Finite Group Databasebeta

← #32016477

Subgroups — #32016477

Order 18176.

984 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×#17986052C211yesmaximal, characteristic, complemented
2C2×#17986021C211yesmaximal, characteristic, complemented
3C42×C22⋊D71C211yesmaximal, characteristic
4#22588950C212yesmaximal, complemented
5#22588895C212yesmaximal, complemented
6#22588634C212yesmaximal
7#22588456C211yesmaximal, characteristic, complemented
8#22588317C211yesmaximal, characteristic, complemented
9#22588195C211yesmaximal, characteristic, complemented
10#22588120C212yesmaximal, complemented
11C71×#1797C211yesmaximal, characteristic, complemented
12C22×#11765848C2212yes—
13C2×C4×C22⋊D71C411yescharacteristic
14C2×C4×C22⋊D71C411yescharacteristic
15C2×C4×C22⋊D71C2211yescharacteristic
16C2×C42×D71C2211yescharacteristic
17#17986052—24no—
18#17986021—24no—
19C2×C4⋊(C71⋊C8)C2211yescharacteristic
20C2×C4⋊(C71⋊C8)C2211yescharacteristic
21C2×Dic71⋊C8C2212yes—
22C2×C22⋊(C71⋊C8)C2212yes—
23C2×(C2×D71)⋊C8C2212yes—
24C2×#14556038C2212yes—
25C2×C4×C71⋊C8C2212yes—
26C4×Dic71⋊C4C411yescharacteristic
27C4×Dic71⋊C4C411yescharacteristic
28C4×Dic71⋊C4C2211yescharacteristic
29C4×(C2×D71)⋊C4C411yescharacteristic
30C4×(C2×D71)⋊C4C411yescharacteristic
31C4×(C2×D71)⋊C4C2211yescharacteristic
32C4×C22⋊Dic71C411yescharacteristic
33C4×C22⋊Dic71C411yescharacteristic
34C4×C22⋊Dic71C2211yescharacteristic
35C42×Dic71C2211yescharacteristic
36#17985753C2212yes—
37#17985753C2212yes—
38#17985753C2212yes—
39#17985753C2212yes—
40C22×C42×C71C2211yescharacteristic
41C2×C4⋊C8×C71C2211yescharacteristic
42C2×C4⋊C8×C71C2211yescharacteristic
43C2×C22⋊C8×C71C2212yes—
44#336×C71C2212yes—
45C22×C22⋊D71C2×C411yescharacteristic
46C22×C4×D71C2×C411yescharacteristic
47C22×C4×D71C2×C411yescharacteristic
48C22×C4×D71C2311yescharacteristic
49C2×#11765848—24no—
50C2×#11765848—24no—
51C2×#11765848—42no—
52C22×C71⋊C8C2312yes—
53C22×C71⋊C8C2312yes—
54C4×C22⋊D71—22no—
55C4×C22⋊D71—22no—
56C4×C22⋊D71—24no—
57C4×C22⋊D71—24no—
58C2×Dic71⋊C4C2×C411yescharacteristic
59C2×Dic71⋊C4C2×C411yescharacteristic
60C2×Dic71⋊C4C2×C411yescharacteristic
61C2×C22⋊Dic71C2×C411yescharacteristic
62C2×C22⋊Dic71C2×C411yescharacteristic
63C2×C22⋊Dic71C2×C411yescharacteristic
64C2×(C2×D71)⋊C4C2×C411yescharacteristic
65C2×(C2×D71)⋊C4C2×C411yescharacteristic
66C2×(C2×D71)⋊C4C2×C411yescharacteristic
67C2×C4×Dic71C2×C411yescharacteristic
68C2×C4×Dic71C2×C411yescharacteristic
69C2×C4×Dic71C2×C411yescharacteristic
70C2×C4×Dic71C2×C411yescharacteristic
71C2×C4×Dic71C2×C411yescharacteristic
72C2×C4×Dic71C2×C411yescharacteristic
73C2×C4×Dic71C2311yescharacteristic
74C2×C4×Dic71C2311yescharacteristic
75C2×C4×Dic71C2311yescharacteristic
76C42×D71—22no—
77C42×D71—22no—
78C4⋊(C71⋊C8)—22no—
79C4⋊(C71⋊C8)D414yescomplemented
80Dic71⋊C8—24no—
81C22⋊(C71⋊C8)—24no—
82(C2×D71)⋊C8—24no—
83#14556038—42no—
84C4×C71⋊C8—24no—
85C71⋊#95C2×C411yescharacteristic
86C71⋊#95C2×C411yescharacteristic
87C71⋊#95C2×C411yescharacteristic
88C71⋊#95C2×C411yescharacteristic
89C71⋊#95C2×C411yescharacteristic
90C71⋊#95C2×C411yescharacteristic
91C23×C4×C71C2×C411yescharacteristic
92C23×C4×C71C2×C411yescharacteristic
93C23×C4×C71C2311yescharacteristic
94C22×C8×C71C2312yes—
95C22×C8×C71C2312yes—
96C2×C42×C71—22no—
97C2×C42×C71—22no—
98C2×C42×C71C2×C411yescharacteristic
99C2×C42×C71C2×C411yescharacteristic
100C2×C42×C71C2311yescharacteristic
101C2×C42×C71C2311yescharacteristic
102C4⋊C8×C71—22no—
103C4⋊C8×C71—22no—
104C22⋊C8×C71—42no—
105C23×D71C22×C411yescharacteristic
106C2×C22⋊D71—22no—
107C2×C22⋊D71—22no—
108C2×C22⋊D71—22no—
109C22×Dic71C22×C411yescharacteristic
110C22×Dic71C22×C411yescharacteristic
111C22×Dic71C22×C411yescharacteristic
112C22×Dic71C22×C411yescharacteristic
113C22×Dic71C22×C411yescharacteristic
114C22×Dic71C22×C411yescharacteristic
115C22×Dic71C22×C411yescharacteristic
116C2×C4×D71—21no—
117C2×C4×D71—21no—
118C2×C4×D71—21no—
119C2×C4×D71—21no—
120C2×C4×D71—22no—
121C2×C4×D71—22no—
122C2×C4×D71—22no—
123C2×C4×D71—22no—
124C2×C4×D71—22no—
125C2×C4×D71—41no—
126C2×C4×D71#4112yes—
127C2×C4×D71#4112yes—
128#11765848—48no—
129C2×C71⋊C8—22no—
130C2×C71⋊C8—22no—
131C2×C71⋊C8—22no—
132C2×C71⋊C8—22no—
133C2×C71⋊C8—22no—
134C2×C71⋊C8C2×D414yes—
135Dic71⋊C4—22no—
136Dic71⋊C4—22no—
137Dic71⋊C4M1614yescomplemented
138C22⋊Dic71—22no—
139C22⋊Dic71—22no—
140C22⋊Dic71—22no—
141(C2×D71)⋊C4—22no—
142(C2×D71)⋊C4—22no—
143(C2×D71)⋊C4—22no—
144C4×Dic71—22no—
145C4×Dic71—22no—
146C4×Dic71—22no—
147C4×Dic71—22no—
148C4×Dic71—21no—
149C4×Dic71—21no—
150C4×Dic71—21no—
151C4×Dic71—21no—
152C4×Dic71—21no—
153C4×Dic71—21no—
154C4×Dic71—22no—
155C4×Dic71—41no—
156C24×C71C22×C411yescharacteristic
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—21no—
164C22×C4×C71—21no—
165C22×C4×C71—21no—
166C22×C4×C71—21no—
167C22×C4×C71—41no—
168C22×C4×C71M1614yescomplemented
169C22×C4×C71C22×C411yescharacteristic
170C22×C4×C71C22×C411yescharacteristic
171C22×C4×C71C22×C411yescharacteristic
172C22×C4×C71C22×C411yescharacteristic
173C22×C4×C71C22×C411yescharacteristic
174C22×C4×C71C2411yescharacteristic
175C2×C8×C71—22no—
176C2×C8×C71—22no—
177C2×C8×C71—22no—
178C2×C8×C71—22no—
179C2×C8×C71—22no—
180C2×C8×C71—22no—
181C42×C71—22no—
182C42×C71—22no—
183C42×C71—21no—
184C42×C71—21no—
185C42×C71C2×D412yes—
186C42×C71#4112yes—
187C23×C71—21no—
188C23×C71—21no—
189C23×C71—21no—
190C23×C71—21no—
191C23×C71—21no—
192C23×C71—21no—
193C23×C71C2×M1612yes—
194C23×C71C23×C411yescharacteristic
195C22×D71—21no—
196C22×D71—21no—
197C22×D71—21no—
198C22×D71—21no—
199C22×D71—21no—
200C22×D71—21no—
201C22×D71#11712yes—
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—28no—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—44no—
213C2×C4×C71—41no—
214C2×C4×C71—41no—
215C2×C4×C71—41no—
216C2×C4×C71—41no—
217C2×C4×C71C4×D412yes—
218C2×C4×C71C4×D412yes—
219C2×C4×C71C2×M1612yes—
220C2×C4×C71C2×M1612yes—
221C2×C4×C71C2×M1612yes—
222C2×C4×C71#13112yes—
223C2×C4×C71C22×D411yescharacteristic
224C2×C4×C71C2×#4111yescharacteristic
225C2×C4×C71C2×#4111yescharacteristic
226C2×C4×C71C2×#4111yescharacteristic
227C2×C4×C71H(5,𝔽2)11yescharacteristic
228C2×C4×C712−1+411yescharacteristic
229C22⋊D71—28no—
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—21no—
241C2×Dic71—21no—
242C2×Dic71—21no—
243C2×Dic71—21no—
244C2×Dic71—21no—
245C2×Dic71—21no—
246C2×Dic71—21no—
247C2×Dic71—21no—
248C2×Dic71#11712yes—
249C2×Dic71C2×M1612yes—
250C2×Dic71C2×M1612yes—
251C4×D71—22no—
252C4×D71—22no—
253C4×D71—22no—
254C4×D71—22no—
255C4×D71—44no—
256C4×D71—44no—
257C8×C71—42no—
258C8×C71—42no—
259C71⋊C8—24no—
260C71⋊C8—42no—
261C22×C71—22no—
262C22×C71—22no—
263C22×C71—22no—
264C22×C71—41no—
265C22×C71—41no—
266C22×C71—41no—
267C22×C71—41no—
268C22×C71C2×#11711yescharacteristic
269C22×C71C2×C4×D411yescharacteristic
270C22×C71#52011yescharacteristic
271C22×C71C22×M1611yescharacteristic
272C22×C71C22×M1611yescharacteristic
273C22×C71C2×#13111yescharacteristic
274C22×C71#56811yescharacteristic
275C2×D71—22no—
276C2×D71—22no—
277C2×D71—22no—
278C2×D71—41no—
279C2×D71—41no—
280C2×D71—41no—
281C2×D71—41no—
282C4×C71—22no—
283C4×C71—22no—
284C4×C71—22no—
285C4×C71—41no—
286C4×C71#43514yes—
287C4×C71#54611yescharacteristic
288C4×C71#54811yescharacteristic
289C4×C71#55211yescharacteristic
290C4×C71#55311yescharacteristic
291Dic71—22no—
292Dic71—22no—
293Dic71—22no—
294Dic71—41no—
295Dic71—41no—
296Dic71—41no—
297Dic71—41no—
298#22749—711nomaximal, Sylow
299C2×C71—42no—
300C2×C71#235211yescharacteristic
301C2×C71#292011yescharacteristic
302C2×C71#292511yescharacteristic
303C2×C71C2×#43511yescharacteristic
304C2×C71D4×M1611yescharacteristic
305C2×C71#298911yescharacteristic
306C2×C71#299111yescharacteristic
307D71—42no—
308C2×#431—711no—
309C2×#423—711no—
310C42×D4—711no—
311#1924—712no—
312#1916—712no—
313#1870—712no—
314#1831—711no—
315#1797—711no—
316#1797D7111yescharacteristic, complemented
317#1767—711no—
318#1753—712no—
319C71#2274911yescharacteristic, complemented, Sylow
320C22×M16—712no—
321C2×C4×D4—711no—
322C2×C4×D4—711no—
323C2×C4×D4—711no—
324C22×C42—711no—
325C22×C42C2×D7111yescharacteristic
326#431—1424no—
327#423—1424no—
328C2×C4⋊C8—711no—
329C2×C4⋊C8—712no—
330C2×C4⋊C8—711no—
331C2×C4⋊C8C2×D7111yescharacteristic
332C2×C4⋊C8C2×D7111yescharacteristic
333C2×C22⋊C8—712no—
334C2×C22⋊C8—712no—
335C2×C22⋊C8C2×D7112yes—
336C2×#97—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—
347C43—711no—
348#336—712no—
349#336—712no—
350#336—712no—
351#336—712no—
352#336C2×D7112yes—
353C22×D4—711no—
354C23×C4—711no—
355C23×C4—711no—
356C23×C4—711no—
357C23×C4C4×D7111yescharacteristic
358C23×C4C4×D7111yescharacteristic
359C23×C4C22×D7111yescharacteristic
360C2×M16—2842no—
361C2×M16—1424no—
362C2×M16—1424no—
363C22×C8—712no—
364C22×C8—712no—
365C22×C8C22×D7112yes—
366C22×C8C22×D7112yes—
367C4×D4—1424no—
368C4×D4—1422no—
369C4×D4—1422no—
370C4×D4—1424no—
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—22no—
381C2×C42—22no—
382C2×C42—711no—
383C2×C42—711no—
384C2×C42—711no—
385C2×C42—711no—
386C2×C42—711no—
387C2×C42—711no—
388C2×C42—711no—
389C2×C42—711no—
390C2×C42—711no—
391C2×C42—1422no—
392C2×C42—1422no—
393C2×C42C4×D7111yescharacteristic
394C2×C42C4×D7111yescharacteristic
395C2×C42C22×D7111yescharacteristic
396C2×C42C22×D7111yescharacteristic
397C4⋊C8—22no—
398C4⋊C8—22no—
399C4⋊C8—714no—
400C4⋊C8—1424no—
401C4⋊C8—1422no—
402C22⋊C8—42no—
403C22⋊C8—1424no—
404C22⋊C8—1424no—
405#97—2842no—
406C4×C8—1424no—
407#95—711no—
408#95—711no—
409#95—711no—
410#95—711no—
411#95—711no—
412#95—711no—
413C24—711no—
414C24C2×C4×D7111yescharacteristic
415C2×D4—1422no—
416C2×D4—1422no—
417C2×D4—1422no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—41no—
429C22×C4—2841no—
430C22×C4—711no—
431C22×C4—711no—
432C22×C4—711no—
433C22×C4—712no—
434C22×C4—712no—
435C22×C4—711no—
436C22×C4—711no—
437C22×C4—711no—
438C22×C4—711no—
439C22×C4—1422no—
440C22×C4—1422no—
441C22×C4—1422no—
442C22×C4—1422no—
443C22×C4—1422no—
444C22×C4—1421no—
445C22×C4—1421no—
446C22×C4—1421no—
447C22×C4—1421no—
448C22×C4#1176584814yescomplemented
449C22×C4C2×C4×D7111yescharacteristic
450C22×C4C2×C4×D7111yescharacteristic
451C22×C4C2×C4×D7111yescharacteristic
452C22×C4C2×C4×D7111yescharacteristic
453C22×C4C2×C4×D7111yescharacteristic
454C22×C4C23×D7111yescharacteristic
455M16—2848no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
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—1422no—
472C22⋊C4—1422no—
473C22⋊C4—1422no—
474C22⋊C4—1422no—
475C22⋊C4—1422no—
476C22⋊C4—1422no—
477C42—21no—
478C42—21no—
479C42—22no—
480C42—22no—
481C42—2841no—
482C42—1422no—
483C42—1421no—
484C42—1421no—
485C42—1421no—
486C42—1421no—
487C42—1421no—
488C42—1421no—
489C42—1422no—
490C42—1422no—
491C42—1422no—
492C42—1422no—
493C42D4×D7112yes—
494C42#1176586412yes—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—712no—
502C23—1421no—
503C23—1421no—
504C23—1421no—
505C23—1421no—
506C23—1421no—
507C23—1421no—
508C23C2×#1176584812yes—
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—28no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—2844no—
527C2×C4—2844no—
528C2×C4—712no—
529C2×C4—712no—
530C2×C4—712no—
531C2×C4—1421no—
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—1421no—
547C2×C4—1421no—
548C2×C4—1421no—
549C2×C4—1422no—
550C2×C4—1422no—
551C2×C4—1422no—
552C2×C4—1422no—
553C2×C4(C22⋊D71)⋊C412yes—
554C2×C4(C22⋊D71)⋊C412yes—
555C2×C4M16×D7112yes—
556C2×C4C2×#1176584812yes—
557C2×C4C2×#1176584812yes—
558C2×C4#1455613612yes—
559C2×C4C2×D4×D7111yescharacteristic
560C2×C4#41×D7111yescharacteristic
561C2×C4#41×D7111yescharacteristic
562C2×C4C2×#1176586411yescharacteristic
563C2×C4#1455617311yescharacteristic
564C2×C4#1455617611yescharacteristic
565C8—42no—
566C8—42no—
567C8—2842no—
568C8—1424no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—2841no—
577C22—2841no—
578C22—2841no—
579C22—2841no—
580C22—1422no—
581C22—1422no—
582C22—1422no—
583C22#117×D7111yescharacteristic
584C22C2×(C22⋊D71)⋊C411yescharacteristic
585C22#1798648011yescharacteristic
586C22C2×M16×D7111yescharacteristic
587C22C22×#1176584811yescharacteristic
588C22C2×#1455613611yescharacteristic
589C22#1798676511yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—41no—
594C4—2841no—
595C4—2841no—
596C4—2841no—
597C4—2841no—
598C4—1422no—
599C4—1422no—
600C4—1422no—
601C4#1798606714yes—
602C4#1798664611yescharacteristic
603C4#1798665911yescharacteristic
604C4#1798668311yescharacteristic
605C4#1798669611yescharacteristic
606C2—42no—
607C2—2842no—
608C2#2259126211yescharacteristic
609C2#2259696811yescharacteristic
610C2#2259700711yescharacteristic
611C2C2×#1798606711yescharacteristic
612C2#2259707211yescharacteristic
613C2#2259745811yescharacteristic
614C2#2259747011yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D71
commutator, socleC23×C71C23×C4
FittingC71×#1797C2
prime radicalC22×C22⋊D71C2×C4
cosocleC22×C4×C71C24
hypercenter#1797D71
hypocenterC71#22749
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ #1797

Complements

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

NH ≅ G/Ntype
1. C2×#17986052C2semidirect
2. C2×#17986021C2semidirect
4. #22588950C2semidirect
5. #22588895C2semidirect
7. #22588456C2semidirect
8. #22588317C2semidirect
9. #22588195C2semidirect
10. #22588120C2semidirect
11. C71×#1797C2semidirect
79. C4⋊(C71⋊C8)D4semidirect
137. Dic71⋊C4M16semidirect
168. C22×C4×C71M16semidirect
316. #1797D71semidirect
319. C71#22749semidirect
448. C22×C4#11765848semidirect

Maximal subgroups

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

subgroupindex
C2×#179860522
C2×#179860212
C42×C22⋊D712
#225889502
#225888952
#225886342
#225884562
#225883172
#225881952
#225881202
C71×#17972
#2274971