Finite Group Databasebeta

← #32056525

Subgroups — #32056525

Order 18176.

678 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22593574C211yesmaximal, characteristic, complemented
2#22589873C211yesmaximal, characteristic
3#22589861C211yesmaximal, characteristic
4#22589728C211yesmaximal, characteristic
5#22589728C211yesmaximal, characteristic
6#22589108C211yesmaximal, characteristic, complemented
7#22589106C211yesmaximal, characteristic, complemented
8#22589098C211yesmaximal, characteristic, complemented
9#22589059C211yesmaximal, characteristic, complemented
10#22588925C211yesmaximal, characteristic, complemented
11#22588918C211yesmaximal, characteristic, complemented
12#22588502C211yesmaximal, characteristic, complemented
13#22588350C211yesmaximal, characteristic
14C4×#14556049C211yesmaximal, characteristic
15C71×#2357C211yesmaximal, characteristic
16C2×#14556061C2211yescharacteristic
17C2×#14556061C2211yescharacteristic
18C2×#14556061C2211yescharacteristic
19C2×#14556059C2211yescharacteristic
20C2×#14556059C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×C4⋊(C71⋊C8)C2211yescharacteristic
23C2×C4⋊(C71⋊C8)C2211yescharacteristic
24C2×C4⋊(C71⋊C8)C2211yescharacteristic
25C2×#14556049C2211yescharacteristic
26C2×#14556049C2211yescharacteristic
27C2×#14556049C2211yescharacteristic
28C2×#14556049C2211yescharacteristic
29C2×C4×C71⋊C8C2211yescharacteristic
30#17985938—21no—
31#17985932—21no—
32#17985897C2211yescharacteristic
33#17985897C2211yescharacteristic
34#17985886—21no—
35#17985882—21no—
36#17985869C2211yescharacteristic
37#17985865C2211yescharacteristic
38#17985865C2211yescharacteristic
39C4×C4⋊Dic71C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985754C2211yescharacteristic
47#17985754C2211yescharacteristic
48C2×C4×D4×C71C2211yescharacteristic
49#390×C71C2211yescharacteristic
50#390×C71C2211yescharacteristic
51#388×C71—21no—
52#388×C71—21no—
53#382×C71C2211yescharacteristic
54#381×C71—21no—
55C4×C4⋊C4×C71C2211yescharacteristic
56C4×C4⋊C4×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C22⋊C4×C71—21no—
59C22×C71⋊C8C2311yescharacteristic
60C22×C71⋊C8C2311yescharacteristic
61C22×C71⋊C8C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C2×C4⋊Dic71—21no—
64C2×C4⋊Dic71—21no—
65C2×C4⋊Dic71C2311yescharacteristic
66C2×C4⋊Dic71C2311yescharacteristic
67C2×C4⋊Dic71C2311yescharacteristic
68C2×C4⋊Dic71C2311yescharacteristic
69C2×Dic71⋊C4—21no—
70C2×Dic71⋊C4—21no—
71C2×C4×Dic71—21no—
72#14556061—41no—
73#14556061—41no—
74#14556061—41no—
75#14556059—22no—
76#14556059—22no—
77#14556059—41no—
78C4⋊(C71⋊C8)—22no—
79C4⋊(C71⋊C8)—22no—
80C4⋊(C71⋊C8)—22no—
81#14556049—22no—
82#14556049—22no—
83#14556049—41no—
84#14556049—41no—
85C4×C71⋊C8—22no—
86C71⋊#95—21no—
87C71⋊#95—21no—
88C71⋊#95—21no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C22×D4×C71C2311yescharacteristic
94C23×C4×C71—21no—
95C4×D4×C71—22no—
96C4×D4×C71Q814yes—
97C2×C4⋊C4×C71D411yescharacteristic
98C2×C4⋊C4×C71D411yescharacteristic
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×C71—21no—
105C2×C22⋊C4×C71—21no—
106C2×C22⋊C4×C71—21no—
107C2×C42×C71—21no—
108C2×C42×C71—21no—
109C2×C42×C71C2311yescharacteristic
110C2×C42×C71C2311yescharacteristic
111C2×C42×C71C2311yescharacteristic
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—
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#4112yes—
133C2×C71⋊C8#4112yes—
134C4⋊Dic71—21no—
135C4⋊Dic71—21no—
136C4⋊Dic71—21no—
137C4⋊Dic71—21no—
138C4⋊Dic71—21no—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—42no—
143C4⋊Dic71—81no—
144Dic71⋊C4—42no—
145Dic71⋊C4—81no—
146C4×Dic71—81no—
147C24×C71—21no—
148C2×D4×C71—21no—
149C2×D4×C71—21no—
150C2×D4×C71—21no—
151C2×D4×C71—21no—
152C2×D4×C71—21no—
153C2×D4×C71C2×Q812yes—
154C22×C4×C71—21no—
155C22×C4×C71—24no—
156C22×C4×C71—21no—
157C22×C4×C71—21no—
158C22×C4×C71—81no—
159C22×C4×C71—81no—
160C22×C4×C71C2×D411yescharacteristic
161C22×C4×C71C2×D411yescharacteristic
162C22×C4×C71C2×D411yescharacteristic
163C22×C4×C71C2×D411yescharacteristic
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71C2411yescharacteristic
167C4⋊C4×C71—22no—
168C4⋊C4×C71—21no—
169C4⋊C4×C71—21no—
170C4⋊C4×C71—21no—
171C4⋊C4×C71—21no—
172C4⋊C4×C71—21no—
173C4⋊C4×C71—41no—
174C4⋊C4×C71C2×Q812yes—
175C22⋊C4×C71—24no—
176C22⋊C4×C71—42no—
177C22⋊C4×C71—42no—
178C22⋊C4×C71—81no—
179C22⋊C4×C71—81no—
180C42×C71—21no—
181C42×C71—21no—
182C42×C71—21no—
183C42×C71—21no—
184C42×C71—81no—
185C42×C71—81no—
186C42×C71C2×Q812yes—
187C42×C71#4112yes—
188C23×C71—22no—
189C23×C71—41no—
190C23×C71—41no—
191C23×C71—41no—
192C23×C71—41no—
193C23×C71—41no—
194C23×C71—41no—
195C23×C71C22×D411yescharacteristic
196D4×C71—22no—
197D4×C71—22no—
198D4×C71—41no—
199D4×C71—41no—
200C2×C4×C71—21no—
201C2×C4×C71—21no—
202C2×C4×C71—21no—
203C2×C4×C71—21no—
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—22no—
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—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—84no—
225C2×C4×C71C22⋊Q812yes—
226C2×C4×C71C22⋊Q812yes—
227C2×C4×C71#13512yes—
228C2×C4×C71C22×Q811yescharacteristic
229C2×C4×C71C2×#4111yescharacteristic
230C2×C4×C71C2×#4111yescharacteristic
231C2×C4×C71C2×#4111yescharacteristic
232C2×C4×C71H(5,𝔽2)11yescharacteristic
233C2×C4×C712−1+411yescharacteristic
234C2×Dic71—41no—
235C2×Dic71—41no—
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—
246C71⋊C8—22no—
247C71⋊C8—22no—
248C71⋊C8—41no—
249C71⋊C8—41no—
250C22×C71—42no—
251C22×C71—42no—
252C22×C71—42no—
253C22×C71—81no—
254C22×C71—81no—
255C22×C71—81no—
256C22×C71—81no—
257C22×C71C2×C22⋊Q811yescharacteristic
258C22×C71#54711yescharacteristic
259C22×C71#55011yescharacteristic
260C22×C71C2×#13511yescharacteristic
261C22×C71#57511yescharacteristic
262C22×C71#57611yescharacteristic
263C22×C71#57711yescharacteristic
264C4×C71—22no—
265C4×C71—22no—
266C4×C71—22no—
267C4×C71—42no—
268C4×C71—41no—
269C4×C71—41no—
270C4×C71—41no—
271C4×C71—81no—
272C4×C71—81no—
273C4×C71#53311yescharacteristic
274C4×C71#54411yescharacteristic
275C4×C71#55211yescharacteristic
276C4×C71#55311yescharacteristic
277Dic71—81no—
278Dic71—81no—
279Dic71—81no—
280Dic71—81no—
281#26096—711nomaximal, Sylow
282C2×C71—82no—
283C2×C71#244111yescharacteristic
284C2×C71#307611yescharacteristic
285C2×C71#308111yescharacteristic
286C2×C71#333911yescharacteristic
287C2×C71#335111yescharacteristic
288C2×C71#339411yescharacteristic
289C2×C71#340311yescharacteristic
290#2570—711no—
291#2357D7111yescharacteristic
292#2094—711no—
293#2092—711no—
294#2067—711no—
295#2067—711no—
296#1949—711no—
297#1949—711no—
298#1947—711no—
299#1939—711no—
300#1920—711no—
301#1919—711no—
302#1846—711no—
303#1804—711no—
304C4×#102—711no—
305C71#2609611yescharacteristic, complemented, Sylow
306C2×C4×D4C2×D7111yescharacteristic
307C2×#107—711no—
308C2×#107—711no—
309C2×#107—711no—
310C2×#106—711no—
311C2×#106—711no—
312C2×#106—711no—
313C2×C4⋊C8—711no—
314C2×C4⋊C8—711no—
315C2×C4⋊C8—711no—
316C2×#102—711no—
317C2×#102—711no—
318C2×#102—711no—
319C2×#102—711no—
320C2×C4×C8—711no—
321#400—1421no—
322#398—1421no—
323#390C2×D7111yescharacteristic
324#390C2×D7111yescharacteristic
325#389—711no—
326#389—711no—
327#388—21no—
328#388—21no—
329#387—1421no—
330#387—1421no—
331#384—711no—
332#382—711no—
333#382—711no—
334#382C2×D7111yescharacteristic
335#381—21no—
336C4×C4⋊C4—711no—
337C4×C4⋊C4C2×D7111yescharacteristic
338C4×C4⋊C4C2×D7111yescharacteristic
339C4×C4⋊C4C2×D7111yescharacteristic
340C4×C22⋊C4—21no—
341#340—711no—
342#340—711no—
343#340—711no—
344#340—711no—
345#340—711no—
346#340—711no—
347#336—711no—
348#336—711no—
349C22×D4C22×D7111yescharacteristic
350C23×C4—21no—
351C22×C8—711no—
352C22×C8—711no—
353C22×C8—711no—
354C22×C8—711no—
355C4×D4—22no—
356C4×D4Dic14214yes—
357C2×C4⋊C4—711no—
358C2×C4⋊C4—711no—
359C2×C4⋊C4—711no—
360C2×C4⋊C4—711no—
361C2×C4⋊C4—1421no—
362C2×C4⋊C4—1421no—
363C2×C4⋊C4—1421no—
364C2×C4⋊C4—1421no—
365C2×C4⋊C4C22⋊D7111yescharacteristic
366C2×C4⋊C4C22⋊D7111yescharacteristic
367C2×C4⋊C4C22×D7111yescharacteristic
368C2×C4⋊C4C22×D7111yescharacteristic
369C2×C4⋊C4C22×D7111yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—1421no—
378C2×C42C22×D7111yescharacteristic
379C2×C42C22×D7111yescharacteristic
380C2×C42C22×D7111yescharacteristic
381#107—2841no—
382#107—2841no—
383#107—2841no—
384#106—2841no—
385#106—1422no—
386#106—1422no—
387C4⋊C8—1422no—
388C4⋊C8—1422no—
389C4⋊C8—1422no—
390#102—2841no—
391#102—2841no—
392#102—1422no—
393#102—1422no—
394C4×C8—1422no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—1421no—
400#95—1421no—
401#95—1421no—
402#95—1421no—
403#95—1421no—
404#95—1421no—
405#95—1421no—
406#95C22⋊D7111yescharacteristic
407#95C22⋊D7111yescharacteristic
408C24—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4C2×Dic14212yes—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—1421no—
422C22×C4—1421no—
423C22×C4—1421no—
424C22×C4—1421no—
425C22×C4C2×C22⋊D7111yescharacteristic
426C22×C4C2×C22⋊D7111yescharacteristic
427C22×C4C2×C22⋊D7111yescharacteristic
428C22×C4C2×C22⋊D7111yescharacteristic
429C22×C4C2×C22⋊D7111yescharacteristic
430C22×C4C2×C22⋊D7111yescharacteristic
431C22×C4C23×D7111yescharacteristic
432C2×C8—712no—
433C2×C8—712no—
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—
443C2×C8—1421no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—41no—
451C4⋊C4—2842no—
452C4⋊C4—2842no—
453C4⋊C4—5681no—
454C4⋊C4—5681no—
455C4⋊C4—1421no—
456C4⋊C4—1421no—
457C4⋊C4—1421no—
458C4⋊C4—1421no—
459C4⋊C4—1421no—
460C4⋊C4—1421no—
461C4⋊C4—1421no—
462C4⋊C4—1421no—
463C4⋊C4C2×Dic14212yes—
464C22⋊C4—24no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—81no—
474C42—81no—
475C42—5681no—
476C42C2×Dic14212yes—
477C42#1176586512yes—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23C22×C22⋊D7111yescharacteristic
486D4—22no—
487D4—22no—
488D4—41no—
489D4—41no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
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—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—84no—
515C2×C4—2841no—
516C2×C4—2841no—
517C2×C4—2841no—
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#1455610712yes—
528C2×C4#1455610712yes—
529C2×C4#1455614812yes—
530C2×C4C22×Dic14211yescharacteristic
531C2×C4#41×D7111yescharacteristic
532C2×C4#41×D7111yescharacteristic
533C2×C4C2×#1176586511yescharacteristic
534C2×C4#1455617411yescharacteristic
535C2×C4#1455617511yescharacteristic
536C8—2841no—
537C8—2841no—
538C8—1422no—
539C8—1422no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C2×#1455610711yescharacteristic
548C22#1798664811yescharacteristic
549C22#1798666811yescharacteristic
550C22C2×#1455614811yescharacteristic
551C22#1798679711yescharacteristic
552C22#1798680211yescharacteristic
553C22#1798681011yescharacteristic
554C4—22no—
555C4—22no—
556C4—22no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—5681no—
564C4—5681no—
565C4—5681no—
566C4—5681no—
567C4#1798655211yescharacteristic
568C4#1798662711yescharacteristic
569C4#1798668811yescharacteristic
570C4#1798669311yescharacteristic
571C2—82no—
572C2#2259204711yescharacteristic
573C2#2259812811yescharacteristic
574C2#2259817111yescharacteristic
575C2#2260073611yescharacteristic
576C2#2260089411yescharacteristic
577C2#2260133211yescharacteristic
578C2#2260142411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2357C2
socleC23×C71C22×D4
prime radicalC22×D4×C71C23
hypercenter#2357D71
hypocenterC71#26096
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. #22593574C2semidirect
6. #22589108C2semidirect
7. #22589106C2semidirect
8. #22589098C2semidirect
9. #22589059C2semidirect
10. #22588925C2semidirect
11. #22588918C2semidirect
12. #22588502C2semidirect
305. C71#26096semidirect

Maximal subgroups

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

subgroupindex
#225935742
#225898732
#225898612
#225897282
#225891082
#225891062
#225890982
#225890592
#225889252
#225889182
#225885022
#225883502
C4×#145560492
C71×#23572
#2609671