Finite Group Databasebeta

← #32033567

Subgroups — #32033567

Order 18176.

1150 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986296C212yesmaximal, complemented
2C4×#14556102C211yesmaximal, characteristic
3#22589670C211yesmaximal, characteristic, complemented
4#22589665C211yesmaximal, characteristic, complemented
5#22589473C211yesmaximal, characteristic, complemented
6#22589473C211yesmaximal, characteristic, complemented
7#22589298C212yesmaximal, complemented
8#22588662C212yesmaximal
9#22588539C212yesmaximal, complemented
10#22588320C211yesmaximal, characteristic, complemented
11C2×#533×C71C211yesmaximal, characteristic, complemented
12C2×#14556102C2211yescharacteristic
13C2×C4×D284C2211yescharacteristic
14C2×C4×C22⋊D71—21no—
15#17986296—28no—
16C2×#14556061C2212yes—
17C2×#14556059C2212yes—
18C2×C4⋊(C71⋊C8)C2212yes—
19C2×#14556048C2212yes—
20C2×#14556048C2212yes—
21C2×C22⋊(C71⋊C8)C2212yes—
22#17985901C2211yescharacteristic
23#17985897C2211yescharacteristic
24#17985890—21no—
25#17985878—21no—
26#17985859C2211yescharacteristic
27C4×C4⋊Dic71C2211yescharacteristic
28C4×(C2×D71)⋊C4—21no—
29#17985761C2212yes—
30#17985761C2212yes—
31#17985761C2212yes—
32#17985761C2212yes—
33#17985754C2211yescharacteristic
34#17985754C2211yescharacteristic
35#533×C71—28no—
36C2×C4⋊Q8×C71C2211yescharacteristic
37C2×C4⋊Q8×C71C2211yescharacteristic
38C2×#125×C71C2212yes—
39C2×C22⋊Q8×C71—21no—
40C2×C22⋊Q8×C71—21no—
41C2×C4×Q8×C71—21no—
42C2×C4×Q8×C71—21no—
43C2×#117×C71C2212yes—
44C22×C42×C71C2211yescharacteristic
45C22×D284C2311yescharacteristic
46C22×C22⋊D71—21no—
47C22×C4×D71—21no—
48C22×C71⋊C8C2312yes—
49C22×C71⋊C8C2312yes—
50#14556102—22no—
51#14556102Q814yescomplemented
52C4×D284—24no—
53C4×C22⋊D71—44no—
54C2×C4⋊Dic71C2311yescharacteristic
55C2×C4⋊Dic71C2311yescharacteristic
56C2×C4⋊Dic71C2311yescharacteristic
57C2×Dic71⋊C4—21no—
58C2×C22⋊Dic71—21no—
59C2×(C2×D71)⋊C4—21no—
60C2×(C2×D71)⋊C4—21no—
61C2×(C2×D71)⋊C4—21no—
62C2×(C2×D71)⋊C4—21no—
63C2×C4×Dic71—21no—
64#14556061—24no—
65#14556059—24no—
66C4⋊(C71⋊C8)—24no—
67#14556048—24no—
68#14556048—24no—
69C22⋊(C71⋊C8)—24no—
70C71⋊#95—21no—
71C71⋊#95—21no—
72C71⋊#95—21no—
73C22×Q8×C71—21no—
74C22×Q8×C71—21no—
75C23×C4×C71D411yescharacteristic
76C23×C4×C71D411yescharacteristic
77C23×C4×C71C2311yescharacteristic
78C4⋊Q8×C71—24no—
79C4⋊Q8×C71—24no—
80#125×C71—24no—
81#125×C71D418yescomplemented
82C22⋊Q8×C71—28no—
83C22⋊Q8×C71—28no—
84C4×Q8×C71—28no—
85C4×Q8×C71—28no—
86#117×C71—24no—
87#117×C71—24no—
88C2×C4⋊C4×C71—22no—
89C2×C4⋊C4×C71—22no—
90C2×C4⋊C4×C71—21no—
91C2×C4⋊C4×C71—21no—
92C2×C4⋊C4×C71C2312yes—
93C2×C4⋊C4×C71C2312yes—
94C2×C22⋊C4×C71—22no—
95C2×C42×C71—22no—
96C2×C42×C71—24no—
97C2×C42×C71D411yescharacteristic
98C2×C42×C71D411yescharacteristic
99C2×C42×C71C2311yescharacteristic
100C2×C42×C71C2311yescharacteristic
101C23×D71—21no—
102C2×D284—22no—
103C2×D284—22no—
104C2×D284—21no—
105C2×D284C2×Q812yes—
106C2×C22⋊D71—24no—
107C2×C22⋊D71—42no—
108C2×C22⋊D71—42no—
109C22×Dic71—21no—
110C22×Dic71—21no—
111C22×Dic71—21no—
112C2×C4×D71—42no—
113C2×C4×D71—42no—
114C2×C4×D71—42no—
115C2×C71⋊C8—22no—
116C2×C71⋊C8—22no—
117C2×C71⋊C8—22no—
118C2×C71⋊C8—22no—
119C2×C71⋊C8—22no—
120C2×C71⋊C8—22no—
121C4⋊Dic71—21no—
122C4⋊Dic71—22no—
123C4⋊Dic71—22no—
124C4⋊Dic71C2×Q812yes—
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—24no—
135C2×Q8×C71—42no—
136C2×Q8×C71—42no—
137C2×Q8×C71—42no—
138C2×Q8×C71—42no—
139C22×C4×C71—22no—
140C22×C4×C71—22no—
141C22×C4×C71—22no—
142C22×C4×C71—22no—
143C22×C4×C71—22no—
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—22no—
147C22×C4×C71—41no—
148C22×C4×C71—41no—
149C22×C4×C71C2×D411yescharacteristic
150C22×C4×C71C2×D411yescharacteristic
151C22×C4×C71C2×D411yescharacteristic
152C22×C4×C71C2×D411yescharacteristic
153C22×C4×C71C2×D411yescharacteristic
154C22×C4×C71C2×Q812yes—
155C22×C4×C71#4112yes—
156C22×C4×C71C2411yescharacteristic
157C4⋊C4×C71—24no—
158C4⋊C4×C71—24no—
159C4⋊C4×C71—28no—
160C4⋊C4×C71—28no—
161C4⋊C4×C71—22no—
162C4⋊C4×C71—22no—
163C4⋊C4×C71C2×D414yes—
164C4⋊C4×C71C2×D414yes—
165C22⋊C4×C71—28no—
166C42×C71—28no—
167C42×C71—22no—
168C42×C71—22no—
169C42×C71—22no—
170C42×C71C2×D414yes—
171C23×C71—21no—
172C23×C71—21no—
173C23×C71—21no—
174C23×C71—21no—
175C23×C71—21no—
176C23×C71—21no—
177C23×C71C22⋊Q812yes—
178C23×C71C22×D411yescharacteristic
179C22×D71—22no—
180C22×D71—41no—
181C22×D71—41no—
182C22×D71—41no—
183C22×D71—41no—
184C22×D71—41no—
185C22×D71—41no—
186Q8×C71—48no—
187Q8×C71—48no—
188C2×C4×C71—21no—
189C2×C4×C71—21no—
190C2×C4×C71—21no—
191C2×C4×C71—21no—
192C2×C4×C71—21no—
193C2×C4×C71—21no—
194C2×C4×C71—21no—
195C2×C4×C71—21no—
196C2×C4×C71—21no—
197C2×C4×C71—21no—
198C2×C4×C71—24no—
199C2×C4×C71—24no—
200C2×C4×C71—24no—
201C2×C4×C71—24no—
202C2×C4×C71—44no—
203C2×C4×C71—44no—
204C2×C4×C71—42no—
205C2×C4×C71—42no—
206C2×C4×C71—42no—
207C2×C4×C71—42no—
208C2×C4×C71#12012yes—
209C2×C4×C71#12012yes—
210C2×C4×C71C22⋊Q812yes—
211C2×C4×C71#13512yes—
212C2×C4×C71AGL(1,ℤ/8)12yes—
213C2×C4×C71C22×D412yes—
214C2×C4×C71C22×Q811yescharacteristic
215C2×C4×C71C2×#4111yescharacteristic
216C2×C4×C712−1+412yes—
217C22⋊D71—48no—
218C2×Dic71—22no—
219C2×Dic71—41no—
220C2×Dic71—41no—
221C2×Dic71—41no—
222C2×Dic71—41no—
223C2×Dic71—41no—
224C2×Dic71—41no—
225C2×Dic71—41no—
226C2×Dic71—41no—
227D284—24no—
228D284—42no—
229C4×D71—84no—
230C71⋊C8—42no—
231C71⋊C8—42no—
232C22×C71—22no—
233C22×C71—22no—
234C22×C71—22no—
235C22×C71—41no—
236C22×C71—41no—
237C22×C71—41no—
238C22×C71—41no—
239C22×C71C2×#12011yescharacteristic
240C22×C71C2×C22⋊Q811yescharacteristic
241C22×C71#53611yescharacteristic
242C22×C71C2×#13511yescharacteristic
243C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
244C22×C71#57511yescharacteristic
245C22×C71#57511yescharacteristic
246C2×D71—42no—
247C2×D71—42no—
248C2×D71—42no—
249C2×D71—81no—
250C2×D71—81no—
251C2×D71—81no—
252C2×D71—81no—
253C4×C71—22no—
254C4×C71—22no—
255C4×C71—22no—
256C4×C71—22no—
257C4×C71—22no—
258C4×C71—44no—
259C4×C71—44no—
260C4×C71#54812yes—
261C4×C71D4×Q812yes—
262Dic71—42no—
263Dic71—81no—
264Dic71—81no—
265#24304—711nomaximal, Sylow
266C2×C71—42no—
267C2×C71#240911yescharacteristic
268C2×C71#300211yescharacteristic
269C2×C71#300611yescharacteristic
270C2×C71#307611yescharacteristic
271C2×C71#307711yescharacteristic
272C2×C71#322811yescharacteristic
273C2×C71#324011yescharacteristic
274D71—82no—
275C2×#533D7111yescharacteristic, complemented
276C2×#478—712no—
277C4×#121—711no—
278#2058—711no—
279#2057—711no—
280#2023—711no—
281#2023—711no—
282#1989—712no—
283#1875—712no—
284#1854—712no—
285#1797—711no—
286C71#2430411yescharacteristic, complemented, Sylow
287#533—28no—
288C2×C4⋊Q8C2×D7111yescharacteristic
289C2×C4⋊Q8C2×D7111yescharacteristic
290C2×#125C2×D7112yes—
291C2×C22⋊Q8—21no—
292C2×C22⋊Q8—21no—
293C2×#121—711no—
294C2×C4×Q8—21no—
295C2×C4×Q8—21no—
296C2×C4×D4—711no—
297C2×C4×D4—1421no—
298C2×#117C2×D7112yes—
299C22×C42C2×D7111yescharacteristic
300#478—1428no—
301C2×#107—712no—
302C2×#106—712no—
303C2×C4⋊C8—712no—
304C2×#102—712no—
305C2×#102—712no—
306C2×C22⋊C8—712no—
307#390—711no—
308#389—711no—
309#388—1421no—
310#386—1421no—
311#380—711no—
312C4×C4⋊C4—711no—
313C4×C22⋊C4—1421no—
314#340—712no—
315#340—712no—
316#340—712no—
317#340—712no—
318#336—711no—
319#336—711no—
320C22×Q8—21no—
321C22×Q8—21no—
322C22×D4—711no—
323C22×D4—1421no—
324C23×C4—1421no—
325C23×C4C22⋊D7111yescharacteristic
326C23×C4C22⋊D7111yescharacteristic
327C23×C4C22×D7111yescharacteristic
328C22×C8—712no—
329C22×C8—712no—
330C4⋊Q8—24no—
331C4⋊Q8—24no—
332#125—24no—
333#125C22⋊D7118yescomplemented
334C22⋊Q8—28no—
335C22⋊Q8—28no—
336#121—714no—
337#121—1422no—
338C4×Q8—28no—
339C4×Q8—28no—
340C4×D4—2844no—
341C4×D4—1424no—
342#117—24no—
343#117—24no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—711no—
349C2×C4⋊C4—711no—
350C2×C4⋊C4—711no—
351C2×C4⋊C4—1421no—
352C2×C4⋊C4C22×D7112yes—
353C2×C4⋊C4C22×D7112yes—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—1421no—
356C2×C22⋊C4—1421no—
357C2×C22⋊C4—1421no—
358C2×C22⋊C4—1421no—
359C2×C22⋊C4—1421no—
360C2×C42—22no—
361C2×C42—24no—
362C2×C42—1421no—
363C2×C42C22⋊D7111yescharacteristic
364C2×C42C22⋊D7111yescharacteristic
365C2×C42C22×D7111yescharacteristic
366C2×C42C22×D7111yescharacteristic
367#107—1424no—
368#106—1424no—
369C4⋊C8—1424no—
370#102—1424no—
371#102—1424no—
372C22⋊C8—1424no—
373#95—1421no—
374#95—1421no—
375#95—1421no—
376C24—1421no—
377C24C2×C22⋊D7111yescharacteristic
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×D4—2842no—
385C2×D4—2842no—
386C2×D4—712no—
387C2×D4—1424no—
388C2×D4—1422no—
389C2×D4—1421no—
390C2×D4—1422no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—2842no—
402C22×C4—2842no—
403C22×C4—2842no—
404C22×C4—1421no—
405C22×C4—1421no—
406C22×C4—1421no—
407C22×C4C2×C22⋊D7111yescharacteristic
408C22×C4C2×C22⋊D7111yescharacteristic
409C22×C4C2×C22⋊D7111yescharacteristic
410C22×C4C2×C22⋊D7111yescharacteristic
411C22×C4C2×C22⋊D7111yescharacteristic
412C22×C4Q8×D7112yes—
413C22×C4#1176586612yes—
414C22×C4C23×D7111yescharacteristic
415C2×C8—1422no—
416C2×C8—1422no—
417C2×C8—1422no—
418C2×C8—1422no—
419C2×C8—1422no—
420C2×C8—1422no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—2842no—
428C4⋊C4—712no—
429C4⋊C4—1422no—
430C4⋊C4—1422no—
431C4⋊C4—1421no—
432C4⋊C4C2×C22⋊D7114yes—
433C4⋊C4C2×C22⋊D7114yes—
434C22⋊C4—28no—
435C22⋊C4—2842no—
436C22⋊C4—2842no—
437C22⋊C4—2842no—
438C22⋊C4—2842no—
439C22⋊C4—1424no—
440C42—28no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—2842no—
445C42C2×C22⋊D7114yes—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—2841no—
453C23—2841no—
454C23—2841no—
455C23—2841no—
456C23—2841no—
457C23—2841no—
458C23—1422no—
459C23#1455610412yes—
460C23C22×C22⋊D7111yescharacteristic
461Q8—48no—
462Q8—48no—
463D4—2848no—
464D4—2842no—
465D4—1424no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—2841no—
487C2×C4—2841no—
488C2×C4—2841no—
489C2×C4—2841no—
490C2×C4—2841no—
491C2×C4—2841no—
492C2×C4—2841no—
493C2×C4—2841no—
494C2×C4—5684no—
495C2×C4—1422no—
496C2×C4#1455609712yes—
497C2×C4#1455609712yes—
498C2×C4#1455610412yes—
499C2×C4#1455614812yes—
500C2×C4#1455615312yes—
501C2×C4C22×C22⋊D7112yes—
502C2×C4C2×Q8×D7111yescharacteristic
503C2×C4C2×#1176586611yescharacteristic
504C2×C4#1455617612yes—
505C8—2842no—
506C8—2842no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—2842no—
515C22—2842no—
516C22—2842no—
517C22—5681no—
518C22—5681no—
519C22—5681no—
520C22—5681no—
521C22C2×#1455609711yescharacteristic
522C22C2×#1455610411yescharacteristic
523C22#1798657311yescharacteristic
524C22C2×#1455614811yescharacteristic
525C22C2×#1455615311yescharacteristic
526C22#1798679711yescharacteristic
527C22#1798679711yescharacteristic
528C4—22no—
529C4—22no—
530C4—22no—
531C4—22no—
532C4—22no—
533C4—44no—
534C4—44no—
535C4—2842no—
536C4—5681no—
537C4—5681no—
538C4#1798666012yes—
539C4Q8×C22⋊D7112yes—
540C2—42no—
541C2—5682no—
542C2#2259169911yescharacteristic
543C2#2259755811yescharacteristic
544C2#2259759211yescharacteristic
545C2#2259812211yescharacteristic
546C2#2259813311yescharacteristic
547C2#2259947611yescharacteristic
548C2#2259958011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×#533×C71C2
socleC23×C71C22×D4
prime radicalC2×#14556102C22
hypercenterC2×#533D71
hypocenterC71#24304
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×#17986296C2semidirect
3. #22589670C2semidirect
4. #22589665C2semidirect
5. #22589473C2semidirect
6. #22589473C2semidirect
7. #22589298C2semidirect
9. #22588539C2semidirect
10. #22588320C2semidirect
11. C2×#533×C71C2semidirect
51. #14556102Q8semidirect
81. #125×C71D4semidirect
275. C2×#533D71semidirect
286. C71#24304semidirect
333. #125C22⋊D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179862962
C4×#145561022
#225896702
#225896652
#225894732
#225892982
#225886622
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