Finite Group Databasebeta

← #31925987

Subgroups — #31925987

Order 18176.

1450 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22586597C212yesmaximal, complemented
2#22586593C212yesmaximal, complemented
3#22586585C212yesmaximal
4#22586577C212yesmaximal
5#22586572C212yesmaximal, complemented
6#22586565C212yesmaximal, complemented
7#22586563C211yesmaximal, characteristic
8#22586563C211yesmaximal, characteristic
9C2×#380×C71C211yesmaximal, characteristic
10C22×C4⋊Dic71C2212yes—
11C22×C22⋊Dic71C2212yes—
12C22×C22⋊Dic71C2211yescharacteristic
13C22×C22⋊Dic71C2211yescharacteristic
14C22×C4×Dic71C2212yes—
15C2×C71⋊#95C2212yes—
16C2×C71⋊#95C2212yes—
17C2×C71⋊#95C2212yes—
18C2×C71⋊#95C2211yescharacteristic
19C2×C71⋊#95C2212yes—
20C2×C71⋊#95C2211yescharacteristic
21C2×C71⋊#95C2212yes—
22C2×C71⋊#95C2212yes—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2212yes—
26C24×C4×C71C2211yescharacteristic
27C22×C4⋊C4×C71C2211yescharacteristic
28C22×C4⋊C4×C71C2211yescharacteristic
29C22×C22⋊C4×C71C2212yes—
30#380×C71C418yescomplemented
31C2×#95×C71C2212yes—
32C23×Dic71C2312yes—
33C23×Dic71C2312yes—
34C23×Dic71C2312yes—
35C23×Dic71C2312yes—
36C2×C4⋊Dic71—42no—
37C2×C4⋊Dic71—42no—
38C2×C4⋊Dic71—42no—
39C2×C22⋊Dic71—24no—
40C2×C22⋊Dic71—24no—
41C2×C22⋊Dic71—22no—
42C2×C22⋊Dic71—41no—
43C2×C22⋊Dic71—42no—
44C2×C22⋊Dic71—41no—
45C2×C22⋊Dic71—41no—
46C2×C22⋊Dic71—41no—
47C2×C22⋊Dic71Q814yes—
48C2×C4×Dic71—24no—
49C2×C4×Dic71—42no—
50C2×C4×Dic71—42no—
51C71⋊#95—24no—
52C71⋊#95—24no—
53C71⋊#95—24no—
54C71⋊#95—42no—
55C71⋊#95—42no—
56C71⋊#95—42no—
57C71⋊#95—42no—
58C71⋊#95—42no—
59C71⋊#95—41no—
60C71⋊#95—41no—
61C71⋊#95—42no—
62C25×C71C2311yescharacteristic
63C23×C4×C71—24no—
64C23×C4×C71—41no—
65C23×C4×C71—41no—
66C23×C4×C71—42no—
67C23×C4×C71C2×C414yes—
68C23×C4×C71C2311yescharacteristic
69C23×C4×C71C2312yes—
70C23×C4×C71C2312yes—
71C23×C4×C71C2311yescharacteristic
72C2×C4⋊C4×C71—41no—
73C2×C4⋊C4×C71—42no—
74C2×C4⋊C4×C71—41no—
75C2×C4⋊C4×C71C2×C414yes—
76C2×C4⋊C4×C71C2×C414yes—
77C2×C22⋊C4×C71—42no—
78C2×C22⋊C4×C71—42no—
79C2×C22⋊C4×C71C2×C418yes—
80#95×C71C2×C418yes—
81C22×Dic71—22no—
82C22×Dic71—22no—
83C22×Dic71—22no—
84C22×Dic71—22no—
85C22×Dic71—22no—
86C22×Dic71—22no—
87C22×Dic71—22no—
88C22×Dic71—22no—
89C22×Dic71—22no—
90C22×Dic71—22no—
91C22×Dic71—22no—
92C22×Dic71—22no—
93C22×Dic71—22no—
94C22×Dic71—22no—
95C22×Dic71—22no—
96C22×Dic71—22no—
97C22×Dic71—22no—
98C22×Dic71—22no—
99C22×Dic71—22no—
100C22×Dic71—22no—
101C22×Dic71—22no—
102C22×Dic71—22no—
103C22×Dic71—22no—
104C22×Dic71—22no—
105C22×Dic71—22no—
106C22×Dic71—22no—
107C22×Dic71C2×Q814yes—
108C22×Dic71#4114yes—
109C4⋊Dic71—48no—
110C22⋊Dic71—44no—
111C22⋊Dic71—48no—
112C22⋊Dic71—44no—
113C4×Dic71—48no—
114C24×C71—22no—
115C24×C71—22no—
116C24×C71—22no—
117C24×C71—21no—
118C24×C71—21no—
119C24×C71—22no—
120C24×C71—21no—
121C24×C71—21no—
122C24×C71C22×C412yes—
123C24×C71C2×Q812yes—
124C24×C71#4112yes—
125C24×C71C2411yescharacteristic
126C22×C4×C71—21no—
127C22×C4×C71—21no—
128C22×C4×C71—22no—
129C22×C4×C71—22no—
130C22×C4×C71—22no—
131C22×C4×C71—22no—
132C22×C4×C71—22no—
133C22×C4×C71—22no—
134C22×C4×C71—22no—
135C22×C4×C71—22no—
136C22×C4×C71—22no—
137C22×C4×C71—22no—
138C22×C4×C71—22no—
139C22×C4×C71—22no—
140C22×C4×C71—21no—
141C22×C4×C71—21no—
142C22×C4×C71—22no—
143C22×C4×C71—22no—
144C22×C4×C71—22no—
145C22×C4×C71—28no—
146C22×C4×C71—44no—
147C22×C4×C71—44no—
148C22×C4×C71—48no—
149C22×C4×C71—48no—
150C22×C4×C71C22×C412yes—
151C22×C4×C71C22×C414yes—
152C22×C4×C71C22×C414yes—
153C22×C4×C71C22×C412yes—
154C22×C4×C71#4114yes—
155C4⋊C4×C71—44no—
156C4⋊C4×C71—44no—
157C22⋊C4×C71—48no—
158C23×C71—22no—
159C23×C71—22no—
160C23×C71—22no—
161C23×C71—22no—
162C23×C71—22no—
163C23×C71—22no—
164C23×C71—22no—
165C23×C71—22no—
166C23×C71—22no—
167C23×C71—22no—
168C23×C71—24no—
169C23×C71—24no—
170C23×C71—24no—
171C23×C71—24no—
172C23×C71—42no—
173C23×C71—42no—
174C23×C71—42no—
175C23×C71—42no—
176C23×C71—42no—
177C23×C71—42no—
178C23×C71—42no—
179C23×C71—42no—
180C23×C71C4×Q814yes—
181C23×C71C23×C411yescharacteristic
182C23×C71C22×Q811yescharacteristic
183C23×C71C2×#4112yes—
184C23×C71C2×#4112yes—
185C23×C71C2×#4111yescharacteristic
186C23×C71H(5,𝔽2)11yescharacteristic
187C23×C71H(5,𝔽2)11yescharacteristic
188C23×C71H(5,𝔽2)12yes—
189C23×C71H(5,𝔽2)12yes—
190C23×C71H(5,𝔽2)11yescharacteristic
191C23×C712−1+411yescharacteristic
192C2×C4×C71—22no—
193C2×C4×C71—22no—
194C2×C4×C71—24no—
195C2×C4×C71—24no—
196C2×C4×C71—22no—
197C2×C4×C71—24no—
198C2×C4×C71—24no—
199C2×C4×C71—24no—
200C2×C4×C71—24no—
201C2×C4×C71—24no—
202C2×C4×C71—22no—
203C2×C4×C71—22no—
204C2×C4×C71—24no—
205C2×C4×C71—416no—
206C2×C4×C71—42no—
207C2×C4×C71—42no—
208C2×C4×C71—42no—
209C2×C4×C71—42no—
210C2×C4×C71—42no—
211C2×C4×C71—42no—
212C2×C4×C71—42no—
213C2×C4×C71—42no—
214C2×C4×C71—42no—
215C2×C4×C71—42no—
216C2×C4×C71#11714yes—
217C2×Dic71—24no—
218C2×Dic71—24no—
219C2×Dic71—24no—
220C2×Dic71—24no—
221C2×Dic71—24no—
222C2×Dic71—24no—
223C2×Dic71—42no—
224C2×Dic71—42no—
225C2×Dic71—42no—
226C2×Dic71—42no—
227C2×Dic71—42no—
228C2×Dic71—42no—
229C2×Dic71—42no—
230C2×Dic71—42no—
231C2×Dic71—42no—
232C2×Dic71—42no—
233C2×Dic71—42no—
234C2×Dic71—42no—
235C2×Dic71—42no—
236C2×Dic71—42no—
237C2×Dic71—42no—
238C2×Dic71—42no—
239C2×Dic71—42no—
240C2×Dic71—42no—
241C2×Dic71—42no—
242C2×Dic71—42no—
243C2×Dic71—42no—
244C2×Dic71—42no—
245C22×C71—24no—
246C22×C71—24no—
247C22×C71—24no—
248C22×C71—42no—
249C22×C71—42no—
250C22×C71—42no—
251C22×C71—42no—
252C22×C71—44no—
253C22×C71—44no—
254C22×C71—44no—
255C22×C71—44no—
256C22×C71C2×#11711yescharacteristic
257C22×C71C2×C4×Q811yescharacteristic
258C22×C71C4×#4111yescharacteristic
259C22×C71#51811yescharacteristic
260C22×C71#51811yescharacteristic
261C22×C71#51811yescharacteristic
262C22×C71#52011yescharacteristic
263C22×C71#53812yes—
264C22×C71#53812yes—
265C22×C71#53812yes—
266C22×C71#53811yescharacteristic
267C22×C71#53911yescharacteristic
268C22×C71#54012yes—
269C22×C71#54311yescharacteristic
270C22×C71#54411yescharacteristic
271C22×C71#55112yes—
272C22×C71#55212yes—
273C22×C71#55312yes—
274C22×C71#55412yes—
275C22×C71#55412yes—
276C22×C71#55512yes—
277C22×C71#56012yes—
278C22×C71#56212yes—
279C4×C71—24no—
280C4×C71—42no—
281C4×C71—44no—
282C4×C71—44no—
283Dic71—44no—
284Dic71—44no—
285Dic71—82no—
286Dic71—82no—
287#14452—711nomaximal, Sylow
288C2×C71—44no—
289C2×C71#231211yescharacteristic
290C2×C71#232911yescharacteristic
291C2×C71#233311yescharacteristic
292C2×C71#234511yescharacteristic
293C2×C71#234611yescharacteristic
294C2×C71#236811yescharacteristic
295C2×C71#237711yescharacteristic
296C2×C71#275012yes—
297C2×C71#275612yes—
298C2×C71#276812yes—
299C2×C71#278412yes—
300C2×#380D7111yescharacteristic
301#1442—712no—
302#1441—712no—
303#1439—712no—
304#1437—712no—
305#1436—712no—
306#1434—712no—
307C22⋊#95—711no—
308C22⋊#95—711no—
309C71#1445211yescharacteristic, complemented, Sylow
310C24×C4C2×D7111yescharacteristic
311C22×C4⋊C4—712no—
312C22×C4⋊C4C2×D7111yescharacteristic
313C22×C4⋊C4C2×D7111yescharacteristic
314C22×C22⋊C4—711no—
315C22×C22⋊C4—711no—
316C22×C22⋊C4—712no—
317C22×C22⋊C4C2×D7112yes—
318C22×C42—712no—
319#380Dic7118yescomplemented
320C2×#95—711no—
321C2×#95—712no—
322C2×#95—711no—
323C2×#95—712no—
324C2×#95—712no—
325C2×#95—712no—
326C2×#95—712no—
327C2×#95—712no—
328C2×#95—712no—
329C2×#95—712no—
330C2×#95—712no—
331C2×#95C2×D7112yes—
332C25C22×D7111yescharacteristic
333C23×C4—24no—
334C23×C4—41no—
335C23×C4—41no—
336C23×C4—42no—
337C23×C4—712no—
338C23×C4—712no—
339C23×C4—712no—
340C23×C4—712no—
341C23×C4C2×Dic7114yes—
342C23×C4C22×D7112yes—
343C23×C4C22×D7112yes—
344C23×C4C22×D7111yescharacteristic
345C23×C4C22×D7111yescharacteristic
346C2×C4⋊C4—41no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4—41no—
349C2×C4⋊C4—2842no—
350C2×C4⋊C4—2842no—
351C2×C4⋊C4—2842no—
352C2×C4⋊C4C2×Dic7114yes—
353C2×C4⋊C4C2×Dic7114yes—
354C2×C22⋊C4—42no—
355C2×C22⋊C4—42no—
356C2×C22⋊C4—2841no—
357C2×C22⋊C4—2841no—
358C2×C22⋊C4—2842no—
359C2×C22⋊C4—2841no—
360C2×C22⋊C4—2841no—
361C2×C22⋊C4—714no—
362C2×C22⋊C4—1424no—
363C2×C22⋊C4—1422no—
364C2×C22⋊C4—1424no—
365C2×C22⋊C4C2×Dic7118yes—
366C2×C42—2842no—
367C2×C42—2842no—
368C2×C42—1424no—
369#95—2842no—
370#95—2842no—
371#95—2842no—
372#95—2842no—
373#95—2841no—
374#95—2842no—
375#95—2841no—
376#95—2842no—
377#95—1424no—
378#95—1424no—
379#95—1424no—
380#95C2×Dic7118yes—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—21no—
385C24—21no—
386C24—22no—
387C24—21no—
388C24—21no—
389C24C22×Dic7112yes—
390C24Q8×D7112yes—
391C24#1176586612yes—
392C24C23×D7111yescharacteristic
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—714no—
418C22×C4—714no—
419C22×C4—1422no—
420C22×C4—1422no—
421C22×C4—1422no—
422C22×C4—1422no—
423C22×C4—1422no—
424C22×C4—1422no—
425C22×C4—1422no—
426C22×C4—1422no—
427C22×C4—1422no—
428C22×C4—1422no—
429C22×C4—1422no—
430C22×C4—1422no—
431C22×C4—1422no—
432C22×C4—1422no—
433C22×C4—1422no—
434C22×C4—1422no—
435C22×C4—1422no—
436C22×C4—1422no—
437C22×C4—1422no—
438C22×C4—1422no—
439C22×C4—1422no—
440C22×C4—1422no—
441C22×C4—1422no—
442C22×C4—1422no—
443C22×C4—1422no—
444C22×C4—1422no—
445C22×C4C22×Dic7112yes—
446C22×C4C22×Dic7114yes—
447C22×C4C22×Dic7114yes—
448C22×C4C22×Dic7112yes—
449C22×C4#1176586514yes—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—2848no—
453C22⋊C4—48no—
454C22⋊C4—2844no—
455C22⋊C4—2848no—
456C22⋊C4—2844no—
457C42—2848no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23Q8×Dic7114yes—
481C23C23×Dic7111yescharacteristic
482C23C2×Q8×D7111yescharacteristic
483C23#41×D7112yes—
484C23C2×#1176586512yes—
485C23C2×#1176586611yescharacteristic
486C23#1455617311yescharacteristic
487C23#1455617312yes—
488C23#1455617311yescharacteristic
489C23#1455617312yes—
490C23#1455617411yescharacteristic
491C23#1455617511yescharacteristic
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—416no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—2842no—
517C2×C4—2842no—
518C2×C4—2842no—
519C2×C4—2842no—
520C2×C4—2842no—
521C2×C4—2842no—
522C2×C4—2842no—
523C2×C4—2842no—
524C2×C4—2842no—
525C2×C4—2842no—
526C2×C4—2842no—
527C2×C4—2842no—
528C2×C4—2842no—
529C2×C4—2842no—
530C2×C4—2842no—
531C2×C4—2842no—
532C2×C4—2842no—
533C2×C4—2842no—
534C2×C4—2842no—
535C2×C4—2842no—
536C2×C4—2842no—
537C2×C4—2842no—
538C2×C4—1424no—
539C2×C4—1424no—
540C2×C4—1424no—
541C2×C4—1424no—
542C2×C4—1424no—
543C2×C4—1424no—
544C2×C4#1455608714yes—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—42no—
549C22—44no—
550C22—42no—
551C22—44no—
552C22—42no—
553C22—44no—
554C22—42no—
555C22—44no—
556C22C2×#1455608711yescharacteristic
557C22C2×Q8×Dic7111yescharacteristic
558C22#41×Dic7111yescharacteristic
559C22#1798647411yescharacteristic
560C22#1798647411yescharacteristic
561C22#1798647411yescharacteristic
562C22#1798648211yescharacteristic
563C22#1798658112yes—
564C22#1798658712yes—
565C22#1798658712yes—
566C22#1798658911yescharacteristic
567C22#1798659811yescharacteristic
568C22#1798659912yes—
569C22#1798662411yescharacteristic
570C22#1798662811yescharacteristic
571C22#1798667612yes—
572C22#1798668812yes—
573C22#1798669712yes—
574C22#1798670312yes—
575C22#1798670412yes—
576C22#1798671212yes—
577C22#1798673712yes—
578C22#1798674412yes—
579C4—24no—
580C4—44no—
581C4—44no—
582C4—42no—
583C4—2844no—
584C4—2844no—
585C4—5682no—
586C4—5682no—
587C2—44no—
588C2#2259093811yescharacteristic
589C2#2259107511yescharacteristic
590C2#2259110711yescharacteristic
591C2#2259121711yescharacteristic
592C2#2259122511yescharacteristic
593C2#2259142511yescharacteristic
594C2#2259149711yescharacteristic
595C2#2259557912yes—
596C2#2259566312yes—
597C2#2259581512yes—
598C2#2259603712yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC2×#380×C71C2
socle, cosocleC24×C71C24
prime radicalC25×C71C23
hypercenterC2×#380D71
hypocenterC71#14452
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. #22586597C2semidirect
2. #22586593C2semidirect
5. #22586572C2semidirect
6. #22586565C2semidirect
30. #380×C71C4semidirect
309. C71#14452semidirect
319. #380Dic71semidirect

Maximal subgroups

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

subgroupindex
#225865972
#225865932
#225865852
#225865772
#225865722
#225865652
#225865632
C2×#380×C712
#1445271