Finite Group Databasebeta

← C2×#22591350 #32156974

Subgroups — C2×#22591350 #32156974

Order 18176.

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×#14556104C211yesmaximal, characteristic
2C22×Q8×Dic71C211yesmaximal, characteristic, complemented
3C22×C4⋊C4×D71C211yesmaximal, characteristic
4#22591350C2116yesmaximal, direct product factor
5C2×#17985896C212yesmaximal, complemented
6C2×#17985895C211yesmaximal, characteristic, complemented
7C2×#17985881C212yesmaximal, complemented
8C2×#17985873C212yesmaximal
9C2×C4×C4⋊Dic71C211yesmaximal, characteristic, complemented
10C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
11C2×#17985846C212yesmaximal
12C2×#391×C71C211yesmaximal, characteristic, complemented
13C23×C4×D71C2212yes—
14C23×C4×D71C2211yescharacteristic
15C2×#14556104C4116yescomplemented
16C2×#14556104C2218yes—
17C2×Q8×Dic71—28no—
18C2×Q8×Dic71C2218yescomplemented
19C22×C4⋊Dic71C2212yes—
20C22×C4⋊Dic71C2211yescharacteristic
21C22×C4⋊Dic71C2211yescharacteristic
22C22×Dic71⋊C4C2212yes—
23C22×Dic71⋊C4C2212yes—
24C22×Dic71⋊C4C2212yes—
25C2×C4⋊C4×D71—28no—
26C2×C4⋊C4×D71C2218yes—
27C22×(C2×D71)⋊C4C2211yescharacteristic
28C22×(C2×D71)⋊C4C2212yes—
29C22×(C2×D71)⋊C4C2211yescharacteristic
30C22×C4×Dic71C2212yes—
31C22×C4×Dic71C2211yescharacteristic
32C22×C4×Dic71C2212yes—
33#17985896C22116yescomplemented
34#17985895C2218yescomplemented
35#17985881C22116yescomplemented
36#17985873C22116yes—
37C4×C4⋊Dic71C2218yescomplemented
38C4×(C2×D71)⋊C4C2218yes—
39#17985846C22116yes—
40C2×C71⋊#95C2212yes—
41C2×C71⋊#95C2211yescharacteristic
42C2×C71⋊#95C2212yes—
43C2×C71⋊#95C2211yescharacteristic
44C23×Q8×C71C2211yescharacteristic
45C22×C4⋊C4×C71C2211yescharacteristic
46C22×C42×C71C2211yescharacteristic
47#391×C71C2218yescomplemented
48C2×#95×C71C2212yes—
49C2×#95×C71C2212yes—
50C24×D71C2311yescharacteristic
51C23×Dic71C2311yescharacteristic
52C23×Dic71C2311yescharacteristic
53C23×Dic71C2312yes—
54C23×Dic71C2312yes—
55C23×Dic71C2311yescharacteristic
56C22×C4×D71—28no—
57C22×C4×D71—28no—
58C22×C4×D71—24no—
59C22×C4×D71—24no—
60C22×C4×D71—44no—
61C22×C4×D71C2×C418yes—
62C22×C4×D71C2318yes—
63C22×C4×D71C2314yes—
64#14556104C2×C4164yescomplemented
65Q8×Dic71—232no—
66C2×C4⋊Dic71—28no—
67C2×C4⋊Dic71C2×C418yescomplemented
68C2×C4⋊Dic71D418yes—
69C2×C4⋊Dic71C2318yes—
70C2×C4⋊Dic71C2314yes—
71C2×C4⋊Dic71C2314yes—
72C2×Dic71⋊C4—28no—
73C2×Dic71⋊C4—44no—
74C2×Dic71⋊C4C2×C4116yescomplemented
75C2×Dic71⋊C4C2318yes—
76C2×Dic71⋊C4C2318yes—
77C2×Dic71⋊C4C2318yes—
78C4⋊C4×D71—232no—
79C2×(C2×D71)⋊C4—24no—
80C2×(C2×D71)⋊C4C2×C4116yes—
81C2×(C2×D71)⋊C4Q818yescomplemented
82C2×(C2×D71)⋊C4C2314yes—
83C2×(C2×D71)⋊C4C2318yes—
84C2×(C2×D71)⋊C4C2314yes—
85C2×C4×Dic71—28no—
86C2×C4×Dic71—24no—
87C2×C4×Dic71—44no—
88C2×C4×Dic71C2318yes—
89C2×C4×Dic71C2314yes—
90C2×C4×Dic71C2318yes—
91C71⋊#95C2318yes—
92C71⋊#95C2314yes—
93C71⋊#95C2314yes—
94C71⋊#95C2318yes—
95C22×Q8×C71—24no—
96C22×Q8×C71—24no—
97C22×Q8×C71C2×C418yescomplemented
98C22×Q8×C71C2314yes—
99C23×C4×C71C2312yes—
100C23×C4×C71C2311yescharacteristic
101C23×C4×C71C2312yes—
102C23×C4×C71C2311yescharacteristic
103C23×C4×C71C2311yescharacteristic
104C2×C4⋊C4×C71—24no—
105C2×C4⋊C4×C71C2314yes—
106C2×C42×C71—24no—
107C2×C42×C71C2314yes—
108#95×C71C2318yes—
109#95×C71C2318yes—
110C23×D71—22no—
111C23×D71—22no—
112C23×D71—22no—
113C23×D71—22no—
114C23×D71C22×C414yes—
115C23×D71C2×Q814yes—
116C23×D71#4114yes—
117C23×D71C2412yes—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—24no—
121C22×Dic71—22no—
122C22×Dic71—24no—
123C22×Dic71—24no—
124C22×Dic71—24no—
125C22×Dic71—22no—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—22no—
129C22×Dic71C22×C414yes—
130C22×Dic71C22×C418yes—
131C22×Dic71C2×D418yes—
132C22×Dic71C2×Q814yes—
133C22×Dic71C2412yes—
134C22×Dic71C2412yes—
135C22×Dic71C2414yes—
136C22×Dic71C2414yes—
137C22×Dic71C2412yes—
138C2×C4×D71—232no—
139C2×C4×D71—28no—
140C2×C4×D71—28no—
141C2×C4×D71—216no—
142C2×C4×D71—216no—
143C2×C4×D71—48no—
144C2×C4×D71—432no—
145C2×C4×D71C22×C4116yes—
146C4⋊Dic71—216no—
147C4⋊Dic71C22×C4116yescomplemented
148C4⋊Dic71C2×D4116yes—
149Dic71⋊C4—216no—
150Dic71⋊C4—48no—
151Dic71⋊C4C22×C4132yescomplemented
152(C2×D71)⋊C4—28no—
153(C2×D71)⋊C4C22×C4132yes—
154(C2×D71)⋊C4C2×Q8116yescomplemented
155C4×Dic71—28no—
156C4×Dic71—216no—
157C4×Dic71—48no—
158C24×C71C2411yescharacteristic
159C2×Q8×C71—232no—
160C2×Q8×C71—28no—
161C2×Q8×C71—28no—
162C2×Q8×C71C22×C4116yescomplemented
163C22×C4×C71—24no—
164C22×C4×C71—22no—
165C22×C4×C71—24no—
166C22×C4×C71—22no—
167C22×C4×C71—24no—
168C22×C4×C71—24no—
169C22×C4×C71—22no—
170C22×C4×C71—22no—
171C22×C4×C71—24no—
172C22×C4×C71—22no—
173C22×C4×C71C22×C418yes—
174C22×C4×C71C22×C414yes—
175C22×C4×C71C2×D414yes—
176C22×C4×C71C2×Q814yes—
177C22×C4×C71#4114yes—
178C22×C4×C71C2414yes—
179C22×C4×C71C2412yes—
180C22×C4×C71C2414yes—
181C22×C4×C71C2412yes—
182C22×C4×C71C2412yes—
183C4⋊C4×C71—28no—
184C42×C71—28no—
185C23×C71C23×C412yes—
186C23×C71C22×D412yes—
187C23×C71C22×Q812yes—
188C23×C71C2×#4112yes—
189C23×C71C2×#4112yes—
190C23×C71H(5,𝔽2)12yes—
191C23×C712−1+412yes—
192C23×C71C2511yescharacteristic
193C22×D71—28no—
194C22×D71—28no—
195C22×D71—28no—
196C22×D71—28no—
197C22×D71—28no—
198C22×D71—28no—
199C22×D71—22no—
200C22×D71—22no—
201C22×D71—22no—
202C22×D71—22no—
203C22×D71C4×Q8116yescomplemented
204C22×D71C23×C414yes—
205C22×D71C22×Q814yes—
206C22×D71C2×#4114yes—
207Q8×C71—232no—
208C2×C4×C71—216no—
209C2×C4×C71—28no—
210C2×C4×C71—24no—
211C2×C4×C71—22no—
212C2×C4×C71—24no—
213C2×C4×C71—22no—
214C2×C4×C71—24no—
215C2×C4×C71—22no—
216C2×C4×C71—22no—
217C2×C4×C71—24no—
218C2×C4×C71—24no—
219C2×C4×C71—22no—
220C2×C4×C71—48no—
221C2×C4×C71—44no—
222C2×C4×C71C4×D4116yes—
223C2×C4×C71C23×C418yes—
224C2×C4×C71C23×C414yes—
225C2×C4×C71C22×D414yes—
226C2×C4×C71C22×Q814yes—
227C2×C4×C71C2×#4114yes—
228C2×Dic71—28no—
229C2×Dic71—216no—
230C2×Dic71—216no—
231C2×Dic71—28no—
232C2×Dic71—22no—
233C2×Dic71—22no—
234C2×Dic71—24no—
235C2×Dic71—22no—
236C2×Dic71—24no—
237C2×Dic71—24no—
238C2×Dic71—24no—
239C2×Dic71—22no—
240C2×Dic71—22no—
241C2×Dic71—22no—
242C2×Dic71—22no—
243C2×Dic71—44no—
244C2×Dic71C23×C418yes—
245C2×Dic71C23×C414yes—
246C2×Dic71C22×D418yes—
247C2×Dic71C22×Q814yes—
248C4×D71—232no—
249C4×D71—432no—
250C22×C71C2×C4×D414yes—
251C22×C71C2×C4×Q814yes—
252C22×C71#52014yes—
253C22×C71#54614yes—
254C22×C71D4×Q814yes—
255C22×C71#55214yes—
256C22×C71#55414yes—
257C22×C71C24×C411yescharacteristic
258C22×C71C23×D411yescharacteristic
259C22×C71C23×Q811yescharacteristic
260C22×C71C22×#4111yescharacteristic
261C22×C71C22×#4111yescharacteristic
262C22×C71C2×H(5,𝔽2)11yescharacteristic
263C22×C71C2×2−1+411yescharacteristic
264C2×D71—232no—
265C2×D71—24no—
266C2×D71—24no—
267C2×D71—24no—
268C2×D71—24no—
269C2×D71—24no—
270C2×D71—24no—
271C2×D71C2×C4×Q818yescomplemented
272C4×C71—28no—
273C4×C71—24no—
274C4×C71—44no—
275C4×C71—42no—
276C4×C71C2×C4×D418yes—
277Dic71—24no—
278Dic71—28no—
279Dic71—28no—
280Dic71—24no—
281Dic71—42no—
282C2×#2360—711nomaximal, Sylow
283C2×C71#236018yescomplemented
284C2×C71C22×C4×D411yescharacteristic
285C2×C71C22×C4×Q811yescharacteristic
286C2×C71C2×#52011yescharacteristic
287C2×C71C2×#54611yescharacteristic
288C2×C71C2×D4×Q811yescharacteristic
289C2×C71C2×#55211yescharacteristic
290C2×C71C2×#55411yescharacteristic
291D71—28no—
292C22×C22⋊Q8—711no—
293C22×C4×Q8—711no—
294C23×C4⋊C4—711no—
295#2360—7116no—
296C2×#391D7111yescharacteristic, complemented
297C2×#389—712no—
298C2×#389—711no—
299C2×#387—712no—
300C2×#385—712no—
301C2×C4×C4⋊C4—711no—
302C2×C4×C22⋊C4—711no—
303C2×C4×C22⋊C4—712no—
304C71C2×#236011yescharacteristic, complemented, Sylow
305C23×Q8C2×D7111yescharacteristic
306C24×C4—712no—
307C24×C4—711no—
308C2×C22⋊Q8—7116no—
309C2×C22⋊Q8—718no—
310C2×C4×Q8—718no—
311C2×C4×Q8—1428no—
312C22×C4⋊C4—712no—
313C22×C4⋊C4—711no—
314C22×C4⋊C4—712no—
315C22×C4⋊C4—711no—
316C22×C4⋊C4—712no—
317C22×C4⋊C4—712no—
318C22×C4⋊C4—718no—
319C22×C4⋊C4—1428no—
320C22×C4⋊C4C2×D7111yescharacteristic
321C22×C22⋊C4—712no—
322C22×C22⋊C4—711no—
323C22×C22⋊C4—711no—
324C22×C42—712no—
325C22×C42—712no—
326C22×C42—711no—
327C22×C42C2×D7111yescharacteristic
328#391C2×D7118yescomplemented
329#389—718no—
330#389—7116no—
331#387—7116no—
332#385—7116no—
333C4×C4⋊C4—718no—
334C4×C22⋊C4—718no—
335C4×C22⋊C4—7116no—
336C2×#95—711no—
337C2×#95—712no—
338C2×#95—711no—
339C2×#95—712no—
340C2×#95C2×D7112yes—
341C2×#95C2×D7112yes—
342C25—711no—
343C22×Q8—24no—
344C22×Q8—24no—
345C22×Q8C4×D7118yescomplemented
346C22×Q8C22×D7114yes—
347C23×C4—2844no—
348C23×C4—718no—
349C23×C4—712no—
350C23×C4—711no—
351C23×C4—711no—
352C23×C4—712no—
353C23×C4—718no—
354C23×C4—714no—
355C23×C4—711no—
356C23×C4—1428no—
357C23×C4—1428no—
358C23×C4—1424no—
359C23×C4—1424no—
360C23×C4C22×D7111yescharacteristic
361C23×C4C22×D7112yes—
362C23×C4C22×D7112yes—
363C23×C4C22×D7111yescharacteristic
364C23×C4C22×D7111yescharacteristic
365C22⋊Q8—7164no—
366C4×Q8—14232no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—2844no—
369C2×C4⋊C4—718no—
370C2×C4⋊C4—7116no—
371C2×C4⋊C4—718no—
372C2×C4⋊C4—718no—
373C2×C4⋊C4—714no—
374C2×C4⋊C4—714no—
375C2×C4⋊C4—718no—
376C2×C4⋊C4—718no—
377C2×C4⋊C4—718no—
378C2×C4⋊C4—14232no—
379C2×C4⋊C4—1428no—
380C2×C4⋊C4—1428no—
381C2×C4⋊C4C22×D7114yes—
382C2×C22⋊C4—7116no—
383C2×C22⋊C4—718no—
384C2×C22⋊C4—718no—
385C2×C22⋊C4—714no—
386C2×C22⋊C4—714no—
387C2×C22⋊C4—1424no—
388C2×C42—24no—
389C2×C42—2844no—
390C2×C42—718no—
391C2×C42—718no—
392C2×C42—714no—
393C2×C42—1428no—
394C2×C42—1424no—
395C2×C42C22×D7114yes—
396#95—714no—
397#95—718no—
398#95—714no—
399#95—718no—
400#95C22×D7118yes—
401#95C22×D7118yes—
402C24—714no—
403C24—714no—
404C24—714no—
405C24—712no—
406C24—1422no—
407C24—1422no—
408C24—1422no—
409C24—1422no—
410C24C23×D7111yescharacteristic
411C2×Q8—28no—
412C2×Q8—28no—
413C2×Q8—232no—
414C2×Q8C2×C4×D71116yescomplemented
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—2848no—
426C22×C4—28432no—
427C22×C4—718no—
428C22×C4—714no—
429C22×C4—718no—
430C22×C4—714no—
431C22×C4—712no—
432C22×C4—714no—
433C22×C4—712no—
434C22×C4—714no—
435C22×C4—7116no—
436C22×C4—712no—
437C22×C4—14216no—
438C22×C4—14216no—
439C22×C4—1422no—
440C22×C4—1422no—
441C22×C4—1422no—
442C22×C4—1422no—
443C22×C4—1422no—
444C22×C4—1424no—
445C22×C4—1422no—
446C22×C4—1424no—
447C22×C4—1424no—
448C22×C4—1422no—
449C22×C4—1424no—
450C22×C4—1428no—
451C22×C4—1428no—
452C22×C4—14232no—
453C22×C4C2×C4×D7118yes—
454C22×C4C2×C4×D7114yes—
455C22×C4D4×D7114yes—
456C22×C4Q8×D7114yes—
457C22×C4#1176586414yes—
458C22×C4C23×D7112yes—
459C22×C4C23×D7114yes—
460C22×C4C23×D7114yes—
461C22×C4C23×D7112yes—
462C22×C4C23×D7112yes—
463C4⋊C4—28no—
464C4⋊C4—2848no—
465C4⋊C4—7132no—
466C4⋊C4—7116no—
467C4⋊C4—7116no—
468C4⋊C4—14216no—
469C4⋊C4—14216no—
470C22⋊C4—7132no—
471C22⋊C4—7116no—
472C22⋊C4—1428no—
473C42—28no—
474C42—2848no—
475C42—14216no—
476C42—1428no—
477C23—714no—
478C23—714no—
479C23—714no—
480C23—7116no—
481C23—1422no—
482C23—1422no—
483C23—1422no—
484C23—1422no—
485C23—1428no—
486C23—1428no—
487C23—1428no—
488C23—1428no—
489C23—1428no—
490C23—1428no—
491C23C22×C4×D7112yes—
492C23C2×D4×D7112yes—
493C23C2×Q8×D7112yes—
494C23#41×D7112yes—
495C23C2×#1176586412yes—
496C23#1455617412yes—
497C23#1455617612yes—
498C23C24×D7111yescharacteristic
499Q8—232no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—28no—
505C2×C4—216no—
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—28432no—
515C2×C4—2844no—
516C2×C4—718no—
517C2×C4—714no—
518C2×C4—718no—
519C2×C4—714no—
520C2×C4—1422no—
521C2×C4—1422no—
522C2×C4—1422no—
523C2×C4—1422no—
524C2×C4—1422no—
525C2×C4—1424no—
526C2×C4—1422no—
527C2×C4—1424no—
528C2×C4—1424no—
529C2×C4—1424no—
530C2×C4—1422no—
531C2×C4—14216no—
532C2×C4—1428no—
533C2×C4—1428no—
534C2×C4—14216no—
535C2×C4—14232no—
536C2×C4(C22⋊D71)⋊C4116yes—
537C2×C4C22×C4×D7118yes—
538C2×C4C22×C4×D7114yes—
539C2×C4C2×D4×D7114yes—
540C2×C4C2×Q8×D7114yes—
541C2×C4C2×#1176586414yes—
542C22—718no—
543C22—1424no—
544C22—1424no—
545C22—1424no—
546C22—1424no—
547C22—14232no—
548C22—1424no—
549C22—1424no—
550C22C2×(C22⋊D71)⋊C414yes—
551C22C4×Q8×D7114yes—
552C22#1798648114yes—
553C22#1798664214yes—
554C22#1798666414yes—
555C22#1798668314yes—
556C22#1798670714yes—
557C22C23×C4×D7111yescharacteristic
558C22C22×D4×D7111yescharacteristic
559C22C22×Q8×D7111yescharacteristic
560C22C2×#41×D7111yescharacteristic
561C22C22×#1176586411yescharacteristic
562C22C2×#1455617411yescharacteristic
563C22C2×#1455617611yescharacteristic
564C4—28no—
565C4—24no—
566C4—44no—
567C4—42no—
568C4—2842no—
569C4—1424no—
570C4—1428no—
571C4—1428no—
572C4—1424no—
573C4C2×(C22⋊D71)⋊C418yes—
574C2—1428no—
575C2#2259135018yesdirect product factor
576C2C22×(C22⋊D71)⋊C411yescharacteristic
577C2C2×C4×Q8×D7111yescharacteristic
578C2C2×#1798648111yescharacteristic
579C2C2×#1798664211yescharacteristic
580C2C2×#1798666411yescharacteristic
581C2C2×#1798668311yescharacteristic
582C2C2×#1798670711yescharacteristic
subgroupG/H
centerC24C23×D71
commutatorC22×C71C24×C4
FrattiniC23C24×D71
FittingC2×#391×C71C2
socleC24×C71C24
prime radicalC24×D71C23
cosocleC23×C71C25
hypercenterC2×#391D71
hypocenterC71C2×#2360
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C22×Q8×Dic71C2semidirect
4. #22591350C2direct (N × H)
5. C2×#17985896C2semidirect
6. C2×#17985895C2semidirect
7. C2×#17985881C2semidirect
9. C2×C4×C4⋊Dic71C2semidirect
12. C2×#391×C71C2semidirect
15. C2×#14556104C4semidirect
18. C2×Q8×Dic71C22semidirect
33. #17985896C22semidirect
34. #17985895C22semidirect
35. #17985881C22semidirect
37. C4×C4⋊Dic71C22semidirect
47. #391×C71C22semidirect
64. #14556104C2×C4semidirect
67. C2×C4⋊Dic71C2×C4semidirect
74. C2×Dic71⋊C4C2×C4semidirect
81. C2×(C2×D71)⋊C4Q8semidirect
97. C22×Q8×C71C2×C4semidirect
147. C4⋊Dic71C22×C4semidirect
151. Dic71⋊C4C22×C4semidirect
154. (C2×D71)⋊C4C2×Q8semidirect
162. C2×Q8×C71C22×C4semidirect
203. C22×D71C4×Q8semidirect
271. C2×D71C2×C4×Q8semidirect
283. C2×C71#2360semidirect
296. C2×#391D71semidirect
304. C71C2×#2360semidirect
328. #391C2×D71semidirect
345. C22×Q8C4×D71semidirect
414. C2×Q8C2×C4×D71semidirect
575. C2#22591350direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561042
C22×Q8×Dic712
C22×C4⋊C4×D712
#225913502
C2×#179858962
C2×#179858952
C2×#179858812
C2×#179858732
C2×C4×C4⋊Dic712
C2×C4×(C2×D71)⋊C42
C2×#179858462
C2×#391×C712
C2×#236071