Finite Group Databasebeta

← #31919441

Subgroups — #31919441

Order 18176.

2222 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D71C211yesmaximal, characteristic
2C2×#17985864C211yesmaximal, characteristic, complemented
3C2×#17985856C211yesmaximal, characteristic
4C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
5#22586598C214yesmaximal, complemented
6#22586570C211yesmaximal, characteristic, complemented
7#22586569C211yesmaximal, characteristic, complemented
8(C2×D71)⋊#95C214yesmaximal
9C2×#382×C71C211yesmaximal, characteristic, complemented
10C23×C4×D71C2212yes—
11C23×C4×D71C2211yescharacteristic
12C22×C4⋊Dic71C2211yescharacteristic
13C22×C4⋊Dic71C2211yescharacteristic
14C22×Dic71⋊C4C2212yes—
15C2×C4⋊C4×D71—24no—
16C2×C4⋊C4×D71—24no—
17C2×C4⋊C4×D71—24no—
18C22×(C2×D71)⋊C4C2212yes—
19C22×(C2×D71)⋊C4C2212yes—
20C22×C4×Dic71C2211yescharacteristic
21#17985864—24no—
22#17985856—24no—
23C4×(C2×D71)⋊C4C418yescomplemented
24C2×C71⋊#95C2214yes—
25C2×C71⋊#95C2212yes—
26C2×C71⋊#95C2214yes—
27C2×C71⋊#95C2214yes—
28C2×C71⋊#95C2212yes—
29C22×C4⋊C4×C71C2211yescharacteristic
30C22×C4⋊C4×C71C2211yescharacteristic
31C22×C42×C71C2211yescharacteristic
32#382×C71—24no—
33C2×#95×C71C2214yes—
34C24×D71C2311yescharacteristic
35C23×Dic71C2312yes—
36C23×Dic71C2312yes—
37C23×Dic71C2312yes—
38C23×Dic71C2311yescharacteristic
39C22×C4×D71—24no—
40C22×C4×D71—24no—
41C22×C4×D71—28no—
42C22×C4×D71—22no—
43C22×C4×D71—22no—
44C22×C4×D71—24no—
45C22×C4×D71—22no—
46C22×C4×D71—22no—
47C22×C4×D71—44no—
48C22×C4×D71C2×C414yes—
49C22×C4×D71D414yes—
50C22×C4×D71Q814yes—
51C2×C4⋊Dic71—22no—
52C2×C4⋊Dic71—22no—
53C2×C4⋊Dic71—22no—
54C2×C4⋊Dic71—22no—
55C2×C4⋊Dic71—22no—
56C2×C4⋊Dic71D414yes—
57C2×Dic71⋊C4—24no—
58C2×Dic71⋊C4—24no—
59C2×Dic71⋊C4—24no—
60C4⋊C4×D71—232no—
61C2×(C2×D71)⋊C4—24no—
62C2×(C2×D71)⋊C4—24no—
63C2×(C2×D71)⋊C4—24no—
64C2×(C2×D71)⋊C4—28no—
65C2×(C2×D71)⋊C4C2×C418yes—
66C2×C4×Dic71—22no—
67C2×C4×Dic71—22no—
68C2×C4×Dic71C2×C414yescomplemented
69C71⋊#95—24no—
70C71⋊#95—28no—
71C71⋊#95—44no—
72C71⋊#95—44no—
73C71⋊#95C2×C418yescomplemented
74C23×C4×C71C2312yes—
75C23×C4×C71C2312yes—
76C23×C4×C71C2312yes—
77C23×C4×C71C2311yescharacteristic
78C2×C4⋊C4×C71—22no—
79C2×C4⋊C4×C71—22no—
80C2×C4⋊C4×C71—22no—
81C2×C4⋊C4×C71—22no—
82C2×C4⋊C4×C71—22no—
83C2×C4⋊C4×C71D414yes—
84C2×C42×C71—22no—
85C2×C42×C71—22no—
86C2×C42×C71C2×C414yescomplemented
87#95×C71—28no—
88C23×D71—21no—
89C23×D71—21no—
90C23×D71—21no—
91C23×D71—21no—
92C23×D71—22no—
93C23×D71—22no—
94C23×D71C22×C412yes—
95C23×D71C2×D412yes—
96C23×D71C2×Q812yes—
97C23×D71#4114yes—
98C23×D71#4114yes—
99C22×Dic71—21no—
100C22×Dic71—21no—
101C22×Dic71—21no—
102C22×Dic71—21no—
103C22×Dic71—22no—
104C22×Dic71—22no—
105C22×Dic71—22no—
106C22×Dic71—24no—
107C22×Dic71—22no—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—24no—
111C22×Dic71—22no—
112C22×Dic71—22no—
113C22×Dic71—22no—
114C22×Dic71—24no—
115C22×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71—24no—
118C22×Dic71C22×C414yes—
119C22×Dic71C22×C412yes—
120C22×Dic71C2×D414yes—
121C22×Dic71C2×D412yes—
122C22×Dic71C2×Q812yes—
123C2×C4×D71—28no—
124C2×C4×D71—28no—
125C2×C4×D71—28no—
126C2×C4×D71—28no—
127C2×C4×D71—28no—
128C2×C4×D71—28no—
129C2×C4×D71—216no—
130C2×C4×D71—416no—
131C2×C4×D71—416no—
132C2×C4×D71—48no—
133C2×C4×D71—48no—
134C2×C4×D71C4⋊C4116yes—
135C4⋊Dic71—28no—
136C4⋊Dic71—44no—
137Dic71⋊C4—216no—
138(C2×D71)⋊C4—216no—
139(C2×D71)⋊C4—48no—
140C4×Dic71—28no—
141C24×C71C2411yescharacteristic
142C22×C4×C71—22no—
143C22×C4×C71—22no—
144C22×C4×C71—22no—
145C22×C4×C71—24no—
146C22×C4×C71—22no—
147C22×C4×C71—22no—
148C22×C4×C71—22no—
149C22×C4×C71—24no—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—24no—
156C22×C4×C71—24no—
157C22×C4×C71C22×C414yes—
158C22×C4×C71C22×C412yes—
159C22×C4×C71C2×D412yes—
160C22×C4×C71C2×D412yes—
161C22×C4×C71C2×D414yes—
162C22×C4×C71C2×D412yes—
163C22×C4×C71C2×Q812yes—
164C22×C4×C71#4112yes—
165C22×C4×C71#4112yes—
166C4⋊C4×C71—28no—
167C4⋊C4×C71—44no—
168C42×C71—28no—
169C23×C71C23×C411yescharacteristic
170C23×C71C22×D411yescharacteristic
171C23×C71C22×D411yescharacteristic
172C23×C71C22×D411yescharacteristic
173C23×C71C22×Q811yescharacteristic
174C23×C71C2×#4112yes—
175C23×C71C2×#4112yes—
176C23×C71C2×#4111yescharacteristic
177C23×C71C2×#4111yescharacteristic
178C23×C71H(5,𝔽2)12yes—
179C23×C712−1+412yes—
180C22×D71—22no—
181C22×D71—22no—
182C22×D71—24no—
183C22×D71—22no—
184C22×D71—22no—
185C22×D71—22no—
186C22×D71—22no—
187C22×D71—22no—
188C22×D71—22no—
189C22×D71—24no—
190C22×D71—24no—
191C22×D71—24no—
192C22×D71—24no—
193C22×D71—24no—
194C22×D71—24no—
195C22×D71—24no—
196C22×D71—24no—
197C22×D71—24no—
198C22×D71C2×C4⋊C414yes—
199C22×D71#11718yes—
200C22×D71#12414yes—
201C22×D71#12414yes—
202C22×D71#12518yes—
203C2×C4×C71—24no—
204C2×C4×C71—24no—
205C2×C4×C71—24no—
206C2×C4×C71—24no—
207C2×C4×C71—28no—
208C2×C4×C71—42no—
209C2×C4×C71—42no—
210C2×C4×C71—42no—
211C2×C4×C71—44no—
212C2×C4×C71—42no—
213C2×C4×C71—42no—
214C2×C4×C71—42no—
215C2×C4×C71—42no—
216C2×C4×C71—42no—
217C2×C4×C71—42no—
218C2×C4×C71—44no—
219C2×C4×C71—44no—
220C2×C4×C71C2×C4⋊C414yes—
221C2×C4×C71C4×D414yes—
222C2×C4×C71C4×D414yes—
223C2×C4×C71#12014yes—
224C2×C4×C71C22⋊Q814yes—
225C2×C4×C71C22⋊Q814yes—
226C2×C4×C71#12314yes—
227C2×Dic71—22no—
228C2×Dic71—22no—
229C2×Dic71—22no—
230C2×Dic71—22no—
231C2×Dic71—22no—
232C2×Dic71—22no—
233C2×Dic71—24no—
234C2×Dic71—24no—
235C2×Dic71—24no—
236C2×Dic71—24no—
237C2×Dic71—28no—
238C2×Dic71—42no—
239C2×Dic71—42no—
240C2×Dic71—44no—
241C2×Dic71—42no—
242C2×Dic71—42no—
243C2×Dic71—42no—
244C2×Dic71—42no—
245C2×Dic71—42no—
246C2×Dic71—42no—
247C2×Dic71—42no—
248C2×Dic71—44no—
249C2×Dic71—44no—
250C2×Dic71C2×C4⋊C414yes—
251C4×D71—232no—
252C4×D71—432no—
253C22×C71C22×C4⋊C411yescharacteristic
254C22×C71C2×#11712yes—
255C22×C71C2×C4×D411yescharacteristic
256C22×C71C2×C4×D411yescharacteristic
257C22×C71#52012yes—
258C22×C71C2×#12011yescharacteristic
259C22×C71C2×C22⋊Q811yescharacteristic
260C22×C71C2×C22⋊Q811yescharacteristic
261C22×C71C2×#12311yescharacteristic
262C22×C71C2×#12411yescharacteristic
263C22×C71C2×#12411yescharacteristic
264C22×C71C2×#12512yes—
265C22×C71#53511yescharacteristic
266C22×C71#53611yescharacteristic
267C22×C71#54412yes—
268C22×C71#54612yes—
269C22×C71#54612yes—
270C22×C71#54812yes—
271C22×C71#54812yes—
272C22×C71#55212yes—
273C22×C71#55212yes—
274C22×C71#55312yes—
275C22×C71#55312yes—
276C2×D71—24no—
277C2×D71—28no—
278C2×D71—28no—
279C2×D71—24no—
280C2×D71—24no—
281C2×D71—28no—
282C2×D71—28no—
283C2×D71—24no—
284C2×D71—24no—
285C2×D71—24no—
286C2×D71#38218yescomplemented
287C4×C71—44no—
288C4×C71—44no—
289C4×C71—82no—
290C4×C71#38518yes—
291Dic71—24no—
292Dic71—44no—
293Dic71—44no—
294Dic71—82no—
295#13881—711nomaximal, Sylow
296C2×C71C2×#38211yescharacteristic
297C2×C71C2×#38511yescharacteristic
298C2×C71#231711yescharacteristic
299C2×C71#235212yes—
300C2×C71#235212yes—
301C2×C71#243511yescharacteristic
302C2×C71#243611yescharacteristic
303C2×C71#244212yes—
304C2×C71#244212yes—
305C2×C71#250311yescharacteristic
306C2×C71#250411yescharacteristic
307D71—28no—
308C23×C4⋊C4—711no—
309C2×#382—711no—
310C2×#382D7111yescharacteristic, complemented
311C2×#380—711no—
312C2×C4×C22⋊C4—711no—
313#1442—714no—
314C4⋊#95—711no—
315C4⋊#95—711no—
316C22⋊#95—714no—
317C71#1388111yescharacteristic, complemented, Sylow
318C24×C4—712no—
319C24×C4—711no—
320C22×C4⋊C4—711no—
321C22×C4⋊C4—712no—
322C22×C4⋊C4—711no—
323C22×C4⋊C4—1424no—
324C22×C4⋊C4—1424no—
325C22×C4⋊C4—1424no—
326C22×C4⋊C4C2×D7111yescharacteristic
327C22×C4⋊C4C2×D7111yescharacteristic
328C22×C22⋊C4—712no—
329C22×C22⋊C4—712no—
330C22×C42—711no—
331C22×C42C2×D7111yescharacteristic
332#382—24no—
333#382—1424no—
334#380—1424no—
335C4×C22⋊C4—718no—
336C2×#95—714no—
337C2×#95—714no—
338C2×#95—712no—
339C2×#95—714no—
340C2×#95—712no—
341C2×#95C2×D7114yes—
342C25—711no—
343C23×C4—2844no—
344C23×C4—711no—
345C23×C4—714no—
346C23×C4—714no—
347C23×C4—714no—
348C23×C4—712no—
349C23×C4—712no—
350C23×C4—712no—
351C23×C4—1428no—
352C23×C4—1424no—
353C23×C4—1424no—
354C23×C4—1424no—
355C23×C4—1422no—
356C23×C4—1422no—
357C23×C4—1422no—
358C23×C4—1422no—
359C23×C4C22×D7112yes—
360C23×C4C22×D7112yes—
361C23×C4C22×D7112yes—
362C23×C4C22×D7111yescharacteristic
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—714no—
369C2×C4⋊C4—14232no—
370C2×C4⋊C4—1422no—
371C2×C4⋊C4—1422no—
372C2×C4⋊C4—1422no—
373C2×C4⋊C4—1424no—
374C2×C4⋊C4—1424no—
375C2×C4⋊C4—1422no—
376C2×C4⋊C4—1424no—
377C2×C4⋊C4—1422no—
378C2×C4⋊C4C22⋊D7114yes—
379C2×C22⋊C4—718no—
380C2×C22⋊C4—1424no—
381C2×C22⋊C4—1428no—
382C2×C22⋊C4—1424no—
383C2×C22⋊C4—1424no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—714no—
387C2×C42—1422no—
388C2×C42—1422no—
389C2×C42C4×D7114yescomplemented
390#95—28no—
391#95—2844no—
392#95—2844no—
393#95—718no—
394#95—1424no—
395#95—1428no—
396C24—712no—
397C24—712no—
398C24—712no—
399C24—714no—
400C24—714no—
401C24—1421no—
402C24—1421no—
403C24—1421no—
404C24—1421no—
405C24—1422no—
406C24—1422no—
407C24C23×D7111yescharacteristic
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—28416no—
424C22×C4—2848no—
425C22×C4—2848no—
426C22×C4—28416no—
427C22×C4—712no—
428C22×C4—712no—
429C22×C4—712no—
430C22×C4—7116no—
431C22×C4—714no—
432C22×C4—714no—
433C22×C4—14216no—
434C22×C4—1422no—
435C22×C4—1424no—
436C22×C4—1422no—
437C22×C4—1422no—
438C22×C4—1424no—
439C22×C4—1422no—
440C22×C4—1422no—
441C22×C4—1422no—
442C22×C4—1422no—
443C22×C4—1422no—
444C22×C4—1424no—
445C22×C4—1422no—
446C22×C4—1422no—
447C22×C4—1424no—
448C22×C4—1422no—
449C22×C4—1421no—
450C22×C4—1421no—
451C22×C4—1421no—
452C22×C4—1421no—
453C22×C4—1428no—
454C22×C4—1428no—
455C22×C4—1428no—
456C22×C4—1428no—
457C22×C4—1428no—
458C22×C4—1428no—
459C22×C4C2×C4×D7114yes—
460C22×C4C2×C4×D7112yes—
461C22×C4D4×D7112yes—
462C22×C4D4×D7112yes—
463C22×C4C2×C22⋊D7114yes—
464C22×C4C2×C22⋊D7112yes—
465C22×C4Q8×D7112yes—
466C22×C4#1176586512yes—
467C22×C4#1176586612yes—
468C4⋊C4—28no—
469C4⋊C4—44no—
470C4⋊C4—2844no—
471C4⋊C4—14216no—
472C4⋊C4—1428no—
473C22⋊C4—2848no—
474C22⋊C4—14216no—
475C42—28no—
476C42—1428no—
477C23—714no—
478C23—718no—
479C23—714no—
480C23—714no—
481C23—718no—
482C23—1422no—
483C23—1422no—
484C23—1424no—
485C23—1424no—
486C23—1424no—
487C23—1424no—
488C23—1424no—
489C23—1424no—
490C23—1424no—
491C23—1424no—
492C23—1424no—
493C23—1422no—
494C23—1422no—
495C23—1424no—
496C23—1422no—
497C23—1422no—
498C23—1422no—
499C23—1422no—
500C23C22×C4×D7111yescharacteristic
501C23C2×D4×D7111yescharacteristic
502C23C2×D4×D7111yescharacteristic
503C23C22×C22⋊D7111yescharacteristic
504C23C2×Q8×D7111yescharacteristic
505C23#41×D7112yes—
506C23#41×D7112yes—
507C23C2×#1176586511yescharacteristic
508C23C2×#1176586611yescharacteristic
509C23#1455617412yes—
510C23#1455617512yes—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—28432no—
529C2×C4—2842no—
530C2×C4—2844no—
531C2×C4—2842no—
532C2×C4—2842no—
533C2×C4—2842no—
534C2×C4—2842no—
535C2×C4—2842no—
536C2×C4—2842no—
537C2×C4—2844no—
538C2×C4—2844no—
539C2×C4—2842no—
540C2×C4—2842no—
541C2×C4—714no—
542C2×C4—1428no—
543C2×C4—1424no—
544C2×C4—1424no—
545C2×C4—1424no—
546C2×C4—1424no—
547C2×C4—1422no—
548C2×C4—1422no—
549C2×C4—1422no—
550C2×C4—1422no—
551C2×C4—1422no—
552C2×C4—1422no—
553C2×C4—14232no—
554C2×C4C4⋊C4×D7114yes—
555C2×C4C4×C22⋊D7114yes—
556C2×C4#1455609014yes—
557C2×C4#1455609614yes—
558C2×C4#1455610414yes—
559C2×C4#1455610614yes—
560C2×C4#1455611314yes—
561C22—718no—
562C22—1424no—
563C22—1428no—
564C22—1428no—
565C22—1424no—
566C22—1428no—
567C22—1424no—
568C22—1428no—
569C22—1424no—
570C22—1424no—
571C22—1424no—
572C22C2×C4⋊C4×D7111yescharacteristic
573C22#117×D7112yes—
574C22C2×C4×C22⋊D7111yescharacteristic
575C22C2×#1455609011yescharacteristic
576C22#1798647912yes—
577C22C2×#1455609611yescharacteristic
578C22C2×#1455610411yescharacteristic
579C22C2×#1455610611yescharacteristic
580C22C2×#1455611311yescharacteristic
581C22#124×D7111yescharacteristic
582C22#124×D7111yescharacteristic
583C22#125×D7112yes—
584C22#1798656711yescharacteristic
585C22#1798656911yescharacteristic
586C22#1798662912yes—
587C22#1798663712yes—
588C22#1798664212yes—
589C22#1798665612yes—
590C22#1798665712yes—
591C22#1798668712yes—
592C22#1798668812yes—
593C22#1798669312yes—
594C22#1798670212yes—
595C4—44no—
596C4—44no—
597C4—82no—
598C4—2844no—
599C4—2844no—
600C4—5682no—
601C4—1424no—
602C4#1798587018yes—
603C2—1428no—
604C2#382×D7111yescharacteristic
605C2C2×#1798587011yescharacteristic
606C2#2259096511yescharacteristic
607C2#2259126612yes—
608C2#2259127012yes—
609C2#2259198811yescharacteristic
610C2#2259199511yescharacteristic
611C2#2259206012yes—
612C2#2259206712yes—
613C2#2259282211yescharacteristic
614C2#2259282911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC2×#382×C71C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC2×#382D71
hypocenterC71#13881
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. C2×#17985864C2semidirect
5. #22586598C2semidirect
6. #22586570C2semidirect
7. #22586569C2semidirect
9. C2×#382×C71C2semidirect
23. C4×(C2×D71)⋊C4C4semidirect
68. C2×C4×Dic71C2×C4semidirect
73. C71⋊#95C2×C4semidirect
86. C2×C42×C71C2×C4semidirect
286. C2×D71#382semidirect
310. C2×#382D71semidirect
317. C71#13881semidirect
389. C2×C42C4×D71semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D712
C2×#179858642
C2×#179858562
C2×C4×(C2×D71)⋊C42
#225865982
#225865702
#225865692
(C2×D71)⋊#952
C2×#382×C712
#1388171