Finite Group Databasebeta

← #32031133

Subgroups — #32031133

Order 18176.

1118 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986332C211yesmaximal, characteristic, complemented
2C22×#14556061C211yesmaximal, characteristic, complemented
3#22592657C211yesmaximal, characteristic
4#22589665C212yesmaximal, complemented
5#22589540C212yesmaximal, complemented
6#22588764C211yesmaximal, characteristic, complemented
7#22588633C211yesmaximal, characteristic
8#22588594C212yesmaximal, complemented
9#22588528C212yesmaximal, complemented
10#22588430C211yesmaximal, characteristic, complemented
11C2×#554×C71C211yesmaximal, characteristic
12C23×C71⋊C8C2211yescharacteristic
13C22×C4⋊Dic71C2211yescharacteristic
14#17986332—24no—
15C2×#14556061—24no—
16C2×#14556061C2212yes—
17C2×#14556061C2212yes—
18C2×#14556061C2212yes—
19C2×#14556052C2212yes—
20C2×#14556052C2212yes—
21C2×C22⋊(C71⋊C8)C2211yescharacteristic
22C2×C22⋊(C71⋊C8)C2212yes—
23#17985938—21no—
24#17985935—21no—
25#17985897C2212yes—
26#17985879—21no—
27#17985871—21no—
28#17985859C2211yescharacteristic
29#17985859C2212yes—
30#17985761C2212yes—
31#17985761C2212yes—
32#17985761C2212yes—
33#17985754C2212yes—
34#554×C71—24no—
35#554×C71—24no—
36C2×C4⋊Q8×C71C2211yescharacteristic
37C2×#125×C71—21no—
38C2×C22⋊Q8×C71—21no—
39C2×C22⋊Q8×C71—21no—
40C2×C22⋊Q8×C71C2211yescharacteristic
41C2×C22⋊Q8×C71C2211yescharacteristic
42C2×C4×Q8×C71C2211yescharacteristic
43C2×C4×D4×C71—21no—
44C2×C4×D4×C71C2211yescharacteristic
45C22×C4⋊C4×C71C2211yescharacteristic
46C22×C4⋊C4×C71C2211yescharacteristic
47C23×Dic71—21no—
48C22×C71⋊C8—24no—
49C22×C71⋊C8—22no—
50C22×C71⋊C8C2312yes—
51C22×C71⋊C8C2312yes—
52C2×C4⋊Dic71—22no—
53C2×C4⋊Dic71—22no—
54C2×C4⋊Dic71—21no—
55C2×C4⋊Dic71C2312yes—
56C2×C4⋊Dic71C2312yes—
57C2×Dic71⋊C4—22no—
58C2×C22⋊Dic71—21no—
59C2×C22⋊Dic71—22no—
60#14556061—24no—
61#14556061—24no—
62#14556061—24no—
63#14556052—24no—
64#14556052—24no—
65C22⋊(C71⋊C8)—24no—
66C22⋊(C71⋊C8)—22no—
67C71⋊#95—22no—
68C71⋊#95—21no—
69C71⋊#95—22no—
70C22×Q8×C71—21no—
71C22×Q8×C71C2311yescharacteristic
72C22×D4×C71D411yescharacteristic
73C23×C4×C71—21no—
74C23×C4×C71—21no—
75C23×C4×C71C2311yescharacteristic
76C23×C4×C71C2311yescharacteristic
77C4⋊Q8×C71—22no—
78C4⋊Q8×C71—22no—
79#125×C71—28no—
80C22⋊Q8×C71—28no—
81C22⋊Q8×C71—28no—
82C22⋊Q8×C71—22no—
83C22⋊Q8×C71—22no—
84C22⋊Q8×C71—22no—
85C22⋊Q8×C71D414yes—
86C4×Q8×C71—22no—
87C4×Q8×C71—22no—
88C4×D4×C71—28no—
89C4×D4×C71—22no—
90C4×D4×C71—22no—
91C2×C4⋊C4×C71—21no—
92C2×C4⋊C4×C71—21no—
93C2×C4⋊C4×C71—22no—
94C2×C4⋊C4×C71—21no—
95C2×C4⋊C4×C71—21no—
96C2×C4⋊C4×C71—21no—
97C2×C4⋊C4×C71—21no—
98C2×C4⋊C4×C71—22no—
99C2×C4⋊C4×C71—22no—
100C2×C4⋊C4×C71D411yescharacteristic
101C2×C4⋊C4×C71D414yes—
102C2×C4⋊C4×C71C2311yescharacteristic
103C2×C4⋊C4×C71C2311yescharacteristic
104C2×C4⋊C4×C71C2311yescharacteristic
105C2×C22⋊C4×C71—21no—
106C2×C22⋊C4×C71—21no—
107C2×C22⋊C4×C71D411yescharacteristic
108C2×C22⋊C4×C71D411yescharacteristic
109C2×C42×C71—21no—
110C2×C42×C71C2311yescharacteristic
111C22×Dic71—22no—
112C22×Dic71—22no—
113C22×Dic71—42no—
114C22×Dic71—82no—
115C2×C71⋊C8—28no—
116C2×C71⋊C8—22no—
117C2×C71⋊C8—22no—
118C2×C71⋊C8—22no—
119C2×C71⋊C8—22no—
120C2×C71⋊C8—22no—
121C2×C71⋊C8#4114yes—
122C4⋊Dic71—22no—
123C4⋊Dic71—22no—
124C4⋊Dic71—22no—
125C4⋊Dic71—22no—
126C4⋊Dic71—42no—
127Dic71⋊C4—82no—
128C22⋊Dic71—42no—
129C22⋊Dic71—82no—
130C24×C71C2×D411yescharacteristic
131C24×C71C2×D411yescharacteristic
132C2×Q8×C71—24no—
133C2×Q8×C71—21no—
134C2×Q8×C71—22no—
135C2×Q8×C71—22no—
136C2×Q8×C71—82no—
137C2×Q8×C71C2×D412yes—
138C2×D4×C71—22no—
139C2×D4×C71—22no—
140C2×D4×C71Q1614yes—
141C22×C4×C71—22no—
142C22×C4×C71—22no—
143C22×C4×C71—21no—
144C22×C4×C71—21no—
145C22×C4×C71—21no—
146C22×C4×C71—21no—
147C22×C4×C71—24no—
148C22×C4×C71—24no—
149C22×C4×C71—21no—
150C22×C4×C71—21no—
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—82no—
154C22×C4×C71—82no—
155C22×C4×C71C2×D411yescharacteristic
156C22×C4×C71C2×D411yescharacteristic
157C22×C4×C71C2×D411yescharacteristic
158C22×C4×C71C2×D412yes—
159C22×C4×C71C2×D412yes—
160C22×C4×C71C2×D411yescharacteristic
161C22×C4×C71C2411yescharacteristic
162C4⋊C4×C71—24no—
163C4⋊C4×C71—24no—
164C4⋊C4×C71—24no—
165C4⋊C4×C71—24no—
166C4⋊C4×C71—24no—
167C4⋊C4×C71—24no—
168C4⋊C4×C71—22no—
169C4⋊C4×C71—21no—
170C4⋊C4×C71—21no—
171C4⋊C4×C71—21no—
172C4⋊C4×C71C2×D412yes—
173C4⋊C4×C71C2×D412yes—
174C4⋊C4×C71C2×D412yes—
175C22⋊C4×C71—24no—
176C22⋊C4×C71—24no—
177C22⋊C4×C71—22no—
178C22⋊C4×C71—22no—
179C42×C71—24no—
180C42×C71—21no—
181C42×C71—21no—
182C23×C71—22no—
183C23×C71—22no—
184C23×C71—21no—
185C23×C71—22no—
186C23×C71—21no—
187C23×C71—22no—
188C23×C71#12012yes—
189C23×C71C2×Q1612yes—
190C23×C71C2×Q1612yes—
191C23×C71#13712yes—
192C23×C71C22×D411yescharacteristic
193Q8×C71—24no—
194Q8×C71—41no—
195Q8×C71—41no—
196Q8×C71—84no—
197D4×C71—28no—
198C2×C4×C71—21no—
199C2×C4×C71—22no—
200C2×C4×C71—21no—
201C2×C4×C71—22no—
202C2×C4×C71—22no—
203C2×C4×C71—24no—
204C2×C4×C71—22no—
205C2×C4×C71—21no—
206C2×C4×C71—22no—
207C2×C4×C71—21no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—42no—
213C2×C4×C71—42no—
214C2×C4×C71—42no—
215C2×C4×C71—42no—
216C2×C4×C71—41no—
217C2×C4×C71—41no—
218C2×C4×C71—42no—
219C2×C4×C71—84no—
220C2×C4×C71—84no—
221C2×C4×C71#12012yes—
222C2×C4×C71C2×Q1612yes—
223C2×C4×C71C22×D411yescharacteristic
224C2×C4×C71C22×D411yescharacteristic
225C2×C4×C71C2×#4112yes—
226C2×C4×C712−1+412yes—
227C2×Dic71—42no—
228C2×Dic71—42no—
229C2×Dic71—42no—
230C2×Dic71—42no—
231C2×Dic71—42no—
232C2×Dic71—42no—
233C2×Dic71—84no—
234C71⋊C8—24no—
235C71⋊C8—42no—
236C22×C71—24no—
237C22×C71—24no—
238C22×C71—22no—
239C22×C71—22no—
240C22×C71—24no—
241C22×C71—22no—
242C22×C71—42no—
243C22×C71—41no—
244C22×C71—41no—
245C22×C71#45114yes—
246C22×C71C2×#12011yescharacteristic
247C22×C71#54712yes—
248C22×C71C22×Q1611yescharacteristic
249C22×C71C2×#13711yescharacteristic
250C22×C71#57612yes—
251C4×C71—22no—
252C4×C71—22no—
253C4×C71—41no—
254C4×C71—42no—
255C4×C71—41no—
256C4×C71—42no—
257C4×C71—42no—
258C4×C71—42no—
259C4×C71#52511yescharacteristic
260C4×C71#53611yescharacteristic
261C4×C71#54712yes—
262Dic71—82no—
263Dic71—82no—
264#24098—711nomaximal, Sylow
265C2×C71—24no—
266C2×C71—42no—
267C2×C71#240611yescharacteristic
268C2×C71C2×#45111yescharacteristic
269C2×C71#300811yescharacteristic
270C2×C71#333812yes—
271C2×C71#334612yes—
272C2×#554D7111yescharacteristic
273C2×#484—711no—
274C22×#107—711no—
275#2487—711no—
276#2057—712no—
277#2036—712no—
278#1892—711no—
279#1869—711no—
280#1863—712no—
281#1852—712no—
282#1824—711no—
283C71#2409811yescharacteristic, complemented, Sylow
284C23×C8—711no—
285#554—24no—
286#554—24no—
287C2×C4⋊Q8C2×D7111yescharacteristic
288C2×#125—21no—
289C2×C22⋊Q8—21no—
290C2×C22⋊Q8—21no—
291C2×C22⋊Q8C2×D7111yescharacteristic
292C2×C22⋊Q8C2×D7111yescharacteristic
293C2×C4×Q8C2×D7111yescharacteristic
294C2×C4×D4—21no—
295C2×C4×D4C2×D7111yescharacteristic
296C22×C4⋊C4—711no—
297C22×C4⋊C4C2×D7111yescharacteristic
298C22×C4⋊C4C2×D7111yescharacteristic
299#484—1424no—
300C2×#107—712no—
301C2×#107—712no—
302C2×#107—712no—
303C2×#107—1424no—
304C2×#103—712no—
305C2×#103—712no—
306C2×C22⋊C8—711no—
307C2×C22⋊C8—712no—
308#400—1421no—
309#399—1421no—
310#389—712no—
311#386—1421no—
312#385—1421no—
313#380—711no—
314#380—712no—
315#340—712no—
316#340—712no—
317#340—712no—
318#336—712no—
319C22×Q8—21no—
320C22×Q8C22×D7111yescharacteristic
321C22×D4C22⋊D7111yescharacteristic
322C23×C4—21no—
323C23×C4—21no—
324C23×C4—1421no—
325C23×C4C22×D7111yescharacteristic
326C23×C4C22×D7111yescharacteristic
327C22×C8—712no—
328C22×C8—712no—
329C22×C8—1424no—
330C22×C8—1422no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333#125—28no—
334C22⋊Q8—28no—
335C22⋊Q8—28no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8C22⋊D7114yes—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×D4—28no—
343C4×D4—22no—
344C4×D4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—712no—
355C2×C4⋊C4—712no—
356C2×C4⋊C4—1421no—
357C2×C4⋊C4—1422no—
358C2×C4⋊C4—1422no—
359C2×C4⋊C4—1422no—
360C2×C4⋊C4C22⋊D7111yescharacteristic
361C2×C4⋊C4C22⋊D7114yes—
362C2×C4⋊C4C22×D7111yescharacteristic
363C2×C4⋊C4C22×D7111yescharacteristic
364C2×C4⋊C4C22×D7111yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1421no—
368C2×C22⋊C4—1422no—
369C2×C22⋊C4C22⋊D7111yescharacteristic
370C2×C22⋊C4C22⋊D7111yescharacteristic
371C2×C42—21no—
372C2×C42C22×D7111yescharacteristic
373#107—1424no—
374#107—1424no—
375#107—1424no—
376#103—1424no—
377#103—1424no—
378C22⋊C8—1424no—
379C22⋊C8—1422no—
380#95—1422no—
381#95—1421no—
382#95—1422no—
383C24C2×C22⋊D7111yescharacteristic
384C24C2×C22⋊D7111yescharacteristic
385C2×Q8—24no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—82no—
390C2×Q8C2×C22⋊D7112yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4#1176585514yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—2842no—
409C22×C4—5682no—
410C22×C4—1422no—
411C22×C4—1422no—
412C22×C4C2×C22⋊D7111yescharacteristic
413C22×C4C2×C22⋊D7111yescharacteristic
414C22×C4C2×C22⋊D7112yes—
415C22×C4C2×C22⋊D7112yes—
416C22×C4C2×C22⋊D7111yescharacteristic
417C22×C4C2×C22⋊D7111yescharacteristic
418C22×C4C23×D7111yescharacteristic
419C2×C8—714no—
420C2×C8—1422no—
421C2×C8—1422no—
422C2×C8—1422no—
423C2×C8—1428no—
424C2×C8—1422no—
425C2×C8—1422no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—2842no—
437C4⋊C4—5682no—
438C4⋊C4—1422no—
439C4⋊C4—1422no—
440C4⋊C4—1422no—
441C4⋊C4—1422no—
442C4⋊C4C2×C22⋊D7112yes—
443C4⋊C4C2×C22⋊D7112yes—
444C4⋊C4C2×C22⋊D7112yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—2842no—
450C22⋊C4—5682no—
451C42—24no—
452C42—21no—
453C42—21no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23#1455609712yes—
461C23C2×#1176585512yes—
462C23C2×#1176585512yes—
463C23#1455615912yes—
464C23C22×C22⋊D7111yescharacteristic
465Q8—24no—
466Q8—41no—
467Q8—41no—
468Q8—84no—
469D4—28no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—84no—
492C2×C4—84no—
493C2×C4—2842no—
494C2×C4—2842no—
495C2×C4—2842no—
496C2×C4—2842no—
497C2×C4—2842no—
498C2×C4—2842no—
499C2×C4—5684no—
500C2×C4#1455609712yes—
501C2×C4C2×#1176585512yes—
502C2×C4C22×C22⋊D7111yescharacteristic
503C2×C4C22×C22⋊D7111yescharacteristic
504C2×C4#41×D7112yes—
505C2×C4#1455617512yes—
506C8—2842no—
507C8—1424no—
508C22—24no—
509C22—24no—
510C22—22no—
511C22—22no—
512C22—24no—
513C22—22no—
514C22—42no—
515C22—41no—
516C22—41no—
517C22#1798615514yes—
518C22C2×#1455609711yescharacteristic
519C22#1798664812yes—
520C22C22×#1176585511yescharacteristic
521C22C2×#1455615911yescharacteristic
522C22#1798680212yes—
523C4—22no—
524C4—22no—
525C4—41no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—5682no—
532C4—5682no—
533C4#1798650611yescharacteristic
534C4#1798657011yescharacteristic
535C4#1798664812yes—
536C2—24no—
537C2—42no—
538C2#2259167111yescharacteristic
539C2C2×#1798615511yescharacteristic
540C2#2259760811yescharacteristic
541C2#2260072812yes—
542C2#2260081912yes—
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×#554×C71C2
socleC23×C71C22×D4
prime radicalC22×D4×C71D4
hypercenterC2×#554D71
hypocenterC71#24098
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#17986332C2semidirect
2. C22×#14556061C2semidirect
4. #22589665C2semidirect
5. #22589540C2semidirect
6. #22588764C2semidirect
8. #22588594C2semidirect
9. #22588528C2semidirect
10. #22588430C2semidirect
283. C71#24098semidirect

Maximal subgroups

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

subgroupindex
C2×#179863322
C22×#145560612
#225926572
#225896652
#225895402
#225887642
#225886332
#225885942
#225885282
#225884302
C2×#554×C712
#2409871