Finite Group Databasebeta

← C2×#23766410 #34447791

Subgroups — C2×#23766410 #34447791

Order 20224.

3614 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579137C211yesmaximal, characteristic
2C22×Q8×Dic79C211yesmaximal, characteristic, complemented
3C22×C4⋊C4×D79C211yesmaximal, characteristic
4#23766410C2116yesmaximal, direct product factor
5C2×#19098460C212yesmaximal, complemented
6C2×#19098459C211yesmaximal, characteristic, complemented
7C2×#19098445C212yesmaximal, complemented
8C2×#19098437C212yesmaximal
9C2×C4×C4⋊Dic79C211yesmaximal, characteristic, complemented
10C2×C4×(C2×D79)⋊C4C211yesmaximal, characteristic
11C2×#19098410C212yesmaximal
12C2×#391×C79C211yesmaximal, characteristic, complemented
13C23×C4×D79C2212yes—
14C23×C4×D79C2211yescharacteristic
15C2×#14579137C4116yescomplemented
16C2×#14579137C2218yes—
17C2×Q8×Dic79—28no—
18C2×Q8×Dic79C2218yescomplemented
19C22×C4⋊Dic79C2212yes—
20C22×C4⋊Dic79C2211yescharacteristic
21C22×C4⋊Dic79C2211yescharacteristic
22C22×Dic79⋊C4C2212yes—
23C22×Dic79⋊C4C2212yes—
24C22×Dic79⋊C4C2212yes—
25C2×C4⋊C4×D79—28no—
26C2×C4⋊C4×D79C2218yes—
27C22×(C2×D79)⋊C4C2211yescharacteristic
28C22×(C2×D79)⋊C4C2212yes—
29C22×(C2×D79)⋊C4C2211yescharacteristic
30C22×C4×Dic79C2212yes—
31C22×C4×Dic79C2211yescharacteristic
32C22×C4×Dic79C2212yes—
33#19098460C22116yescomplemented
34#19098459C2218yescomplemented
35#19098445C22116yescomplemented
36#19098437C22116yes—
37C4×C4⋊Dic79C2218yescomplemented
38C4×(C2×D79)⋊C4C2218yes—
39#19098410C22116yes—
40C2×C79⋊#95C2212yes—
41C2×C79⋊#95C2211yescharacteristic
42C2×C79⋊#95C2212yes—
43C2×C79⋊#95C2211yescharacteristic
44C23×Q8×C79C2211yescharacteristic
45C22×C4⋊C4×C79C2211yescharacteristic
46C22×C42×C79C2211yescharacteristic
47#391×C79C2218yescomplemented
48C2×#95×C79C2212yes—
49C2×#95×C79C2212yes—
50C24×D79C2311yescharacteristic
51C23×Dic79C2311yescharacteristic
52C23×Dic79C2312yes—
53C23×Dic79C2311yescharacteristic
54C23×Dic79C2312yes—
55C23×Dic79C2311yescharacteristic
56C22×C4×D79—24no—
57C22×C4×D79—24no—
58C22×C4×D79—28no—
59C22×C4×D79—28no—
60C22×C4×D79—44no—
61C22×C4×D79C2×C418yes—
62C22×C4×D79C2318yes—
63C22×C4×D79C2314yes—
64#14579137C2×C4164yescomplemented
65Q8×Dic79—232no—
66C2×C4⋊Dic79—28no—
67C2×C4⋊Dic79C2×C418yescomplemented
68C2×C4⋊Dic79D418yes—
69C2×C4⋊Dic79C2318yes—
70C2×C4⋊Dic79C2314yes—
71C2×C4⋊Dic79C2314yes—
72C2×Dic79⋊C4—28no—
73C2×Dic79⋊C4—44no—
74C2×Dic79⋊C4C2×C4116yescomplemented
75C2×Dic79⋊C4C2318yes—
76C2×Dic79⋊C4C2318yes—
77C2×Dic79⋊C4C2318yes—
78C4⋊C4×D79—232no—
79C2×(C2×D79)⋊C4—24no—
80C2×(C2×D79)⋊C4C2×C4116yes—
81C2×(C2×D79)⋊C4Q818yescomplemented
82C2×(C2×D79)⋊C4C2314yes—
83C2×(C2×D79)⋊C4C2318yes—
84C2×(C2×D79)⋊C4C2314yes—
85C2×C4×Dic79—24no—
86C2×C4×Dic79—28no—
87C2×C4×Dic79—44no—
88C2×C4×Dic79C2318yes—
89C2×C4×Dic79C2314yes—
90C2×C4×Dic79C2318yes—
91C79⋊#95C2318yes—
92C79⋊#95C2314yes—
93C79⋊#95C2314yes—
94C79⋊#95C2318yes—
95C22×Q8×C79—24no—
96C22×Q8×C79—24no—
97C22×Q8×C79C2×C418yescomplemented
98C22×Q8×C79C2314yes—
99C23×C4×C79C2311yescharacteristic
100C23×C4×C79C2312yes—
101C23×C4×C79C2312yes—
102C23×C4×C79C2311yescharacteristic
103C23×C4×C79C2311yescharacteristic
104C2×C4⋊C4×C79—24no—
105C2×C4⋊C4×C79C2314yes—
106C2×C42×C79—24no—
107C2×C42×C79C2314yes—
108#95×C79C2318yes—
109#95×C79C2318yes—
110C23×D79—22no—
111C23×D79—22no—
112C23×D79—22no—
113C23×D79—22no—
114C23×D79C22×C414yes—
115C23×D79C2×Q814yes—
116C23×D79#4114yes—
117C23×D79C2412yes—
118C22×Dic79—22no—
119C22×Dic79—22no—
120C22×Dic79—22no—
121C22×Dic79—24no—
122C22×Dic79—22no—
123C22×Dic79—22no—
124C22×Dic79—24no—
125C22×Dic79—24no—
126C22×Dic79—22no—
127C22×Dic79—24no—
128C22×Dic79—22no—
129C22×Dic79C22×C418yes—
130C22×Dic79C22×C414yes—
131C22×Dic79C2×D418yes—
132C22×Dic79C2×Q814yes—
133C22×Dic79C2412yes—
134C22×Dic79C2414yes—
135C22×Dic79C2412yes—
136C22×Dic79C2414yes—
137C22×Dic79C2412yes—
138C2×C4×D79—216no—
139C2×C4×D79—216no—
140C2×C4×D79—232no—
141C2×C4×D79—28no—
142C2×C4×D79—28no—
143C2×C4×D79—48no—
144C2×C4×D79—432no—
145C2×C4×D79C22×C4116yes—
146C4⋊Dic79—216no—
147C4⋊Dic79C22×C4116yescomplemented
148C4⋊Dic79C2×D4116yes—
149Dic79⋊C4—216no—
150Dic79⋊C4—48no—
151Dic79⋊C4C22×C4132yescomplemented
152(C2×D79)⋊C4—28no—
153(C2×D79)⋊C4C22×C4132yes—
154(C2×D79)⋊C4C2×Q8116yescomplemented
155C4×Dic79—216no—
156C4×Dic79—28no—
157C4×Dic79—48no—
158C24×C79C2411yescharacteristic
159C2×Q8×C79—232no—
160C2×Q8×C79—28no—
161C2×Q8×C79—28no—
162C2×Q8×C79C22×C4116yescomplemented
163C22×C4×C79—24no—
164C22×C4×C79—22no—
165C22×C4×C79—24no—
166C22×C4×C79—22no—
167C22×C4×C79—24no—
168C22×C4×C79—22no—
169C22×C4×C79—22no—
170C22×C4×C79—24no—
171C22×C4×C79—24no—
172C22×C4×C79—22no—
173C22×C4×C79C22×C418yes—
174C22×C4×C79C22×C414yes—
175C22×C4×C79C2×D414yes—
176C22×C4×C79C2×Q814yes—
177C22×C4×C79#4114yes—
178C22×C4×C79C2412yes—
179C22×C4×C79C2414yes—
180C22×C4×C79C2414yes—
181C22×C4×C79C2412yes—
182C22×C4×C79C2412yes—
183C4⋊C4×C79—28no—
184C42×C79—28no—
185C23×C79C23×C412yes—
186C23×C79C22×D412yes—
187C23×C79C22×Q812yes—
188C23×C79C2×#4112yes—
189C23×C79C2×#4112yes—
190C23×C79H(5,𝔽2)12yes—
191C23×C792−1+412yes—
192C23×C79C2511yescharacteristic
193C22×D79—28no—
194C22×D79—28no—
195C22×D79—28no—
196C22×D79—28no—
197C22×D79—28no—
198C22×D79—28no—
199C22×D79—22no—
200C22×D79—22no—
201C22×D79—22no—
202C22×D79—22no—
203C22×D79C4×Q8116yescomplemented
204C22×D79C23×C414yes—
205C22×D79C22×Q814yes—
206C22×D79C2×#4114yes—
207Q8×C79—232no—
208C2×C4×C79—216no—
209C2×C4×C79—28no—
210C2×C4×C79—24no—
211C2×C4×C79—22no—
212C2×C4×C79—24no—
213C2×C4×C79—22no—
214C2×C4×C79—24no—
215C2×C4×C79—22no—
216C2×C4×C79—24no—
217C2×C4×C79—22no—
218C2×C4×C79—24no—
219C2×C4×C79—22no—
220C2×C4×C79—48no—
221C2×C4×C79—44no—
222C2×C4×C79C4×D4116yes—
223C2×C4×C79C23×C418yes—
224C2×C4×C79C23×C414yes—
225C2×C4×C79C22×D414yes—
226C2×C4×C79C22×Q814yes—
227C2×C4×C79C2×#4114yes—
228C2×Dic79—28no—
229C2×Dic79—216no—
230C2×Dic79—216no—
231C2×Dic79—28no—
232C2×Dic79—22no—
233C2×Dic79—22no—
234C2×Dic79—24no—
235C2×Dic79—22no—
236C2×Dic79—24no—
237C2×Dic79—24no—
238C2×Dic79—24no—
239C2×Dic79—22no—
240C2×Dic79—22no—
241C2×Dic79—22no—
242C2×Dic79—22no—
243C2×Dic79—44no—
244C2×Dic79C23×C418yes—
245C2×Dic79C23×C414yes—
246C2×Dic79C22×D418yes—
247C2×Dic79C22×Q814yes—
248C4×D79—232no—
249C4×D79—432no—
250C22×C79C2×C4×D414yes—
251C22×C79C2×C4×Q814yes—
252C22×C79#52014yes—
253C22×C79#54614yes—
254C22×C79D4×Q814yes—
255C22×C79#55214yes—
256C22×C79#55414yes—
257C22×C79C24×C411yescharacteristic
258C22×C79C23×D411yescharacteristic
259C22×C79C23×Q811yescharacteristic
260C22×C79C22×#4111yescharacteristic
261C22×C79C22×#4111yescharacteristic
262C22×C79C2×H(5,𝔽2)11yescharacteristic
263C22×C79C2×2−1+411yescharacteristic
264C2×D79—232no—
265C2×D79—24no—
266C2×D79—24no—
267C2×D79—24no—
268C2×D79—24no—
269C2×D79—24no—
270C2×D79—24no—
271C2×D79C2×C4×Q818yescomplemented
272C4×C79—28no—
273C4×C79—24no—
274C4×C79—44no—
275C4×C79—42no—
276C4×C79C2×C4×D418yes—
277Dic79—24no—
278Dic79—28no—
279Dic79—28no—
280Dic79—24no—
281Dic79—42no—
282C2×#2360—791nomaximal, Sylow
283C2×C79#236018yescomplemented
284C2×C79C22×C4×D411yescharacteristic
285C2×C79C22×C4×Q811yescharacteristic
286C2×C79C2×#52011yescharacteristic
287C2×C79C2×#54611yescharacteristic
288C2×C79C2×D4×Q811yescharacteristic
289C2×C79C2×#55211yescharacteristic
290C2×C79C2×#55411yescharacteristic
291D79—28no—
292C22×C22⋊Q8—791no—
293C22×C4×Q8—791no—
294C23×C4⋊C4—791no—
295#2360—7916no—
296C2×#391D7911yescharacteristic, complemented
297C2×#389—792no—
298C2×#389—791no—
299C2×#387—792no—
300C2×#385—792no—
301C2×C4×C4⋊C4—791no—
302C2×C4×C22⋊C4—791no—
303C2×C4×C22⋊C4—792no—
304C79C2×#236011yescharacteristic, complemented, Sylow
305C23×Q8C2×D7911yescharacteristic
306C24×C4—792no—
307C24×C4—791no—
308C2×C22⋊Q8—798no—
309C2×C22⋊Q8—7916no—
310C2×C4×Q8—798no—
311C2×C4×Q8—1588no—
312C22×C4⋊C4—792no—
313C22×C4⋊C4—791no—
314C22×C4⋊C4—792no—
315C22×C4⋊C4—791no—
316C22×C4⋊C4—792no—
317C22×C4⋊C4—792no—
318C22×C4⋊C4—798no—
319C22×C4⋊C4—1588no—
320C22×C4⋊C4C2×D7911yescharacteristic
321C22×C22⋊C4—792no—
322C22×C22⋊C4—791no—
323C22×C22⋊C4—791no—
324C22×C42—792no—
325C22×C42—792no—
326C22×C42—791no—
327C22×C42C2×D7911yescharacteristic
328#391C2×D7918yescomplemented
329#389—798no—
330#389—7916no—
331#387—7916no—
332#385—7916no—
333C4×C4⋊C4—798no—
334C4×C22⋊C4—798no—
335C4×C22⋊C4—7916no—
336C2×#95—791no—
337C2×#95—792no—
338C2×#95—791no—
339C2×#95—792no—
340C2×#95C2×D7912yes—
341C2×#95C2×D7912yes—
342C25—791no—
343C22×Q8—24no—
344C22×Q8—24no—
345C22×Q8C4×D7918yescomplemented
346C22×Q8C22×D7914yes—
347C23×C4—3164no—
348C23×C4—798no—
349C23×C4—792no—
350C23×C4—791no—
351C23×C4—791no—
352C23×C4—792no—
353C23×C4—798no—
354C23×C4—794no—
355C23×C4—791no—
356C23×C4—1588no—
357C23×C4—1588no—
358C23×C4—1584no—
359C23×C4—1584no—
360C23×C4C22×D7911yescharacteristic
361C23×C4C22×D7912yes—
362C23×C4C22×D7912yes—
363C23×C4C22×D7911yescharacteristic
364C23×C4C22×D7911yescharacteristic
365C22⋊Q8—7964no—
366C4×Q8—15832no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—3164no—
369C2×C4⋊C4—798no—
370C2×C4⋊C4—7916no—
371C2×C4⋊C4—798no—
372C2×C4⋊C4—798no—
373C2×C4⋊C4—794no—
374C2×C4⋊C4—794no—
375C2×C4⋊C4—798no—
376C2×C4⋊C4—798no—
377C2×C4⋊C4—798no—
378C2×C4⋊C4—15832no—
379C2×C4⋊C4—1588no—
380C2×C4⋊C4—1588no—
381C2×C4⋊C4C22×D7914yes—
382C2×C22⋊C4—7916no—
383C2×C22⋊C4—798no—
384C2×C22⋊C4—798no—
385C2×C22⋊C4—794no—
386C2×C22⋊C4—794no—
387C2×C22⋊C4—1584no—
388C2×C42—24no—
389C2×C42—3164no—
390C2×C42—798no—
391C2×C42—798no—
392C2×C42—794no—
393C2×C42—1588no—
394C2×C42—1584no—
395C2×C42C22×D7914yes—
396#95—794no—
397#95—798no—
398#95—794no—
399#95—798no—
400#95C22×D7918yes—
401#95C22×D7918yes—
402C24—794no—
403C24—794no—
404C24—792no—
405C24—794no—
406C24—1582no—
407C24—1582no—
408C24—1582no—
409C24—1582no—
410C24C23×D7911yescharacteristic
411C2×Q8—28no—
412C2×Q8—232no—
413C2×Q8—28no—
414C2×Q8C2×C4×D79116yescomplemented
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—3168no—
426C22×C4—31632no—
427C22×C4—794no—
428C22×C4—792no—
429C22×C4—794no—
430C22×C4—792no—
431C22×C4—794no—
432C22×C4—798no—
433C22×C4—794no—
434C22×C4—798no—
435C22×C4—7916no—
436C22×C4—792no—
437C22×C4—15816no—
438C22×C4—15816no—
439C22×C4—1582no—
440C22×C4—1582no—
441C22×C4—1582no—
442C22×C4—1584no—
443C22×C4—1582no—
444C22×C4—1584no—
445C22×C4—1584no—
446C22×C4—1582no—
447C22×C4—1584no—
448C22×C4—1582no—
449C22×C4—1582no—
450C22×C4—1588no—
451C22×C4—15832no—
452C22×C4—1588no—
453C22×C4C2×C4×D7918yes—
454C22×C4C2×C4×D7914yes—
455C22×C4D4×D7914yes—
456C22×C4Q8×D7914yes—
457C22×C4#1192962314yes—
458C22×C4C23×D7912yes—
459C22×C4C23×D7914yes—
460C22×C4C23×D7912yes—
461C22×C4C23×D7914yes—
462C22×C4C23×D7912yes—
463C4⋊C4—28no—
464C4⋊C4—3168no—
465C4⋊C4—7932no—
466C4⋊C4—7916no—
467C4⋊C4—7916no—
468C4⋊C4—15816no—
469C4⋊C4—15816no—
470C22⋊C4—7916no—
471C22⋊C4—7932no—
472C22⋊C4—1588no—
473C42—28no—
474C42—3168no—
475C42—15816no—
476C42—1588no—
477C23—7916no—
478C23—794no—
479C23—794no—
480C23—794no—
481C23—1582no—
482C23—1582no—
483C23—1588no—
484C23—1588no—
485C23—1588no—
486C23—1588no—
487C23—1588no—
488C23—1588no—
489C23—1582no—
490C23—1582no—
491C23C22×C4×D7912yes—
492C23C2×D4×D7912yes—
493C23C2×Q8×D7912yes—
494C23#41×D7912yes—
495C23C2×#1192962312yes—
496C23#1457920712yes—
497C23#1457920912yes—
498C23C24×D7911yescharacteristic
499Q8—232no—
500C2×C4—22no—
501C2×C4—28no—
502C2×C4—216no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—48no—
513C2×C4—44no—
514C2×C4—31632no—
515C2×C4—3164no—
516C2×C4—798no—
517C2×C4—794no—
518C2×C4—798no—
519C2×C4—794no—
520C2×C4—1582no—
521C2×C4—1582no—
522C2×C4—1582no—
523C2×C4—1584no—
524C2×C4—1584no—
525C2×C4—1584no—
526C2×C4—1582no—
527C2×C4—15816no—
528C2×C4—1588no—
529C2×C4—1588no—
530C2×C4—15816no—
531C2×C4—1582no—
532C2×C4—1582no—
533C2×C4—1584no—
534C2×C4—1582no—
535C2×C4—15832no—
536C2×C4(C22⋊D79)⋊C4116yes—
537C2×C4C22×C4×D7918yes—
538C2×C4C22×C4×D7914yes—
539C2×C4C2×D4×D7914yes—
540C2×C4C2×Q8×D7914yes—
541C2×C4C2×#1192962314yes—
542C22—798no—
543C22—1584no—
544C22—1584no—
545C22—1584no—
546C22—1584no—
547C22—15832no—
548C22—1584no—
549C22—1584no—
550C22C2×(C22⋊D79)⋊C414yes—
551C22C4×Q8×D7914yes—
552C22#1909904514yes—
553C22#1909920614yes—
554C22#1909922814yes—
555C22#1909924714yes—
556C22#1909927114yes—
557C22C23×C4×D7911yescharacteristic
558C22C22×D4×D7911yescharacteristic
559C22C22×Q8×D7911yescharacteristic
560C22C2×#41×D7911yescharacteristic
561C22C22×#1192962311yescharacteristic
562C22C2×#1457920711yescharacteristic
563C22C2×#1457920911yescharacteristic
564C4—28no—
565C4—24no—
566C4—44no—
567C4—42no—
568C4—3162no—
569C4—1584no—
570C4—1588no—
571C4—1588no—
572C4—1584no—
573C4C2×(C22⋊D79)⋊C418yes—
574C2—1588no—
575C2#2376641018yesdirect product factor
576C2C22×(C22⋊D79)⋊C411yescharacteristic
577C2C2×C4×Q8×D7911yescharacteristic
578C2C2×#1909904511yescharacteristic
579C2C2×#1909920611yescharacteristic
580C2C2×#1909922811yescharacteristic
581C2C2×#1909924711yescharacteristic
582C2C2×#1909927111yescharacteristic
subgroupG/H
centerC24C23×D79
commutatorC22×C79C24×C4
FrattiniC23C24×D79
FittingC2×#391×C79C2
socleC24×C79C24
prime radicalC24×D79C23
cosocleC23×C79C25
hypercenterC2×#391D79
hypocenterC79C2×#2360
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C22×Q8×Dic79C2semidirect
4. #23766410C2direct (N × H)
5. C2×#19098460C2semidirect
6. C2×#19098459C2semidirect
7. C2×#19098445C2semidirect
9. C2×C4×C4⋊Dic79C2semidirect
12. C2×#391×C79C2semidirect
15. C2×#14579137C4semidirect
18. C2×Q8×Dic79C22semidirect
33. #19098460C22semidirect
34. #19098459C22semidirect
35. #19098445C22semidirect
37. C4×C4⋊Dic79C22semidirect
47. #391×C79C22semidirect
64. #14579137C2×C4semidirect
67. C2×C4⋊Dic79C2×C4semidirect
74. C2×Dic79⋊C4C2×C4semidirect
81. C2×(C2×D79)⋊C4Q8semidirect
97. C22×Q8×C79C2×C4semidirect
147. C4⋊Dic79C22×C4semidirect
151. Dic79⋊C4C22×C4semidirect
154. (C2×D79)⋊C4C2×Q8semidirect
162. C2×Q8×C79C22×C4semidirect
203. C22×D79C4×Q8semidirect
271. C2×D79C2×C4×Q8semidirect
283. C2×C79#2360semidirect
296. C2×#391D79semidirect
304. C79C2×#2360semidirect
328. #391C2×D79semidirect
345. C22×Q8C4×D79semidirect
414. C2×Q8C2×C4×D79semidirect
575. C2#23766410direct (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×Q8×Dic792
C22×C4⋊C4×D792
#237664102
C2×#190984602
C2×#190984592
C2×#190984452
C2×#190984372
C2×C4×C4⋊Dic792
C2×C4×(C2×D79)⋊C42
C2×#190984102
C2×#391×C792
C2×#236079