Finite Group Databasebeta

← #32002339

Subgroups — #32002339

Order 18176.

982 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986054C211yesmaximal, characteristic
2#22590744C211yesmaximal, characteristic
3C4×C4⋊C4×D71C211yesmaximal, characteristic
4#22589102C211yesmaximal, characteristic, complemented
5#22589100C211yesmaximal, characteristic, complemented
6#22589097C211yesmaximal, characteristic, complemented
7#22589095C211yesmaximal, characteristic, complemented
8#22588534C211yesmaximal, characteristic
9#22588531C211yesmaximal, characteristic
10#22588529C211yesmaximal, characteristic
11#22588526C211yesmaximal, characteristic
12#22588238C211yesmaximal, characteristic, complemented
13#22588238C211yesmaximal, characteristic, complemented
14#22588237C211yesmaximal, characteristic, complemented
15C71×#1776C211yesmaximal, characteristic, complemented
16C2×#14556085—21no—
17C2×C4⋊C4×D71C2211yescharacteristic
18C2×C4⋊C4×D71C2211yescharacteristic
19C2×C42×D71—21no—
20C2×C42×D71C2211yescharacteristic
21#17986054C418yescomplemented
22C2×#14556062C2211yescharacteristic
23C2×#14556061C2211yescharacteristic
24C2×#14556061C2211yescharacteristic
25C2×#14556060C2211yescharacteristic
26C2×#14556059C2211yescharacteristic
27C2×#14556059C2211yescharacteristic
28C2×Dic71⋊C8C2211yescharacteristic
29C2×Dic71⋊C8C2211yescharacteristic
30C2×(C2×D71)⋊C8C2211yescharacteristic
31C2×(C2×D71)⋊C8C2211yescharacteristic
32C2×#14556039C2211yescharacteristic
33#17985868C2211yescharacteristic
34#17985868C2211yescharacteristic
35#17985863—21no—
36#17985858—21no—
37#17985856C2211yescharacteristic
38#17985856C2211yescharacteristic
39C4⋊C4×Dic71C2211yescharacteristic
40C4⋊C4×Dic71C2211yescharacteristic
41C4×Dic71⋊C4—21no—
42C4×Dic71⋊C4—21no—
43C4×C4⋊Dic71C2211yescharacteristic
44C4×C4⋊Dic71C2211yescharacteristic
45#17985850—21no—
46#95×D71—21no—
47#17985762C2211yescharacteristic
48#17985762C2211yescharacteristic
49#17985761C2211yescharacteristic
50#17985761C2211yescharacteristic
51#17985761C2211yescharacteristic
52#17985761C2211yescharacteristic
53C2×#107×C71C2211yescharacteristic
54C2×#106×C71C2211yescharacteristic
55C2×#97×C71C2211yescharacteristic
56C4×C4⋊C4×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58#340×C71C2211yescharacteristic
59#340×C71C2211yescharacteristic
60C22×C4×D71—21no—
61C22×C4×D71—21no—
62C22×C4×D71D411yescharacteristic
63C22×C4×D71D411yescharacteristic
64C22×C4×D71C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67#14556085—44no—
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×Dic71⋊C4—21no—
73C2×Dic71⋊C4—21no—
74C2×Dic71⋊C4—21no—
75C2×Dic71⋊C4—21no—
76C2×Dic71⋊C4—21no—
77C2×Dic71⋊C4—21no—
78C4⋊C4×D71—22no—
79C4⋊C4×D71—22no—
80C4⋊C4×D71—22no—
81C4⋊C4×D71—22no—
82C2×(C2×D71)⋊C4—21no—
83C2×(C2×D71)⋊C4—21no—
84C2×C4×Dic71—21no—
85C2×C4×Dic71—21no—
86C2×C4×Dic71—21no—
87C2×C4×Dic71—21no—
88C2×C4×Dic71D411yescharacteristic
89C2×C4×Dic71D411yescharacteristic
90C2×C4×Dic71C2311yescharacteristic
91C42×D71—44no—
92C42×D71C2×C414yes—
93C42×D71Q814yes—
94#14556062—22no—
95#14556061—22no—
96#14556061—22no—
97#14556060—22no—
98#14556059—22no—
99#14556059—22no—
100Dic71⋊C8C2×C414yescomplemented
101Dic71⋊C8C2×C414yescomplemented
102(C2×D71)⋊C8C2×C414yes—
103(C2×D71)⋊C8C2×C414yes—
104#14556039C2×C414yescomplemented
105C71⋊#95—21no—
106C71⋊#95—21no—
107C71⋊#95—21no—
108C71⋊#95—21no—
109C71⋊#95—21no—
110C71⋊#95—21no—
111C22×C8×C71C2311yescharacteristic
112C22×C8×C71C2311yescharacteristic
113C2×C4⋊C4×C71C2311yescharacteristic
114C2×C4⋊C4×C71C2311yescharacteristic
115C2×C4⋊C4×C71C2311yescharacteristic
116C2×C4⋊C4×C71C2311yescharacteristic
117C2×C42×C71—21no—
118C2×C42×C71—21no—
119C2×C42×C71C2311yescharacteristic
120#107×C71—22no—
121#106×C71—22no—
122#97×C71C2×C414yescomplemented
123#95×C71—21no—
124#95×C71—21no—
125C23×D71C2×D411yescharacteristic
126C22×Dic71—21no—
127C22×Dic71—21no—
128C22×Dic71—21no—
129C22×Dic71—21no—
130C22×Dic71C2×D411yescharacteristic
131C22×Dic71C2×D411yescharacteristic
132C22×Dic71C2×D411yescharacteristic
133C2×C4×D71—22no—
134C2×C4×D71—22no—
135C2×C4×D71—21no—
136C2×C4×D71—21no—
137C2×C4×D71—21no—
138C2×C4×D71—21no—
139C2×C4×D71—22no—
140C2×C4×D71—22no—
141C2×C4×D71—24no—
142C2×C4×D71—24no—
143C2×C4×D71—42no—
144C2×C4×D71—42no—
145C2×C4×D71—42no—
146C2×C4×D71—42no—
147C2×C4×D71C4⋊C414yes—
148C2×C4×D71C4⋊C414yes—
149C2×C4×D71C22×C412yes—
150C2×C4×D71C2×Q812yes—
151C2×C71⋊C8—21no—
152C2×C71⋊C8—21no—
153C2×C71⋊C8—21no—
154C2×C71⋊C8—21no—
155C2×C71⋊C8C22×C412yes—
156C2×C71⋊C8C22×C412yes—
157C4⋊Dic71—21no—
158C4⋊Dic71—21no—
159C4⋊Dic71—21no—
160C4⋊Dic71—21no—
161C4⋊Dic71—21no—
162C4⋊Dic71—21no—
163C4⋊Dic71#4112yes—
164C4⋊Dic71#4112yes—
165Dic71⋊C4—24no—
166Dic71⋊C4—24no—
167Dic71⋊C4—42no—
168Dic71⋊C4—42no—
169Dic71⋊C4—42no—
170Dic71⋊C4—42no—
171(C2×D71)⋊C4—42no—
172(C2×D71)⋊C4—42no—
173C4×Dic71—42no—
174C4×Dic71—42no—
175C4×Dic71—42no—
176C4×Dic71—42no—
177C4×Dic71C4⋊C414yes—
178C4×Dic71C4⋊C414yes—
179C4×Dic71C22×C412yes—
180C4×Dic71C2×Q812yes—
181C22×C4×C71—21no—
182C22×C4×C71—21no—
183C22×C4×C71—21no—
184C22×C4×C71—21no—
185C22×C4×C71C2×D411yescharacteristic
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2411yescharacteristic
188C2×C8×C71—21no—
189C2×C8×C71—21no—
190C2×C8×C71—21no—
191C2×C8×C71—21no—
192C2×C8×C71C22×C412yes—
193C2×C8×C71C22×C412yes—
194C4⋊C4×C71—21no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—21no—
197C4⋊C4×C71—21no—
198C4⋊C4×C71—21no—
199C4⋊C4×C71—21no—
200C4⋊C4×C71#4112yes—
201C4⋊C4×C71#4112yes—
202C42×C71—42no—
203C42×C71—42no—
204C42×C71C22×C412yes—
205C42×C71C2×Q812yes—
206C23×C71C22×D411yescharacteristic
207C22×D71—21no—
208C22×D71—21no—
209C22×D71—21no—
210C22×D71—21no—
211C22×D71C2×C4⋊C412yes—
212C22×D71AGL(1,ℤ/8)12yes—
213C22×D71#13712yes—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—22no—
219C2×C4×C71—22no—
220C2×C4×C71—41no—
221C2×C4×C71—41no—
222C2×C4×C71—41no—
223C2×C4×C71—41no—
224C2×C4×C71—41no—
225C2×C4×C71—41no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71C2×C4⋊C412yes—
231C2×C4×C71C2×C4⋊C412yes—
232C2×C4×C71C23×C411yescharacteristic
233C2×C4×C71C22×Q811yescharacteristic
234C2×C4×C71C2×#4111yescharacteristic
235C2×C4×C71C2×#4111yescharacteristic
236C2×C4×C71C2×#4111yescharacteristic
237C2×C4×C71C2×#4111yescharacteristic
238C2×Dic71—21no—
239C2×Dic71—21no—
240C2×Dic71—21no—
241C2×Dic71—21no—
242C2×Dic71—22no—
243C2×Dic71—22no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71C2×C4⋊C412yes—
255C2×Dic71C2×C4⋊C412yes—
256C2×Dic71C2×C4⋊C412yes—
257C2×Dic71AGL(1,ℤ/8)12yes—
258C2×Dic71#13712yes—
259C4×D71—28no—
260C4×D71—22no—
261C4×D71—22no—
262C4×D71—22no—
263C4×D71—22no—
264C4×D71—28no—
265C4×D71—48no—
266C4×D71—48no—
267C8×C71—22no—
268C8×C71—22no—
269C71⋊C8—22no—
270C71⋊C8—22no—
271C22×C71C22×C4⋊C411yescharacteristic
272C22×C71#52511yescharacteristic
273C22×C71#52511yescharacteristic
274C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C71C2×#13711yescharacteristic
277C22×C71C2×#13711yescharacteristic
278C2×D71—22no—
279C2×D71—22no—
280C2×D71—22no—
281C2×D71—22no—
282C2×D71—22no—
283C2×D71—22no—
284C2×D71#42814yes—
285C4×C71—22no—
286C4×C71—22no—
287C4×C71—42no—
288C4×C71—42no—
289C4×C71—81no—
290C4×C71—81no—
291C4×C71C4×#4111yescharacteristic
292C4×C71C4×#4111yescharacteristic
293C4×C71#53311yescharacteristic
294C4×C71#53311yescharacteristic
295Dic71—22no—
296Dic71—22no—
297Dic71—42no—
298Dic71—42no—
299Dic71—81no—
300Dic71—81no—
301Dic71#42814yes—
302#21365—711nomaximal, Sylow
303C2×C71#229611yescharacteristic
304C2×C71C2×#42811yescharacteristic
305C2×C71C2×#42811yescharacteristic
306C2×C71#310911yescharacteristic
307C2×C71#311011yescharacteristic
308C2×C71#311211yescharacteristic
309C2×C71#311311yescharacteristic
310D71—24no—
311C2×#432—711no—
312#2285—711no—
313C2×C4×C4⋊C4—711no—
314#1948—711no—
315#1948—711no—
316#1947—711no—
317#1947—711no—
318#1853—711no—
319#1853—711no—
320#1852—711no—
321#1852—711no—
322#1776—711no—
323#1776—711no—
324#1776—711no—
325#1776D7111yescharacteristic, complemented
326C71#2136511yescharacteristic, complemented, Sylow
327C2×#117—1421no—
328C22×C4⋊C4—711no—
329C22×C4⋊C4—711no—
330C22×C42—711no—
331C22×C42—1421no—
332#432—718no—
333C2×#107—711no—
334C2×#107—711no—
335C2×#107—711no—
336C2×#107C2×D7111yescharacteristic
337C2×#106—711no—
338C2×#106—711no—
339C2×#106—711no—
340C2×#106C2×D7111yescharacteristic
341C2×C4⋊C8—711no—
342C2×C4⋊C8—711no—
343C2×C22⋊C8—711no—
344C2×C22⋊C8—711no—
345C2×#97—711no—
346C2×#97C2×D7111yescharacteristic
347#384—711no—
348#384—711no—
349#382—1421no—
350#380—711no—
351#380—711no—
352#380—1421no—
353C4×C4⋊C4—711no—
354C4×C4⋊C4—711no—
355C4×C4⋊C4—711no—
356C4×C4⋊C4—711no—
357C4×C4⋊C4—1421no—
358C4×C4⋊C4—1421no—
359C4×C4⋊C4—1421no—
360C4×C4⋊C4C2×D7111yescharacteristic
361C4×C4⋊C4C2×D7111yescharacteristic
362C2×#95—1421no—
363#340—711no—
364#340—711no—
365#340—711no—
366#340—711no—
367#340—711no—
368#340—711no—
369#340C2×D7111yescharacteristic
370#340C2×D7111yescharacteristic
371C23×C4—711no—
372C23×C4—711no—
373C23×C4—711no—
374C23×C4—1421no—
375C23×C4—1421no—
376C22×C8—711no—
377C22×C8—711no—
378C22×C8C22×D7111yescharacteristic
379C22×C8C22×D7111yescharacteristic
380#117—2844no—
381C2×C4⋊C4—711no—
382C2×C4⋊C4—711no—
383C2×C4⋊C4—711no—
384C2×C4⋊C4—711no—
385C2×C4⋊C4—1422no—
386C2×C4⋊C4—1422no—
387C2×C4⋊C4—1421no—
388C2×C4⋊C4—1421no—
389C2×C4⋊C4—1421no—
390C2×C4⋊C4—1421no—
391C2×C4⋊C4—1421no—
392C2×C4⋊C4—1421no—
393C2×C4⋊C4—1422no—
394C2×C4⋊C4—1422no—
395C2×C4⋊C4C22×D7111yescharacteristic
396C2×C4⋊C4C22×D7111yescharacteristic
397C2×C4⋊C4C22×D7111yescharacteristic
398C2×C4⋊C4C22×D7111yescharacteristic
399C2×C22⋊C4—1421no—
400C2×C22⋊C4—1421no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—2844no—
404C2×C42—711no—
405C2×C42—711no—
406C2×C42—711no—
407C2×C42—714no—
408C2×C42—714no—
409C2×C42—1421no—
410C2×C42—1421no—
411C2×C42—1421no—
412C2×C42—1421no—
413C2×C42C22×D7111yescharacteristic
414#107—22no—
415#107—1422no—
416#107—1422no—
417#107—1422no—
418#106—22no—
419#106—1422no—
420#106—1422no—
421#106—1422no—
422C4⋊C8—714no—
423C4⋊C8—714no—
424C22⋊C8—714no—
425C22⋊C8—714no—
426#97—714no—
427#97C4×D7114yescomplemented
428#95—21no—
429#95—21no—
430#95—1421no—
431#95—1421no—
432#95—1421no—
433#95—1421no—
434#95—1421no—
435#95—1421no—
436C24—711no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—2842no—
442C22×C4—2842no—
443C22×C4—2842no—
444C22×C4—2842no—
445C22×C4—711no—
446C22×C4—711no—
447C22×C4—711no—
448C22×C4—712no—
449C22×C4—712no—
450C22×C4—714no—
451C22×C4—714no—
452C22×C4—1424no—
453C22×C4—1424no—
454C22×C4—1422no—
455C22×C4—1422no—
456C22×C4—1422no—
457C22×C4—1422no—
458C22×C4—1421no—
459C22×C4—1421no—
460C22×C4—1421no—
461C22×C4—1421no—
462C22×C4—1421no—
463C22×C4—1421no—
464C22×C4—1421no—
465C22×C4—1421no—
466C22×C4D4×D7111yescharacteristic
467C22×C4D4×D7111yescharacteristic
468C22×C4C23×D7111yescharacteristic
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—712no—
474C2×C8—712no—
475C2×C8—1421no—
476C2×C8—1421no—
477C2×C8—1421no—
478C2×C8—1421no—
479C2×C8C2×C4×D7112yes—
480C2×C8C2×C4×D7112yes—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—2842no—
488C4⋊C4—2842no—
489C4⋊C4—2842no—
490C4⋊C4—2842no—
491C4⋊C4—712no—
492C4⋊C4—712no—
493C4⋊C4—1424no—
494C4⋊C4—1424no—
495C4⋊C4—1421no—
496C4⋊C4—1421no—
497C4⋊C4—1421no—
498C4⋊C4—1421no—
499C4⋊C4—1421no—
500C4⋊C4—1421no—
501C4⋊C4#1176586512yes—
502C4⋊C4#1176586512yes—
503C22⋊C4—2842no—
504C22⋊C4—2842no—
505C42—42no—
506C42—42no—
507C42—2842no—
508C42—2842no—
509C42—2842no—
510C42—2842no—
511C42—712no—
512C42—712no—
513C42—714no—
514C42—714no—
515C42C2×C4×D7112yes—
516C42Q8×D7112yes—
517C23—712no—
518C23—712no—
519C23—712no—
520C23—1421no—
521C23—1421no—
522C23—1421no—
523C23—1421no—
524C23C2×D4×D7111yescharacteristic
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—2848no—
542C2×C4—2848no—
543C2×C4—2841no—
544C2×C4—2841no—
545C2×C4—2841no—
546C2×C4—2841no—
547C2×C4—2841no—
548C2×C4—2841no—
549C2×C4—2841no—
550C2×C4—2841no—
551C2×C4—2841no—
552C2×C4—2841no—
553C2×C4—712no—
554C2×C4—712no—
555C2×C4—712no—
556C2×C4—712no—
557C2×C4—712no—
558C2×C4—1428no—
559C2×C4—1428no—
560C2×C4—1422no—
561C2×C4—1422no—
562C2×C4—1421no—
563C2×C4—1421no—
564C2×C4—1421no—
565C2×C4—1421no—
566C2×C4—1422no—
567C2×C4—1422no—
568C2×C4—1422no—
569C2×C4—1422no—
570C2×C4C4⋊C4×D7112yes—
571C2×C4C4⋊C4×D7112yes—
572C2×C4C22×C4×D7111yescharacteristic
573C2×C4C2×Q8×D7111yescharacteristic
574C2×C4#41×D7111yescharacteristic
575C2×C4#41×D7111yescharacteristic
576C2×C4C2×#1176586511yescharacteristic
577C2×C4C2×#1176586511yescharacteristic
578C8—22no—
579C8—22no—
580C8—1422no—
581C8—1422no—
582C22—714no—
583C22—1422no—
584C22—1422no—
585C22—1422no—
586C22—1422no—
587C22—1422no—
588C22—1422no—
589C22C2×C4⋊C4×D7111yescharacteristic
590C22#1798650811yescharacteristic
591C22#1798650811yescharacteristic
592C22AGL(1,ℤ/8)×D7111yescharacteristic
593C22AGL(1,ℤ/8)×D7111yescharacteristic
594C22#137×D7111yescharacteristic
595C22#137×D7111yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—81no—
601C4—81no—
602C4—2842no—
603C4—2842no—
604C4—5681no—
605C4—5681no—
606C4—714no—
607C4—1422no—
608C4—1422no—
609C4#1798646711yescharacteristic
610C4C4×#1176586511yescharacteristic
611C4#1798655311yescharacteristic
612C4#1798655311yescharacteristic
613C2—1424no—
614C2#2259082011yescharacteristic
615C2#428×D7111yescharacteristic
616C2#428×D7111yescharacteristic
617C2#2259842911yescharacteristic
618C2#2259843711yescharacteristic
619C2#2259846011yescharacteristic
620C2#2259846811yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1776C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1776D71
hypocenterC71#21365
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #22589102C2semidirect
5. #22589100C2semidirect
6. #22589097C2semidirect
7. #22589095C2semidirect
12. #22588238C2semidirect
13. #22588238C2semidirect
14. #22588237C2semidirect
15. C71×#1776C2semidirect
21. #17986054C4semidirect
100. Dic71⋊C8C2×C4semidirect
101. Dic71⋊C8C2×C4semidirect
104. #14556039C2×C4semidirect
122. #97×C71C2×C4semidirect
325. #1776D71semidirect
326. C71#21365semidirect
427. #97C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860542
#225907442
C4×C4⋊C4×D712
#225891022
#225891002
#225890972
#225890952
#225885342
#225885312
#225885292
#225885262
#225882382
#225882372
C71×#17762
#2136571