Finite Group Databasebeta

← #32007732

Subgroups — #32007732

Order 18176.

1234 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985963C211yesmaximal, characteristic
2C2×#17985859C211yesmaximal, characteristic
3#22588533C212yesmaximal, complemented
4#22588532C212yesmaximal, complemented
5#22588528C212yesmaximal, complemented
6#22588527C212yesmaximal, complemented
7#22588371C214yesmaximal, complemented
8C2×#380×C71C211yesmaximal, characteristic
9C22×#11765849C2212yes—
10C22×C4⋊Dic71C2211yescharacteristic
11C22×C4⋊Dic71C2211yescharacteristic
12C22×C22⋊Dic71—21no—
13C2×#14556061C2214yescomplemented
14C2×#14556059C2214yescomplemented
15#17985963C418yescomplemented
16C2×C22⋊(C71⋊C8)C2212yes—
17C2×C22⋊(C71⋊C8)C2212yes—
18#17985859—21no—
19#17985859—21no—
20#17985859C2212yes—
21#17985859C2212yes—
22C2×C71⋊#95—21no—
23#17985761C2214yescomplemented
24#17985761C2214yescomplemented
25C24×C4×C71C2211yescharacteristic
26C22×C4⋊C4×C71C2211yescharacteristic
27C22×C4⋊C4×C71C2211yescharacteristic
28C22×C22⋊C4×C71—21no—
29#380×C71—21no—
30#380×C71—21no—
31#380×C71C2212yes—
32#380×C71C2212yes—
33C2×#95×C71—21no—
34C23×Dic71—21no—
35C23×Dic71—21no—
36C2×#11765849—42no—
37C2×#11765849—42no—
38C2×#11765849C2×C418yescomplemented
39C22×C71⋊C8C2314yes—
40C2×C4⋊Dic71—21no—
41C2×C4⋊Dic71—21no—
42C2×C4⋊Dic71—42no—
43C2×C4⋊Dic71—42no—
44C2×C4⋊Dic71C2312yes—
45C2×C4⋊Dic71C2312yes—
46C2×C22⋊Dic71—22no—
47C2×C22⋊Dic71—22no—
48C2×C22⋊Dic71—44no—
49#14556061—44no—
50#14556059—44no—
51C22⋊(C71⋊C8)C2×C418yescomplemented
52C22⋊(C71⋊C8)C2×C418yescomplemented
53C71⋊#95—22no—
54C71⋊#95—22no—
55C25×C71D411yescharacteristic
56C23×C4×C71—21no—
57C23×C4×C71—21no—
58C23×C4×C71—42no—
59C23×C4×C71—42no—
60C23×C4×C71C2×C414yes—
61C23×C4×C71D411yescharacteristic
62C23×C4×C71D411yescharacteristic
63C23×C4×C71D411yescharacteristic
64C23×C4×C71Q814yes—
65C23×C4×C71C2311yescharacteristic
66C23×C4×C71C2311yescharacteristic
67C23×C4×C71C2311yescharacteristic
68C2×C4⋊C4×C71—21no—
69C2×C4⋊C4×C71—21no—
70C2×C4⋊C4×C71—42no—
71C2×C4⋊C4×C71—42no—
72C2×C4⋊C4×C71C2312yes—
73C2×C4⋊C4×C71C2312yes—
74C2×C22⋊C4×C71—22no—
75C2×C22⋊C4×C71—22no—
76C2×C22⋊C4×C71—44no—
77#95×C71—22no—
78#95×C71—22no—
79C22×Dic71—22no—
80C22×Dic71—22no—
81C22×Dic71—42no—
82C22×Dic71—42no—
83C22×Dic71—42no—
84C22×Dic71—42no—
85C22×Dic71—42no—
86C22×Dic71—42no—
87#11765849—48no—
88C2×C71⋊C8—24no—
89C2×C71⋊C8—24no—
90C2×C71⋊C8C22×C418yes—
91C4⋊Dic71—24no—
92C4⋊Dic71—24no—
93C4⋊Dic71—42no—
94C4⋊Dic71—42no—
95C22⋊Dic71—44no—
96C22⋊Dic71—44no—
97C24×C71—22no—
98C24×C71—22no—
99C24×C71—22no—
100C24×C71—22no—
101C24×C71—41no—
102C24×C71—41no—
103C24×C71C4⋊C414yes—
104C24×C71C2×D411yescharacteristic
105C24×C71C2×D411yescharacteristic
106C24×C71C2×D411yescharacteristic
107C22×C4×C71—22no—
108C22×C4×C71—22no—
109C22×C4×C71—22no—
110C22×C4×C71—22no—
111C22×C4×C71—22no—
112C22×C4×C71—22no—
113C22×C4×C71—22no—
114C22×C4×C71—22no—
115C22×C4×C71—48no—
116C22×C4×C71—48no—
117C22×C4×C71—41no—
118C22×C4×C71—41no—
119C22×C4×C71—41no—
120C22×C4×C71—41no—
121C22×C4×C71—41no—
122C22×C4×C71—41no—
123C22×C4×C71—42no—
124C22×C4×C71—42no—
125C22×C4×C71—42no—
126C22×C4×C71—42no—
127C22×C4×C71—42no—
128C22×C4×C71—42no—
129C22×C4×C71C4⋊C414yes—
130C22×C4×C71C4⋊C414yes—
131C22×C4×C71C4⋊C414yes—
132C22×C4×C71C22×C412yes—
133C22×C4×C71C22×C412yes—
134C22×C4×C71C22×C412yes—
135C22×C4×C71C2×D411yescharacteristic
136C22×C4×C71C2×D411yescharacteristic
137C22×C4×C71C2×D411yescharacteristic
138C22×C4×C71C2×Q812yes—
139C22×C4×C71C2×Q812yes—
140C22×C4×C71C2×Q812yes—
141C22×C4×C71C2411yescharacteristic
142C4⋊C4×C71—24no—
143C4⋊C4×C71—24no—
144C4⋊C4×C71—42no—
145C4⋊C4×C71—42no—
146C22⋊C4×C71—44no—
147C22⋊C4×C71—44no—
148C23×C71—21no—
149C23×C71—21no—
150C23×C71—21no—
151C23×C71—21no—
152C23×C71—21no—
153C23×C71—21no—
154C23×C71—21no—
155C23×C71—21no—
156C23×C71—21no—
157C23×C71—21no—
158C23×C71—21no—
159C23×C71—21no—
160C23×C71—21no—
161C23×C71—21no—
162C23×C71—28no—
163C23×C71—28no—
164C23×C71—44no—
165C23×C71—44no—
166C23×C71—44no—
167C23×C71—44no—
168C23×C71—44no—
169C23×C71C2×C4⋊C412yes—
170C23×C71C2×C4⋊C412yes—
171C23×C71C2×C4⋊C412yes—
172C23×C71AGL(1,ℤ/8)12yes—
173C23×C71AGL(1,ℤ/8)12yes—
174C23×C71#13712yes—
175C23×C71#13712yes—
176C23×C71C22×D411yescharacteristic
177C2×C4×C71—21no—
178C2×C4×C71—21no—
179C2×C4×C71—21no—
180C2×C4×C71—21no—
181C2×C4×C71—21no—
182C2×C4×C71—21no—
183C2×C4×C71—24no—
184C2×C4×C71—24no—
185C2×C4×C71—24no—
186C2×C4×C71—24no—
187C2×C4×C71—24no—
188C2×C4×C71—24no—
189C2×C4×C71—44no—
190C2×C4×C71—44no—
191C2×C4×C71—44no—
192C2×C4×C71—42no—
193C2×C4×C71—42no—
194C2×C4×C71—42no—
195C2×C4×C71—42no—
196C2×C4×C71—42no—
197C2×C4×C71—42no—
198C2×C4×C71—84no—
199C2×C4×C71—84no—
200C2×C4×C71C2×C4⋊C412yes—
201C2×C4×C71C2×C4⋊C412yes—
202C2×C4×C71C2×C4⋊C412yes—
203C2×C4×C71C23×C411yescharacteristic
204C2×C4×C71C22×Q811yescharacteristic
205C2×C4×C71H(5,𝔽2)12yes—
206C2×C4×C71H(5,𝔽2)12yes—
207C2×Dic71—42no—
208C2×Dic71—42no—
209C2×Dic71—42no—
210C2×Dic71—42no—
211C2×Dic71—42no—
212C2×Dic71—42no—
213C2×Dic71—84no—
214C2×Dic71—84no—
215C71⋊C8—28no—
216C22×C71—22no—
217C22×C71—22no—
218C22×C71—22no—
219C22×C71—22no—
220C22×C71—22no—
221C22×C71—22no—
222C22×C71—22no—
223C22×C71—22no—
224C22×C71—22no—
225C22×C71—22no—
226C22×C71—22no—
227C22×C71—22no—
228C22×C71—22no—
229C22×C71—22no—
230C22×C71—22no—
231C22×C71—416no—
232C22×C71—41no—
233C22×C71—41no—
234C22×C71—41no—
235C22×C71—41no—
236C22×C71#42814yes—
237C22×C71#42814yes—
238C22×C71C22×C4⋊C411yescharacteristic
239C22×C71#53411yescharacteristic
240C22×C71#53411yescharacteristic
241C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
242C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
243C22×C71C2×#13711yescharacteristic
244C22×C71C2×#13711yescharacteristic
245C4×C71—22no—
246C4×C71—22no—
247C4×C71—22no—
248C4×C71—82no—
249C4×C71—82no—
250C4×C71#51812yes—
251C4×C71#54312yes—
252Dic71—82no—
253Dic71—82no—
254#21942—711nomaximal, Sylow
255C2×C71—24no—
256C2×C71—24no—
257C2×C71—42no—
258C2×C71#231511yescharacteristic
259C2×C71C2×#42811yescharacteristic
260C2×C71C2×#42811yescharacteristic
261C2×C71#319211yescharacteristic
262C2×C71#319311yescharacteristic
263C2×C71#319311yescharacteristic
264C2×C71#319511yescharacteristic
265C2×#407—711no—
266C2×#380—711no—
267C2×#380D7111yescharacteristic
268#1853—712no—
269#1853—712no—
270#1852—712no—
271#1852—712no—
272#1809—714no—
273C71#2194211yescharacteristic, complemented, Sylow
274C24×C4C2×D7111yescharacteristic
275C22×M16—712no—
276C22×C4⋊C4—711no—
277C22×C4⋊C4—711no—
278C22×C4⋊C4C2×D7111yescharacteristic
279C22×C4⋊C4C2×D7111yescharacteristic
280C22×C22⋊C4—21no—
281C22×C22⋊C4—1421no—
282C2×#107—714no—
283C2×#106—714no—
284#407—718no—
285C2×C22⋊C8—712no—
286C2×C22⋊C8—712no—
287#380—21no—
288#380—21no—
289#380—712no—
290#380—712no—
291#380—1421no—
292#380—1421no—
293#380C2×D7112yes—
294#380C2×D7112yes—
295C2×#95—21no—
296C2×#95—1421no—
297#340—714no—
298#340—714no—
299C25C22⋊D7111yescharacteristic
300C23×C4—21no—
301C23×C4—21no—
302C23×C4—42no—
303C23×C4—42no—
304C23×C4—1421no—
305C23×C4—1421no—
306C23×C4Dic14214yes—
307C23×C4C4×D7114yes—
308C23×C4C22⋊D7111yescharacteristic
309C23×C4C22⋊D7111yescharacteristic
310C23×C4C22⋊D7111yescharacteristic
311C23×C4C22×D7111yescharacteristic
312C23×C4C22×D7111yescharacteristic
313C23×C4C22×D7111yescharacteristic
314C2×M16—2842no—
315C2×M16—2842no—
316C2×M16—718no—
317C22×C8—714no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—2842no—
323C2×C4⋊C4—2842no—
324C2×C4⋊C4—712no—
325C2×C4⋊C4—712no—
326C2×C4⋊C4—1421no—
327C2×C4⋊C4—1421no—
328C2×C4⋊C4C22×D7112yes—
329C2×C4⋊C4C22×D7112yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—2844no—
334C2×C22⋊C4—1422no—
335C2×C22⋊C4—1422no—
336#107—2844no—
337#106—2844no—
338C22⋊C8—718no—
339C22⋊C8—718no—
340#95—22no—
341#95—22no—
342#95—1422no—
343#95—1422no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—41no—
349C24—41no—
350C24Dic71⋊C414yes—
351C24C2×C22⋊D7111yescharacteristic
352C24C2×C22⋊D7111yescharacteristic
353C24C2×C22⋊D7111yescharacteristic
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—2842no—
377C22×C4—2842no—
378C22×C4—2842no—
379C22×C4—2842no—
380C22×C4—2842no—
381C22×C4—2842no—
382C22×C4—1422no—
383C22×C4—1422no—
384C22×C4Dic71⋊C414yes—
385C22×C4Dic71⋊C414yes—
386C22×C4Dic71⋊C414yes—
387C22×C4C2×C4×D7112yes—
388C22×C4C2×C4×D7112yes—
389C22×C4C2×C4×D7112yes—
390C22×C4C2×C22⋊D7111yescharacteristic
391C22×C4C2×C22⋊D7111yescharacteristic
392C22×C4C2×C22⋊D7111yescharacteristic
393C22×C4C2×Dic14212yes—
394C22×C4C2×Dic14212yes—
395C22×C4C2×Dic14212yes—
396C22×C4C23×D7111yescharacteristic
397M16—2848no—
398C2×C8—718no—
399C2×C8—1424no—
400C2×C8—1424no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—2842no—
406C4⋊C4—2842no—
407C4⋊C4—1424no—
408C4⋊C4—1424no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—2844no—
412C22⋊C4—2844no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—28no—
426C23—28no—
427C23—21no—
428C23—21no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23C2×Dic71⋊C412yes—
435C23C2×Dic71⋊C412yes—
436C23C2×Dic71⋊C412yes—
437C23#1455615312yes—
438C23#1455615412yes—
439C23#1455615812yes—
440C23#1455615912yes—
441C23C22×C22⋊D7111yescharacteristic
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4—2842no—
466C2×C4—2842no—
467C2×C4—2842no—
468C2×C4—2842no—
469C2×C4—2842no—
470C2×C4—2842no—
471C2×C4—5684no—
472C2×C4—5684no—
473C2×C4C2×Dic71⋊C412yes—
474C2×C4C2×Dic71⋊C412yes—
475C2×C4C2×Dic71⋊C412yes—
476C2×C4C22×C4×D7111yescharacteristic
477C2×C4C22×Dic14211yescharacteristic
478C2×C4#1455617312yes—
479C2×C4#1455617312yes—
480C8—1428no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—416no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22#1798603914yes—
502C22#1798604214yes—
503C22C22×Dic71⋊C411yescharacteristic
504C22#1798655811yescharacteristic
505C22#1798655811yescharacteristic
506C22C2×#1455615311yescharacteristic
507C22C2×#1455615411yescharacteristic
508C22C2×#1455615811yescharacteristic
509C22C2×#1455615911yescharacteristic
510C4—22no—
511C4—22no—
512C4—22no—
513C4—82no—
514C4—82no—
515C4—5682no—
516C4—5682no—
517C4#1798647212yes—
518C4#1798662512yes—
519C2—24no—
520C2—24no—
521C2—42no—
522C2#2259095211yescharacteristic
523C2C2×#1798603911yescharacteristic
524C2C2×#1798604211yescharacteristic
525C2#2259913711yescharacteristic
526C2#2259914211yescharacteristic
527C2#2259914611yescharacteristic
528C2#2259916111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#380×C71C2
socleC23×C71C22×D4
prime radicalC25×C71D4
cosocleC22×C4×C71C24
hypercenterC2×#380D71
hypocenterC71#21942
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
3. #22588533C2semidirect
4. #22588532C2semidirect
5. #22588528C2semidirect
6. #22588527C2semidirect
7. #22588371C2semidirect
13. C2×#14556061C22semidirect
14. C2×#14556059C22semidirect
15. #17985963C4semidirect
23. #17985761C22semidirect
24. #17985761C22semidirect
38. C2×#11765849C2×C4semidirect
51. C22⋊(C71⋊C8)C2×C4semidirect
52. C22⋊(C71⋊C8)C2×C4semidirect
273. C71#21942semidirect

Maximal subgroups

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

subgroupindex
C2×#179859632
C2×#179858592
#225885332
#225885322
#225885282
#225885272
#225883712
C2×#380×C712
#2194271