Finite Group Databasebeta

← #32008668

Subgroups — #32008668

Order 18176.

1080 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986138C211yesmaximal, characteristic
2C4×#14556102C211yesmaximal, characteristic
3#22589224C212yesmaximal, complemented
4#22589033C212yesmaximal, complemented
5#22588978C212yesmaximal, complemented
6#22588677C212yesmaximal, complemented
7#22588657C211yesmaximal, characteristic
8#22588528C211yesmaximal, characteristic, complemented
9#22588275C211yesmaximal, characteristic, complemented
10#22588102C211yesmaximal, characteristic, complemented
11C2×#391×C71C211yesmaximal, characteristic, complemented
12C22×#11765852C2212yes—
13C2×#14556102C2211yescharacteristic
14C2×C4×D284C2211yescharacteristic
15C2×C4×C22⋊D71—21no—
16#17986138C418yescomplemented
17C2×#14556061C2212yes—
18C2×#14556052C2212yes—
19C2×#14556052C2212yes—
20C2×#14556052C2212yes—
21C2×#14556048C2212yes—
22C2×C22⋊(C71⋊C8)C2211yescharacteristic
23C2×C22⋊(C71⋊C8)C2211yescharacteristic
24C2×#14556039C2212yes—
25#17985901C2211yescharacteristic
26#17985897C2211yescharacteristic
27#17985890—21no—
28#17985878—21no—
29#17985859C2211yescharacteristic
30C4×C4⋊Dic71C2211yescharacteristic
31C4×(C2×D71)⋊C4—21no—
32#17985761C2212yes—
33#17985761C2212yes—
34#17985754C2212yes—
35C23×Q8×C71C2211yescharacteristic
36C22×C4⋊C4×C71C2211yescharacteristic
37C22×C42×C71C2211yescharacteristic
38#391×C71—21no—
39#391×C71—21no—
40#391×C71C2212yes—
41#391×C71C2212yes—
42C2×#95×C71—21no—
43C2×#95×C71—21no—
44C22×D284C2311yescharacteristic
45C22×C22⋊D71—21no—
46C22×C4×D71—21no—
47C2×#11765852—42no—
48C2×#11765852—42no—
49C2×#11765852C2×C418yescomplemented
50C22×C71⋊C8C2312yes—
51C22×C71⋊C8C2312yes—
52#14556102—22no—
53#14556102C2×C414yes—
54C4×D284—42no—
55C4×C22⋊D71—44no—
56C2×C4⋊Dic71C2311yescharacteristic
57C2×C4⋊Dic71C2311yescharacteristic
58C2×C4⋊Dic71C2311yescharacteristic
59C2×Dic71⋊C4—21no—
60C2×C22⋊Dic71—21no—
61C2×(C2×D71)⋊C4—21no—
62C2×(C2×D71)⋊C4—21no—
63C2×(C2×D71)⋊C4—21no—
64C2×(C2×D71)⋊C4—21no—
65C2×C4×Dic71—21no—
66#14556061—42no—
67#14556052—42no—
68#14556052—42no—
69#14556052C2×C418yescomplemented
70#14556048—42no—
71C22⋊(C71⋊C8)—22no—
72C22⋊(C71⋊C8)C2×C414yescomplemented
73#14556039—42no—
74C71⋊#95—21no—
75C71⋊#95—21no—
76C71⋊#95—21no—
77C22×Q8×C71—21no—
78C22×Q8×C71—22no—
79C22×Q8×C71—42no—
80C22×Q8×C71—42no—
81C22×Q8×C71C2×C414yescomplemented
82C22×Q8×C71C2312yes—
83C23×C4×C71—21no—
84C23×C4×C71—21no—
85C23×C4×C71D411yescharacteristic
86C23×C4×C71D411yescharacteristic
87C23×C4×C71C2311yescharacteristic
88C2×C4⋊C4×C71—21no—
89C2×C4⋊C4×C71—42no—
90C2×C4⋊C4×C71C2312yes—
91C2×C42×C71—42no—
92C2×C42×C71D411yescharacteristic
93C2×C42×C71D411yescharacteristic
94C2×C42×C71C2311yescharacteristic
95C2×C42×C71C2311yescharacteristic
96#95×C71—22no—
97#95×C71—22no—
98#95×C71—22no—
99#95×C71—22no—
100C23×D71—21no—
101C2×D284—22no—
102C2×D284—22no—
103C2×D284—21no—
104C2×D284C22×C412yes—
105C2×C22⋊D71—24no—
106C2×C22⋊D71—42no—
107C2×C22⋊D71—42no—
108C22×Dic71—21no—
109C22×Dic71—21no—
110C22×Dic71—21no—
111C2×C4×D71—42no—
112C2×C4×D71—42no—
113C2×C4×D71—42no—
114#11765852—48no—
115C2×C71⋊C8—22no—
116C2×C71⋊C8—22no—
117C2×C71⋊C8—22no—
118C2×C71⋊C8—22no—
119C2×C71⋊C8—22no—
120C2×C71⋊C8C22×C414yes—
121C4⋊Dic71—21no—
122C4⋊Dic71—22no—
123C4⋊Dic71—22no—
124C4⋊Dic71C22×C412yes—
125Dic71⋊C4—42no—
126C22⋊Dic71—42no—
127(C2×D71)⋊C4—24no—
128(C2×D71)⋊C4—42no—
129(C2×D71)⋊C4—42no—
130(C2×D71)⋊C4—42no—
131C4×Dic71—42no—
132C24×C71C2×D411yescharacteristic
133C2×Q8×C71—24no—
134C2×Q8×C71—22no—
135C2×Q8×C71—22no—
136C2×Q8×C71—22no—
137C2×Q8×C71—22no—
138C2×Q8×C71—22no—
139C2×Q8×C71—48no—
140C2×Q8×C71—42no—
141C2×Q8×C71—42no—
142C2×Q8×C71—48no—
143C2×Q8×C71C22×C414yes—
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—24no—
147C22×C4×C71—22no—
148C22×C4×C71—22no—
149C22×C4×C71—22no—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—41no—
153C22×C4×C71—41no—
154C22×C4×C71—41no—
155C22×C4×C71—42no—
156C22×C4×C71—42no—
157C22×C4×C71—42no—
158C22×C4×C71—42no—
159C22×C4×C71—42no—
160C22×C4×C71C22×C412yes—
161C22×C4×C71C2×D411yescharacteristic
162C22×C4×C71C2×D411yescharacteristic
163C22×C4×C71C2×D411yescharacteristic
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71#4112yes—
167C22×C4×C71C2411yescharacteristic
168C4⋊C4×C71—22no—
169C4⋊C4×C71—22no—
170C4⋊C4×C71—42no—
171C42×C71—22no—
172C42×C71—22no—
173C42×C71—41no—
174C42×C71—41no—
175C23×C71—21no—
176C23×C71—21no—
177C23×C71—21no—
178C23×C71—21no—
179C23×C71C4×D412yes—
180C23×C71AGL(1,ℤ/8)12yes—
181C23×C71#13712yes—
182C23×C71C22×D411yescharacteristic
183C22×D71—22no—
184C22×D71—41no—
185C22×D71—41no—
186C22×D71—41no—
187C22×D71—41no—
188C22×D71—41no—
189C22×D71—41no—
190Q8×C71—24no—
191Q8×C71—24no—
192Q8×C71—48no—
193Q8×C71—42no—
194Q8×C71—42no—
195C2×C4×C71—21no—
196C2×C4×C71—21no—
197C2×C4×C71—21no—
198C2×C4×C71—21no—
199C2×C4×C71—24no—
200C2×C4×C71—24no—
201C2×C4×C71—24no—
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—21no—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—44no—
212C2×C4×C71—44no—
213C2×C4×C71—48no—
214C2×C4×C71—42no—
215C2×C4×C71—42no—
216C2×C4×C71—42no—
217C2×C4×C71—42no—
218C2×C4×C71—42no—
219C2×C4×C71—84no—
220C2×C4×C71C4×D412yes—
221C2×C4×C71#13712yes—
222C2×C4×C71C23×C411yescharacteristic
223C2×C4×C71C2×#4111yescharacteristic
224C2×C4×C71H(5,𝔽2)12yes—
225C2×C4×C71H(5,𝔽2)12yes—
226C22⋊D71—48no—
227C2×Dic71—22no—
228C2×Dic71—41no—
229C2×Dic71—41no—
230C2×Dic71—41no—
231C2×Dic71—41no—
232C2×Dic71—41no—
233C2×Dic71—41no—
234C2×Dic71—41no—
235C2×Dic71—41no—
236D284—24no—
237D284—42no—
238C4×D71—84no—
239C71⋊C8—24no—
240C71⋊C8—42no—
241C22×C71—22no—
242C22×C71—22no—
243C22×C71—22no—
244C22×C71—22no—
245C22×C71—22no—
246C22×C71—22no—
247C22×C71#44014yes—
248C22×C71C2×C4×D411yescharacteristic
249C22×C71#53411yescharacteristic
250C22×C71#53511yescharacteristic
251C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
252C22×C71C2×#13711yescharacteristic
253C22×C71C2×#13711yescharacteristic
254C22×C71#57511yescharacteristic
255C2×D71—42no—
256C2×D71—42no—
257C2×D71—42no—
258C2×D71—81no—
259C2×D71—81no—
260C2×D71—81no—
261C2×D71—81no—
262C4×C71—22no—
263C4×C71—22no—
264C4×C71—44no—
265C4×C71—41no—
266C4×C71—41no—
267C4×C71—41no—
268C4×C71—82no—
269C4×C71#51812yes—
270C4×C71#53812yes—
271Dic71—42no—
272Dic71—81no—
273Dic71—81no—
274#22047—711nomaximal, Sylow
275C2×C71—24no—
276C2×C71#232111yescharacteristic
277C2×C71C2×#44011yescharacteristic
278C2×C71C4×#13711yescharacteristic
279C2×C71#319311yescharacteristic
280C2×C71#319611yescharacteristic
281C2×C71#321311yescharacteristic
282C2×C71#321811yescharacteristic
283D71—82no—
284C2×#448—711no—
285C4×#121—711no—
286C2×#391D7111yescharacteristic, complemented
287#1973—712no—
288#1935—712no—
289#1928—712no—
290#1877—712no—
291#1874—711no—
292#1852—711no—
293#1787—711no—
294#1749—711no—
295C71#2204711yescharacteristic, complemented, Sylow
296C23×Q8C2×D7111yescharacteristic
297C22×SD16—712no—
298C2×#121—711no—
299C2×C4×D4—711no—
300C2×C4×D4—1421no—
301C22×C4⋊C4C2×D7111yescharacteristic
302C22×C42C2×D7111yescharacteristic
303#448—718no—
304C2×#107—712no—
305C2×#103—712no—
306C2×#103—712no—
307C2×#103—712no—
308C2×#102—712no—
309C2×C22⋊C8—711no—
310C2×C22⋊C8—711no—
311C2×#97—712no—
312#391—21no—
313#391—21no—
314#391C2×D7112yes—
315#391C2×D7112yes—
316#390—711no—
317#389—711no—
318#388—1421no—
319#386—1421no—
320#380—711no—
321C4×C4⋊C4—711no—
322C4×C22⋊C4—1421no—
323C2×#95—21no—
324C2×#95—21no—
325#340—712no—
326#340—712no—
327#336—712no—
328C22×Q8—21no—
329C22×Q8—22no—
330C22×Q8—42no—
331C22×Q8—42no—
332C22×Q8C4×D7114yescomplemented
333C22×Q8C22×D7112yes—
334C22×D4—711no—
335C22×D4—1421no—
336C23×C4—21no—
337C23×C4—21no—
338C23×C4—1421no—
339C23×C4C22⋊D7111yescharacteristic
340C23×C4C22⋊D7111yescharacteristic
341C23×C4C22×D7111yescharacteristic
342C2×SD16—2842no—
343C2×SD16—2842no—
344C2×SD16—718no—
345C22×C8—712no—
346C22×C8—712no—
347#121—714no—
348#121—1422no—
349C4×D4—2842no—
350C4×D4—2844no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—711no—
354C2×C4⋊C4—711no—
355C2×C4⋊C4—711no—
356C2×C4⋊C4—1421no—
357C2×C4⋊C4C22×D7112yes—
358C2×C22⋊C4—1421no—
359C2×C22⋊C4—1421no—
360C2×C22⋊C4—1421no—
361C2×C22⋊C4—1421no—
362C2×C22⋊C4—1421no—
363C2×C42—42no—
364C2×C42—1421no—
365C2×C42C22⋊D7111yescharacteristic
366C2×C42C22⋊D7111yescharacteristic
367C2×C42C22×D7111yescharacteristic
368C2×C42C22×D7111yescharacteristic
369#107—2842no—
370#103—2842no—
371#103—2842no—
372#103—718no—
373#102—2842no—
374C22⋊C8—714no—
375C22⋊C8—1422no—
376#97—2842no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—1421no—
382#95—1421no—
383#95—1421no—
384C24—1421no—
385C24C2×C22⋊D7111yescharacteristic
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—48no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—48no—
396C2×Q8C2×C4×D7114yes—
397C2×D4—2842no—
398C2×D4—2842no—
399C2×D4—712no—
400C2×D4—1424no—
401C2×D4—1421no—
402C2×D4—1422no—
403C2×D4—1422no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—2842no—
421C22×C4—2842no—
422C22×C4—2842no—
423C22×C4—1421no—
424C22×C4—1421no—
425C22×C4—1421no—
426C22×C4C2×C4×D7112yes—
427C22×C4C2×C22⋊D7111yescharacteristic
428C22×C4C2×C22⋊D7111yescharacteristic
429C22×C4C2×C22⋊D7111yescharacteristic
430C22×C4C2×C22⋊D7111yescharacteristic
431C22×C4C2×C22⋊D7111yescharacteristic
432C22×C4#1176586512yes—
433C22×C4C23×D7111yescharacteristic
434SD16—2848no—
435C2×C8—714no—
436C2×C8—1422no—
437C2×C8—1422no—
438C2×C8—1422no—
439C2×C8—1422no—
440C2×C8—1422no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—2842no—
445C4⋊C4—712no—
446C4⋊C4—1422no—
447C4⋊C4—1422no—
448C4⋊C4—1421no—
449C22⋊C4—2842no—
450C22⋊C4—2842no—
451C22⋊C4—2842no—
452C22⋊C4—2842no—
453C22⋊C4—1424no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—2842no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—2841no—
464C23—2841no—
465C23—2841no—
466C23—2841no—
467C23—2841no—
468C23—2841no—
469C23—1422no—
470C23C4×C22⋊D7112yes—
471C23#1455615312yes—
472C23#1455615912yes—
473C23C22×C22⋊D7111yescharacteristic
474Q8—24no—
475Q8—24no—
476Q8—48no—
477Q8—42no—
478Q8—42no—
479D4—2848no—
480D4—2842no—
481D4—1424no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—44no—
499C2×C4—48no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—84no—
507C2×C4—2841no—
508C2×C4—2841no—
509C2×C4—2841no—
510C2×C4—2841no—
511C2×C4—2841no—
512C2×C4—2841no—
513C2×C4—2841no—
514C2×C4—2841no—
515C2×C4—5684no—
516C2×C4—1422no—
517C2×C4C4×C22⋊D7112yes—
518C2×C4#1455615912yes—
519C2×C4C22×C4×D7111yescharacteristic
520C2×C4C2×#1176586511yescharacteristic
521C2×C4#1455617312yes—
522C2×C4#1455617312yes—
523C8—2842no—
524C8—1424no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—2842no—
532C22—2842no—
533C22—2842no—
534C22—5681no—
535C22—5681no—
536C22—5681no—
537C22—5681no—
538C22#1798609714yes—
539C22C2×C4×C22⋊D7111yescharacteristic
540C22#1798655811yescharacteristic
541C22#1798656611yescharacteristic
542C22C2×#1455615311yescharacteristic
543C22C2×#1455615911yescharacteristic
544C22C2×#1455615911yescharacteristic
545C22#1798679711yescharacteristic
546C4—22no—
547C4—22no—
548C4—41no—
549C4—41no—
550C4—44no—
551C4—41no—
552C4—82no—
553C4—2842no—
554C4—5681no—
555C4—5681no—
556C4#1798647212yes—
557C4#1798658712yes—
558C2—24no—
559C2—5682no—
560C2#2259099811yescharacteristic
561C2C2×#1798609711yescharacteristic
562C2C4×#1455615911yescharacteristic
563C2#2259914211yescharacteristic
564C2#2259917011yescharacteristic
565C2#2259932711yescharacteristic
566C2#2259936711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#391×C71C2
socleC23×C71C22×D4
prime radicalC2×#14556102C22
cosocleC22×C4×C71C24
hypercenterC2×#391D71
hypocenterC71#22047
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. #22589224C2semidirect
4. #22589033C2semidirect
5. #22588978C2semidirect
6. #22588677C2semidirect
8. #22588528C2semidirect
9. #22588275C2semidirect
10. #22588102C2semidirect
11. C2×#391×C71C2semidirect
16. #17986138C4semidirect
49. C2×#11765852C2×C4semidirect
69. #14556052C2×C4semidirect
72. C22⋊(C71⋊C8)C2×C4semidirect
81. C22×Q8×C71C2×C4semidirect
286. C2×#391D71semidirect
295. C71#22047semidirect
332. C22×Q8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179861382
C4×#145561022
#225892242
#225890332
#225889782
#225886772
#225886572
#225885282
#225882752
#225881022
C2×#391×C712
#2204771