Finite Group Databasebeta

← #34347120

Subgroups — #34347120

Order 20224.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098971C211yesmaximal, characteristic, complemented
2#23767107C211yesmaximal, characteristic
3#23767096C211yesmaximal, characteristic
4#23764879C211yesmaximal, characteristic, complemented
5#23764874C211yesmaximal, characteristic, complemented
6#23764736C211yesmaximal, characteristic, complemented
7#23764731C211yesmaximal, characteristic, complemented
8#23764136C211yesmaximal, characteristic, complemented
9#23764134C211yesmaximal, characteristic, complemented
10#23763599C211yesmaximal, characteristic
11#23763593C211yesmaximal, characteristic
12#23763555C211yesmaximal, characteristic, complemented
13#23763433C211yesmaximal, characteristic
14#23763181C211yesmaximal, characteristic
15C2×#428×C79C211yesmaximal, characteristic
16C2×#14579160C2211yescharacteristic
17C2×#14579152C2211yescharacteristic
18C2×#14579140—21no—
19#19098971—24no—
20C2×#14579095C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579093C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×#14579092C2211yescharacteristic
26C2×C22⋊(C79⋊C8)C2211yescharacteristic
27C2×#14579071C2211yescharacteristic
28C2×C8×Dic79C2211yescharacteristic
29#19098499—21no—
30#19098496—21no—
31#19098486—21no—
32#19098483—21no—
33#19098462C2211yescharacteristic
34#19098462C2211yescharacteristic
35#19098461—21no—
36#19098459C2211yescharacteristic
37#19098459C2211yescharacteristic
38#19098447—21no—
39#19098436—21no—
40#19098432C2211yescharacteristic
41#19098432C2211yescharacteristic
42#19098423C2211yescharacteristic
43#19098423C2211yescharacteristic
44C4×C22⋊Dic79C2211yescharacteristic
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098317C2211yescharacteristic
52#19098317C2211yescharacteristic
53C22×M16×C79C2211yescharacteristic
54C2×#117×C79C2211yescharacteristic
55C22×C4⋊C4×C79C2211yescharacteristic
56#428×C79—24no—
57C2×#107×C79C2211yescharacteristic
58C2×#107×C79C2211yescharacteristic
59C2×#106×C79C2211yescharacteristic
60C2×#106×C79C2211yescharacteristic
61C22×Dic158—21no—
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64#14579160—22no—
65#14579160—41no—
66#14579152—22no—
67#14579152—41no—
68#14579140—44no—
69C2×C4⋊Dic79—21no—
70C2×C4⋊Dic79—21no—
71C2×C4⋊Dic79—21no—
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81C2×Dic79⋊C4—21no—
82C2×Dic79⋊C4—21no—
83C2×Dic79⋊C4—21no—
84C2×C22⋊Dic79—21no—
85C2×C22⋊Dic79—21no—
86C2×C22⋊Dic79D411yescharacteristic
87C2×C22⋊Dic79D411yescharacteristic
88C2×C4×Dic79C2311yescharacteristic
89C2×C4×Dic79C2311yescharacteristic
90#14579095—22no—
91#14579094—22no—
92#14579093—22no—
93#14579092—22no—
94#14579092—41no—
95#14579092—41no—
96C22⋊(C79⋊C8)Q814yescomplemented
97#14579071—22no—
98C8×Dic79—22no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95D411yescharacteristic
106C79⋊#95D411yescharacteristic
107C23×C4×C79—21no—
108C23×C4×C79C2311yescharacteristic
109C2×M16×C79—22no—
110C2×M16×C79—22no—
111C2×M16×C79Q814yescomplemented
112C22×C8×C79C2311yescharacteristic
113C22×C8×C79C2311yescharacteristic
114#117×C79—22no—
115#117×C79—41no—
116C2×C4⋊C4×C79—22no—
117C2×C4⋊C4×C79—21no—
118C2×C4⋊C4×C79—41no—
119C2×C4⋊C4×C79C2311yescharacteristic
120C2×C4⋊C4×C79C2311yescharacteristic
121C2×C4⋊C4×C79C2311yescharacteristic
122C2×C4⋊C4×C79C2311yescharacteristic
123C2×C22⋊C4×C79—21no—
124C2×C42×C79—21no—
125#107×C79—22no—
126#107×C79D414yes—
127#106×C79—22no—
128#106×C79—22no—
129C2×Dic158—42no—
130C2×Dic158—42no—
131C2×Dic158—81no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C22×Dic79C2×D411yescharacteristic
137C22×Dic79C2×D411yescharacteristic
138C22×Dic79C2×D411yescharacteristic
139C22×Dic79C2×D411yescharacteristic
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8C2×Q812yes—
145C2×C79⋊C8C2×Q812yes—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—42no—
155C4⋊Dic79—42no—
156C4⋊Dic79—42no—
157Dic79⋊C4—42no—
158Dic79⋊C4—42no—
159Dic79⋊C4—42no—
160Dic79⋊C4—42no—
161Dic79⋊C4—42no—
162Dic79⋊C4—81no—
163Dic79⋊C4—81no—
164Dic79⋊C4—81no—
165C22⋊Dic79—22no—
166C22⋊Dic79—22no—
167C22⋊Dic79—42no—
168C22⋊Dic79—81no—
169C4×Dic79—21no—
170C4×Dic79—21no—
171C4×Dic79—21no—
172C4×Dic79—21no—
173C24×C79C2×D411yescharacteristic
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79—21no—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79—21no—
182C22×C4×C79—42no—
183C22×C4×C79—42no—
184C22×C4×C79—81no—
185C22×C4×C79C2×D411yescharacteristic
186C22×C4×C79C2×Q812yes—
187C22×C4×C79C2×Q812yes—
188C22×C4×C79C2411yescharacteristic
189M16×C79—28no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79C2×D412yes—
193C2×C8×C79C2×Q812yes—
194C2×C8×C79C2×Q812yes—
195C2×C8×C79#4112yes—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79—21no—
200C4⋊C4×C79—21no—
201C4⋊C4×C79—21no—
202C4⋊C4×C79—42no—
203C4⋊C4×C79C2×D412yes—
204C4⋊C4×C79C2×D412yes—
205C22⋊C4×C79—42no—
206C42×C79—42no—
207C23×C79—21no—
208C23×C79—21no—
209C23×C79—21no—
210C23×C79—21no—
211C23×C79—21no—
212C23×C79—21no—
213C23×C79C4⋊Q812yes—
214C23×C79C22×D411yescharacteristic
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—22no—
218C2×C4×C79—22no—
219C2×C4×C79—22no—
220C2×C4×C79—22no—
221C2×C4×C79—22no—
222C2×C4×C79—22no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79—41no—
233C2×C4×C79—84no—
234C2×C4×C79C4⋊Q812yes—
235C2×C4×C79C22×D411yescharacteristic
236C2×C4×C79C22×Q811yescharacteristic
237C2×C4×C79C22×Q811yescharacteristic
238C2×C4×C79C2×#4111yescharacteristic
239C2×C4×C79H(5,𝔽2)11yescharacteristic
240C2×C4×C792−1+411yescharacteristic
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—21no—
244C2×Dic79—21no—
245C2×Dic79—21no—
246C2×Dic79—21no—
247C2×Dic79—21no—
248C2×Dic79—21no—
249C2×Dic79—21no—
250C2×Dic79—21no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79—41no—
262C2×Dic79—41no—
263C2×Dic79AGL(1,ℤ/8)12yes—
264C2×Dic79#13712yes—
265Dic158—84no—
266C8×C79—22no—
267C8×C79C22⋊Q814yescomplemented
268C79⋊C8—22no—
269C79⋊C8—22no—
270C22×C79—22no—
271C22×C79—22no—
272C22×C79—22no—
273C22×C79—41no—
274C22×C79—41no—
275C22×C79—41no—
276C22×C79—41no—
277C22×C79C2×C4⋊Q811yescharacteristic
278C22×C79D4211yescharacteristic
279C22×C79#54711yescharacteristic
280C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C79C2×#13711yescharacteristic
282C22×C79#57711yescharacteristic
283C22×C79#57711yescharacteristic
284C4×C79—22no—
285C4×C79—42no—
286C4×C79—42no—
287C4×C79—81no—
288C4×C79—81no—
289C4×C79C2×C22⋊Q811yescharacteristic
290C4×C79#54411yescharacteristic
291C4×C79D4×Q811yescharacteristic
292C4×C79#55411yescharacteristic
293Dic79—22no—
294Dic79—22no—
295Dic79—41no—
296Dic79—41no—
297Dic79—81no—
298Dic79—81no—
299Dic79—81no—
300Dic79—81no—
301#26078—791nomaximal, Sylow
302C2×C79—42no—
303C2×C79#244011yescharacteristic
304C2×C79#316311yescharacteristic
305C2×C79#316511yescharacteristic
306C2×C79#327211yescharacteristic
307C2×C79#327811yescharacteristic
308C2×C79#334411yescharacteristic
309C2×C79#334511yescharacteristic
310C2×#501—791no—
311C2×#428D7911yescharacteristic
312#2441—791no—
313#2440—791no—
314#2084—791no—
315#2083—791no—
316#2059—791no—
317#2058—791no—
318#1943—791no—
319#1942—791no—
320#1854—791no—
321#1853—791no—
322#1844—791no—
323#1809—791no—
324#1753—791no—
325C79#2607811yescharacteristic, complemented, Sylow
326C22×M16C2×D7911yescharacteristic
327C2×C4⋊Q8—791no—
328C2×#125—791no—
329C2×C22⋊Q8—1581no—
330C2×#117C2×D7911yescharacteristic
331C22×C4⋊C4C2×D7911yescharacteristic
332#501—1584no—
333#428—24no—
334C2×#107—791no—
335C2×#107—791no—
336C2×#107C2×D7911yescharacteristic
337C2×#107C2×D7911yescharacteristic
338C2×#106—791no—
339C2×#106—791no—
340C2×#106—791no—
341C2×#106—791no—
342C2×#106C2×D7911yescharacteristic
343C2×#106C2×D7911yescharacteristic
344C2×C22⋊C8—791no—
345C2×#97—791no—
346C2×C4×C8—791no—
347#399—1581no—
348#398—1581no—
349#396—1581no—
350#395—1581no—
351#389—791no—
352#389—791no—
353#389—791no—
354#389—791no—
355#389—1581no—
356#387—1581no—
357#385—1581no—
358#384—791no—
359#384—791no—
360#380—791no—
361#380—791no—
362C4×C22⋊C4—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#340—791no—
368#340—791no—
369#336—791no—
370#336—791no—
371C22×Q8—1581no—
372C23×C4—21no—
373C23×C4C22×D7911yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16Dic15814yescomplemented
377C22×C8—791no—
378C22×C8—791no—
379C22×C8C22×D7911yescharacteristic
380C22×C8C22×D7911yescharacteristic
381C4⋊Q8—3161no—
382C4⋊Q8—1582no—
383#125—3161no—
384#125—1582no—
385C22⋊Q8—3164no—
386#117—22no—
387#117—41no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—41no—
391C2×C4⋊C4—791no—
392C2×C4⋊C4—791no—
393C2×C4⋊C4—791no—
394C2×C4⋊C4—791no—
395C2×C4⋊C4—1581no—
396C2×C4⋊C4—1581no—
397C2×C4⋊C4—1581no—
398C2×C4⋊C4—1581no—
399C2×C4⋊C4—1581no—
400C2×C4⋊C4—1581no—
401C2×C4⋊C4—1581no—
402C2×C4⋊C4—1581no—
403C2×C4⋊C4—1581no—
404C2×C4⋊C4—1581no—
405C2×C4⋊C4—1581no—
406C2×C4⋊C4C22×D7911yescharacteristic
407C2×C4⋊C4C22×D7911yescharacteristic
408C2×C4⋊C4C22×D7911yescharacteristic
409C2×C4⋊C4C22×D7911yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—791no—
412C2×C22⋊C4—791no—
413C2×C22⋊C4—1581no—
414C2×C22⋊C4—1581no—
415C2×C42—21no—
416C2×C42—791no—
417C2×C42—791no—
418#107—22no—
419#107—1582no—
420#107—1582no—
421#107C22⋊D7914yes—
422#106—22no—
423#106—22no—
424#106—3161no—
425#106—3161no—
426#106—1582no—
427#106—1582no—
428C22⋊C8—794no—
429#97—1582no—
430C4×C8—1582no—
431#95—791no—
432#95—791no—
433#95—1581no—
434#95—1581no—
435#95—1581no—
436#95—1581no—
437#95—1581no—
438#95—1581no—
439C24D4×D7911yescharacteristic
440C2×Q8—3162no—
441C2×Q8—3162no—
442C2×Q8—6321no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—81no—
454C22×C4—791no—
455C22×C4—791no—
456C22×C4—791no—
457C22×C4—791no—
458C22×C4—1581no—
459C22×C4—1581no—
460C22×C4—1581no—
461C22×C4—1581no—
462C22×C4D4×D7911yescharacteristic
463C22×C4Q8×D7912yes—
464C22×C4C2×Dic15812yes—
465C22×C4C23×D7911yescharacteristic
466M16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—792no—
470C2×C8—792no—
471C2×C8—1581no—
472C2×C8—1581no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8C2×C22⋊D7912yes—
476C2×C8C2×Dic15812yes—
477C2×C8C2×Dic15812yes—
478C2×C8#1192962412yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—3162no—
487C4⋊C4—3162no—
488C4⋊C4—3162no—
489C4⋊C4—3162no—
490C4⋊C4—3162no—
491C4⋊C4—3162no—
492C4⋊C4—3162no—
493C4⋊C4—3162no—
494C4⋊C4—6321no—
495C4⋊C4—6321no—
496C4⋊C4—6321no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C4⋊C4—1581no—
501C4⋊C4—1581no—
502C4⋊C4—1581no—
503C4⋊C4—1581no—
504C4⋊C4—1581no—
505C4⋊C4C2×C22⋊D7912yes—
506C4⋊C4C2×C22⋊D7912yes—
507C22⋊C4—42no—
508C22⋊C4—3162no—
509C22⋊C4—6321no—
510C22⋊C4—1582no—
511C22⋊C4—1582no—
512C42—42no—
513C42—1581no—
514C42—1581no—
515C42—1581no—
516C42—1581no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1457916012yes—
524C23C2×D4×D7911yescharacteristic
525Q8—6324no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—84no—
545C2×C4—3161no—
546C2×C4—3161no—
547C2×C4—3161no—
548C2×C4—3161no—
549C2×C4—3161no—
550C2×C4—3161no—
551C2×C4—3161no—
552C2×C4—3161no—
553C2×C4—3161no—
554C2×C4—3161no—
555C2×C4—3161no—
556C2×C4—3161no—
557C2×C4—792no—
558C2×C4—792no—
559C2×C4—1581no—
560C2×C4—1581no—
561C2×C4—1581no—
562C2×C4—1581no—
563C2×C4—1581no—
564C2×C4—1581no—
565C2×C4—1581no—
566C2×C4—1581no—
567C2×C4—1581no—
568C2×C4—1581no—
569C2×C4#1457916012yes—
570C2×C4C22×C22⋊D7911yescharacteristic
571C2×C4C2×Q8×D7911yescharacteristic
572C2×C4C22×Dic15811yescharacteristic
573C2×C4C2×#1192962411yescharacteristic
574C2×C4#1457920611yescharacteristic
575C2×C4#1457920911yescharacteristic
576C8—22no—
577C8—1582no—
578C8—1582no—
579C8#1457914014yescomplemented
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1457916011yescharacteristic
588C22D4×C22⋊D7911yescharacteristic
589C22#1909921411yescharacteristic
590C22AGL(1,ℤ/8)×D7911yescharacteristic
591C22#137×D7911yescharacteristic
592C22#1909937211yescharacteristic
593C22#1909937311yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—3161no—
600C4—3161no—
601C4—6321no—
602C4—6321no—
603C4—6321no—
604C4—6321no—
605C4—1582no—
606C4—1582no—
607C4C2×#1457914011yescharacteristic
608C4#1909919411yescharacteristic
609C4Q8×C22⋊D7911yescharacteristic
610C4#1909926811yescharacteristic
611C2—42no—
612C2#2376709611yescharacteristic
613C2#2377393011yescharacteristic
614C2#2377394411yescharacteristic
615C2#2377499411yescharacteristic
616C2#2377506011yescharacteristic
617C2#2377585611yescharacteristic
618C2#2377587111yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC2×#428×C79C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenterC2×#428D79
hypocenterC79#26078
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#19098971C2semidirect
4. #23764879C2semidirect
5. #23764874C2semidirect
6. #23764736C2semidirect
7. #23764731C2semidirect
8. #23764136C2semidirect
9. #23764134C2semidirect
12. #23763555C2semidirect
96. C22⋊(C79⋊C8)Q8semidirect
111. C2×M16×C79Q8semidirect
267. C8×C79C22⋊Q8semidirect
325. C79#26078semidirect
376. C2×M16Dic158semidirect
579. C8#14579140semidirect

Maximal subgroups

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

subgroupindex
C2×#190989712
#237671072
#237670962
#237648792
#237648742
#237647362
#237647312
#237641362
#237641342
#237635992
#237635932
#237635552
#237634332
#237631812
C2×#428×C792
#2607879