Finite Group Databasebeta

← #34346262

Subgroups — #34346262

Order 20224.

732 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23767107C211yesmaximal, characteristic
2#23766269C211yesmaximal, characteristic
3#23764966C211yesmaximal, characteristic, complemented
4#23764903C211yesmaximal, characteristic, complemented
5#23764890C211yesmaximal, characteristic, complemented
6#23764737C211yesmaximal, characteristic, complemented
7#23764354C211yesmaximal, characteristic, complemented
8#23764162C211yesmaximal, characteristic, complemented
9#23763991C211yesmaximal, characteristic, complemented
10#23763974C211yesmaximal, characteristic, complemented
11#23763599C211yesmaximal, characteristic
12#23763583C211yesmaximal, characteristic
13#23763490C211yesmaximal, characteristic
14#23763433C211yesmaximal, characteristic
15C79×#1736C211yesmaximal, characteristic
16C2×#14579152C2211yescharacteristic
17C2×#14579120—21no—
18C2×#14579095C2211yescharacteristic
19C2×#14579094C2211yescharacteristic
20C2×#14579093C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×Dic79⋊C8C2211yescharacteristic
23C2×Dic79⋊C8C2211yescharacteristic
24C2×C22⋊(C79⋊C8)C2211yescharacteristic
25#19098499—21no—
26#19098486—21no—
27#19098462C2211yescharacteristic
28#19098462C2211yescharacteristic
29#19098462C2211yescharacteristic
30#19098461—21no—
31#19098459C2211yescharacteristic
32#19098449—21no—
33#19098449—21no—
34#19098447—21no—
35#19098432C2211yescharacteristic
36#19098426—21no—
37#19098423C2211yescharacteristic
38#19098423C2211yescharacteristic
39C4⋊C4×Dic79C2211yescharacteristic
40C4⋊C4×Dic79C2211yescharacteristic
41C4×C22⋊Dic79C2211yescharacteristic
42#19098326C2211yescharacteristic
43#19098326C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098317C2211yescharacteristic
51#19098317C2211yescharacteristic
52C22×M16×C79C2211yescharacteristic
53C2×#117×C79C2211yescharacteristic
54C22×C4⋊C4×C79C2211yescharacteristic
55#340×C79C2211yescharacteristic
56#340×C79C2211yescharacteristic
57#340×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59C22×C79⋊C8C2311yescharacteristic
60C22×C79⋊C8C2311yescharacteristic
61#14579152—22no—
62#14579152—41no—
63#14579120—44no—
64C2×C4⋊Dic79—21no—
65C2×C4⋊Dic79—21no—
66C2×C4⋊Dic79—21no—
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×Dic79⋊C4—21no—
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×C22⋊Dic79—21no—
77C2×C22⋊Dic79—21no—
78C2×C22⋊Dic79D411yescharacteristic
79C2×C22⋊Dic79D411yescharacteristic
80C2×C4×Dic79—21no—
81C2×C4×Dic79—21no—
82C2×C4×Dic79C2311yescharacteristic
83C2×C4×Dic79C2311yescharacteristic
84#14579095—22no—
85#14579094—22no—
86#14579093—41no—
87#14579092—41no—
88Dic79⋊C8—22no—
89Dic79⋊C8—41no—
90C22⋊(C79⋊C8)—22no—
91C79⋊#95—21no—
92C79⋊#95—21no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95D411yescharacteristic
100C79⋊#95D411yescharacteristic
101C23×C4×C79—21no—
102C23×C4×C79C2311yescharacteristic
103C2×M16×C79—22no—
104C2×M16×C79—22no—
105C2×M16×C79—41no—
106C22×C8×C79C2311yescharacteristic
107C22×C8×C79C2311yescharacteristic
108#117×C79—22no—
109#117×C79—41no—
110C2×C4⋊C4×C79—21no—
111C2×C4⋊C4×C79—22no—
112C2×C4⋊C4×C79Q814yes—
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C22⋊C4×C79—21no—
118C2×C42×C79—21no—
119C22×Dic79—21no—
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79—21no—
123C22×Dic79C2×D411yescharacteristic
124C22×Dic79C2×D411yescharacteristic
125C22×Dic79C2×D411yescharacteristic
126C22×Dic79C2×D411yescharacteristic
127C2×C79⋊C8—21no—
128C2×C79⋊C8—21no—
129C2×C79⋊C8—21no—
130C2×C79⋊C8—21no—
131C2×C79⋊C8—21no—
132C2×C79⋊C8—21no—
133C4⋊Dic79—21no—
134C4⋊Dic79—21no—
135C4⋊Dic79—21no—
136C4⋊Dic79—21no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—42no—
142C4⋊Dic79—42no—
143C4⋊Dic79—42no—
144C4⋊Dic79—42no—
145Dic79⋊C4—42no—
146Dic79⋊C4—42no—
147Dic79⋊C4—81no—
148Dic79⋊C4—81no—
149C22⋊Dic79—22no—
150C22⋊Dic79—41no—
151C22⋊Dic79—42no—
152C22⋊Dic79—81no—
153C4×Dic79—21no—
154C4×Dic79—21no—
155C4×Dic79—21no—
156C4×Dic79—21no—
157C4×Dic79—42no—
158C4×Dic79—42no—
159C24×C79C2×D411yescharacteristic
160C22×C4×C79—21no—
161C22×C4×C79—21no—
162C22×C4×C79—21no—
163C22×C4×C79—21no—
164C22×C4×C79—24no—
165C22×C4×C79—21no—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79—21no—
169C22×C4×C79—42no—
170C22×C4×C79—42no—
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2×Q812yes—
173C22×C4×C79#4112yes—
174C22×C4×C79C2411yescharacteristic
175M16×C79—44no—
176C2×C8×C79—21no—
177C2×C8×C79—21no—
178C2×C8×C79—21no—
179C2×C8×C79—21no—
180C2×C8×C79—21no—
181C2×C8×C79#4112yes—
182C4⋊C4×C79—24no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79C2×Q812yes—
189C4⋊C4×C79C2×Q812yes—
190C4⋊C4×C79#4112yes—
191C22⋊C4×C79—42no—
192C42×C79—42no—
193C23×C79—21no—
194C23×C79—21no—
195C23×C79—21no—
196C23×C79—21no—
197C23×C79—21no—
198C23×C79—21no—
199C23×C79C22⋊Q812yes—
200C23×C79C22×D411yescharacteristic
201C2×C4×C79—21no—
202C2×C4×C79—21no—
203C2×C4×C79—22no—
204C2×C4×C79—22no—
205C2×C4×C79—22no—
206C2×C4×C79—22no—
207C2×C4×C79—22no—
208C2×C4×C79—22no—
209C2×C4×C79—22no—
210C2×C4×C79—48no—
211C2×C4×C79—41no—
212C2×C4×C79—41no—
213C2×C4×C79—41no—
214C2×C4×C79—41no—
215C2×C4×C79—41no—
216C2×C4×C79—41no—
217C2×C4×C79—41no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79C22⋊Q812yes—
221C2×C4×C79C22×Q811yescharacteristic
222C2×C4×C79C2×#4111yescharacteristic
223C2×C4×C79C2×#4111yescharacteristic
224C2×C4×C79C2×#4111yescharacteristic
225C2×C4×C79H(5,𝔽2)11yescharacteristic
226C2×C4×C792−1+411yescharacteristic
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79—21no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79AGL(1,ℤ/8)12yes—
251C8×C79—22no—
252C8×C79—41no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C22×C79—22no—
256C22×C79—22no—
257C22×C79—22no—
258C22×C79—41no—
259C22×C79—41no—
260C22×C79—41no—
261C22×C79—41no—
262C22×C79C2×C22⋊Q811yescharacteristic
263C22×C79#54611yescharacteristic
264C22×C79#54711yescharacteristic
265C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C79#57511yescharacteristic
267C22×C79#57711yescharacteristic
268C22×C79#57811yescharacteristic
269C4×C79—22no—
270C4×C79—42no—
271C4×C79—42no—
272C4×C79—42no—
273C4×C79—81no—
274C4×C79#53311yescharacteristic
275C4×C79#54411yescharacteristic
276C4×C79#55111yescharacteristic
277C4×C79#55211yescharacteristic
278Dic79—22no—
279Dic79—41no—
280Dic79—41no—
281Dic79—41no—
282Dic79—81no—
283Dic79—81no—
284Dic79—81no—
285Dic79—81no—
286#26014—791nomaximal, Sylow
287C2×C79—42no—
288C2×C79#243811yescharacteristic
289C2×C79#307711yescharacteristic
290C2×C79#308211yescharacteristic
291C2×C79#331711yescharacteristic
292C2×C79#333011yescharacteristic
293C2×C79#334511yescharacteristic
294C2×C79#335211yescharacteristic
295#2441—791no—
296#2344—791no—
297#2100—791no—
298#2088—791no—
299#2086—791no—
300#2059—791no—
301#1989—791no—
302#1948—791no—
303#1921—791no—
304#1918—791no—
305#1854—791no—
306#1851—791no—
307#1824—791no—
308#1809—791no—
309#1736D7911yescharacteristic
310C79#2601411yescharacteristic, complemented, Sylow
311C22×M16C2×D7911yescharacteristic
312C2×#125—791no—
313C2×#117—1581no—
314C2×#117C2×D7911yescharacteristic
315C22×C4⋊C4C2×D7911yescharacteristic
316C2×#107—791no—
317C2×#107—791no—
318C2×#106—791no—
319C2×#106—791no—
320C2×C4⋊C8—791no—
321C2×C4⋊C8—791no—
322C2×C22⋊C8—791no—
323#399—1581no—
324#396—1581no—
325#389—791no—
326#389—791no—
327#389—791no—
328#389—791no—
329#389—1581no—
330#387—1581no—
331#387—1581no—
332#387—1581no—
333#384—791no—
334#381—1581no—
335#380—791no—
336#380—791no—
337C4×C4⋊C4—791no—
338C4×C4⋊C4—791no—
339C4×C22⋊C4—791no—
340#340—791no—
341#340—791no—
342#340—791no—
343#340—791no—
344#340—791no—
345#340—791no—
346#340—791no—
347#340—791no—
348#340C2×D7911yescharacteristic
349#340C2×D7911yescharacteristic
350#340C2×D7911yescharacteristic
351#340C2×D7911yescharacteristic
352#336—791no—
353#336—791no—
354C23×C4—21no—
355C23×C4C22×D7911yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16—41no—
359C22×C8—791no—
360C22×C8—791no—
361C22×C8C22×D7911yescharacteristic
362C22×C8C22×D7911yescharacteristic
363#125—3161no—
364#125—1582no—
365#117—22no—
366#117—41no—
367#117—3164no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—791no—
372C2×C4⋊C4—791no—
373C2×C4⋊C4—791no—
374C2×C4⋊C4—1581no—
375C2×C4⋊C4—1581no—
376C2×C4⋊C4—1581no—
377C2×C4⋊C4—1581no—
378C2×C4⋊C4—1581no—
379C2×C4⋊C4—1581no—
380C2×C4⋊C4—1581no—
381C2×C4⋊C4—1581no—
382C2×C4⋊C4Dic15814yes—
383C2×C4⋊C4C22×D7911yescharacteristic
384C2×C4⋊C4C22×D7911yescharacteristic
385C2×C4⋊C4C22×D7911yescharacteristic
386C2×C4⋊C4C22×D7911yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—791no—
389C2×C22⋊C4—791no—
390C2×C22⋊C4—1581no—
391C2×C22⋊C4—1581no—
392C2×C42—21no—
393C2×C42—791no—
394C2×C42—791no—
395C2×C42—1581no—
396C2×C42—1581no—
397#107—1582no—
398#107—1582no—
399#106—3161no—
400#106—3161no—
401C4⋊C8—3161no—
402C4⋊C8—1582no—
403C22⋊C8—1582no—
404#95—791no—
405#95—791no—
406#95—1581no—
407#95—1581no—
408#95—1581no—
409#95—1581no—
410#95—1581no—
411#95—1581no—
412#95—1581no—
413#95—1581no—
414C24D4×D7911yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—791no—
427C22×C4—791no—
428C22×C4—791no—
429C22×C4—791no—
430C22×C4—1581no—
431C22×C4—1581no—
432C22×C4—1581no—
433C22×C4—1581no—
434C22×C4D4×D7911yescharacteristic
435C22×C4C2×Dic15812yes—
436C22×C4#1192962312yes—
437C22×C4C23×D7911yescharacteristic
438M16—44no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—1581no—
445C2×C8—1581no—
446C2×C8—1581no—
447C2×C8—1581no—
448C2×C8—1581no—
449C2×C8—1581no—
450C2×C8#1192962412yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—3162no—
458C4⋊C4—3162no—
459C4⋊C4—3162no—
460C4⋊C4—3162no—
461C4⋊C4—3162no—
462C4⋊C4—3162no—
463C4⋊C4—6321no—
464C4⋊C4—6321no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4C2×Dic15812yes—
474C4⋊C4C2×Dic15812yes—
475C4⋊C4#1192962412yes—
476C22⋊C4—42no—
477C22⋊C4—3162no—
478C22⋊C4—3161no—
479C22⋊C4—6321no—
480C22⋊C4—1582no—
481C42—42no—
482C42—3162no—
483C42—3162no—
484C42—1581no—
485C42—1581no—
486C42—1581no—
487C42—1581no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23#1457913812yes—
495C23C2×D4×D7911yescharacteristic
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—48no—
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—41no—
515C2×C4—3161no—
516C2×C4—3161no—
517C2×C4—3161no—
518C2×C4—3161no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—3161no—
523C2×C4—3161no—
524C2×C4—3161no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—792no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4—1581no—
537C2×C4—1581no—
538C2×C4—1581no—
539C2×C4#1457913812yes—
540C2×C4C22×Dic15811yescharacteristic
541C2×C4C2×#1192962311yescharacteristic
542C2×C4C2×#1192962411yescharacteristic
543C2×C4C2×#1192962411yescharacteristic
544C2×C4#1457920611yescharacteristic
545C2×C4#1457920911yescharacteristic
546C8—22no—
547C8—41no—
548C8—3161no—
549C8—3161no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×#1457913811yescharacteristic
558C22#1909920311yescharacteristic
559C22#1909921411yescharacteristic
560C22AGL(1,ℤ/8)×D7911yescharacteristic
561C22#1909935911yescharacteristic
562C22#1909937211yescharacteristic
563C22#1909937911yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—81no—
569C4—3161no—
570C4—3161no—
571C4—3161no—
572C4—6321no—
573C4—6321no—
574C4—6321no—
575C4—6321no—
576C4—1582no—
577C4#1909912011yescharacteristic
578C4#1909919411yescharacteristic
579C4#1909923911yescharacteristic
580C4#1909924811yescharacteristic
581C2—42no—
582C2#2376707611yescharacteristic
583C2#2377319411yescharacteristic
584C2#2377324411yescharacteristic
585C2#2377552711yescharacteristic
586C2#2377570111yescharacteristic
587C2#2377587111yescharacteristic
588C2#2377597111yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#1736C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#1736D79
hypocenterC79#26014
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
3. #23764966C2semidirect
4. #23764903C2semidirect
5. #23764890C2semidirect
6. #23764737C2semidirect
7. #23764354C2semidirect
8. #23764162C2semidirect
9. #23763991C2semidirect
10. #23763974C2semidirect
310. C79#26014semidirect

Maximal subgroups

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

subgroupindex
#237671072
#237662692
#237649662
#237649032
#237648902
#237647372
#237643542
#237641622
#237639912
#237639742
#237635992
#237635832
#237634902
#237634332
C79×#17362
#2601479