Finite Group Databasebeta

← C2×#22586845 #31964593

Subgroups — C2×#22586845 #31964593

Order 18176.

1834 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986510C211yesmaximal, characteristic
2C22×C71⋊#95C211yesmaximal, characteristic, complemented
3#22586845C218yesmaximal, direct product factor
4C2×#17985737C212yesmaximal, complemented
5C2×#17985734C212yesmaximal, complemented
6C2×#407×C71C211yesmaximal, characteristic, complemented
7C22×#11765865C411yescharacteristic, complemented
8C24×Dic71—21no—
9#17986510—22no—
10#17986510—22no—
11#17986510C414yescomplemented
12#17986510C2214yes—
13C2×#14556111—21no—
14C2×#14556107C411yescharacteristic, complemented
15C2×#14556107C2211yescharacteristic
16C2×#14556102C411yescharacteristic, complemented
17C2×#14556102C2211yescharacteristic
18C2×#14556097—21no—
19C2×C4×C22⋊D71—21no—
20C2×C4×C22⋊D71—21no—
21C2×#14556087C411yescharacteristic, complemented
22C2×C71⋊#95—21no—
23C2×C71⋊#95—22no—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2214yescomplemented
26#17985737C2218yescomplemented
27#17985734C2218yescomplemented
28C24×C4×C71C2211yescharacteristic
29C22×M16×C71—21no—
30#407×C71—22no—
31#407×C71C2214yescomplemented
32C2×C22⋊C8×C71—21no—
33C2×C22⋊C8×C71C2212yes—
34C2×#11765865—22no—
35C2×#11765865—44no—
36C2×#11765865C2×C414yescomplemented
37C22×Dic142C2×C411yescharacteristic, complemented
38C22×D284C2×C411yescharacteristic, complemented
39C22×C22⋊D71—21no—
40C22×C22⋊D71—21no—
41C22×C22⋊D71—21no—
42C23×Dic71—28no—
43C23×Dic71—22no—
44C23×Dic71—24no—
45C23×Dic71—44no—
46C23×Dic71—44no—
47C22×C4×D71—21no—
48#14556111—24no—
49#14556111—24no—
50#14556107—22no—
51#14556107—21no—
52#14556107—21no—
53#14556107C2×C414yescomplemented
54#14556107C2×C412yescomplemented
55#14556107C2312yes—
56#14556102—22no—
57#14556102—21no—
58#14556102—21no—
59#14556102C2×C414yescomplemented
60#14556102C2×C412yescomplemented
61#14556102C2312yes—
62#14556097—24no—
63#14556097—24no—
64C4×C22⋊D71—24no—
65C4×C22⋊D71—24no—
66C4×C22⋊D71—24no—
67C4×C22⋊D71—24no—
68#14556087—22no—
69#14556087C2×C414yescomplemented
70C2×C4⋊Dic71C2×C411yescharacteristic, complemented
71C2×C4⋊Dic71C2×C411yescharacteristic, complemented
72C2×Dic71⋊C4—21no—
73C2×Dic71⋊C4—21no—
74C2×C22⋊Dic71—21no—
75C2×C22⋊Dic71—21no—
76C2×C22⋊Dic71—21no—
77C2×(C2×D71)⋊C4—21no—
78C2×(C2×D71)⋊C4—21no—
79C2×C4×Dic71—21no—
80C71⋊#95—22no—
81C71⋊#95—24no—
82C71⋊#95C2314yes—
83C25×C71D411yescharacteristic
84C23×C4×C71—44no—
85C23×C4×C71C2×C411yescharacteristic
86C23×C4×C71C2×C411yescharacteristic
87C23×C4×C71C2×C412yes—
88C23×C4×C71D411yescharacteristic
89C23×C4×C71D411yescharacteristic
90C23×C4×C71D412yes—
91C23×C4×C71D411yescharacteristic
92C23×C4×C71D412yes—
93C23×C4×C71C2311yescharacteristic
94C23×C4×C71C2312yes—
95C2×M16×C71—24no—
96C2×M16×C71—44no—
97C22×C8×C71—22no—
98C22⋊C8×C71—24no—
99C22⋊C8×C71—22no—
100C22⋊C8×C71C2314yes—
101C23×D71—21no—
102#11765865—48no—
103#11765865—48no—
104C2×Dic142—21no—
105C2×Dic142—42no—
106C2×Dic142C22×C412yescomplemented
107C2×D284—21no—
108C2×D284—42no—
109C2×D284C22×C412yescomplemented
110C2×C22⋊D71—24no—
111C2×C22⋊D71—22no—
112C2×C22⋊D71—24no—
113C2×C22⋊D71—22no—
114C2×C22⋊D71—44no—
115C2×C22⋊D71—44no—
116C2×C22⋊D71—42no—
117C2×C22⋊D71—42no—
118C22×Dic71—28no—
119C22×Dic71—216no—
120C22×Dic71—21no—
121C22×Dic71—21no—
122C22×Dic71—21no—
123C22×Dic71—24no—
124C22×Dic71—432no—
125C22×Dic71—44no—
126C22×Dic71—48no—
127C22×Dic71—48no—
128C22×Dic71—44no—
129C2×C4×D71—24no—
130C2×C4×D71—44no—
131C4⋊Dic71—21no—
132C4⋊Dic71—21no—
133C4⋊Dic71C22×C412yescomplemented
134C4⋊Dic71C22×C412yescomplemented
135Dic71⋊C4—24no—
136Dic71⋊C4—22no—
137Dic71⋊C4—22no—
138C22⋊Dic71—24no—
139C22⋊Dic71—22no—
140C22⋊Dic71—24no—
141C22⋊Dic71—22no—
142(C2×D71)⋊C4—24no—
143(C2×D71)⋊C4—22no—
144(C2×D71)⋊C4—22no—
145C4×Dic71—24no—
146C24×C71—24no—
147C24×C71—24no—
148C24×C71—22no—
149C24×C71—22no—
150C24×C71C22⋊C412yes—
151C24×C71C22⋊C411yescharacteristic
152C24×C71C22⋊C411yescharacteristic
153C24×C71C2×D412yes—
154C24×C71C2×D411yescharacteristic
155C22×C4×C71—24no—
156C22×C4×C71—24no—
157C22×C4×C71—42no—
158C22×C4×C71—44no—
159C22×C4×C71—42no—
160C22×C4×C71—44no—
161C22×C4×C71—42no—
162C22×C4×C71—44no—
163C22×C4×C71—42no—
164C22×C4×C71—44no—
165C22×C4×C71C22⋊C411yescharacteristic
166C22×C4×C71C22⋊C412yes—
167C22×C4×C71C22⋊C411yescharacteristic
168C22×C4×C71C22⋊C412yes—
169C22×C4×C71C22⋊C412yes—
170C22×C4×C71C22⋊C412yes—
171C22×C4×C71C22⋊C412yes—
172C22×C4×C71C22×C411yescharacteristic
173C22×C4×C71C22×C412yes—
174C22×C4×C71C22×C411yescharacteristic
175C22×C4×C71C22×C412yes—
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D412yes—
178C22×C4×C71C2×D412yes—
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D412yes—
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2411yescharacteristic
183M16×C71—48no—
184C2×C8×C71—24no—
185C2×C8×C71—44no—
186C23×C71—216no—
187C23×C71—216no—
188C23×C71—22no—
189C23×C71—22no—
190C23×C71—24no—
191C23×C71—22no—
192C23×C71—24no—
193C23×C71—22no—
194C23×C71—24no—
195C23×C71—22no—
196C23×C71—22no—
197C23×C71—24no—
198C23×C71—22no—
199C23×C71—24no—
200C23×C71—22no—
201C23×C71#9914yes—
202C23×C71#10414yes—
203C23×C71C2×C22⋊C412yes—
204C23×C71C2×C22⋊C411yescharacteristic
205C23×C71C2×C22⋊C412yes—
206C23×C71C2×C22⋊C411yescharacteristic
207C23×C71AGL(1,ℤ/8)12yes—
208C23×C71#13712yes—
209C23×C71C22×D411yescharacteristic
210C22×D71—22no—
211C22×D71—42no—
212C22×D71—42no—
213C22×D71—42no—
214C2×C4×C71—22no—
215C2×C4×C71—24no—
216C2×C4×C71—24no—
217C2×C4×C71—22no—
218C2×C4×C71—22no—
219C2×C4×C71—24no—
220C2×C4×C71—22no—
221C2×C4×C71—24no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71—41no—
226C2×C4×C71—42no—
227C2×C4×C71—41no—
228C2×C4×C71—42no—
229C2×C4×C71—41no—
230C2×C4×C71—42no—
231C2×C4×C71—41no—
232C2×C4×C71C2×C22⋊C412yes—
233C2×C4×C71C2×C22⋊C412yes—
234C2×C4×C71C2×C22⋊C411yescharacteristic
235C2×C4×C71C2×C22⋊C411yescharacteristic
236C2×C4×C71C23×C411yescharacteristic
237C2×C4×C71C22×D411yescharacteristic
238C22⋊D71—42no—
239C22⋊D71—42no—
240C22⋊D71—48no—
241C22⋊D71—44no—
242C22⋊D71—48no—
243C2×Dic71—28no—
244C2×Dic71—22no—
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—432no—
248C2×Dic71—44no—
249C2×Dic71—416no—
250C2×Dic71—44no—
251C2×Dic71—42no—
252C2×Dic71—42no—
253C2×Dic71—42no—
254D284—42no—
255D284—42no—
256C4×D71—48no—
257Dic142—42no—
258Dic142—42no—
259C8×C71—44no—
260C22×C71—28no—
261C22×C71—28no—
262C22×C71—216no—
263C22×C71—21no—
264C22×C71—21no—
265C22×C71—22no—
266C22×C71—22no—
267C22×C71—22no—
268C22×C71—21no—
269C22×C71—22no—
270C22×C71—21no—
271C22×C71—24no—
272C22×C71—28no—
273C22×C71—28no—
274C22×C71—24no—
275C22×C71C2×#9914yes—
276C22×C71C2×#9912yes—
277C22×C71#41712yes—
278C22×C71#41712yes—
279C22×C71C2×#10414yes—
280C22×C71C2×#10412yes—
281C22×C71C22×C22⋊C411yescharacteristic
282C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C71C2×#13711yescharacteristic
284C2×D71—42no—
285C2×D71—42no—
286C2×D71—42no—
287C2×D71—84no—
288C4×C71—21no—
289C4×C71—21no—
290C4×C71—21no—
291C4×C71—22no—
292C4×C71—21no—
293C4×C71—22no—
294Dic71—48no—
295Dic71—42no—
296Dic71—42no—
297Dic71—42no—
298C2×#1504—711nomaximal, Sylow
299C2×C71—21no—
300C2×C71—21no—
301C2×C71—24no—
302C2×C71—22no—
303C2×C71—24no—
304C2×C71#150414yescomplemented
305C2×C71C22×#9911yescharacteristic
306C2×C71C2×#41711yescharacteristic
307C2×C71C22×#10411yescharacteristic
308D71—82no—
309C2×#525—711no—
310C2×#407D7111yescharacteristic, complemented
311C22×#95—711no—
312#1504—718no—
313C2×#328—712no—
314C2×#327—712no—
315C71C2×#150411yescharacteristic, complemented, Sylow
316C22×#41—711no—
317C24×C4—1421no—
318C24×C4C2×D7111yescharacteristic
319C22×M16—21no—
320#525—714no—
321#525—714no—
322#525—1422no—
323#525—1422no—
324C2×#123—1421no—
325C2×C22⋊Q8—711no—
326C2×C22⋊Q8—711no—
327C2×#121—711no—
328C2×#121—711no—
329C2×#120—1421no—
330C2×C4×D4—1421no—
331C2×C4×D4—1421no—
332C2×#117—711no—
333#407—22no—
334#407C2×D7114yescomplemented
335C2×C22⋊C8—21no—
336C2×C22⋊C8C2×D7112yes—
337C2×#95—712no—
338C2×#95—714no—
339C2×#95—1421no—
340C2×#95—1422no—
341#328—718no—
342#327—718no—
343C25D28411yescharacteristic
344C2×#41—2844no—
345C2×#41—714no—
346C2×#41—1422no—
347C22×Q8—711no—
348C22×D4—711no—
349C22×D4—1421no—
350C22×D4—1421no—
351C22×D4—1421no—
352C23×C4—44no—
353C23×C4—2844no—
354C23×C4—2844no—
355C23×C4—1428no—
356C23×C4—1424no—
357C23×C4—1422no—
358C23×C4—1421no—
359C23×C4C4×D7111yescharacteristic
360C23×C4C4×D7111yescharacteristic
361C23×C4C4×D7112yes—
362C23×C4D28411yescharacteristic
363C23×C4D28412yes—
364C23×C4C22⋊D7111yescharacteristic
365C23×C4C22⋊D7111yescharacteristic
366C23×C4C22⋊D7112yes—
367C23×C4C22×D7111yescharacteristic
368C23×C4C22×D7112yes—
369C2×M16—24no—
370C2×M16—44no—
371C22×C8—22no—
372#123—1424no—
373#123—1424no—
374C22⋊Q8—714no—
375C22⋊Q8—712no—
376C22⋊Q8—712no—
377C22⋊Q8—1422no—
378C22⋊Q8—1421no—
379C22⋊Q8—1421no—
380#121—714no—
381#121—712no—
382#121—712no—
383#121—1422no—
384#121—1421no—
385#121—1421no—
386#120—1424no—
387#120—1424no—
388C4×D4—1424no—
389C4×D4—1424no—
390C4×D4—1424no—
391C4×D4—1424no—
392#117—714no—
393#117—1422no—
394C2×C4⋊C4—711no—
395C2×C4⋊C4—711no—
396C2×C4⋊C4—1421no—
397C2×C4⋊C4—1421no—
398C2×C22⋊C4—1421no—
399C2×C22⋊C4—1421no—
400C2×C22⋊C4—1421no—
401C2×C22⋊C4—1421no—
402C2×C22⋊C4—1421no—
403C2×C42—1421no—
404C22⋊C8—24no—
405C22⋊C8—22no—
406C22⋊C8C22×D7114yes—
407#95—714no—
408#95—1424no—
409#95—1422no—
410C24—22no—
411C24—24no—
412C24—24no—
413C24—22no—
414C24—1421no—
415C24(C2×D71)⋊C412yes—
416C24(C2×D71)⋊C411yescharacteristic
417C24(C2×D71)⋊C411yescharacteristic
418C24C2×D28412yes—
419C24C2×D28411yescharacteristic
420#41—2848no—
421#41—2848no—
422C2×Q8—2842no—
423C2×Q8—712no—
424C2×Q8—1421no—
425C2×D4—2842no—
426C2×D4—2842no—
427C2×D4—2844no—
428C2×D4—2844no—
429C2×D4—2842no—
430C2×D4—712no—
431C2×D4—1422no—
432C2×D4—1422no—
433C2×D4—1424no—
434C2×D4—1424no—
435C2×D4—1421no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—2844no—
447C22×C4—28432no—
448C22×C4—2848no—
449C22×C4—2844no—
450C22×C4—2848no—
451C22×C4—2844no—
452C22×C4—14216no—
453C22×C4—1428no—
454C22×C4—1424no—
455C22×C4—1421no—
456C22×C4—1421no—
457C22×C4—1421no—
458C22×C4—1424no—
459C22×C4(C2×D71)⋊C412yes—
460C22×C4(C2×D71)⋊C412yes—
461C22×C4(C2×D71)⋊C411yescharacteristic
462C22×C4(C2×D71)⋊C411yescharacteristic
463C22×C4(C2×D71)⋊C412yes—
464C22×C4(C2×D71)⋊C412yes—
465C22×C4(C2×D71)⋊C412yes—
466C22×C4C2×C4×D7112yes—
467C22×C4C2×C4×D7111yescharacteristic
468C22×C4C2×C4×D7111yescharacteristic
469C22×C4C2×C4×D7112yes—
470C22×C4C2×C22⋊D7112yes—
471C22×C4C2×C22⋊D7111yescharacteristic
472C22×C4C2×C22⋊D7112yes—
473C22×C4C2×C22⋊D7111yescharacteristic
474C22×C4C2×D28412yes—
475C22×C4C2×D28411yescharacteristic
476C22×C4C23×D7111yescharacteristic
477M16—48no—
478C2×C8—24no—
479C2×C8—44no—
480C4⋊C4—712no—
481C4⋊C4—712no—
482C4⋊C4—1422no—
483C4⋊C4—1424no—
484C4⋊C4—1422no—
485C4⋊C4—1421no—
486C4⋊C4—1421no—
487C22⋊C4—1422no—
488C22⋊C4—1424no—
489C22⋊C4—1422no—
490C22⋊C4—1424no—
491C22⋊C4—1422no—
492C22⋊C4—1424no—
493C22⋊C4—1422no—
494C42—1424no—
495C23—216no—
496C23—216no—
497C23—24no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—2842no—
511C23—2842no—
512C23—2842no—
513C23—1422no—
514C23#1455604214yes—
515C23#1455605414yes—
516C23C2×(C2×D71)⋊C412yes—
517C23C2×(C2×D71)⋊C411yescharacteristic
518C23C2×(C2×D71)⋊C412yes—
519C23C2×(C2×D71)⋊C411yescharacteristic
520C23#1455615612yes—
521C23#1455616112yes—
522C23C22×D28411yescharacteristic
523Q8—2842no—
524Q8—2842no—
525D4—2842no—
526D4—2842no—
527D4—2848no—
528D4—2844no—
529D4—2848no—
530D4—2842no—
531D4—2842no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—2848no—
551C2×C4—28432no—
552C2×C4—2844no—
553C2×C4—28416no—
554C2×C4—2842no—
555C2×C4—2842no—
556C2×C4—2842no—
557C2×C4—2844no—
558C2×C4—1428no—
559C2×C4—1422no—
560C2×C4—1422no—
561C2×C4—1422no—
562C2×C4C2×(C2×D71)⋊C412yes—
563C2×C4C2×(C2×D71)⋊C411yescharacteristic
564C2×C4C2×(C2×D71)⋊C412yes—
565C2×C4C2×(C2×D71)⋊C411yescharacteristic
566C2×C4C22×C4×D7111yescharacteristic
567C2×C4C22×C22⋊D7111yescharacteristic
568C8—44no—
569C22—28no—
570C22—216no—
571C22—28no—
572C22—21no—
573C22—22no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—22no—
578C22—24no—
579C22—24no—
580C22—28no—
581C22—28no—
582C22—22no—
583C22—22no—
584C22—2842no—
585C22—2842no—
586C22—2842no—
587C22—5684no—
588C22C2×#1455604214yes—
589C22C2×#1455604212yes—
590C22#1798600112yes—
591C22#1798600112yes—
592C22C2×#1455605414yes—
593C22C2×#1455605412yes—
594C22C22×(C2×D71)⋊C411yescharacteristic
595C22C2×#1455615611yescharacteristic
596C22C2×#1455616111yescharacteristic
597C4—21no—
598C4—22no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—22no—
603C4—2848no—
604C4—2842no—
605C4—2842no—
606C4—2842no—
607C2—21no—
608C2—24no—
609C2—22no—
610C2—24no—
611C2—21no—
612C2—5682no—
613C2#2258684514yesdirect product factor
614C2C22×#1455604211yescharacteristic
615C2C2×#1798600111yescharacteristic
616C2C22×#1455605411yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#407×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC2×#17986510C2
cosocleC22×C4×C71C24
hypercenterC2×#407D71
hypocenterC71C2×#1504
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C22×C71⋊#95C2semidirect
3. #22586845C2direct (N × H)
4. C2×#17985737C2semidirect
5. C2×#17985734C2semidirect
6. C2×#407×C71C2semidirect
7. C22×#11765865C4semidirect
11. #17986510C4semidirect
14. C2×#14556107C4semidirect
16. C2×#14556102C4semidirect
21. C2×#14556087C4semidirect
25. C2×C71⋊#95C22semidirect
26. #17985737C22semidirect
27. #17985734C22semidirect
31. #407×C71C22semidirect
36. C2×#11765865C2×C4semidirect
37. C22×Dic142C2×C4semidirect
38. C22×D284C2×C4semidirect
53. #14556107C2×C4semidirect
54. #14556107C2×C4semidirect
59. #14556102C2×C4semidirect
60. #14556102C2×C4semidirect
69. #14556087C2×C4semidirect
70. C2×C4⋊Dic71C2×C4semidirect
71. C2×C4⋊Dic71C2×C4semidirect
106. C2×Dic142C22×C4semidirect
109. C2×D284C22×C4semidirect
133. C4⋊Dic71C22×C4semidirect
134. C4⋊Dic71C22×C4semidirect
304. C2×C71#1504semidirect
310. C2×#407D71semidirect
315. C71C2×#1504semidirect
334. #407C2×D71semidirect
613. C2#22586845direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179865102
C22×C71⋊#952
#225868452
C2×#179857372
C2×#179857342
C2×#407×C712
C2×#150471