Finite Group Databasebeta

← #32052172

Subgroups — #32052172

Order 18176.

1038 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986409C211yesmaximal, characteristic, complemented
2C2×#17986284C211yesmaximal, characteristic, complemented
3#22593513C211yesmaximal, characteristic
4#22589540C212yesmaximal, complemented
5#22589533C212yesmaximal, complemented
6#22589298C211yesmaximal, characteristic, complemented
7#22589139C211yesmaximal, characteristic
8#22588539C212yesmaximal, complemented
9#22588533C212yesmaximal, complemented
10#22588170C211yesmaximal, characteristic, complemented
11C2×#552×C71C211yesmaximal, characteristic
12C2×#14556128C2211yescharacteristic
13#17986409—24no—
14#17986284—24no—
15C2×#14556061C2212yes—
16C2×#14556059C2212yes—
17C2×#14556059C2212yes—
18C2×C4⋊(C71⋊C8)C2211yescharacteristic
19C2×#14556052C2212yes—
20C2×#14556052C2212yes—
21C2×C22⋊(C71⋊C8)C2212yes—
22C2×C22⋊(C71⋊C8)C2212yes—
23C2×#14556039C2211yescharacteristic
24#17985911—21no—
25#17985911—21no—
26#17985902—21no—
27#17985864—21no—
28#17985859C2212yes—
29#17985859C2212yes—
30C4×C4⋊Dic71C2211yescharacteristic
31#17985761C2212yes—
32#17985761C2212yes—
33#17985761C2212yes—
34#552×C71—24no—
35#552×C71—24no—
36C2×#126×C71—21no—
37C2×#125×C71—21no—
38C2×#125×C71C2211yescharacteristic
39C2×#123×C71—21no—
40C2×C22⋊Q8×C71C2211yescharacteristic
41C2×C22⋊Q8×C71C2211yescharacteristic
42C2×C4×Q8×C71C2211yescharacteristic
43C2×C4×D4×C71C2211yescharacteristic
44C2×#117×C71—21no—
45C2×#117×C71C2211yescharacteristic
46C22×C4⋊C4×C71C2211yescharacteristic
47C22×Dic142—21no—
48C22×C71⋊C8C2312yes—
49C22×C71⋊C8C2312yes—
50#14556128—22no—
51#14556128—22no—
52C2×C4⋊Dic71—21no—
53C2×C4⋊Dic71C2312yes—
54C2×C4⋊Dic71C2312yes—
55C2×C22⋊Dic71—22no—
56C2×C22⋊Dic71—22no—
57C2×C4×Dic71—21no—
58#14556061—24no—
59#14556059—24no—
60#14556059—24no—
61C4⋊(C71⋊C8)—22no—
62#14556052—24no—
63#14556052—24no—
64C22⋊(C71⋊C8)—24no—
65C22⋊(C71⋊C8)—24no—
66#14556039—22no—
67C71⋊#95—21no—
68C71⋊#95—22no—
69C71⋊#95—22no—
70C22×Q8×C71C2311yescharacteristic
71C22×D4×C71D411yescharacteristic
72C23×C4×C71—21no—
73C23×C4×C71C2311yescharacteristic
74C23×C4×C71C2311yescharacteristic
75#126×C71—28no—
76#125×C71—28no—
77#125×C71—22no—
78#125×C71D414yes—
79#123×C71—28no—
80C22⋊Q8×C71—22no—
81C22⋊Q8×C71—22no—
82C22⋊Q8×C71—22no—
83C22⋊Q8×C71—22no—
84C4×Q8×C71—22no—
85C4×Q8×C71D414yes—
86C4×D4×C71—22no—
87C4×D4×C71—22no—
88#117×C71—28no—
89#117×C71—22no—
90#117×C71—22no—
91C2×C4⋊C4×C71—21no—
92C2×C4⋊C4×C71—21no—
93C2×C4⋊C4×C71—21no—
94C2×C4⋊C4×C71—21no—
95C2×C4⋊C4×C71—21no—
96C2×C4⋊C4×C71—21no—
97C2×C4⋊C4×C71—22no—
98C2×C4⋊C4×C71—22no—
99C2×C4⋊C4×C71D411yescharacteristic
100C2×C4⋊C4×C71C2311yescharacteristic
101C2×C4⋊C4×C71C2311yescharacteristic
102C2×C4⋊C4×C71C2311yescharacteristic
103C2×C22⋊C4×C71—21no—
104C2×C22⋊C4×C71—21no—
105C2×C22⋊C4×C71—21no—
106C2×C22⋊C4×C71D411yescharacteristic
107C2×C22⋊C4×C71D411yescharacteristic
108C2×C42×C71—21no—
109C2×C42×C71—21no—
110C2×C42×C71C2311yescharacteristic
111C2×Dic142—42no—
112C2×Dic142—82no—
113C22×Dic71—22no—
114C22×Dic71—22no—
115C2×C71⋊C8—22no—
116C2×C71⋊C8—22no—
117C2×C71⋊C8—22no—
118C2×C71⋊C8—22no—
119C2×C71⋊C8—22no—
120C2×C71⋊C8#4114yes—
121C4⋊Dic71—22no—
122C4⋊Dic71—22no—
123C4⋊Dic71—22no—
124C4⋊Dic71—22no—
125C4⋊Dic71—42no—
126C22⋊Dic71—82no—
127C22⋊Dic71—82no—
128C4×Dic71—42no—
129C24×C71C2×D411yescharacteristic
130C24×C71C2×D411yescharacteristic
131C2×Q8×C71—21no—
132C2×Q8×C71—22no—
133C2×Q8×C71—22no—
134C2×Q8×C71C2×D412yes—
135C2×D4×C71—22no—
136C2×D4×C71—22no—
137C2×D4×C71—41no—
138C22×C4×C71—24no—
139C22×C4×C71—21no—
140C22×C4×C71—21no—
141C22×C4×C71—21no—
142C22×C4×C71—21no—
143C22×C4×C71—21no—
144C22×C4×C71—21no—
145C22×C4×C71—21no—
146C22×C4×C71—21no—
147C22×C4×C71—22no—
148C22×C4×C71—22no—
149C22×C4×C71—22no—
150C22×C4×C71—22no—
151C22×C4×C71—42no—
152C22×C4×C71—42no—
153C22×C4×C71C2×D411yescharacteristic
154C22×C4×C71C2×D411yescharacteristic
155C22×C4×C71C2×D411yescharacteristic
156C22×C4×C71C2×D411yescharacteristic
157C22×C4×C71C2411yescharacteristic
158C4⋊C4×C71—24no—
159C4⋊C4×C71—24no—
160C4⋊C4×C71—24no—
161C4⋊C4×C71—24no—
162C4⋊C4×C71—24no—
163C4⋊C4×C71—24no—
164C4⋊C4×C71—22no—
165C4⋊C4×C71—21no—
166C4⋊C4×C71—21no—
167C4⋊C4×C71—21no—
168C4⋊C4×C71C2×D412yes—
169C4⋊C4×C71C2×D412yes—
170C4⋊C4×C71C2×D412yes—
171C22⋊C4×C71—24no—
172C22⋊C4×C71—24no—
173C22⋊C4×C71—24no—
174C22⋊C4×C71—22no—
175C22⋊C4×C71—22no—
176C42×C71—24no—
177C42×C71—24no—
178C42×C71C2×D412yes—
179C42×C71C2×D412yes—
180C23×C71—21no—
181C23×C71—22no—
182C23×C71—21no—
183C23×C71—22no—
184C23×C71—21no—
185C23×C71—22no—
186C23×C71—21no—
187C23×C71—21no—
188C23×C71—22no—
189C23×C71#13712yes—
190C23×C71C22×D411yescharacteristic
191Q8×C71—22no—
192Q8×C71—22no—
193Q8×C71—42no—
194D4×C71—44no—
195C2×C4×C71—21no—
196C2×C4×C71—21no—
197C2×C4×C71—22no—
198C2×C4×C71—21no—
199C2×C4×C71—22no—
200C2×C4×C71—22no—
201C2×C4×C71—22no—
202C2×C4×C71—21no—
203C2×C4×C71—22no—
204C2×C4×C71—22no—
205C2×C4×C71—22no—
206C2×C4×C71—48no—
207C2×C4×C71—42no—
208C2×C4×C71—41no—
209C2×C4×C71—41no—
210C2×C4×C71—41no—
211C2×C4×C71—41no—
212C2×C4×C71—42no—
213C2×C4×C71—41no—
214C2×C4×C71—41no—
215C2×C4×C71—41no—
216C2×C4×C71—41no—
217C2×C4×C71—42no—
218C2×C4×C71—42no—
219C2×C4×C71#12012yes—
220C2×C4×C71#12012yes—
221C2×C4×C71C22×D411yescharacteristic
222C2×C4×C71C22×D411yescharacteristic
223C2×C4×C71C2×#4112yes—
224C2×C4×C712−1+412yes—
225C2×Dic71—42no—
226C2×Dic71—42no—
227C2×Dic71—42no—
228C2×Dic71—42no—
229C2×Dic71—42no—
230C2×Dic71—42no—
231Dic142—84no—
232C71⋊C8—24no—
233C71⋊C8—42no—
234C22×C71—24no—
235C22×C71—22no—
236C22×C71—42no—
237C22×C71—41no—
238C22×C71—41no—
239C22×C71—42no—
240C22×C71—42no—
241C22×C71—41no—
242C22×C71—41no—
243C22×C71—41no—
244C22×C71C2×#12011yescharacteristic
245C22×C71#54712yes—
246C22×C71C2×#13711yescharacteristic
247C22×C71#57511yescharacteristic
248C22×C71#57712yes—
249C4×C71—22no—
250C4×C71—22no—
251C4×C71—41no—
252C4×C71—41no—
253C4×C71—42no—
254C4×C71—42no—
255C4×C71—42no—
256C4×C71—42no—
257C4×C71#53211yescharacteristic
258C4×C71#53611yescharacteristic
259C4×C71#54812yes—
260Dic71—82no—
261Dic71—82no—
262#25715—711nomaximal, Sylow
263C2×C71—42no—
264C2×C71—81no—
265C2×C71#243211yescharacteristic
266C2×C71#300711yescharacteristic
267C2×C71#301011yescharacteristic
268C2×C71#334412yes—
269C2×C71#334812yes—
270C2×#552D7111yescharacteristic
271C2×#501—711no—
272C2×#475—711no—
273#2563—711no—
274#2036—712no—
275#2035—712no—
276#1989—711no—
277#1955—711no—
278#1854—712no—
279#1853—712no—
280#1763—711no—
281C71#2571511yescharacteristic, complemented, Sylow
282#552—24no—
283#552—24no—
284C2×C4⋊Q8—711no—
285C2×#126—21no—
286C2×#125—21no—
287C2×#125C2×D7111yescharacteristic
288C2×#123—21no—
289C2×C22⋊Q8C2×D7111yescharacteristic
290C2×C22⋊Q8C2×D7111yescharacteristic
291C2×C4×Q8C2×D7111yescharacteristic
292C2×C4×D4C2×D7111yescharacteristic
293C2×#117—21no—
294C2×#117C2×D7111yescharacteristic
295C22×C4⋊C4C2×D7111yescharacteristic
296#501—1424no—
297#475—1424no—
298C2×#107—712no—
299C2×#106—712no—
300C2×#106—712no—
301C2×C4⋊C8—711no—
302C2×#103—712no—
303C2×#103—712no—
304C2×C22⋊C8—712no—
305C2×C22⋊C8—712no—
306C2×#97—711no—
307#393—1421no—
308#393—1421no—
309#390—1421no—
310#382—1421no—
311#380—712no—
312#380—712no—
313C4×C4⋊C4—711no—
314#340—712no—
315#340—712no—
316#340—712no—
317C22×Q8—1421no—
318C22×Q8C22×D7111yescharacteristic
319C22×D4C22⋊D7111yescharacteristic
320C23×C4—21no—
321C23×C4C22×D7111yescharacteristic
322C23×C4C22×D7111yescharacteristic
323C22×C8—712no—
324C22×C8—712no—
325C4⋊Q8—1422no—
326C4⋊Q8—1422no—
327#126—28no—
328#125—28no—
329#125—22no—
330#125C22⋊D7114yes—
331#123—28no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C4×Q8—22no—
337C4×Q8C22⋊D7114yes—
338C4×D4—22no—
339C4×D4—22no—
340#117—28no—
341#117—22no—
342#117—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—712no—
352C2×C4⋊C4—712no—
353C2×C4⋊C4—1421no—
354C2×C4⋊C4C22⋊D7111yescharacteristic
355C2×C4⋊C4C22×D7111yescharacteristic
356C2×C4⋊C4C22×D7111yescharacteristic
357C2×C4⋊C4C22×D7111yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—1422no—
362C2×C22⋊C4—1422no—
363C2×C22⋊C4C22⋊D7111yescharacteristic
364C2×C22⋊C4C22⋊D7111yescharacteristic
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—1421no—
368C2×C42C22×D7111yescharacteristic
369#107—1424no—
370#106—1424no—
371#106—1424no—
372C4⋊C8—1422no—
373#103—1424no—
374#103—1424no—
375C22⋊C8—1424no—
376C22⋊C8—1424no—
377#97—1422no—
378#95—1421no—
379#95—1422no—
380#95—1422no—
381C24C2×C22⋊D7111yescharacteristic
382C24C2×C22⋊D7111yescharacteristic
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—2842no—
387C2×Q8—5682no—
388C2×Q8C2×C22⋊D7112yes—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—41no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—1422no—
408C22×C4—1422no—
409C22×C4C2×C22⋊D7111yescharacteristic
410C22×C4C2×C22⋊D7111yescharacteristic
411C22×C4C2×C22⋊D7111yescharacteristic
412C22×C4C2×C22⋊D7111yescharacteristic
413C22×C4C23×D7111yescharacteristic
414C2×C8—714no—
415C2×C8—1422no—
416C2×C8—1422no—
417C2×C8—1422no—
418C2×C8—1422no—
419C2×C8—1422no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—2842no—
431C4⋊C4—1422no—
432C4⋊C4—1422no—
433C4⋊C4—1422no—
434C4⋊C4—1422no—
435C4⋊C4C2×C22⋊D7112yes—
436C4⋊C4C2×C22⋊D7112yes—
437C4⋊C4C2×C22⋊D7112yes—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—5682no—
444C22⋊C4—5682no—
445C42—24no—
446C42—24no—
447C42—2842no—
448C42C2×C22⋊D7112yes—
449C42C2×C22⋊D7112yes—
450C23—22no—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—22no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—22no—
459C23#1455615812yes—
460C23C22×C22⋊D7111yescharacteristic
461Q8—22no—
462Q8—22no—
463Q8—42no—
464Q8—5684no—
465D4—44no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—48no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—2842no—
491C2×C4—2842no—
492C2×C4—2842no—
493C2×C4—2842no—
494C2×C4—2842no—
495C2×C4—2842no—
496C2×C4#1455609712yes—
497C2×C4#1455609712yes—
498C2×C4C22×C22⋊D7111yescharacteristic
499C2×C4C22×C22⋊D7111yescharacteristic
500C2×C4#41×D7112yes—
501C2×C4#1455617512yes—
502C8—2842no—
503C8—1424no—
504C22—24no—
505C22—22no—
506C22—42no—
507C22—41no—
508C22—41no—
509C22—42no—
510C22—42no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22C2×#1455609711yescharacteristic
515C22#1798664812yes—
516C22C2×#1455615811yescharacteristic
517C22#1798679711yescharacteristic
518C22#1798680712yes—
519C4—22no—
520C4—22no—
521C4—41no—
522C4—41no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—5682no—
528C4—5682no—
529C4#1798654511yescharacteristic
530C4#1798657011yescharacteristic
531C4#1798665612yes—
532C2—42no—
533C2—81no—
534C2#2259196411yescharacteristic
535C2#2259760011yescharacteristic
536C2#2259762611yescharacteristic
537C2#2260079512yes—
538C2#2260084912yes—
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×#552×C71C2
socleC23×C71C22×D4
prime radicalC22×D4×C71D4
hypercenterC2×#552D71
hypocenterC71#25715
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#17986409C2semidirect
2. C2×#17986284C2semidirect
4. #22589540C2semidirect
5. #22589533C2semidirect
6. #22589298C2semidirect
8. #22588539C2semidirect
9. #22588533C2semidirect
10. #22588170C2semidirect
281. C71#25715semidirect

Maximal subgroups

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

subgroupindex
C2×#179864092
C2×#179862842
#225935132
#225895402
#225895332
#225892982
#225891392
#225885392
#225885332
#225881702
C2×#552×C712
#2571571