Finite Group Databasebeta

← #32002352

Subgroups — #32002352

Order 18176.

826 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986057C211yesmaximal, characteristic
2#22590947C211yesmaximal, characteristic
3#22590697C211yesmaximal, characteristic
4#22589105C211yesmaximal, characteristic, complemented
5#22589105C211yesmaximal, characteristic, complemented
6#22589102C211yesmaximal, characteristic, complemented
7#22589097C211yesmaximal, characteristic, complemented
8#22588541C211yesmaximal, characteristic
9#22588537C211yesmaximal, characteristic
10#22588533C211yesmaximal, characteristic
11#22588239C211yesmaximal, characteristic, complemented
12#22588236C211yesmaximal, characteristic, complemented
13#107×Dic71C211yesmaximal, characteristic, complemented
14#22588228C211yesmaximal, characteristic, complemented
15C71×#1852C211yesmaximal, characteristic
16C2×#14556087C2211yescharacteristic
17C2×#14556087C2211yescharacteristic
18#17986057C418yescomplemented
19C2×#14556062C2211yescharacteristic
20C2×#14556061C2211yescharacteristic
21C2×#14556060C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×#14556059C2211yescharacteristic
25C2×Dic71⋊C8C2211yescharacteristic
26C2×Dic71⋊C8C2211yescharacteristic
27C2×C22⋊(C71⋊C8)C2211yescharacteristic
28C2×#14556038C2211yescharacteristic
29C2×C8×Dic71C2211yescharacteristic
30#17985893—21no—
31#17985887—21no—
32#17985884—21no—
33#17985882—21no—
34#17985868C2211yescharacteristic
35#17985864C2211yescharacteristic
36#17985859C2211yescharacteristic
37C4⋊C4×Dic71—21no—
38C4⋊C4×Dic71—21no—
39C4⋊C4×Dic71C2211yescharacteristic
40C4⋊C4×Dic71C2211yescharacteristic
41C4⋊C4×Dic71C2211yescharacteristic
42C4⋊C4×Dic71C2211yescharacteristic
43C4⋊C4×Dic71C2211yescharacteristic
44C4⋊C4×Dic71C2211yescharacteristic
45C4×C22⋊Dic71—21no—
46C4×C22⋊Dic71—21no—
47#17985762C2211yescharacteristic
48#17985762C2211yescharacteristic
49#17985761C2211yescharacteristic
50#17985761C2211yescharacteristic
51#17985761C2211yescharacteristic
52#17985761C2211yescharacteristic
53C22×C4⋊C4×C71C2211yescharacteristic
54C2×#107×C71C2211yescharacteristic
55C2×#107×C71C2211yescharacteristic
56C2×C22⋊C8×C71C2211yescharacteristic
57#380×C71C2211yescharacteristic
58#340×C71C2211yescharacteristic
59#340×C71C2211yescharacteristic
60C22×C71⋊C8C2311yescharacteristic
61C22×C71⋊C8C2311yescharacteristic
62#14556087—22no—
63#14556087—22no—
64#14556087C2×C414yes—
65#14556087Q814yes—
66C2×C4⋊Dic71D411yescharacteristic
67C2×C4⋊Dic71D411yescharacteristic
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×Dic71⋊C4—21no—
73C2×Dic71⋊C4—21no—
74C2×C22⋊Dic71—21no—
75C2×C22⋊Dic71—21no—
76C2×C22⋊Dic71D411yescharacteristic
77C2×C22⋊Dic71D411yescharacteristic
78C2×C4×Dic71—21no—
79C2×C4×Dic71—21no—
80C2×C4×Dic71—21no—
81C2×C4×Dic71—21no—
82C2×C4×Dic71—21no—
83C2×C4×Dic71—21no—
84C2×C4×Dic71—21no—
85C2×C4×Dic71C2311yescharacteristic
86C2×C4×Dic71C2311yescharacteristic
87#14556062—22no—
88#14556061—22no—
89#14556060—22no—
90#14556059—22no—
91#14556059—22no—
92#14556059—22no—
93Dic71⋊C8C2×C414yescomplemented
94Dic71⋊C8C2×C414yescomplemented
95C22⋊(C71⋊C8)C2×C414yes—
96#14556038C2×C414yescomplemented
97C8×Dic71C2×C414yescomplemented
98C71⋊#95—21no—
99C71⋊#95—21no—
100C71⋊#95—21no—
101C71⋊#95—21no—
102C71⋊#95—21no—
103C71⋊#95—21no—
104C71⋊#95—21no—
105C71⋊#95—21no—
106C71⋊#95—21no—
107C23×C4×C71—21no—
108C23×C4×C71C2311yescharacteristic
109C22×C8×C71C2311yescharacteristic
110C22×C8×C71C2311yescharacteristic
111C2×C4⋊C4×C71—21no—
112C2×C4⋊C4×C71—22no—
113C2×C4⋊C4×C71—22no—
114C2×C4⋊C4×C71C2311yescharacteristic
115C2×C4⋊C4×C71C2311yescharacteristic
116C2×C4⋊C4×C71C2311yescharacteristic
117C2×C4⋊C4×C71C2311yescharacteristic
118C2×C22⋊C4×C71—21no—
119#107×C71—22no—
120#107×C71—22no—
121C22⋊C8×C71C2×C414yes—
122#95×C71—21no—
123C22×Dic71—21no—
124C22×Dic71—21no—
125C22×Dic71—21no—
126C22×Dic71—21no—
127C22×Dic71C2×D411yescharacteristic
128C22×Dic71C2×D411yescharacteristic
129C22×Dic71C2×D411yescharacteristic
130C22×Dic71C2×D411yescharacteristic
131C2×C71⋊C8—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8C22×C412yes—
136C2×C71⋊C8C22×C412yes—
137C4⋊Dic71—21no—
138C4⋊Dic71—21no—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71C4⋊C414yes—
144C4⋊Dic71C4⋊C414yes—
145C4⋊Dic71#4112yes—
146C4⋊Dic71#4112yes—
147Dic71⋊C4—81no—
148Dic71⋊C4—81no—
149C22⋊Dic71—24no—
150C22⋊Dic71—81no—
151C22⋊Dic71C4⋊C414yes—
152C22⋊Dic71C4⋊C414yes—
153C4×Dic71—24no—
154C4×Dic71—81no—
155C4×Dic71—81no—
156C4×Dic71—81no—
157C4×Dic71—81no—
158C4×Dic71—81no—
159C4×Dic71—81no—
160C4×Dic71C22×C412yes—
161C4×Dic71C22×C412yes—
162C4×Dic71C2×Q812yes—
163C4×Dic71C2×Q812yes—
164C24×C71C2×D411yescharacteristic
165C22×C4×C71—21no—
166C22×C4×C71—21no—
167C22×C4×C71—21no—
168C22×C4×C71—21no—
169C22×C4×C71—21no—
170C22×C4×C71—21no—
171C22×C4×C71—24no—
172C22×C4×C71—21no—
173C22×C4×C71—21no—
174C22×C4×C71—81no—
175C22×C4×C71—81no—
176C22×C4×C71C22×C412yes—
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×Q812yes—
179C22×C4×C71C2411yescharacteristic
180C2×C8×C71—21no—
181C2×C8×C71—21no—
182C2×C8×C71—21no—
183C2×C8×C71—21no—
184C2×C8×C71C22×C412yes—
185C2×C8×C71C22×C412yes—
186C4⋊C4×C71—24no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71#4112yes—
194C4⋊C4×C71#4112yes—
195C22⋊C4×C71—81no—
196C23×C71—21no—
197C23×C71—21no—
198C23×C71—21no—
199C23×C71—21no—
200C23×C71—21no—
201C23×C71—21no—
202C23×C71C2×C4⋊C412yes—
203C23×C71C22×D411yescharacteristic
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—22no—
207C2×C4×C71—22no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—41no—
213C2×C4×C71—41no—
214C2×C4×C71—41no—
215C2×C4×C71—41no—
216C2×C4×C71—41no—
217C2×C4×C71—41no—
218C2×C4×C71—41no—
219C2×C4×C71—41no—
220C2×C4×C71—41no—
221C2×C4×C71—41no—
222C2×C4×C71—84no—
223C2×C4×C71C2×C4⋊C412yes—
224C2×C4×C71C23×C411yescharacteristic
225C2×C4×C71C22×Q811yescharacteristic
226C2×C4×C71C2×#4111yescharacteristic
227C2×C4×C71C2×#4111yescharacteristic
228C2×C4×C71C2×#4111yescharacteristic
229C2×C4×C71C2×#4111yescharacteristic
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—22no—
239C2×Dic71—22no—
240C2×Dic71—41no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71C2×C4⋊C412yes—
251C2×Dic71C2×C4⋊C412yes—
252C2×Dic71C2×C4⋊C412yes—
253C2×Dic71C2×C4⋊C412yes—
254C8×C71—22no—
255C8×C71—22no—
256C71⋊C8—22no—
257C71⋊C8—22no—
258C22×C71—22no—
259C22×C71—22no—
260C22×C71—22no—
261C22×C71—41no—
262C22×C71—41no—
263C22×C71—41no—
264C22×C71—41no—
265C22×C71C22×C4⋊C411yescharacteristic
266C22×C71#52511yescharacteristic
267C22×C71#52511yescharacteristic
268C22×C71#57611yescharacteristic
269C22×C71#57711yescharacteristic
270C22×C71#57711yescharacteristic
271C22×C71#57811yescharacteristic
272C4×C71—22no—
273C4×C71—42no—
274C4×C71—42no—
275C4×C71—81no—
276C4×C71—81no—
277C4×C71C4×#4111yescharacteristic
278C4×C71C4×#4111yescharacteristic
279C4×C71#53311yescharacteristic
280C4×C71#53311yescharacteristic
281Dic71—22no—
282Dic71—22no—
283Dic71—22no—
284Dic71—22no—
285Dic71—42no—
286Dic71—42no—
287Dic71—81no—
288Dic71—81no—
289#21367—711nomaximal, Sylow
290C2×C71—42no—
291C2×C71#229611yescharacteristic
292C2×C71#291211yescharacteristic
293C2×C71#291311yescharacteristic
294C2×C71#311811yescharacteristic
295C2×C71#311911yescharacteristic
296C2×C71#312011yescharacteristic
297C2×C71#312111yescharacteristic
298C2×#433—711no—
299#2314—711no—
300#2274—711no—
301#1949—711no—
302#1949—711no—
303#1948—711no—
304#1947—711no—
305#1854—711no—
306#1854—711no—
307#1853—711no—
308#1852D7111yescharacteristic
309#1776—711no—
310#1776—711no—
311C4×#107—711no—
312C4×#106—711no—
313C71#2136711yescharacteristic, complemented, Sylow
314C2×#117—711no—
315C2×#117—711no—
316C22×C4⋊C4C2×D7111yescharacteristic
317#433—718no—
318C2×#107—711no—
319C2×#107—711no—
320C2×#107C2×D7111yescharacteristic
321C2×#107C2×D7111yescharacteristic
322C2×#106—711no—
323C2×#106—711no—
324C2×#106—711no—
325C2×#106—711no—
326C2×C4⋊C8—711no—
327C2×C4⋊C8—711no—
328C2×C22⋊C8—711no—
329C2×C22⋊C8C2×D7111yescharacteristic
330C2×#97—711no—
331C2×C4×C8—711no—
332#388—1421no—
333#388—1421no—
334#387—1421no—
335#387—1421no—
336#384—711no—
337#382—711no—
338#380—711no—
339#380C2×D7111yescharacteristic
340C4×C4⋊C4—711no—
341C4×C4⋊C4—711no—
342C4×C4⋊C4—711no—
343C4×C4⋊C4—711no—
344C4×C4⋊C4—711no—
345C4×C4⋊C4—711no—
346C4×C4⋊C4—1421no—
347C4×C4⋊C4—1421no—
348C4×C22⋊C4—1421no—
349C4×C22⋊C4—1421no—
350#340—711no—
351#340—711no—
352#340—711no—
353#340—711no—
354#340—711no—
355#340—711no—
356#340C2×D7111yescharacteristic
357#340C2×D7111yescharacteristic
358C23×C4—21no—
359C23×C4C22×D7111yescharacteristic
360C22×C8—711no—
361C22×C8—711no—
362C22×C8C22×D7111yescharacteristic
363C22×C8C22×D7111yescharacteristic
364#117—714no—
365#117—714no—
366#117—1422no—
367#117—1422no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—711no—
372C2×C4⋊C4—711no—
373C2×C4⋊C4—711no—
374C2×C4⋊C4—711no—
375C2×C4⋊C4—711no—
376C2×C4⋊C4—711no—
377C2×C4⋊C4—1421no—
378C2×C4⋊C4—1421no—
379C2×C4⋊C4C22×D7111yescharacteristic
380C2×C4⋊C4C22×D7111yescharacteristic
381C2×C4⋊C4C22×D7111yescharacteristic
382C2×C4⋊C4C22×D7111yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—711no—
385C2×C22⋊C4—711no—
386C2×C22⋊C4—1421no—
387C2×C22⋊C4—1421no—
388C2×C42—711no—
389C2×C42—711no—
390C2×C42—1421no—
391C2×C42—1421no—
392C2×C42—1421no—
393C2×C42—1421no—
394C2×C42—1421no—
395C2×C42—1421no—
396C2×C42—1421no—
397#107—22no—
398#107—22no—
399#107—1422no—
400#107—1422no—
401#106—1422no—
402#106—1422no—
403#106—1422no—
404#106—1422no—
405C4⋊C8—714no—
406C4⋊C8—714no—
407C22⋊C8—714no—
408C22⋊C8C4×D7114yes—
409#97—714no—
410C4×C8—714no—
411#95—21no—
412#95—1421no—
413#95—1421no—
414#95—1421no—
415#95—1421no—
416#95—1421no—
417#95—1421no—
418#95—1421no—
419#95—1421no—
420#95—1421no—
421C24D4×D7111yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—711no—
434C22×C4—711no—
435C22×C4—711no—
436C22×C4—711no—
437C22×C4—1421no—
438C22×C4—1421no—
439C22×C4—1421no—
440C22×C4—1421no—
441C22×C4C2×C4×D7112yes—
442C22×C4D4×D7111yescharacteristic
443C22×C4Q8×D7112yes—
444C22×C4C23×D7111yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—712no—
450C2×C8—712no—
451C2×C8—1421no—
452C2×C8—1421no—
453C2×C8—1421no—
454C2×C8—1421no—
455C2×C8C2×C4×D7112yes—
456C2×C8C2×C4×D7112yes—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—5681no—
465C4⋊C4—5681no—
466C4⋊C4—714no—
467C4⋊C4—714no—
468C4⋊C4—712no—
469C4⋊C4—712no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4—1421no—
473C4⋊C4—1421no—
474C4⋊C4—1421no—
475C4⋊C4—1421no—
476C4⋊C4#1176586412yes—
477C4⋊C4#1176586412yes—
478C22⋊C4—81no—
479C22⋊C4—5681no—
480C22⋊C4—714no—
481C22⋊C4—714no—
482C22⋊C4—1424no—
483C42—5681no—
484C42—5681no—
485C42—5681no—
486C42—5681no—
487C42—5681no—
488C42—5681no—
489C42—712no—
490C42—712no—
491C42—712no—
492C42—712no—
493C42—1424no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C4⋊C4×D7112yes—
501C23C2×D4×D7111yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—84no—
521C2×C4—2841no—
522C2×C4—2841no—
523C2×C4—2841no—
524C2×C4—2841no—
525C2×C4—2841no—
526C2×C4—2841no—
527C2×C4—2841no—
528C2×C4—2841no—
529C2×C4—2841no—
530C2×C4—2841no—
531C2×C4—712no—
532C2×C4—712no—
533C2×C4—712no—
534C2×C4—712no—
535C2×C4—1422no—
536C2×C4—1422no—
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×C4C4⋊C4×D7112yes—
546C2×C4C22×C4×D7111yescharacteristic
547C2×C4C2×Q8×D7111yescharacteristic
548C2×C4#41×D7111yescharacteristic
549C2×C4#41×D7111yescharacteristic
550C2×C4C2×#1176586411yescharacteristic
551C2×C4C2×#1176586411yescharacteristic
552C8—22no—
553C8—22no—
554C8—1422no—
555C8—1422no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C2×C4⋊C4×D7111yescharacteristic
564C22#1798650511yescharacteristic
565C22#1798650711yescharacteristic
566C22#1798680011yescharacteristic
567C22#1798680811yescharacteristic
568C22#1798680911yescharacteristic
569C22#1798681211yescharacteristic
570C4—22no—
571C4—42no—
572C4—42no—
573C4—81no—
574C4—81no—
575C4—2842no—
576C4—2842no—
577C4—5681no—
578C4—5681no—
579C4—1422no—
580C4—1422no—
581C4—1422no—
582C4—1422no—
583C4C4×#1176586411yescharacteristic
584C4C4×#1176586411yescharacteristic
585C4#1798655111yescharacteristic
586C4#1798655111yescharacteristic
587C2—42no—
588C2#2259081911yescharacteristic
589C2#2259693411yescharacteristic
590C2#2259693611yescharacteristic
591C2#2259851711yescharacteristic
592C2#2259854211yescharacteristic
593C2#2259854911yescharacteristic
594C2#2259857011yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1852C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
cosocleC22×C4×C71C24
hypercenter#1852D71
hypocenterC71#21367
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
4. #22589105C2semidirect
5. #22589105C2semidirect
6. #22589102C2semidirect
7. #22589097C2semidirect
11. #22588239C2semidirect
12. #22588236C2semidirect
13. #107×Dic71C2semidirect
14. #22588228C2semidirect
18. #17986057C4semidirect
93. Dic71⋊C8C2×C4semidirect
94. Dic71⋊C8C2×C4semidirect
96. #14556038C2×C4semidirect
97. C8×Dic71C2×C4semidirect
313. C71#21367semidirect

Maximal subgroups

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

subgroupindex
C2×#179860572
#225909472
#225906972
#225891052
#225891022
#225890972
#225885412
#225885372
#225885332
#225882392
#225882362
#107×Dic712
#225882282
C71×#18522
#2136771