Finite Group Databasebeta

← #32020681

Subgroups — #32020681

Order 18176.

798 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986296C211yesmaximal, characteristic
2#22591287C211yesmaximal, characteristic
3#22589310C211yesmaximal, characteristic, complemented
4#22589103C211yesmaximal, characteristic, complemented
5#22589036C211yesmaximal, characteristic
6#22589022C211yesmaximal, characteristic
7#22588967C211yesmaximal, characteristic
8#22588967C211yesmaximal, characteristic
9#22588915C211yesmaximal, characteristic, complemented
10#22588908C211yesmaximal, characteristic, complemented
11#22588238C211yesmaximal, characteristic, complemented
12C4×#14556059C211yesmaximal, characteristic, complemented
13#22588215C211yesmaximal, characteristic, complemented
14#22588168C211yesmaximal, characteristic
15C71×#2370C211yesmaximal, characteristic, complemented
16C2×C4×D284C2211yescharacteristic
17C2×#14556086—21no—
18#17986296C418yescomplemented
19C2×#14556061C2211yescharacteristic
20C2×#14556059C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×C4⋊(C71⋊C8)C2211yescharacteristic
24C2×#14556048C2211yescharacteristic
25C2×#14556048C2211yescharacteristic
26C2×#14556048C2211yescharacteristic
27C2×#14556048C2211yescharacteristic
28C2×#14556039C2211yescharacteristic
29C2×#14556039C2211yescharacteristic
30C2×C4×C71⋊C8C2211yescharacteristic
31#17985901C2211yescharacteristic
32#17985901C2211yescharacteristic
33#17985891—21no—
34#17985891—21no—
35#17985869C2211yescharacteristic
36#17985856—21no—
37C4×C4⋊Dic71C2211yescharacteristic
38C4×C4⋊Dic71C2211yescharacteristic
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×#125×C71C2211yescharacteristic
50#389×C71—21no—
51#389×C71C2211yescharacteristic
52#389×C71C2211yescharacteristic
53#387×C71—21no—
54#387×C71—21no—
55C4×C4⋊C4×C71—21no—
56C4×C4⋊C4×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C4⋊C4×C71C2211yescharacteristic
59C4×C4⋊C4×C71C2211yescharacteristic
60C22×D284C2311yescharacteristic
61C22×C4×D71—21no—
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C4×D284—22no—
67C4×D284C2×C414yes—
68#14556086—44no—
69C2×C4⋊Dic71—21no—
70C2×C4⋊Dic71—21no—
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×(C2×D71)⋊C4—21no—
75C2×(C2×D71)⋊C4—21no—
76C2×(C2×D71)⋊C4—21no—
77C2×(C2×D71)⋊C4—21no—
78C2×(C2×D71)⋊C4—21no—
79C2×C4×Dic71—21no—
80C2×C4×Dic71—21no—
81#14556061C2×C414yes—
82#14556059—22no—
83#14556059—22no—
84#14556059C2×C414yes—
85C4⋊(C71⋊C8)C2×C414yes—
86#14556048—22no—
87#14556048—22no—
88#14556048C2×C414yes—
89#14556048C2×C414yes—
90#14556039—22no—
91#14556039—22no—
92C4×C71⋊C8—22no—
93C71⋊#95—21no—
94C71⋊#95—21no—
95#125×C71—22no—
96#125×C71C2×C414yes—
97C2×C4⋊C4×C71—21no—
98C2×C4⋊C4×C71—21no—
99C2×C4⋊C4×C71—21no—
100C2×C4⋊C4×C71—21no—
101C2×C4⋊C4×C71D411yescharacteristic
102C2×C4⋊C4×C71D411yescharacteristic
103C2×C4⋊C4×C71C2311yescharacteristic
104C2×C4⋊C4×C71C2311yescharacteristic
105C2×C4⋊C4×C71C2311yescharacteristic
106C2×C4⋊C4×C71C2311yescharacteristic
107C2×C42×C71—21no—
108C2×C42×C71—21no—
109C2×C42×C71—21no—
110C2×C42×C71C2311yescharacteristic
111C2×C42×C71C2311yescharacteristic
112C2×C42×C71C2311yescharacteristic
113#95×C71—21no—
114#95×C71—21no—
115#95×C71—21no—
116#95×C71—21no—
117#95×C71—21no—
118#95×C71D411yescharacteristic
119#95×C71D411yescharacteristic
120C23×D71—21no—
121C2×D284—21no—
122C2×D284—21no—
123C2×D284—21no—
124C2×D284—21no—
125C2×D284—21no—
126C2×D284C22×C412yes—
127C22×Dic71—21no—
128C22×Dic71—21no—
129C22×Dic71—21no—
130C2×C4×D71—24no—
131C2×C4×D71—42no—
132C2×C4×D71—42no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8—21no—
136C2×C71⋊C8—21no—
137C2×C71⋊C8—21no—
138C2×C71⋊C8—21no—
139C2×C71⋊C8—21no—
140C2×C71⋊C8—21no—
141C2×C71⋊C8—21no—
142C2×C71⋊C8C22×C412yes—
143C2×C71⋊C8C22×C412yes—
144C2×C71⋊C8#4112yes—
145C4⋊Dic71—21no—
146C4⋊Dic71—21no—
147C4⋊Dic71—21no—
148C4⋊Dic71—21no—
149C4⋊Dic71—21no—
150C4⋊Dic71—42no—
151C4⋊Dic71—81no—
152C4⋊Dic71C22×C412yes—
153(C2×D71)⋊C4—24no—
154(C2×D71)⋊C4—42no—
155(C2×D71)⋊C4—42no—
156(C2×D71)⋊C4—81no—
157(C2×D71)⋊C4—81no—
158C4×Dic71—42no—
159C4×Dic71—42no—
160C22×C4×C71—21no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71—21no—
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71C2×D411yescharacteristic
167C22×C4×C71C2×D411yescharacteristic
168C22×C4×C71C2×D411yescharacteristic
169C22×C4×C71C2×D411yescharacteristic
170C22×C4×C71C2411yescharacteristic
171C4⋊C4×C71—24no—
172C4⋊C4×C71—24no—
173C4⋊C4×C71—22no—
174C4⋊C4×C71—21no—
175C4⋊C4×C71—21no—
176C4⋊C4×C71—21no—
177C4⋊C4×C71—21no—
178C4⋊C4×C71—21no—
179C4⋊C4×C71—41no—
180C4⋊C4×C71—42no—
181C4⋊C4×C71—81no—
182C4⋊C4×C71C22×C412yes—
183C4⋊C4×C71C22×C412yes—
184C4⋊C4×C71#4112yes—
185C42×C71—21no—
186C42×C71—21no—
187C42×C71—21no—
188C42×C71—21no—
189C42×C71—42no—
190C42×C71—81no—
191C42×C71—81no—
192C42×C71C22×C412yes—
193C42×C71#4112yes—
194C23×C71C22×D411yescharacteristic
195C22×D71—22no—
196C22×D71—41no—
197C22×D71—41no—
198C22×D71—41no—
199C22×D71—41no—
200C22×D71—41no—
201C22×D71—41no—
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—22no—
209C2×C4×C71—22no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—22no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
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×C71C4×D412yes—
230C2×C4×C71C4×D412yes—
231C2×C4×C71AGL(1,ℤ/8)12yes—
232C2×C4×C71C23×C411yescharacteristic
233C2×C4×C71C2×#4111yescharacteristic
234C2×C4×C71C2×#4111yescharacteristic
235C2×C4×C71C2×#4111yescharacteristic
236C2×C4×C712−1+411yescharacteristic
237C2×C4×C712−1+411yescharacteristic
238C2×Dic71—22no—
239C2×Dic71—41no—
240C2×Dic71—41no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247D284—22no—
248D284—22no—
249D284—41no—
250D284—41no—
251C4×D71—48no—
252C71⋊C8—22no—
253C71⋊C8—22no—
254C71⋊C8—22no—
255C71⋊C8—41no—
256C22×C71C2×C4×D411yescharacteristic
257C22×C71#54711yescharacteristic
258C22×C71#54811yescharacteristic
259C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C71#57511yescharacteristic
261C22×C71#57711yescharacteristic
262C22×C71#57711yescharacteristic
263C2×D71—42no—
264C2×D71—42no—
265C2×D71—42no—
266C2×D71—81no—
267C2×D71—81no—
268C2×D71—81no—
269C2×D71—81no—
270C4×C71—22no—
271C4×C71—22no—
272C4×C71—22no—
273C4×C71—41no—
274C4×C71—41no—
275C4×C71—41no—
276C4×C71—42no—
277C4×C71—42no—
278C4×C71—42no—
279C4×C71—81no—
280C4×C71C4×#4111yescharacteristic
281C4×C71#51911yescharacteristic
282C4×C71#55211yescharacteristic
283C4×C71#55611yescharacteristic
284Dic71—42no—
285Dic71—81no—
286Dic71—81no—
287#23072—711nomaximal, Sylow
288C2×C71#236211yescharacteristic
289C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
290C2×C71#295211yescharacteristic
291C2×C71#334511yescharacteristic
292C2×C71#334811yescharacteristic
293C2×C71#336811yescharacteristic
294C2×C71#337611yescharacteristic
295D71—82no—
296C2×#478—711no—
297#2370D7111yescharacteristic, complemented
298#2354—711no—
299#1992—711no—
300#1948—711no—
301#1936—711no—
302#1934—711no—
303#1926—711no—
304#1926—711no—
305#1919—711no—
306#1918—711no—
307#1776—711no—
308C4×#106—711no—
309#1771—711no—
310#1762—711no—
311C71#2307211yescharacteristic, complemented, Sylow
312C2×#125C2×D7111yescharacteristic
313C2×C4×D4—711no—
314C2×#117—1421no—
315#478—718no—
316C2×#107—711no—
317C2×#106—711no—
318C2×#106—711no—
319C2×#106—711no—
320C2×C4⋊C8—711no—
321C2×#102—711no—
322C2×#102—711no—
323C2×#102—711no—
324C2×#102—711no—
325C2×#97—711no—
326C2×#97—711no—
327C2×C4×C8—711no—
328#390—711no—
329#390—711no—
330#389—21no—
331#389C2×D7111yescharacteristic
332#389C2×D7111yescharacteristic
333#388—1421no—
334#388—1421no—
335#387—21no—
336#387—21no—
337#384—711no—
338#380—1421no—
339C4×C4⋊C4—21no—
340C4×C4⋊C4—711no—
341C4×C4⋊C4—711no—
342C4×C4⋊C4C2×D7111yescharacteristic
343C4×C4⋊C4C2×D7111yescharacteristic
344C4×C4⋊C4C2×D7111yescharacteristic
345C4×C4⋊C4C2×D7111yescharacteristic
346#340—711no—
347#340—711no—
348#340—711no—
349#340—711no—
350#340—711no—
351#340—711no—
352#340—711no—
353#340—711no—
354#336—711no—
355#336—711no—
356C22×D4—711no—
357C23×C4—1421no—
358C22×C8—711no—
359C22×C8—711no—
360C22×C8—711no—
361C22×C8—711no—
362#125—22no—
363#125C4×D7114yes—
364C4×D4—714no—
365C4×D4—1422no—
366#117—2844no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—711no—
372C2×C4⋊C4—711no—
373C2×C4⋊C4—711no—
374C2×C4⋊C4—1421no—
375C2×C4⋊C4—1421no—
376C2×C4⋊C4C22⋊D7111yescharacteristic
377C2×C4⋊C4C22⋊D7111yescharacteristic
378C2×C4⋊C4C22×D7111yescharacteristic
379C2×C4⋊C4C22×D7111yescharacteristic
380C2×C4⋊C4C22×D7111yescharacteristic
381C2×C4⋊C4C22×D7111yescharacteristic
382C2×C22⋊C4—1421no—
383C2×C22⋊C4—1421no—
384C2×C22⋊C4—1421no—
385C2×C22⋊C4—1421no—
386C2×C22⋊C4—1421no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—1421no—
391C2×C42—1421no—
392C2×C42C22×D7111yescharacteristic
393C2×C42C22×D7111yescharacteristic
394C2×C42C22×D7111yescharacteristic
395#107—714no—
396#106—714no—
397#106—1422no—
398#106—1422no—
399C4⋊C8—714no—
400#102—714no—
401#102—714no—
402#102—1422no—
403#102—1422no—
404#97—1422no—
405#97—1422no—
406C4×C8—1422no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—1421no—
413#95—1421no—
414#95C22⋊D7111yescharacteristic
415#95C22⋊D7111yescharacteristic
416C24—1421no—
417C2×D4—712no—
418C2×D4—1421no—
419C2×D4—1421no—
420C2×D4—1421no—
421C2×D4—1421no—
422C2×D4—1421no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—2842no—
428C22×C4—2842no—
429C22×C4—1421no—
430C22×C4—1421no—
431C22×C4—1421no—
432C22×C4—1424no—
433C22×C4C2×C22⋊D7111yescharacteristic
434C22×C4C2×C22⋊D7111yescharacteristic
435C22×C4C2×C22⋊D7111yescharacteristic
436C22×C4C2×C22⋊D7111yescharacteristic
437C22×C4C2×C22⋊D7111yescharacteristic
438C22×C4C2×C22⋊D7111yescharacteristic
439C22×C4C23×D7111yescharacteristic
440C2×C8—712no—
441C2×C8—712no—
442C2×C8—712no—
443C2×C8—1421no—
444C2×C8—1421no—
445C2×C8—1421no—
446C2×C8—1421no—
447C2×C8—1421no—
448C2×C8—1421no—
449C2×C8—1421no—
450C2×C8—1421no—
451C2×C8—1421no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—42no—
462C4⋊C4—81no—
463C4⋊C4—2842no—
464C4⋊C4—5681no—
465C4⋊C4—712no—
466C4⋊C4—1421no—
467C4⋊C4—1421no—
468C4⋊C4—1421no—
469C4⋊C4—1421no—
470C4⋊C4—1421no—
471C4⋊C4C2×C4×D7112yes—
472C4⋊C4C2×C4×D7112yes—
473C4⋊C4#1176586612yes—
474C22⋊C4—2842no—
475C22⋊C4—2842no—
476C22⋊C4—5681no—
477C22⋊C4—5681no—
478C22⋊C4—1424no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—42no—
484C42—81no—
485C42—81no—
486C42—2842no—
487C42—2842no—
488C42C2×C4×D7112yes—
489C42#1176586512yes—
490C23—2841no—
491C23—2841no—
492C23—2841no—
493C23—2841no—
494C23—2841no—
495C23—2841no—
496C23—1422no—
497C23C22×C22⋊D7111yescharacteristic
498D4—2841no—
499D4—2841no—
500D4—1422no—
501D4—1422no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—2848no—
530C2×C4—2841no—
531C2×C4—2841no—
532C2×C4—2841no—
533C2×C4—2841no—
534C2×C4—2841no—
535C2×C4—2841no—
536C2×C4—2841no—
537C2×C4—2841no—
538C2×C4—1422no—
539C2×C4C4×C22⋊D7112yes—
540C2×C4C4×C22⋊D7112yes—
541C2×C4#1455615312yes—
542C2×C4C22×C4×D7111yescharacteristic
543C2×C4#41×D7111yescharacteristic
544C2×C4C2×#1176586511yescharacteristic
545C2×C4C2×#1176586611yescharacteristic
546C2×C4#1455617511yescharacteristic
547C2×C4#1455617611yescharacteristic
548C8—2841no—
549C8—1422no—
550C8—1422no—
551C8—1422no—
552C22—2842no—
553C22—2842no—
554C22—2842no—
555C22—5681no—
556C22—5681no—
557C22—5681no—
558C22—5681no—
559C22C2×C4×C22⋊D7111yescharacteristic
560C22#1798664811yescharacteristic
561C22#1798666011yescharacteristic
562C22C2×#1455615311yescharacteristic
563C22#1798679711yescharacteristic
564C22#1798680711yescharacteristic
565C22#1798681011yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—42no—
570C4—41no—
571C4—42no—
572C4—41no—
573C4—41no—
574C4—42no—
575C4—81no—
576C4—2842no—
577C4—5681no—
578C4—5681no—
579C4C4×#1176586611yescharacteristic
580C4#1798647511yescharacteristic
581C4#1798669011yescharacteristic
582C4#1798672211yescharacteristic
583C2—5682no—
584C2#2259136311yescharacteristic
585C2C4×#1455615311yescharacteristic
586C2#2259722111yescharacteristic
587C2#2260080411yescharacteristic
588C2#2260084911yescharacteristic
589C2#2260106911yescharacteristic
590C2#2260116711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2370C2
socleC23×C71C22×D4
prime radicalC22×D284C23
cosocleC22×C4×C71C24
hypercenter#2370D71
hypocenterC71#23072
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

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

NH ≅ G/Ntype
3. #22589310C2semidirect
4. #22589103C2semidirect
9. #22588915C2semidirect
10. #22588908C2semidirect
11. #22588238C2semidirect
12. C4×#14556059C2semidirect
13. #22588215C2semidirect
15. C71×#2370C2semidirect
18. #17986296C4semidirect
297. #2370D71semidirect
311. C71#23072semidirect

Maximal subgroups

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

subgroupindex
C2×#179862962
#225912872
#225893102
#225891032
#225890362
#225890222
#225889672
#225889152
#225889082
#225882382
C4×#145560592
#225882152
#225881682
C71×#23702
#2307271