Finite Group Databasebeta

← #13515013

Subgroups — #13515013

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
2C7×#2273C211yesmaximal, characteristic
3#11739540C211yesmaximal, characteristic
4C2×#89722C211yesmaximal, characteristic
5C2×#89718C211yesmaximal, characteristic
6#11739528C211yesmaximal, characteristic
7C2×C22⋊C4×Dic7C211yesmaximal, characteristic
8#11730919C214yesmaximal, complemented
9#11730918C214yesmaximal, complemented
10#11730911C214yesmaximal, complemented
11#11730910C214yesmaximal, complemented
12#11730892C214yesmaximal, complemented
13C4×#6441C214yesmaximal, complemented
14C24×Dic7C2211yescharacteristic
15C22×C7×C22⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C7×C22⋊C4C2211yescharacteristic
18C22×C22⋊Dic7C2211yescharacteristic
19C22×C22⋊Dic7C2211yescharacteristic
20C22×C22⋊Dic7C2211yescharacteristic
21C22×C4⋊Dic7C2212yes—
22C2×#6487C2214yes—
23C22×Dic7⋊C4C2212yes—
24C22×C4×Dic7C2212yes—
25#90174C4116yescomplemented
26C2×#6441C2214yes—
27C2×#6441C2214yes—
28C2×C4×Dic14C2214yescomplemented
29C4×C7×C22⋊C4C2214yes—
30C4×C7×C22⋊C4C2214yes—
31C4×C7×C22⋊C4C2212yes—
32C4×C7×C22⋊C4C2212yes—
33#89982C2212yes—
34#89982C2212yes—
35C4×C22⋊Dic7C2214yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38C2×C7⋊#95C2212yes—
39#89722C2212yes—
40#89722C2212yes—
41#89722C2214yes—
42#89718C2212yes—
43#89718C2212yes—
44#89718C2214yes—
45#89718C2212yes—
46#89718C2212yes—
47#89714C2214yes—
48C22⋊C4×Dic7C2214yes—
49C22⋊C4×Dic7C2214yes—
50C22⋊C4×Dic7C2212yes—
51C22⋊C4×Dic7C2214yes—
52C22⋊C4×Dic7C2212yes—
53C4×C4⋊Dic7C2214yescomplemented
54#89705C2214yescomplemented
55#89705C2214yescomplemented
56C4×Dic7⋊C4C2214yescomplemented
57C4×Dic7⋊C4C2214yescomplemented
58#89701C2214yescomplemented
59#89700C2214yescomplemented
60#89427C2214yescomplemented
61#89427C2214yescomplemented
62#89426C2214yescomplemented
63#89426C2214yescomplemented
64#89419C2214yescomplemented
65#89419C2214yescomplemented
66#89418C2214yescomplemented
67#89418C2214yescomplemented
68#34270—71nomaximal, Sylow
69C25×C7C2311yescharacteristic
70C23×C4×C7C2311yescharacteristic
71C23×C4×C7C2311yescharacteristic
72C23×C4×C7C2312yes—
73C23×C4×C7C2312yes—
74C23×Dic7—24no—
75C23×Dic7—44no—
76C23×Dic7C2311yescharacteristic
77C23×Dic7C2312yes—
78C23×Dic7C2312yes—
79C23×Dic7C2311yescharacteristic
80C23×Dic7C2311yescharacteristic
81C23×Dic7C2311yescharacteristic
82C23×Dic7C2312yes—
83C23×Dic7C2312yes—
84C22×Dic14C2314yes—
85C2×C7×C22⋊C4—24no—
86C2×C7×C22⋊C4C2×C418yes—
87C2×C7×C22⋊C4Q818yescomplemented
88C2×C7×C22⋊C4C2312yes—
89C2×C7×C22⋊C4C2312yes—
90C2×C7×C22⋊C4C2312yes—
91C2×C7×C22⋊C4C2312yes—
92C2×C7×C22⋊C4C2312yes—
93C2×C7×C22⋊C4C2312yes—
94C2×C42×C7C2314yes—
95C2×C42×C7C2314yes—
96C2×C22⋊Dic7—24no—
97C2×C22⋊Dic7C2×C418yes—
98C2×C22⋊Dic7C2×C418yes—
99C2×C22⋊Dic7C2312yes—
100C2×C22⋊Dic7C2312yes—
101C2×C22⋊Dic7C2312yes—
102C2×C22⋊Dic7C2312yes—
103C2×C22⋊Dic7C2312yes—
104C2×C22⋊Dic7C2312yes—
105C2×C4⋊Dic7—44no—
106C2×C4⋊Dic7C2314yes—
107C2×C4⋊Dic7C2314yes—
108C2×C4⋊Dic7C2314yes—
109#6487C2×C4132yescomplemented
110C2×Dic7⋊C4—44no—
111C2×Dic7⋊C4C2314yes—
112C2×Dic7⋊C4C2314yes—
113C2×Dic7⋊C4C2314yes—
114C2×Dic7⋊C4C2314yes—
115C2×Dic7⋊C4C2314yes—
116C2×Dic7⋊C4C2314yes—
117C2×C4×Dic7—44no—
118C2×C4×Dic7C2314yes—
119C2×C4×Dic7C2314yes—
120C2×C4×Dic7C2314yes—
121C2×C4×Dic7C2314yes—
122#6441C2×C4132yescomplemented
123#6441C2×C4132yescomplemented
124C4×Dic14—48no—
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×#385—71no—
147#2273—71no—
148#2273D711yescharacteristic
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—22no—
160C22×C4×C7—42no—
161C22×C4×C7—42no—
162C22×C4×C7—42no—
163C22×C4×C7—42no—
164C22×C4×C7—42no—
165C22×C4×C7—42no—
166C22×C4×C7C22×C418yes—
167C22×C4×C7C2×Q818yes—
168C22×C4×C7C2412yes—
169C22×C4×C7C2412yes—
170C22×C4×C7C2414yes—
171C22×C4×C7C2414yes—
172C22×Dic7—22no—
173C22×Dic7—24no—
174C22×Dic7—24no—
175C22×Dic7—24no—
176C22×Dic7—22no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—24no—
180C22×Dic7—24no—
181C22×Dic7—42no—
182C22×Dic7—42no—
183C22×Dic7—42no—
184C22×Dic7—42no—
185C22×Dic7—42no—
186C22×Dic7—42no—
187C22×Dic7—42no—
188C22×Dic7—42no—
189C22×Dic7—42no—
190C22×Dic7—416no—
191C22×Dic7C22×C418yes—
192C22×Dic7C22×C418yes—
193C22×Dic7C2412yes—
194C22×Dic7C2412yes—
195C22×Dic7C2412yes—
196C22×Dic7C2414yes—
197C22×Dic7C2414yes—
198C22×Dic7C2412yes—
199C2×Dic14—44no—
200C2×Dic14—44no—
201C2×Dic14C22×C4116yescomplemented
202C7×C22⋊C4—24no—
203C7×C22⋊C4—24no—
204C7×C22⋊C4C22×C418yes—
205C7×C22⋊C4C22×C418yes—
206C7×C22⋊C4C2×Q818yes—
207C7×C22⋊C4C2×Q818yes—
208C42×C7—44no—
209C42×C7—44no—
210C22⋊Dic7—24no—
211C22⋊Dic7—24no—
212C22⋊Dic7C22×C418yes—
213C22⋊Dic7C22×C418yes—
214C22⋊Dic7C22×C418yes—
215C22⋊Dic7C22×C418yes—
216C4⋊Dic7—28no—
217C4⋊Dic7—44no—
218C4⋊Dic7C22×C4116yescomplemented
219Dic7⋊C4—28no—
220Dic7⋊C4—44no—
221Dic7⋊C4—44no—
222Dic7⋊C4—44no—
223Dic7⋊C4C22×C4116yescomplemented
224Dic7⋊C4C22×C4116yescomplemented
225C4×Dic7—28no—
226C4×Dic7—44no—
227C4×Dic7—44no—
228C4×Dic7—44no—
229C24×C4—71no—
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⋊C4—72no—
237C22×C22⋊C4—71no—
238C22×C22⋊C4—71no—
239C22×C22⋊C4—71no—
240C22×C22⋊C4C2×D711yescharacteristic
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—72no—
260#385—72no—
261#385—72no—
262#385—74no—
263#385—74no—
264#385—72no—
265#385—72no—
266#380—74no—
267C4×C4⋊C4—74no—
268C4×C4⋊C4—74no—
269C4×C4⋊C4—74no—
270C4×C22⋊C4—74no—
271C4×C22⋊C4—72no—
272C4×C22⋊C4—74no—
273C4×C22⋊C4—74no—
274C4×C22⋊C4—74no—
275C4×C22⋊C4—72no—
276C4×C22⋊C4C2×D714yes—
277C4×C22⋊C4C2×D712yes—
278C4×C22⋊C4C2×D714yes—
279C4×C22⋊C4C2×D712yes—
280C2×#95—72no—
281C2×#95—72no—
282C2×#95—72no—
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—24no—
305C2×C4×C7—24no—
306C2×C4×C7—24no—
307C2×C4×C7—24no—
308C2×C4×C7—22no—
309C2×C4×C7—22no—
310C2×C4×C7—24no—
311C2×C4×C7—24no—
312C2×C4×C7—48no—
313C2×C4×C7—48no—
314C2×C4×C7—82no—
315C2×C4×C7—82no—
316C2×C4×C7C23×C418yes—
317C2×C4×C7C22×Q818yes—
318C2×Dic7—28no—
319C2×Dic7—22no—
320C2×Dic7—24no—
321C2×Dic7—22no—
322C2×Dic7—24no—
323C2×Dic7—24no—
324C2×Dic7—24no—
325C2×Dic7—22no—
326C2×Dic7—22no—
327C2×Dic7—22no—
328C2×Dic7—24no—
329C2×Dic7—22no—
330C2×Dic7—22no—
331C2×Dic7—24no—
332C2×Dic7—48no—
333C2×Dic7—48no—
334C2×Dic7—48no—
335C2×Dic7—48no—
336C2×Dic7—44no—
337C2×Dic7—82no—
338C2×Dic7—82no—
339C2×Dic7C23×C418yes—
340C2×Dic7C23×C418yes—
341C2×Dic7H(5,𝔽2)14yes—
342Dic14—416no—
343C25C22×D711yescharacteristic
344C22×Q8—74no—
345C23×C4—72no—
346C23×C4—72no—
347C23×C4—71no—
348C23×C4—72no—
349C23×C4—72no—
350C23×C4—71no—
351C23×C4—71no—
352C23×C4—71no—
353C23×C4—144no—
354C23×C4—284no—
355C23×C4C22×D711yescharacteristic
356C23×C4C22×D711yescharacteristic
357C23×C4C22×D712yes—
358C23×C4C22×D712yes—
359C22⋊Q8—732no—
360C22⋊Q8—732no—
361C22⋊Q8—732no—
362C4×Q8—288no—
363C2×C4⋊C4—74no—
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—74no—
372C2×C4⋊C4—284no—
373C2×C4⋊C4—284no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—78no—
376C2×C22⋊C4—78no—
377C2×C22⋊C4—72no—
378C2×C22⋊C4—72no—
379C2×C22⋊C4—72no—
380C2×C22⋊C4—72no—
381C2×C22⋊C4—72no—
382C2×C22⋊C4—72no—
383C2×C22⋊C4—144no—
384C2×C22⋊C4Dic1418yescomplemented
385C2×C22⋊C4C4×D718yes—
386C2×C22⋊C4C22×D712yes—
387C2×C22⋊C4C22×D712yes—
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—284no—
397C2×C42C22×D714yes—
398C2×C42C22×D714yes—
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—28no—
434C4×C7—42no—
435C4×C7—42no—
436Dic7—28no—
437Dic7—28no—
438Dic7—24no—
439Dic7—42no—
440Dic7—42no—
441Dic7—42no—
442C24—42no—
443C24—42no—
444C24C2×Dic1414yes—
445C24C2×C4×D714yes—
446C24#115714yes—
447C24C23×D711yescharacteristic
448C24C23×D711yescharacteristic
449C24C23×D711yescharacteristic
450C2×Q8—716no—
451C2×Q8—284no—
452C2×Q8—284no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—72no—
462C22×C4—74no—
463C22×C4—74no—
464C22×C4—78no—
465C22×C4—78no—
466C22×C4—72no—
467C22×C4—72no—
468C22×C4—72no—
469C22×C4—144no—
470C22×C4—144no—
471C22×C4—144no—
472C22×C4—142no—
473C22×C4—144no—
474C22×C4—144no—
475C22×C4—142no—
476C22×C4—142no—
477C22×C4—142no—
478C22×C4—282no—
479C22×C4—282no—
480C22×C4—282no—
481C22×C4—282no—
482C22×C4—282no—
483C22×C4—282no—
484C22×C4—282no—
485C22×C4—282no—
486C22×C4—282no—
487C22×C4—2816no—
488C22×C4C2×Dic1418yes—
489C22×C4C2×C4×D718yes—
490C22×C4C23×D712yes—
491C22×C4C23×D712yes—
492C22×C4C23×D714yes—
493C22×C4C23×D714yes—
494C4⋊C4—716no—
495C4⋊C4—716no—
496C4⋊C4—716no—
497C4⋊C4—148no—
498C4⋊C4—148no—
499C4⋊C4—284no—
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—144no—
510C22⋊C4—144no—
511C22⋊C4C2×Dic1418yes—
512C22⋊C4C2×Dic1418yes—
513C22⋊C4C2×C4×D718yes—
514C22⋊C4C2×C4×D718yes—
515C42—44no—
516C42—44no—
517C42—148no—
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—48no—
538C23—44no—
539C23C4×Dic14116yescomplemented
540C23C22×Dic1412yes—
541C23C22×Dic1412yes—
542C23C22×Dic1412yes—
543C23C22×C4×D712yes—
544C23C22×C4×D712yes—
545C23C22×C4×D712yes—
546C23C2×#115712yes—
547C23C2×#115712yes—
548C23C2×#115712yes—
549C23C24×D711yescharacteristic
550Q8—2816no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—78no—
562C2×C4—78no—
563C2×C4—74no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—148no—
567C2×C4—142no—
568C2×C4—144no—
569C2×C4—142no—
570C2×C4—144no—
571C2×C4—144no—
572C2×C4—144no—
573C2×C4—142no—
574C2×C4—142no—
575C2×C4—142no—
576C2×C4—144no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—144no—
580C2×C4—288no—
581C2×C4—288no—
582C2×C4—288no—
583C2×C4—288no—
584C2×C4—284no—
585C2×C4—562no—
586C2×C4—562no—
587C2×C4C22×Dic1418yes—
588C2×C4C22×C4×D718yes—
589C7#3427011yescharacteristic, complemented, Sylow
590C22—24no—
591C22—24no—
592C22—24no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—416no—
603C22C2×C4×Dic1414yes—
604C22C2×C4×Dic1414yes—
605C22C2×C4×Dic1414yes—
606C22C23×Dic1411yescharacteristic
607C22C23×C4×D711yescharacteristic
608C22C22×#115711yescharacteristic
609C22D7×H(5,𝔽2)12yes—
610C22#9061812yes—
611C4—28no—
612C4—28no—
613C4—42no—
614C4—42no—
615C4—148no—
616C4—148no—
617C4—144no—
618C4—282no—
619C4—282no—
620C4—282no—
621C2—24no—
622C2—24no—
623C2—24no—
624C2C22×C4×Dic1411yescharacteristic
625C2#1174292512yes—
626C2#1174292711yescharacteristic
627C2#1174292812yes—
628C2#1174293611yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2273C2
socle, cosocleC23×C7C25
prime radicalC25×C7C23
hypercenter#2273D7
hypocenterC7#34270
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2273

Complements

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

NH ≅ G/Ntype
8. #11730919C2semidirect
9. #11730918C2semidirect
10. #11730911C2semidirect
11. #11730910C2semidirect
12. #11730892C2semidirect
13. C4×#6441C2semidirect
25. #90174C4semidirect
28. C2×C4×Dic14C22semidirect
53. C4×C4⋊Dic7C22semidirect
54. #89705C22semidirect
55. #89705C22semidirect
56. C4×Dic7⋊C4C22semidirect
57. C4×Dic7⋊C4C22semidirect
58. #89701C22semidirect
59. #89700C22semidirect
60. #89427C22semidirect
61. #89427C22semidirect
62. #89426C22semidirect
63. #89426C22semidirect
64. #89419C22semidirect
65. #89419C22semidirect
66. #89418C22semidirect
67. #89418C22semidirect
87. C2×C7×C22⋊C4Q8semidirect
109. #6487C2×C4semidirect
122. #6441C2×C4semidirect
123. #6441C2×C4semidirect
201. C2×Dic14C22×C4semidirect
218. C4⋊Dic7C22×C4semidirect
223. Dic7⋊C4C22×C4semidirect
224. Dic7⋊C4C22×C4semidirect
293. C23×C7C4×Q8semidirect
384. C2×C22⋊C4Dic14semidirect
539. C23C4×Dic14semidirect
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
C7×#22732
#117395402
C2×#897222
C2×#897182
#117395282
C2×C22⋊C4×Dic72
#117309192
#117309182
#117309112
#117309102
#117308922
C4×#64412
#342707