Finite Group Databasebeta

← #34292916

Subgroups — #34292916

Order 20224.

1338 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×#19099188C211yesmaximal, characteristic
2#23766013C211yesmaximal, characteristic
3#23765752C211yesmaximal, characteristic
4#23764386C211yesmaximal, characteristic, complemented
5#23764385C211yesmaximal, characteristic, complemented
6#23764384C211yesmaximal, characteristic, complemented
7#23764383C214yesmaximal, complemented
8#342×D79C214yesmaximal
9C79×#1996C211yesmaximal, characteristic, complemented
10#19099188—22no—
11#19099188—22no—
12#19099188—22no—
13#19099188—22no—
14C2×#14579141C412yescomplemented
15C2×#14579140C2211yescharacteristic
16C2×#14579139C412yescomplemented
17C2×#14579139C412yescomplemented
18C2×#14579138C2212yes—
19C2×#14579137C412yescomplemented
20C22×(C2×D79)⋊C4C2211yescharacteristic
21C2×C22⋊C4×D79C2212yes—
22C2×C22⋊C4×D79C2212yes—
23C2×C22⋊C4×D79C2212yes—
24#19098437—22no—
25#19098434—22no—
26#19098423C2211yescharacteristic
27#19098421C2212yes—
28C22⋊C4×Dic79C2212yes—
29C4×(C2×D79)⋊C4—22no—
30C4×C22⋊Dic79C2211yescharacteristic
31#19098410—22no—
32#19098330C2214yes—
33#19098329C2214yes—
34#19098329C2214yes—
35C2×C22⋊Q8×C79C2211yescharacteristic
36#380×C79C2211yescharacteristic
37C4×C22⋊C4×C79C2211yescharacteristic
38#342×C79C2214yes—
39C24×D79C2311yescharacteristic
40C22×Dic158C2×C412yescomplemented
41C22×Dic158C2×C411yescharacteristic, complemented
42C23×Dic79C2311yescharacteristic
43C22×C4×D79—22no—
44C22×C4×D79—22no—
45C22×C4×D79C2×C412yes—
46C22×C4×D79C2×C412yes—
47#14579141—24no—
48#14579141—24no—
49#14579140—21no—
50#14579140—21no—
51#14579140—21no—
52#14579140—21no—
53#14579139—24no—
54#14579139—24no—
55#14579139—24no—
56#14579139—24no—
57#14579138—24no—
58#14579138—24no—
59#14579137—24no—
60#14579137—24no—
61C2×C4⋊Dic79—21no—
62C2×C4⋊Dic79C2×C412yescomplemented
63C2×C4⋊Dic79C2×C411yescharacteristic, complemented
64C2×Dic79⋊C4—22no—
65C2×Dic79⋊C4C2×C412yescomplemented
66C2×Dic79⋊C4C2×C412yescomplemented
67C2×Dic79⋊C4C2×C412yescomplemented
68C2×C22⋊Dic79C2312yes—
69C2×C22⋊Dic79C2312yes—
70C2×C22⋊Dic79C2312yes—
71C2×(C2×D79)⋊C4—22no—
72C2×(C2×D79)⋊C4—22no—
73C2×(C2×D79)⋊C4C2×C412yes—
74C2×(C2×D79)⋊C4C2×C412yes—
75C2×(C2×D79)⋊C4C2×C412yes—
76C2×(C2×D79)⋊C4C2×C412yes—
77C2×(C2×D79)⋊C4D412yes—
78C2×(C2×D79)⋊C4D412yes—
79C2×(C2×D79)⋊C4D412yes—
80C2×(C2×D79)⋊C4D412yes—
81C22⋊C4×D79—24no—
82C22⋊C4×D79—24no—
83C22⋊C4×D79—24no—
84C22⋊C4×D79—28no—
85C22⋊C4×D79Q818yescomplemented
86C2×C4×Dic79—21no—
87C2×C4×Dic79—22no—
88C79⋊#95—21no—
89C79⋊#95—21no—
90C79⋊#95—22no—
91C79⋊#95—22no—
92C22×Q8×C79C2×C411yescharacteristic, complemented
93C23×C4×C79C2311yescharacteristic
94C22⋊Q8×C79—21no—
95C22⋊Q8×C79—21no—
96C22⋊Q8×C79—21no—
97C22⋊Q8×C79—21no—
98C2×C4⋊C4×C79—21no—
99C2×C4⋊C4×C79C2×C411yescharacteristic, complemented
100C2×C4⋊C4×C79C2×C412yescomplemented
101C2×C22⋊C4×C79C2312yes—
102C2×C22⋊C4×C79C2312yes—
103C2×C22⋊C4×C79C2312yes—
104C2×C42×C79—21no—
105#95×C79—21no—
106#95×C79—21no—
107C23×D79—81no—
108C23×D79—81no—
109C23×D79C22×C412yes—
110C23×D79C2×D412yes—
111C23×D79C2×D412yes—
112C23×D79C2×Q814yes—
113C23×D79#4114yes—
114C2×Dic158—24no—
115C2×Dic158—21no—
116C2×Dic158—21no—
117C2×Dic158—81no—
118C2×Dic158—82no—
119C22×Dic79—22no—
120C22×Dic79—22no—
121C22×Dic79—81no—
122C22×Dic79C22×C412yes—
123C22×Dic79C22×C412yes—
124C22×Dic79C22×C412yes—
125C22×Dic79C2×D412yes—
126C22×Dic79C2×D412yes—
127C2×C4×D79—24no—
128C2×C4×D79—24no—
129C2×C4×D79—28no—
130C2×C4×D79—44no—
131C2×C4×D79—82no—
132C2×C4×D79—82no—
133C2×C4×D79—82no—
134C2×C4×D79—82no—
135C2×C4×D79—82no—
136C2×C4×D79—82no—
137C4⋊Dic79—24no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—81no—
141Dic79⋊C4—24no—
142Dic79⋊C4—24no—
143Dic79⋊C4—22no—
144Dic79⋊C4—22no—
145Dic79⋊C4—82no—
146C22⋊Dic79—22no—
147C22⋊Dic79—22no—
148C22⋊Dic79—22no—
149C22⋊Dic79—24no—
150C22⋊Dic79C2×Q814yes—
151(C2×D79)⋊C4—24no—
152(C2×D79)⋊C4—24no—
153(C2×D79)⋊C4—28no—
154(C2×D79)⋊C4—44no—
155(C2×D79)⋊C4C22⋊C414yes—
156(C2×D79)⋊C4C22⋊C414yes—
157(C2×D79)⋊C4C22⋊C414yes—
158(C2×D79)⋊C4C22⋊C414yes—
159C4×Dic79—81no—
160C4×Dic79—82no—
161C24×C79C2411yescharacteristic
162C2×Q8×C79—21no—
163C2×Q8×C79—21no—
164C2×Q8×C79—81no—
165C22×C4×C79—22no—
166C22×C4×C79—22no—
167C22×C4×C79—81no—
168C22×C4×C79C22×C412yes—
169C22×C4×C79C22×C412yes—
170C22×C4×C79C22×C411yescharacteristic
171C22×C4×C79C22×C411yescharacteristic
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C4⋊C4×C79—21no—
177C4⋊C4×C79—21no—
178C4⋊C4×C79—22no—
179C4⋊C4×C79—22no—
180C4⋊C4×C79—81no—
181C22⋊C4×C79—22no—
182C22⋊C4×C79—22no—
183C22⋊C4×C79—22no—
184C22⋊C4×C79—24no—
185C22⋊C4×C79C2×Q814yes—
186C42×C79—81no—
187C23×C79—41no—
188C23×C79—41no—
189C23×C79C23×C411yescharacteristic
190C23×C79C22×D411yescharacteristic
191C23×C79C22×D411yescharacteristic
192C23×C79C22×Q812yes—
193C23×C79C2×#4112yes—
194C22×D79—42no—
195C22×D79—42no—
196C22×D79—42no—
197C22×D79—42no—
198C22×D79—42no—
199C22×D79—42no—
200C22×D79—84no—
201C22×D79—84no—
202C22×D79C2×C22⋊C414yes—
203C22×D79C4×Q818yes—
204C22×D79C22⋊Q814yes—
205C22×D79C22⋊Q814yes—
206C22×D79#12414yes—
207C22×D79C4⋊Q814yes—
208Q8×C79—81no—
209Q8×C79—81no—
210C2×C4×C79—22no—
211C2×C4×C79—22no—
212C2×C4×C79—24no—
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—42no—
220C2×C4×C79—42no—
221C2×C4×C79—42no—
222C2×C4×C79—42no—
223C2×C4×C79—42no—
224C2×C4×C79—42no—
225C2×C4×C79—42no—
226C2×C4×C79C2×C22⋊C411yescharacteristic
227C2×C4×C79C2×C22⋊C411yescharacteristic
228C2×C4×C79C2×C22⋊C411yescharacteristic
229C2×C4×C79C2×C22⋊C411yescharacteristic
230C2×Dic79—22no—
231C2×Dic79—22no—
232C2×Dic79—24no—
233C2×Dic79—42no—
234C2×Dic79—42no—
235C2×Dic79—42no—
236C2×Dic79—42no—
237C2×Dic79—42no—
238C2×Dic79—42no—
239C2×Dic79—42no—
240C2×Dic79—42no—
241C2×Dic79—42no—
242C2×Dic79—42no—
243C2×Dic79C2×C22⋊C414yes—
244C4×D79—84no—
245C4×D79—84no—
246C4×D79—88no—
247C4×D79—88no—
248Dic158—84no—
249Dic158—81no—
250Dic158—81no—
251C22×C79—21no—
252C22×C79—21no—
253C22×C79—21no—
254C22×C79—21no—
255C22×C79—21no—
256C22×C79—21no—
257C22×C79—42no—
258C22×C79—42no—
259C22×C79C22×C22⋊C411yescharacteristic
260C22×C79C2×C4×Q812yes—
261C22×C79C2×C22⋊Q812yes—
262C22×C79C2×#12411yescharacteristic
263C22×C79C2×C4⋊Q811yescharacteristic
264C2×D79—24no—
265C2×D79—24no—
266C2×D79—48no—
267C2×D79—44no—
268C2×D79—44no—
269C2×D79—44no—
270C2×D79—44no—
271C2×D79#39118yes—
272C4×C79—21no—
273C4×C79—21no—
274C4×C79—21no—
275C4×C79—21no—
276C4×C79—44no—
277C4×C79—42no—
278C4×C79—42no—
279C4×C79—82no—
280Dic79—24no—
281Dic79—44no—
282Dic79—42no—
283Dic79—42no—
284Dic79—82no—
285#21333—791nomaximal, Sylow
286C2×C79—22no—
287C2×C79—21no—
288C2×C79—22no—
289C2×C79—21no—
290C2×C79C2×#39111yescharacteristic
291C2×C79#288311yescharacteristic
292C2×C79#288311yescharacteristic
293D79—28no—
294C2×#543—791no—
295#2315—791no—
296#2273—791no—
297#1996—791no—
298#1996—791no—
299#1996—794no—
300#1996—791no—
301#1996D7911yescharacteristic, complemented
302C2×#342—794no—
303C79#2133311yescharacteristic, complemented, Sylow
304#543—1582no—
305#543—1582no—
306#543—1582no—
307#543—1582no—
308C2×C22⋊Q8—792no—
309C2×C22⋊Q8—791no—
310C2×C22⋊Q8—792no—
311C2×C22⋊Q8—792no—
312C2×C22⋊Q8—792no—
313C2×C22⋊Q8—792no—
314C2×C22⋊Q8C2×D7911yescharacteristic
315C22×C22⋊C4—791no—
316C22×C22⋊C4—792no—
317C22×C22⋊C4—792no—
318C22×C22⋊C4—792no—
319#385—1582no—
320#385—1582no—
321#380—791no—
322#380—792no—
323#380C2×D7911yescharacteristic
324C4×C22⋊C4—791no—
325C4×C22⋊C4—792no—
326C4×C22⋊C4—1582no—
327C4×C22⋊C4—1582no—
328C4×C22⋊C4C2×D7911yescharacteristic
329#342—794no—
330#342—794no—
331#342—794no—
332#342C2×D7914yes—
333C25—791no—
334C22×Q8—792no—
335C22×Q8—791no—
336C22×Q8C4×D7911yescharacteristic, complemented
337C23×C4—791no—
338C23×C4—792no—
339C23×C4—792no—
340C23×C4—1582no—
341C23×C4—1582no—
342C23×C4C22×D7911yescharacteristic
343C22⋊Q8—21no—
344C22⋊Q8—21no—
345C22⋊Q8—21no—
346C22⋊Q8—21no—
347C22⋊Q8—1584no—
348C22⋊Q8—1584no—
349C22⋊Q8—1584no—
350C22⋊Q8—1584no—
351C22⋊Q8—1584no—
352C22⋊Q8—1584no—
353C22⋊Q8—1584no—
354C22⋊Q8—1584no—
355C22⋊Q8—1584no—
356C22⋊Q8—1584no—
357C22⋊Q8—1581no—
358C22⋊Q8—1581no—
359C22⋊Q8—1581no—
360C22⋊Q8—1581no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—792no—
363C2×C4⋊C4—792no—
364C2×C4⋊C4—792no—
365C2×C4⋊C4—792no—
366C2×C4⋊C4—791no—
367C2×C4⋊C4—1582no—
368C2×C4⋊C4—1581no—
369C2×C4⋊C4C4×D7912yescomplemented
370C2×C4⋊C4C4×D7911yescharacteristic, complemented
371C2×C22⋊C4—792no—
372C2×C22⋊C4—792no—
373C2×C22⋊C4—792no—
374C2×C22⋊C4—792no—
375C2×C22⋊C4—792no—
376C2×C22⋊C4—792no—
377C2×C22⋊C4—792no—
378C2×C22⋊C4—798no—
379C2×C22⋊C4—792no—
380C2×C22⋊C4—792no—
381C2×C22⋊C4—792no—
382C2×C22⋊C4—792no—
383C2×C22⋊C4—1588no—
384C2×C22⋊C4—1584no—
385C2×C22⋊C4—1584no—
386C2×C22⋊C4—1584no—
387C2×C22⋊C4—1582no—
388C2×C22⋊C4—1582no—
389C2×C22⋊C4C22×D7912yes—
390C2×C22⋊C4C22×D7912yes—
391C2×C22⋊C4C22×D7912yes—
392C2×C42—21no—
393C2×C42—1582no—
394C2×C42—1581no—
395#95—21no—
396#95—21no—
397#95—1582no—
398#95—1582no—
399#95—1581no—
400#95—1581no—
401C24—792no—
402C24—792no—
403C24—792no—
404C24—794no—
405C24—794no—
406C24—6321no—
407C24—6321no—
408C24C23×D7911yescharacteristic
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—81no—
412C2×Q8—6321no—
413C2×Q8—6322no—
414C2×Q8—1584no—
415C2×Q8—1581no—
416C2×Q8—1581no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—81no—
420C22×C4—3164no—
421C22×C4—792no—
422C22×C4—792no—
423C22×C4—792no—
424C22×C4—792no—
425C22×C4—792no—
426C22×C4—6322no—
427C22×C4—6322no—
428C22×C4—6322no—
429C22×C4—6322no—
430C22×C4—6322no—
431C22×C4—6322no—
432C22×C4—6321no—
433C22×C4—1588no—
434C22×C4—1584no—
435C22×C4—1584no—
436C22×C4—1582no—
437C22×C4—1582no—
438C22×C4C2×C4×D7912yes—
439C22×C4C2×C4×D7912yes—
440C22×C4C2×C4×D7911yescharacteristic
441C22×C4C2×C4×D7911yescharacteristic
442C22×C4D4×D7911yescharacteristic
443C22×C4D4×D7911yescharacteristic
444C22×C4D4×D7911yescharacteristic
445C22×C4D4×D7911yescharacteristic
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—81no—
451C4⋊C4—6322no—
452C4⋊C4—6321no—
453C4⋊C4—1584no—
454C4⋊C4—1584no—
455C4⋊C4—1584no—
456C4⋊C4—1582no—
457C4⋊C4—1582no—
458C4⋊C4—1581no—
459C4⋊C4—1581no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—24no—
464C22⋊C4—3164no—
465C22⋊C4—794no—
466C22⋊C4—794no—
467C22⋊C4—794no—
468C22⋊C4—794no—
469C22⋊C4—794no—
470C22⋊C4—1588no—
471C22⋊C4—1584no—
472C22⋊C4—1584no—
473C22⋊C4—1584no—
474C22⋊C4—1582no—
475C22⋊C4—1582no—
476C22⋊C4—1582no—
477C22⋊C4Q8×D7914yes—
478C42—81no—
479C42—6321no—
480C42—6322no—
481C23—41no—
482C23—41no—
483C23—3162no—
484C23—3162no—
485C23—3162no—
486C23—3162no—
487C23—3162no—
488C23—3162no—
489C23—798no—
490C23—794no—
491C23—794no—
492C23—794no—
493C23—794no—
494C23—794no—
495C23—6324no—
496C23—6324no—
497C23C22×C4×D7911yescharacteristic
498C23C2×D4×D7911yescharacteristic
499C23C2×D4×D7911yescharacteristic
500C23C2×Q8×D7912yes—
501C23#41×D7912yes—
502Q8—81no—
503Q8—81no—
504Q8—6321no—
505Q8—6321no—
506Q8—6324no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—3162no—
524C2×C4—3162no—
525C2×C4—3162no—
526C2×C4—3162no—
527C2×C4—3162no—
528C2×C4—3162no—
529C2×C4—3162no—
530C2×C4—3162no—
531C2×C4—3162no—
532C2×C4—3162no—
533C2×C4—794no—
534C2×C4—6328no—
535C2×C4—6324no—
536C2×C4—6324no—
537C2×C4—6328no—
538C2×C4—1582no—
539C2×C4—1582no—
540C2×C4—1584no—
541C2×C4C22⋊C4×D7911yescharacteristic
542C2×C4C22⋊C4×D7911yescharacteristic
543C2×C4C22⋊C4×D7911yescharacteristic
544C2×C4C22⋊C4×D7911yescharacteristic
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—42no—
552C22—42no—
553C22—3164no—
554C22—3164no—
555C22—3164no—
556C22—3168no—
557C22—3164no—
558C22—798no—
559C22—1584no—
560C22—1584no—
561C22C2×C22⋊C4×D7911yescharacteristic
562C22C4×Q8×D7912yes—
563C22C22⋊Q8×D7912yes—
564C22#124×D7911yescharacteristic
565C22C4⋊Q8×D7911yescharacteristic
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—82no—
574C4—3164no—
575C4—3162no—
576C4—3162no—
577C4—6322no—
578C4—1584no—
579C2—22no—
580C2—21no—
581C2—22no—
582C2—21no—
583C2—1588no—
584C2#391×D7911yescharacteristic
585C2#2377185811yescharacteristic
586C2#2377185911yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D79
commutatorC23×C79C23×C4
FrattiniC24C23×D79
FittingC79×#1996C2
socleC22×C79C22×C22⋊C4
prime radicalC24×D79C23
cosocleC24×C79C24
hypercenter#1996D79
hypocenterC79#21333
LCSG ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
4. #23764386C2semidirect
5. #23764385C2semidirect
6. #23764384C2semidirect
7. #23764383C2semidirect
9. C79×#1996C2semidirect
14. C2×#14579141C4semidirect
16. C2×#14579139C4semidirect
17. C2×#14579139C4semidirect
19. C2×#14579137C4semidirect
40. C22×Dic158C2×C4semidirect
41. C22×Dic158C2×C4semidirect
62. C2×C4⋊Dic79C2×C4semidirect
63. C2×C4⋊Dic79C2×C4semidirect
65. C2×Dic79⋊C4C2×C4semidirect
66. C2×Dic79⋊C4C2×C4semidirect
67. C2×Dic79⋊C4C2×C4semidirect
85. C22⋊C4×D79Q8semidirect
92. C22×Q8×C79C2×C4semidirect
99. C2×C4⋊C4×C79C2×C4semidirect
100. C2×C4⋊C4×C79C2×C4semidirect
301. #1996D79semidirect
303. C79#21333semidirect
336. C22×Q8C4×D79semidirect
369. C2×C4⋊C4C4×D79semidirect
370. C2×C4⋊C4C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190991882
#237660132
#237657522
#237643862
#237643852
#237643842
#237643832
#342×D792
C79×#19962
#2133379