Finite Group Databasebeta

← #32009648

Subgroups — #32009648

Order 18176.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986286C211yesmaximal, characteristic
2#22590984C211yesmaximal, characteristic
3#22589293C211yesmaximal, characteristic, complemented
4#22589106C211yesmaximal, characteristic, complemented
5#22589080C211yesmaximal, characteristic, complemented
6#22589065C211yesmaximal, characteristic, complemented
7#22589036C211yesmaximal, characteristic
8#22589022C211yesmaximal, characteristic
9#22588967C211yesmaximal, characteristic
10#22588960C211yesmaximal, characteristic
11#22588925C211yesmaximal, characteristic, complemented
12#22588918C211yesmaximal, characteristic, complemented
13#22588347C211yesmaximal, characteristic
14#22588197C211yesmaximal, characteristic, complemented
15C71×#2331C211yesmaximal, characteristic, complemented
16C2×C4×D284C2211yescharacteristic
17#17986286C418yescomplemented
18C2×#14556061C2211yescharacteristic
19C2×#14556059C2211yescharacteristic
20C2×#14556059C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×C4⋊(C71⋊C8)C2211yescharacteristic
23C2×C4⋊(C71⋊C8)C2211yescharacteristic
24C2×#14556048C2211yescharacteristic
25C2×#14556048C2211yescharacteristic
26C2×#14556048C2211yescharacteristic
27C2×#14556048C2211yescharacteristic
28C2×#14556039C2211yescharacteristic
29C2×C4×C71⋊C8C2211yescharacteristic
30#17985901C2211yescharacteristic
31#17985901C2211yescharacteristic
32#17985897C2211yescharacteristic
33#17985897C2211yescharacteristic
34#17985888—21no—
35#17985888—21no—
36#17985861—21no—
37#17985858—21no—
38C4×C4⋊Dic71C2211yescharacteristic
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×C4⋊Q8×C71C2211yescharacteristic
50#391×C71—21no—
51#391×C71—21no—
52#389×C71C2211yescharacteristic
53#389×C71C2211yescharacteristic
54#387×C71—21no—
55#387×C71—21no—
56#382×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C4⋊C4×C71C2211yescharacteristic
59#376×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—
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×Dic71⋊C4—21no—
72C2×Dic71⋊C4—21no—
73C2×(C2×D71)⋊C4—21no—
74C2×(C2×D71)⋊C4—21no—
75C2×(C2×D71)⋊C4—21no—
76C2×(C2×D71)⋊C4—21no—
77C2×(C2×D71)⋊C4—21no—
78C2×C4×Dic71—21no—
79#14556061—41no—
80#14556059—41no—
81#14556059C2×C414yes—
82#14556059C2×C414yes—
83C4⋊(C71⋊C8)—22no—
84C4⋊(C71⋊C8)C2×C414yes—
85#14556048—41no—
86#14556048—41no—
87#14556048C2×C414yes—
88#14556048C2×C414yes—
89#14556039—41no—
90C4×C71⋊C8—41no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C22×Q8×C71—21no—
95C4⋊Q8×C71—22no—
96C4⋊Q8×C71C2×C414yes—
97C2×C4⋊C4×C71—21no—
98C2×C4⋊C4×C71—21no—
99C2×C4⋊C4×C71—21no—
100C2×C4⋊C4×C71C2311yescharacteristic
101C2×C4⋊C4×C71C2311yescharacteristic
102C2×C4⋊C4×C71C2311yescharacteristic
103C2×C4⋊C4×C71C2311yescharacteristic
104C2×C42×C71—21no—
105C2×C42×C71—21no—
106C2×C42×C71C2311yescharacteristic
107C2×C42×C71C2311yescharacteristic
108C2×C42×C71C2311yescharacteristic
109#95×C71—21no—
110#95×C71—21no—
111#95×C71—21no—
112#95×C71—21no—
113#95×C71—21no—
114#95×C71—21no—
115#95×C71D411yescharacteristic
116#95×C71D411yescharacteristic
117#95×C71D411yescharacteristic
118#95×C71D411yescharacteristic
119C23×D71—21no—
120C2×D284—21no—
121C2×D284—21no—
122C2×D284—21no—
123C2×D284—21no—
124C2×D284—21no—
125C2×D284C22×C412yes—
126C22×Dic71—21no—
127C22×Dic71—21no—
128C22×Dic71—21no—
129C2×C4×D71—24no—
130C2×C4×D71—81no—
131C2×C4×D71—81no—
132C2×C71⋊C8—21no—
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—
144C4⋊Dic71—21no—
145C4⋊Dic71—21no—
146C4⋊Dic71—21no—
147C4⋊Dic71—21no—
148C4⋊Dic71—21no—
149C4⋊Dic71C22×C412yes—
150Dic71⋊C4—81no—
151Dic71⋊C4—81no—
152(C2×D71)⋊C4—24no—
153(C2×D71)⋊C4—81no—
154(C2×D71)⋊C4—81no—
155(C2×D71)⋊C4—81no—
156(C2×D71)⋊C4—81no—
157C4×Dic71—81no—
158C2×Q8×C71—24no—
159C2×Q8×C71—81no—
160C2×Q8×C71—81no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71—21no—
164C22×C4×C71—21no—
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71C2×D411yescharacteristic
167C22×C4×C71C2×D411yescharacteristic
168C22×C4×C71C2×D411yescharacteristic
169C22×C4×C71C2×D411yescharacteristic
170C22×C4×C71C2×D411yescharacteristic
171C22×C4×C71C2411yescharacteristic
172C4⋊C4×C71—24no—
173C4⋊C4×C71—21no—
174C4⋊C4×C71—21no—
175C4⋊C4×C71—21no—
176C4⋊C4×C71—21no—
177C4⋊C4×C71—21no—
178C4⋊C4×C71—21no—
179C4⋊C4×C71—81no—
180C4⋊C4×C71—81no—
181C4⋊C4×C71C22×C412yes—
182C4⋊C4×C71C22×C412yes—
183C42×C71—21no—
184C42×C71—21no—
185C42×C71—21no—
186C42×C71—21no—
187C42×C71—81no—
188C42×C71—81no—
189C42×C71C22×C412yes—
190C42×C71#4112yes—
191C23×C71C22×D411yescharacteristic
192C22×D71—22no—
193C22×D71—41no—
194C22×D71—41no—
195C22×D71—41no—
196C22×D71—41no—
197C22×D71—41no—
198C22×D71—41no—
199Q8×C71—84no—
200C2×C4×C71—21no—
201C2×C4×C71—21no—
202C2×C4×C71—21no—
203C2×C4×C71—21no—
204C2×C4×C71—22no—
205C2×C4×C71—22no—
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×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—41no—
219C2×C4×C71—41no—
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×C71C4×D412yes—
229C2×C4×C71C4×D412yes—
230C2×C4×C71C23×C411yescharacteristic
231C2×C4×C71C2×#4111yescharacteristic
232C2×C4×C71H(5,𝔽2)11yescharacteristic
233C2×C4×C712−1+411yescharacteristic
234C2×C4×C712−1+411yescharacteristic
235C2×C4×C712−1+411yescharacteristic
236C2×Dic71—22no—
237C2×Dic71—41no—
238C2×Dic71—41no—
239C2×Dic71—41no—
240C2×Dic71—41no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245D284—22no—
246D284—22no—
247D284—41no—
248D284—41no—
249C4×D71—84no—
250C71⋊C8—22no—
251C71⋊C8—22no—
252C71⋊C8—41no—
253C71⋊C8—41no—
254C22×C71C2×C4×D411yescharacteristic
255C22×C71#53611yescharacteristic
256C22×C71#53711yescharacteristic
257C22×C71#57611yescharacteristic
258C22×C71#57711yescharacteristic
259C22×C71#57711yescharacteristic
260C22×C71#57711yescharacteristic
261C2×D71—42no—
262C2×D71—42no—
263C2×D71—42no—
264C2×D71—81no—
265C2×D71—81no—
266C2×D71—81no—
267C2×D71—81no—
268C4×C71—22no—
269C4×C71—22no—
270C4×C71—42no—
271C4×C71—42no—
272C4×C71—41no—
273C4×C71—41no—
274C4×C71—41no—
275C4×C71—41no—
276C4×C71—81no—
277C4×C71—81no—
278C4×C71#51911yescharacteristic
279C4×C71#52011yescharacteristic
280C4×C71#54111yescharacteristic
281C4×C71#54211yescharacteristic
282Dic71—42no—
283Dic71—81no—
284Dic71—81no—
285#22123—711nomaximal, Sylow
286C2×C71#232311yescharacteristic
287C2×C71#294611yescharacteristic
288C2×C71#294911yescharacteristic
289C2×C71#324111yescharacteristic
290C2×C71#324411yescharacteristic
291C2×C71#325711yescharacteristic
292C2×C71#326311yescharacteristic
293D71—82no—
294C2×#476—711no—
295#2331D7111yescharacteristic, complemented
296#2319—711no—
297#1988—711no—
298#1949—711no—
299#1944—711no—
300#1941—711no—
301#1936—711no—
302#1934—711no—
303#1926—711no—
304#1925—711no—
305#1920—711no—
306#1919—711no—
307#1804—711no—
308#1768—711no—
309C71#2212311yescharacteristic, complemented, Sylow
310C2×C4⋊Q8C2×D7111yescharacteristic
311C2×C4×D4—711no—
312#476—718no—
313C2×#107—711no—
314C2×#106—711no—
315C2×#106—711no—
316C2×#106—711no—
317C2×C4⋊C8—711no—
318C2×C4⋊C8—711no—
319C2×#102—711no—
320C2×#102—711no—
321C2×#102—711no—
322C2×#102—711no—
323C2×#97—711no—
324C2×C4×C8—711no—
325#391—21no—
326#391—21no—
327#390—711no—
328#390—711no—
329#389—711no—
330#389—711no—
331#389C2×D7111yescharacteristic
332#389C2×D7111yescharacteristic
333#388—1421no—
334#388—1421no—
335#387—21no—
336#387—21no—
337#382C2×D7111yescharacteristic
338#381—1421no—
339#380—1421no—
340C4×C4⋊C4—711no—
341C4×C4⋊C4C2×D7111yescharacteristic
342C4×C4⋊C4C2×D7111yescharacteristic
343#376C2×D7111yescharacteristic
344#340—711no—
345#340—711no—
346#340—711no—
347#340—711no—
348#340—711no—
349#340—711no—
350#340—711no—
351#340—711no—
352#336—711no—
353#336—711no—
354C22×Q8—21no—
355C22×D4—711no—
356C23×C4—1421no—
357C22×C8—711no—
358C22×C8—711no—
359C22×C8—711no—
360C22×C8—711no—
361C4⋊Q8—22no—
362C4⋊Q8C4×D7114yes—
363C4×D4—714no—
364C4×D4—1422no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—711no—
369C2×C4⋊C4—711no—
370C2×C4⋊C4—711no—
371C2×C4⋊C4—1421no—
372C2×C4⋊C4—1421no—
373C2×C4⋊C4C22×D7111yescharacteristic
374C2×C4⋊C4C22×D7111yescharacteristic
375C2×C4⋊C4C22×D7111yescharacteristic
376C2×C4⋊C4C22×D7111yescharacteristic
377C2×C22⋊C4—1421no—
378C2×C22⋊C4—1421no—
379C2×C22⋊C4—1421no—
380C2×C22⋊C4—1421no—
381C2×C22⋊C4—1421no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—1421no—
385C2×C42C22×D7111yescharacteristic
386C2×C42C22×D7111yescharacteristic
387C2×C42C22×D7111yescharacteristic
388#107—2841no—
389#106—2841no—
390#106—714no—
391#106—714no—
392C4⋊C8—714no—
393C4⋊C8—1422no—
394#102—2841no—
395#102—2841no—
396#102—714no—
397#102—714no—
398#97—2841no—
399C4×C8—2841no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—1421no—
407#95—1421no—
408#95—1421no—
409#95C22⋊D7111yescharacteristic
410#95C22⋊D7111yescharacteristic
411#95C22⋊D7111yescharacteristic
412#95C22⋊D7111yescharacteristic
413C24—1421no—
414C2×Q8—24no—
415C2×Q8—81no—
416C2×Q8—81no—
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—5681no—
428C22×C4—5681no—
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—1421no—
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—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—5681no—
462C4⋊C4—5681no—
463C4⋊C4—712no—
464C4⋊C4—1421no—
465C4⋊C4—1421no—
466C4⋊C4—1421no—
467C4⋊C4—1421no—
468C4⋊C4—1421no—
469C4⋊C4C2×C4×D7112yes—
470C4⋊C4C2×C4×D7112yes—
471C22⋊C4—5681no—
472C22⋊C4—5681no—
473C22⋊C4—5681no—
474C22⋊C4—5681no—
475C22⋊C4—1424no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—81no—
481C42—81no—
482C42—5681no—
483C42C2×C4×D7112yes—
484C42#1176586512yes—
485C23—2841no—
486C23—2841no—
487C23—2841no—
488C23—2841no—
489C23—2841no—
490C23—2841no—
491C23—1422no—
492C23C22×C22⋊D7111yescharacteristic
493Q8—84no—
494D4—2841no—
495D4—2841no—
496D4—1422no—
497D4—1422no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—2841no—
527C2×C4—2841no—
528C2×C4—2841no—
529C2×C4—2841no—
530C2×C4—2841no—
531C2×C4—2841no—
532C2×C4—2841no—
533C2×C4—2841no—
534C2×C4—5684no—
535C2×C4—1422no—
536C2×C4C4×C22⋊D7112yes—
537C2×C4C4×C22⋊D7112yes—
538C2×C4C22×C4×D7111yescharacteristic
539C2×C4C2×#1176586511yescharacteristic
540C2×C4#1455617411yescharacteristic
541C2×C4#1455617511yescharacteristic
542C2×C4#1455617611yescharacteristic
543C2×C4#1455617611yescharacteristic
544C8—2841no—
545C8—2841no—
546C8—1422no—
547C8—1422no—
548C22—2842no—
549C22—2842no—
550C22—2842no—
551C22—5681no—
552C22—5681no—
553C22—5681no—
554C22—5681no—
555C22C2×C4×C22⋊D7111yescharacteristic
556C22#1798657311yescharacteristic
557C22#1798657711yescharacteristic
558C22#1798680211yescharacteristic
559C22#1798680711yescharacteristic
560C22#1798680711yescharacteristic
561C22#1798681011yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—2842no—
573C4—5681no—
574C4—5681no—
575C4#1798647511yescharacteristic
576C4#1798648111yescharacteristic
577C4#1798660711yescharacteristic
578C4#1798662111yescharacteristic
579C2—5682no—
580C2#2259101611yescharacteristic
581C2#2259717211yescharacteristic
582C2#2259719911yescharacteristic
583C2#2259959311yescharacteristic
584C2#2259963811yescharacteristic
585C2#2259975111yescharacteristic
586C2#2259980511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2331C2
socleC23×C71C22×D4
prime radicalC22×D284C23
cosocleC22×C4×C71C24
hypercenter#2331D71
hypocenterC71#22123
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ #376 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #22589293C2semidirect
4. #22589106C2semidirect
5. #22589080C2semidirect
6. #22589065C2semidirect
11. #22588925C2semidirect
12. #22588918C2semidirect
14. #22588197C2semidirect
15. C71×#2331C2semidirect
17. #17986286C4semidirect
295. #2331D71semidirect
309. C71#22123semidirect

Maximal subgroups

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

subgroupindex
C2×#179862862
#225909842
#225892932
#225891062
#225890802
#225890652
#225890362
#225890222
#225889672
#225889602
#225889252
#225889182
#225883472
#225881972
C71×#23312
#2212371