Finite Group Databasebeta

← #32055216

Subgroups — #32055216

Order 18176.

694 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22591231C211yesmaximal, characteristic
2#22589897C211yesmaximal, characteristic, complemented
3#22589842C211yesmaximal, characteristic, complemented
4#22589841C211yesmaximal, characteristic, complemented
5#22589675C211yesmaximal, characteristic, complemented
6#22589298C211yesmaximal, characteristic, complemented
7#22589101C211yesmaximal, characteristic, complemented
8#22588929C211yesmaximal, characteristic, complemented
9#22588922C211yesmaximal, characteristic, complemented
10#22588539C211yesmaximal, characteristic
11#22588520C211yesmaximal, characteristic
12#22588432C211yesmaximal, characteristic
13#22588363C211yesmaximal, characteristic
14#22588056C211yesmaximal, characteristic
15C71×#2441C211yesmaximal, characteristic
16C23×C71⋊C8C2211yescharacteristic
17C2×#14556087C2211yescharacteristic
18C2×#14556061C2211yescharacteristic
19C2×#14556059C2211yescharacteristic
20C2×#14556059C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×C4⋊(C71⋊C8)C2211yescharacteristic
23C2×C4⋊(C71⋊C8)C2211yescharacteristic
24C2×C22⋊(C71⋊C8)C2211yescharacteristic
25#17985897C2211yescharacteristic
26#17985897C2211yescharacteristic
27#17985893—21no—
28#17985893—21no—
29#17985866—21no—
30#17985859C2211yescharacteristic
31C4×C4⋊Dic71C2211yescharacteristic
32C4×C4⋊Dic71C2211yescharacteristic
33#17985850—21no—
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×#125×C71C2211yescharacteristic
49C2×#117×C71C2211yescharacteristic
50#399×C71—21no—
51#396×C71—21no—
52#389×C71—21no—
53#389×C71C2211yescharacteristic
54#389×C71C2211yescharacteristic
55#387×C71—21no—
56#384×C71C2211yescharacteristic
57#380×C71C2211yescharacteristic
58C4×C22⋊C4×C71C2211yescharacteristic
59C22×C71⋊C8—22no—
60C22×C71⋊C8—22no—
61C22×C71⋊C8—41no—
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66#14556087—22no—
67#14556087Q814yes—
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×Dic71⋊C4—21no—
73C2×Dic71⋊C4—21no—
74C2×C22⋊Dic71—21no—
75C2×C22⋊Dic71—21no—
76C2×C4×Dic71—21no—
77C2×C4×Dic71—21no—
78C2×C4×Dic71—21no—
79#14556061—22no—
80#14556059—22no—
81#14556059—41no—
82#14556059—41no—
83C4⋊(C71⋊C8)—22no—
84C4⋊(C71⋊C8)—41no—
85C22⋊(C71⋊C8)—22no—
86C71⋊#95—21no—
87C71⋊#95—21no—
88C71⋊#95—21no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C23×C4×C71C2311yescharacteristic
92#125×C71—22no—
93#125×C71—41no—
94#117×C71—22no—
95#117×C71—41no—
96C2×C4⋊C4×C71—21no—
97C2×C4⋊C4×C71—21no—
98C2×C4⋊C4×C71—21no—
99C2×C4⋊C4×C71—21no—
100C2×C4⋊C4×C71—21no—
101C2×C4⋊C4×C71—21no—
102C2×C4⋊C4×C71C2311yescharacteristic
103C2×C4⋊C4×C71C2311yescharacteristic
104C2×C4⋊C4×C71C2311yescharacteristic
105C2×C4⋊C4×C71C2311yescharacteristic
106C2×C22⋊C4×C71—21no—
107C2×C22⋊C4×C71—21no—
108C2×C22⋊C4×C71D411yescharacteristic
109C2×C22⋊C4×C71D411yescharacteristic
110C2×C42×C71—21no—
111C2×C42×C71C2311yescharacteristic
112C2×C42×C71C2311yescharacteristic
113#95×C71—21no—
114#95×C71—21no—
115#95×C71—21no—
116#95×C71D411yescharacteristic
117#95×C71D411yescharacteristic
118C22×Dic71—21no—
119C22×Dic71—21no—
120C22×Dic71—21no—
121C22×Dic71—21no—
122C2×C71⋊C8—21no—
123C2×C71⋊C8—21no—
124C2×C71⋊C8—21no—
125C2×C71⋊C8—21no—
126C2×C71⋊C8—21no—
127C2×C71⋊C8—21no—
128C2×C71⋊C8—21no—
129C2×C71⋊C8—21no—
130C2×C71⋊C8—21no—
131C2×C71⋊C8—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—44no—
134C2×C71⋊C8#4112yes—
135C4⋊Dic71—21no—
136C4⋊Dic71—21no—
137C4⋊Dic71—21no—
138C4⋊Dic71—21no—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71C2×Q812yes—
142C4⋊Dic71C2×Q812yes—
143Dic71⋊C4—42no—
144Dic71⋊C4—42no—
145C22⋊Dic71—24no—
146C22⋊Dic71—81no—
147C4×Dic71—24no—
148C4×Dic71—81no—
149C4×Dic71—81no—
150C24×C71C2×D411yescharacteristic
151C22×C4×C71—21no—
152C22×C4×C71—21no—
153C22×C4×C71—21no—
154C22×C4×C71—21no—
155C22×C4×C71—21no—
156C22×C4×C71—21no—
157C22×C4×C71—21no—
158C22×C4×C71—21no—
159C22×C4×C71C2×D411yescharacteristic
160C22×C4×C71C2×D411yescharacteristic
161C22×C4×C71C2×D411yescharacteristic
162C22×C4×C71C2×D411yescharacteristic
163C22×C4×C71C2×D411yescharacteristic
164C22×C4×C71C2×Q812yes—
165C22×C4×C71#4112yes—
166C22×C4×C71C2411yescharacteristic
167C4⋊C4×C71—21no—
168C4⋊C4×C71—21no—
169C4⋊C4×C71—21no—
170C4⋊C4×C71—21no—
171C4⋊C4×C71—21no—
172C4⋊C4×C71—21no—
173C4⋊C4×C71—21no—
174C4⋊C4×C71—42no—
175C4⋊C4×C71—42no—
176C4⋊C4×C71—42no—
177C4⋊C4×C71—42no—
178C4⋊C4×C71—81no—
179C4⋊C4×C71—81no—
180C4⋊C4×C71#4112yes—
181C22⋊C4×C71—22no—
182C22⋊C4×C71—41no—
183C22⋊C4×C71—42no—
184C22⋊C4×C71—81no—
185C42×C71—21no—
186C42×C71—21no—
187C42×C71—21no—
188C42×C71—21no—
189C42×C71—42no—
190C23×C71—21no—
191C23×C71—21no—
192C23×C71—21no—
193C23×C71—21no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71C22⋊Q812yes—
197C23×C71C22×D411yescharacteristic
198C2×C4×C71—21no—
199C2×C4×C71—21no—
200C2×C4×C71—21no—
201C2×C4×C71—21no—
202C2×C4×C71—22no—
203C2×C4×C71—22no—
204C2×C4×C71—22no—
205C2×C4×C71—22no—
206C2×C4×C71—21no—
207C2×C4×C71—21no—
208C2×C4×C71—22no—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—41no—
217C2×C4×C71—41no—
218C2×C4×C71—41no—
219C2×C4×C71—41no—
220C2×C4×C71—41no—
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×C71C22⋊Q812yes—
228C2×C4×C71#13512yes—
229C2×C4×C71C22×Q811yescharacteristic
230C2×C4×C71C2×#4111yescharacteristic
231C2×C4×C71C2×#4111yescharacteristic
232C2×C4×C71C2×#4111yescharacteristic
233C2×C4×C71H(5,𝔽2)11yescharacteristic
234C2×C4×C712−1+411yescharacteristic
235C2×Dic71—22no—
236C2×Dic71—22no—
237C2×Dic71—41no—
238C2×Dic71—41no—
239C2×Dic71—41no—
240C2×Dic71—41no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247C71⋊C8—22no—
248C71⋊C8—41no—
249C71⋊C8—41no—
250C71⋊C8—41no—
251C22×C71—22no—
252C22×C71—22no—
253C22×C71—22no—
254C22×C71—41no—
255C22×C71—41no—
256C22×C71—41no—
257C22×C71—41no—
258C22×C71C2×C22⋊Q811yescharacteristic
259C22×C71#54611yescharacteristic
260C22×C71#54711yescharacteristic
261C22×C71C2×#13511yescharacteristic
262C22×C71#57511yescharacteristic
263C22×C71#57711yescharacteristic
264C22×C71#57711yescharacteristic
265C4×C71—22no—
266C4×C71—22no—
267C4×C71—41no—
268C4×C71—41no—
269C4×C71—41no—
270C4×C71—42no—
271C4×C71—81no—
272C4×C71—81no—
273C4×C71—81no—
274C4×C71#53311yescharacteristic
275C4×C71#54411yescharacteristic
276C4×C71#55111yescharacteristic
277C4×C71#55211yescharacteristic
278Dic71—42no—
279Dic71—42no—
280Dic71—81no—
281Dic71—81no—
282#25999—711nomaximal, Sylow
283C2×C71—42no—
284C2×C71#243811yescharacteristic
285C2×C71#307611yescharacteristic
286C2×C71#308411yescharacteristic
287C2×C71#330411yescharacteristic
288C2×C71#330711yescharacteristic
289C2×C71#333711yescharacteristic
290C2×C71#335011yescharacteristic
291#2441D7111yescharacteristic
292#2347—711no—
293#2099—711no—
294#2088—711no—
295#2088—711no—
296#2059—711no—
297#1989—711no—
298#1948—711no—
299#1921—711no—
300#1920—711no—
301#1854—711no—
302#1850—711no—
303#1825—711no—
304#1807—711no—
305#1735—711no—
306C71#2599911yescharacteristic, complemented, Sylow
307C23×C8—711no—
308C2×#125C2×D7111yescharacteristic
309C2×#117—711no—
310C2×#117C2×D7111yescharacteristic
311C2×#107—711no—
312C2×#106—711no—
313C2×#106—711no—
314C2×#106—711no—
315C2×C4⋊C8—711no—
316C2×C4⋊C8—711no—
317C2×C22⋊C8—711no—
318#399—21no—
319#396—21no—
320#389—21no—
321#389—711no—
322#389—711no—
323#389C2×D7111yescharacteristic
324#389C2×D7111yescharacteristic
325#388—1421no—
326#388—1421no—
327#387—21no—
328#384C2×D7111yescharacteristic
329#383—1421no—
330#380—711no—
331#380C2×D7111yescharacteristic
332C4×C4⋊C4—711no—
333C4×C4⋊C4—711no—
334C4×C4⋊C4—1421no—
335C4×C22⋊C4C2×D7111yescharacteristic
336#340—711no—
337#340—711no—
338#340—711no—
339#340—711no—
340#340—711no—
341#340—711no—
342#340—711no—
343#340—711no—
344#340—711no—
345#340—711no—
346#340—711no—
347#340—711no—
348#336—711no—
349#336—711no—
350C23×C4C22×D7111yescharacteristic
351C22×C8—2841no—
352C22×C8—711no—
353C22×C8—711no—
354C22×C8—711no—
355C22×C8—711no—
356C22×C8—1422no—
357C22×C8—1422no—
358#125—22no—
359#125—41no—
360#117—22no—
361#117—41no—
362#117—714no—
363#117—1422no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—711no—
371C2×C4⋊C4—711no—
372C2×C4⋊C4—711no—
373C2×C4⋊C4—711no—
374C2×C4⋊C4—1421no—
375C2×C4⋊C4—1421no—
376C2×C4⋊C4C22×D7111yescharacteristic
377C2×C4⋊C4C22×D7111yescharacteristic
378C2×C4⋊C4C22×D7111yescharacteristic
379C2×C4⋊C4C22×D7111yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1421no—
383C2×C22⋊C4—1421no—
384C2×C22⋊C4C22⋊D7111yescharacteristic
385C2×C22⋊C4C22⋊D7111yescharacteristic
386C2×C42—21no—
387C2×C42—1421no—
388C2×C42—1421no—
389C2×C42—1421no—
390C2×C42C22×D7111yescharacteristic
391C2×C42C22×D7111yescharacteristic
392#107—1422no—
393#106—2841no—
394#106—2841no—
395#106—1422no—
396C4⋊C8—2841no—
397C4⋊C8—1422no—
398C22⋊C8—1422no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—1421no—
403#95—1421no—
404#95—1421no—
405#95—1421no—
406#95—1421no—
407#95C22⋊D7111yescharacteristic
408#95C22⋊D7111yescharacteristic
409C24C2×C22⋊D7111yescharacteristic
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—1421no—
419C22×C4—1421no—
420C22×C4—1421no—
421C22×C4—1421no—
422C22×C4C2×C22⋊D7111yescharacteristic
423C22×C4C2×C22⋊D7111yescharacteristic
424C22×C4C2×C22⋊D7111yescharacteristic
425C22×C4C2×C22⋊D7111yescharacteristic
426C22×C4C2×C22⋊D7111yescharacteristic
427C22×C4Q8×D7112yes—
428C22×C4#1176586612yes—
429C22×C4C23×D7111yescharacteristic
430C2×C8—2844no—
431C2×C8—712no—
432C2×C8—1421no—
433C2×C8—1421no—
434C2×C8—1421no—
435C2×C8—1421no—
436C2×C8—1421no—
437C2×C8—1421no—
438C2×C8—1421no—
439C2×C8—1421no—
440C2×C8—1421no—
441C2×C8—1421no—
442C2×C8—1421no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—2842no—
457C4⋊C4—2842no—
458C4⋊C4—712no—
459C4⋊C4—712no—
460C4⋊C4—1421no—
461C4⋊C4—1421no—
462C4⋊C4—1421no—
463C4⋊C4—1421no—
464C4⋊C4—1421no—
465C4⋊C4—1421no—
466C4⋊C4#1176586412yes—
467C22⋊C4—22no—
468C22⋊C4—41no—
469C22⋊C4—42no—
470C22⋊C4—81no—
471C22⋊C4—5681no—
472C22⋊C4—1424no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—42no—
478C42—5681no—
479C42—5681no—
480C42—1424no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23#1455610412yes—
488C23C22×C22⋊D7111yescharacteristic
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—2841no—
519C2×C4—2841no—
520C2×C4—2841no—
521C2×C4—2841no—
522C2×C4—2841no—
523C2×C4—2841no—
524C2×C4—2841no—
525C2×C4—2841no—
526C2×C4—2841no—
527C2×C4—2841no—
528C2×C4—1422no—
529C2×C4—1422no—
530C2×C4#1455610412yes—
531C2×C4#1455614812yes—
532C2×C4C2×Q8×D7111yescharacteristic
533C2×C4#41×D7111yescharacteristic
534C2×C4C2×#1176586411yescharacteristic
535C2×C4C2×#1176586611yescharacteristic
536C2×C4#1455617311yescharacteristic
537C2×C4#1455617511yescharacteristic
538C8—2841no—
539C8—2841no—
540C8—2841no—
541C8—1422no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22C2×#1455610411yescharacteristic
550C22#1798664311yescharacteristic
551C22#1798664811yescharacteristic
552C22C2×#1455614811yescharacteristic
553C22#1798679711yescharacteristic
554C22#1798680711yescharacteristic
555C22#1798681011yescharacteristic
556C4—22no—
557C4—22no—
558C4—41no—
559C4—41no—
560C4—42no—
561C4—41no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—2842no—
566C4—2842no—
567C4—5681no—
568C4—5681no—
569C4#1798655111yescharacteristic
570C4#1798662811yescharacteristic
571C4#1798667611yescharacteristic
572C4#1798669111yescharacteristic
573C2—42no—
574C2#2259201211yescharacteristic
575C2#2259812211yescharacteristic
576C2#2259819811yescharacteristic
577C2#2260027911yescharacteristic
578C2#2260031611yescharacteristic
579C2#2260071911yescharacteristic
580C2#2260087911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2441C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenter#2441D71
hypocenterC71#25999
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
2. #22589897C2semidirect
3. #22589842C2semidirect
4. #22589841C2semidirect
5. #22589675C2semidirect
6. #22589298C2semidirect
7. #22589101C2semidirect
8. #22588929C2semidirect
9. #22588922C2semidirect
306. C71#25999semidirect

Maximal subgroups

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

subgroupindex
#225912312
#225898972
#225898422
#225898412
#225896752
#225892982
#225891012
#225889292
#225889222
#225885392
#225885202
#225884322
#225883632
#225880562
C71×#24412
#2599971