Finite Group Databasebeta

← C2×#23767100 #34456099

Subgroups — C2×#23767100 #34456099

Order 20224.

3550 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579160C211yesmaximal, characteristic, complemented
2C22×#14579141C212yesmaximal
3C22×C4⋊C4×D79C211yesmaximal, characteristic
4#23767100C2116yesmaximal, direct product factor
5C2×#19098493C212yesmaximal, complemented
6C2×#19098484C212yesmaximal, complemented
7C2×#19098462C211yesmaximal, characteristic, complemented
8C2×#19098437C212yesmaximal
9C2×#19098433C211yesmaximal, characteristic, complemented
10C2×#19098420C211yesmaximal, characteristic
11C2×C4×(C2×D79)⋊C4C211yesmaximal, characteristic
12C2×#389×C79C211yesmaximal, characteristic, complemented
13C23×Dic158C2212yes—
14C23×C4×D79C2212yes—
15C23×C4×D79C2211yescharacteristic
16C2×#14579160—28no—
17C2×#14579160C2218yescomplemented
18C2×#14579141—216no—
19C2×#14579141C22116yes—
20C22×C4⋊Dic79C2211yescharacteristic
21C22×C4⋊Dic79C2211yescharacteristic
22C22×C4⋊Dic79C2212yes—
23C22×C4⋊Dic79C2212yes—
24C22×C4⋊Dic79C2212yes—
25C22×Dic79⋊C4C2212yes—
26C22×Dic79⋊C4C2212yes—
27C2×C4⋊C4×D79—28no—
28C2×C4⋊C4×D79C2218yes—
29C22×(C2×D79)⋊C4C2212yes—
30C22×(C2×D79)⋊C4C2211yescharacteristic
31C22×(C2×D79)⋊C4C2211yescharacteristic
32C22×C4×Dic79C2211yescharacteristic
33#19098493C22116yescomplemented
34#19098484C22116yescomplemented
35#19098462C2218yescomplemented
36#19098437C22116yes—
37#19098433C2218yescomplemented
38#19098420C2218yes—
39C4×(C2×D79)⋊C4C2218yes—
40C2×C79⋊#95C2212yes—
41C2×C79⋊#95C2212yes—
42C2×C79⋊#95C2211yescharacteristic
43C2×C79⋊#95C2211yescharacteristic
44C22×C4⋊C4×C79C2211yescharacteristic
45C22×C4⋊C4×C79C2212yes—
46C22×C4⋊C4×C79C2211yescharacteristic
47C22×C42×C79C2211yescharacteristic
48#389×C79C2218yescomplemented
49C2×#95×C79C2212yes—
50C24×D79C2311yescharacteristic
51C22×Dic158—28no—
52C22×Dic158—28no—
53C22×Dic158—44no—
54C22×Dic158C2318yes—
55C23×Dic79C2312yes—
56C23×Dic79C2312yes—
57C23×Dic79C2311yescharacteristic
58C23×Dic79C2311yescharacteristic
59C23×Dic79C2311yescharacteristic
60C22×C4×D79—24no—
61C22×C4×D79—24no—
62C22×C4×D79—28no—
63C22×C4×D79—28no—
64C22×C4×D79—44no—
65C22×C4×D79Q818yescomplemented
66C22×C4×D79C2318yes—
67C22×C4×D79C2314yes—
68#14579160—232no—
69#14579141—264no—
70C2×C4⋊Dic79—24no—
71C2×C4⋊Dic79—28no—
72C2×C4⋊Dic79—24no—
73C2×C4⋊Dic79—28no—
74C2×C4⋊Dic79—44no—
75C2×C4⋊Dic79C2314yes—
76C2×C4⋊Dic79C2314yes—
77C2×C4⋊Dic79C2318yes—
78C2×C4⋊Dic79C2318yes—
79C2×C4⋊Dic79C2318yes—
80C2×Dic79⋊C4—28no—
81C2×Dic79⋊C4—28no—
82C2×Dic79⋊C4C2318yes—
83C2×Dic79⋊C4C2318yes—
84C4⋊C4×D79—232no—
85C2×(C2×D79)⋊C4—28no—
86C2×(C2×D79)⋊C4D418yes—
87C2×(C2×D79)⋊C4Q818yescomplemented
88C2×(C2×D79)⋊C4C2318yes—
89C2×(C2×D79)⋊C4C2314yes—
90C2×(C2×D79)⋊C4C2314yes—
91C2×C4×Dic79—24no—
92C2×C4×Dic79C2314yes—
93C79⋊#95C2314yes—
94C79⋊#95C2314yes—
95C79⋊#95C2318yes—
96C79⋊#95C2318yes—
97C23×C4×C79C2312yes—
98C23×C4×C79C2311yescharacteristic
99C23×C4×C79C2312yes—
100C23×C4×C79C2311yescharacteristic
101C23×C4×C79C2311yescharacteristic
102C2×C4⋊C4×C79—28no—
103C2×C4⋊C4×C79—24no—
104C2×C4⋊C4×C79D418yes—
105C2×C4⋊C4×C79C2314yes—
106C2×C4⋊C4×C79C2318yes—
107C2×C4⋊C4×C79C2314yes—
108C2×C42×C79—24no—
109C2×C42×C79C2314yes—
110#95×C79C2318yes—
111C23×D79—22no—
112C23×D79—22no—
113C23×D79—22no—
114C23×D79—22no—
115C23×D79C2×D414yes—
116C23×D79C2×Q814yes—
117C23×D79C2×Q814yes—
118C23×D79C2412yes—
119C2×Dic158—216no—
120C2×Dic158—216no—
121C2×Dic158—48no—
122C2×Dic158—432no—
123C22×Dic79—22no—
124C22×Dic79—22no—
125C22×Dic79—22no—
126C22×Dic79—24no—
127C22×Dic79—24no—
128C22×Dic79—24no—
129C22×Dic79—24no—
130C22×Dic79—22no—
131C22×Dic79—22no—
132C22×Dic79—24no—
133C22×Dic79—24no—
134C22×Dic79—22no—
135C22×Dic79C2×D414yes—
136C22×Dic79C2×Q814yes—
137C22×Dic79C2×Q814yes—
138C22×Dic79C2414yes—
139C22×Dic79C2414yes—
140C22×Dic79C2412yes—
141C22×Dic79C2412yes—
142C22×Dic79C2412yes—
143C2×C4×D79—216no—
144C2×C4×D79—232no—
145C2×C4×D79—28no—
146C2×C4×D79—216no—
147C2×C4×D79—28no—
148C2×C4×D79—48no—
149C2×C4×D79—432no—
150C2×C4×D79C2×Q8116yescomplemented
151C4⋊Dic79—216no—
152C4⋊Dic79—28no—
153C4⋊Dic79—216no—
154C4⋊Dic79—28no—
155C4⋊Dic79—48no—
156Dic79⋊C4—216no—
157Dic79⋊C4—216no—
158(C2×D79)⋊C4—216no—
159(C2×D79)⋊C4C2×D4116yes—
160(C2×D79)⋊C4C2×Q8116yescomplemented
161C4×Dic79—28no—
162C24×C79C2411yescharacteristic
163C22×C4×C79—24no—
164C22×C4×C79—22no—
165C22×C4×C79—22no—
166C22×C4×C79—24no—
167C22×C4×C79—22no—
168C22×C4×C79—22no—
169C22×C4×C79—24no—
170C22×C4×C79—24no—
171C22×C4×C79—24no—
172C22×C4×C79C2×D414yes—
173C22×C4×C79C2×D418yes—
174C22×C4×C79C2×D414yes—
175C22×C4×C79C2×Q814yes—
176C22×C4×C79C2×Q814yes—
177C22×C4×C79#4114yes—
178C22×C4×C79C2414yes—
179C22×C4×C79C2414yes—
180C22×C4×C79C2412yes—
181C22×C4×C79C2412yes—
182C22×C4×C79C2412yes—
183C4⋊C4×C79—216no—
184C4⋊C4×C79—28no—
185C4⋊C4×C79C2×D4116yes—
186C42×C79—28no—
187C23×C79C22×D412yes—
188C23×C79C22×D412yes—
189C23×C79C22×Q812yes—
190C23×C79C22×Q812yes—
191C23×C79C2×#4112yes—
192C23×C79H(5,𝔽2)12yes—
193C23×C792−1+412yes—
194C23×C79C2511yescharacteristic
195C22×D79—28no—
196C22×D79—28no—
197C22×D79—28no—
198C22×D79—28no—
199C22×D79—28no—
200C22×D79—28no—
201C22×D79—22no—
202C22×D79—22no—
203C22×D79—22no—
204C22×D79—22no—
205C22×D79C4⋊Q8116yescomplemented
206C22×D79C22×D414yes—
207C22×D79C22×Q814yes—
208C22×D79C22×Q814yes—
209C2×C4×C79—216no—
210C2×C4×C79—28no—
211C2×C4×C79—28no—
212C2×C4×C79—24no—
213C2×C4×C79—22no—
214C2×C4×C79—22no—
215C2×C4×C79—22no—
216C2×C4×C79—24no—
217C2×C4×C79—22no—
218C2×C4×C79—24no—
219C2×C4×C79—24no—
220C2×C4×C79—24no—
221C2×C4×C79—48no—
222C2×C4×C79C22⋊Q8116yescomplemented
223C2×C4×C79C22×D414yes—
224C2×C4×C79C22×D418yes—
225C2×C4×C79C22×D414yes—
226C2×C4×C79C22×Q814yes—
227C2×C4×C79C22×Q814yes—
228C2×C4×C79C2×#4114yes—
229C2×Dic79—28no—
230C2×Dic79—28no—
231C2×Dic79—28no—
232C2×Dic79—22no—
233C2×Dic79—22no—
234C2×Dic79—22no—
235C2×Dic79—22no—
236C2×Dic79—22no—
237C2×Dic79—22no—
238C2×Dic79—24no—
239C2×Dic79—24no—
240C2×Dic79—24no—
241C2×Dic79—24no—
242C2×Dic79—24no—
243C2×Dic79—24no—
244C2×Dic79—48no—
245C2×Dic79—48no—
246C2×Dic79C22×D414yes—
247C2×Dic79C22×Q814yes—
248C2×Dic79C22×Q814yes—
249C4×D79—232no—
250C4×D79—432no—
251Dic158—432no—
252C22×C79C2×C22⋊Q814yes—
253C22×C79C2×C4⋊Q814yes—
254C22×C79#54414yes—
255C22×C79D4214yes—
256C22×C79#54714yes—
257C22×C79D4×Q814yes—
258C22×C79#55414yes—
259C22×C79C23×D411yescharacteristic
260C22×C79C23×D411yescharacteristic
261C22×C79C23×Q811yescharacteristic
262C22×C79C23×Q811yescharacteristic
263C22×C79C22×#4111yescharacteristic
264C22×C79C2×H(5,𝔽2)11yescharacteristic
265C22×C79C2×2−1+411yescharacteristic
266C2×D79—24no—
267C2×D79—24no—
268C2×D79—24no—
269C2×D79—232no—
270C2×D79—24no—
271C2×D79—24no—
272C2×D79—24no—
273C2×D79C2×C4⋊Q818yescomplemented
274C4×C79—28no—
275C4×C79—24no—
276C4×C79—24no—
277C4×C79—44no—
278C4×C79C2×C22⋊Q818yescomplemented
279Dic79—24no—
280Dic79—24no—
281Dic79—24no—
282Dic79—44no—
283Dic79—44no—
284C2×#2440—791nomaximal, Sylow
285C2×C79#244018yescomplemented
286C2×C79C22×C22⋊Q811yescharacteristic
287C2×C79C22×C4⋊Q811yescharacteristic
288C2×C79C2×#54411yescharacteristic
289C2×C79C2×D4211yescharacteristic
290C2×C79C2×#54711yescharacteristic
291C2×C79C2×D4×Q811yescharacteristic
292C2×C79C2×#55411yescharacteristic
293D79—28no—
294C22×C4⋊Q8—791no—
295C22×C22⋊Q8—792no—
296C23×C4⋊C4—791no—
297#2440—7916no—
298C2×#398—792no—
299C2×#395—792no—
300C2×#389—791no—
301C2×#389D7911yescharacteristic, complemented
302C2×#385—792no—
303C2×#384—791no—
304C2×#380—791no—
305C2×C4×C22⋊C4—791no—
306C79C2×#244011yescharacteristic, complemented, Sylow
307C23×Q8—792no—
308C24×C4—792no—
309C24×C4—791no—
310C2×C4⋊Q8—798no—
311C2×C4⋊Q8—1588no—
312C2×C22⋊Q8—7916no—
313C2×C22⋊Q8—15816no—
314C22×C4⋊C4—792no—
315C22×C4⋊C4—792no—
316C22×C4⋊C4—792no—
317C22×C4⋊C4—792no—
318C22×C4⋊C4—792no—
319C22×C4⋊C4—791no—
320C22×C4⋊C4—791no—
321C22×C4⋊C4—798no—
322C22×C4⋊C4—1588no—
323C22×C4⋊C4C2×D7912yes—
324C22×C4⋊C4C2×D7911yescharacteristic
325C22×C4⋊C4C2×D7911yescharacteristic
326C22×C22⋊C4—791no—
327C22×C22⋊C4—791no—
328C22×C22⋊C4—792no—
329C22×C42—791no—
330C22×C42C2×D7911yescharacteristic
331#398—7916no—
332#395—7916no—
333#389—798no—
334#389C2×D7918yescomplemented
335#385—7916no—
336#384—798no—
337#380—798no—
338C4×C22⋊C4—798no—
339C2×#95—791no—
340C2×#95—792no—
341C2×#95—791no—
342C2×#95—792no—
343C2×#95C2×D7912yes—
344C25—791no—
345C22×Q8—3164no—
346C22×Q8—798no—
347C22×Q8—1588no—
348C22×Q8—1588no—
349C23×C4—3164no—
350C23×C4—798no—
351C23×C4—791no—
352C23×C4—791no—
353C23×C4—791no—
354C23×C4—798no—
355C23×C4—794no—
356C23×C4—792no—
357C23×C4—792no—
358C23×C4—1588no—
359C23×C4—1588no—
360C23×C4—1584no—
361C23×C4—1584no—
362C23×C4C22×D7912yes—
363C23×C4C22×D7911yescharacteristic
364C23×C4C22×D7912yes—
365C23×C4C22×D7911yescharacteristic
366C23×C4C22×D7911yescharacteristic
367C4⋊Q8—15832no—
368C22⋊Q8—15864no—
369C2×C4⋊C4—28no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—3164no—
372C2×C4⋊C4—798no—
373C2×C4⋊C4—794no—
374C2×C4⋊C4—798no—
375C2×C4⋊C4—798no—
376C2×C4⋊C4—798no—
377C2×C4⋊C4—794no—
378C2×C4⋊C4—798no—
379C2×C4⋊C4—1588no—
380C2×C4⋊C4—1588no—
381C2×C4⋊C4—1588no—
382C2×C4⋊C4—1584no—
383C2×C4⋊C4—1588no—
384C2×C4⋊C4—1584no—
385C2×C4⋊C4—15832no—
386C2×C4⋊C4D31618yes—
387C2×C4⋊C4C22×D7918yes—
388C2×C4⋊C4C22×D7914yes—
389C2×C4⋊C4C22×D7914yes—
390C2×C22⋊C4—798no—
391C2×C22⋊C4—798no—
392C2×C22⋊C4—794no—
393C2×C22⋊C4—794no—
394C2×C22⋊C4—798no—
395C2×C22⋊C4—1588no—
396C2×C42—24no—
397C2×C42—794no—
398C2×C42—1584no—
399C2×C42C22×D7914yes—
400#95—794no—
401#95—794no—
402#95—798no—
403#95—798no—
404#95C22×D7918yes—
405C24—794no—
406C24—794no—
407C24—792no—
408C24—794no—
409C24—1582no—
410C24—1582no—
411C24—1582no—
412C24—1582no—
413C24C23×D7911yescharacteristic
414C2×Q8—31632no—
415C2×Q8—3168no—
416C2×Q8—15816no—
417C2×Q8—15816no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—31632no—
428C22×C4—3168no—
429C22×C4—794no—
430C22×C4—794no—
431C22×C4—792no—
432C22×C4—792no—
433C22×C4—792no—
434C22×C4—794no—
435C22×C4—7916no—
436C22×C4—794no—
437C22×C4—794no—
438C22×C4—15816no—
439C22×C4—1584no—
440C22×C4—1584no—
441C22×C4—1584no—
442C22×C4—1584no—
443C22×C4—1584no—
444C22×C4—1584no—
445C22×C4—15816no—
446C22×C4—1582no—
447C22×C4—1582no—
448C22×C4—1582no—
449C22×C4—1582no—
450C22×C4—1582no—
451C22×C4—1582no—
452C22×C4—1588no—
453C22×C4—15832no—
454C22×C4—1588no—
455C22×C4D4×D7914yes—
456C22×C4C2×D31618yes—
457C22×C4C2×D31614yes—
458C22×C4Q8×D7914yes—
459C22×C4Q8×D7914yes—
460C22×C4#1192962314yes—
461C22×C4C23×D7914yes—
462C22×C4C23×D7912yes—
463C22×C4C23×D7912yes—
464C22×C4C23×D7914yes—
465C22×C4C23×D7912yes—
466C4⋊C4—28no—
467C4⋊C4—216no—
468C4⋊C4—3168no—
469C4⋊C4—15816no—
470C4⋊C4—15816no—
471C4⋊C4—15816no—
472C4⋊C4—15816no—
473C4⋊C4—1588no—
474C4⋊C4—1588no—
475C4⋊C4C2×D316116yes—
476C22⋊C4—7916no—
477C22⋊C4—7916no—
478C22⋊C4—15816no—
479C42—28no—
480C42—1588no—
481C23—7916no—
482C23—794no—
483C23—794no—
484C23—794no—
485C23—1582no—
486C23—1582no—
487C23—1588no—
488C23—1588no—
489C23—1588no—
490C23—1588no—
491C23—1588no—
492C23—1588no—
493C23—1582no—
494C23—1582no—
495C23C2×D4×D7912yes—
496C23C22×D31612yes—
497C23C2×Q8×D7912yes—
498C23C2×Q8×D7912yes—
499C23C2×#1192962312yes—
500C23#1457920712yes—
501C23#1457920812yes—
502C23C24×D7911yescharacteristic
503Q8—31632no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—216no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—48no—
517C2×C4—31632no—
518C2×C4—3168no—
519C2×C4—3168no—
520C2×C4—794no—
521C2×C4—794no—
522C2×C4—794no—
523C2×C4—1584no—
524C2×C4—1584no—
525C2×C4—1584no—
526C2×C4—1584no—
527C2×C4—1584no—
528C2×C4—1584no—
529C2×C4—1582no—
530C2×C4—1582no—
531C2×C4—1588no—
532C2×C4—1588no—
533C2×C4—1588no—
534C2×C4—1582no—
535C2×C4—1582no—
536C2×C4—1582no—
537C2×C4—1582no—
538C2×C4—15832no—
539C2×C4#14579141116yescomplemented
540C2×C4C2×D4×D7914yes—
541C2×C4C22×D31618yes—
542C2×C4C22×D31614yes—
543C2×C4C2×Q8×D7914yes—
544C2×C4C2×Q8×D7914yes—
545C2×C4C2×#1192962314yes—
546C22—798no—
547C22—1584no—
548C22—15832no—
549C22—1584no—
550C22—1584no—
551C22—1584no—
552C22—1584no—
553C22—1584no—
554C22C2×#1457914114yes—
555C22C4⋊Q8×D7914yes—
556C22#1909919314yes—
557C22D4×D31614yes—
558C22#1909921314yes—
559C22Q8×D31614yes—
560C22#1909927114yes—
561C22C22×D4×D7911yescharacteristic
562C22C23×D31611yescharacteristic
563C22C22×Q8×D7911yescharacteristic
564C22C22×Q8×D7911yescharacteristic
565C22C22×#1192962311yescharacteristic
566C22C2×#1457920711yescharacteristic
567C22C2×#1457920811yescharacteristic
568C4—28no—
569C4—24no—
570C4—24no—
571C4—44no—
572C4—3164no—
573C4—3164no—
574C4—1584no—
575C4—1584no—
576C4—1584no—
577C4C2×#1457914118yescomplemented
578C2—1588no—
579C2#2376710018yesdirect product factor
580C2C22×#1457914111yescharacteristic
581C2C2×C4⋊Q8×D7911yescharacteristic
582C2C2×#1909919311yescharacteristic
583C2C2×D4×D31611yescharacteristic
584C2C2×#1909921311yescharacteristic
585C2C2×Q8×D31611yescharacteristic
586C2C2×#1909927111yescharacteristic
subgroupG/H
centerC24C23×D79
commutator, cosocleC23×C79C25
FrattiniC23C24×D79
FittingC2×#389×C79C2
socleC24×C79C24
prime radicalC24×D79C23
hypercenterC2×#389D79
hypocenterC79C2×#2440
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#14579160C2semidirect
4. #23767100C2direct (N × H)
5. C2×#19098493C2semidirect
6. C2×#19098484C2semidirect
7. C2×#19098462C2semidirect
9. C2×#19098433C2semidirect
12. C2×#389×C79C2semidirect
17. C2×#14579160C22semidirect
33. #19098493C22semidirect
34. #19098484C22semidirect
35. #19098462C22semidirect
37. #19098433C22semidirect
48. #389×C79C22semidirect
65. C22×C4×D79Q8semidirect
87. C2×(C2×D79)⋊C4Q8semidirect
150. C2×C4×D79C2×Q8semidirect
160. (C2×D79)⋊C4C2×Q8semidirect
205. C22×D79C4⋊Q8semidirect
222. C2×C4×C79C22⋊Q8semidirect
273. C2×D79C2×C4⋊Q8semidirect
278. C4×C79C2×C22⋊Q8semidirect
285. C2×C79#2440semidirect
301. C2×#389D79semidirect
306. C79C2×#2440semidirect
334. #389C2×D79semidirect
539. C2×C4#14579141semidirect
577. C4C2×#14579141semidirect
579. C2#23767100direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145791602
C22×#145791412
C22×C4⋊C4×D792
#237671002
C2×#190984932
C2×#190984842
C2×#190984622
C2×#190984372
C2×#190984332
C2×#190984202
C2×C4×(C2×D79)⋊C42
C2×#389×C792
C2×#244079