Finite Group Databasebeta

← #32055364

Subgroups — #32055364

Order 18176.

740 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22592042C211yesmaximal, characteristic
2#22589903C211yesmaximal, characteristic, complemented
3#22589842C211yesmaximal, characteristic, complemented
4#22589836C211yesmaximal, characteristic, complemented
5#22589673C211yesmaximal, characteristic, complemented
6#22589292C211yesmaximal, characteristic, complemented
7#22589103C211yesmaximal, characteristic, complemented
8#22588925C211yesmaximal, characteristic, complemented
9#22588911C211yesmaximal, characteristic, complemented
10#22588532C211yesmaximal, characteristic
11#22588525C211yesmaximal, characteristic
12#22588435C211yesmaximal, characteristic
13#22588371C211yesmaximal, characteristic
14#22588061C211yesmaximal, characteristic
15C71×#2346C211yesmaximal, characteristic
16C22×#11765849C2211yescharacteristic
17C2×#14556128C2211yescharacteristic
18C2×#14556105—21no—
19C22×C4⋊Dic71C2211yescharacteristic
20C2×#14556061C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×C4⋊(C71⋊C8)C2211yescharacteristic
25C2×C4⋊(C71⋊C8)C2211yescharacteristic
26C2×C22⋊(C71⋊C8)C2211yescharacteristic
27#17985931—21no—
28#17985920—21no—
29#17985897C2211yescharacteristic
30#17985897C2211yescharacteristic
31#17985874—21no—
32#17985869C2211yescharacteristic
33#17985859C2211yescharacteristic
34#17985761C2211yescharacteristic
35#17985761C2211yescharacteristic
36#17985761C2211yescharacteristic
37#17985761C2211yescharacteristic
38#17985761C2211yescharacteristic
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985754C2211yescharacteristic
47#17985754C2211yescharacteristic
48C2×#117×C71C2211yescharacteristic
49#389×C71C2211yescharacteristic
50#389×C71C2211yescharacteristic
51#388×C71—21no—
52#388×C71—21no—
53#382×C71—21no—
54#380×C71C2211yescharacteristic
55C4×C4⋊C4×C71C2211yescharacteristic
56C4×C4⋊C4×C71C2211yescharacteristic
57C4×C22⋊C4×C71C2211yescharacteristic
58#376×C71—21no—
59C22×Dic142—21no—
60C23×Dic71—21no—
61C2×#11765849—22no—
62C2×#11765849—22no—
63C2×#11765849—41no—
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68#14556128—22no—
69#14556128—41no—
70#14556105—44no—
71C2×C4⋊Dic71—22no—
72C2×C4⋊Dic71—21no—
73C2×C4⋊Dic71—21no—
74C2×C4⋊Dic71—21no—
75C2×C4⋊Dic71—21no—
76C2×C4⋊Dic71—41no—
77C2×C4⋊Dic71C2311yescharacteristic
78C2×C4⋊Dic71C2311yescharacteristic
79C2×C4⋊Dic71C2311yescharacteristic
80C2×C4⋊Dic71C2311yescharacteristic
81C2×Dic71⋊C4—21no—
82C2×Dic71⋊C4—21no—
83C2×C22⋊Dic71—21no—
84#14556061—22no—
85#14556059—22no—
86#14556059—41no—
87#14556059—41no—
88C4⋊(C71⋊C8)—22no—
89C4⋊(C71⋊C8)—41no—
90C22⋊(C71⋊C8)—22no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C23×C4×C71C2311yescharacteristic
95#117×C71—22no—
96#117×C71Q814yescomplemented
97C2×C4⋊C4×C71—21no—
98C2×C4⋊C4×C71—21no—
99C2×C4⋊C4×C71C2311yescharacteristic
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×C71D411yescharacteristic
106C2×C22⋊C4×C71D411yescharacteristic
107C2×C42×C71—21no—
108C2×C42×C71—21no—
109C2×C42×C71—21no—
110C2×C42×C71C2311yescharacteristic
111C2×C42×C71C2311yescharacteristic
112#95×C71—21no—
113#95×C71—21no—
114#95×C71—21no—
115#95×C71—21no—
116#95×C71—21no—
117#95×C71D411yescharacteristic
118#95×C71D411yescharacteristic
119C2×Dic142—42no—
120C2×Dic142—42no—
121C2×Dic142—81no—
122C22×Dic71—21no—
123C22×Dic71—21no—
124C22×Dic71—21no—
125C22×Dic71—21no—
126C22×Dic71—42no—
127C22×Dic71—42no—
128C22×Dic71—81no—
129#11765849—44no—
130C2×C71⋊C8—21no—
131C2×C71⋊C8—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8—21no—
136C2×C71⋊C8—21no—
137C2×C71⋊C8—21no—
138C2×C71⋊C8—21no—
139C2×C71⋊C8—21no—
140C2×C71⋊C8—21no—
141C2×C71⋊C8#4112yes—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71—21no—
146C4⋊Dic71—21no—
147C4⋊Dic71—21no—
148C4⋊Dic71—21no—
149C4⋊Dic71—42no—
150C4⋊Dic71—42no—
151C4⋊Dic71—42no—
152C4⋊Dic71—81no—
153C4⋊Dic71#4112yes—
154Dic71⋊C4—42no—
155Dic71⋊C4—42no—
156C22⋊Dic71—42no—
157C24×C71C2×D411yescharacteristic
158C22×C4×C71—21no—
159C22×C4×C71—21no—
160C22×C4×C71—21no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71—21no—
164C22×C4×C71—21no—
165C22×C4×C71—21no—
166C22×C4×C71C2×D411yescharacteristic
167C22×C4×C71C2×D411yescharacteristic
168C22×C4×C71C2×D411yescharacteristic
169C22×C4×C71C2×D411yescharacteristic
170C22×C4×C71C2×D411yescharacteristic
171C22×C4×C71C2×Q812yes—
172C22×C4×C71#4112yes—
173C22×C4×C71C2411yescharacteristic
174C4⋊C4×C71—21no—
175C4⋊C4×C71—21no—
176C4⋊C4×C71—21no—
177C4⋊C4×C71—21no—
178C4⋊C4×C71—21no—
179C4⋊C4×C71—21no—
180C4⋊C4×C71—81no—
181C4⋊C4×C71—81no—
182C4⋊C4×C71C2×Q812yes—
183C4⋊C4×C71C2×Q812yes—
184C22⋊C4×C71—24no—
185C22⋊C4×C71—22no—
186C22⋊C4×C71—22no—
187C22⋊C4×C71—81no—
188C42×C71—24no—
189C42×C71—21no—
190C42×C71—21no—
191C42×C71—21no—
192C42×C71—21no—
193C42×C71—42no—
194C42×C71—42no—
195C23×C71—21no—
196C23×C71—21no—
197C23×C71—21no—
198C23×C71—21no—
199C23×C71—21no—
200C23×C71—21no—
201C23×C71C22⋊Q812yes—
202C23×C71C22×D411yescharacteristic
203C2×C4×C71—21no—
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—21no—
207C2×C4×C71—21no—
208C2×C4×C71—21no—
209C2×C4×C71—22no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—41no—
222C2×C4×C71—41no—
223C2×C4×C71—41no—
224C2×C4×C71—41no—
225C2×C4×C71—41no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71C22⋊Q812yes—
232C2×C4×C71AGL(1,ℤ/8)12yes—
233C2×C4×C71#13712yes—
234C2×C4×C71C22×Q811yescharacteristic
235C2×C4×C71C2×#4111yescharacteristic
236C2×C4×C71C2×#4111yescharacteristic
237C2×C4×C71C2×#4111yescharacteristic
238C2×C4×C71H(5,𝔽2)11yescharacteristic
239C2×C4×C712−1+411yescharacteristic
240C2×Dic71—22no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—84no—
253Dic142—84no—
254C71⋊C8—22no—
255C71⋊C8—41no—
256C71⋊C8—41no—
257C71⋊C8—41no—
258C22×C71—22no—
259C22×C71—22no—
260C22×C71—22no—
261C22×C71—41no—
262C22×C71—41no—
263C22×C71—41no—
264C22×C71—41no—
265C22×C71C2×C22⋊Q811yescharacteristic
266C22×C71#54611yescharacteristic
267C22×C71#54711yescharacteristic
268C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C71C2×#13711yescharacteristic
270C22×C71#57711yescharacteristic
271C22×C71#57711yescharacteristic
272C4×C71—22no—
273C4×C71—22no—
274C4×C71—22no—
275C4×C71—41no—
276C4×C71—42no—
277C4×C71—42no—
278C4×C71—41no—
279C4×C71—81no—
280C4×C71—81no—
281C4×C71#53311yescharacteristic
282C4×C71#54411yescharacteristic
283C4×C71#55111yescharacteristic
284C4×C71#55211yescharacteristic
285Dic71—42no—
286Dic71—81no—
287Dic71—81no—
288Dic71—81no—
289#26009—711nomaximal, Sylow
290C2×C71—42no—
291C2×C71#243811yescharacteristic
292C2×C71#307911yescharacteristic
293C2×C71#308411yescharacteristic
294C2×C71#331011yescharacteristic
295C2×C71#332911yescharacteristic
296C2×C71#334411yescharacteristic
297C2×C71#335411yescharacteristic
298#2440—711no—
299#2346D7111yescharacteristic
300#2100—711no—
301#2088—711no—
302#2087—711no—
303#2058—711no—
304#1988—711no—
305#1948—711no—
306#1920—711no—
307#1918—711no—
308#1853—711no—
309#1851—711no—
310#1825—711no—
311#1809—711no—
312#1736—711no—
313C71#2600911yescharacteristic, complemented, Sylow
314C22×M16—711no—
315C2×C4⋊Q8—711no—
316C2×C22⋊Q8—1421no—
317C2×#117C2×D7111yescharacteristic
318C22×C4⋊C4—711no—
319C2×#107—711no—
320C2×#106—711no—
321C2×#106—711no—
322C2×#106—711no—
323C2×C4⋊C8—711no—
324C2×C4⋊C8—711no—
325C2×C22⋊C8—711no—
326#398—1421no—
327#395—1421no—
328#389—711no—
329#389—711no—
330#389C2×D7111yescharacteristic
331#389C2×D7111yescharacteristic
332#388—21no—
333#388—21no—
334#385—1421no—
335#384—711no—
336#382—21no—
337#380—711no—
338#380C2×D7111yescharacteristic
339C4×C4⋊C4C2×D7111yescharacteristic
340C4×C4⋊C4C2×D7111yescharacteristic
341C4×C22⋊C4C2×D7111yescharacteristic
342#376—21no—
343#340—711no—
344#340—711no—
345#340—711no—
346#340—711no—
347#340—711no—
348#340—711no—
349#340—711no—
350#340—711no—
351#340—711no—
352#340—711no—
353#340—711no—
354#340—711no—
355#336—711no—
356#336—711no—
357C22×Q8—1421no—
358C23×C4—1421no—
359C23×C4C22×D7111yescharacteristic
360C2×M16—2841no—
361C2×M16—1422no—
362C2×M16—1422no—
363C22×C8—711no—
364C22×C8—711no—
365C22×C8—711no—
366C22×C8—711no—
367C4⋊Q8—2841no—
368C4⋊Q8—1422no—
369C22⋊Q8—2844no—
370#117—22no—
371#117Dic14214yescomplemented
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—2841no—
375C2×C4⋊C4—711no—
376C2×C4⋊C4—711no—
377C2×C4⋊C4—711no—
378C2×C4⋊C4—711no—
379C2×C4⋊C4—1421no—
380C2×C4⋊C4—1421no—
381C2×C4⋊C4—1421no—
382C2×C4⋊C4—1421no—
383C2×C4⋊C4—1421no—
384C2×C4⋊C4—1422no—
385C2×C4⋊C4—1421no—
386C2×C4⋊C4C22×D7111yescharacteristic
387C2×C4⋊C4C22×D7111yescharacteristic
388C2×C4⋊C4C22×D7111yescharacteristic
389C2×C4⋊C4C22×D7111yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1421no—
393C2×C22⋊C4C22⋊D7111yescharacteristic
394C2×C22⋊C4C22⋊D7111yescharacteristic
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42C22×D7111yescharacteristic
399C2×C42C22×D7111yescharacteristic
400#107—1422no—
401#106—2841no—
402#106—2841no—
403#106—1422no—
404C4⋊C8—2841no—
405C4⋊C8—1422no—
406C22⋊C8—1422no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—1421no—
413#95—1421no—
414#95—1421no—
415#95C22⋊D7111yescharacteristic
416#95C22⋊D7111yescharacteristic
417C24C2×C22⋊D7111yescharacteristic
418C2×Q8—2842no—
419C2×Q8—2842no—
420C2×Q8—5681no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—2842no—
430C22×C4—2842no—
431C22×C4—5681no—
432C22×C4—1421no—
433C22×C4—1421no—
434C22×C4—1421no—
435C22×C4—1421no—
436C22×C4C2×C22⋊D7111yescharacteristic
437C22×C4C2×C22⋊D7111yescharacteristic
438C22×C4C2×C22⋊D7111yescharacteristic
439C22×C4C2×C22⋊D7111yescharacteristic
440C22×C4C2×C22⋊D7111yescharacteristic
441C22×C4C2×Dic14212yes—
442C22×C4#1176586512yes—
443C22×C4C23×D7111yescharacteristic
444M16—2844no—
445C2×C8—712no—
446C2×C8—1421no—
447C2×C8—1421no—
448C2×C8—1421no—
449C2×C8—1421no—
450C2×C8—1421no—
451C2×C8—1421no—
452C2×C8—1421no—
453C2×C8—1421no—
454C2×C8—1421no—
455C2×C8—1421no—
456C2×C8—1421no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—2842no—
466C4⋊C4—2842no—
467C4⋊C4—2842no—
468C4⋊C4—2842no—
469C4⋊C4—2842no—
470C4⋊C4—5681no—
471C4⋊C4—712no—
472C4⋊C4—1421no—
473C4⋊C4—1421no—
474C4⋊C4—1421no—
475C4⋊C4—1421no—
476C4⋊C4—1421no—
477C4⋊C4—1421no—
478C4⋊C4—1421no—
479C4⋊C4C2×Dic14212yes—
480C4⋊C4C2×Dic14212yes—
481C22⋊C4—24no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—81no—
485C22⋊C4—2842no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—24no—
491C42—42no—
492C42—42no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23#1455610712yes—
500C23C22×C22⋊D7111yescharacteristic
501Q8—5684no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—2841no—
531C2×C4—2841no—
532C2×C4—2841no—
533C2×C4—2841no—
534C2×C4—2841no—
535C2×C4—2841no—
536C2×C4—2841no—
537C2×C4—2841no—
538C2×C4—2841no—
539C2×C4—2841no—
540C2×C4—2841no—
541C2×C4—5684no—
542C2×C4—1422no—
543C2×C4#1455610712yes—
544C2×C4#1455615312yes—
545C2×C4#1455615812yes—
546C2×C4C22×Dic14211yescharacteristic
547C2×C4#41×D7111yescharacteristic
548C2×C4#41×D7111yescharacteristic
549C2×C4C2×#1176586511yescharacteristic
550C2×C4#1455617411yescharacteristic
551C2×C4#1455617511yescharacteristic
552C8—2841no—
553C8—2841no—
554C8—2841no—
555C8—1422no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C2×#1455610711yescharacteristic
564C22#1798663711yescharacteristic
565C22#1798664811yescharacteristic
566C22C2×#1455615311yescharacteristic
567C22C2×#1455615811yescharacteristic
568C22#1798680711yescharacteristic
569C22#1798681011yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—42no—
574C4—42no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—2842no—
580C4—5681no—
581C4—5681no—
582C4—5681no—
583C4#1798655211yescharacteristic
584C4#1798662711yescharacteristic
585C4#1798667311yescharacteristic
586C4#1798668211yescharacteristic
587C2—42no—
588C2#2259201011yescharacteristic
589C2#2259815511yescharacteristic
590C2#2259820211yescharacteristic
591C2#2260035511yescharacteristic
592C2#2260062811yescharacteristic
593C2#2260079511yescharacteristic
594C2#2260092911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2346C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenter#2346D71
hypocenterC71#26009
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2346

Complements

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

NH ≅ G/Ntype
2. #22589903C2semidirect
3. #22589842C2semidirect
4. #22589836C2semidirect
5. #22589673C2semidirect
6. #22589292C2semidirect
7. #22589103C2semidirect
8. #22588925C2semidirect
9. #22588911C2semidirect
96. #117×C71Q8semidirect
313. C71#26009semidirect
371. #117Dic142semidirect

Maximal subgroups

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

subgroupindex
#225920422
#225899032
#225898422
#225898362
#225896732
#225892922
#225891032
#225889252
#225889112
#225885322
#225885252
#225884352
#225883712
#225880612
C71×#23462
#2600971