Finite Group Databasebeta

← C2×#23765881 #34446649

Subgroups — C2×#23765881 #34446649

Order 20224.

3854 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579119C211yesmaximal, characteristic
2C22×#14579119C211yesmaximal, characteristic
3C22×C42×D79C211yesmaximal, characteristic
4#23765881C2116yesmaximal, direct product factor
5C2×#19098462C211yesmaximal, characteristic, complemented
6C2×#19098461C211yesmaximal, characteristic, complemented
7C2×#19098459C211yesmaximal, characteristic, complemented
8C2×#19098437C212yesmaximal
9C2×#19098434C212yesmaximal
10C2×C4⋊C4×Dic79C212yesmaximal, complemented
11C2×#19098414C212yesmaximal, complemented
12C2×#389×C79C211yesmaximal, characteristic, complemented
13C23×C4×D79C2211yescharacteristic
14C23×C4×D79C2211yescharacteristic
15C23×C4×D79C2211yescharacteristic
16C2×#14579119—28no—
17C2×#14579119C4116yescomplemented
18C2×#14579119C2218yes—
19C2×#14579119C2218yes—
20C22×C4⋊Dic79C2212yes—
21C22×C4⋊Dic79C2211yescharacteristic
22C22×C4⋊Dic79C2211yescharacteristic
23C22×Dic79⋊C4C2212yes—
24C22×Dic79⋊C4C2212yes—
25C22×(C2×D79)⋊C4C2212yes—
26C22×(C2×D79)⋊C4C2212yes—
27C22×C4×Dic79C2211yescharacteristic
28C22×C4×Dic79C2211yescharacteristic
29C22×C4×Dic79C2211yescharacteristic
30C22×C4×Dic79C2212yes—
31C22×C4×Dic79C2212yes—
32C2×C42×D79—28no—
33C2×C42×D79C2218yes—
34#19098462C2218yescomplemented
35#19098461C2218yescomplemented
36#19098459C2218yescomplemented
37#19098437C22116yes—
38#19098434C22116yes—
39C4⋊C4×Dic79C22116yescomplemented
40#19098414C22116yescomplemented
41C2×C79⋊#95C2212yes—
42C2×C79⋊#95C2212yes—
43C2×C79⋊#95C2212yes—
44C22×C4⋊C4×C79C2212yes—
45C22×C4⋊C4×C79C2211yescharacteristic
46C22×C4⋊C4×C79C2211yescharacteristic
47C22×C42×C79C2211yescharacteristic
48#389×C79C2218yescomplemented
49C2×#95×C79C2212yes—
50C24×D79C2311yescharacteristic
51C23×Dic79C2312yes—
52C23×Dic79C2312yes—
53C23×Dic79C2311yescharacteristic
54C23×Dic79C2311yescharacteristic
55C23×Dic79C2311yescharacteristic
56C22×C4×D79—24no—
57C22×C4×D79—24no—
58C22×C4×D79—24no—
59C22×C4×D79—24no—
60C22×C4×D79—24no—
61C22×C4×D79—24no—
62C22×C4×D79C2×C418yes—
63C22×C4×D79D418yes—
64C22×C4×D79Q818yes—
65C22×C4×D79C2314yes—
66C22×C4×D79C2314yes—
67C22×C4×D79C2314yes—
68#14579119—232no—
69#14579119C2×C4164yescomplemented
70C2×C4⋊Dic79—24no—
71C2×C4⋊Dic79—28no—
72C2×C4⋊Dic79C2×C418yescomplemented
73C2×C4⋊Dic79C2318yes—
74C2×C4⋊Dic79C2314yes—
75C2×C4⋊Dic79C2314yes—
76C2×Dic79⋊C4—28no—
77C2×Dic79⋊C4—28no—
78C2×Dic79⋊C4C2318yes—
79C2×Dic79⋊C4C2318yes—
80C2×(C2×D79)⋊C4—28no—
81C2×(C2×D79)⋊C4C2×C4116yes—
82C2×(C2×D79)⋊C4C2318yes—
83C2×(C2×D79)⋊C4C2318yes—
84C2×C4×Dic79—24no—
85C2×C4×Dic79—24no—
86C2×C4×Dic79—24no—
87C2×C4×Dic79—28no—
88C2×C4×Dic79C2×C4116yescomplemented
89C2×C4×Dic79C2314yes—
90C2×C4×Dic79C2318yes—
91C2×C4×Dic79C2314yes—
92C2×C4×Dic79C2314yes—
93C2×C4×Dic79C2318yes—
94C42×D79—232no—
95C79⋊#95C2318yes—
96C79⋊#95C2318yes—
97C79⋊#95C2318yes—
98C23×C4×C79C2312yes—
99C23×C4×C79C2312yes—
100C23×C4×C79C2311yescharacteristic
101C23×C4×C79C2311yescharacteristic
102C23×C4×C79C2311yescharacteristic
103C2×C4⋊C4×C79—24no—
104C2×C4⋊C4×C79—28no—
105C2×C4⋊C4×C79C2×C418yescomplemented
106C2×C4⋊C4×C79C2318yes—
107C2×C4⋊C4×C79C2314yes—
108C2×C4⋊C4×C79C2314yes—
109C2×C42×C79—24no—
110C2×C42×C79C2314yes—
111#95×C79C2318yes—
112C23×D79—22no—
113C23×D79—22no—
114C23×D79—22no—
115C23×D79—22no—
116C23×D79C22×C414yes—
117C23×D79C2×D414yes—
118C23×D79C2×Q814yes—
119C23×D79C2412yes—
120C22×Dic79—22no—
121C22×Dic79—22no—
122C22×Dic79—24no—
123C22×Dic79—24no—
124C22×Dic79—24no—
125C22×Dic79—22no—
126C22×Dic79—22no—
127C22×Dic79—24no—
128C22×Dic79—24no—
129C22×Dic79C22×C418yes—
130C22×Dic79C22×C414yes—
131C22×Dic79C2×D414yes—
132C22×Dic79C2×Q814yes—
133C22×Dic79#4114yes—
134C22×Dic79#4114yes—
135C22×Dic79C2414yes—
136C22×Dic79C2414yes—
137C22×Dic79C2412yes—
138C22×Dic79C2412yes—
139C22×Dic79C2412yes—
140C2×C4×D79—232no—
141C2×C4×D79—232no—
142C2×C4×D79—28no—
143C2×C4×D79—28no—
144C2×C4×D79—28no—
145C2×C4×D79—28no—
146C2×C4×D79—28no—
147C2×C4×D79—28no—
148C2×C4×D79C4⋊C4164yescomplemented
149C2×C4×D79C22×C4116yes—
150C2×C4×D79C2×D4116yes—
151C2×C4×D79C2×Q8116yes—
152C4⋊Dic79—216no—
153C4⋊Dic79—28no—
154C4⋊Dic79C22×C4116yescomplemented
155Dic79⋊C4—216no—
156Dic79⋊C4—216no—
157(C2×D79)⋊C4—216no—
158(C2×D79)⋊C4C22×C4132yes—
159C4×Dic79—28no—
160C4×Dic79—28no—
161C4×Dic79—216no—
162C4×Dic79—28no—
163C4×Dic79C22×C4132yescomplemented
164C24×C79C2411yescharacteristic
165C22×C4×C79—22no—
166C22×C4×C79—22no—
167C22×C4×C79—24no—
168C22×C4×C79—24no—
169C22×C4×C79—22no—
170C22×C4×C79—22no—
171C22×C4×C79—24no—
172C22×C4×C79—24no—
173C22×C4×C79—24no—
174C22×C4×C79C22×C418yes—
175C22×C4×C79C22×C414yes—
176C22×C4×C79C2×D414yes—
177C22×C4×C79C2×Q814yes—
178C22×C4×C79#4114yes—
179C22×C4×C79#4114yes—
180C22×C4×C79C2414yes—
181C22×C4×C79C2414yes—
182C22×C4×C79C2412yes—
183C22×C4×C79C2412yes—
184C22×C4×C79C2412yes—
185C4⋊C4×C79—28no—
186C4⋊C4×C79—216no—
187C4⋊C4×C79C22×C4116yescomplemented
188C42×C79—28no—
189C23×C79C23×C412yes—
190C23×C79C22×D412yes—
191C23×C79C22×Q812yes—
192C23×C79C2×#4112yes—
193C23×C79C2×#4112yes—
194C23×C79C2×#4112yes—
195C23×C79C2×#4112yes—
196C23×C79C2511yescharacteristic
197C22×D79—28no—
198C22×D79—28no—
199C22×D79—28no—
200C22×D79—28no—
201C22×D79—28no—
202C22×D79—28no—
203C22×D79—22no—
204C22×D79—22no—
205C22×D79—22no—
206C22×D79—22no—
207C22×D79C2×C4⋊C4116yes—
208C22×D79C23×C414yes—
209C22×D79C22×D414yes—
210C22×D79C22×Q814yes—
211C2×C4×C79—216no—
212C2×C4×C79—28no—
213C2×C4×C79—28no—
214C2×C4×C79—22no—
215C2×C4×C79—22no—
216C2×C4×C79—24no—
217C2×C4×C79—22no—
218C2×C4×C79—22no—
219C2×C4×C79—24no—
220C2×C4×C79—24no—
221C2×C4×C79—24no—
222C2×C4×C79—24no—
223C2×C4×C79—48no—
224C2×C4×C79C2×C4⋊C4116yes—
225C2×C4×C79C23×C418yes—
226C2×C4×C79C23×C414yes—
227C2×C4×C79C22×D414yes—
228C2×C4×C79C22×Q814yes—
229C2×C4×C79C2×#4114yes—
230C2×C4×C79C2×#4114yes—
231C2×Dic79—216no—
232C2×Dic79—28no—
233C2×Dic79—28no—
234C2×Dic79—22no—
235C2×Dic79—22no—
236C2×Dic79—24no—
237C2×Dic79—24no—
238C2×Dic79—22no—
239C2×Dic79—22no—
240C2×Dic79—24no—
241C2×Dic79—24no—
242C2×Dic79—24no—
243C2×Dic79—48no—
244C2×Dic79C2×C4⋊C4116yes—
245C2×Dic79C23×C418yes—
246C2×Dic79C23×C414yes—
247C2×Dic79C22×D414yes—
248C2×Dic79C22×Q814yes—
249C2×Dic79C2×#4114yes—
250C2×Dic79C2×#4114yes—
251C4×D79—232no—
252C4×D79—232no—
253C4×D79C2×C4⋊C4164yescomplemented
254C22×C79C22×C4⋊C414yes—
255C22×C79C4×#4114yes—
256C22×C79C4×#4114yes—
257C22×C79#52514yes—
258C22×C79#52514yes—
259C22×C79#53314yes—
260C22×C79#53314yes—
261C22×C79C24×C411yescharacteristic
262C22×C79C23×D411yescharacteristic
263C22×C79C23×Q811yescharacteristic
264C22×C79C22×#4111yescharacteristic
265C22×C79C22×#4111yescharacteristic
266C22×C79C22×#4111yescharacteristic
267C22×C79C22×#4111yescharacteristic
268C2×D79—24no—
269C2×D79—24no—
270C2×D79—24no—
271C2×D79—24no—
272C2×D79—24no—
273C2×D79—24no—
274C2×D79—232no—
275C2×D79C22×C4⋊C418yes—
276C4×C79—28no—
277C4×C79—24no—
278C4×C79—24no—
279C4×C79—44no—
280C4×C79C22×C4⋊C418yes—
281Dic79—28no—
282Dic79—24no—
283Dic79—24no—
284Dic79—44no—
285Dic79C22×C4⋊C418yes—
286C2×#2296—791nomaximal, Sylow
287C2×C79#229618yescomplemented
288C2×C79C23×C4⋊C411yescharacteristic
289C2×C79C2×C4×#4111yescharacteristic
290C2×C79C2×C4×#4111yescharacteristic
291C2×C79C2×#52511yescharacteristic
292C2×C79C2×#52511yescharacteristic
293C2×C79C2×#53311yescharacteristic
294C2×C79C2×#53311yescharacteristic
295D79—28no—
296C22×#117—791no—
297C22×#117—791no—
298C23×C42—791no—
299#2296—7916no—
300C2×#389—791no—
301C2×#389—791no—
302C2×#389—791no—
303C2×#389D7911yescharacteristic, complemented
304C2×#385—792no—
305C2×#385—792no—
306C2×C4×C4⋊C4—792no—
307C2×C4×C4⋊C4—792no—
308C79C2×#229611yescharacteristic, complemented, Sylow
309C24×C4—791no—
310C24×C4—791no—
311C24×C4—791no—
312C2×#117—798no—
313C2×#117—798no—
314C2×#117—7916no—
315C2×#117—1588no—
316C22×C4⋊C4—791no—
317C22×C4⋊C4—791no—
318C22×C4⋊C4—792no—
319C22×C4⋊C4—792no—
320C22×C4⋊C4—792no—
321C22×C4⋊C4C2×D7911yescharacteristic
322C22×C4⋊C4C2×D7912yes—
323C22×C4⋊C4C2×D7911yescharacteristic
324C22×C22⋊C4—792no—
325C22×C22⋊C4—792no—
326C22×C42—792no—
327C22×C42—791no—
328C22×C42—791no—
329C22×C42—791no—
330C22×C42—792no—
331C22×C42—798no—
332C22×C42—1588no—
333C22×C42C2×D7911yescharacteristic
334#389—798no—
335#389—798no—
336#389—798no—
337#389C2×D7918yescomplemented
338#385—7916no—
339#385—7916no—
340C4×C4⋊C4—7916no—
341C4×C4⋊C4—7916no—
342C2×#95—792no—
343C2×#95—792no—
344C2×#95—792no—
345C2×#95C2×D7912yes—
346C25—791no—
347C23×C4—791no—
348C23×C4—798no—
349C23×C4—798no—
350C23×C4—798no—
351C23×C4—794no—
352C23×C4—792no—
353C23×C4—791no—
354C23×C4—791no—
355C23×C4—792no—
356C23×C4—794no—
357C23×C4—794no—
358C23×C4—1584no—
359C23×C4—1584no—
360C23×C4—1584no—
361C23×C4—1584no—
362C23×C4—1584no—
363C23×C4—1584no—
364C23×C4C22×D7912yes—
365C23×C4C22×D7912yes—
366C23×C4C22×D7911yescharacteristic
367C23×C4C22×D7911yescharacteristic
368C23×C4C22×D7911yescharacteristic
369#117—7964no—
370#117—15832no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—28no—
373C2×C4⋊C4—798no—
374C2×C4⋊C4—794no—
375C2×C4⋊C4—794no—
376C2×C4⋊C4—798no—
377C2×C4⋊C4—798no—
378C2×C4⋊C4—798no—
379C2×C4⋊C4—1588no—
380C2×C4⋊C4—1588no—
381C2×C4⋊C4—1584no—
382C2×C4⋊C4—1588no—
383C2×C4⋊C4C4×D7918yescomplemented
384C2×C4⋊C4C22×D7914yes—
385C2×C4⋊C4C22×D7918yes—
386C2×C4⋊C4C22×D7914yes—
387C2×C22⋊C4—7916no—
388C2×C22⋊C4—798no—
389C2×C22⋊C4—798no—
390C2×C22⋊C4—1588no—
391C2×C42—24no—
392C2×C42—7916no—
393C2×C42—798no—
394C2×C42—798no—
395C2×C42—794no—
396C2×C42—794no—
397C2×C42—794no—
398C2×C42—15832no—
399C2×C42—1588no—
400C2×C42—1584no—
401C2×C42—1584no—
402C2×C42—1584no—
403C2×C42C22×D7914yes—
404#95—798no—
405#95—798no—
406#95—798no—
407#95C22×D7918yes—
408C24—794no—
409C24—792no—
410C24—794no—
411C24—794no—
412C24—1582no—
413C24—1582no—
414C24—1582no—
415C24—1582no—
416C24C23×D7911yescharacteristic
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—794no—
427C22×C4—792no—
428C22×C4—794no—
429C22×C4—794no—
430C22×C4—7916no—
431C22×C4—7964no—
432C22×C4—7916no—
433C22×C4—7916no—
434C22×C4—794no—
435C22×C4—794no—
436C22×C4—792no—
437C22×C4—792no—
438C22×C4—794no—
439C22×C4—798no—
440C22×C4—794no—
441C22×C4—1588no—
442C22×C4—1588no—
443C22×C4—1588no—
444C22×C4—1588no—
445C22×C4—15832no—
446C22×C4—15832no—
447C22×C4—1584no—
448C22×C4—1584no—
449C22×C4—1584no—
450C22×C4—1584no—
451C22×C4—1582no—
452C22×C4—1584no—
453C22×C4—1582no—
454C22×C4—1582no—
455C22×C4—1582no—
456C22×C4—1588no—
457C22×C4—1588no—
458C22×C4C2×C4×D7918yes—
459C22×C4C2×C4×D7914yes—
460C22×C4D4×D7914yes—
461C22×C4Q8×D7914yes—
462C22×C4#1192962314yes—
463C22×C4#1192962514yes—
464C22×C4C23×D7914yes—
465C22×C4C23×D7912yes—
466C22×C4C23×D7912yes—
467C22×C4C23×D7914yes—
468C22×C4C23×D7912yes—
469C4⋊C4—216no—
470C4⋊C4—28no—
471C4⋊C4—7916no—
472C4⋊C4—15816no—
473C4⋊C4—15816no—
474C4⋊C4—15816no—
475C4⋊C4—1588no—
476C4⋊C4C2×C4×D79116yescomplemented
477C22⋊C4—7932no—
478C22⋊C4—15816no—
479C42—28no—
480C42—7932no—
481C42—1588no—
482C42—1588no—
483C42—15816no—
484C42—1588no—
485C23—7916no—
486C23—794no—
487C23—794no—
488C23—794no—
489C23—1588no—
490C23—1588no—
491C23—1588no—
492C23—1588no—
493C23—1588no—
494C23—1588no—
495C23—1582no—
496C23—1582no—
497C23—1582no—
498C23—1582no—
499C23C22×C4×D7912yes—
500C23C2×D4×D7912yes—
501C23C2×Q8×D7912yes—
502C23#41×D7912yes—
503C23#41×D7912yes—
504C23C2×#1192962312yes—
505C23C2×#1192962512yes—
506C23C24×D7911yescharacteristic
507C2×C4—216no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—48no—
520C2×C4—3168no—
521C2×C4—7916no—
522C2×C4—794no—
523C2×C4—794no—
524C2×C4—794no—
525C2×C4—7964no—
526C2×C4—798no—
527C2×C4—794no—
528C2×C4—794no—
529C2×C4—15832no—
530C2×C4—15832no—
531C2×C4—1584no—
532C2×C4—1584no—
533C2×C4—1584no—
534C2×C4—15816no—
535C2×C4—1588no—
536C2×C4—1588no—
537C2×C4—1582no—
538C2×C4—1582no—
539C2×C4—1584no—
540C2×C4—1582no—
541C2×C4—1584no—
542C2×C4—1582no—
543C2×C4C4⋊C4×D79116yes—
544C2×C4C22×C4×D7918yes—
545C2×C4C22×C4×D7914yes—
546C2×C4C2×D4×D7914yes—
547C2×C4C2×Q8×D7914yes—
548C2×C4C2×#1192962314yes—
549C2×C4C2×#1192962514yes—
550C22—798no—
551C22—1584no—
552C22—1584no—
553C22—15832no—
554C22—1584no—
555C22—1584no—
556C22—1584no—
557C22—1584no—
558C22C2×C4⋊C4×D7914yes—
559C22C4×#1192962314yes—
560C22C4×#1192962514yes—
561C22#1909907314yes—
562C22#1909907614yes—
563C22#1909911814yes—
564C22#1909911914yes—
565C22C23×C4×D7911yescharacteristic
566C22C22×D4×D7911yescharacteristic
567C22C22×Q8×D7911yescharacteristic
568C22C2×#41×D7911yescharacteristic
569C22C2×#41×D7911yescharacteristic
570C22C22×#1192962311yescharacteristic
571C22C22×#1192962511yescharacteristic
572C4—28no—
573C4—24no—
574C4—24no—
575C4—44no—
576C4—3164no—
577C4—798no—
578C4—1588no—
579C4—1584no—
580C4—1584no—
581C4C2×C4⋊C4×D7918yes—
582C2—1588no—
583C2#2376588118yesdirect product factor
584C2C22×C4⋊C4×D7911yescharacteristic
585C2C2×C4×#1192962311yescharacteristic
586C2C2×C4×#1192962511yescharacteristic
587C2C2×#1909907311yescharacteristic
588C2C2×#1909907611yescharacteristic
589C2C2×#1909911811yescharacteristic
590C2C2×#1909911911yescharacteristic
subgroupG/H
centerC24C23×D79
commutatorC22×C79C24×C4
FrattiniC23C24×D79
FittingC2×#389×C79C2
socleC24×C79C24
prime radicalC24×D79C23
cosocleC23×C79C25
hypercenterC2×#389D79
hypocenterC79C2×#2296
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
4. #23765881C2direct (N × H)
5. C2×#19098462C2semidirect
6. C2×#19098461C2semidirect
7. C2×#19098459C2semidirect
10. C2×C4⋊C4×Dic79C2semidirect
11. C2×#19098414C2semidirect
12. C2×#389×C79C2semidirect
17. C2×#14579119C4semidirect
34. #19098462C22semidirect
35. #19098461C22semidirect
36. #19098459C22semidirect
39. C4⋊C4×Dic79C22semidirect
40. #19098414C22semidirect
48. #389×C79C22semidirect
69. #14579119C2×C4semidirect
72. C2×C4⋊Dic79C2×C4semidirect
88. C2×C4×Dic79C2×C4semidirect
105. C2×C4⋊C4×C79C2×C4semidirect
148. C2×C4×D79C4⋊C4semidirect
154. C4⋊Dic79C22×C4semidirect
163. C4×Dic79C22×C4semidirect
187. C4⋊C4×C79C22×C4semidirect
253. C4×D79C2×C4⋊C4semidirect
287. C2×C79#2296semidirect
303. C2×#389D79semidirect
308. C79C2×#2296semidirect
337. #389C2×D79semidirect
383. C2×C4⋊C4C4×D79semidirect
476. C4⋊C4C2×C4×D79semidirect
583. C2#23765881direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145791192
C22×C42×D792
#237658812
C2×#190984622
C2×#190984612
C2×#190984592
C2×#190984372
C2×#190984342
C2×C4⋊C4×Dic792
C2×#190984142
C2×#389×C792
C2×#229679