Finite Group Databasebeta

← #34351186

Subgroups — #34351186

Order 20224.

896 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098860C211yesmaximal, characteristic, complemented
2C2×#19098606C211yesmaximal, characteristic, complemented
3#23766414C211yesmaximal, characteristic
4#23764934C211yesmaximal, characteristic, complemented
5#23764901C211yesmaximal, characteristic, complemented
6#23764785C211yesmaximal, characteristic, complemented
7#23764735C211yesmaximal, characteristic, complemented
8#23764161C211yesmaximal, characteristic, complemented
9#23763968C211yesmaximal, characteristic, complemented
10#23763836C211yesmaximal, characteristic
11#23763722C211yesmaximal, characteristic
12#23763660C211yesmaximal, characteristic, complemented
13#23763592C211yesmaximal, characteristic, complemented
14#23763585C211yesmaximal, characteristic, complemented
15C79×#2441C211yesmaximal, characteristic, complemented
16C22×#11929608C2211yescharacteristic
17C2×#14579135C2211yescharacteristic
18C2×C4×D316C2211yescharacteristic
19C2×(C22⋊D79)⋊C4—21no—
20C22×C4⋊Dic79C2211yescharacteristic
21#19098860—24no—
22#19098606—24no—
23C2×#14579094C2211yescharacteristic
24C2×#14579094C2211yescharacteristic
25C2×#14579092C2211yescharacteristic
26C2×#14579092C2211yescharacteristic
27C2×#14579092C2211yescharacteristic
28C2×#14579092C2211yescharacteristic
29C2×C4⋊(C79⋊C8)C2211yescharacteristic
30C2×C4⋊(C79⋊C8)C2211yescharacteristic
31C2×#14579081C2211yescharacteristic
32C2×#14579081C2211yescharacteristic
33C2×#14579081C2211yescharacteristic
34C2×#14579081C2211yescharacteristic
35C2×C22⋊(C79⋊C8)C2211yescharacteristic
36#19098465C2211yescharacteristic
37#19098461C2211yescharacteristic
38#19098455—21no—
39#19098440—21no—
40#19098437—21no—
41C4×C4⋊Dic79C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098318C2211yescharacteristic
49#19098318C2211yescharacteristic
50C2×#125×C79C2211yescharacteristic
51C2×#117×C79C2211yescharacteristic
52#399×C79—21no—
53#396×C79—21no—
54#389×C79—21no—
55#389×C79C2211yescharacteristic
56#389×C79C2211yescharacteristic
57#387×C79—21no—
58#384×C79C2211yescharacteristic
59#380×C79C2211yescharacteristic
60C4×C22⋊C4×C79C2211yescharacteristic
61C22×D316C2311yescharacteristic
62C22×C22⋊D79—21no—
63C23×Dic79—21no—
64C22×C4×D79—21no—
65C2×#11929608—22no—
66C2×#11929608—22no—
67C2×#11929608—22no—
68C22×C79⋊C8C2311yescharacteristic
69C22×C79⋊C8C2311yescharacteristic
70C22×C79⋊C8C2311yescharacteristic
71C22×C79⋊C8C2311yescharacteristic
72#14579135—22no—
73#14579135Q814yes—
74C4×D316—22no—
75C4×D316—41no—
76(C22⋊D79)⋊C4—44no—
77C2×C4⋊Dic79—22no—
78C2×C4⋊Dic79—21no—
79C2×C4⋊Dic79—41no—
80C2×C4⋊Dic79C2311yescharacteristic
81C2×C4⋊Dic79C2311yescharacteristic
82C2×C4⋊Dic79C2311yescharacteristic
83C2×Dic79⋊C4—21no—
84C2×(C2×D79)⋊C4—21no—
85C2×(C2×D79)⋊C4—21no—
86C2×(C2×D79)⋊C4—21no—
87C2×(C2×D79)⋊C4—21no—
88C2×C4×Dic79—21no—
89#14579094—22no—
90#14579094D414yescomplemented
91#14579092—22no—
92#14579092—22no—
93#14579092—22no—
94#14579092—41no—
95C4⋊(C79⋊C8)—22no—
96C4⋊(C79⋊C8)—22no—
97#14579081—22no—
98#14579081—22no—
99#14579081—22no—
100#14579081—41no—
101C22⋊(C79⋊C8)—22no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C23×C4×C79C2311yescharacteristic
105#125×C79—22no—
106#125×C79—41no—
107#117×C79—22no—
108#117×C79—41no—
109C2×C4⋊C4×C79—21no—
110C2×C4⋊C4×C79—21no—
111C2×C4⋊C4×C79—21no—
112C2×C4⋊C4×C79—21no—
113C2×C4⋊C4×C79—21no—
114C2×C4⋊C4×C79—21no—
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C4⋊C4×C79C2311yescharacteristic
119C2×C22⋊C4×C79—21no—
120C2×C22⋊C4×C79—21no—
121C2×C22⋊C4×C79D411yescharacteristic
122C2×C22⋊C4×C79D411yescharacteristic
123C2×C42×C79—21no—
124C2×C42×C79C2311yescharacteristic
125C2×C42×C79C2311yescharacteristic
126#95×C79—21no—
127#95×C79—21no—
128#95×C79—21no—
129#95×C79D411yescharacteristic
130#95×C79D411yescharacteristic
131C23×D79—21no—
132C2×D316—21no—
133C2×D316—21no—
134C2×D316—21no—
135C2×D316—21no—
136C2×D316—21no—
137C2×D316C2×Q812yes—
138C2×C22⋊D79—24no—
139C2×C22⋊D79—42no—
140C2×C22⋊D79—42no—
141C22×Dic79—21no—
142C22×Dic79—21no—
143C22×Dic79—21no—
144C22×Dic79—42no—
145C22×Dic79—42no—
146C22×Dic79—81no—
147C2×C4×D79—42no—
148C2×C4×D79—42no—
149C2×C4×D79—81no—
150#11929608—28no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8—21no—
155C2×C79⋊C8—21no—
156C2×C79⋊C8—21no—
157C2×C79⋊C8—21no—
158C2×C79⋊C8—21no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8C2×D412yes—
162C2×C79⋊C8#4112yes—
163C4⋊Dic79—21no—
164C4⋊Dic79—21no—
165C4⋊Dic79—21no—
166C4⋊Dic79—21no—
167C4⋊Dic79—42no—
168C4⋊Dic79C2×D412yes—
169C4⋊Dic79C2×Q812yes—
170Dic79⋊C4—42no—
171(C2×D79)⋊C4—24no—
172(C2×D79)⋊C4—42no—
173(C2×D79)⋊C4—42no—
174(C2×D79)⋊C4—81no—
175C4×Dic79—42no—
176C24×C79C2×D411yescharacteristic
177C22×C4×C79—21no—
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×C79C2×D411yescharacteristic
186C22×C4×C79C2×D411yescharacteristic
187C22×C4×C79C2×D411yescharacteristic
188C22×C4×C79C2×D411yescharacteristic
189C22×C4×C79C2×D411yescharacteristic
190C22×C4×C79C2×Q812yes—
191C22×C4×C79#4112yes—
192C22×C4×C79C2411yescharacteristic
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79—21no—
200C4⋊C4×C79—42no—
201C4⋊C4×C79—42no—
202C4⋊C4×C79—42no—
203C4⋊C4×C79—42no—
204C4⋊C4×C79—81no—
205C4⋊C4×C79—81no—
206C4⋊C4×C79C2×D412yes—
207C22⋊C4×C79—22no—
208C22⋊C4×C79—41no—
209C22⋊C4×C79—42no—
210C22⋊C4×C79—81no—
211C42×C79—21no—
212C42×C79—21no—
213C42×C79—21no—
214C42×C79—21no—
215C42×C79—42no—
216C23×C79—21no—
217C23×C79—21no—
218C23×C79—21no—
219C23×C79—21no—
220C23×C79—21no—
221C23×C79—21no—
222C23×C79C22⋊Q812yes—
223C23×C79C22×D411yescharacteristic
224C22×D79—22no—
225C22×D79—41no—
226C22×D79—41no—
227C22×D79—41no—
228C22×D79—41no—
229C22×D79—41no—
230C22×D79—41no—
231C2×C4×C79—21no—
232C2×C4×C79—21no—
233C2×C4×C79—21no—
234C2×C4×C79—21no—
235C2×C4×C79—22no—
236C2×C4×C79—22no—
237C2×C4×C79—22no—
238C2×C4×C79—22no—
239C2×C4×C79—21no—
240C2×C4×C79—21no—
241C2×C4×C79—22no—
242C2×C4×C79—21no—
243C2×C4×C79—21no—
244C2×C4×C79—21no—
245C2×C4×C79—21no—
246C2×C4×C79—21no—
247C2×C4×C79—21no—
248C2×C4×C79—21no—
249C2×C4×C79—41no—
250C2×C4×C79—41no—
251C2×C4×C79—41no—
252C2×C4×C79—41no—
253C2×C4×C79—41no—
254C2×C4×C79—41no—
255C2×C4×C79—41no—
256C2×C4×C79—41no—
257C2×C4×C79—41no—
258C2×C4×C79—41no—
259C2×C4×C79—41no—
260C2×C4×C79C22⋊Q812yes—
261C2×C4×C79AGL(1,ℤ/8)12yes—
262C2×C4×C79C22×D411yescharacteristic
263C2×C4×C79C22×Q811yescharacteristic
264C2×C4×C79C2×#4111yescharacteristic
265C2×C4×C79C2×#4111yescharacteristic
266C2×C4×C79H(5,𝔽2)11yescharacteristic
267C2×C4×C792−1+411yescharacteristic
268C22⋊D79—48no—
269C2×Dic79—22no—
270C2×Dic79—22no—
271C2×Dic79—41no—
272C2×Dic79—41no—
273C2×Dic79—41no—
274C2×Dic79—41no—
275C2×Dic79—41no—
276C2×Dic79—41no—
277C2×Dic79—41no—
278C2×Dic79—84no—
279D316—22no—
280D316—22no—
281D316—41no—
282D316—41no—
283C4×D79—84no—
284C79⋊C8—22no—
285C79⋊C8—22no—
286C79⋊C8—41no—
287C79⋊C8—41no—
288C22×C79—22no—
289C22×C79—22no—
290C22×C79—22no—
291C22×C79—41no—
292C22×C79—41no—
293C22×C79—41no—
294C22×C79—41no—
295C22×C79C2×C22⋊Q811yescharacteristic
296C22×C79D4211yescharacteristic
297C22×C79#54711yescharacteristic
298C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C79#57511yescharacteristic
300C22×C79#57711yescharacteristic
301C22×C79#57711yescharacteristic
302C2×D79—42no—
303C2×D79—42no—
304C2×D79—42no—
305C2×D79—81no—
306C2×D79—81no—
307C2×D79—81no—
308C2×D79—81no—
309C4×C79—22no—
310C4×C79—22no—
311C4×C79—41no—
312C4×C79—41no—
313C4×C79—41no—
314C4×C79—42no—
315C4×C79—81no—
316C4×C79—81no—
317C4×C79—81no—
318C4×C79#52511yescharacteristic
319C4×C79#53911yescharacteristic
320C4×C79D4×Q811yescharacteristic
321C4×C79#55411yescharacteristic
322Dic79—42no—
323Dic79—42no—
324Dic79—81no—
325#26366—791nomaximal, Sylow
326C2×C79—42no—
327C2×C79#244911yescharacteristic
328C2×C79#307711yescharacteristic
329C2×C79#308411yescharacteristic
330C2×C79#327111yescharacteristic
331C2×C79#327211yescharacteristic
332C2×C79#334511yescharacteristic
333C2×C79#335111yescharacteristic
334D79—82no—
335C2×#478—791no—
336C2×#428—791no—
337#2441D7911yescharacteristic, complemented
338#2361—791no—
339#2094—791no—
340#2088—791no—
341#2067—791no—
342#2059—791no—
343#1948—791no—
344#1918—791no—
345#1895—791no—
346#1875—791no—
347#1864—791no—
348#1853—791no—
349#1851—791no—
350C79#2636611yescharacteristic, complemented, Sylow
351C22×M16—791no—
352C2×#125C2×D7911yescharacteristic
353C2×#121—791no—
354C2×C4×D4—791no—
355C2×C4×D4—1581no—
356C2×#117C2×D7911yescharacteristic
357C22×C4⋊C4—791no—
358#478—1584no—
359#428—1584no—
360C2×#107—791no—
361C2×#107—791no—
362C2×#106—791no—
363C2×#106—791no—
364C2×#106—791no—
365C2×#106—791no—
366C2×C4⋊C8—791no—
367C2×C4⋊C8—791no—
368C2×#102—791no—
369C2×#102—791no—
370C2×#102—791no—
371C2×#102—791no—
372C2×C22⋊C8—791no—
373#399—21no—
374#396—21no—
375#390—791no—
376#389—21no—
377#389—791no—
378#389C2×D7911yescharacteristic
379#389C2×D7911yescharacteristic
380#388—1581no—
381#387—21no—
382#386—1581no—
383#385—1581no—
384#384C2×D7911yescharacteristic
385#380C2×D7911yescharacteristic
386C4×C4⋊C4—791no—
387C4×C22⋊C4C2×D7911yescharacteristic
388#340—791no—
389#340—791no—
390#340—791no—
391#340—791no—
392#340—791no—
393#340—791no—
394#336—791no—
395#336—791no—
396C22×D4—791no—
397C22×D4—1581no—
398C23×C4—1581no—
399C23×C4—1581no—
400C23×C4C22×D7911yescharacteristic
401C2×M16—1582no—
402C2×M16—1582no—
403C2×M16—1582no—
404C22×C8—791no—
405C22×C8—791no—
406C22×C8—791no—
407C22×C8—791no—
408#125—22no—
409#125—41no—
410#121—794no—
411#121—1582no—
412C4×D4—3161no—
413C4×D4—3164no—
414C4×D4—1582no—
415#117—22no—
416#117—41no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—3161no—
424C2×C4⋊C4—791no—
425C2×C4⋊C4—791no—
426C2×C4⋊C4—791no—
427C2×C4⋊C4—1581no—
428C2×C4⋊C4—1581no—
429C2×C4⋊C4—1582no—
430C2×C4⋊C4C22×D7911yescharacteristic
431C2×C4⋊C4C22×D7911yescharacteristic
432C2×C4⋊C4C22×D7911yescharacteristic
433C2×C4⋊C4C22×D7911yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1581no—
437C2×C22⋊C4—1581no—
438C2×C22⋊C4—1581no—
439C2×C22⋊C4—1581no—
440C2×C22⋊C4C22⋊D7911yescharacteristic
441C2×C22⋊C4C22⋊D7911yescharacteristic
442C2×C42—21no—
443C2×C42—1581no—
444C2×C42C22×D7911yescharacteristic
445C2×C42C22×D7911yescharacteristic
446#107—794no—
447#107—1582no—
448#106—3161no—
449#106—1582no—
450#106—1582no—
451#106—1582no—
452C4⋊C8—1582no—
453C4⋊C8—1582no—
454#102—3161no—
455#102—1582no—
456#102—1582no—
457#102—1582no—
458C22⋊C8—1582no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—1581no—
463#95—1581no—
464#95C22⋊D7911yescharacteristic
465#95C22⋊D7911yescharacteristic
466C24—1581no—
467C24C2×C22⋊D7911yescharacteristic
468C2×D4—3162no—
469C2×D4—3162no—
470C2×D4—792no—
471C2×D4—1584no—
472C2×D4—1581no—
473C2×D4—1581no—
474C2×D4—1581no—
475C2×D4—1581no—
476C2×D4—1581no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—3162no—
486C22×C4—3162no—
487C22×C4—3162no—
488C22×C4—3162no—
489C22×C4—6321no—
490C22×C4—6321no—
491C22×C4—1581no—
492C22×C4—1581no—
493C22×C4—1581no—
494C22×C4C2×C22⋊D7911yescharacteristic
495C22×C4C2×C22⋊D7911yescharacteristic
496C22×C4C2×C22⋊D7911yescharacteristic
497C22×C4C2×C22⋊D7911yescharacteristic
498C22×C4C2×C22⋊D7911yescharacteristic
499C22×C4Q8×D7912yes—
500C22×C4#1192962512yes—
501C22×C4C23×D7911yescharacteristic
502M16—1588no—
503C2×C8—792no—
504C2×C8—792no—
505C2×C8—1581no—
506C2×C8—1581no—
507C2×C8—1581no—
508C2×C8—1581no—
509C2×C8—1581no—
510C2×C8—1581no—
511C2×C8—1581no—
512C2×C8—1581no—
513C2×C8—1581no—
514C2×C8—1581no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—81no—
527C4⋊C4—81no—
528C4⋊C4—3162no—
529C4⋊C4—3162no—
530C4⋊C4—792no—
531C4⋊C4—792no—
532C4⋊C4—1581no—
533C4⋊C4—1581no—
534C4⋊C4—1581no—
535C4⋊C4—1581no—
536C4⋊C4D4×D7912yes—
537C22⋊C4—22no—
538C22⋊C4—41no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C22⋊C4—3162no—
542C22⋊C4—3162no—
543C22⋊C4—6321no—
544C22⋊C4—1584no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—42no—
550C42—3162no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—3161no—
558C23—3161no—
559C23—3161no—
560C23—3161no—
561C23—3161no—
562C23—3161no—
563C23—1582no—
564C23#1457913712yes—
565C23C22×C22⋊D7911yescharacteristic
566D4—3168no—
567D4—3161no—
568D4—3161no—
569D4—1582no—
570D4—1582no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—3161no—
601C2×C4—3161no—
602C2×C4—3161no—
603C2×C4—3161no—
604C2×C4—3161no—
605C2×C4—3161no—
606C2×C4—3161no—
607C2×C4—6324no—
608C2×C4—6324no—
609C2×C4—1582no—
610C2×C4—1582no—
611C2×C4#1457913712yes—
612C2×C4#1457918612yes—
613C2×C4C2×D4×D7911yescharacteristic
614C2×C4C2×Q8×D7911yescharacteristic
615C2×C4#41×D7911yescharacteristic
616C2×C4C2×#1192962511yescharacteristic
617C2×C4#1457920611yescharacteristic
618C2×C4#1457920811yescharacteristic
619C8—3161no—
620C8—3161no—
621C8—1582no—
622C8—1582no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—3162no—
631C22—3162no—
632C22—3162no—
633C22—6321no—
634C22—6321no—
635C22—6321no—
636C22—6321no—
637C22C2×#1457913711yescharacteristic
638C22D4×C22⋊D7911yescharacteristic
639C22#1909921211yescharacteristic
640C22C2×#1457918611yescharacteristic
641C22#1909936111yescharacteristic
642C22#1909937111yescharacteristic
643C22#1909937411yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—41no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—3162no—
654C4—3162no—
655C4—6321no—
656C4#1909907611yescharacteristic
657C4#1909916211yescharacteristic
658C4#1909922611yescharacteristic
659C4#1909926711yescharacteristic
660C2—42no—
661C2—6322no—
662C2#2376720511yescharacteristic
663C2#2377319311yescharacteristic
664C2#2377325811yescharacteristic
665C2#2377498111yescharacteristic
666C2#2377500011yescharacteristic
667C2#2377586411yescharacteristic
668C2#2377596011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2441C2
socleC23×C79C22×D4
prime radicalC2×#14579135C22
hypercenter#2441D79
hypocenterC79#26366
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
1. C2×#19098860C2semidirect
2. C2×#19098606C2semidirect
4. #23764934C2semidirect
5. #23764901C2semidirect
6. #23764785C2semidirect
7. #23764735C2semidirect
8. #23764161C2semidirect
9. #23763968C2semidirect
12. #23763660C2semidirect
13. #23763592C2semidirect
14. #23763585C2semidirect
15. C79×#2441C2semidirect
90. #14579094D4semidirect
337. #2441D79semidirect
350. C79#26366semidirect

Maximal subgroups

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

subgroupindex
C2×#190988602
C2×#190986062
#237664142
#237649342
#237649012
#237647852
#237647352
#237641612
#237639682
#237638362
#237637222
#237636602
#237635922
#237635852
C79×#24412
#2636679