Finite Group Databasebeta

← C2×#22588798 #31999239

Subgroups — C2×#22588798 #31999239

Order 18176.

2066 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556111C211yesmaximal, characteristic
2C22×#14556042C212yesmaximal
3C2×#17985862C211yesmaximal, characteristic, complemented
4#22588798C218yesmaximal, direct product factor
5C2×#17985765C212yesmaximal, complemented
6C22×#117×C71C211yesmaximal, characteristic, complemented
7C23×C22⋊D71C2211yescharacteristic
8C2×#14556111—22no—
9C2×#14556111—22no—
10C2×#14556111—21no—
11C2×#14556111—21no—
12C2×#14556111—22no—
13C2×#14556111C414yescomplemented
14C2×#14556111C2214yes—
15C22×Dic71⋊C4C412yescomplemented
16C22×C22⋊Dic71C2212yes—
17C22×C22⋊Dic71C2211yescharacteristic
18C22×(C2×D71)⋊C4C412yescomplemented
19C2×#14556042—28no—
20C2×#14556042C2218yes—
21#17985862—22no—
22#17985862C2214yescomplemented
23C2×C71⋊#95—22no—
24#17985765C2218yescomplemented
25C24×C4×C71C2211yescharacteristic
26C2×#117×C71—22no—
27C2×#117×C71—28no—
28C2×#117×C71C2214yescomplemented
29C22×C4⋊C4×C71—21no—
30C22×C22⋊C4×C71C2212yes—
31C22×C42×C71—21no—
32C24×D71C2×C411yescharacteristic
33C22×C22⋊D71—24no—
34C22×C22⋊D71—21no—
35C22×C22⋊D71—21no—
36C22×C22⋊D71—21no—
37C22×C22⋊D71—21no—
38C22×C22⋊D71—24no—
39C22×C22⋊D71C2×C412yes—
40C22×C22⋊D71C2312yes—
41C23×Dic71—22no—
42C23×Dic71C2×C411yescharacteristic
43C23×Dic71C2×C412yes—
44#14556111—28no—
45#14556111—24no—
46#14556111—24no—
47#14556111—24no—
48#14556111—24no—
49#14556111—28no—
50C2×Dic71⋊C4—24no—
51C2×Dic71⋊C4—22no—
52C2×Dic71⋊C4—22no—
53C2×Dic71⋊C4C2×C418yescomplemented
54C2×C22⋊Dic71—21no—
55C2×C22⋊Dic71—21no—
56C2×C22⋊Dic71—21no—
57C2×C22⋊Dic71—21no—
58C2×C22⋊Dic71—24no—
59C2×C22⋊Dic71—22no—
60C2×C22⋊Dic71—24no—
61C2×C22⋊Dic71C2×C412yes—
62C2×C22⋊Dic71C2314yes—
63C2×C22⋊Dic71C2312yes—
64C2×(C2×D71)⋊C4—24no—
65C2×(C2×D71)⋊C4—22no—
66C2×(C2×D71)⋊C4—22no—
67C2×(C2×D71)⋊C4C2×C418yescomplemented
68#14556042—216no—
69C71⋊#95—28no—
70C25×C71C2311yescharacteristic
71C23×C4×C71—24no—
72C23×C4×C71—24no—
73C23×C4×C71—22no—
74C23×C4×C71C2×C412yes—
75C23×C4×C71C2×C412yes—
76C23×C4×C71D412yes—
77C23×C4×C71D412yes—
78C23×C4×C71D411yescharacteristic
79C23×C4×C71D411yescharacteristic
80C23×C4×C71D412yes—
81C23×C4×C71C2312yes—
82#117×C71—216no—
83#117×C71—216no—
84C2×C4⋊C4×C71—28no—
85C2×C4⋊C4×C71—24no—
86C2×C22⋊C4×C71—22no—
87C2×C22⋊C4×C71—24no—
88C2×C22⋊C4×C71D418yes—
89C2×C22⋊C4×C71C2314yes—
90C2×C42×C71—28no—
91C2×C42×C71—24no—
92C23×D71—21no—
93C23×D71—21no—
94C23×D71—24no—
95C23×D71—44no—
96C23×D71C22×C412yes—
97C2×C22⋊D71—22no—
98C2×C22⋊D71—22no—
99C2×C22⋊D71—28no—
100C2×C22⋊D71—24no—
101C2×C22⋊D71—22no—
102C2×C22⋊D71—22no—
103C2×C22⋊D71—24no—
104C2×C22⋊D71—42no—
105C2×C22⋊D71—42no—
106C2×C22⋊D71—44no—
107C2×C22⋊D71—44no—
108C2×C22⋊D71—42no—
109C2×C22⋊D71—42no—
110C22×Dic71—21no—
111C22×Dic71—21no—
112C22×Dic71—22no—
113C22×Dic71—22no—
114C22×Dic71—24no—
115C22×Dic71—24no—
116C22×Dic71—44no—
117C22×Dic71—48no—
118C22×Dic71—44no—
119C22×Dic71C22×C412yes—
120C22×Dic71C22×C414yes—
121Dic71⋊C4—28no—
122Dic71⋊C4—28no—
123C22⋊Dic71—22no—
124C22⋊Dic71—22no—
125C22⋊Dic71—22no—
126C22⋊Dic71—22no—
127C22⋊Dic71—24no—
128C22⋊Dic71—24no—
129C22⋊Dic71—44no—
130(C2×D71)⋊C4—28no—
131(C2×D71)⋊C4—28no—
132C24×C71—44no—
133C24×C71C22×C411yescharacteristic
134C24×C71C22×C411yescharacteristic
135C24×C71C2×D411yescharacteristic
136C24×C71C2×D411yescharacteristic
137C24×C71C2×D414yes—
138C24×C71C2×D411yescharacteristic
139C24×C71C2×D411yescharacteristic
140C24×C71#4114yes—
141C24×C71C2411yescharacteristic
142C22×C4×C71—28no—
143C22×C4×C71—28no—
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—24no—
147C22×C4×C71—28no—
148C22×C4×C71—28no—
149C22×C4×C71—22no—
150C22×C4×C71—22no—
151C22×C4×C71—24no—
152C22×C4×C71—24no—
153C22×C4×C71—24no—
154C22×C4×C71—28no—
155C22×C4×C71—24no—
156C22×C4×C71—48no—
157C22×C4×C71C22⋊C414yes—
158C22×C4×C71C22⋊C412yes—
159C22×C4×C71C22⋊C412yes—
160C22×C4×C71C22⋊C414yes—
161C22×C4×C71C22⋊C412yes—
162C22×C4×C71C22⋊C412yes—
163C22×C4×C71C22×C414yes—
164C22×C4×C71C2×D412yes—
165C22×C4×C71C2×D412yes—
166C22×C4×C71C2×D414yes—
167C4⋊C4×C71—216no—
168C22⋊C4×C71—28no—
169C22⋊C4×C71—24no—
170C22⋊C4×C71C2×D418yes—
171C42×C71—216no—
172C23×C71—24no—
173C23×C71—24no—
174C23×C71—24no—
175C23×C71—42no—
176C23×C71—42no—
177C23×C71—48no—
178C23×C71—42no—
179C23×C71—48no—
180C23×C71—42no—
181C23×C71C2×C22⋊C411yescharacteristic
182C23×C71C2×C22⋊C411yescharacteristic
183C23×C71C2×C22⋊C411yescharacteristic
184C23×C71C2×C22⋊C411yescharacteristic
185C23×C71C4×D414yes—
186C23×C71C4×D414yes—
187C23×C71#12012yes—
188C23×C71#12012yes—
189C23×C71#12114yes—
190C23×C71#12114yes—
191C23×C71#12312yes—
192C23×C71#12312yes—
193C23×C71C23×C411yescharacteristic
194C23×C71C22×D411yescharacteristic
195C23×C71C22×D412yes—
196C23×C71C22×D411yescharacteristic
197C23×C71C2×#4112yes—
198C22×D71—22no—
199C22×D71—24no—
200C22×D71—44no—
201C22×D71—42no—
202C22×D71—42no—
203C22×D71—44no—
204C22×D71—44no—
205C22×D71—88no—
206C2×C4×C71—28no—
207C2×C4×C71—22no—
208C2×C4×C71—22no—
209C2×C4×C71—28no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—28no—
213C2×C4×C71—24no—
214C2×C4×C71—24no—
215C2×C4×C71—28no—
216C2×C4×C71—24no—
217C2×C4×C71—24no—
218C2×C4×C71—28no—
219C2×C4×C71—48no—
220C2×C4×C71—416no—
221C2×C4×C71C2×C22⋊C414yes—
222C2×C4×C71C2×C22⋊C414yes—
223C22⋊D71—42no—
224C22⋊D71—44no—
225C22⋊D71—42no—
226C22⋊D71—44no—
227C22⋊D71—44no—
228C2×Dic71—24no—
229C2×Dic71—22no—
230C2×Dic71—24no—
231C2×Dic71—44no—
232C2×Dic71—42no—
233C2×Dic71—42no—
234C2×Dic71—44no—
235C2×Dic71—44no—
236C2×Dic71—48no—
237C2×Dic71—44no—
238C2×Dic71—88no—
239C22×C71—22no—
240C22×C71—22no—
241C22×C71—22no—
242C22×C71—22no—
243C22×C71—22no—
244C22×C71—22no—
245C22×C71—28no—
246C22×C71—24no—
247C22×C71—28no—
248C22×C71—24no—
249C22×C71—44no—
250C22×C71—42no—
251C22×C71—44no—
252C22×C71—42no—
253C22×C71—42no—
254C22×C71—42no—
255C22×C71#38612yes—
256C22×C71#38612yes—
257C22×C71#38612yes—
258C22×C71#38612yes—
259C22×C71#41014yes—
260C22×C71C22×C22⋊C411yescharacteristic
261C22×C71C2×C4×D412yes—
262C22×C71C2×#12011yescharacteristic
263C22×C71C2×#12112yes—
264C22×C71C2×#12311yescharacteristic
265C2×D71—44no—
266C2×D71—42no—
267C2×D71—84no—
268C2×D71—84no—
269C2×D71—84no—
270C4×C71—24no—
271C4×C71—24no—
272C4×C71—48no—
273Dic71—44no—
274Dic71—42no—
275Dic71—84no—
276C2×#1899—711nomaximal, Sylow
277C2×C71—22no—
278C2×C71—21no—
279C2×C71—22no—
280C2×C71—22no—
281C2×C71—21no—
282C2×C71—22no—
283C2×C71—22no—
284C2×C71#189914yescomplemented
285C2×C71C2×#38611yescharacteristic
286C2×C71C2×#41012yes—
287D71—82no—
288C22×#123—711no—
289C22×#117D7111yescharacteristic, complemented
290C22×#99—712no—
291C2×#381—711no—
292#1899—718no—
293C2×#342—712no—
294C71C2×#189911yescharacteristic, complemented, Sylow
295C23×D4—711no—
296C24×C4C2×D7111yescharacteristic
297C2×#123—714no—
298C2×#123—714no—
299C2×#123—1422no—
300C2×#123—1422no—
301C2×#123—1421no—
302C2×#123—1422no—
303C2×#123—1421no—
304C2×#117—22no—
305C2×#117—28no—
306C2×#117C2×D7114yescomplemented
307C22×C4⋊C4—21no—
308C22×C4⋊C4—712no—
309C22×C22⋊C4—711no—
310C22×C22⋊C4—712no—
311C22×C22⋊C4—712no—
312C22×C22⋊C4C2×D7112yes—
313C22×C42—21no—
314C2×#99—718no—
315C2×#99—1428no—
316#381—714no—
317#381—1422no—
318C2×#95—1422no—
319#342—718no—
320C25—711no—
321C25C22×D7111yescharacteristic
322C22×D4—712no—
323C22×D4—712no—
324C22×D4—1421no—
325C22×D4—1421no—
326C22×D4—1424no—
327C22×D4—1421no—
328C22×D4—1421no—
329C22×D4—1424no—
330C23×C4—24no—
331C23×C4—24no—
332C23×C4—22no—
333C23×C4—712no—
334C23×C4—711no—
335C23×C4—1422no—
336C23×C4C4×D7112yes—
337C23×C4C4×D7112yes—
338C23×C4D28412yes—
339C23×C4D28412yes—
340C23×C4C22⋊D7111yescharacteristic
341C23×C4C22⋊D7112yes—
342C23×C4C22⋊D7111yescharacteristic
343C23×C4C22×D7112yes—
344#123—1428no—
345#123—1424no—
346#123—1424no—
347#123—1424no—
348#123—1424no—
349#123—1428no—
350#117—216no—
351#117—216no—
352C2×C4⋊C4—28no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—718no—
355C2×C4⋊C4—1424no—
356C2×C4⋊C4—1422no—
357C2×C4⋊C4—1422no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—712no—
361C2×C22⋊C4—712no—
362C2×C22⋊C4—718no—
363C2×C22⋊C4—714no—
364C2×C22⋊C4—1422no—
365C2×C22⋊C4—1424no—
366C2×C22⋊C4—1424no—
367C2×C22⋊C4—1421no—
368C2×C22⋊C4—1421no—
369C2×C22⋊C4—1421no—
370C2×C22⋊C4—1421no—
371C2×C22⋊C4—1424no—
372C2×C22⋊C4—1422no—
373C2×C22⋊C4—1422no—
374C2×C22⋊C4C22⋊D7118yes—
375C2×C22⋊C4C22×D7114yes—
376C2×C42—28no—
377C2×C42—24no—
378#99—14216no—
379#95—1428no—
380C24—44no—
381C24—2844no—
382C24—712no—
383C24—1421no—
384C24—1421no—
385C24—1424no—
386C24C2×C4×D7111yescharacteristic
387C24C2×C4×D7111yescharacteristic
388C24C2×C22⋊D7111yescharacteristic
389C24C2×C22⋊D7111yescharacteristic
390C24C2×C22⋊D7114yes—
391C24C2×D28411yescharacteristic
392C24C2×D28411yescharacteristic
393C24#1176586514yes—
394C24C23×D7111yescharacteristic
395C2×D4—2842no—
396C2×D4—2842no—
397C2×D4—2844no—
398C2×D4—2844no—
399C2×D4—2842no—
400C2×D4—2842no—
401C2×D4—1424no—
402C2×D4—1422no—
403C2×D4—1422no—
404C2×D4—1424no—
405C2×D4—1422no—
406C2×D4—1422no—
407C2×D4—1428no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—48no—
423C22×C4—2844no—
424C22×C4—2848no—
425C22×C4—2844no—
426C22×C4—714no—
427C22×C4—712no—
428C22×C4—1424no—
429C22×C4—1422no—
430C22×C4—1421no—
431C22×C4—1422no—
432C22×C4—1421no—
433C22×C4—1424no—
434C22×C4(C2×D71)⋊C414yes—
435C22×C4(C2×D71)⋊C414yes—
436C22×C4(C2×D71)⋊C412yes—
437C22×C4(C2×D71)⋊C412yes—
438C22×C4(C2×D71)⋊C412yes—
439C22×C4(C2×D71)⋊C412yes—
440C22×C4C2×C4×D7114yes—
441C22×C4C2×C22⋊D7112yes—
442C22×C4C2×C22⋊D7112yes—
443C22×C4C2×D28414yes—
444C4⋊C4—216no—
445C4⋊C4—1428no—
446C4⋊C4—1428no—
447C22⋊C4—24no—
448C22⋊C4—28no—
449C22⋊C4—2844no—
450C22⋊C4—1424no—
451C22⋊C4—1424no—
452C22⋊C4—1422no—
453C22⋊C4—1422no—
454C22⋊C4—1422no—
455C22⋊C4—1422no—
456C22⋊C4—1428no—
457C22⋊C4—1428no—
458C22⋊C4C2×C22⋊D7118yes—
459C42—216no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—42no—
464C23—48no—
465C23—42no—
466C23—42no—
467C23—48no—
468C23—42no—
469C23—2844no—
470C23—2844no—
471C23—2842no—
472C23—2842no—
473C23—2844no—
474C23—5688no—
475C23—1422no—
476C23—1424no—
477C23C2×(C2×D71)⋊C411yescharacteristic
478C23C2×(C2×D71)⋊C411yescharacteristic
479C23C2×(C2×D71)⋊C411yescharacteristic
480C23C2×(C2×D71)⋊C411yescharacteristic
481C23C4×C22⋊D7114yes—
482C23C4×C22⋊D7114yes—
483C23#1455609712yes—
484C23#1455609712yes—
485C23#1455610214yes—
486C23#1455610214yes—
487C23#1455611112yes—
488C23#1455611112yes—
489C23C22×C4×D7111yescharacteristic
490C23C22×C22⋊D7111yescharacteristic
491C23C22×C22⋊D7112yes—
492C23C22×D28411yescharacteristic
493C23C2×#1176586512yes—
494D4—2844no—
495D4—2844no—
496D4—2842no—
497D4—2844no—
498D4—2842no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—416no—
513C2×C4—48no—
514C2×C4—2844no—
515C2×C4—2842no—
516C2×C4—2842no—
517C2×C4—2844no—
518C2×C4—2844no—
519C2×C4—2848no—
520C2×C4—2844no—
521C2×C4—5688no—
522C2×C4—1424no—
523C2×C4—1422no—
524C2×C4—1424no—
525C2×C4C2×(C2×D71)⋊C414yes—
526C2×C4C2×(C2×D71)⋊C414yes—
527C22—24no—
528C22—28no—
529C22—28no—
530C22—24no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—42no—
539C22—44no—
540C22—42no—
541C22—44no—
542C22—42no—
543C22—2842no—
544C22—2844no—
545C22—5684no—
546C22—5684no—
547C22—5684no—
548C22#1798587812yes—
549C22#1798587812yes—
550C22#1798587812yes—
551C22#1798587812yes—
552C22#1798597214yes—
553C22C22×(C2×D71)⋊C411yescharacteristic
554C22C2×C4×C22⋊D7112yes—
555C22C2×#1455609711yescharacteristic
556C22C2×#1455610212yes—
557C22C2×#1455611111yescharacteristic
558C4—24no—
559C4—24no—
560C4—48no—
561C4—2842no—
562C4—2844no—
563C4—5684no—
564C2—21no—
565C2—22no—
566C2—22no—
567C2—22no—
568C2—22no—
569C2—22no—
570C2—21no—
571C2—5682no—
572C2#2258879814yesdirect product factor
573C2C2×#1798587811yescharacteristic
574C2C2×#1798597212yes—
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC23×C71C23×C4
FrattiniC24C23×D71
FittingC22×#117×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC23×C22⋊D71C22
cosocleC24×C71C24
hypercenterC22×#117D71
hypocenterC71C2×#1899
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. C2×#17985862C2semidirect
4. #22588798C2direct (N × H)
5. C2×#17985765C2semidirect
6. C22×#117×C71C2semidirect
13. C2×#14556111C4semidirect
15. C22×Dic71⋊C4C4semidirect
18. C22×(C2×D71)⋊C4C4semidirect
22. #17985862C22semidirect
24. #17985765C22semidirect
28. C2×#117×C71C22semidirect
53. C2×Dic71⋊C4C2×C4semidirect
67. C2×(C2×D71)⋊C4C2×C4semidirect
284. C2×C71#1899semidirect
289. C22×#117D71semidirect
294. C71C2×#1899semidirect
306. C2×#117C2×D71semidirect
572. C2#22588798direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561112
C22×#145560422
C2×#179858622
#225887982
C2×#179857652
C22×#117×C712
C2×#189971