Finite Group Databasebeta

← #34347133

Subgroups — #34347133

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
2C2×#19098606C211yesmaximal, characteristic
3#23767107C211yesmaximal, characteristic
4#23767103C211yesmaximal, characteristic
5#23764880C211yesmaximal, characteristic, complemented
6#23764874C211yesmaximal, characteristic, complemented
7#23764737C211yesmaximal, characteristic, complemented
8#23764731C211yesmaximal, characteristic, complemented
9#23764137C211yesmaximal, characteristic, complemented
10#23764132C211yesmaximal, characteristic, complemented
11#23763595C211yesmaximal, characteristic
12#23763555C211yesmaximal, characteristic, complemented
13#23763431C211yesmaximal, characteristic
14#23763186C211yesmaximal, characteristic
15C79×#1854C211yesmaximal, characteristic
16C22×#11929608C2211yescharacteristic
17C2×#14579160C2211yescharacteristic
18C2×#14579152C2211yescharacteristic
19C2×#14579138—21no—
20C22×C4⋊Dic79C2211yescharacteristic
21#19098971—24no—
22#19098606—24no—
23C2×#14579094C2211yescharacteristic
24C2×#14579094C2211yescharacteristic
25C2×#14579094C2211yescharacteristic
26C2×#14579093C2211yescharacteristic
27C2×#14579093C2211yescharacteristic
28C2×#14579092C2211yescharacteristic
29C2×#14579092C2211yescharacteristic
30C2×#14579092C2211yescharacteristic
31C2×#14579071C2211yescharacteristic
32C2×C8×Dic79C2211yescharacteristic
33#19098499—21no—
34#19098497—21no—
35#19098486—21no—
36#19098484—21no—
37#19098462C2211yescharacteristic
38#19098462C2211yescharacteristic
39#19098461—21no—
40#19098459C2211yescharacteristic
41#19098459C2211yescharacteristic
42#19098447—21no—
43#19098435—21no—
44#19098432C2211yescharacteristic
45#19098432C2211yescharacteristic
46#19098423C2211yescharacteristic
47C4×C22⋊Dic79C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098326C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098317C2211yescharacteristic
53#19098317C2211yescharacteristic
54C2×#117×C79C2211yescharacteristic
55C2×#107×C79C2211yescharacteristic
56C2×#106×C79C2211yescharacteristic
57C2×C22⋊C8×C79C2211yescharacteristic
58#380×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60#340×C79C2211yescharacteristic
61C22×Dic158—21no—
62C23×Dic79—21no—
63C2×#11929608—22no—
64C2×#11929608—22no—
65C2×#11929608Q814yescomplemented
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68#14579160—22no—
69#14579160—41no—
70#14579152—22no—
71#14579152—41no—
72#14579138—44no—
73C2×C4⋊Dic79—22no—
74C2×C4⋊Dic79—21no—
75C2×C4⋊Dic79—21no—
76C2×C4⋊Dic79—21no—
77C2×C4⋊Dic79—21no—
78C2×C4⋊Dic79—41no—
79C2×C4⋊Dic79C2311yescharacteristic
80C2×C4⋊Dic79C2311yescharacteristic
81C2×C4⋊Dic79C2311yescharacteristic
82C2×C4⋊Dic79C2311yescharacteristic
83C2×Dic79⋊C4—21no—
84C2×Dic79⋊C4—21no—
85C2×Dic79⋊C4—21no—
86C2×Dic79⋊C4—21no—
87C2×Dic79⋊C4—21no—
88C2×Dic79⋊C4—21no—
89C2×Dic79⋊C4—21no—
90C2×Dic79⋊C4—21no—
91C2×C22⋊Dic79—21no—
92C2×C22⋊Dic79D411yescharacteristic
93C2×C22⋊Dic79D411yescharacteristic
94C2×C4×Dic79C2311yescharacteristic
95C2×C4×Dic79C2311yescharacteristic
96#14579094—22no—
97#14579094—22no—
98#14579094D414yes—
99#14579093—22no—
100#14579093—41no—
101#14579092—22no—
102#14579092—22no—
103#14579092—22no—
104#14579071—22no—
105C8×Dic79—22no—
106C79⋊#95—21no—
107C79⋊#95—21no—
108C79⋊#95—21no—
109C79⋊#95—21no—
110C79⋊#95—21no—
111C79⋊#95D411yescharacteristic
112C79⋊#95D411yescharacteristic
113C23×C4×C79C2311yescharacteristic
114C22×C8×C79C2311yescharacteristic
115C22×C8×C79C2311yescharacteristic
116#117×C79—22no—
117#117×C79—41no—
118C2×C4⋊C4×C79C2311yescharacteristic
119C2×C4⋊C4×C79C2311yescharacteristic
120C2×C4⋊C4×C79C2311yescharacteristic
121C2×C4⋊C4×C79C2311yescharacteristic
122C2×C22⋊C4×C79—21no—
123C2×C22⋊C4×C79—21no—
124C2×C42×C79—21no—
125#107×C79—22no—
126#106×C79—41no—
127C22⋊C8×C79Q814yescomplemented
128#95×C79—21no—
129C2×Dic158—42no—
130C2×Dic158—42no—
131C2×Dic158—81no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C22×Dic79—42no—
137C22×Dic79—42no—
138C22×Dic79—81no—
139C22×Dic79C2×D411yescharacteristic
140C22×Dic79C2×D411yescharacteristic
141C22×Dic79C2×D411yescharacteristic
142C22×Dic79C2×D411yescharacteristic
143#11929608—28no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8C2×D412yes—
147C2×C79⋊C8C2×Q812yes—
148C2×C79⋊C8C2×Q812yes—
149C2×C79⋊C8#4112yes—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79—42no—
158C4⋊Dic79—42no—
159C4⋊Dic79—42no—
160C4⋊Dic79—42no—
161C4⋊Dic79C2×D412yes—
162Dic79⋊C4—42no—
163Dic79⋊C4—42no—
164Dic79⋊C4—42no—
165Dic79⋊C4—42no—
166Dic79⋊C4—42no—
167Dic79⋊C4—81no—
168Dic79⋊C4—81no—
169Dic79⋊C4—81no—
170C22⋊Dic79—22no—
171C22⋊Dic79—22no—
172C22⋊Dic79—81no—
173C4×Dic79—21no—
174C4×Dic79—21no—
175C4×Dic79—21no—
176C4×Dic79—21no—
177C24×C79C2×D411yescharacteristic
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79—21no—
182C22×C4×C79—21no—
183C22×C4×C79—21no—
184C22×C4×C79—21no—
185C22×C4×C79—21no—
186C22×C4×C79C2×D411yescharacteristic
187C22×C4×C79C2×Q812yes—
188C22×C4×C79C2×Q812yes—
189C22×C4×C79C2411yescharacteristic
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79—21no—
193C2×C8×C79—21no—
194C2×C8×C79C2×Q812yes—
195C2×C8×C79C2×Q812yes—
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—21no—
203C4⋊C4×C79C2×D412yes—
204C22⋊C4×C79—42no—
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—41no—
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×C79C4⋊Q812yes—
234C2×C4×C79C22×D411yescharacteristic
235C2×C4×C79C22×Q811yescharacteristic
236C2×C4×C79C22×Q811yescharacteristic
237C2×C4×C79C2×#4111yescharacteristic
238C2×C4×C79H(5,𝔽2)11yescharacteristic
239C2×C4×C792−1+411yescharacteristic
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—21no—
244C2×Dic79—21no—
245C2×Dic79—21no—
246C2×Dic79—22no—
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—84no—
263C2×Dic79AGL(1,ℤ/8)12yes—
264C2×Dic79#13712yes—
265Dic158—84no—
266C8×C79—22no—
267C8×C79—22no—
268C79⋊C8—22no—
269C79⋊C8C22⋊Q814yescomplemented
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—81no—
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—42no—
297Dic79—41no—
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×#428—791no—
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#1854D7911yescharacteristic
321#1853—791no—
322#1844—791no—
323#1809—791no—
324#1753—791no—
325C79#2607811yescharacteristic, complemented, Sylow
326C22×M16—791no—
327C2×C4⋊Q8—791no—
328C2×#125—791no—
329C2×C22⋊Q8—1581no—
330C2×#117C2×D7911yescharacteristic
331C22×C4⋊C4—791no—
332#501—1584no—
333#428—1584no—
334C2×#107—791no—
335C2×#107—791no—
336C2×#107—791no—
337C2×#107C2×D7911yescharacteristic
338C2×#106—791no—
339C2×#106—791no—
340C2×#106—791no—
341C2×#106—791no—
342C2×#106—791no—
343C2×#106C2×D7911yescharacteristic
344C2×C22⋊C8C2×D7911yescharacteristic
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#380C2×D7911yescharacteristic
362C4×C22⋊C4—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#340C2×D7911yescharacteristic
368#340C2×D7911yescharacteristic
369#336—791no—
370#336—791no—
371C22×Q8—1581no—
372C23×C4—1581no—
373C23×C4C22×D7911yescharacteristic
374C2×M16—794no—
375C2×M16—1582no—
376C2×M16—1582no—
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—3161no—
389C2×C4⋊C4—791no—
390C2×C4⋊C4—791no—
391C2×C4⋊C4—791no—
392C2×C4⋊C4—791no—
393C2×C4⋊C4—1581no—
394C2×C4⋊C4—1581no—
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—1582no—
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—21no—
412C2×C22⋊C4—791no—
413C2×C22⋊C4—791no—
414C2×C22⋊C4—1581no—
415C2×C42—21no—
416C2×C42—791no—
417C2×C42—791no—
418#107—22no—
419#107—794no—
420#107—1582no—
421#107—1582no—
422#106—41no—
423#106—3161no—
424#106—1582no—
425#106—1582no—
426#106—1582no—
427#106—1582no—
428C22⋊C8Dic15814yescomplemented
429#97—1582no—
430C4×C8—1582no—
431#95—21no—
432#95—791no—
433#95—791no—
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—3162no—
452C22×C4—3162no—
453C22×C4—791no—
454C22×C4—791no—
455C22×C4—791no—
456C22×C4—791no—
457C22×C4—6321no—
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—1588no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—792no—
472C2×C8—792no—
473C2×C8—792no—
474C2×C8—792no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8C2×Dic15812yes—
478C2×C8C2×Dic15812yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
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—3162no—
495C4⋊C4—792no—
496C4⋊C4—6321no—
497C4⋊C4—6321no—
498C4⋊C4—6321no—
499C4⋊C4—1581no—
500C4⋊C4—1581no—
501C4⋊C4—1581no—
502C4⋊C4—1581no—
503C4⋊C4—1581no—
504C4⋊C4—1581no—
505C4⋊C4—1581no—
506C4⋊C4D4×D7912yes—
507C22⋊C4—42no—
508C22⋊C4—42no—
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—41no—
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—3161no—
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—792no—
556C2×C4—792no—
557C2×C4—6324no—
558C2×C4—1581no—
559C2×C4—1581no—
560C2×C4—1582no—
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×C4C2×D4×D7911yescharacteristic
571C2×C4C2×Q8×D7911yescharacteristic
572C2×C4C22×Dic15811yescharacteristic
573C2×C4#41×D7911yescharacteristic
574C2×C4#1457920611yescharacteristic
575C2×C4#1457920811yescharacteristic
576C8—22no—
577C8—22no—
578C8—794no—
579C8—1582no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1457916011yescharacteristic
588C22#1909919511yescharacteristic
589C22#1909921111yescharacteristic
590C22AGL(1,ℤ/8)×D7911yescharacteristic
591C22#137×D7911yescharacteristic
592C22#1909937011yescharacteristic
593C22#1909937011yescharacteristic
594C4—22no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—3162no—
600C4—3161no—
601C4—3161no—
602C4—6321no—
603C4—6321no—
604C4—6321no—
605C4—1582no—
606C4—1582no—
607C4C22⋊Q8×D7911yescharacteristic
608C4#1909919211yescharacteristic
609C4D4×Dic15811yescharacteristic
610C4#1909927211yescharacteristic
611C2—42no—
612C2#2376710111yescharacteristic
613C2#2377393211yescharacteristic
614C2#2377394411yescharacteristic
615C2#2377499311yescharacteristic
616C2#2377505911yescharacteristic
617C2#2377584811yescharacteristic
618C2#2377587311yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#1854C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#1854D79
hypocenterC79#26078
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
1. C2×#19098971C2semidirect
5. #23764880C2semidirect
6. #23764874C2semidirect
7. #23764737C2semidirect
8. #23764731C2semidirect
9. #23764137C2semidirect
10. #23764132C2semidirect
12. #23763555C2semidirect
65. C2×#11929608Q8semidirect
127. C22⋊C8×C79Q8semidirect
269. C79⋊C8C22⋊Q8semidirect
325. C79#26078semidirect
428. C22⋊C8Dic158semidirect

Maximal subgroups

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

subgroupindex
C2×#190989712
C2×#190986062
#237671072
#237671032
#237648802
#237648742
#237647372
#237647312
#237641372
#237641322
#237635952
#237635552
#237634312
#237631862
C79×#18542
#2607879