Finite Group Databasebeta

← C2×#23766832 #34456686

Subgroups — C2×#23766832 #34456686

Order 20224.

3526 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579137C211yesmaximal, characteristic, complemented
2C22×#14579137C211yesmaximal, characteristic, complemented
3C22×C4×D316C211yesmaximal, characteristic
4C22×#14579119C211yesmaximal, characteristic, complemented
5#23766832C2116yesmaximal, direct product factor
6C2×#19098492C212yesmaximal, complemented
7C2×#19098481C212yesmaximal
8C2×#19098461C211yesmaximal, characteristic, complemented
9C2×#19098460C212yesmaximal, complemented
10C2×#19098452C212yesmaximal, complemented
11C2×#19098420C211yesmaximal, characteristic, complemented
12C2×#391×C79C211yesmaximal, characteristic, complemented
13C23×D316C2211yescharacteristic
14C23×C4×D79C2211yescharacteristic
15C23×C4×D79C2211yescharacteristic
16C2×#14579137—28no—
17C2×#14579137—28no—
18C2×#14579137C2218yescomplemented
19C2×#14579137C2218yescomplemented
20C2×C4×D316—28no—
21C2×C4×D316C2218yes—
22C2×#14579119—28no—
23C2×#14579119C2218yescomplemented
24C22×C4⋊Dic79C2211yescharacteristic
25C22×C4⋊Dic79C2211yescharacteristic
26C22×C4⋊Dic79C2211yescharacteristic
27C22×Dic79⋊C4C2212yes—
28C22×Dic79⋊C4C2212yes—
29C22×Dic79⋊C4C2212yes—
30C22×(C2×D79)⋊C4C2212yes—
31C22×(C2×D79)⋊C4C2212yes—
32C22×(C2×D79)⋊C4C2211yescharacteristic
33C22×(C2×D79)⋊C4C2211yescharacteristic
34C22×(C2×D79)⋊C4C2212yes—
35C22×(C2×D79)⋊C4C2212yes—
36C22×C4×Dic79C2212yes—
37#19098492C22116yescomplemented
38#19098481C22116yes—
39#19098461C2218yescomplemented
40#19098460C22116yescomplemented
41#19098452C22116yescomplemented
42#19098420C2218yescomplemented
43C2×C79⋊#95C2212yes—
44C2×C79⋊#95C2212yes—
45C23×Q8×C79C2211yescharacteristic
46C22×C4⋊C4×C79C2211yescharacteristic
47C22×C42×C79C2211yescharacteristic
48#391×C79C2218yescomplemented
49C2×#95×C79C2212yes—
50C2×#95×C79C2212yes—
51C24×D79C2311yescharacteristic
52C24×D79C2311yescharacteristic
53C22×D316—24no—
54C22×D316—24no—
55C22×D316—24no—
56C22×D316C2314yes—
57C23×Dic79C2312yes—
58C23×Dic79C2311yescharacteristic
59C23×Dic79C2312yes—
60C23×Dic79C2311yescharacteristic
61C22×C4×D79—24no—
62C22×C4×D79—24no—
63C22×C4×D79—24no—
64C22×C4×D79—24no—
65C22×C4×D79—24no—
66C22×C4×D79D418yes—
67C22×C4×D79C2314yes—
68C22×C4×D79C2314yes—
69#14579137—232no—
70#14579137—232no—
71C4×D316—232no—
72#14579119—232no—
73C2×C4⋊Dic79—24no—
74C2×C4⋊Dic79—24no—
75C2×C4⋊Dic79D418yescomplemented
76C2×C4⋊Dic79C2314yes—
77C2×C4⋊Dic79C2314yes—
78C2×C4⋊Dic79C2314yes—
79C2×Dic79⋊C4—28no—
80C2×Dic79⋊C4—28no—
81C2×Dic79⋊C4—28no—
82C2×Dic79⋊C4C2318yes—
83C2×Dic79⋊C4C2318yes—
84C2×Dic79⋊C4C2318yes—
85C2×(C2×D79)⋊C4—24no—
86C2×(C2×D79)⋊C4—28no—
87C2×(C2×D79)⋊C4—28no—
88C2×(C2×D79)⋊C4—28no—
89C2×(C2×D79)⋊C4—24no—
90C2×(C2×D79)⋊C4—28no—
91C2×(C2×D79)⋊C4C2318yes—
92C2×(C2×D79)⋊C4C2314yes—
93C2×(C2×D79)⋊C4C2318yes—
94C2×(C2×D79)⋊C4C2318yes—
95C2×(C2×D79)⋊C4C2318yes—
96C2×(C2×D79)⋊C4C2314yes—
97C2×C4×Dic79—28no—
98C2×C4×Dic79C2318yes—
99C79⋊#95C2318yes—
100C79⋊#95C2318yes—
101C22×Q8×C79—24no—
102C22×Q8×C79—24no—
103C22×Q8×C79—24no—
104C22×Q8×C79C2314yes—
105C23×C4×C79C2311yescharacteristic
106C23×C4×C79C2312yes—
107C23×C4×C79C2312yes—
108C23×C4×C79C2311yescharacteristic
109C23×C4×C79C2311yescharacteristic
110C2×C4⋊C4×C79—24no—
111C2×C4⋊C4×C79C2314yes—
112C2×C42×C79—24no—
113C2×C42×C79C2314yes—
114#95×C79C2318yes—
115#95×C79C2318yes—
116C23×D79—22no—
117C23×D79—22no—
118C23×D79—22no—
119C23×D79—22no—
120C23×D79—22no—
121C23×D79—22no—
122C23×D79—22no—
123C23×D79—22no—
124C23×D79—22no—
125C23×D79—22no—
126C23×D79—22no—
127C23×D79—22no—
128C23×D79C2×D414yes—
129C23×D79#4114yes—
130C23×D79C2412yes—
131C23×D79C2412yes—
132C2×D316—232no—
133C2×D316—28no—
134C2×D316—28no—
135C2×D316—28no—
136C22×Dic79—22no—
137C22×Dic79—24no—
138C22×Dic79—22no—
139C22×Dic79—22no—
140C22×Dic79—24no—
141C22×Dic79—22no—
142C22×Dic79—24no—
143C22×Dic79—24no—
144C22×Dic79—22no—
145C22×Dic79—24no—
146C22×Dic79C2×D418yes—
147C22×Dic79C2×D414yes—
148C22×Dic79C2414yes—
149C22×Dic79C2412yes—
150C22×Dic79C2414yes—
151C22×Dic79C2412yes—
152C2×C4×D79—232no—
153C2×C4×D79—28no—
154C2×C4×D79—28no—
155C2×C4×D79—28no—
156C2×C4×D79—28no—
157C2×C4×D79—28no—
158C2×C4×D79—416no—
159C2×C4×D79C2×D4116yes—
160C4⋊Dic79—28no—
161C4⋊Dic79—28no—
162C4⋊Dic79C2×D4116yescomplemented
163Dic79⋊C4—216no—
164Dic79⋊C4—216no—
165Dic79⋊C4—216no—
166(C2×D79)⋊C4—216no—
167(C2×D79)⋊C4—28no—
168(C2×D79)⋊C4—216no—
169(C2×D79)⋊C4—28no—
170(C2×D79)⋊C4—216no—
171(C2×D79)⋊C4—216no—
172C4×Dic79—216no—
173C24×C79C2411yescharacteristic
174C2×Q8×C79—28no—
175C2×Q8×C79—28no—
176C2×Q8×C79—28no—
177C2×Q8×C79—416no—
178C22×C4×C79—22no—
179C22×C4×C79—22no—
180C22×C4×C79—22no—
181C22×C4×C79—24no—
182C22×C4×C79—24no—
183C22×C4×C79—24no—
184C22×C4×C79—24no—
185C22×C4×C79—24no—
186C22×C4×C79—22no—
187C22×C4×C79—22no—
188C22×C4×C79—24no—
189C22×C4×C79C2×D414yes—
190C22×C4×C79C2×D414yes—
191C22×C4×C79#4114yes—
192C22×C4×C79#4114yes—
193C22×C4×C79C2412yes—
194C22×C4×C79C2414yes—
195C22×C4×C79C2414yes—
196C22×C4×C79C2412yes—
197C22×C4×C79C2412yes—
198C4⋊C4×C79—28no—
199C42×C79—28no—
200C23×C79C22×D412yes—
201C23×C79C22×D412yes—
202C23×C79C2×#4112yes—
203C23×C79C2×#4112yes—
204C23×C79C2×#4112yes—
205C23×C792−1+412yes—
206C23×C792−1+412yes—
207C23×C79C2511yescharacteristic
208C22×D79—28no—
209C22×D79—28no—
210C22×D79—28no—
211C22×D79—28no—
212C22×D79—28no—
213C22×D79—28no—
214C22×D79—22no—
215C22×D79—22no—
216C22×D79—22no—
217C22×D79—22no—
218C22×D79—22no—
219C22×D79—22no—
220C22×D79—22no—
221C22×D79—22no—
222C22×D79—22no—
223C22×D79—22no—
224C22×D79—22no—
225C22×D79—22no—
226C22×D79—44no—
227C22×D79—44no—
228C22×D79—44no—
229C22×D79—44no—
230C22×D79—44no—
231C22×D79—44no—
232C22×D79—44no—
233C22×D79—44no—
234C22×D79C22×D414yes—
235C22×D79C2×#4114yes—
236Q8×C79—416no—
237C2×C4×C79—28no—
238C2×C4×C79—22no—
239C2×C4×C79—22no—
240C2×C4×C79—24no—
241C2×C4×C79—22no—
242C2×C4×C79—24no—
243C2×C4×C79—24no—
244C2×C4×C79—24no—
245C2×C4×C79—24no—
246C2×C4×C79—22no—
247C2×C4×C79—24no—
248C2×C4×C79—22no—
249C2×C4×C79—48no—
250C2×C4×C79—44no—
251C2×C4×C79—48no—
252C2×C4×C79#121116yes—
253C2×C4×C79C22×D414yes—
254C2×C4×C79C22×D414yes—
255C2×C4×C79C2×#4114yes—
256C2×C4×C79C2×#4114yes—
257C2×Dic79—28no—
258C2×Dic79—216no—
259C2×Dic79—22no—
260C2×Dic79—24no—
261C2×Dic79—22no—
262C2×Dic79—24no—
263C2×Dic79—24no—
264C2×Dic79—22no—
265C2×Dic79—22no—
266C2×Dic79—24no—
267C2×Dic79—24no—
268C2×Dic79—22no—
269C2×Dic79—44no—
270C2×Dic79—48no—
271C2×Dic79C22×D418yes—
272C2×Dic79C22×D414yes—
273D316—232no—
274C4×D79—232no—
275C4×D79—416no—
276C22×C79C2×#12114yes—
277C22×C79#52514yes—
278C22×C79#53614yes—
279C22×C79#54714yes—
280C22×C79#54814yes—
281C22×C79#55214yes—
282C22×C79#55614yes—
283C22×C79C23×D411yescharacteristic
284C22×C79C23×D411yescharacteristic
285C22×C79C22×#4111yescharacteristic
286C22×C79C22×#4111yescharacteristic
287C22×C79C22×#4111yescharacteristic
288C22×C79C2×2−1+411yescharacteristic
289C22×C79C2×2−1+411yescharacteristic
290C2×D79—24no—
291C2×D79—24no—
292C2×D79—24no—
293C2×D79—24no—
294C2×D79—24no—
295C2×D79—24no—
296C2×D79—42no—
297C2×D79—416no—
298C2×D79—416no—
299C2×D79—42no—
300C2×D79—42no—
301C2×D79—42no—
302C2×D79—42no—
303C2×D79—42no—
304C2×D79—42no—
305C2×D79—42no—
306C4×C79—24no—
307C4×C79—44no—
308C4×C79—42no—
309C4×C79—44no—
310C4×C79C2×#12118yes—
311Dic79—24no—
312Dic79—28no—
313Dic79—42no—
314Dic79—44no—
315C2×#2415—791nomaximal, Sylow
316C2×C79#241518yescomplemented
317C2×C79C22×#12111yescharacteristic
318C2×C79C2×#52511yescharacteristic
319C2×C79C2×#53611yescharacteristic
320C2×C79C2×#54711yescharacteristic
321C2×C79C2×#54811yescharacteristic
322C2×C79C2×#55211yescharacteristic
323C2×C79C2×#55611yescharacteristic
324D79—44no—
325D79—44no—
326C22×C22⋊Q8—791no—
327C22×C22⋊Q8—791no—
328C22×C4×D4—791no—
329C22×#117—791no—
330#2415—7916no—
331C2×#397—792no—
332C2×#394—792no—
333C2×#391D7911yescharacteristic, complemented
334C2×#389—791no—
335C2×#389—792no—
336C2×#388—792no—
337C2×#380—791no—
338C79C2×#241511yescharacteristic, complemented, Sylow
339C23×Q8C2×D7911yescharacteristic
340C23×D4—791no—
341C24×C4—791no—
342C24×C4—791no—
343C2×C22⋊Q8—798no—
344C2×C22⋊Q8—798no—
345C2×C22⋊Q8—1588no—
346C2×C22⋊Q8—1588no—
347C2×C4×D4—798no—
348C2×C4×D4—1588no—
349C2×#117—798no—
350C2×#117—1588no—
351C22×C4⋊C4—791no—
352C22×C4⋊C4—792no—
353C22×C4⋊C4—792no—
354C22×C4⋊C4—791no—
355C22×C4⋊C4—792no—
356C22×C4⋊C4—791no—
357C22×C4⋊C4C2×D7911yescharacteristic
358C22×C22⋊C4—792no—
359C22×C22⋊C4—792no—
360C22×C22⋊C4—792no—
361C22×C22⋊C4—792no—
362C22×C22⋊C4—791no—
363C22×C22⋊C4—791no—
364C22×C42—792no—
365C22×C42C2×D7911yescharacteristic
366#397—7916no—
367#394—7916no—
368#391C2×D7918yescomplemented
369#389—798no—
370#389—7916no—
371#388—7916no—
372#380—798no—
373C2×#95—792no—
374C2×#95—792no—
375C2×#95C2×D7912yes—
376C2×#95C2×D7912yes—
377C25—791no—
378C25—791no—
379C22×Q8—24no—
380C22×Q8—24no—
381C22×Q8—24no—
382C22×Q8C22×D7914yes—
383C22×D4—794no—
384C22×D4—1584no—
385C22×D4—1584no—
386C22×D4—1584no—
387C23×C4—792no—
388C23×C4—791no—
389C23×C4—798no—
390C23×C4—794no—
391C23×C4—792no—
392C23×C4—791no—
393C23×C4—794no—
394C23×C4—1584no—
395C23×C4—1584no—
396C23×C4—1584no—
397C23×C4—1584no—
398C23×C4—1584no—
399C23×C4C22×D7912yes—
400C23×C4C22×D7912yes—
401C23×C4C22×D7911yescharacteristic
402C23×C4C22×D7911yescharacteristic
403C23×C4C22×D7911yescharacteristic
404C22⋊Q8—15832no—
405C22⋊Q8—15832no—
406C4×D4—15832no—
407#117—15832no—
408C2×C4⋊C4—24no—
409C2×C4⋊C4—798no—
410C2×C4⋊C4—794no—
411C2×C4⋊C4—794no—
412C2×C4⋊C4—794no—
413C2×C4⋊C4—798no—
414C2×C4⋊C4—798no—
415C2×C4⋊C4—798no—
416C2×C4⋊C4—1584no—
417C2×C4⋊C4—1588no—
418C2×C4⋊C4—1588no—
419C2×C4⋊C4—1584no—
420C2×C4⋊C4—1588no—
421C2×C4⋊C4C22×D7914yes—
422C2×C22⋊C4—798no—
423C2×C22⋊C4—794no—
424C2×C22⋊C4—794no—
425C2×C22⋊C4—798no—
426C2×C22⋊C4—798no—
427C2×C22⋊C4—798no—
428C2×C22⋊C4—1588no—
429C2×C22⋊C4—1584no—
430C2×C22⋊C4—1588no—
431C2×C22⋊C4—1584no—
432C2×C22⋊C4—1588no—
433C2×C22⋊C4—1588no—
434C2×C42—24no—
435C2×C42—798no—
436C2×C42—1588no—
437C2×C42C22×D7914yes—
438#95—798no—
439#95—798no—
440#95C22×D7918yes—
441#95C22×D7918yes—
442C24—794no—
443C24—792no—
444C24—792no—
445C24—794no—
446C24—1582no—
447C24—1582no—
448C24—1582no—
449C24—1582no—
450C24—1582no—
451C24—1582no—
452C24—1582no—
453C24—1582no—
454C24—1582no—
455C24—1582no—
456C24—1582no—
457C24—1582no—
458C24C23×D7911yescharacteristic
459C2×Q8—28no—
460C2×Q8—28no—
461C2×Q8—28no—
462C2×Q8—416no—
463C2×D4—1588no—
464C2×D4—1588no—
465C2×D4—15832no—
466C2×D4—1588no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—31616no—
479C22×C4—794no—
480C22×C4—792no—
481C22×C4—798no—
482C22×C4—794no—
483C22×C4—7916no—
484C22×C4—794no—
485C22×C4—792no—
486C22×C4—1584no—
487C22×C4—1582no—
488C22×C4—1584no—
489C22×C4—1584no—
490C22×C4—1582no—
491C22×C4—1582no—
492C22×C4—1588no—
493C22×C4—1588no—
494C22×C4—1588no—
495C22×C4—1582no—
496C22×C4—1584no—
497C22×C4—1584no—
498C22×C4—1582no—
499C22×C4—1588no—
500C22×C4—15832no—
501C22×C4—1588no—
502C22×C4D4×D7914yes—
503C22×C4D4×D7914yes—
504C22×C4#1192962414yes—
505C22×C4#1192962514yes—
506C22×C4C23×D7912yes—
507C22×C4C23×D7914yes—
508C22×C4C23×D7914yes—
509C22×C4C23×D7912yes—
510C22×C4C23×D7912yes—
511C4⋊C4—28no—
512C4⋊C4—7916no—
513C4⋊C4—15816no—
514C4⋊C4—15816no—
515C4⋊C4—1588no—
516C4⋊C4—15816no—
517C4⋊C4—1588no—
518C22⋊C4—15816no—
519C22⋊C4—1588no—
520C22⋊C4—15816no—
521C22⋊C4—15816no—
522C22⋊C4—1588no—
523C22⋊C4—15816no—
524C42—28no—
525C42—15816no—
526C23—3164no—
527C23—3164no—
528C23—3164no—
529C23—3164no—
530C23—3164no—
531C23—3164no—
532C23—3164no—
533C23—3164no—
534C23—794no—
535C23—794no—
536C23—1588no—
537C23—1582no—
538C23—1582no—
539C23—1582no—
540C23—1582no—
541C23—1588no—
542C23—1588no—
543C23—1588no—
544C23—1588no—
545C23—1588no—
546C23—1582no—
547C23—1582no—
548C23—1582no—
549C23—1582no—
550C23—1582no—
551C23—1582no—
552C23—1582no—
553C23—1582no—
554C23C2×D4×D7912yes—
555C23C2×D4×D7912yes—
556C23#41×D7912yes—
557C23C2×#1192962412yes—
558C23C2×#1192962512yes—
559C23#1457920812yes—
560C23#1457920912yes—
561C23C24×D7911yescharacteristic
562Q8—416no—
563D4—15832no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—3164no—
580C2×C4—3168no—
581C2×C4—31616no—
582C2×C4—798no—
583C2×C4—794no—
584C2×C4—1584no—
585C2×C4—1582no—
586C2×C4—1582no—
587C2×C4—1584no—
588C2×C4—1582no—
589C2×C4—1584no—
590C2×C4—1584no—
591C2×C4—15816no—
592C2×C4—1588no—
593C2×C4—1582no—
594C2×C4—1582no—
595C2×C4—1584no—
596C2×C4—15832no—
597C2×C4#14579134116yes—
598C2×C4C2×D4×D7914yes—
599C2×C4C2×D4×D7914yes—
600C2×C4C2×#1192962414yes—
601C2×C4C2×#1192962514yes—
602C22—31616no—
603C22—31616no—
604C22—3162no—
605C22—3162no—
606C22—3162no—
607C22—3162no—
608C22—3162no—
609C22—3162no—
610C22—3162no—
611C22—3162no—
612C22—1584no—
613C22—1584no—
614C22—1584no—
615C22—1584no—
616C22—1584no—
617C22—1584no—
618C22C2×#1457913414yes—
619C22#1909907614yes—
620C22#1909913614yes—
621C22#1909921614yes—
622C22#1909922114yes—
623C22#1909925414yes—
624C22#1909928614yes—
625C22C22×D4×D7911yescharacteristic
626C22C22×D4×D7911yescharacteristic
627C22C2×#41×D7911yescharacteristic
628C22C22×#1192962411yescharacteristic
629C22C22×#1192962511yescharacteristic
630C22C2×#1457920811yescharacteristic
631C22C2×#1457920911yescharacteristic
632C4—24no—
633C4—44no—
634C4—44no—
635C4—42no—
636C4—3162no—
637C4—3164no—
638C4—1588no—
639C4—1584no—
640C4C2×#1457913418yes—
641C2—3164no—
642C2—3164no—
643C2#2376683218yesdirect product factor
644C2C22×#1457913411yescharacteristic
645C2C2×#1909907611yescharacteristic
646C2C2×#1909913611yescharacteristic
647C2C2×#1909921611yescharacteristic
648C2C2×#1909922111yescharacteristic
649C2C2×#1909925411yescharacteristic
650C2C2×#1909928611yescharacteristic
subgroupG/H
centerC24C23×D79
commutator, cosocleC23×C79C25
FrattiniC23C24×D79
FittingC2×#391×C79C2
socleC24×C79C24
prime radicalC23×D316C22
hypercenterC2×#391D79
hypocenterC79C2×#2415
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
1. C22×#14579137C2semidirect
2. C22×#14579137C2semidirect
4. C22×#14579119C2semidirect
5. #23766832C2direct (N × H)
6. C2×#19098492C2semidirect
8. C2×#19098461C2semidirect
9. C2×#19098460C2semidirect
10. C2×#19098452C2semidirect
11. C2×#19098420C2semidirect
12. C2×#391×C79C2semidirect
18. C2×#14579137C22semidirect
19. C2×#14579137C22semidirect
23. C2×#14579119C22semidirect
37. #19098492C22semidirect
39. #19098461C22semidirect
40. #19098460C22semidirect
41. #19098452C22semidirect
42. #19098420C22semidirect
48. #391×C79C22semidirect
75. C2×C4⋊Dic79D4semidirect
162. C4⋊Dic79C2×D4semidirect
316. C2×C79#2415semidirect
333. C2×#391D79semidirect
338. C79C2×#2415semidirect
368. #391C2×D79semidirect
643. C2#23766832direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145791372
C22×C4×D3162
C22×#145791192
#237668322
C2×#190984922
C2×#190984812
C2×#190984612
C2×#190984602
C2×#190984522
C2×#190984202
C2×#391×C792
C2×#241579