Finite Group Databasebeta

← #31922747

Subgroups — #31922747

Order 18176.

2062 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985910C211yesmaximal, characteristic
2C2×#17985870C212yesmaximal
3C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
4#22586611C211yesmaximal, characteristic, complemented
5#22586603C211yesmaximal, characteristic, complemented
6#22586582C211yesmaximal, characteristic
7#22586578C212yesmaximal
8#22586573C212yesmaximal, complemented
9#22586572C212yesmaximal, complemented
10#22586567C211yesmaximal, characteristic, complemented
11C22×C4×Q8×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
19#17985910C418yescomplemented
20#17985870—28no—
21C4×(C2×D71)⋊C4—24no—
22C2×C71⋊#95C2212yes—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2211yescharacteristic
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2211yescharacteristic
28C2×C71⋊#95C2211yescharacteristic
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2212yes—
31C2×C71⋊#95C2211yescharacteristic
32C2×C71⋊#95C2212yes—
33C23×Q8×C71C2211yescharacteristic
34C2×C4×Q8×C71—28no—
35C2×C4×Q8×C71—24no—
36C22×C4⋊C4×C71C2211yescharacteristic
37C22×C4⋊C4×C71C2212yes—
38C22×C42×C71C2212yes—
39C22×C42×C71C2211yescharacteristic
40C24×D71C2311yescharacteristic
41C23×Dic71C2312yes—
42C23×Dic71C2311yescharacteristic
43C23×Dic71C2312yes—
44C23×Dic71C2311yescharacteristic
45C23×Dic71C2311yescharacteristic
46C22×C4×D71—22no—
47C22×C4×D71—22no—
48C22×C4×D71—22no—
49C22×C4×D71—24no—
50C22×C4×D71—42no—
51C2×Dic71⋊C4—24no—
52C2×Dic71⋊C4—24no—
53C2×Dic71⋊C4—42no—
54C2×(C2×D71)⋊C4—24no—
55C2×(C2×D71)⋊C4—22no—
56C2×(C2×D71)⋊C4—22no—
57C2×(C2×D71)⋊C4—24no—
58C2×(C2×D71)⋊C4—24no—
59C2×(C2×D71)⋊C4—24no—
60C2×(C2×D71)⋊C4—22no—
61C2×(C2×D71)⋊C4—22no—
62C2×(C2×D71)⋊C4—42no—
63C2×(C2×D71)⋊C4C2×C418yes—
64C2×(C2×D71)⋊C4C2×C414yes—
65C2×(C2×D71)⋊C4Q814yescomplemented
66C2×C4×Dic71—22no—
67C2×C4×Dic71—42no—
68C71⋊#95—22no—
69C71⋊#95—22no—
70C71⋊#95—24no—
71C71⋊#95—24no—
72C71⋊#95—24no—
73C71⋊#95—22no—
74C71⋊#95—24no—
75C71⋊#95—42no—
76C71⋊#95—42no—
77C71⋊#95C2×C414yescomplemented
78C71⋊#95C2×C418yescomplemented
79C22×Q8×C71—24no—
80C22×Q8×C71—22no—
81C22×Q8×C71—22no—
82C22×Q8×C71—24no—
83C22×Q8×C71C2×C414yescomplemented
84C23×C4×C71C2312yes—
85C23×C4×C71C2312yes—
86C23×C4×C71C2311yescharacteristic
87C23×C4×C71C2311yescharacteristic
88C23×C4×C71C2311yescharacteristic
89C4×Q8×C71—232no—
90C2×C4⋊C4×C71—24no—
91C2×C4⋊C4×C71—22no—
92C2×C4⋊C4×C71—24no—
93C2×C4⋊C4×C71—24no—
94C2×C4⋊C4×C71D418yes—
95C2×C42×C71—24no—
96C2×C42×C71—24no—
97C2×C42×C71—24no—
98C2×C42×C71—24no—
99C2×C42×C71D414yes—
100C23×D71—21no—
101C23×D71—22no—
102C23×D71—21no—
103C23×D71—21no—
104C23×D71—22no—
105C23×D71—21no—
106C23×D71—22no—
107C23×D71—22no—
108C23×D71C22×C412yes—
109C23×D71C2×Q812yes—
110C23×D71#4112yes—
111C22×Dic71—21no—
112C22×Dic71—22no—
113C22×Dic71—21no—
114C22×Dic71—21no—
115C22×Dic71—22no—
116C22×Dic71—21no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—21no—
120C22×Dic71—22no—
121C22×Dic71—21no—
122C22×Dic71—22no—
123C22×Dic71—22no—
124C22×Dic71—22no—
125C22×Dic71—22no—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—22no—
129C22×Dic71—22no—
130C22×Dic71—22no—
131C22×Dic71—22no—
132C22×Dic71—22no—
133C22×Dic71—22no—
134C22×Dic71—22no—
135C22×Dic71—22no—
136C22×Dic71C22×C414yes—
137C22×Dic71C22×C412yes—
138C22×Dic71C2×Q812yes—
139C22×Dic71#4112yes—
140C2×C4×D71—28no—
141C2×C4×D71—48no—
142C2×C4×D71—48no—
143C2×C4×D71—44no—
144C2×C4×D71—44no—
145Dic71⋊C4—48no—
146(C2×D71)⋊C4—28no—
147(C2×D71)⋊C4—216no—
148(C2×D71)⋊C4—28no—
149(C2×D71)⋊C4—48no—
150C4×Dic71—44no—
151C24×C71C2411yescharacteristic
152C2×Q8×C71—28no—
153C2×Q8×C71—28no—
154C2×Q8×C71—216no—
155C2×Q8×C71—28no—
156C2×Q8×C71—216no—
157C22×C4×C71—22no—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
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×C71C22×C414yes—
175C22×C4×C71C22×C412yes—
176C22×C4×C71C2×D414yes—
177C22×C4×C71C2×D414yes—
178C22×C4×C71C2×D412yes—
179C22×C4×C71C2×D414yes—
180C22×C4×C71C2×D412yes—
181C22×C4×C71C2×D412yes—
182C22×C4×C71C2×Q812yes—
183C22×C4×C71#4112yes—
184C22×C4×C71#4114yes—
185C22×C4×C71#4112yes—
186C4⋊C4×C71—28no—
187C4⋊C4×C71—216no—
188C42×C71—216no—
189C42×C71—28no—
190C23×C71C23×C411yescharacteristic
191C23×C71C22×D411yescharacteristic
192C23×C71C22×D412yes—
193C23×C71C22×Q811yescharacteristic
194C23×C71C2×#4111yescharacteristic
195C23×C71C2×#4111yescharacteristic
196C23×C71C2×#4111yescharacteristic
197C23×C71C2×#4112yes—
198C23×C71C2×#4111yescharacteristic
199C23×C71H(5,𝔽2)12yes—
200C23×C712−1+412yes—
201C22×D71—24no—
202C22×D71—22no—
203C22×D71—22no—
204C22×D71—24no—
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—41no—
214C22×D71—41no—
215C22×D71—42no—
216C22×D71—41no—
217C22×D71—41no—
218C22×D71—42no—
219C22×D71—42no—
220C22×D71—42no—
221C22×D71—42no—
222C22×D71—42no—
223C22×D71C4×Q814yescomplemented
224Q8×C71—232no—
225C2×C4×C71—24no—
226C2×C4×C71—24no—
227C2×C4×C71—24no—
228C2×C4×C71—22no—
229C2×C4×C71—24no—
230C2×C4×C71—22no—
231C2×C4×C71—22no—
232C2×C4×C71—24no—
233C2×C4×C71—22no—
234C2×C4×C71—24no—
235C2×C4×C71—24no—
236C2×C4×C71—24no—
237C2×C4×C71—22no—
238C2×C4×C71—22no—
239C2×C4×C71—24no—
240C2×C4×C71—24no—
241C2×C4×C71—24no—
242C2×C4×C71—24no—
243C2×C4×C71—24no—
244C2×C4×C71—24no—
245C2×C4×C71—24no—
246C2×C4×C71—42no—
247C2×C4×C71—42no—
248C2×C4×C71—42no—
249C2×C4×C71—42no—
250C2×C4×C71C4×D414yes—
251C2×C4×C71C4×D418yes—
252C2×C4×C71#12014yes—
253C2×C4×C71C22⋊Q814yes—
254C2×Dic71—22no—
255C2×Dic71—24no—
256C2×Dic71—24no—
257C2×Dic71—24no—
258C2×Dic71—24no—
259C2×Dic71—22no—
260C2×Dic71—22no—
261C2×Dic71—24no—
262C2×Dic71—24no—
263C2×Dic71—41no—
264C2×Dic71—41no—
265C2×Dic71—41no—
266C2×Dic71—41no—
267C2×Dic71—41no—
268C2×Dic71—41no—
269C2×Dic71—42no—
270C2×Dic71—42no—
271C2×Dic71—42no—
272C2×Dic71—42no—
273C2×Dic71—42no—
274C2×Dic71—42no—
275C2×Dic71—42no—
276C2×Dic71—42no—
277C2×Dic71—42no—
278C2×Dic71—42no—
279C2×Dic71—42no—
280C2×Dic71—42no—
281C2×Dic71—42no—
282C2×Dic71—42no—
283C4×D71—416no—
284C22×C71C2×C4×D411yescharacteristic
285C22×C71C2×C4×D412yes—
286C22×C71C2×C4×Q811yescharacteristic
287C22×C71C4×#4111yescharacteristic
288C22×C71#52012yes—
289C22×C71C2×#12011yescharacteristic
290C22×C71C2×C22⋊Q811yescharacteristic
291C22×C71#52511yescharacteristic
292C22×C71#52511yescharacteristic
293C22×C71#52512yes—
294C22×C71#52911yescharacteristic
295C22×C71#53311yescharacteristic
296C22×C71#53511yescharacteristic
297C22×C71#53611yescharacteristic
298C22×C71#54212yes—
299C22×C71#54612yes—
300C22×C71#54612yes—
301C22×C71#54812yes—
302C22×C71D4×Q812yes—
303C22×C71#55212yes—
304C22×C71#55412yes—
305C22×C71#55612yes—
306C22×C71#55912yes—
307C2×D71—28no—
308C2×D71—24no—
309C2×D71—42no—
310C2×D71—42no—
311C2×D71—42no—
312C2×D71—44no—
313C2×D71—42no—
314C2×D71—42no—
315C2×D71—44no—
316C2×D71—44no—
317C2×D71—42no—
318C4×C71—24no—
319C4×C71—24no—
320C4×C71—28no—
321C4×C71—24no—
322C4×C71—44no—
323Dic71—42no—
324Dic71—42no—
325Dic71—44no—
326Dic71—42no—
327Dic71—82no—
328#14172—711nomaximal, Sylow
329C2×C71C4×#12011yescharacteristic
330C2×C71C4×C22⋊Q811yescharacteristic
331C2×C71#232611yescharacteristic
332C2×C71#236012yes—
333C2×C71#236712yes—
334C2×C71#240411yescharacteristic
335C2×C71#240911yescharacteristic
336C2×C71#245812yes—
337C2×C71#246512yes—
338C2×C71#258011yescharacteristic
339C2×C71#259911yescharacteristic
340D71—44no—
341C22×C4×Q8D7111yescharacteristic, complemented
342C2×#393—711no—
343C2×#385—712no—
344C2×C4×C22⋊C4—711no—
345#1446—711no—
346#1444—711no—
347#1439—711no—
348#1437—712no—
349#1436—712no—
350#1436—712no—
351#1434—711no—
352C71#1417211yescharacteristic, complemented, Sylow
353C23×Q8C2×D7111yescharacteristic
354C24×C4—711no—
355C2×C4×Q8—28no—
356C2×C4×Q8—24no—
357C22×C4⋊C4—712no—
358C22×C4⋊C4C2×D7111yescharacteristic
359C22×C4⋊C4C2×D7112yes—
360C22×C22⋊C4—712no—
361C22×C22⋊C4—712no—
362C22×C22⋊C4—711no—
363C22×C22⋊C4—711no—
364C22×C42—711no—
365C22×C42C2×D7112yes—
366C22×C42C2×D7111yescharacteristic
367#393—718no—
368#385—1428no—
369C4×C22⋊C4—1424no—
370C2×#95—712no—
371C2×#95—712no—
372C2×#95—711no—
373C2×#95—712no—
374C2×#95—712no—
375C2×#95—712no—
376C2×#95—711no—
377C2×#95—711no—
378C2×#95—712no—
379C2×#95—712no—
380C2×#95—711no—
381C25—711no—
382C22×Q8—24no—
383C22×Q8—22no—
384C22×Q8—22no—
385C22×Q8—24no—
386C22×Q8C4×D7114yescomplemented
387C23×C4—2842no—
388C23×C4—712no—
389C23×C4—711no—
390C23×C4—711no—
391C23×C4—711no—
392C23×C4—712no—
393C23×C4—1422no—
394C23×C4—1422no—
395C23×C4—1424no—
396C23×C4—1422no—
397C23×C4C22×D7112yes—
398C23×C4C22×D7112yes—
399C23×C4C22×D7111yescharacteristic
400C23×C4C22×D7111yescharacteristic
401C23×C4C22×D7111yescharacteristic
402C4×Q8—232no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—24no—
407C2×C4⋊C4—2842no—
408C2×C4⋊C4—1424no—
409C2×C4⋊C4—1424no—
410C2×C4⋊C4C22⋊D7118yes—
411C2×C22⋊C4—2842no—
412C2×C22⋊C4—718no—
413C2×C22⋊C4—714no—
414C2×C22⋊C4—714no—
415C2×C22⋊C4—1424no—
416C2×C22⋊C4—1424no—
417C2×C22⋊C4—1422no—
418C2×C22⋊C4—1422no—
419C2×C22⋊C4—1424no—
420C2×C22⋊C4—1424no—
421C2×C22⋊C4—1422no—
422C2×C22⋊C4—1422no—
423C2×C42—24no—
424C2×C42—24no—
425C2×C42—24no—
426C2×C42—24no—
427C2×C42—2842no—
428C2×C42—1422no—
429C2×C42C22⋊D7114yes—
430#95—2842no—
431#95—2842no—
432#95—714no—
433#95—718no—
434#95—1422no—
435#95—1422no—
436#95—1424no—
437#95—1424no—
438#95—1422no—
439#95—1424no—
440#95—1424no—
441C24—712no—
442C24—712no—
443C24—712no—
444C24—1421no—
445C24—1422no—
446C24—1421no—
447C24—1421no—
448C24—1422no—
449C24—1421no—
450C24—1422no—
451C24—1422no—
452C24C23×D7111yescharacteristic
453C2×Q8—28no—
454C2×Q8—216no—
455C2×Q8—28no—
456C2×Q8—28no—
457C2×Q8—216no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
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—22no—
474C22×C4—22no—
475C22×C4—2844no—
476C22×C4—2848no—
477C22×C4—2848no—
478C22×C4—2844no—
479C22×C4—714no—
480C22×C4—712no—
481C22×C4—712no—
482C22×C4—712no—
483C22×C4—1428no—
484C22×C4—1422no—
485C22×C4—1422no—
486C22×C4—1422no—
487C22×C4—1422no—
488C22×C4—1422no—
489C22×C4—1422no—
490C22×C4—1422no—
491C22×C4—1421no—
492C22×C4—1421no—
493C22×C4—1421no—
494C22×C4—1422no—
495C22×C4—1422no—
496C22×C4—1422no—
497C22×C4—1421no—
498C22×C4—1422no—
499C22×C4—1421no—
500C22×C4—1422no—
501C22×C4—1422no—
502C22×C4—1421no—
503C22×C4—1422no—
504C22×C4—1422no—
505C22×C4—1422no—
506C22×C4—1422no—
507C22×C4—1422no—
508C22×C4—1422no—
509C22×C4C2×C4×D7114yes—
510C22×C4C2×C4×D7112yes—
511C22×C4C2×C22⋊D7114yes—
512C22×C4C2×C22⋊D7114yes—
513C22×C4C2×C22⋊D7112yes—
514C22×C4C2×C22⋊D7114yes—
515C22×C4C2×C22⋊D7112yes—
516C22×C4C2×C22⋊D7112yes—
517C22×C4Q8×D7112yes—
518C22×C4#1176586512yes—
519C22×C4#1176586514yes—
520C22×C4#1176586612yes—
521C4⋊C4—28no—
522C4⋊C4—216no—
523C4⋊C4—2848no—
524C22⋊C4—2848no—
525C22⋊C4—1428no—
526C22⋊C4—14216no—
527C22⋊C4—1428no—
528C42—216no—
529C42—28no—
530C42—2844no—
531C23—2841no—
532C23—2841no—
533C23—2842no—
534C23—2842no—
535C23—2842no—
536C23—2842no—
537C23—2842no—
538C23—2841no—
539C23—2842no—
540C23—2841no—
541C23—714no—
542C23—1424no—
543C23—1424no—
544C23—1424no—
545C23—1424no—
546C23—1422no—
547C23—1424no—
548C23—1422no—
549C23—1424no—
550C23—1422no—
551C23—1422no—
552C23—1422no—
553C23—1422no—
554C23C22×C4×D7111yescharacteristic
555C23C22×C22⋊D7111yescharacteristic
556C23C22×C22⋊D7112yes—
557C23C2×Q8×D7111yescharacteristic
558C23#41×D7111yescharacteristic
559C23#41×D7111yescharacteristic
560C23C2×#1176586511yescharacteristic
561C23C2×#1176586512yes—
562C23C2×#1176586611yescharacteristic
563C23#1455617412yes—
564C23#1455617612yes—
565Q8—232no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—28416no—
592C2×C4—2841no—
593C2×C4—2841no—
594C2×C4—2841no—
595C2×C4—2841no—
596C2×C4—2842no—
597C2×C4—2842no—
598C2×C4—2842no—
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—2842no—
611C2×C4—2842no—
612C2×C4—1424no—
613C2×C4—1424no—
614C2×C4—1422no—
615C2×C4—1424no—
616C2×C4—1424no—
617C2×C4—1424no—
618C2×C4—1424no—
619C2×C4—1422no—
620C2×C4—1422no—
621C2×C4C4×C22⋊D7114yes—
622C2×C4C4×C22⋊D7118yes—
623C2×C4#1455609714yes—
624C2×C4#1455610414yes—
625C22—2844no—
626C22—2842no—
627C22—2844no—
628C22—2842no—
629C22—2842no—
630C22—2842no—
631C22—2842no—
632C22—2844no—
633C22—2842no—
634C22—1428no—
635C22—1424no—
636C22C2×C4×C22⋊D7111yescharacteristic
637C22C2×C4×C22⋊D7112yes—
638C22C4×Q8×D7111yescharacteristic
639C22C4×#1176586611yescharacteristic
640C22#1798648112yes—
641C22C2×#1455609711yescharacteristic
642C22C2×#1455610411yescharacteristic
643C22#1798650611yescharacteristic
644C22#1798651012yes—
645C22#1798651011yescharacteristic
646C22#1798653611yescharacteristic
647C22#1798655311yescharacteristic
648C22#1798656311yescharacteristic
649C22#1798657311yescharacteristic
650C22#1798662112yes—
651C22#1798663712yes—
652C22#1798663712yes—
653C22#1798666012yes—
654C22Q8×C22⋊D7112yes—
655C22#1798669012yes—
656C22#1798670912yes—
657C22#1798672112yes—
658C22#1798673412yes—
659C4—28no—
660C4—24no—
661C4—24no—
662C4—24no—
663C4—44no—
664C4—2842no—
665C4—2842no—
666C4—2842no—
667C4—2844no—
668C4—5682no—
669C2—2844no—
670C2C4×#1455609711yescharacteristic
671C2C4×#1455610411yescharacteristic
672C2#2259104711yescharacteristic
673C2#2259134112yes—
674C2#2259140712yes—
675C2#2259165511yescharacteristic
676C2#2259169911yescharacteristic
677C2#2259225612yes—
678C2#2259235512yes—
679C2#2259369811yescharacteristic
680C2#2259391111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC22×C4×Q8×C71C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC22×C4×Q8D71
hypocenterC71#14172
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
4. #22586611C2semidirect
5. #22586603C2semidirect
8. #22586573C2semidirect
9. #22586572C2semidirect
10. #22586567C2semidirect
11. C22×C4×Q8×C71C2semidirect
19. #17985910C4semidirect
65. C2×(C2×D71)⋊C4Q8semidirect
77. C71⋊#95C2×C4semidirect
78. C71⋊#95C2×C4semidirect
83. C22×Q8×C71C2×C4semidirect
223. C22×D71C4×Q8semidirect
341. C22×C4×Q8D71semidirect
352. C71#14172semidirect
386. C22×Q8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179859102
C2×#179858702
C2×C4×(C2×D71)⋊C42
#225866112
#225866032
#225865822
#225865782
#225865732
#225865722
#225865672
C22×C4×Q8×C712
#1417271