Finite Group Databasebeta

← #32055514

Subgroups — #32055514

Order 18176.

696 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22592064C211yesmaximal, characteristic
2#22589908C211yesmaximal, characteristic, complemented
3#22589841C211yesmaximal, characteristic, complemented
4#22589841C211yesmaximal, characteristic, complemented
5#22589678C211yesmaximal, characteristic, complemented
6#22589297C211yesmaximal, characteristic, complemented
7#22589108C211yesmaximal, characteristic, complemented
8#22588918C211yesmaximal, characteristic, complemented
9#22588918C211yesmaximal, characteristic, complemented
10#22588532C211yesmaximal, characteristic
11#22588525C211yesmaximal, characteristic
12#22588435C211yesmaximal, characteristic
13#22588371C211yesmaximal, characteristic
14#22588061C211yesmaximal, characteristic
15C71×#2346C211yesmaximal, characteristic
16C22×#11765849C2211yescharacteristic
17C2×#14556120C2211yescharacteristic
18C22×C4⋊Dic71C2211yescharacteristic
19C2×#14556061C2211yescharacteristic
20C2×#14556059C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×C4⋊(C71⋊C8)C2211yescharacteristic
24C2×C4⋊(C71⋊C8)C2211yescharacteristic
25C2×C22⋊(C71⋊C8)C2211yescharacteristic
26#17985941—21no—
27#17985933—21no—
28#17985897C2211yescharacteristic
29#17985897C2211yescharacteristic
30#17985871—21no—
31#17985865C2211yescharacteristic
32#17985859C2211yescharacteristic
33#17985857—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×#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—
59C23×Dic71—21no—
60C2×#11765849—22no—
61C2×#11765849—22no—
62C2×#11765849—41no—
63C22×C71⋊C8C2311yescharacteristic
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67#14556120—22no—
68#14556120—41no—
69C2×C4⋊Dic71—22no—
70C2×C4⋊Dic71—21no—
71C2×C4⋊Dic71—41no—
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×C4⋊Dic71C2311yescharacteristic
75C2×C4⋊Dic71C2311yescharacteristic
76C2×Dic71⋊C4—21no—
77C2×Dic71⋊C4—21no—
78C2×Dic71⋊C4—21no—
79C2×Dic71⋊C4—21no—
80C2×C22⋊Dic71—21no—
81#14556061—41no—
82#14556059—22no—
83#14556059—22no—
84#14556059—41no—
85C4⋊(C71⋊C8)—22no—
86C4⋊(C71⋊C8)—41no—
87C22⋊(C71⋊C8)—22no—
88C71⋊#95—21no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C23×C4×C71C2311yescharacteristic
94#117×C71—22no—
95#117×C71Q814yes—
96C2×C4⋊C4×C71—21no—
97C2×C4⋊C4×C71—21no—
98C2×C4⋊C4×C71C2311yescharacteristic
99C2×C4⋊C4×C71C2311yescharacteristic
100C2×C4⋊C4×C71C2311yescharacteristic
101C2×C4⋊C4×C71C2311yescharacteristic
102C2×C22⋊C4×C71—21no—
103C2×C22⋊C4×C71—21no—
104C2×C22⋊C4×C71D411yescharacteristic
105C2×C22⋊C4×C71D411yescharacteristic
106C2×C42×C71—21no—
107C2×C42×C71—21no—
108C2×C42×C71—21no—
109C2×C42×C71C2311yescharacteristic
110C2×C42×C71C2311yescharacteristic
111#95×C71—21no—
112#95×C71—21no—
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—
122C22×Dic71—42no—
123C22×Dic71—42no—
124C22×Dic71—81no—
125#11765849—44no—
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—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8—21no—
136C2×C71⋊C8—21no—
137C2×C71⋊C8#4112yes—
138C4⋊Dic71—21no—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71—42no—
146C4⋊Dic71#4112yes—
147Dic71⋊C4—42no—
148Dic71⋊C4—42no—
149Dic71⋊C4—81no—
150Dic71⋊C4—81no—
151C22⋊Dic71—42no—
152C24×C71C2×D411yescharacteristic
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×C71—21no—
160C22×C4×C71—21no—
161C22×C4×C71C2×D411yescharacteristic
162C22×C4×C71C2×D411yescharacteristic
163C22×C4×C71C2×D411yescharacteristic
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71C2×Q812yes—
167C22×C4×C71#4112yes—
168C22×C4×C71C2411yescharacteristic
169C4⋊C4×C71—21no—
170C4⋊C4×C71—21no—
171C4⋊C4×C71—21no—
172C4⋊C4×C71—21no—
173C4⋊C4×C71—21no—
174C4⋊C4×C71—21no—
175C4⋊C4×C71—81no—
176C4⋊C4×C71—81no—
177C4⋊C4×C71C2×Q812yes—
178C4⋊C4×C71C2×Q812yes—
179C22⋊C4×C71—24no—
180C22⋊C4×C71—41no—
181C22⋊C4×C71—41no—
182C22⋊C4×C71—81no—
183C42×C71—24no—
184C42×C71—21no—
185C42×C71—21no—
186C42×C71—21no—
187C42×C71—21no—
188C42×C71—42no—
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—22no—
201C2×C4×C71—22no—
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—21no—
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—21no—
217C2×C4×C71—21no—
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×C71—41no—
228C2×C4×C71C22⋊Q812yes—
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—41no—
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—
247C2×Dic71—84no—
248C71⋊C8—22no—
249C71⋊C8—41no—
250C71⋊C8—41no—
251C71⋊C8—41no—
252C22×C71—22no—
253C22×C71—22no—
254C22×C71—22no—
255C22×C71—41no—
256C22×C71—41no—
257C22×C71—41no—
258C22×C71—41no—
259C22×C71C2×C22⋊Q811yescharacteristic
260C22×C71#54611yescharacteristic
261C22×C71#54711yescharacteristic
262C22×C71#57511yescharacteristic
263C22×C71#57511yescharacteristic
264C22×C71#57711yescharacteristic
265C22×C71#57711yescharacteristic
266C4×C71—22no—
267C4×C71—42no—
268C4×C71—42no—
269C4×C71—41no—
270C4×C71—41no—
271C4×C71—41no—
272C4×C71—41no—
273C4×C71—81no—
274C4×C71—81no—
275C4×C71#53311yescharacteristic
276C4×C71#54411yescharacteristic
277C4×C71#55111yescharacteristic
278C4×C71#55211yescharacteristic
279Dic71—42no—
280Dic71—81no—
281Dic71—81no—
282Dic71—81no—
283#26019—711nomaximal, Sylow
284C2×C71—42no—
285C2×C71#243811yescharacteristic
286C2×C71#308011yescharacteristic
287C2×C71#308411yescharacteristic
288C2×C71#332111yescharacteristic
289C2×C71#332311yescharacteristic
290C2×C71#334811yescharacteristic
291C2×C71#335111yescharacteristic
292#2442—711no—
293#2346D7111yescharacteristic
294#2101—711no—
295#2088—711no—
296#2088—711no—
297#2059—711no—
298#1989—711no—
299#1949—711no—
300#1919—711no—
301#1919—711no—
302#1853—711no—
303#1851—711no—
304#1825—711no—
305#1809—711no—
306#1736—711no—
307C71#2601911yescharacteristic, complemented, Sylow
308C22×M16—711no—
309C2×#125—711no—
310C2×#117C2×D7111yescharacteristic
311C22×C4⋊C4—711no—
312C2×#107—711no—
313C2×#106—711no—
314C2×#106—711no—
315C2×#106—711no—
316C2×C4⋊C8—711no—
317C2×C4⋊C8—711no—
318C2×C22⋊C8—711no—
319#400—1421no—
320#398—1421no—
321#389—711no—
322#389—711no—
323#389C2×D7111yescharacteristic
324#389C2×D7111yescharacteristic
325#388—21no—
326#388—21no—
327#385—1421no—
328#382—21no—
329#382—711no—
330#380—711no—
331#380—1421no—
332#380C2×D7111yescharacteristic
333C4×C4⋊C4C2×D7111yescharacteristic
334C4×C4⋊C4C2×D7111yescharacteristic
335C4×C22⋊C4C2×D7111yescharacteristic
336#376—21no—
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#340—711no—
349#336—711no—
350#336—711no—
351C23×C4—1421no—
352C23×C4C22×D7111yescharacteristic
353C2×M16—2841no—
354C2×M16—1422no—
355C2×M16—1422no—
356C22×C8—711no—
357C22×C8—711no—
358C22×C8—711no—
359C22×C8—711no—
360#125—2841no—
361#125—1422no—
362#117—22no—
363#117Dic14214yes—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—2841no—
367C2×C4⋊C4—711no—
368C2×C4⋊C4—711no—
369C2×C4⋊C4—711no—
370C2×C4⋊C4—711no—
371C2×C4⋊C4—1421no—
372C2×C4⋊C4—1421no—
373C2×C4⋊C4—1421no—
374C2×C4⋊C4—1422no—
375C2×C4⋊C4—1421no—
376C2×C4⋊C4—1421no—
377C2×C4⋊C4C22×D7111yescharacteristic
378C2×C4⋊C4C22×D7111yescharacteristic
379C2×C4⋊C4C22×D7111yescharacteristic
380C2×C4⋊C4C22×D7111yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1421no—
384C2×C22⋊C4C22⋊D7111yescharacteristic
385C2×C22⋊C4C22⋊D7111yescharacteristic
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42C22×D7111yescharacteristic
390C2×C42C22×D7111yescharacteristic
391#107—2841no—
392#106—2841no—
393#106—1422no—
394#106—1422no—
395C4⋊C8—2841no—
396C4⋊C8—1422no—
397C22⋊C8—1422no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—1421no—
404#95—1421no—
405#95—1421no—
406#95—1421no—
407#95—1421no—
408#95C22⋊D7111yescharacteristic
409#95C22⋊D7111yescharacteristic
410C24C2×C22⋊D7111yescharacteristic
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—2842no—
420C22×C4—2842no—
421C22×C4—5681no—
422C22×C4—1421no—
423C22×C4—1421no—
424C22×C4—1421no—
425C22×C4—1421no—
426C22×C4C2×C22⋊D7111yescharacteristic
427C22×C4C2×C22⋊D7111yescharacteristic
428C22×C4C2×C22⋊D7111yescharacteristic
429C22×C4C2×C22⋊D7111yescharacteristic
430C22×C4C2×C22⋊D7111yescharacteristic
431C22×C4C2×Dic14212yes—
432C22×C4#1176586512yes—
433C22×C4C23×D7111yescharacteristic
434M16—2844no—
435C2×C8—712no—
436C2×C8—1421no—
437C2×C8—1421no—
438C2×C8—1421no—
439C2×C8—1421no—
440C2×C8—1421no—
441C2×C8—1421no—
442C2×C8—1421no—
443C2×C8—1421no—
444C2×C8—1421no—
445C2×C8—1421no—
446C2×C8—1421no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—2842no—
456C4⋊C4—2842no—
457C4⋊C4—2842no—
458C4⋊C4—5681no—
459C4⋊C4—5681no—
460C4⋊C4—712no—
461C4⋊C4—1421no—
462C4⋊C4—1421no—
463C4⋊C4—1421no—
464C4⋊C4—1421no—
465C4⋊C4—1421no—
466C4⋊C4—1421no—
467C4⋊C4—1421no—
468C4⋊C4C2×Dic14212yes—
469C4⋊C4C2×Dic14212yes—
470C22⋊C4—24no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—81no—
474C22⋊C4—2842no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—24no—
480C42—42no—
481C42—42no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23#1455610712yes—
489C23C22×C22⋊D7111yescharacteristic
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
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—41no—
519C2×C4—41no—
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—2841no—
529C2×C4—2841no—
530C2×C4—2841no—
531C2×C4—5684no—
532C2×C4—1422no—
533C2×C4#1455610712yes—
534C2×C4C22×Dic14211yescharacteristic
535C2×C4#41×D7111yescharacteristic
536C2×C4#41×D7111yescharacteristic
537C2×C4C2×#1176586511yescharacteristic
538C2×C4#1455617411yescharacteristic
539C2×C4#1455617511yescharacteristic
540C8—2841no—
541C8—2841no—
542C8—2841no—
543C8—1422no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22C2×#1455610711yescharacteristic
552C22#1798663711yescharacteristic
553C22#1798664811yescharacteristic
554C22#1798679711yescharacteristic
555C22#1798679711yescharacteristic
556C22#1798680711yescharacteristic
557C22#1798681011yescharacteristic
558C4—22no—
559C4—42no—
560C4—42no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—2842no—
568C4—5681no—
569C4—5681no—
570C4—5681no—
571C4#1798655211yescharacteristic
572C4#1798662711yescharacteristic
573C4#1798667311yescharacteristic
574C4#1798668211yescharacteristic
575C2—42no—
576C2#2259201011yescharacteristic
577C2#2259816211yescharacteristic
578C2#2259820211yescharacteristic
579C2#2260052011yescharacteristic
580C2#2260055011yescharacteristic
581C2#2260085511yescharacteristic
582C2#2260090011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2346C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenter#2346D71
hypocenterC71#26019
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. #22589908C2semidirect
3. #22589841C2semidirect
4. #22589841C2semidirect
5. #22589678C2semidirect
6. #22589297C2semidirect
7. #22589108C2semidirect
8. #22588918C2semidirect
9. #22588918C2semidirect
307. C71#26019semidirect

Maximal subgroups

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

subgroupindex
#225920642
#225899082
#225898412
#225896782
#225892972
#225891082
#225889182
#225885322
#225885252
#225884352
#225883712
#225880612
C71×#23462
#2601971