Finite Group Databasebeta

← #31919147

Subgroups — #31919147

Order 18176.

1814 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
1C2×C4×Dic71⋊C4C211yesmaximal, characteristic, complemented
2C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
3C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
4C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
5#22586600C212yesmaximal, complemented
6#22586595C212yesmaximal, complemented
7#22586588C212yesmaximal
8#22586584C212yesmaximal
9C4×C71⋊#95C211yesmaximal, characteristic, complemented
10C4×C71⋊#95C211yesmaximal, characteristic, complemented
11C2×#376×C71C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C22×Dic71⋊C4C2212yes—
14C22×(C2×D71)⋊C4C2212yes—
15C22×(C2×D71)⋊C4C2212yes—
16C22×(C2×D71)⋊C4C2211yescharacteristic
17C22×(C2×D71)⋊C4C2211yescharacteristic
18C22×C4×Dic71C2211yescharacteristic
19C22×C4×Dic71C2211yescharacteristic
20C22×C4×Dic71C2211yescharacteristic
21C4×Dic71⋊C4—24no—
22C4×(C2×D71)⋊C4—24no—
23C4×(C2×D71)⋊C4—24no—
24C4×(C2×D71)⋊C4C418yescomplemented
25C2×C71⋊#95C2212yes—
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2212yes—
31C2×C71⋊#95C2212yes—
32C2×C71⋊#95C2211yescharacteristic
33C2×C71⋊#95C2211yescharacteristic
34C22×C42×C71C2211yescharacteristic
35C22×C42×C71C2211yescharacteristic
36C22×C42×C71C2211yescharacteristic
37#376×C71—24no—
38C2×#95×C71C2212yes—
39C2×#95×C71C2212yes—
40C24×D71C2311yescharacteristic
41C23×Dic71C2311yescharacteristic
42C23×Dic71C2311yescharacteristic
43C23×Dic71C2312yes—
44C23×Dic71C2312yes—
45C23×Dic71C2311yescharacteristic
46C22×C4×D71—22no—
47C22×C4×D71—22no—
48C22×C4×D71—22no—
49C22×C4×D71—22no—
50C22×C4×D71—22no—
51C22×C4×D71—22no—
52C22×C4×D71C2×C414yes—
53C2×Dic71⋊C4—24no—
54C2×Dic71⋊C4—24no—
55C2×Dic71⋊C4—24no—
56C2×(C2×D71)⋊C4—24no—
57C2×(C2×D71)⋊C4—24no—
58C2×(C2×D71)⋊C4—24no—
59C2×(C2×D71)⋊C4—24no—
60C2×(C2×D71)⋊C4—24no—
61C2×(C2×D71)⋊C4—24no—
62C2×(C2×D71)⋊C4—24no—
63C2×(C2×D71)⋊C4—24no—
64C2×(C2×D71)⋊C4C2×C414yes—
65C2×(C2×D71)⋊C4C2×C414yes—
66C2×C4×Dic71—22no—
67C2×C4×Dic71—22no—
68C2×C4×Dic71—22no—
69C2×C4×Dic71—22no—
70C2×C4×Dic71—22no—
71C2×C4×Dic71—22no—
72C2×C4×Dic71—22no—
73C2×C4×Dic71—22no—
74C2×C4×Dic71C2×C414yes—
75C71⋊#95—24no—
76C71⋊#95—24no—
77C71⋊#95—24no—
78C71⋊#95—24no—
79C71⋊#95—24no—
80C71⋊#95—24no—
81C71⋊#95—24no—
82C71⋊#95C2×C414yes—
83C71⋊#95C2×C414yes—
84C23×C4×C71C2311yescharacteristic
85C23×C4×C71C2311yescharacteristic
86C23×C4×C71C2312yes—
87C23×C4×C71C2312yes—
88C23×C4×C71C2311yescharacteristic
89C2×C42×C71—22no—
90C2×C42×C71—22no—
91C2×C42×C71—22no—
92C2×C42×C71—22no—
93C2×C42×C71—22no—
94C2×C42×C71—22no—
95C2×C42×C71C2×C414yes—
96C2×C42×C71D414yes—
97C2×C42×C71D414yes—
98#95×C71—24no—
99#95×C71—24no—
100C23×D71—21no—
101C23×D71—21no—
102C23×D71—21no—
103C23×D71—21no—
104C23×D71—21no—
105C23×D71—21no—
106C23×D71—22no—
107C23×D71—22no—
108C23×D71—22no—
109C23×D71C22×C412yes—
110C23×D71#4114yes—
111C22×Dic71—21no—
112C22×Dic71—21no—
113C22×Dic71—22no—
114C22×Dic71—21no—
115C22×Dic71—21no—
116C22×Dic71—21no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—21no—
121C22×Dic71—21no—
122C22×Dic71—21no—
123C22×Dic71—21no—
124C22×Dic71—22no—
125C22×Dic71—22no—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—21no—
129C22×Dic71—22no—
130C22×Dic71—22no—
131C22×Dic71—22no—
132C22×Dic71—22no—
133C22×Dic71—22no—
134C22×Dic71—22no—
135C22×Dic71—22no—
136C22×Dic71C22×C412yes—
137C22×Dic71C22×C412yes—
138C22×Dic71C22×C412yes—
139C22×Dic71#4114yes—
140C22×Dic71#4114yes—
141C22×Dic71#4114yes—
142C2×C4×D71—28no—
143C2×C4×D71—28no—
144C2×C4×D71—28no—
145C2×C4×D71—44no—
146C2×C4×D71—44no—
147C2×C4×D71—44no—
148C2×C4×D71—44no—
149Dic71⋊C4—216no—
150(C2×D71)⋊C4—216no—
151(C2×D71)⋊C4—216no—
152(C2×D71)⋊C4—28no—
153(C2×D71)⋊C4—28no—
154C4×Dic71—28no—
155C4×Dic71—44no—
156C4×Dic71—44no—
157C24×C71C2411yescharacteristic
158C22×C4×C71—21no—
159C22×C4×C71—21no—
160C22×C4×C71—22no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71—22no—
168C22×C4×C71—22no—
169C22×C4×C71—22no—
170C22×C4×C71—22no—
171C22×C4×C71—22no—
172C22×C4×C71—22no—
173C22×C4×C71—22no—
174C22×C4×C71—22no—
175C22×C4×C71—22no—
176C22×C4×C71—22no—
177C22×C4×C71—22no—
178C22×C4×C71C22×C412yes—
179C22×C4×C71C22×C412yes—
180C22×C4×C71C22×C412yes—
181C22×C4×C71C2×D414yes—
182C22×C4×C71C2×D412yes—
183C22×C4×C71C2×D414yes—
184C22×C4×C71C2×D412yes—
185C22×C4×C71#4112yes—
186C22×C4×C71#4112yes—
187C22×C4×C71#4112yes—
188C22×C4×C71#4112yes—
189C42×C71—28no—
190C42×C71—28no—
191C42×C71—28no—
192C23×C71C23×C411yescharacteristic
193C23×C71C22×D411yescharacteristic
194C23×C71C22×D411yescharacteristic
195C23×C71C2×#4112yes—
196C23×C71C2×#4112yes—
197C23×C71C2×#4112yes—
198C23×C71C2×#4112yes—
199C23×C71C2×#4111yescharacteristic
200C23×C71C2×#4111yescharacteristic
201C23×C71C2×#4111yescharacteristic
202C23×C71C2×#4111yescharacteristic
203C22×D71—22no—
204C22×D71—22no—
205C22×D71—22no—
206C22×D71—22no—
207C22×D71—22no—
208C22×D71—22no—
209C22×D71—24no—
210C22×D71—24no—
211C22×D71—24no—
212C22×D71—24no—
213C22×D71—24no—
214C22×D71—24no—
215C22×D71—41no—
216C22×D71—41no—
217C22×D71—42no—
218C22×D71—41no—
219C22×D71—41no—
220C22×D71—42no—
221C22×D71—42no—
222C22×D71—42no—
223C22×D71—42no—
224C22×D71—42no—
225C22×D71#11714yes—
226C2×C4×C71—22no—
227C2×C4×C71—22no—
228C2×C4×C71—22no—
229C2×C4×C71—22no—
230C2×C4×C71—22no—
231C2×C4×C71—22no—
232C2×C4×C71—24no—
233C2×C4×C71—24no—
234C2×C4×C71—24no—
235C2×C4×C71—24no—
236C2×C4×C71—24no—
237C2×C4×C71—24no—
238C2×C4×C71—42no—
239C2×C4×C71—42no—
240C2×C4×C71—42no—
241C2×C4×C71—42no—
242C2×C4×C71—42no—
243C2×C4×C71—42no—
244C2×C4×C71—42no—
245C2×C4×C71—42no—
246C2×C4×C71—42no—
247C2×C4×C71—42no—
248C2×C4×C71—42no—
249C2×C4×C71—42no—
250C2×C4×C71#11714yes—
251C2×C4×C71C4×D414yes—
252C2×C4×C71C4×D414yes—
253C2×C4×C71#12114yes—
254C2×C4×C71#12314yes—
255C2×C4×C71#12414yes—
256C2×C4×C71#12614yes—
257C2×Dic71—22no—
258C2×Dic71—22no—
259C2×Dic71—24no—
260C2×Dic71—24no—
261C2×Dic71—22no—
262C2×Dic71—22no—
263C2×Dic71—22no—
264C2×Dic71—22no—
265C2×Dic71—22no—
266C2×Dic71—22no—
267C2×Dic71—24no—
268C2×Dic71—24no—
269C2×Dic71—24no—
270C2×Dic71—24no—
271C2×Dic71—24no—
272C2×Dic71—24no—
273C2×Dic71—42no—
274C2×Dic71—41no—
275C2×Dic71—41no—
276C2×Dic71—42no—
277C2×Dic71—42no—
278C2×Dic71—42no—
279C2×Dic71—42no—
280C2×Dic71—41no—
281C2×Dic71—41no—
282C2×Dic71—42no—
283C2×Dic71—42no—
284C2×Dic71—42no—
285C2×Dic71—42no—
286C2×Dic71—42no—
287C2×Dic71#11714yes—
288C4×D71—416no—
289C22×C71C2×#11711yescharacteristic
290C22×C71C2×#11711yescharacteristic
291C22×C71C2×#11711yescharacteristic
292C22×C71C2×C4×D411yescharacteristic
293C22×C71C2×C4×D411yescharacteristic
294C22×C71C4×#4111yescharacteristic
295C22×C71C4×#4111yescharacteristic
296C22×C71C2×#12111yescharacteristic
297C22×C71C2×#12311yescharacteristic
298C22×C71#52512yes—
299C22×C71#52512yes—
300C22×C71#52512yes—
301C22×C71C2×#12411yescharacteristic
302C22×C71C2×#12611yescharacteristic
303C22×C71#52912yes—
304C22×C71#52912yes—
305C22×C71#52912yes—
306C22×C71#52912yes—
307C22×C71#52912yes—
308C22×C71#52912yes—
309C22×C71#52912yes—
310C22×C71#52912yes—
311C22×C71#53212yes—
312C2×D71—24no—
313C2×D71—24no—
314C2×D71—24no—
315C2×D71—44no—
316C2×D71—44no—
317C2×D71—44no—
318C2×D71—44no—
319C2×D71—42no—
320C2×D71—42no—
321C2×D71—42no—
322C2×D71—42no—
323C4×C71—44no—
324C4×C71—44no—
325C4×C71—42no—
326C4×C71—42no—
327C4×C71#38818yes—
328Dic71—24no—
329Dic71—44no—
330Dic71—44no—
331Dic71—42no—
332Dic71—42no—
333#13842—711nomaximal, Sylow
334C2×C71#228311yescharacteristic
335C2×C71C2×#38811yescharacteristic
336C2×C71#229011yescharacteristic
337C2×C71#229011yescharacteristic
338C2×C71#229011yescharacteristic
339C2×C71#229311yescharacteristic
340C2×C71#229811yescharacteristic
341C2×C71#239212yes—
342C2×C71#239512yes—
343C2×C71#239812yes—
344C2×C71#240112yes—
345D71—44no—
346C2×C4×C4⋊C4—711no—
347C2×C4×C22⋊C4—711no—
348C2×C4×C22⋊C4—711no—
349C2×C4×C22⋊C4—711no—
350C2×#376D7111yescharacteristic, complemented
351#1443—712no—
352#1442—712no—
353#1440—712no—
354#1439—712no—
355C4×#95—711no—
356C4×#95—711no—
357C71#1384211yescharacteristic, complemented, Sylow
358C24×C4—711no—
359C22×C4⋊C4—712no—
360C22×C22⋊C4—711no—
361C22×C22⋊C4—711no—
362C22×C22⋊C4—712no—
363C22×C22⋊C4—712no—
364C22×C42—711no—
365C22×C42—711no—
366C22×C42—711no—
367C22×C42C2×D7111yescharacteristic
368C22×C42C2×D7111yescharacteristic
369C22×C42C2×D7111yescharacteristic
370C4×C4⋊C4—1424no—
371C4×C22⋊C4—718no—
372C4×C22⋊C4—1424no—
373C4×C22⋊C4—1424no—
374#376—24no—
375C2×#95—711no—
376C2×#95—711no—
377C2×#95—712no—
378C2×#95—712no—
379C2×#95—712no—
380C2×#95—712no—
381C2×#95—712no—
382C2×#95—712no—
383C2×#95—712no—
384C2×#95C2×D7112yes—
385C2×#95C2×D7112yes—
386C25—711no—
387C23×C4—711no—
388C23×C4—712no—
389C23×C4—711no—
390C23×C4—712no—
391C23×C4—711no—
392C23×C4—714no—
393C23×C4—1422no—
394C23×C4—1422no—
395C23×C4—1422no—
396C23×C4—1422no—
397C23×C4—1422no—
398C23×C4—1422no—
399C23×C4C22×D7111yescharacteristic
400C23×C4C22×D7111yescharacteristic
401C23×C4C22×D7112yes—
402C23×C4C22×D7112yes—
403C23×C4C22×D7111yescharacteristic
404C2×C4⋊C4—1424no—
405C2×C4⋊C4—1424no—
406C2×C4⋊C4—1424no—
407C2×C22⋊C4—714no—
408C2×C22⋊C4—714no—
409C2×C22⋊C4—1424no—
410C2×C22⋊C4—1424no—
411C2×C22⋊C4—1424no—
412C2×C22⋊C4—1424no—
413C2×C22⋊C4—1424no—
414C2×C22⋊C4—1424no—
415C2×C22⋊C4—1424no—
416C2×C22⋊C4—1424no—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42—22no—
420C2×C42—22no—
421C2×C42—22no—
422C2×C42—22no—
423C2×C42—714no—
424C2×C42—1422no—
425C2×C42—1422no—
426C2×C42—1422no—
427C2×C42—1422no—
428C2×C42—1422no—
429C2×C42—1422no—
430C2×C42—1422no—
431C2×C42—1422no—
432C2×C42C4×D7114yes—
433C2×C42D28414yes—
434C2×C42C22⋊D7114yes—
435#95—24no—
436#95—24no—
437#95—714no—
438#95—714no—
439#95—1424no—
440#95—1424no—
441#95—1424no—
442#95—1424no—
443#95—1424no—
444#95—1424no—
445#95—1424no—
446C24—712no—
447C24—714no—
448C24—1421no—
449C24—1421no—
450C24—1422no—
451C24—1422no—
452C24—1422no—
453C24—1421no—
454C24—1421no—
455C24—1421no—
456C24—1421no—
457C24C23×D7111yescharacteristic
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—2844no—
479C22×C4—2844no—
480C22×C4—2844no—
481C22×C4—2844no—
482C22×C4—712no—
483C22×C4—714no—
484C22×C4—712no—
485C22×C4—712no—
486C22×C4—714no—
487C22×C4—714no—
488C22×C4—1421no—
489C22×C4—1422no—
490C22×C4—1421no—
491C22×C4—1421no—
492C22×C4—1422no—
493C22×C4—1422no—
494C22×C4—1422no—
495C22×C4—1422no—
496C22×C4—1422no—
497C22×C4—1422no—
498C22×C4—1422no—
499C22×C4—1422no—
500C22×C4—1422no—
501C22×C4—1422no—
502C22×C4—1422no—
503C22×C4—1422no—
504C22×C4—1422no—
505C22×C4—1421no—
506C22×C4—1421no—
507C22×C4—1421no—
508C22×C4—1421no—
509C22×C4—1421no—
510C22×C4—1428no—
511C22×C4—1428no—
512C22×C4—1428no—
513C22×C4—1421no—
514C22×C4—1422no—
515C22×C4—1421no—
516C22×C4C2×C4×D7112yes—
517C22×C4C2×C4×D7112yes—
518C22×C4C2×C4×D7112yes—
519C22×C4C2×C22⋊D7114yes—
520C22×C4C2×C22⋊D7112yes—
521C22×C4C2×D28414yes—
522C22×C4C2×D28412yes—
523C22×C4#1176586512yes—
524C22×C4#1176586512yes—
525C22×C4#1176586512yes—
526C22×C4#1176586512yes—
527C4⋊C4—14216no—
528C22⋊C4—1428no—
529C22⋊C4—1428no—
530C22⋊C4—14216no—
531C22⋊C4—14216no—
532C42—28no—
533C42—28no—
534C42—28no—
535C42—2844no—
536C42—2844no—
537C42—1428no—
538C23—2841no—
539C23—2841no—
540C23—2842no—
541C23—2841no—
542C23—2841no—
543C23—2842no—
544C23—2842no—
545C23—2842no—
546C23—2842no—
547C23—2842no—
548C23—714no—
549C23—1422no—
550C23—1422no—
551C23—1422no—
552C23—1422no—
553C23—1422no—
554C23—1422no—
555C23—1424no—
556C23—1424no—
557C23—1424no—
558C23—1424no—
559C23—1424no—
560C23—1424no—
561C23C22×C4×D7111yescharacteristic
562C23C22×C22⋊D7111yescharacteristic
563C23C22×D28411yescharacteristic
564C23#41×D7112yes—
565C23#41×D7112yes—
566C23#41×D7112yes—
567C23#41×D7112yes—
568C23C2×#1176586511yescharacteristic
569C23C2×#1176586511yescharacteristic
570C23C2×#1176586511yescharacteristic
571C23C2×#1176586511yescharacteristic
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—2842no—
597C2×C4—2841no—
598C2×C4—2841no—
599C2×C4—2842no—
600C2×C4—2842no—
601C2×C4—2842no—
602C2×C4—2842no—
603C2×C4—2841no—
604C2×C4—2841no—
605C2×C4—2842no—
606C2×C4—2842no—
607C2×C4—2842no—
608C2×C4—2842no—
609C2×C4—2842no—
610C2×C4—28416no—
611C2×C4—714no—
612C2×C4—1422no—
613C2×C4—1422no—
614C2×C4—1422no—
615C2×C4—1422no—
616C2×C4—1422no—
617C2×C4—1422no—
618C2×C4—1424no—
619C2×C4—1424no—
620C2×C4—1424no—
621C2×C4—1424no—
622C2×C4—1424no—
623C2×C4—1424no—
624C2×C4—1422no—
625C2×C4—1422no—
626C2×C4—1424no—
627C2×C4—1424no—
628C2×C4#1455608514yes—
629C2×C4C4×C22⋊D7114yes—
630C2×C4C4×D28414yes—
631C2×C4#1455610214yes—
632C2×C4#1455611114yes—
633C2×C4#1455611614yes—
634C2×C4#1455612214yes—
635C22—2842no—
636C22—2842no—
637C22—2842no—
638C22—2844no—
639C22—2844no—
640C22—2842no—
641C22—2844no—
642C22—2844no—
643C22—1424no—
644C22—1424no—
645C22—1424no—
646C22#117×D7111yescharacteristic
647C22#117×D7111yescharacteristic
648C22C2×#1455608511yescharacteristic
649C22C2×C4×C22⋊D7111yescharacteristic
650C22C2×C4×D28411yescharacteristic
651C22#1798646711yescharacteristic
652C22#1798646711yescharacteristic
653C22C2×#1455610211yescharacteristic
654C22C2×#1455611111yescharacteristic
655C22#1798650612yes—
656C22#1798651112yes—
657C22#1798651112yes—
658C22C2×#1455611611yescharacteristic
659C22C2×#1455612211yescharacteristic
660C22#1798652712yes—
661C22#1798652712yes—
662C22#1798652712yes—
663C22#1798652712yes—
664C22#1798652712yes—
665C22#1798652712yes—
666C22#1798652712yes—
667C22#1798652712yes—
668C22#1798654512yes—
669C4—42no—
670C4—42no—
671C4—44no—
672C4—44no—
673C4—2842no—
674C4—2842no—
675C4—2844no—
676C4—2844no—
677C4—1424no—
678C4#1798589018yes—
679C2—2844no—
680C2#2259073211yescharacteristic
681C2C2×#1798589011yescharacteristic
682C2#2259077711yescharacteristic
683C2#2259078411yescharacteristic
684C2#2259078411yescharacteristic
685C2#2259079911yescharacteristic
686C2#2259083511yescharacteristic
687C2#2259157612yes—
688C2#2259159712yes—
689C2#2259162612yes—
690C2#2259163812yes—
subgroupG/H
centerC23×C4C22×D71
commutatorC23×C71C23×C4
FrattiniC24C23×D71
FittingC2×#376×C71C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC2×#376D71
hypocenterC71#13842
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23×C4 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic71⋊C4C2semidirect
5. #22586600C2semidirect
6. #22586595C2semidirect
9. C4×C71⋊#95C2semidirect
10. C4×C71⋊#95C2semidirect
11. C2×#376×C71C2semidirect
24. C4×(C2×D71)⋊C4C4semidirect
350. C2×#376D71semidirect
357. C71#13842semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic71⋊C42
C2×C4×(C2×D71)⋊C42
#225866002
#225865952
#225865882
#225865842
C4×C71⋊#952
C2×#376×C712
#1384271