Finite Group Databasebeta

← #32016470

Subgroups — #32016470

Order 18176.

1052 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
1C2×#17986051C211yesmaximal, characteristic, complemented
2C2×C4⋊C8×D71C211yesmaximal, characteristic, complemented
3C42×D284C211yesmaximal, characteristic
4#22588944C212yesmaximal, complemented
5#22588896C212yesmaximal, complemented
6#22588631C212yesmaximal
7#22588317C211yesmaximal, characteristic, complemented
8#22588317C211yesmaximal, characteristic, complemented
9C4×C4⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
10#22588110C212yesmaximal, complemented
11C71×#1831C211yesmaximal, characteristic, complemented
12C22×C8×D71C2212yes—
13C2×C4×D284C411yescharacteristic
14C2×C4×D284C411yescharacteristic
15C2×C4×D284C2211yescharacteristic
16C2×C42×D71C2211yescharacteristic
17C2×C42×D71C2211yescharacteristic
18#17986051—24no—
19C2×C4⋊(C71⋊C8)C2211yescharacteristic
20C2×C4⋊(C71⋊C8)C2211yescharacteristic
21C2×Dic71⋊C8C2212yes—
22C4⋊C8×D71—24no—
23C2×(C2×D71)⋊C8C2212yes—
24C2×(C2×D71)⋊C8C2212yes—
25C2×(C2×D71)⋊C8C2212yes—
26C2×C4×C71⋊C8C2212yes—
27C2×C8×Dic71C2212yes—
28C4×C4⋊Dic71C411yescharacteristic
29C4×C4⋊Dic71C411yescharacteristic
30C4×C4⋊Dic71C2211yescharacteristic
31C4×(C2×D71)⋊C4C411yescharacteristic
32C4×(C2×D71)⋊C4C411yescharacteristic
33C4×(C2×D71)⋊C4C411yescharacteristic
34C4×(C2×D71)⋊C4C411yescharacteristic
35C4×(C2×D71)⋊C4C2211yescharacteristic
36C4×(C2×D71)⋊C4C2211yescharacteristic
37#17985754C2212yes—
38#17985753C2212yes—
39#17985753C2212yes—
40C2×C4⋊C8×C71C2211yescharacteristic
41C2×C4⋊C8×C71C2211yescharacteristic
42C43×C71C2211yescharacteristic
43#336×C71C2212yes—
44#336×C71C2212yes—
45C22×D284C2×C411yescharacteristic
46C22×C4×D71C2×C411yescharacteristic
47C22×C4×D71C2×C411yescharacteristic
48C22×C4×D71C2×C411yescharacteristic
49C22×C4×D71C2×C411yescharacteristic
50C22×C4×D71C2311yescharacteristic
51C22×C4×D71C2311yescharacteristic
52C22×C71⋊C8C2312yes—
53C22×C71⋊C8C2312yes—
54C2×C8×D71—24no—
55C2×C8×D71—24no—
56C2×C8×D71—42no—
57C4×D284—22no—
58C4×D284—22no—
59C4×D284—24no—
60C4×D284C818yescomplemented
61C2×C4⋊Dic71C2×C411yescharacteristic
62C2×C4⋊Dic71C2×C411yescharacteristic
63C2×C4⋊Dic71C2×C411yescharacteristic
64C2×(C2×D71)⋊C4C2×C411yescharacteristic
65C2×(C2×D71)⋊C4C2×C411yescharacteristic
66C2×(C2×D71)⋊C4C2×C411yescharacteristic
67C2×(C2×D71)⋊C4C2×C411yescharacteristic
68C2×(C2×D71)⋊C4C2×C411yescharacteristic
69C2×(C2×D71)⋊C4C2×C411yescharacteristic
70C2×C4×Dic71C2×C411yescharacteristic
71C2×C4×Dic71C2×C411yescharacteristic
72C2×C4×Dic71C2×C411yescharacteristic
73C2×C4×Dic71C2×C411yescharacteristic
74C2×C4×Dic71C2311yescharacteristic
75C2×C4×Dic71C2311yescharacteristic
76C42×D71—22no—
77C42×D71—22no—
78C42×D71—22no—
79C42×D71—22no—
80C4⋊(C71⋊C8)—22no—
81C4⋊(C71⋊C8)D414yescomplemented
82Dic71⋊C8—24no—
83(C2×D71)⋊C8—24no—
84(C2×D71)⋊C8—24no—
85(C2×D71)⋊C8—42no—
86C4×C71⋊C8—42no—
87C8×Dic71—24no—
88C71⋊#95C2×C411yescharacteristic
89C71⋊#95C2×C411yescharacteristic
90C71⋊#95C2×C411yescharacteristic
91C71⋊#95C2×C411yescharacteristic
92C71⋊#95C2×C411yescharacteristic
93C71⋊#95C2×C411yescharacteristic
94C22×C8×C71C2312yes—
95C22×C8×C71C2312yes—
96C2×C42×C71C2×C411yescharacteristic
97C2×C42×C71C2×C411yescharacteristic
98C2×C42×C71C2×C411yescharacteristic
99C2×C42×C71C2×C411yescharacteristic
100C2×C42×C71C2311yescharacteristic
101C2×C42×C71C2311yescharacteristic
102C2×C42×C71C2311yescharacteristic
103C4⋊C8×C71—22no—
104C4⋊C8×C71—22no—
105C23×D71C22×C411yescharacteristic
106C23×D71C22×C411yescharacteristic
107C2×D284—22no—
108C2×D284—22no—
109C2×D284C2×C814yes—
110C22×Dic71C22×C411yescharacteristic
111C22×Dic71C22×C411yescharacteristic
112C22×Dic71C22×C411yescharacteristic
113C22×Dic71C22×C411yescharacteristic
114C22×Dic71C22×C411yescharacteristic
115C22×Dic71C22×C411yescharacteristic
116C2×C4×D71—22no—
117C2×C4×D71—22no—
118C2×C4×D71—22no—
119C2×C4×D71—22no—
120C2×C4×D71—21no—
121C2×C4×D71—21no—
122C2×C4×D71—21no—
123C2×C4×D71—21no—
124C2×C4×D71—22no—
125C2×C4×D71—22no—
126C2×C4×D71—22no—
127C2×C4×D71—21no—
128C2×C4×D71—21no—
129C2×C4×D71—21no—
130C2×C4×D71—21no—
131C2×C4×D71—21no—
132C2×C4×D71—21no—
133C2×C4×D71—22no—
134C2×C4×D71—41no—
135C2×C4×D71—41no—
136C2×C4×D71C2×C814yes—
137C2×C4×D71C2×C814yescomplemented
138C2×C4×D71#4112yes—
139C2×C4×D71#4112yes—
140C2×C71⋊C8—22no—
141C2×C71⋊C8—22no—
142C2×C71⋊C8—22no—
143C2×C71⋊C8—22no—
144C2×C71⋊C8—22no—
145C2×C71⋊C8C2×D414yes—
146C8×D71—48no—
147C4⋊Dic71—22no—
148C4⋊Dic71—22no—
149C4⋊Dic71C2×C814yescomplemented
150(C2×D71)⋊C4—22no—
151(C2×D71)⋊C4—22no—
152(C2×D71)⋊C4—22no—
153(C2×D71)⋊C4C2×C814yes—
154(C2×D71)⋊C4C2×C814yescomplemented
155(C2×D71)⋊C4M1614yes—
156C4×Dic71—22no—
157C4×Dic71—22no—
158C4×Dic71—22no—
159C4×Dic71—22no—
160C4×Dic71—21no—
161C4×Dic71—21no—
162C4×Dic71—21no—
163C4×Dic71—21no—
164C22×C4×C71C22×C411yescharacteristic
165C22×C4×C71C22×C411yescharacteristic
166C22×C4×C71C22×C411yescharacteristic
167C22×C4×C71C22×C411yescharacteristic
168C22×C4×C71C22×C411yescharacteristic
169C22×C4×C71C22×C411yescharacteristic
170C22×C4×C71C2411yescharacteristic
171C2×C8×C71—22no—
172C2×C8×C71—22no—
173C2×C8×C71—22no—
174C2×C8×C71—22no—
175C2×C8×C71—22no—
176C2×C8×C71—22no—
177C42×C71—22no—
178C42×C71—22no—
179C42×C71—21no—
180C42×C71—21no—
181C42×C71—21no—
182C42×C71—21no—
183C42×C71—41no—
184C42×C71C2×C814yescomplemented
185C42×C71C2×D412yes—
186C42×C71#4112yes—
187C23×C71C23×C411yescharacteristic
188C22×D71—21no—
189C22×D71—21no—
190C22×D71—21no—
191C22×D71—21no—
192C22×D71—21no—
193C22×D71—21no—
194C22×D71—21no—
195C22×D71—21no—
196C22×D71—21no—
197C22×D71—21no—
198C22×D71#11712yes—
199C22×D71C22×C812yes—
200C22×D71C22×C812yes—
201C22×D71C2×M1612yes—
202C2×C4×C71—21no—
203C2×C4×C71—21no—
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—21no—
207C2×C4×C71—21no—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71C4×D412yes—
214C2×C4×C71C4×D412yes—
215C2×C4×C71C22×C812yes—
216C2×C4×C71C22×C812yes—
217C2×C4×C71C22×C812yes—
218C2×C4×C71C2×M1612yes—
219C2×C4×C71#13112yes—
220C2×C4×C71C22×D411yescharacteristic
221C2×C4×C71C2×#4111yescharacteristic
222C2×C4×C71C2×#4111yescharacteristic
223C2×C4×C71C2×#4111yescharacteristic
224C2×C4×C71H(5,𝔽2)11yescharacteristic
225C2×C4×C712−1+411yescharacteristic
226C2×Dic71—21no—
227C2×Dic71—21no—
228C2×Dic71—21no—
229C2×Dic71—21no—
230C2×Dic71—21no—
231C2×Dic71—21no—
232C2×Dic71—21no—
233C2×Dic71—21no—
234C2×Dic71—21no—
235C2×Dic71—21no—
236C2×Dic71—21no—
237C2×Dic71—21no—
238C2×Dic71—21no—
239C2×Dic71—21no—
240C2×Dic71#11712yes—
241C2×Dic71C22×C812yes—
242C2×Dic71C22×C812yes—
243C2×Dic71C2×M1612yes—
244D284—28no—
245C4×D71—22no—
246C4×D71—22no—
247C4×D71—22no—
248C4×D71—22no—
249C4×D71—28no—
250C4×D71—28no—
251C4×D71—44no—
252C4×D71—44no—
253C4×D71—41no—
254C4×D71—41no—
255C4×D71—41no—
256C4×D71—41no—
257C8×C71—42no—
258C8×C71—42no—
259C71⋊C8—24no—
260C71⋊C8—42no—
261C22×C71C2×#11711yescharacteristic
262C22×C71C2×C4×D411yescharacteristic
263C22×C71#52011yescharacteristic
264C22×C71C23×C811yescharacteristic
265C22×C71C22×M1611yescharacteristic
266C22×C71C2×#13111yescharacteristic
267C22×C71#56811yescharacteristic
268C2×D71—22no—
269C2×D71—22no—
270C2×D71—22no—
271C2×D71—22no—
272C2×D71—22no—
273C2×D71—22no—
274C2×D71—22no—
275C2×D71—22no—
276C2×D71—22no—
277C2×D71—41no—
278C2×D71—41no—
279C2×D71—41no—
280C2×D71—41no—
281C2×D71#43114yescomplemented
282C4×C71—22no—
283C4×C71—22no—
284C4×C71—22no—
285C4×C71—22no—
286C4×C71—41no—
287C4×C71C8×D414yes—
288C4×C71#54611yescharacteristic
289C4×C71#54811yescharacteristic
290C4×C71#55211yescharacteristic
291C4×C71#55311yescharacteristic
292Dic71—22no—
293Dic71—22no—
294Dic71—22no—
295Dic71—22no—
296Dic71—41no—
297Dic71—41no—
298#22748—711nomaximal, Sylow
299C2×C71#235211yescharacteristic
300C2×C71C2×#43111yescharacteristic
301C2×C71#292511yescharacteristic
302C2×C71C2×C8×D411yescharacteristic
303C2×C71D4×M1611yescharacteristic
304C2×C71#298811yescharacteristic
305C2×C71#299011yescharacteristic
306D71—24no—
307D71—42no—
308C2×#431—711no—
309C22×C4⋊C8—711no—
310C42×D4—711no—
311#1923—712no—
312#1916—712no—
313#1869—712no—
314#1831D7111yescharacteristic, complemented
315#1797—711no—
316#1797—711no—
317C4×C4⋊C8—711no—
318C8×C22⋊C4—712no—
319C71#2274811yescharacteristic, complemented, Sylow
320C23×C8—712no—
321C2×C4×D4—711no—
322C2×C4×D4—711no—
323C2×C4×D4—711no—
324C22×C42—711no—
325C22×C42—711no—
326#431—1424no—
327C2×C4⋊C8—711no—
328C2×C4⋊C8—712no—
329C2×C4⋊C8—711no—
330C2×C4⋊C8—1424no—
331C2×C4⋊C8C2×D7111yescharacteristic
332C2×C4⋊C8C2×D7111yescharacteristic
333C2×C22⋊C8—712no—
334C2×C22⋊C8—712no—
335C2×C22⋊C8—712no—
336C2×C4×C8—712no—
337C2×C4×C8—712no—
338C4×C4⋊C4—711no—
339C4×C4⋊C4—711no—
340C4×C4⋊C4—711no—
341C4×C22⋊C4—711no—
342C4×C22⋊C4—711no—
343C4×C22⋊C4—711no—
344C4×C22⋊C4—711no—
345C4×C22⋊C4—711no—
346C4×C22⋊C4—711no—
347C43C2×D7111yescharacteristic
348#336—712no—
349#336—712no—
350#336—712no—
351#336C2×D7112yes—
352#336C2×D7112yes—
353C22×D4—711no—
354C23×C4—711no—
355C23×C4—711no—
356C23×C4—711no—
357C23×C4—711no—
358C23×C4—711no—
359C23×C4—711no—
360C22×C8—2842no—
361C22×C8—712no—
362C22×C8—712no—
363C22×C8—1424no—
364C22×C8—1424no—
365C22×C8C22×D7112yes—
366C22×C8C22×D7112yes—
367C4×D4—718no—
368C4×D4—1424no—
369C4×D4—1422no—
370C4×D4—1422no—
371C2×C4⋊C4—711no—
372C2×C4⋊C4—711no—
373C2×C4⋊C4—711no—
374C2×C22⋊C4—711no—
375C2×C22⋊C4—711no—
376C2×C22⋊C4—711no—
377C2×C22⋊C4—711no—
378C2×C22⋊C4—711no—
379C2×C22⋊C4—711no—
380C2×C42—711no—
381C2×C42—711no—
382C2×C42—711no—
383C2×C42—711no—
384C2×C42—711no—
385C2×C42—711no—
386C2×C42—1422no—
387C2×C42—1422no—
388C2×C42—1422no—
389C2×C42—1422no—
390C2×C42C4×D7111yescharacteristic
391C2×C42C4×D7111yescharacteristic
392C2×C42C4×D7111yescharacteristic
393C2×C42C4×D7111yescharacteristic
394C2×C42C22×D7111yescharacteristic
395C2×C42C22×D7111yescharacteristic
396C2×C42C22×D7111yescharacteristic
397C4⋊C8—22no—
398C4⋊C8—22no—
399C4⋊C8—714no—
400C4⋊C8—1424no—
401C4⋊C8—1422no—
402C22⋊C8—2842no—
403C22⋊C8—1424no—
404C22⋊C8—1424no—
405C4×C8—2842no—
406C4×C8—1424no—
407#95—711no—
408#95—711no—
409#95—711no—
410#95—711no—
411#95—711no—
412#95—711no—
413C24—711no—
414C24—711no—
415C2×D4—714no—
416C2×D4—1422no—
417C2×D4—1422no—
418C22×C4—2841no—
419C22×C4—2841no—
420C22×C4—711no—
421C22×C4—711no—
422C22×C4—711no—
423C22×C4—711no—
424C22×C4—712no—
425C22×C4—712no—
426C22×C4—714no—
427C22×C4—714no—
428C22×C4—711no—
429C22×C4—711no—
430C22×C4—1422no—
431C22×C4—1422no—
432C22×C4—1421no—
433C22×C4—1421no—
434C22×C4—1421no—
435C22×C4—1421no—
436C22×C4—1421no—
437C22×C4—1421no—
438C22×C4—1422no—
439C22×C4—1422no—
440C22×C4—1421no—
441C22×C4—1421no—
442C22×C4—1421no—
443C22×C4—1421no—
444C22×C4—1422no—
445C22×C4—1422no—
446C22×C4—1422no—
447C22×C4—1422no—
448C22×C4C2×C4×D7111yescharacteristic
449C22×C4C2×C4×D7111yescharacteristic
450C22×C4C2×C4×D7111yescharacteristic
451C22×C4C2×C4×D7111yescharacteristic
452C22×C4C2×C4×D7111yescharacteristic
453C22×C4C2×C4×D7111yescharacteristic
454C22×C4C23×D7111yescharacteristic
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—2848no—
462C2×C8—714no—
463C2×C8—1422no—
464C2×C8—1422no—
465C2×C8—1422no—
466C2×C8—1422no—
467C2×C8—1422no—
468C4⋊C4—714no—
469C4⋊C4—1422no—
470C4⋊C4—1422no—
471C22⋊C4—714no—
472C22⋊C4—714no—
473C22⋊C4—714no—
474C22⋊C4—1422no—
475C22⋊C4—1422no—
476C22⋊C4—1422no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—22no—
482C42—22no—
483C42—41no—
484C42—1421no—
485C42—1421no—
486C42—1421no—
487C42—1421no—
488C42—1422no—
489C42—1422no—
490C42—1422no—
491C42—1422no—
492C42C8×D7114yescomplemented
493C42D4×D7112yes—
494C42#1176586612yes—
495C23—712no—
496C23—712no—
497C23—712no—
498C23—712no—
499C23—1421no—
500C23—1421no—
501C23—1421no—
502C23—1421no—
503C23—1421no—
504C23—1421no—
505C23—1421no—
506C23—1421no—
507C23—1421no—
508C23—1421no—
509C23C22×C4×D7111yescharacteristic
510D4—1428no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—2844no—
523C2×C4—2844no—
524C2×C4—2841no—
525C2×C4—2841no—
526C2×C4—2841no—
527C2×C4—2841no—
528C2×C4—712no—
529C2×C4—712no—
530C2×C4—712no—
531C2×C4—712no—
532C2×C4—1421no—
533C2×C4—1421no—
534C2×C4—1421no—
535C2×C4—1421no—
536C2×C4—1421no—
537C2×C4—1421no—
538C2×C4—1421no—
539C2×C4—1421no—
540C2×C4—1421no—
541C2×C4—1421no—
542C2×C4—1421no—
543C2×C4—1421no—
544C2×C4—1421no—
545C2×C4—1421no—
546C2×C4—1428no—
547C2×C4—1428no—
548C2×C4—1422no—
549C2×C4—1422no—
550C2×C4—1422no—
551C2×C4—1422no—
552C2×C4#1455609012yes—
553C2×C4#1455609012yes—
554C2×C4C2×C8×D7112yes—
555C2×C4C2×C8×D7112yes—
556C2×C4C2×C8×D7112yes—
557C2×C4M16×D7112yes—
558C2×C4#1455613612yes—
559C2×C4C2×D4×D7111yescharacteristic
560C2×C4#41×D7111yescharacteristic
561C2×C4#41×D7111yescharacteristic
562C2×C4C2×#1176586611yescharacteristic
563C2×C4#1455617411yescharacteristic
564C2×C4#1455617511yescharacteristic
565C8—42no—
566C8—42no—
567C8—2842no—
568C8—1424no—
569C22—2841no—
570C22—2841no—
571C22—2841no—
572C22—2841no—
573C22—714no—
574C22—1422no—
575C22—1422no—
576C22—1422no—
577C22—1422no—
578C22—1422no—
579C22—1422no—
580C22—1422no—
581C22—1422no—
582C22—1422no—
583C22#117×D7111yescharacteristic
584C22C2×#1455609011yescharacteristic
585C22#1798647911yescharacteristic
586C22C22×C8×D7111yescharacteristic
587C22C2×M16×D7111yescharacteristic
588C22C2×#1455613611yescharacteristic
589C22#1798676511yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—41no—
595C4—2841no—
596C4—2841no—
597C4—1422no—
598C4—1422no—
599C4—1422no—
600C4—1422no—
601C4#1798606414yes—
602C4#1798664211yescharacteristic
603C4#1798665711yescharacteristic
604C4#1798668711yescharacteristic
605C4#1798670211yescharacteristic
606C2—2842no—
607C2—1424no—
608C2#2259126611yescharacteristic
609C2#431×D7111yescharacteristic
610C2#2259700711yescharacteristic
611C2C2×#1798606411yescharacteristic
612C2#2259707511yescharacteristic
613C2#2259744811yescharacteristic
614C2#2259746711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D71
commutatorC22×C71C23×C8
FittingC71×#1831C2
socleC23×C71C23×C4
prime radicalC22×D284C2×C4
cosocleC22×C4×C71C24
hypercenter#1831D71
hypocenterC71#22748
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ #1831

Complements

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

NH ≅ G/Ntype
1. C2×#17986051C2semidirect
2. C2×C4⋊C8×D71C2semidirect
4. #22588944C2semidirect
5. #22588896C2semidirect
7. #22588317C2semidirect
8. #22588317C2semidirect
9. C4×C4⋊(C71⋊C8)C2semidirect
10. #22588110C2semidirect
11. C71×#1831C2semidirect
60. C4×D284C8semidirect
81. C4⋊(C71⋊C8)D4semidirect
137. C2×C4×D71C2×C8semidirect
149. C4⋊Dic71C2×C8semidirect
154. (C2×D71)⋊C4C2×C8semidirect
184. C42×C71C2×C8semidirect
281. C2×D71#431semidirect
314. #1831D71semidirect
319. C71#22748semidirect
492. C42C8×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860512
C2×C4⋊C8×D712
C42×D2842
#225889442
#225888962
#225886312
#225883172
C4×C4⋊(C71⋊C8)2
#225881102
C71×#18312
#2274871