Finite Group Databasebeta

← C2×#23767706 #34457321

Subgroups — C2×#23767706 #34457321

Order 20224.

3678 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579146C212yesmaximal
2C22×C4×D316C211yesmaximal, characteristic
3C22×C4⋊C4×D79C211yesmaximal, characteristic, complemented
4#23767706C2116yesmaximal, direct product factor
5C2×#19098496C212yesmaximal, complemented
6C2×#19098492C212yesmaximal, complemented
7C2×#19098461C211yesmaximal, characteristic, complemented
8C2×#19098425C212yesmaximal, complemented
9C2×#19098420C211yesmaximal, characteristic, complemented
10C2×#19098420C211yesmaximal, characteristic, complemented
11C2×#19098420C211yesmaximal, characteristic, complemented
12C2×#389×C79C211yesmaximal, characteristic, complemented
13C23×D316C2211yescharacteristic
14C23×C4×D79C2211yescharacteristic
15C23×C4×D79C2212yes—
16C23×C4×D79C2211yescharacteristic
17C2×#14579146—216no—
18C2×#14579146C22116yes—
19C2×C4×D316—28no—
20C2×C4×D316C2218yes—
21C22×C4⋊Dic79C2211yescharacteristic
22C22×C4⋊Dic79C2211yescharacteristic
23C22×C4⋊Dic79C2211yescharacteristic
24C22×Dic79⋊C4C2212yes—
25C22×Dic79⋊C4C2212yes—
26C2×C4⋊C4×D79—28no—
27C2×C4⋊C4×D79C2218yescomplemented
28C22×(C2×D79)⋊C4C2212yes—
29C22×(C2×D79)⋊C4C2212yes—
30C22×(C2×D79)⋊C4C2212yes—
31C22×(C2×D79)⋊C4C2211yescharacteristic
32C22×(C2×D79)⋊C4C2211yescharacteristic
33#19098496C22116yescomplemented
34#19098492C22116yescomplemented
35#19098461C2218yescomplemented
36#19098425C22116yescomplemented
37#19098420C2218yescomplemented
38#19098420C2218yescomplemented
39#19098420C2218yescomplemented
40C2×C79⋊#95C2212yes—
41C2×C79⋊#95C2212yes—
42C2×C79⋊#95C2212yes—
43C2×C79⋊#95C2212yes—
44C22×C4⋊C4×C79C2211yescharacteristic
45C22×C4⋊C4×C79C2211yescharacteristic
46C22×C4⋊C4×C79C2212yes—
47C22×C42×C79C2211yescharacteristic
48#389×C79C2218yescomplemented
49C2×#95×C79C2212yes—
50C24×D79C2311yescharacteristic
51C24×D79C2311yescharacteristic
52C22×D316—24no—
53C22×D316—24no—
54C22×D316D418yes—
55C22×D316C2314yes—
56C23×Dic79C2311yescharacteristic
57C23×Dic79C2312yes—
58C23×Dic79C2312yes—
59C23×Dic79C2311yescharacteristic
60C22×C4×D79—24no—
61C22×C4×D79—24no—
62C22×C4×D79—24no—
63C22×C4×D79—24no—
64C22×C4×D79—24no—
65C22×C4×D79—28no—
66C22×C4×D79—28no—
67C22×C4×D79—28no—
68C22×C4×D79D418yes—
69C22×C4×D79C2314yes—
70C22×C4×D79C2318yes—
71C22×C4×D79C2314yes—
72#14579146—264no—
73C4×D316—232no—
74C2×C4⋊Dic79—24no—
75C2×C4⋊Dic79—24no—
76C2×C4⋊Dic79—24no—
77C2×C4⋊Dic79C2314yes—
78C2×C4⋊Dic79C2314yes—
79C2×C4⋊Dic79C2314yes—
80C2×Dic79⋊C4—28no—
81C2×Dic79⋊C4—28no—
82C2×Dic79⋊C4C2318yes—
83C2×Dic79⋊C4C2318yes—
84C4⋊C4×D79—232no—
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)⋊C4C2318yes—
91C2×(C2×D79)⋊C4C2318yes—
92C2×(C2×D79)⋊C4C2318yes—
93C2×(C2×D79)⋊C4C2314yes—
94C2×(C2×D79)⋊C4C2314yes—
95C79⋊#95C2318yes—
96C79⋊#95C2318yes—
97C79⋊#95C2318yes—
98C79⋊#95C2318yes—
99C23×C4×C79C2311yescharacteristic
100C23×C4×C79C2312yes—
101C23×C4×C79C2312yes—
102C23×C4×C79C2311yescharacteristic
103C23×C4×C79C2311yescharacteristic
104C2×C4⋊C4×C79—24no—
105C2×C4⋊C4×C79—28no—
106C2×C4⋊C4×C79—24no—
107C2×C4⋊C4×C79C2314yes—
108C2×C4⋊C4×C79C2314yes—
109C2×C4⋊C4×C79C2318yes—
110C2×C42×C79—24no—
111C2×C42×C79C2314yes—
112#95×C79C2318yes—
113C23×D79—22no—
114C23×D79—22no—
115C23×D79—22no—
116C23×D79—22no—
117C23×D79—22no—
118C23×D79—22no—
119C23×D79—22no—
120C23×D79—22no—
121C23×D79—22no—
122C23×D79—22no—
123C23×D79C2×D414yes—
124C23×D79C2×D414yes—
125C23×D79C2×D414yes—
126C23×D79#4114yes—
127C23×D79C2412yes—
128C23×D79C2412yes—
129C2×D316—232no—
130C2×D316—28no—
131C2×D316—28no—
132C2×D316C2×D4116yes—
133C22×Dic79—24no—
134C22×Dic79—24no—
135C22×Dic79—22no—
136C22×Dic79—24no—
137C22×Dic79—22no—
138C22×Dic79—22no—
139C22×Dic79—24no—
140C22×Dic79—24no—
141C22×Dic79—22no—
142C22×Dic79—22no—
143C22×Dic79—24no—
144C22×Dic79C2×D414yes—
145C22×Dic79C2412yes—
146C22×Dic79C2414yes—
147C22×Dic79C2414yes—
148C22×Dic79C2412yes—
149C2×C4×D79—216no—
150C2×C4×D79—216no—
151C2×C4×D79—216no—
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—432no—
159C2×C4×D79—416no—
160C2×C4×D79C2×D4116yes—
161C4⋊Dic79—28no—
162C4⋊Dic79—28no—
163C4⋊Dic79—28no—
164Dic79⋊C4—216no—
165Dic79⋊C4—216no—
166(C2×D79)⋊C4—28no—
167(C2×D79)⋊C4—28no—
168(C2×D79)⋊C4—216no—
169(C2×D79)⋊C4—216no—
170(C2×D79)⋊C4—216no—
171C24×C79C2411yescharacteristic
172C22×C4×C79—22no—
173C22×C4×C79—22no—
174C22×C4×C79—22no—
175C22×C4×C79—24no—
176C22×C4×C79—24no—
177C22×C4×C79—24no—
178C22×C4×C79—24no—
179C22×C4×C79—24no—
180C22×C4×C79—22no—
181C22×C4×C79—22no—
182C22×C4×C79—24no—
183C22×C4×C79C2×D414yes—
184C22×C4×C79C2×D414yes—
185C22×C4×C79#4114yes—
186C22×C4×C79#4114yes—
187C22×C4×C79C2412yes—
188C22×C4×C79C2414yes—
189C22×C4×C79C2414yes—
190C22×C4×C79C2412yes—
191C22×C4×C79C2412yes—
192C4⋊C4×C79—28no—
193C4⋊C4×C79—216no—
194C4⋊C4×C79—28no—
195C42×C79—28no—
196C23×C79C22×D412yes—
197C23×C79C22×D412yes—
198C23×C79C2×#4112yes—
199C23×C79C2×#4112yes—
200C23×C79C2×#4112yes—
201C23×C792−1+412yes—
202C23×C792−1+412yes—
203C23×C79C2511yescharacteristic
204C22×D79—28no—
205C22×D79—28no—
206C22×D79—28no—
207C22×D79—28no—
208C22×D79—28no—
209C22×D79—28no—
210C22×D79—22no—
211C22×D79—22no—
212C22×D79—22no—
213C22×D79—22no—
214C22×D79—28no—
215C22×D79—28no—
216C22×D79—28no—
217C22×D79—22no—
218C22×D79—22no—
219C22×D79—22no—
220C22×D79—22no—
221C22×D79—22no—
222C22×D79—22no—
223C22×D79—44no—
224C22×D79—44no—
225C22×D79—44no—
226C22×D79—44no—
227C22×D79#121116yes—
228C22×D79C22×D414yes—
229C22×D79C22×D414yes—
230C22×D79C22×D414yes—
231C22×D79C2×#4114yes—
232C2×C4×C79—28no—
233C2×C4×C79—22no—
234C2×C4×C79—22no—
235C2×C4×C79—22no—
236C2×C4×C79—24no—
237C2×C4×C79—24no—
238C2×C4×C79—24no—
239C2×C4×C79—24no—
240C2×C4×C79—24no—
241C2×C4×C79—22no—
242C2×C4×C79—24no—
243C2×C4×C79—22no—
244C2×C4×C79—48no—
245C2×C4×C79—44no—
246C2×C4×C79—48no—
247C2×C4×C79#123116yescomplemented
248C2×C4×C79C22×D414yes—
249C2×C4×C79C22×D414yes—
250C2×C4×C79C2×#4114yes—
251C2×C4×C79C2×#4114yes—
252C2×Dic79—28no—
253C2×Dic79—22no—
254C2×Dic79—22no—
255C2×Dic79—24no—
256C2×Dic79—24no—
257C2×Dic79—22no—
258C2×Dic79—24no—
259C2×Dic79—24no—
260C2×Dic79—24no—
261C2×Dic79—22no—
262C2×Dic79—22no—
263C2×Dic79—24no—
264C2×Dic79—48no—
265C2×Dic79—44no—
266C2×Dic79—48no—
267C2×Dic79C22×D414yes—
268D316—232no—
269C4×D79—232no—
270C4×D79—432no—
271C4×D79—416no—
272C22×C79C2×#12114yes—
273C22×C79C2×#12314yes—
274C22×C79#53614yes—
275C22×C79#54714yes—
276C22×C79#54714yes—
277C22×C79#55214yes—
278C22×C79#55214yes—
279C22×C79C23×D411yescharacteristic
280C22×C79C23×D411yescharacteristic
281C22×C79C22×#4111yescharacteristic
282C22×C79C22×#4111yescharacteristic
283C22×C79C22×#4111yescharacteristic
284C22×C79C2×2−1+411yescharacteristic
285C22×C79C2×2−1+411yescharacteristic
286C2×D79—24no—
287C2×D79—24no—
288C2×D79—24no—
289C2×D79—24no—
290C2×D79—24no—
291C2×D79—24no—
292C2×D79—232no—
293C2×D79—24no—
294C2×D79—24no—
295C2×D79—24no—
296C2×D79—42no—
297C2×D79—42no—
298C2×D79—42no—
299C2×D79—42no—
300C2×D79—416no—
301C2×D79C2×#12118yes—
302C4×C79—24no—
303C4×C79—44no—
304C4×C79—42no—
305C4×C79—44no—
306C4×C79C2×#12318yescomplemented
307Dic79—24no—
308Dic79—44no—
309Dic79—42no—
310Dic79—44no—
311C2×#2486—791nomaximal, Sylow
312C2×C79#248618yescomplemented
313C2×C79C22×#12111yescharacteristic
314C2×C79C22×#12311yescharacteristic
315C2×C79C2×#53611yescharacteristic
316C2×C79C2×#54711yescharacteristic
317C2×C79C2×#54711yescharacteristic
318C2×C79C2×#55211yescharacteristic
319C2×C79C2×#55211yescharacteristic
320D79—28no—
321D79—44no—
322C22×#123—792no—
323C22×C4×D4—791no—
324C23×C4⋊C4—791no—
325#2486—7916no—
326C2×#398—792no—
327C2×#397—792no—
328C2×#389—791no—
329C2×#389D7911yescharacteristic, complemented
330C2×#381—792no—
331C2×#380—791no—
332C2×#380—791no—
333C2×#380—791no—
334C79C2×#248611yescharacteristic, complemented, Sylow
335C23×D4—791no—
336C24×C4—791no—
337C24×C4—792no—
338C24×C4—791no—
339C2×#123—7916no—
340C2×#123—15816no—
341C2×C4×D4—798no—
342C2×C4×D4—1588no—
343C22×C4⋊C4—792no—
344C22×C4⋊C4—791no—
345C22×C4⋊C4—792no—
346C22×C4⋊C4—791no—
347C22×C4⋊C4—791no—
348C22×C4⋊C4—798no—
349C22×C4⋊C4—1588no—
350C22×C4⋊C4C2×D7912yes—
351C22×C4⋊C4C2×D7911yescharacteristic
352C22×C4⋊C4C2×D7911yescharacteristic
353C22×C22⋊C4—791no—
354C22×C22⋊C4—792no—
355C22×C22⋊C4—792no—
356C22×C22⋊C4—791no—
357C22×C22⋊C4—792no—
358C22×C42C2×D7911yescharacteristic
359#398—7916no—
360#397—7916no—
361#389—798no—
362#389C2×D7918yescomplemented
363#381—7916no—
364#380—798no—
365#380—798no—
366#380—798no—
367C2×#95—792no—
368C2×#95—792no—
369C2×#95—792no—
370C2×#95—792no—
371C2×#95C2×D7912yes—
372C25—791no—
373C25—791no—
374C22×D4—798no—
375C22×D4—794no—
376C22×D4—1584no—
377C22×D4—1584no—
378C23×C4—791no—
379C23×C4—798no—
380C23×C4—794no—
381C23×C4—792no—
382C23×C4—792no—
383C23×C4—791no—
384C23×C4—798no—
385C23×C4—794no—
386C23×C4—1588no—
387C23×C4—1588no—
388C23×C4—1588no—
389C23×C4—1584no—
390C23×C4—1584no—
391C23×C4—1584no—
392C23×C4—1584no—
393C23×C4—1584no—
394C23×C4C22×D7912yes—
395C23×C4C22×D7911yescharacteristic
396C23×C4C22×D7912yes—
397C23×C4C22×D7911yescharacteristic
398C23×C4C22×D7911yescharacteristic
399#123—15864no—
400C4×D4—15832no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—28no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—794no—
405C2×C4⋊C4—798no—
406C2×C4⋊C4—794no—
407C2×C4⋊C4—798no—
408C2×C4⋊C4—794no—
409C2×C4⋊C4—15832no—
410C2×C4⋊C4—1588no—
411C2×C4⋊C4—1584no—
412C2×C4⋊C4—1584no—
413C2×C4⋊C4—1588no—
414C2×C4⋊C4—1584no—
415C2×C4⋊C4C22×D7914yes—
416C2×C4⋊C4C22×D7918yes—
417C2×C4⋊C4C22×D7914yes—
418C2×C22⋊C4—794no—
419C2×C22⋊C4—794no—
420C2×C22⋊C4—798no—
421C2×C22⋊C4—798no—
422C2×C22⋊C4—798no—
423C2×C22⋊C4—1584no—
424C2×C22⋊C4—1584no—
425C2×C22⋊C4—1588no—
426C2×C22⋊C4—1588no—
427C2×C22⋊C4—1588no—
428C2×C42—24no—
429C2×C42C22×D7914yes—
430#95—798no—
431#95—798no—
432#95—798no—
433#95—798no—
434#95C22×D7918yes—
435C24—794no—
436C24—792no—
437C24—794no—
438C24—794no—
439C24—792no—
440C24—794no—
441C24—1582no—
442C24—1582no—
443C24—1582no—
444C24—1582no—
445C24—1582no—
446C24—1582no—
447C24—1582no—
448C24—1582no—
449C24—1582no—
450C24—1582no—
451C24C23×D7911yescharacteristic
452C2×D4—7916no—
453C2×D4—1588no—
454C2×D4—15832no—
455C2×D4—1588no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—31632no—
468C22×C4—31616no—
469C22×C4—792no—
470C22×C4—794no—
471C22×C4—7916no—
472C22×C4—792no—
473C22×C4—794no—
474C22×C4—794no—
475C22×C4—15816no—
476C22×C4—15816no—
477C22×C4—15816no—
478C22×C4—1584no—
479C22×C4—1582no—
480C22×C4—1584no—
481C22×C4—1582no—
482C22×C4—1584no—
483C22×C4—1584no—
484C22×C4—1584no—
485C22×C4—1582no—
486C22×C4—1584no—
487C22×C4—1588no—
488C22×C4—1588no—
489C22×C4—1588no—
490C22×C4—1582no—
491C22×C4—1582no—
492C22×C4—1588no—
493C22×C4—1588no—
494C22×C4—15832no—
495C22×C4D4×D7914yes—
496C22×C4D4×D7914yes—
497C22×C4#1192962414yes—
498C22×C4#1192962514yes—
499C22×C4C23×D7912yes—
500C22×C4C23×D7914yes—
501C22×C4C23×D7914yes—
502C22×C4C23×D7912yes—
503C22×C4C23×D7912yes—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—216no—
507C4⋊C4—15816no—
508C4⋊C4—1588no—
509C4⋊C4—1588no—
510C4⋊C4—15816no—
511C4⋊C4—1588no—
512C22⋊C4—1588no—
513C22⋊C4—1588no—
514C22⋊C4—15816no—
515C22⋊C4—15816no—
516C22⋊C4—15816no—
517C42—28no—
518C23—3164no—
519C23—3164no—
520C23—3164no—
521C23—3164no—
522C23—7916no—
523C23—794no—
524C23—794no—
525C23—794no—
526C23—794no—
527C23—1582no—
528C23—1582no—
529C23—1588no—
530C23—1588no—
531C23—1588no—
532C23—1582no—
533C23—1582no—
534C23—1582no—
535C23—1582no—
536C23—1588no—
537C23—1582no—
538C23—1582no—
539C23—1588no—
540C23—1588no—
541C23—1588no—
542C23—1588no—
543C23—1588no—
544C23—1582no—
545C23—1582no—
546C23C2×D4×D7912yes—
547C23C2×D4×D7912yes—
548C23#41×D7912yes—
549C23C2×#1192962412yes—
550C23C2×#1192962512yes—
551C23#1457920812yes—
552C23#1457920912yes—
553C23C24×D7911yescharacteristic
554D4—15832no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—31632no—
571C2×C4—3168no—
572C2×C4—3168no—
573C2×C4—3164no—
574C2×C4—31616no—
575C2×C4—794no—
576C2×C4—1584no—
577C2×C4—1584no—
578C2×C4—1584no—
579C2×C4—1582no—
580C2×C4—1584no—
581C2×C4—1584no—
582C2×C4—1582no—
583C2×C4—1584no—
584C2×C4—1582no—
585C2×C4—1588no—
586C2×C4—1582no—
587C2×C4—1582no—
588C2×C4—15832no—
589C2×C4#14579146116yescomplemented
590C2×C4C2×D4×D7914yes—
591C2×C4C2×D4×D7914yes—
592C2×C4C2×#1192962414yes—
593C2×C4C2×#1192962514yes—
594C22—3162no—
595C22—3162no—
596C22—3162no—
597C22—31616no—
598C22—3162no—
599C22—798no—
600C22—1584no—
601C22—1584no—
602C22—1584no—
603C22—1584no—
604C22—1584no—
605C22—1584no—
606C22—1584no—
607C22—1584no—
608C22—15832no—
609C22—1584no—
610C22#121×D7914yes—
611C22C2×#1457914614yes—
612C22#1909913614yes—
613C22#1909921614yes—
614C22#1909921814yes—
615C22#1909925114yes—
616C22#1909925414yes—
617C22C22×D4×D7911yescharacteristic
618C22C22×D4×D7911yescharacteristic
619C22C2×#41×D7911yescharacteristic
620C22C22×#1192962411yescharacteristic
621C22C22×#1192962511yescharacteristic
622C22C2×#1457920811yescharacteristic
623C22C2×#1457920911yescharacteristic
624C4—24no—
625C4—44no—
626C4—44no—
627C4—42no—
628C4—3164no—
629C4—3164no—
630C4—3162no—
631C4—1584no—
632C4C2×#1457914618yescomplemented
633C2—3164no—
634C2—1588no—
635C2#2376770618yesdirect product factor
636C2C2×#121×D7911yescharacteristic
637C2C22×#1457914611yescharacteristic
638C2C2×#1909913611yescharacteristic
639C2C2×#1909921611yescharacteristic
640C2C2×#1909921811yescharacteristic
641C2C2×#1909925111yescharacteristic
642C2C2×#1909925411yescharacteristic
subgroupG/H
centerC24C23×D79
commutator, cosocleC23×C79C25
FrattiniC23C24×D79
FittingC2×#389×C79C2
socleC24×C79C24
prime radicalC23×D316C22
hypercenterC2×#389D79
hypocenterC79C2×#2486
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊C4×D79C2semidirect
4. #23767706C2direct (N × H)
5. C2×#19098496C2semidirect
6. C2×#19098492C2semidirect
7. C2×#19098461C2semidirect
8. C2×#19098425C2semidirect
9. C2×#19098420C2semidirect
10. C2×#19098420C2semidirect
11. C2×#19098420C2semidirect
12. C2×#389×C79C2semidirect
27. C2×C4⋊C4×D79C22semidirect
33. #19098496C22semidirect
34. #19098492C22semidirect
35. #19098461C22semidirect
36. #19098425C22semidirect
37. #19098420C22semidirect
38. #19098420C22semidirect
39. #19098420C22semidirect
48. #389×C79C22semidirect
247. C2×C4×C79#123semidirect
306. C4×C79C2×#123semidirect
312. C2×C79#2486semidirect
329. C2×#389D79semidirect
334. C79C2×#2486semidirect
362. #389C2×D79semidirect
589. C2×C4#14579146semidirect
632. C4C2×#14579146semidirect
635. C2#23767706direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145791462
C22×C4×D3162
C22×C4⋊C4×D792
#237677062
C2×#190984962
C2×#190984922
C2×#190984612
C2×#190984252
C2×#190984202
C2×#389×C792
C2×#248679