Finite Group Databasebeta

← #32015234

Subgroups — #32015234

Order 18176.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986042C211yesmaximal, characteristic
2#22591207C211yesmaximal, characteristic
3#22589080C211yesmaximal, characteristic, complemented
4#22589065C211yesmaximal, characteristic, complemented
5#22588932C211yesmaximal, characteristic, complemented
6#22588922C211yesmaximal, characteristic, complemented
7#22588915C211yesmaximal, characteristic, complemented
8#22588911C211yesmaximal, characteristic, complemented
9#22588435C211yesmaximal, characteristic
10#22588428C211yesmaximal, characteristic
11#22588371C211yesmaximal, characteristic
12#22588238C211yesmaximal, characteristic, complemented
13C4×#14556061C211yesmaximal, characteristic, complemented
14#22588124C211yesmaximal, characteristic
15C71×#2346C211yesmaximal, characteristic
16C22×#11765849C2211yescharacteristic
17C2×#14556084—21no—
18C22×C4⋊Dic71C2211yescharacteristic
19#17986042C418yescomplemented
20C2×#14556061C2211yescharacteristic
21C2×#14556061C2211yescharacteristic
22C2×#14556061C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×#14556059C2211yescharacteristic
25C2×#14556059C2211yescharacteristic
26C2×C22⋊(C71⋊C8)C2211yescharacteristic
27C2×#14556039C2211yescharacteristic
28C2×C4×C71⋊C8C2211yescharacteristic
29#17985897C2211yescharacteristic
30#17985897C2211yescharacteristic
31#17985884—21no—
32#17985884—21no—
33#17985860—21no—
34#17985859C2211yescharacteristic
35C4×C4⋊Dic71C2211yescharacteristic
36C4×C4⋊Dic71C2211yescharacteristic
37#17985761C2211yescharacteristic
38#17985761C2211yescharacteristic
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×#117×C71C2211yescharacteristic
50#389×C71C2211yescharacteristic
51#389×C71C2211yescharacteristic
52#388×C71—21no—
53#388×C71—21no—
54#382×C71—21no—
55#380×C71C2211yescharacteristic
56C4×C4⋊C4×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C22⋊C4×C71C2211yescharacteristic
59#376×C71—21no—
60C23×Dic71—21no—
61C2×#11765849—22no—
62C2×#11765849—22no—
63C2×#11765849C2×C414yes—
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68#14556084—44no—
69C2×C4⋊Dic71—21no—
70C2×C4⋊Dic71—22no—
71C2×C4⋊Dic71C2×C414yes—
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×C4⋊Dic71C2311yescharacteristic
75C2×C4⋊Dic71C2311yescharacteristic
76C2×Dic71⋊C4—21no—
77C2×Dic71⋊C4—21no—
78C2×C22⋊Dic71—21no—
79C2×C4×Dic71—21no—
80C2×C4×Dic71—21no—
81#14556061—22no—
82#14556061C2×C414yes—
83#14556061C2×C414yes—
84#14556059—22no—
85#14556059C2×C414yes—
86#14556059C2×C414yes—
87C22⋊(C71⋊C8)—22no—
88#14556039—22no—
89C4×C71⋊C8—22no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C71⋊#95—21no—
95C23×C4×C71C2311yescharacteristic
96#117×C71—22no—
97#117×C71C2×C414yes—
98C2×C4⋊C4×C71—21no—
99C2×C4⋊C4×C71—21no—
100C2×C4⋊C4×C71C2311yescharacteristic
101C2×C4⋊C4×C71C2311yescharacteristic
102C2×C4⋊C4×C71C2311yescharacteristic
103C2×C4⋊C4×C71C2311yescharacteristic
104C2×C22⋊C4×C71—21no—
105C2×C22⋊C4×C71—21no—
106C2×C22⋊C4×C71D411yescharacteristic
107C2×C22⋊C4×C71D411yescharacteristic
108C2×C42×C71—21no—
109C2×C42×C71—21no—
110C2×C42×C71—21no—
111C2×C42×C71C2311yescharacteristic
112C2×C42×C71C2311yescharacteristic
113#95×C71—21no—
114#95×C71—21no—
115#95×C71—21no—
116#95×C71—21no—
117#95×C71—21no—
118#95×C71D411yescharacteristic
119#95×C71D411yescharacteristic
120C22×Dic71—24no—
121C22×Dic71—21no—
122C22×Dic71—21no—
123C22×Dic71—21no—
124C22×Dic71—21no—
125C22×Dic71—42no—
126C22×Dic71—42no—
127#11765849—28no—
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⋊C8C22×C412yes—
137C2×C71⋊C8C22×C412yes—
138C2×C71⋊C8#4112yes—
139C2×C71⋊C8#4112yes—
140C4⋊Dic71—24no—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71—21no—
146C4⋊Dic71—21no—
147C4⋊Dic71C22×C412yes—
148C4⋊Dic71C22×C412yes—
149Dic71⋊C4—42no—
150Dic71⋊C4—42no—
151C22⋊Dic71—42no—
152C4×Dic71—42no—
153C4×Dic71—42no—
154C24×C71C2×D411yescharacteristic
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×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71C22×C412yes—
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71C2×D411yescharacteristic
167C22×C4×C71C2×D411yescharacteristic
168C22×C4×C71C2×D411yescharacteristic
169C22×C4×C71#4112yes—
170C22×C4×C71C2411yescharacteristic
171C4⋊C4×C71—21no—
172C4⋊C4×C71—21no—
173C4⋊C4×C71—21no—
174C4⋊C4×C71—21no—
175C4⋊C4×C71—21no—
176C4⋊C4×C71—21no—
177C4⋊C4×C71—81no—
178C4⋊C4×C71—81no—
179C4⋊C4×C71C22×C412yes—
180C4⋊C4×C71C22×C412yes—
181C22⋊C4×C71—24no—
182C22⋊C4×C71—22no—
183C22⋊C4×C71—41no—
184C22⋊C4×C71—81no—
185C42×C71—24no—
186C42×C71—21no—
187C42×C71—21no—
188C42×C71—21no—
189C42×C71—21no—
190C42×C71—42no—
191C42×C71—42no—
192C23×C71—21no—
193C23×C71—21no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71—21no—
197C23×C71—21no—
198C23×C71C4×D412yes—
199C23×C71C22×D411yescharacteristic
200C2×C4×C71—21no—
201C2×C4×C71—21no—
202C2×C4×C71—21no—
203C2×C4×C71—21no—
204C2×C4×C71—22no—
205C2×C4×C71—22no—
206C2×C4×C71—22no—
207C2×C4×C71—22no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
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—21no—
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×C71—41no—
229C2×C4×C71C4×D412yes—
230C2×C4×C71AGL(1,ℤ/8)12yes—
231C2×C4×C71C23×C411yescharacteristic
232C2×C4×C71C2×#4111yescharacteristic
233C2×C4×C71C2×#4111yescharacteristic
234C2×C4×C71C2×#4111yescharacteristic
235C2×C4×C712−1+411yescharacteristic
236C2×C4×C712−1+411yescharacteristic
237C2×Dic71—22no—
238C2×Dic71—22no—
239C2×Dic71—48no—
240C2×Dic71—41no—
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—
250C71⋊C8—22no—
251C71⋊C8—41no—
252C71⋊C8—41no—
253C71⋊C8#11714yescomplemented
254C22×C71—22no—
255C22×C71—22no—
256C22×C71—22no—
257C22×C71—41no—
258C22×C71—41no—
259C22×C71—41no—
260C22×C71—41no—
261C22×C71C2×C4×D411yescharacteristic
262C22×C71#54711yescharacteristic
263C22×C71#54711yescharacteristic
264C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C71#57511yescharacteristic
266C22×C71#57711yescharacteristic
267C22×C71#57811yescharacteristic
268C4×C71—22no—
269C4×C71—22no—
270C4×C71—41no—
271C4×C71—41no—
272C4×C71—41no—
273C4×C71—42no—
274C4×C71—42no—
275C4×C71—81no—
276C4×C71—81no—
277C4×C71C2×#11711yescharacteristic
278C4×C71#51911yescharacteristic
279C4×C71#55211yescharacteristic
280C4×C71#55211yescharacteristic
281Dic71—42no—
282Dic71—42no—
283Dic71—81no—
284Dic71—81no—
285#22624—711nomaximal, Sylow
286C2×C71—42no—
287C2×C71#234411yescharacteristic
288C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
289C2×C71#294711yescharacteristic
290C2×C71#334311yescharacteristic
291C2×C71#334811yescharacteristic
292C2×C71#335211yescharacteristic
293C2×C71#335411yescharacteristic
294C2×#428—711no—
295#2346D7111yescharacteristic
296#2344—711no—
297#1944—711no—
298#1941—711no—
299#1921—711no—
300#1920—711no—
301#1919—711no—
302#1918—711no—
303#1825—711no—
304#1824—711no—
305#1809—711no—
306#1776—711no—
307C4×#107—711no—
308#1753—711no—
309C71#2262411yescharacteristic, complemented, Sylow
310C22×M16—711no—
311C2×#117—1421no—
312C2×#117C2×D7111yescharacteristic
313C22×C4⋊C4—711no—
314#428—718no—
315C2×#107—711no—
316C2×#107—711no—
317C2×#107—711no—
318C2×#106—711no—
319C2×#106—711no—
320C2×#106—711no—
321C2×C22⋊C8—711no—
322C2×#97—711no—
323C2×C4×C8—711no—
324#389—711no—
325#389—711no—
326#389C2×D7111yescharacteristic
327#389C2×D7111yescharacteristic
328#388—21no—
329#388—21no—
330#387—1421no—
331#387—1421no—
332#382—21no—
333#381—1421no—
334#380—711no—
335#380C2×D7111yescharacteristic
336C4×C4⋊C4—711no—
337C4×C4⋊C4—711no—
338C4×C4⋊C4C2×D7111yescharacteristic
339C4×C4⋊C4C2×D7111yescharacteristic
340C4×C22⋊C4C2×D7111yescharacteristic
341#376—21no—
342#340—711no—
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#336—711no—
353#336—711no—
354C23×C4—1421no—
355C23×C4C22×D7111yescharacteristic
356C2×M16—714no—
357C2×M16—1422no—
358C2×M16—1422no—
359C22×C8—711no—
360C22×C8—711no—
361C22×C8—711no—
362C22×C8—711no—
363#117—22no—
364#117—2844no—
365#117C4×D7114yes—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—714no—
369C2×C4⋊C4—711no—
370C2×C4⋊C4—711no—
371C2×C4⋊C4—711no—
372C2×C4⋊C4—711no—
373C2×C4⋊C4—1421no—
374C2×C4⋊C4—1421no—
375C2×C4⋊C4—1421no—
376C2×C4⋊C4—1422no—
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×C42—1421no—
390C2×C42—1421no—
391C2×C42C22×D7111yescharacteristic
392C2×C42C22×D7111yescharacteristic
393#107—714no—
394#107—714no—
395#107—1422no—
396#106—714no—
397#106—714no—
398#106—1422no—
399C22⋊C8—1422no—
400#97—1422no—
401C4×C8—1422no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—1421no—
408#95—1421no—
409#95—1421no—
410#95—1421no—
411#95—1421no—
412#95C22⋊D7111yescharacteristic
413#95C22⋊D7111yescharacteristic
414C24C2×C22⋊D7111yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—2842no—
424C22×C4—2842no—
425C22×C4—1421no—
426C22×C4—1421no—
427C22×C4—1421no—
428C22×C4—1421no—
429C22×C4—1424no—
430C22×C4C2×C4×D7112yes—
431C22×C4C2×C22⋊D7111yescharacteristic
432C22×C4C2×C22⋊D7111yescharacteristic
433C22×C4C2×C22⋊D7111yescharacteristic
434C22×C4C2×C22⋊D7111yescharacteristic
435C22×C4C2×C22⋊D7111yescharacteristic
436C22×C4#1176586512yes—
437C22×C4C23×D7111yescharacteristic
438M16—1428no—
439C2×C8—712no—
440C2×C8—712no—
441C2×C8—712no—
442C2×C8—712no—
443C2×C8—1421no—
444C2×C8—1421no—
445C2×C8—1421no—
446C2×C8—1421no—
447C2×C8—1421no—
448C2×C8—1421no—
449C2×C8—1421no—
450C2×C8—1421no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—2842no—
460C4⋊C4—2842no—
461C4⋊C4—712no—
462C4⋊C4—712no—
463C4⋊C4—1421no—
464C4⋊C4—1421no—
465C4⋊C4—1421no—
466C4⋊C4—1421no—
467C4⋊C4—1421no—
468C4⋊C4—1421no—
469C4⋊C4—1424no—
470C4⋊C4C2×C4×D7112yes—
471C4⋊C4C2×C4×D7112yes—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—41no—
475C22⋊C4—81no—
476C22⋊C4—2842no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—24no—
482C42—42no—
483C42—42no—
484C42—2842no—
485C42—2842no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C4×C22⋊D7112yes—
493C23C22×C22⋊D7111yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—2848no—
524C2×C4—2841no—
525C2×C4—2841no—
526C2×C4—2841no—
527C2×C4—2841no—
528C2×C4—2841no—
529C2×C4—2841no—
530C2×C4—2841no—
531C2×C4—2841no—
532C2×C4—2841no—
533C2×C4—2841no—
534C2×C4—1422no—
535C2×C4—1422no—
536C2×C4C4×C22⋊D7112yes—
537C2×C4#1455615312yes—
538C2×C4C22×C4×D7111yescharacteristic
539C2×C4#41×D7111yescharacteristic
540C2×C4#41×D7111yescharacteristic
541C2×C4C2×#1176586511yescharacteristic
542C2×C4#1455617511yescharacteristic
543C2×C4#1455617511yescharacteristic
544C8—2841no—
545C8—2841no—
546C8—714no—
547C8—1422no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×C4×C22⋊D7111yescharacteristic
556C22#1798664811yescharacteristic
557C22#1798664811yescharacteristic
558C22C2×#1455615311yescharacteristic
559C22#1798679711yescharacteristic
560C22#1798681011yescharacteristic
561C22#1798681411yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—2842no—
572C4—2842no—
573C4—5681no—
574C4—5681no—
575C4#117×D7111yescharacteristic
576C4#1798647611yescharacteristic
577C4#1798668211yescharacteristic
578C4#1798668211yescharacteristic
579C2—42no—
580C2#2259121011yescharacteristic
581C2C4×#1455615311yescharacteristic
582C2#2259717911yescharacteristic
583C2#2260078011yescharacteristic
584C2#2260085511yescharacteristic
585C2#2260090911yescharacteristic
586C2#2260092911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2346C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
cosocleC22×C4×C71C24
hypercenter#2346D71
hypocenterC71#22624
LCSG ▷ C2×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
3. #22589080C2semidirect
4. #22589065C2semidirect
5. #22588932C2semidirect
6. #22588922C2semidirect
7. #22588915C2semidirect
8. #22588911C2semidirect
12. #22588238C2semidirect
13. C4×#14556061C2semidirect
19. #17986042C4semidirect
253. C71⋊C8#117semidirect
309. C71#22624semidirect

Maximal subgroups

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

subgroupindex
C2×#179860422
#225912072
#225890802
#225890652
#225889322
#225889222
#225889152
#225889112
#225884352
#225884282
#225883712
#225882382
C4×#145560612
#225881242
C71×#23462
#2262471