22⋊Dic7C22×C418yes—
211C22⋊Dic7C22×C418yes—
212C22⋊Dic7C22×C418yes—
213C22⋊Dic7C22×C418yes—
214C22⋊Dic7C2×Q818yes—
215C22⋊Dic7C2×Q818yes—
216C4⋊Dic7C22×C4116yescomplemented
217Dic7⋊C4—28no—
218Dic7⋊C4—44no—
219Dic7⋊C4—44no—
220Dic7⋊C4—44no—
221Dic7⋊C4C22×C4116yescomplemented
222Dic7⋊C4C22×C4116yescomplemented
223C4×Dic7—28no—
224C4×Dic7—44no—
225C4×Dic7—44no—
226C4×Dic7—44no—
227C4×Dic7—44no—
228C4×Dic7—44no—
229C24×C4C2×D711yescharacteristic
230#543—716no—
231C2×C22⋊Q8—74no—
232C2×C22⋊Q8—74no—
233C2×C22⋊Q8—74no—
234C2×C4×Q8—74no—
235C22×C4⋊C4—72no—
236C22×C4⋊C4C2×D712yes—
237C22×C22⋊C4—71no—
238C22×C22⋊C4—71no—
239C22×C22⋊C4—71no—
240C22×C22⋊C4—71no—
241C22×C22⋊C4C2×D711yescharacteristic
242C22×C22⋊C4C2×D711yescharacteristic
243C22×C42—72no—
244#391—74no—
245#391—74no—
246#391—74no—
247#389—74no—
248#389—74no—
249#389—74no—
250#387—74no—
251#387—74no—
252#387—74no—
253#387—74no—
254#387—74no—
255#387—74no—
256#385—72no—
257#385—72no—
258#385—72no—
259#385—74no—
260#385—72no—
261#385—72no—
262#385—72no—
263#385C2×D712yes—
264#385C2×D714yes—
265#385C2×D712yes—
266#380—74no—
267C4×C4⋊C4—74no—
268C4×C4⋊C4—74no—
269C4×C4⋊C4—74no—
270C4×C22⋊C4—72no—
271C4×C22⋊C4—74no—
272C4×C22⋊C4—74no—
273C4×C22⋊C4—74no—
274C4×C22⋊C4—72no—
275C4×C22⋊C4—74no—
276C4×C22⋊C4—74no—
277C4×C22⋊C4—72no—
278C4×C22⋊C4—74no—
279C4×C22⋊C4—72no—
280C2×#95—72no—
281C2×#95—72no—
282C2×#95C2×D712yes—
283C23×C7—22no—
284C23×C7—22no—
285C23×C7—22no—
286C23×C7—22no—
287C23×C7—22no—
288C23×C7—22no—
289C23×C7—44no—
290C23×C7—48no—
291C23×C7—48no—
292C23×C7—44no—
293C23×C7C4×Q8116yescomplemented
294C23×C7C23×C412yes—
295C23×C7C23×C412yes—
296C23×C7C23×C412yes—
297C23×C7C22×Q812yes—
298C23×C7C22×Q812yes—
299C23×C7C22×Q812yes—
300C23×C7C2×#4112yes—
301C23×C7C2×#4112yes—
302C23×C7C2×#4112yes—
303C23×C7C2511yescharacteristic
304C2×C4×C7—28no—
305C2×C4×C7—22no—
306C2×C4×C7—22no—
307C2×C4×C7—24no—
308C2×C4×C7—24no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—22no—
312C2×C4×C7—24no—
313C2×C4×C7—22no—
314C2×C4×C7—22no—
315C2×C4×C7—24no—
316C2×C4×C7—48no—
317C2×C4×C7—48no—
318C2×C4×C7—48no—
319C2×C4×C7—44no—
320C2×C4×C7—82no—
321C2×C4×C7—82no—
322C2×C4×C7C23×C418yes—
323C2×C4×C7H(5,𝔽2)14yes—
324C2×Dic7—24no—
325C2×Dic7—24no—
326C2×Dic7—24no—
327C2×Dic7—24no—
328C2×Dic7—24no—
329C2×Dic7—24no—
330C2×Dic7—22no—
331C2×Dic7—22no—
332C2×Dic7—24no—
333C2×Dic7—24no—
334C2×Dic7—48no—
335C2×Dic7—48no—
336C2×Dic7—48no—
337C2×Dic7—82no—
338C2×Dic7—82no—
339C2×Dic7C23×C418yes—
340C2×Dic7C23×C418yes—
341C2×Dic7C22×Q818yes—
342Dic14—416no—
343C25C22×D711yescharacteristic
344C22×Q8—74no—
345C23×C4—24no—
346C23×C4—44no—
347C23×C4—72no—
348C23×C4—72no—
349C23×C4—72no—
350C23×C4—71no—
351C23×C4—71no—
352C23×C4C22×D711yescharacteristic
353C23×C4C22×D712yes—
354C23×C4C22×D712yes—
355C23×C4C22×D711yescharacteristic
356C23×C4C22×D711yescharacteristic
357C23×C4C22×D711yescharacteristic
358C23×C4C22×D712yes—
359C22⋊Q8—732no—
360C22⋊Q8—732no—
361C22⋊Q8—732no—
362C4×Q8—288no—
363C2×C4⋊C4—44no—
364C2×C4⋊C4—74no—
365C2×C4⋊C4—74no—
366C2×C4⋊C4—74no—
367C2×C4⋊C4—74no—
368C2×C4⋊C4—74no—
369C2×C4⋊C4—74no—
370C2×C4⋊C4—74no—
371C2×C4⋊C4—284no—
372C2×C4⋊C4C22×D714yes—
373C2×C4⋊C4C22×D714yes—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—78no—
376C2×C22⋊C4—78no—
377C2×C22⋊C4—78no—
378C2×C22⋊C4—72no—
379C2×C22⋊C4—72no—
380C2×C22⋊C4—72no—
381C2×C22⋊C4—72no—
382C2×C22⋊C4—72no—
383C2×C22⋊C4—72no—
384C2×C22⋊C4—72no—
385C2×C22⋊C4—72no—
386C2×C22⋊C4—144no—
387C2×C22⋊C4C4×D718yes—
388C2×C22⋊C4C22×D712yes—
389C2×C22⋊C4C22×D712yes—
390C2×C22⋊C4C22×D712yes—
391C2×C22⋊C4C22×D712yes—
392C2×C42—74no—
393C2×C42—74no—
394C2×C42—74no—
395C2×C42—74no—
396C2×C42—74no—
397C2×C42—74no—
398C2×C42—284no—
399#95—74no—
400#95—74no—
401#95—74no—
402#95—74no—
403#95—74no—
404#95—74no—
405#95—74no—
406#95—74no—
407#95—74no—
408#95—74no—
409#95C22×D714yes—
410#95C22×D714yes—
411C22×C7—24no—
412C22×C7—24no—
413C22×C7—24no—
414C22×C7—22no—
415C22×C7—22no—
416C22×C7—22no—
417C22×C7—22no—
418C22×C7—22no—
419C22×C7—22no—
420C22×C7—24no—
421C22×C7—24no—
422C22×C7—24no—
423C22×C7—416no—
424C22×C7C2×C4×Q814yes—
425C22×C7C2×C4×Q814yes—
426C22×C7C2×C4×Q814yes—
427C22×C7C24×C411yescharacteristic
428C22×C7C23×Q811yescharacteristic
429C22×C7C22×#4111yescharacteristic
430C22×C7C2×H(5,𝔽2)12yes—
431C22×C7#58512yes—
432C4×C7—28no—
433C4×C7—24no—
434C4×C7—42no—
435C4×C7—42no—
436C4×C7—42no—
437Dic7—28no—
438Dic7—28no—
439Dic7—28no—
440Dic7—42no—
441Dic7—42no—
442C24—42no—
443C24—42no—
444C24C2×C4×D714yes—
445C24#115914yes—
446C24Q8×D714yes—
447C24C23×D711yescharacteristic
448C24C23×D711yescharacteristic
449C24C23×D711yescharacteristic
450C2×Q8—716no—
451C2×Q8—284no—
452C2×Q8—284no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—416no—
470C22×C4—74no—
471C22×C4—74no—
472C22×C4—74no—
473C22×C4—78no—
474C22×C4—78no—
475C22×C4—78no—
476C22×C4—72no—
477C22×C4—72no—
478C22×C4—142no—
479C22×C4—142no—
480C22×C4—282no—
481C22×C4—282no—
482C22×C4—282no—
483C22×C4—282no—
484C22×C4—282no—
485C22×C4—282no—
486C22×C4—282no—
487C22×C4—282no—
488C22×C4C2×C4×D718yes—
489C22×C4C23×D712yes—
490C22×C4C23×D712yes—
491C22×C4C23×D712yes—
492C22×C4C23×D714yes—
493C22×C4C23×D712yes—
494C4⋊C4—28no—
495C4⋊C4—44no—
496C4⋊C4—716no—
497C4⋊C4—716no—
498C4⋊C4—716no—
499C4⋊C4—148no—
500C4⋊C4—284no—
501C4⋊C4—284no—
502C4⋊C4—284no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—78no—
506C22⋊C4—78no—
507C22⋊C4—78no—
508C22⋊C4—78no—
509C22⋊C4—78no—
510C22⋊C4—78no—
511C22⋊C4—144no—
512C22⋊C4—144no—
513C22⋊C4C2×C4×D718yes—
514C22⋊C4C2×C4×D718yes—
515C42—148no—
516C42—284no—
517C42—284no—
518C42—284no—
519C42—284no—
520C42—284no—
521C2×C7—24no—
522C2×C7—24no—
523C2×C7—24no—
524C2×C7C22×C4×Q811yescharacteristic
525C2×C7C4×H(5,𝔽2)12yes—
526C2×C7#349011yescharacteristic
527C2×C7#349111yescharacteristic
528C2×C7#350112yes—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—44no—
536C23—48no—
537C23—44no—
538C23—48no—
539C23#6450116yescomplemented
540C23C22×C4×D712yes—
541C23C22×C4×D712yes—
542C23C22×C4×D712yes—
543C23C2×#115912yes—
544C23C2×#115912yes—
545C23C2×#115912yes—
546C23C2×Q8×D712yes—
547C23C2×Q8×D712yes—
548C23C2×Q8×D712yes—
549C23C24×D711yescharacteristic
550Q8—2816no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—44no—
567C2×C4—78no—
568C2×C4—78no—
569C2×C4—78no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—144no—
573C2×C4—144no—
574C2×C4—144no—
575C2×C4—144no—
576C2×C4—144no—
577C2×C4—144no—
578C2×C4—142no—
579C2×C4—142no—
580C2×C4—144no—
581C2×C4—144no—
582C2×C4—288no—
583C2×C4—288no—
584C2×C4—288no—
585C2×C4—562no—
586C2×C4—562no—
587C2×C4C22×C4×D718yes—
588C2×C4#654814yes—
589C7#3427011yescharacteristic, complemented, Sylow
590C22—22no—
591C22—22no—
592C22—22no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—416no—
603C22C2×#645014yes—
604C22C2×#645014yes—
605C22C2×#645014yes—
606C22C23×C4×D711yescharacteristic
607C22C22×#115911yescharacteristic
608C22C2×#654812yes—
609C22C22×Q8×D711yescharacteristic
610C22#9061612yes—
611C4—24no—
612C4—28no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—148no—
617C4—148no—
618C4—148no—
619C4—282no—
620C4—282no—
621C2—24no—
622C2—24no—
623C2—24no—
624C2C22×#645011yescharacteristic
625C2C4×#654812yes—
626C2#1174298111yescharacteristic
627C2#1174305412yes—
628C2#1174306911yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC2×C7×#385C2
socle, cosocleC23×C7C25
prime radicalC25×C7C23
hypercenterC2×#385D7
hypocenterC7#34270
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
5. #11739607C2semidirect
9. #11731844C2semidirect
10. #11731841C2semidirect
11. #11731832C2semidirect
12. #11731826C2semidirect
13. #11730784C2semidirect
25. C2×#6450C22semidirect
26. #90174C4semidirect
45. #89750C22semidirect
46. #89750C22semidirect
47. C4⋊C4×Dic7C22semidirect
48. #89746C22semidirect
49. #89745C22semidirect
50. #89744C22semidirect
51. #89744C22semidirect
60. #89425C22semidirect
61. #89425C22semidirect
62. #89424C22semidirect
63. #89424C22semidirect
64. #89421C22semidirect
65. #89421C22semidirect
66. #89418C22semidirect
67. #89418C22semidirect
97. C2×C22⋊Dic7Q8semidirect
107. #6487C2×C4semidirect
123. #6441C2×C4semidirect
124. #6441C2×C4semidirect
201. C2×Dic14C22×C4semidirect
216. C4⋊Dic7C22×C4semidirect
221. Dic7⋊C4C22×C4semidirect
222. Dic7⋊C4C22×C4semidirect
293. C23×C7C4×Q8semidirect
539. C23#6450semidirect
589. C7#34270semidirect

Maximal subgroups

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

subgroupindex
C2×#901742
C2×C7×#3852
C2×#899822
C2×C4×C22⋊Dic72
#117396072
#117395402
#117395282
#117318442
#117318412
#117318322
#117318262
#117307842
#342707