Finite Group Databasebeta

← #32007973

Subgroups — #32007973

Order 18176.

966 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986051C211yesmaximal, characteristic
2#22590744C211yesmaximal, characteristic
3#384×D71C211yesmaximal, characteristic
4#22589099C211yesmaximal, characteristic, complemented
5#22589098C211yesmaximal, characteristic, complemented
6#22589096C211yesmaximal, characteristic, complemented
7#22588529C212yesmaximal
8#22588526C212yesmaximal
9#22588498C212yesmaximal, complemented
10#22588466C212yesmaximal, complemented
11C71×#1947C211yesmaximal, characteristic, complemented
12C2×#14556085—21no—
13C2×C4⋊C4×D71—21no—
14C2×C4⋊C4×D71—21no—
15C2×C4⋊C4×D71C2212yes—
16C2×C42×D71C2211yescharacteristic
17#17986051C418yescomplemented
18C2×#14556062C2212yes—
19C2×#14556061C2212yes—
20C2×#14556061C2212yes—
21C2×C4⋊(C71⋊C8)C2211yescharacteristic
22C2×(C2×D71)⋊C8C2212yes—
23C2×C8×Dic71C2212yes—
24#17985869C2211yescharacteristic
25#17985868—21no—
26#17985868—21no—
27#17985868C2212yes—
28#17985868C2211yescharacteristic
29#17985868C2211yescharacteristic
30#17985863—21no—
31#17985858—21no—
32#17985856C2211yescharacteristic
33#17985856C2211yescharacteristic
34C4×Dic71⋊C4—21no—
35C4×C4⋊Dic71C2211yescharacteristic
36#17985762C2212yes—
37#17985761C2212yes—
38#17985761C2212yes—
39C2×#107×C71C2212yes—
40C2×C4⋊C8×C71C2211yescharacteristic
41#384×C71C2211yescharacteristic
42C4×C4⋊C4×C71C2211yescharacteristic
43#340×C71C2212yes—
44C22×C4×D71—22no—
45C22×C4×D71D411yescharacteristic
46C22×C4×D71D411yescharacteristic
47C22×C4×D71C2311yescharacteristic
48C22×C71⋊C8C2312yes—
49#14556085—44no—
50C2×C4⋊Dic71—21no—
51C2×C4⋊Dic71—21no—
52C2×C4⋊Dic71C2311yescharacteristic
53C2×C4⋊Dic71C2311yescharacteristic
54C2×C4⋊Dic71C2312yes—
55C2×Dic71⋊C4—21no—
56C2×Dic71⋊C4—21no—
57C2×Dic71⋊C4—21no—
58C2×Dic71⋊C4—21no—
59C2×Dic71⋊C4—22no—
60C2×Dic71⋊C4—22no—
61C2×Dic71⋊C4—22no—
62C4⋊C4×D71—42no—
63C4⋊C4×D71—42no—
64C4⋊C4×D71—44no—
65C4⋊C4×D71—44no—
66C2×(C2×D71)⋊C4—21no—
67C2×(C2×D71)⋊C4—21no—
68C2×C4×Dic71—21no—
69C2×C4×Dic71D411yescharacteristic
70C2×C4×Dic71D411yescharacteristic
71C2×C4×Dic71C2311yescharacteristic
72C42×D71C2×C414yes—
73C42×D71Q814yes—
74#14556062—42no—
75#14556061—42no—
76#14556061—42no—
77C4⋊(C71⋊C8)C2×C414yescomplemented
78(C2×D71)⋊C8C2×C418yes—
79C8×Dic71C2×C418yescomplemented
80C71⋊#95—21no—
81C71⋊#95—21no—
82C71⋊#95—21no—
83C22×C8×C71C2312yes—
84C2×C4⋊C4×C71—21no—
85C2×C4⋊C4×C71—21no—
86C2×C4⋊C4×C71C2311yescharacteristic
87C2×C4⋊C4×C71C2312yes—
88C2×C4⋊C4×C71C2311yescharacteristic
89C2×C42×C71—21no—
90C2×C42×C71C2311yescharacteristic
91#107×C71—42no—
92C4⋊C8×C71C2×C414yescomplemented
93#95×C71—21no—
94C23×D71C2×D411yescharacteristic
95C22×Dic71—22no—
96C22×Dic71—21no—
97C22×Dic71—21no—
98C22×Dic71C2×D411yescharacteristic
99C22×Dic71C2×D411yescharacteristic
100C22×Dic71C2×D411yescharacteristic
101C2×C4×D71—21no—
102C2×C4×D71—21no—
103C2×C4×D71—21no—
104C2×C4×D71—21no—
105C2×C4×D71—22no—
106C2×C4×D71—22no—
107C2×C4×D71—44no—
108C2×C4×D71—44no—
109C2×C4×D71—44no—
110C2×C4×D71C4⋊C414yes—
111C2×C4×D71C4⋊C414yes—
112C2×C4×D71D814yes—
113C2×C4×D71Q1614yes—
114C2×C4×D71C22×C412yes—
115C2×C4×D71C2×Q812yes—
116C2×C71⋊C8—22no—
117C2×C71⋊C8—22no—
118C2×C71⋊C8C22×C414yes—
119C4⋊Dic71—21no—
120C4⋊Dic71—22no—
121C4⋊Dic71—21no—
122C4⋊Dic71—22no—
123C4⋊Dic71—21no—
124C4⋊Dic71—21no—
125C4⋊Dic71—42no—
126C4⋊Dic71—42no—
127Dic71⋊C4—44no—
128Dic71⋊C4—44no—
129Dic71⋊C4—42no—
130Dic71⋊C4—42no—
131Dic71⋊C4—44no—
132Dic71⋊C4—42no—
133Dic71⋊C4—42no—
134(C2×D71)⋊C4—42no—
135(C2×D71)⋊C4—42no—
136C4×Dic71—42no—
137C4×Dic71C4⋊C414yes—
138C4×Dic71C4⋊C414yes—
139C4×Dic71C22×C412yes—
140C4×Dic71C2×Q812yes—
141C22×C4×C71—22no—
142C22×C4×C71—21no—
143C22×C4×C71—21no—
144C22×C4×C71C2×D411yescharacteristic
145C22×C4×C71C2×D411yescharacteristic
146C22×C4×C71C2411yescharacteristic
147C2×C8×C71—22no—
148C2×C8×C71—22no—
149C2×C8×C71C22×C414yes—
150C4⋊C4×C71—21no—
151C4⋊C4×C71—22no—
152C4⋊C4×C71—21no—
153C4⋊C4×C71—22no—
154C4⋊C4×C71—21no—
155C4⋊C4×C71—21no—
156C4⋊C4×C71—42no—
157C4⋊C4×C71—42no—
158C42×C71—42no—
159C42×C71C22×C412yes—
160C42×C71C2×Q812yes—
161C23×C71C22×D411yescharacteristic
162C22×D71—21no—
163C22×D71—21no—
164C22×D71—21no—
165C22×D71—21no—
166C22×D71C2×C4⋊C412yes—
167C22×D71C2×D812yes—
168C22×D71C2×Q1612yes—
169C2×C4×C71—21no—
170C2×C4×C71—21no—
171C2×C4×C71—41no—
172C2×C4×C71—42no—
173C2×C4×C71—41no—
174C2×C4×C71—41no—
175C2×C4×C71—41no—
176C2×C4×C71—42no—
177C2×C4×C71—42no—
178C2×C4×C71—41no—
179C2×C4×C71—41no—
180C2×C4×C71C2×C4⋊C412yes—
181C2×C4×C71C2×C4⋊C412yes—
182C2×C4×C71C2×D812yes—
183C2×C4×C71C2×Q1612yes—
184C2×C4×C71C23×C411yescharacteristic
185C2×C4×C71C22×Q811yescharacteristic
186C2×C4×C71H(5,𝔽2)11yescharacteristic
187C2×C4×C71H(5,𝔽2)11yescharacteristic
188C2×C4×C712−1+411yescharacteristic
189C2×C4×C712−1+411yescharacteristic
190C2×Dic71—21no—
191C2×Dic71—21no—
192C2×Dic71—42no—
193C2×Dic71—41no—
194C2×Dic71—41no—
195C2×Dic71—41no—
196C2×Dic71—41no—
197C2×Dic71—42no—
198C2×Dic71—42no—
199C2×Dic71—41no—
200C2×Dic71—41no—
201C2×Dic71C2×C4⋊C412yes—
202C2×Dic71C2×C4⋊C412yes—
203C2×Dic71C2×C4⋊C412yes—
204C2×Dic71C2×D812yes—
205C2×Dic71C2×Q1612yes—
206C2×Dic71AGL(1,ℤ/8)12yes—
207C2×Dic71#13712yes—
208C4×D71—22no—
209C4×D71—22no—
210C4×D71—22no—
211C4×D71—22no—
212C4×D71—28no—
213C4×D71—88no—
214C4×D71#107116yescomplemented
215C8×C71—24no—
216C71⋊C8—24no—
217C22×C71C22×C4⋊C411yescharacteristic
218C22×C71#53511yescharacteristic
219C22×C71#53611yescharacteristic
220C22×C71C22×D811yescharacteristic
221C22×C71C22×Q1611yescharacteristic
222C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
223C22×C71C2×#13711yescharacteristic
224C2×D71—22no—
225C2×D71—22no—
226C2×D71—22no—
227C2×D71—22no—
228C2×D71—22no—
229C2×D71—22no—
230C2×D71C2×#10714yes—
231C4×C71—22no—
232C4×C71—82no—
233C4×C71—81no—
234C4×C71—81no—
235C4×C71C2×#10714yes—
236C4×C71#52011yescharacteristic
237C4×C71#52011yescharacteristic
238C4×C71#54411yescharacteristic
239C4×C71#54411yescharacteristic
240Dic71—22no—
241Dic71—82no—
242Dic71—81no—
243Dic71—81no—
244Dic71C2×#10714yes—
245Dic71#42814yes—
246#21982—711nomaximal, Sylow
247C2×C71#231711yescharacteristic
248C2×C71C22×#10711yescharacteristic
249C2×C71C2×#42811yescharacteristic
250C2×C71#320511yescharacteristic
251C2×C71#320911yescharacteristic
252C2×C71#322611yescharacteristic
253C2×C71#323011yescharacteristic
254D71—24no—
255C2×#431—711no—
256#2285—711no—
257C2×#384—711no—
258#1947—711no—
259#1947—711no—
260#1947—711no—
261#1947D7111yescharacteristic, complemented
262#1852—712no—
263#1852—712no—
264#1845—712no—
265#1834—712no—
266C71#2198211yescharacteristic, complemented, Sylow
267C2×#117—1421no—
268C22×C4⋊C4—712no—
269C22×C4⋊C4—1421no—
270C22×C4⋊C4—1421no—
271C22×C42—711no—
272#431—718no—
273C2×#107—712no—
274C2×#107—712no—
275C2×#107—712no—
276C2×#107C2×D7112yes—
277C2×C4⋊C8—711no—
278C2×C4⋊C8C2×D7111yescharacteristic
279C2×C22⋊C8—712no—
280C2×C4×C8—712no—
281#384—711no—
282#384—712no—
283#384—711no—
284#384—711no—
285#384—1421no—
286#384—1421no—
287#384C2×D7111yescharacteristic
288#382—1421no—
289#380—711no—
290#380—711no—
291#380—1421no—
292C4×C4⋊C4—711no—
293C4×C4⋊C4—1421no—
294C4×C4⋊C4C2×D7111yescharacteristic
295#340—712no—
296#340—712no—
297#340—712no—
298#340C2×D7112yes—
299C23×C4—711no—
300C23×C4—711no—
301C23×C4—711no—
302C23×C4—1422no—
303C22×C8—712no—
304C22×C8C22×D7112yes—
305#117—2844no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—2842no—
309C2×C4⋊C4—2842no—
310C2×C4⋊C4—2844no—
311C2×C4⋊C4—2844no—
312C2×C4⋊C4—712no—
313C2×C4⋊C4—711no—
314C2×C4⋊C4—711no—
315C2×C4⋊C4—1422no—
316C2×C4⋊C4—1422no—
317C2×C4⋊C4—1422no—
318C2×C4⋊C4—1421no—
319C2×C4⋊C4—1421no—
320C2×C4⋊C4—1421no—
321C2×C4⋊C4—1421no—
322C2×C4⋊C4—1421no—
323C2×C4⋊C4—1421no—
324C2×C4⋊C4C22×D7112yes—
325C2×C4⋊C4C22×D7111yescharacteristic
326C2×C4⋊C4C22×D7111yescharacteristic
327C2×C22⋊C4—1421no—
328C2×C22⋊C4—1421no—
329C2×C42—21no—
330C2×C42—711no—
331C2×C42—711no—
332C2×C42—711no—
333C2×C42—714no—
334C2×C42—714no—
335C2×C42—1421no—
336C2×C42C22×D7111yescharacteristic
337#107—42no—
338#107—2842no—
339#107—2842no—
340#107—2842no—
341C4⋊C8—714no—
342C4⋊C8C4×D7114yescomplemented
343C22⋊C8—718no—
344C4×C8—718no—
345#95—21no—
346#95—1421no—
347#95—1421no—
348#95—1421no—
349C24—711no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—2844no—
354C22×C4—2844no—
355C22×C4—2844no—
356C22×C4—711no—
357C22×C4—711no—
358C22×C4—714no—
359C22×C4—714no—
360C22×C4—714no—
361C22×C4—711no—
362C22×C4—712no—
363C22×C4—712no—
364C22×C4—714no—
365C22×C4—1422no—
366C22×C4—1422no—
367C22×C4—1421no—
368C22×C4—1421no—
369C22×C4—1422no—
370C22×C4—1421no—
371C22×C4—1421no—
372C22×C4—1421no—
373C22×C4—1421no—
374C22×C4D4×D7111yescharacteristic
375C22×C4D4×D7111yescharacteristic
376C22×C4C23×D7111yescharacteristic
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—714no—
380C2×C8—1422no—
381C2×C8—1422no—
382C2×C8C2×C4×D7114yes—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—2842no—
392C4⋊C4—2844no—
393C4⋊C4—2844no—
394C4⋊C4—2842no—
395C4⋊C4—2842no—
396C4⋊C4—2842no—
397C4⋊C4—2842no—
398C4⋊C4—2844no—
399C4⋊C4—2842no—
400C4⋊C4—1421no—
401C4⋊C4—1422no—
402C4⋊C4—1421no—
403C4⋊C4—1422no—
404C4⋊C4—1421no—
405C4⋊C4—1421no—
406C22⋊C4—2842no—
407C22⋊C4—2842no—
408C42—42no—
409C42—2842no—
410C42—712no—
411C42—712no—
412C42—714no—
413C42—714no—
414C42C2×C4×D7112yes—
415C42Q8×D7112yes—
416C23—712no—
417C23—712no—
418C23—712no—
419C23—1421no—
420C23—1421no—
421C23—1421no—
422C23—1421no—
423C23C2×D4×D7111yescharacteristic
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—2842no—
436C2×C4—2841no—
437C2×C4—2841no—
438C2×C4—2841no—
439C2×C4—2841no—
440C2×C4—2842no—
441C2×C4—2842no—
442C2×C4—2841no—
443C2×C4—2841no—
444C2×C4—5688no—
445C2×C4—712no—
446C2×C4—712no—
447C2×C4—712no—
448C2×C4—712no—
449C2×C4—712no—
450C2×C4—712no—
451C2×C4—7116no—
452C2×C4—712no—
453C2×C4—1428no—
454C2×C4—1421no—
455C2×C4—1421no—
456C2×C4—1422no—
457C2×C4—1422no—
458C2×C4—1422no—
459C2×C4—1422no—
460C2×C4C4⋊C4×D7112yes—
461C2×C4C4⋊C4×D7112yes—
462C2×C4D8×D7112yes—
463C2×C4Q16×D7112yes—
464C2×C4C22×C4×D7111yescharacteristic
465C2×C4C2×Q8×D7111yescharacteristic
466C2×C4#1455617311yescharacteristic
467C2×C4#1455617411yescharacteristic
468C2×C4#1455617511yescharacteristic
469C2×C4#1455617611yescharacteristic
470C8—24no—
471C8—1424no—
472C22—714no—
473C22—1422no—
474C22—1422no—
475C22—1422no—
476C22—1422no—
477C22—1422no—
478C22—1422no—
479C22C2×C4⋊C4×D7111yescharacteristic
480C22#1798656511yescharacteristic
481C22#1798657211yescharacteristic
482C22C2×D8×D7111yescharacteristic
483C22C2×Q16×D7111yescharacteristic
484C22AGL(1,ℤ/8)×D7111yescharacteristic
485C22#137×D7111yescharacteristic
486C4—22no—
487C4—81no—
488C4—82no—
489C4—81no—
490C4—5681no—
491C4—5682no—
492C4—5681no—
493C4—714no—
494C4—714no—
495C4—1422no—
496C4#107×D7114yes—
497C4#1798647811yescharacteristic
498C4#1798648111yescharacteristic
499C4#1798662611yescharacteristic
500C4#1798662911yescharacteristic
501C2—1424no—
502C2#2259096211yescharacteristic
503C2C2×#107×D7111yescharacteristic
504C2#428×D7111yescharacteristic
505C2#2259925711yescharacteristic
506C2#2259928711yescharacteristic
507C2#2259946011yescharacteristic
508C2#2259949011yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1947C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1947D71
hypocenterC71#21982
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
4. #22589099C2semidirect
5. #22589098C2semidirect
6. #22589096C2semidirect
9. #22588498C2semidirect
10. #22588466C2semidirect
11. C71×#1947C2semidirect
17. #17986051C4semidirect
77. C4⋊(C71⋊C8)C2×C4semidirect
79. C8×Dic71C2×C4semidirect
92. C4⋊C8×C71C2×C4semidirect
214. C4×D71#107semidirect
261. #1947D71semidirect
266. C71#21982semidirect
342. C4⋊C8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860512
#225907442
#384×D712
#225890992
#225890982
#225890962
#225885292
#225885262
#225884982
#225884662
C71×#19472
#2198271