Finite Group Databasebeta

← #31998673

Subgroups — #31998673

Order 18176.

1986 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986510C211yesmaximal, characteristic
2C2×#17985993C212yesmaximal, complemented
3C22×C22⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
4#22588662C212yesmaximal, complemented
5#22588657C212yesmaximal, complemented
6#22588652C212yesmaximal, complemented
7#22588647C212yesmaximal, complemented
8#22588056C212yesmaximal, complemented
9C22×#117×C71C211yesmaximal, characteristic, complemented
10C22×#11765865C2211yescharacteristic
11C23×C71⋊C8C2212yes—
12#17986510—24no—
13#17986510C418yescomplemented
14C2×#14556111—21no—
15C2×#14556107C2211yescharacteristic
16C2×#14556107C2211yescharacteristic
17C2×#14556102C2211yescharacteristic
18C2×#14556102C2211yescharacteristic
19C2×#14556097—21no—
20C2×C4×C22⋊D71—21no—
21C2×C4×C22⋊D71—21no—
22C2×#14556087C2211yescharacteristic
23#17985993—28no—
24C2×#14556051C2214yescomplemented
25C2×#14556048C2214yescomplemented
26C2×C22⋊(C71⋊C8)—24no—
27C2×C22⋊(C71⋊C8)C2212yes—
28C2×C22⋊(C71⋊C8)C2212yes—
29#17985761C2214yescomplemented
30#17985761C2214yescomplemented
31C24×C4×C71C2211yescharacteristic
32C2×#117×C71—21no—
33C2×#117×C71—21no—
34C2×#117×C71—28no—
35C2×#117×C71C2212yes—
36C2×#117×C71C2212yes—
37C22×C4⋊C4×C71C2212yes—
38C22×C22⋊C4×C71—21no—
39C22×C42×C71—21no—
40C2×#11765865—22no—
41C2×#11765865—24no—
42C2×#11765865—24no—
43C2×#11765865C2×C414yescomplemented
44C22×Dic142C2311yescharacteristic
45C22×D284C2311yescharacteristic
46C22×C22⋊D71—21no—
47C22×C22⋊D71—21no—
48C22×C22⋊D71—21no—
49C22×C4×D71—21no—
50C22×C71⋊C8—24no—
51C22×C71⋊C8—24no—
52C22×C71⋊C8—24no—
53C22×C71⋊C8C2314yes—
54#14556111—28no—
55#14556107—22no—
56#14556107—22no—
57#14556107C2×C414yes—
58#14556107C2×C414yescomplemented
59#14556102—22no—
60#14556102—22no—
61#14556102C2×C414yes—
62#14556102C2×C414yescomplemented
63#14556097—28no—
64C4×C22⋊D71—28no—
65C4×C22⋊D71—28no—
66#14556087—22no—
67#14556087C2×C414yescomplemented
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×Dic71⋊C4—21no—
71C2×Dic71⋊C4—21no—
72C2×C22⋊Dic71—21no—
73C2×C22⋊Dic71—21no—
74C2×C22⋊Dic71—21no—
75C2×(C2×D71)⋊C4—21no—
76C2×(C2×D71)⋊C4—21no—
77C2×C4×Dic71—21no—
78#14556051—28no—
79#14556048—28no—
80C22⋊(C71⋊C8)—24no—
81C22⋊(C71⋊C8)—24no—
82C25×C71D411yescharacteristic
83C23×C4×C71—24no—
84C23×C4×C71—24no—
85C23×C4×C71—22no—
86C23×C4×C71C2×C414yes—
87C23×C4×C71D411yescharacteristic
88C23×C4×C71D411yescharacteristic
89C23×C4×C71D411yescharacteristic
90C23×C4×C71D414yes—
91C23×C4×C71C2311yescharacteristic
92C23×C4×C71C2311yescharacteristic
93C23×C4×C71C2311yescharacteristic
94#117×C71—28no—
95#117×C71—28no—
96#117×C71—24no—
97#117×C71—28no—
98#117×C71D418yescomplemented
99C2×C4⋊C4×C71—22no—
100C2×C4⋊C4×C71—28no—
101C2×C4⋊C4×C71C2314yes—
102C2×C22⋊C4×C71—28no—
103C2×C22⋊C4×C71—22no—
104C2×C22⋊C4×C71—22no—
105C2×C42×C71—28no—
106C2×C42×C71—22no—
107C2×C42×C71—22no—
108C23×D71—21no—
109#11765865—216no—
110#11765865—48no—
111C2×Dic142—22no—
112C2×Dic142—22no—
113C2×Dic142—21no—
114C2×Dic142C22×C412yes—
115C2×D284—22no—
116C2×D284—22no—
117C2×D284—21no—
118C2×D284C22×C412yes—
119C2×C22⋊D71—24no—
120C2×C22⋊D71—24no—
121C2×C22⋊D71—24no—
122C2×C22⋊D71—44no—
123C2×C22⋊D71—42no—
124C2×C22⋊D71—42no—
125C2×C22⋊D71—44no—
126C22×Dic71—21no—
127C22×Dic71—21no—
128C22×Dic71—21no—
129C2×C4×D71—24no—
130C2×C4×D71—42no—
131C2×C4×D71—42no—
132C2×C71⋊C8—24no—
133C2×C71⋊C8—24no—
134C2×C71⋊C8—24no—
135C2×C71⋊C8—48no—
136C4⋊Dic71—21no—
137C4⋊Dic71—21no—
138C4⋊Dic71C22×C412yes—
139C4⋊Dic71C22×C412yes—
140Dic71⋊C4—24no—
141Dic71⋊C4—24no—
142C22⋊Dic71—24no—
143C22⋊Dic71—24no—
144C22⋊Dic71—24no—
145(C2×D71)⋊C4—24no—
146(C2×D71)⋊C4—24no—
147C4×Dic71—24no—
148C24×C71—22no—
149C24×C71—22no—
150C24×C71—24no—
151C24×C71—42no—
152C24×C71C22⋊C414yes—
153C24×C71C2×D411yescharacteristic
154C24×C71C2×D411yescharacteristic
155C24×C71C2×D411yescharacteristic
156C22×C4×C71—22no—
157C22×C4×C71—22no—
158C22×C4×C71—216no—
159C22×C4×C71—22no—
160C22×C4×C71—22no—
161C22×C4×C71—216no—
162C22×C4×C71—22no—
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71—22no—
168C22×C4×C71—28no—
169C22×C4×C71—24no—
170C22×C4×C71—44no—
171C22×C4×C71—44no—
172C22×C4×C71C22⋊C414yes—
173C22×C4×C71C22⋊C414yes—
174C22×C4×C71C22⋊C414yes—
175C22×C4×C71C22×C412yes—
176C22×C4×C71C22×C412yes—
177C22×C4×C71C22×C412yes—
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D414yes—
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D412yes—
183C22×C4×C71C2×D412yes—
184C22×C4×C71C2×D412yes—
185C22×C4×C71#4114yes—
186C22×C4×C71C2411yescharacteristic
187C4⋊C4×C71—28no—
188C4⋊C4×C71—24no—
189C4⋊C4×C71C2×D418yes—
190C22⋊C4×C71—28no—
191C22⋊C4×C71—28no—
192C42×C71—28no—
193C42×C71—28no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71—22no—
197C23×C71—22no—
198C23×C71—22no—
199C23×C71—21no—
200C23×C71—21no—
201C23×C71—22no—
202C23×C71—22no—
203C23×C71—28no—
204C23×C71—28no—
205C23×C71—44no—
206C23×C71—48no—
207C23×C71—48no—
208C23×C71C2×C22⋊C412yes—
209C23×C71C2×C22⋊C412yes—
210C23×C71C2×C22⋊C412yes—
211C23×C71#12012yes—
212C23×C71#12312yes—
213C23×C71#13514yes—
214C23×C71C22×D411yescharacteristic
215C22×D71—22no—
216C22×D71—41no—
217C22×D71—41no—
218C22×D71—42no—
219C22×D71—42no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—28no—
225C2×C4×C71—24no—
226C2×C4×C71—24no—
227C2×C4×C71—24no—
228C2×C4×C71—24no—
229C2×C4×C71—28no—
230C2×C4×C71—28no—
231C2×C4×C71—28no—
232C2×C4×C71—416no—
233C2×C4×C71—44no—
234C2×C4×C71—44no—
235C2×C4×C71C2×C22⋊C412yes—
236C2×C4×C71C2×C22⋊C412yes—
237C2×C4×C71C2×C22⋊C412yes—
238C2×C4×C71C4×D418yes—
239C2×C4×C71#12012yes—
240C2×C4×C71#12118yes—
241C2×C4×C71#12312yes—
242C2×C4×C71C23×C411yescharacteristic
243C2×C4×C71C22×D412yes—
244C2×C4×C71C22×D411yescharacteristic
245C2×C4×C71C2×#4112yes—
246C22⋊D71—48no—
247C22⋊D71—48no—
248C22⋊D71—48no—
249C2×Dic71—22no—
250C2×Dic71—22no—
251C2×Dic71—22no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—42no—
255C2×Dic71—42no—
256D284—24no—
257D284—42no—
258C4×D71—48no—
259Dic142—24no—
260Dic142—42no—
261C71⋊C8—44no—
262C22×C71—24no—
263C22×C71—24no—
264C22×C71—22no—
265C22×C71—22no—
266C22×C71—22no—
267C22×C71—24no—
268C22×C71—24no—
269C22×C71—22no—
270C22×C71—22no—
271C22×C71—22no—
272C22×C71—22no—
273C22×C71—416no—
274C22×C71—42no—
275C22×C71—42no—
276C22×C71#38614yes—
277C22×C71#41614yescomplemented
278C22×C71C22×C22⋊C411yescharacteristic
279C22×C71C2×#12011yescharacteristic
280C22×C71C2×#12311yescharacteristic
281C22×C71C2×#13512yes—
282C22×C71#57512yes—
283C2×D71—42no—
284C2×D71—42no—
285C2×D71—42no—
286C2×D71—82no—
287C2×D71—82no—
288C4×C71—22no—
289C4×C71—22no—
290C4×C71—48no—
291C4×C71#38614yes—
292C4×C71C2×C4×D412yes—
293C4×C71C2×#12112yes—
294Dic71—42no—
295Dic71—42no—
296Dic71—42no—
297#20908—711nomaximal, Sylow
298C2×C71—24no—
299C2×C71—24no—
300C2×C71—42no—
301C2×C71C2×#38611yescharacteristic
302C2×C71C2×#41611yescharacteristic
303C2×C71#289611yescharacteristic
304C2×C71#300212yes—
305C2×C71#309112yes—
306D71—82no—
307C2×#525—711no—
308C22×#117D7111yescharacteristic, complemented
309C2×#416—712no—
310C22×C22⋊C8—711no—
311#1875—712no—
312#1874—712no—
313#1873—712no—
314#1872—712no—
315#1735—712no—
316C71#2090811yescharacteristic, complemented, Sylow
317C22×#41—711no—
318C24×C4C2×D7111yescharacteristic
319C23×C8—712no—
320#525—718no—
321#525—1424no—
322C2×#123—1421no—
323C2×C22⋊Q8—711no—
324C2×C22⋊Q8—711no—
325C2×#121—711no—
326C2×#121—711no—
327C2×#120—1421no—
328C2×C4×D4—1421no—
329C2×C4×D4—1421no—
330C2×#117—21no—
331C2×#117—21no—
332C2×#117—28no—
333C2×#117—711no—
334C2×#117C2×D7112yes—
335C2×#117C2×D7112yes—
336C22×C4⋊C4C2×D7112yes—
337C22×C22⋊C4—21no—
338C22×C42—21no—
339#416—1428no—
340C2×#103—714no—
341C2×#102—714no—
342C2×C22⋊C8—712no—
343C2×C22⋊C8—712no—
344C2×C22⋊C8—1424no—
345#340—714no—
346#340—714no—
347C25C22⋊D7111yescharacteristic
348C2×#41—714no—
349C2×#41—1424no—
350C2×#41—1424no—
351C2×#41—1422no—
352C22×Q8—711no—
353C22×D4—711no—
354C22×D4—1421no—
355C22×D4—1421no—
356C22×D4—1421no—
357C23×C4—24no—
358C23×C4—24no—
359C23×C4—22no—
360C23×C4—1421no—
361C23×C4C4×D7114yes—
362C23×C4D28414yes—
363C23×C4C22⋊D7111yescharacteristic
364C23×C4C22⋊D7111yescharacteristic
365C23×C4C22⋊D7111yescharacteristic
366C23×C4C22×D7111yescharacteristic
367C23×C4C22×D7111yescharacteristic
368C23×C4C22×D7111yescharacteristic
369C22×C8—714no—
370C22×C8—1424no—
371C22×C8—1424no—
372C22×C8—1424no—
373#123—1428no—
374C22⋊Q8—714no—
375C22⋊Q8—714no—
376C22⋊Q8—1422no—
377C22⋊Q8—1422no—
378#121—714no—
379#121—714no—
380#121—1422no—
381#121—1422no—
382#120—1428no—
383C4×D4—1428no—
384C4×D4—1428no—
385#117—28no—
386#117—28no—
387#117—24no—
388#117—28no—
389#117—714no—
390#117—1422no—
391#117C22⋊D7118yescomplemented
392C2×C4⋊C4—22no—
393C2×C4⋊C4—28no—
394C2×C4⋊C4—711no—
395C2×C4⋊C4—711no—
396C2×C4⋊C4—1421no—
397C2×C4⋊C4—1421no—
398C2×C4⋊C4C22×D7114yes—
399C2×C22⋊C4—28no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—1421no—
403C2×C22⋊C4—1421no—
404C2×C22⋊C4—1421no—
405C2×C22⋊C4—1421no—
406C2×C22⋊C4—1421no—
407C2×C42—28no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42—1421no—
411#103—1428no—
412#102—1428no—
413C22⋊C8—1424no—
414C22⋊C8—1424no—
415C24—22no—
416C24—22no—
417C24—24no—
418C24—42no—
419C24—1421no—
420C24(C2×D71)⋊C414yes—
421C24C2×C22⋊D7111yescharacteristic
422C24C2×C22⋊D7111yescharacteristic
423C24C2×C22⋊D7111yescharacteristic
424#41—2848no—
425#41—14216no—
426C2×Q8—712no—
427C2×Q8—1422no—
428C2×Q8—1422no—
429C2×Q8—1421no—
430C2×D4—2844no—
431C2×D4—2842no—
432C2×D4—2842no—
433C2×D4—2844no—
434C2×D4—712no—
435C2×D4—1424no—
436C2×D4—1424no—
437C2×D4—1424no—
438C2×D4—1422no—
439C2×D4—1421no—
440C2×D4—1422no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—216no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—216no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—2842no—
458C22×C4—2842no—
459C22×C4—1421no—
460C22×C4—1421no—
461C22×C4—1421no—
462C22×C4—1424no—
463C22×C4(C2×D71)⋊C414yes—
464C22×C4(C2×D71)⋊C414yes—
465C22×C4(C2×D71)⋊C414yes—
466C22×C4C2×C4×D7112yes—
467C22×C4C2×C4×D7112yes—
468C22×C4C2×C4×D7112yes—
469C22×C4C2×C22⋊D7111yescharacteristic
470C22×C4C2×C22⋊D7111yescharacteristic
471C22×C4C2×C22⋊D7114yes—
472C22×C4C2×C22⋊D7111yescharacteristic
473C22×C4C2×D28412yes—
474C22×C4C2×D28412yes—
475C22×C4C2×D28412yes—
476C22×C4#1176586514yes—
477C22×C4C23×D7111yescharacteristic
478C2×C8—2848no—
479C2×C8—1424no—
480C2×C8—1424no—
481C2×C8—1424no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—712no—
485C4⋊C4—712no—
486C4⋊C4—1421no—
487C4⋊C4—1421no—
488C4⋊C4—1424no—
489C4⋊C4—1424no—
490C4⋊C4C2×C22⋊D7118yes—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—1424no—
494C22⋊C4—1424no—
495C22⋊C4—1424no—
496C22⋊C4—1424no—
497C22⋊C4—1424no—
498C42—28no—
499C42—28no—
500C42—1424no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—28no—
507C23—21no—
508C23—21no—
509C23—28no—
510C23—22no—
511C23—22no—
512C23—44no—
513C23—48no—
514C23—48no—
515C23—2841no—
516C23—2841no—
517C23—2842no—
518C23—2842no—
519C23—1422no—
520C23C2×(C2×D71)⋊C412yes—
521C23C2×(C2×D71)⋊C412yes—
522C23C2×(C2×D71)⋊C412yes—
523C23#1455609712yes—
524C23#1455611112yes—
525C23#1455614814yes—
526C23C22×C22⋊D7111yescharacteristic
527Q8—2842no—
528Q8—1424no—
529D4—2848no—
530D4—2848no—
531D4—2848no—
532D4—2842no—
533D4—1424no—
534C2×C4—28no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—416no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—2841no—
550C2×C4—2841no—
551C2×C4—2842no—
552C2×C4—2842no—
553C2×C4—2848no—
554C2×C4—1422no—
555C2×C4—1422no—
556C2×C4—1422no—
557C2×C4C2×(C2×D71)⋊C412yes—
558C2×C4C2×(C2×D71)⋊C412yes—
559C2×C4C2×(C2×D71)⋊C412yes—
560C2×C4C4×C22⋊D7118yes—
561C2×C4#1455609712yes—
562C2×C4#1455610218yes—
563C2×C4#1455611112yes—
564C2×C4C22×C4×D7111yescharacteristic
565C2×C4C22×C22⋊D7112yes—
566C2×C4C22×D28411yescharacteristic
567C2×C4C2×#1176586512yes—
568C8—2844no—
569C22—22no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—416no—
581C22—42no—
582C22—42no—
583C22—2842no—
584C22—2842no—
585C22—2842no—
586C22—5682no—
587C22—5682no—
588C22#1798587814yes—
589C22#1798599314yescomplemented
590C22C22×(C2×D71)⋊C411yescharacteristic
591C22C2×#1455609711yescharacteristic
592C22C2×#1455611111yescharacteristic
593C22C2×#1455614812yes—
594C22#1798679712yes—
595C4—22no—
596C4—22no—
597C4—48no—
598C4—2842no—
599C4—2842no—
600C4—2842no—
601C4#1798587814yes—
602C4C2×C4×C22⋊D7112yes—
603C4C2×#1455610212yes—
604C2—24no—
605C2—24no—
606C2—42no—
607C2—5682no—
608C2C2×#1798587811yescharacteristic
609C2C2×#1798599311yescharacteristic
610C2#2259685711yescharacteristic
611C2#2259755812yes—
612C2#2259825312yes—
subgroupG/H
centerC22×C4C2×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC22×#117×C71C2
socleC23×C71C22×D4
prime radicalC2×#17986510C2
cosocleC22×C4×C71C24
hypercenterC22×#117D71
hypocenterC71#20908
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C2×#17985993C2semidirect
3. C22×C22⋊(C71⋊C8)C2semidirect
4. #22588662C2semidirect
5. #22588657C2semidirect
6. #22588652C2semidirect
7. #22588647C2semidirect
8. #22588056C2semidirect
9. C22×#117×C71C2semidirect
13. #17986510C4semidirect
24. C2×#14556051C22semidirect
25. C2×#14556048C22semidirect
29. #17985761C22semidirect
30. #17985761C22semidirect
43. C2×#11765865C2×C4semidirect
58. #14556107C2×C4semidirect
62. #14556102C2×C4semidirect
67. #14556087C2×C4semidirect
98. #117×C71D4semidirect
277. C22×C71#416semidirect
308. C22×#117D71semidirect
316. C71#20908semidirect
391. #117C22⋊D71semidirect
589. C22#17985993semidirect

Maximal subgroups

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

subgroupindex
C2×#179865102
C2×#179859932
C22×C22⋊(C71⋊C8)2
#225886622
#225886572
#225886522
#225886472
#225880562
C22×#117×C712
#2090871