Finite Group Databasebeta

← C2×#23766022 #34446888

Subgroups — C2×#23766022 #34446888

Order 20224.

3502 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579119C211yesmaximal, characteristic
2C22×#14579118C211yesmaximal, characteristic
3C22×C4⋊C4×D79C211yesmaximal, characteristic
4#23766022C2116yesmaximal, direct product factor
5C2×#19098462C211yesmaximal, characteristic, complemented
6C2×#19098461C211yesmaximal, characteristic, complemented
7C2×#19098459C211yesmaximal, characteristic, complemented
8C2×#19098437C212yesmaximal
9C2×#19098434C212yesmaximal
10C2×#19098432C212yesmaximal, complemented
11C2×#19098427C212yesmaximal, complemented
12C2×#389×C79C211yesmaximal, characteristic, complemented
13C23×C4×D79C2212yes—
14C23×C4×D79C2211yescharacteristic
15C2×#14579119C4116yescomplemented
16C2×#14579119C2218yes—
17C2×#14579118—28no—
18C2×#14579118C2218yes—
19C22×C4⋊Dic79C2211yescharacteristic
20C22×C4⋊Dic79C2212yes—
21C22×C4⋊Dic79C2211yescharacteristic
22C22×Dic79⋊C4C2211yescharacteristic
23C22×Dic79⋊C4C2212yes—
24C22×Dic79⋊C4C2211yescharacteristic
25C22×Dic79⋊C4C2212yes—
26C22×Dic79⋊C4C2212yes—
27C2×C4⋊C4×D79—28no—
28C2×C4⋊C4×D79C2218yes—
29C22×(C2×D79)⋊C4C2211yescharacteristic
30C22×(C2×D79)⋊C4C2211yescharacteristic
31C22×(C2×D79)⋊C4C2212yes—
32C22×C4×Dic79C2211yescharacteristic
33C22×C4×Dic79C2212yes—
34#19098462C2218yescomplemented
35#19098461C2218yescomplemented
36#19098459C2218yescomplemented
37#19098437C22116yes—
38#19098434C22116yes—
39#19098432C22116yescomplemented
40#19098427C22116yescomplemented
41C2×C79⋊#95C2212yes—
42C2×C79⋊#95C2212yes—
43C2×C79⋊#95C2212yes—
44C22×C4⋊C4×C79C2211yescharacteristic
45C22×C4⋊C4×C79C2212yes—
46C22×C4⋊C4×C79C2211yescharacteristic
47C22×C42×C79C2211yescharacteristic
48#389×C79C2218yescomplemented
49C2×#95×C79C2212yes—
50C24×D79C2311yescharacteristic
51C23×Dic79C2311yescharacteristic
52C23×Dic79C2312yes—
53C23×Dic79C2311yescharacteristic
54C23×Dic79C2312yes—
55C23×Dic79C2311yescharacteristic
56C22×C4×D79—28no—
57C22×C4×D79—44no—
58C22×C4×D79—44no—
59C22×C4×D79C2×C418yes—
60C22×C4×D79D418yes—
61C22×C4×D79Q818yes—
62C22×C4×D79C2318yes—
63C22×C4×D79C2314yes—
64#14579119C2×C4164yescomplemented
65#14579118—232no—
66C2×C4⋊Dic79—24no—
67C2×C4⋊Dic79—44no—
68C2×C4⋊Dic79C2×C418yescomplemented
69C2×C4⋊Dic79C2314yes—
70C2×C4⋊Dic79C2318yes—
71C2×C4⋊Dic79C2314yes—
72C2×Dic79⋊C4—28no—
73C2×Dic79⋊C4—24no—
74C2×Dic79⋊C4—24no—
75C2×Dic79⋊C4—44no—
76C2×Dic79⋊C4—44no—
77C2×Dic79⋊C4C2314yes—
78C2×Dic79⋊C4C2318yes—
79C2×Dic79⋊C4C2314yes—
80C2×Dic79⋊C4C2318yes—
81C2×Dic79⋊C4C2318yes—
82C4⋊C4×D79—232no—
83C2×(C2×D79)⋊C4—24no—
84C2×(C2×D79)⋊C4—24no—
85C2×(C2×D79)⋊C4C2×C4116yes—
86C2×(C2×D79)⋊C4C2314yes—
87C2×(C2×D79)⋊C4C2314yes—
88C2×(C2×D79)⋊C4C2318yes—
89C2×C4×Dic79—24no—
90C2×C4×Dic79C2×C4116yescomplemented
91C2×C4×Dic79C2314yes—
92C2×C4×Dic79C2318yes—
93C79⋊#95C2318yes—
94C79⋊#95C2318yes—
95C79⋊#95C2318yes—
96C23×C4×C79C2311yescharacteristic
97C23×C4×C79C2312yes—
98C23×C4×C79C2311yescharacteristic
99C23×C4×C79C2312yes—
100C23×C4×C79C2311yescharacteristic
101C2×C4⋊C4×C79—24no—
102C2×C4⋊C4×C79—44no—
103C2×C4⋊C4×C79C2×C418yescomplemented
104C2×C4⋊C4×C79C2314yes—
105C2×C4⋊C4×C79C2318yes—
106C2×C4⋊C4×C79C2314yes—
107C2×C42×C79—24no—
108C2×C42×C79C2314yes—
109#95×C79C2318yes—
110C23×D79—22no—
111C23×D79—22no—
112C23×D79—22no—
113C23×D79—22no—
114C23×D79C22×C414yes—
115C23×D79C2×D414yes—
116C23×D79C2×Q814yes—
117C23×D79C2412yes—
118C22×Dic79—22no—
119C22×Dic79—22no—
120C22×Dic79—24no—
121C22×Dic79—24no—
122C22×Dic79—22no—
123C22×Dic79—24no—
124C22×Dic79—22no—
125C22×Dic79—24no—
126C22×Dic79—24no—
127C22×Dic79—22no—
128C22×Dic79—22no—
129C22×Dic79C22×C418yes—
130C22×Dic79C22×C414yes—
131C22×Dic79C2×D414yes—
132C22×Dic79C2×Q814yes—
133C22×Dic79C2412yes—
134C22×Dic79C2414yes—
135C22×Dic79C2412yes—
136C22×Dic79C2414yes—
137C22×Dic79C2412yes—
138C2×C4×D79—216no—
139C2×C4×D79—48no—
140C2×C4×D79—48no—
141C2×C4×D79—432no—
142C2×C4×D79C4⋊C4164yescomplemented
143C2×C4×D79C22×C4116yes—
144C2×C4×D79C2×D4116yes—
145C2×C4×D79C2×Q8116yes—
146C4⋊Dic79—28no—
147C4⋊Dic79—48no—
148C4⋊Dic79C22×C4116yescomplemented
149Dic79⋊C4—216no—
150Dic79⋊C4—28no—
151Dic79⋊C4—28no—
152Dic79⋊C4—48no—
153Dic79⋊C4—48no—
154(C2×D79)⋊C4—28no—
155(C2×D79)⋊C4—28no—
156(C2×D79)⋊C4C22×C4132yes—
157C4×Dic79—28no—
158C4×Dic79C22×C4132yescomplemented
159C24×C79C2411yescharacteristic
160C22×C4×C79—24no—
161C22×C4×C79—24no—
162C22×C4×C79—22no—
163C22×C4×C79—22no—
164C22×C4×C79—24no—
165C22×C4×C79—22no—
166C22×C4×C79—24no—
167C22×C4×C79—22no—
168C22×C4×C79—24no—
169C22×C4×C79—22no—
170C22×C4×C79—22no—
171C22×C4×C79C22×C418yes—
172C22×C4×C79C22×C414yes—
173C22×C4×C79C2×D414yes—
174C22×C4×C79C2×Q814yes—
175C22×C4×C79C2412yes—
176C22×C4×C79C2414yes—
177C22×C4×C79C2412yes—
178C22×C4×C79C2414yes—
179C22×C4×C79C2412yes—
180C4⋊C4×C79—28no—
181C4⋊C4×C79—48no—
182C4⋊C4×C79C22×C4116yescomplemented
183C42×C79—28no—
184C23×C79C23×C412yes—
185C23×C79C22×D412yes—
186C23×C79C22×Q812yes—
187C23×C79H(5,𝔽2)12yes—
188C23×C79H(5,𝔽2)12yes—
189C23×C792−1+412yes—
190C23×C792−1+412yes—
191C23×C79C2511yescharacteristic
192C22×D79—28no—
193C22×D79—28no—
194C22×D79—28no—
195C22×D79—28no—
196C22×D79—28no—
197C22×D79—28no—
198C22×D79—22no—
199C22×D79—22no—
200C22×D79—22no—
201C22×D79—22no—
202C22×D79C2×C4⋊C4116yes—
203C22×D79C23×C414yes—
204C22×D79C22×D414yes—
205C22×D79C22×Q814yes—
206C2×C4×C79—216no—
207C2×C4×C79—24no—
208C2×C4×C79—24no—
209C2×C4×C79—22no—
210C2×C4×C79—22no—
211C2×C4×C79—24no—
212C2×C4×C79—22no—
213C2×C4×C79—24no—
214C2×C4×C79—24no—
215C2×C4×C79—22no—
216C2×C4×C79—22no—
217C2×C4×C79—22no—
218C2×C4×C79—48no—
219C2×C4×C79—44no—
220C2×C4×C79—44no—
221C2×C4×C79C2×C4⋊C4116yes—
222C2×C4×C79C23×C418yes—
223C2×C4×C79C23×C414yes—
224C2×C4×C79C22×D414yes—
225C2×C4×C79C22×Q814yes—
226C2×Dic79—216no—
227C2×Dic79—24no—
228C2×Dic79—24no—
229C2×Dic79—22no—
230C2×Dic79—22no—
231C2×Dic79—24no—
232C2×Dic79—22no—
233C2×Dic79—24no—
234C2×Dic79—22no—
235C2×Dic79—24no—
236C2×Dic79—22no—
237C2×Dic79—22no—
238C2×Dic79—48no—
239C2×Dic79—44no—
240C2×Dic79—44no—
241C2×Dic79C2×C4⋊C4116yes—
242C2×Dic79C23×C418yes—
243C2×Dic79C23×C414yes—
244C2×Dic79C22×D414yes—
245C2×Dic79C22×Q814yes—
246C4×D79—432no—
247C4×D79C2×C4⋊C4164yescomplemented
248C22×C79C22×C4⋊C414yes—
249C22×C79#52014yes—
250C22×C79#52014yes—
251C22×C79#53514yes—
252C22×C79#53614yes—
253C22×C79#54414yes—
254C22×C79#54414yes—
255C22×C79C24×C411yescharacteristic
256C22×C79C23×D411yescharacteristic
257C22×C79C23×Q811yescharacteristic
258C22×C79C2×H(5,𝔽2)11yescharacteristic
259C22×C79C2×H(5,𝔽2)11yescharacteristic
260C22×C79C2×2−1+411yescharacteristic
261C22×C79C2×2−1+411yescharacteristic
262C2×D79—232no—
263C2×D79—24no—
264C2×D79—24no—
265C2×D79—24no—
266C2×D79—24no—
267C2×D79—24no—
268C2×D79—24no—
269C2×D79C22×C4⋊C418yes—
270C4×C79—28no—
271C4×C79—42no—
272C4×C79—44no—
273C4×C79—42no—
274C4×C79C22×C4⋊C418yes—
275Dic79—28no—
276Dic79—44no—
277Dic79—42no—
278Dic79—42no—
279Dic79C22×C4⋊C418yes—
280C2×#2317—791nomaximal, Sylow
281C2×C79#231718yescomplemented
282C2×C79C23×C4⋊C411yescharacteristic
283C2×C79C2×#52011yescharacteristic
284C2×C79C2×#52011yescharacteristic
285C2×C79C2×#53511yescharacteristic
286C2×C79C2×#53611yescharacteristic
287C2×C79C2×#54411yescharacteristic
288C2×C79C2×#54411yescharacteristic
289D79—28no—
290C22×#117—791no—
291C22×#117—791no—
292C23×C4⋊C4—791no—
293#2317—7916no—
294C2×#389—791no—
295C2×#389—791no—
296C2×#389—791no—
297C2×#389D7911yescharacteristic, complemented
298C2×#385—792no—
299C2×#385—792no—
300C2×#384—792no—
301C2×#382—792no—
302C79C2×#231711yescharacteristic, complemented, Sylow
303C24×C4—792no—
304C24×C4—791no—
305C2×#117—798no—
306C2×#117—7916no—
307C2×#117—798no—
308C2×#117—1588no—
309C22×C4⋊C4—792no—
310C22×C4⋊C4—792no—
311C22×C4⋊C4—792no—
312C22×C4⋊C4—791no—
313C22×C4⋊C4—792no—
314C22×C4⋊C4—791no—
315C22×C4⋊C4—791no—
316C22×C4⋊C4—791no—
317C22×C4⋊C4—798no—
318C22×C4⋊C4—1588no—
319C22×C4⋊C4C2×D7911yescharacteristic
320C22×C4⋊C4C2×D7912yes—
321C22×C4⋊C4C2×D7911yescharacteristic
322C22×C22⋊C4—792no—
323C22×C22⋊C4—791no—
324C22×C22⋊C4—791no—
325C22×C42—792no—
326C22×C42—791no—
327C22×C42C2×D7911yescharacteristic
328#389—798no—
329#389—798no—
330#389—798no—
331#389C2×D7918yescomplemented
332#385—7916no—
333#385—7916no—
334#384—7916no—
335#382—7916no—
336C2×#95—792no—
337C2×#95—792no—
338C2×#95—792no—
339C2×#95C2×D7912yes—
340C25—791no—
341C23×C4—3164no—
342C23×C4—3164no—
343C23×C4—798no—
344C23×C4—791no—
345C23×C4—798no—
346C23×C4—798no—
347C23×C4—798no—
348C23×C4—794no—
349C23×C4—792no—
350C23×C4—791no—
351C23×C4—791no—
352C23×C4—792no—
353C23×C4—1588no—
354C23×C4C22×D7911yescharacteristic
355C23×C4C22×D7911yescharacteristic
356C23×C4C22×D7912yes—
357C23×C4C22×D7912yes—
358C23×C4C22×D7911yescharacteristic
359#117—7964no—
360#117—15832no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—44no—
363C2×C4⋊C4—3164no—
364C2×C4⋊C4—3164no—
365C2×C4⋊C4—3164no—
366C2×C4⋊C4—798no—
367C2×C4⋊C4—798no—
368C2×C4⋊C4—798no—
369C2×C4⋊C4—798no—
370C2×C4⋊C4—794no—
371C2×C4⋊C4—798no—
372C2×C4⋊C4—794no—
373C2×C4⋊C4—794no—
374C2×C4⋊C4—794no—
375C2×C4⋊C4—15832no—
376C2×C4⋊C4—1588no—
377C2×C4⋊C4—1584no—
378C2×C4⋊C4—1584no—
379C2×C4⋊C4—1584no—
380C2×C4⋊C4C4×D7918yescomplemented
381C2×C4⋊C4C22×D7914yes—
382C2×C4⋊C4C22×D7918yes—
383C2×C4⋊C4C22×D7914yes—
384C2×C22⋊C4—7916no—
385C2×C22⋊C4—798no—
386C2×C22⋊C4—794no—
387C2×C22⋊C4—794no—
388C2×C22⋊C4—1584no—
389C2×C22⋊C4—1584no—
390C2×C42—24no—
391C2×C42—7916no—
392C2×C42—798no—
393C2×C42—794no—
394C2×C42—1584no—
395C2×C42C22×D7914yes—
396#95—798no—
397#95—798no—
398#95—798no—
399#95C22×D7918yes—
400C24—794no—
401C24—792no—
402C24—794no—
403C24—794no—
404C24—1582no—
405C24—1582no—
406C24—1582no—
407C24—1582no—
408C24C23×D7911yescharacteristic
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—3168no—
421C22×C4—3168no—
422C22×C4—31632no—
423C22×C4—794no—
424C22×C4—792no—
425C22×C4—794no—
426C22×C4—794no—
427C22×C4—7916no—
428C22×C4—7964no—
429C22×C4—7916no—
430C22×C4—7916no—
431C22×C4—794no—
432C22×C4—798no—
433C22×C4—792no—
434C22×C4—794no—
435C22×C4—792no—
436C22×C4—15816no—
437C22×C4—1582no—
438C22×C4—1584no—
439C22×C4—1582no—
440C22×C4—1584no—
441C22×C4—1582no—
442C22×C4—1582no—
443C22×C4—1582no—
444C22×C4—1582no—
445C22×C4—1584no—
446C22×C4—1584no—
447C22×C4—1584no—
448C22×C4C2×C4×D7918yes—
449C22×C4C2×C4×D7914yes—
450C22×C4D4×D7914yes—
451C22×C4Q8×D7914yes—
452C22×C4C23×D7912yes—
453C22×C4C23×D7914yes—
454C22×C4C23×D7912yes—
455C22×C4C23×D7914yes—
456C22×C4C23×D7912yes—
457C4⋊C4—28no—
458C4⋊C4—48no—
459C4⋊C4—3168no—
460C4⋊C4—3168no—
461C4⋊C4—3168no—
462C4⋊C4—7916no—
463C4⋊C4—15816no—
464C4⋊C4—1588no—
465C4⋊C4—1588no—
466C4⋊C4—1588no—
467C4⋊C4C2×C4×D79116yescomplemented
468C22⋊C4—7932no—
469C22⋊C4—1588no—
470C22⋊C4—1588no—
471C42—28no—
472C42—7932no—
473C42—1588no—
474C23—7916no—
475C23—794no—
476C23—794no—
477C23—794no—
478C23—1588no—
479C23—1582no—
480C23—1582no—
481C23—1582no—
482C23—1582no—
483C23—1588no—
484C23—1588no—
485C23—1588no—
486C23—1588no—
487C23—1588no—
488C23C22×C4×D7912yes—
489C23C2×D4×D7912yes—
490C23C2×Q8×D7912yes—
491C23#1457920612yes—
492C23#1457920712yes—
493C23#1457920812yes—
494C23#1457920912yes—
495C23C24×D7911yescharacteristic
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—216no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—31632no—
512C2×C4—3164no—
513C2×C4—3168no—
514C2×C4—3164no—
515C2×C4—7916no—
516C2×C4—794no—
517C2×C4—794no—
518C2×C4—794no—
519C2×C4—7964no—
520C2×C4—798no—
521C2×C4—1582no—
522C2×C4—1584no—
523C2×C4—1582no—
524C2×C4—1584no—
525C2×C4—1582no—
526C2×C4—1582no—
527C2×C4—1582no—
528C2×C4—1582no—
529C2×C4—1584no—
530C2×C4—1584no—
531C2×C4—1584no—
532C2×C4—15816no—
533C2×C4C4⋊C4×D79116yes—
534C2×C4C22×C4×D7918yes—
535C2×C4C22×C4×D7914yes—
536C2×C4C2×D4×D7914yes—
537C2×C4C2×Q8×D7914yes—
538C22—798no—
539C22—15832no—
540C22—1584no—
541C22—1584no—
542C22—1584no—
543C22—1584no—
544C22—1584no—
545C22—1584no—
546C22C2×C4⋊C4×D7914yes—
547C22#1909904214yes—
548C22#1909904514yes—
549C22#1909912914yes—
550C22#1909913614yes—
551C22#1909919014yes—
552C22#1909919314yes—
553C22C23×C4×D7911yescharacteristic
554C22C22×D4×D7911yescharacteristic
555C22C22×Q8×D7911yescharacteristic
556C22C2×#1457920611yescharacteristic
557C22C2×#1457920711yescharacteristic
558C22C2×#1457920811yescharacteristic
559C22C2×#1457920911yescharacteristic
560C4—28no—
561C4—44no—
562C4—42no—
563C4—42no—
564C4—3164no—
565C4—3162no—
566C4—3162no—
567C4—798no—
568C4—1588no—
569C4C2×C4⋊C4×D7918yes—
570C2—1588no—
571C2#2376602218yesdirect product factor
572C2C22×C4⋊C4×D7911yescharacteristic
573C2C2×#1909904211yescharacteristic
574C2C2×#1909904511yescharacteristic
575C2C2×#1909912911yescharacteristic
576C2C2×#1909913611yescharacteristic
577C2C2×#1909919011yescharacteristic
578C2C2×#1909919311yescharacteristic
subgroupG/H
centerC24C23×D79
commutatorC22×C79C24×C4
FrattiniC23C24×D79
FittingC2×#389×C79C2
socleC24×C79C24
prime radicalC24×D79C23
cosocleC23×C79C25
hypercenterC2×#389D79
hypocenterC79C2×#2317
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
4. #23766022C2direct (N × H)
5. C2×#19098462C2semidirect
6. C2×#19098461C2semidirect
7. C2×#19098459C2semidirect
10. C2×#19098432C2semidirect
11. C2×#19098427C2semidirect
12. C2×#389×C79C2semidirect
15. C2×#14579119C4semidirect
34. #19098462C22semidirect
35. #19098461C22semidirect
36. #19098459C22semidirect
39. #19098432C22semidirect
40. #19098427C22semidirect
48. #389×C79C22semidirect
64. #14579119C2×C4semidirect
68. C2×C4⋊Dic79C2×C4semidirect
90. C2×C4×Dic79C2×C4semidirect
103. C2×C4⋊C4×C79C2×C4semidirect
142. C2×C4×D79C4⋊C4semidirect
148. C4⋊Dic79C22×C4semidirect
158. C4×Dic79C22×C4semidirect
182. C4⋊C4×C79C22×C4semidirect
247. C4×D79C2×C4⋊C4semidirect
281. C2×C79#2317semidirect
297. C2×#389D79semidirect
302. C79C2×#2317semidirect
331. #389C2×D79semidirect
380. C2×C4⋊C4C4×D79semidirect
467. C4⋊C4C2×C4×D79semidirect
571. C2#23766022direct (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×#145791182
C22×C4⋊C4×D792
#237660222
C2×#190984622
C2×#190984612
C2×#190984592
C2×#190984372
C2×#190984342
C2×#190984322
C2×#190984272
C2×#389×C792
C2×#231779