Finite Group Databasebeta

← C2×#22588127 #31995097

Subgroups — C2×#22588127 #31995097

Order 18176.

2242 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×(C22⋊D71)⋊C4C211yesmaximal, characteristic
2C22×#14556042C212yesmaximal
3C2×C22⋊C4×Dic71C212yesmaximal, complemented
4#22588127C218yesmaximal, direct product factor
5C2×#17985766C211yesmaximal, characteristic, complemented
6C2×#342×C71C211yesmaximal, characteristic, complemented
7C23×C22⋊D71C2211yescharacteristic
8C24×Dic71C2211yescharacteristic
9C23×C4×D71C411yescharacteristic, complemented
10C2×(C22⋊D71)⋊C4—22no—
11C2×(C22⋊D71)⋊C4—22no—
12C2×(C22⋊D71)⋊C4—24no—
13C2×(C22⋊D71)⋊C4C414yescomplemented
14C2×(C22⋊D71)⋊C4C2214yes—
15C22×Dic71⋊C4C411yescharacteristic, complemented
16C22×C22⋊Dic71C2212yes—
17C22×(C2×D71)⋊C4C411yescharacteristic, complemented
18C22×C4×Dic71—22no—
19C22×C4×Dic71C411yescharacteristic, complemented
20C2×#14556042C4116yescomplemented
21C2×#14556042C2218yes—
22C22⋊C4×Dic71—24no—
23C22⋊C4×Dic71C2218yescomplemented
24C2×C71⋊#95—22no—
25#17985766C2214yescomplemented
26C22×C22⋊C4×C71C2211yescharacteristic
27C22×C22⋊C4×C71C2212yes—
28#342×C71C2214yescomplemented
29C24×D71C2×C411yescharacteristic
30C22×C22⋊D71—22no—
31C22×C22⋊D71—22no—
32C22×C22⋊D71—22no—
33C22×C22⋊D71—22no—
34C22×C22⋊D71C2×C418yes—
35C22×C22⋊D71C2×C412yes—
36C22×C22⋊D71C2312yes—
37C23×Dic71—22no—
38C23×Dic71—44no—
39C23×Dic71C2×C411yescharacteristic
40C23×Dic71C2×C411yescharacteristic
41C23×Dic71C2×C411yescharacteristic
42C23×Dic71C2×C412yes—
43C23×Dic71D412yes—
44C23×Dic71D412yes—
45C23×Dic71D414yes—
46C23×Dic71C2312yes—
47C22×C4×D71—22no—
48C22×C4×D71—21no—
49C22×C4×D71—21no—
50C22×C4×D71—44no—
51C22×C4×D71C2×C414yescomplemented
52(C22⋊D71)⋊C4—28no—
53(C22⋊D71)⋊C4—28no—
54(C22⋊D71)⋊C4—28no—
55(C22⋊D71)⋊C4—28no—
56C2×Dic71⋊C4—22no—
57C2×Dic71⋊C4—22no—
58C2×Dic71⋊C4C2×C414yescomplemented
59C2×C22⋊Dic71—22no—
60C2×C22⋊Dic71—24no—
61C2×C22⋊Dic71C2×C418yescomplemented
62C2×C22⋊Dic71C2314yes—
63C2×(C2×D71)⋊C4—22no—
64C2×(C2×D71)⋊C4—22no—
65C2×(C2×D71)⋊C4C2×C414yescomplemented
66C2×C4×Dic71—22no—
67C2×C4×Dic71—21no—
68C2×C4×Dic71—21no—
69C2×C4×Dic71—28no—
70C2×C4×Dic71—48no—
71C2×C4×Dic71C2×C414yescomplemented
72#14556042C2×C4132yescomplemented
73C71⋊#95—28no—
74C25×C71C2311yescharacteristic
75C23×C4×C71—22no—
76C23×C4×C71C2×C411yescharacteristic
77C23×C4×C71C2×C411yescharacteristic
78C2×C22⋊C4×C71—22no—
79C2×C22⋊C4×C71—22no—
80C2×C22⋊C4×C71—22no—
81C2×C22⋊C4×C71—24no—
82C2×C22⋊C4×C71C2×C412yes—
83C2×C22⋊C4×C71C2×C418yescomplemented
84C2×C22⋊C4×C71C2312yes—
85C2×C22⋊C4×C71C2314yes—
86C23×D71—21no—
87C23×D71—21no—
88C23×D71—22no—
89C23×D71—42no—
90C23×D71—42no—
91C23×D71C4214yescomplemented
92C23×D71C22×C412yes—
93C2×C22⋊D71—24no—
94C2×C22⋊D71—24no—
95C2×C22⋊D71—28no—
96C2×C22⋊D71—28no—
97C2×C22⋊D71—28no—
98C2×C22⋊D71—28no—
99C2×C22⋊D71—24no—
100C2×C22⋊D71—22no—
101C2×C22⋊D71—22no—
102C2×C22⋊D71C4218yescomplemented
103C2×C22⋊D71C22×C418yes—
104C22×Dic71—21no—
105C22×Dic71—21no—
106C22×Dic71—22no—
107C22×Dic71—24no—
108C22×Dic71—28no—
109C22×Dic71—28no—
110C22×Dic71—24no—
111C22×Dic71—44no—
112C22×Dic71—48no—
113C22×Dic71—44no—
114C22×Dic71—42no—
115C22×Dic71—44no—
116C22×Dic71—44no—
117C22×Dic71—44no—
118C22×Dic71C4214yescomplemented
119C22×Dic71C22⋊C412yes—
120C22×Dic71C22⋊C411yescharacteristic
121C22×Dic71C22⋊C412yes—
122C22×Dic71C22⋊C414yes—
123C22×Dic71C22⋊C412yes—
124C22×Dic71C22⋊C414yes—
125C22×Dic71C22⋊C411yescharacteristic
126C22×Dic71C22×C412yes—
127C22×Dic71C22×C412yes—
128C22×Dic71C22×C412yes—
129C22×Dic71C2×D414yes—
130C22×Dic71C2×D414yes—
131C2×C4×D71—24no—
132C2×C4×D71—24no—
133C2×C4×D71—44no—
134C2×C4×D71—48no—
135C2×C4×D71—44no—
136C2×C4×D71—48no—
137Dic71⋊C4—28no—
138C22⋊Dic71—28no—
139C22⋊Dic71—24no—
140C22⋊Dic71C22×C418yescomplemented
141(C2×D71)⋊C4—28no—
142C4×Dic71—24no—
143C4×Dic71—24no—
144C4×Dic71—416no—
145C24×C71—42no—
146C24×C71—42no—
147C24×C71C22×C411yescharacteristic
148C24×C71C22×C411yescharacteristic
149C24×C71C22×C414yes—
150C24×C71C2×D412yes—
151C24×C71C2×D412yes—
152C24×C71#4112yes—
153C24×C71#4112yes—
154C24×C71C2411yescharacteristic
155C22×C4×C71—21no—
156C22×C4×C71—21no—
157C22×C4×C71—21no—
158C22×C4×C71—21no—
159C22×C4×C71—28no—
160C22×C4×C71—24no—
161C22×C4×C71—44no—
162C22×C4×C71—44no—
163C22×C4×C71—42no—
164C22×C4×C71—42no—
165C22×C4×C71C22×C412yes—
166C22×C4×C71C22×C412yes—
167C22⋊C4×C71—24no—
168C22⋊C4×C71—28no—
169C22⋊C4×C71—24no—
170C22⋊C4×C71—24no—
171C22⋊C4×C71C22×C418yescomplemented
172C23×C71—22no—
173C23×C71—24no—
174C23×C71—22no—
175C23×C71—24no—
176C23×C71—41no—
177C23×C71—48no—
178C23×C71—41no—
179C23×C71—42no—
180C23×C71—42no—
181C23×C71—44no—
182C23×C71—44no—
183C23×C71—41no—
184C23×C71—41no—
185C23×C71C2×C4212yes—
186C23×C71C2×C4212yes—
187C23×C71C2×C22⋊C412yes—
188C23×C71C2×C22⋊C411yescharacteristic
189C23×C71C2×C22⋊C411yescharacteristic
190C23×C71#11712yes—
191C23×C71#11712yes—
192C23×C71C4×D414yes—
193C23×C71C4×D414yes—
194C23×C71C4×D414yes—
195C23×C71C4×D414yes—
196C23×C71C23×C411yescharacteristic
197C23×C71C23×C412yes—
198C23×C71C22×D412yes—
199C23×C71C2×#4111yescharacteristic
200C23×C71C2×#4111yescharacteristic
201C22×D71—22no—
202C22×D71—22no—
203C22×D71—24no—
204C22×D71—24no—
205C22×D71—42no—
206C22×D71—42no—
207C22×D71—48no—
208C22×D71—42no—
209C22×D71—42no—
210C22×D71—42no—
211C22×D71—48no—
212C22×D71—42no—
213C22×D71C2×C4214yescomplemented
214C2×C4×C71—22no—
215C2×C4×C71—22no—
216C2×C4×C71—28no—
217C2×C4×C71—44no—
218C2×C4×C71—416no—
219C2×C4×C71—42no—
220C2×C4×C71—44no—
221C2×C4×C71—42no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71—42no—
226C22⋊D71—28no—
227C22⋊D71—28no—
228C22⋊D71—216no—
229C2×Dic71—22no—
230C2×Dic71—22no—
231C2×Dic71—24no—
232C2×Dic71—24no—
233C2×Dic71—22no—
234C2×Dic71—24no—
235C2×Dic71—28no—
236C2×Dic71—24no—
237C2×Dic71—24no—
238C2×Dic71—28no—
239C2×Dic71—28no—
240C2×Dic71—44no—
241C2×Dic71—42no—
242C2×Dic71—42no—
243C2×Dic71—44no—
244C2×Dic71—44no—
245C2×Dic71—44no—
246C2×Dic71—416no—
247C2×Dic71#9914yescomplemented
248C2×Dic71C2×C4214yescomplemented
249C2×Dic71C2×C22⋊C412yes—
250C2×Dic71C2×C22⋊C414yes—
251C2×Dic71C2×C22⋊C412yes—
252C4×D71—48no—
253C4×D71—48no—
254C22×C71—22no—
255C22×C71—21no—
256C22×C71—21no—
257C22×C71—22no—
258C22×C71—22no—
259C22×C71—21no—
260C22×C71—21no—
261C22×C71—22no—
262C22×C71—28no—
263C22×C71—28no—
264C22×C71—24no—
265C22×C71—24no—
266C22×C71—42no—
267C22×C71—42no—
268C22×C71—44no—
269C22×C71—44no—
270C22×C71—42no—
271C22×C71—41no—
272C22×C71—41no—
273C22×C71C4×C22⋊C412yes—
274C22×C71C4×C22⋊C414yes—
275C22×C71C4×C22⋊C412yes—
276C22×C71C2×#9912yes—
277C22×C71#41012yes—
278C22×C71C22×C4211yescharacteristic
279C22×C71C22×C22⋊C411yescharacteristic
280C22×C71C2×#11711yescharacteristic
281C22×C71C2×C4×D412yes—
282C22×C71C2×C4×D412yes—
283C2×D71—24no—
284C2×D71—44no—
285C2×D71—48no—
286C2×D71—42no—
287C2×D71—42no—
288C2×D71—48no—
289C2×D71—44no—
290C4×C71—48no—
291C4×C71—42no—
292C4×C71—42no—
293Dic71—22no—
294Dic71—24no—
295Dic71—24no—
296Dic71—48no—
297Dic71C2×#9914yescomplemented
298C2×C4×#99—711nomaximal, Sylow
299C2×C71—22no—
300C2×C71—22no—
301C2×C71—21no—
302C2×C71—22no—
303C2×C71—21no—
304C2×C71—24no—
305C2×C71C4×#9914yescomplemented
306C2×C71C2×C4×C22⋊C411yescharacteristic
307C2×C71C22×#9911yescharacteristic
308C2×C71C2×#41011yescharacteristic
309D71—44no—
310C22×C4×D4—711no—
311C22×#99—712no—
312C2×C4×C22⋊C4—712no—
313C4×#99—718no—
314C2×#342—711no—
315C2×#342D7111yescharacteristic, complemented
316C71C2×C4×#9911yescharacteristic, complemented, Sylow
317C23×D4—711no—
318C24×C4—711no—
319C24×C4—711no—
320C2×C4×D4—714no—
321C2×C4×D4—714no—
322C2×C4×D4—1422no—
323C2×C4×D4—1424no—
324C2×C4×D4—1422no—
325C22×C4⋊C4—711no—
326C22×C22⋊C4—712no—
327C22×C22⋊C4—711no—
328C22×C22⋊C4C2×D7111yescharacteristic
329C22×C22⋊C4C2×D7112yes—
330C22×C42—711no—
331C22×C42—1422no—
332C2×#99—7116no—
333C2×#99—718no—
334C4×C22⋊C4—718no—
335C4×C22⋊C4—1424no—
336C2×#95—1422no—
337#342—714no—
338#342C2×D7114yescomplemented
339C25—711no—
340C25C22×D7111yescharacteristic
341C22×D4—712no—
342C22×D4—718no—
343C22×D4—712no—
344C22×D4—1422no—
345C22×D4—1422no—
346C22×D4—1422no—
347C22×D4—1422no—
348C23×C4—22no—
349C23×C4—2844no—
350C23×C4—2844no—
351C23×C4—711no—
352C23×C4—712no—
353C23×C4—712no—
354C23×C4—714no—
355C23×C4—712no—
356C23×C4—714no—
357C23×C4—711no—
358C23×C4—711no—
359C23×C4—712no—
360C23×C4—1422no—
361C23×C4—1421no—
362C23×C4—1421no—
363C23×C4—1422no—
364C23×C4C4×D7111yescharacteristic
365C23×C4C4×D7111yescharacteristic
366C4×D4—1428no—
367C4×D4—1428no—
368C4×D4—1428no—
369C4×D4—1428no—
370C2×C4⋊C4—714no—
371C2×C4⋊C4—1422no—
372C2×C4⋊C4—1422no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—718no—
378C2×C22⋊C4—714no—
379C2×C22⋊C4—714no—
380C2×C22⋊C4—1422no—
381C2×C22⋊C4—1422no—
382C2×C22⋊C4—1422no—
383C2×C22⋊C4—1424no—
384C2×C22⋊C4C4×D7112yes—
385C2×C22⋊C4C4×D7118yescomplemented
386C2×C22⋊C4C22×D7112yes—
387C2×C22⋊C4C22×D7114yes—
388C2×C42—2848no—
389C2×C42—714no—
390C2×C42—1428no—
391C2×C42—1422no—
392C2×C42—1421no—
393C2×C42—1421no—
394#99—7132no—
395#95—1428no—
396C24—42no—
397C24—42no—
398C24—2842no—
399C24—2842no—
400C24—712no—
401C24—714no—
402C24—1421no—
403C24—1421no—
404C24—1422no—
405C24C2×C4×D7111yescharacteristic
406C24C2×C4×D7111yescharacteristic
407C24C2×C4×D7114yes—
408C24D4×D7112yes—
409C24D4×D7112yes—
410C24#1176586412yes—
411C24#1176586612yes—
412C24C23×D7111yescharacteristic
413C2×D4—718no—
414C2×D4—718no—
415C2×D4—1424no—
416C2×D4—1422no—
417C2×D4—1422no—
418C2×D4—1424no—
419C2×D4—1424no—
420C2×D4—1428no—
421C2×D4—1428no—
422C2×D4—1428no—
423C2×D4—1428no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—28no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—2844no—
435C22×C4—2848no—
436C22×C4—2844no—
437C22×C4—2848no—
438C22×C4—2844no—
439C22×C4—2844no—
440C22×C4—2844no—
441C22×C4—2842no—
442C22×C4—2844no—
443C22×C4—2848no—
444C22×C4—2844no—
445C22×C4—712no—
446C22×C4—714no—
447C22×C4—712no—
448C22×C4—714no—
449C22×C4—711no—
450C22×C4—712no—
451C22×C4—712no—
452C22×C4—714no—
453C22×C4—714no—
454C22×C4—712no—
455C22×C4—714no—
456C22×C4—711no—
457C22×C4—712no—
458C22×C4—1424no—
459C22×C4—1424no—
460C22×C4—1424no—
461C22×C4—1428no—
462C22×C4—1428no—
463C22×C4—1424no—
464C22×C4—1421no—
465C22×C4—1421no—
466C22×C4—1422no—
467C22×C4C2×C4×D7112yes—
468C22×C4C2×C4×D7112yes—
469C4⋊C4—1428no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—28no—
474C22⋊C4—718no—
475C22⋊C4—1428no—
476C22⋊C4—1424no—
477C22⋊C4—1428no—
478C22⋊C4C2×C4×D7118yescomplemented
479C42—28416no—
480C42—1424no—
481C42—1424no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—41no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—48no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—41no—
495C23—2842no—
496C23—2842no—
497C23—2848no—
498C23—2842no—
499C23—2842no—
500C23—2848no—
501C23—2842no—
502C23—2842no—
503C23—714no—
504C23—1424no—
505C23—1424no—
506C23—1422no—
507C23—1422no—
508C23C42×D7112yes—
509C23C42×D7112yes—
510C23C22⋊C4×D7112yes—
511C23C22⋊C4×D7111yescharacteristic
512C23C22⋊C4×D7111yescharacteristic
513C23#1455608612yes—
514C23#1455608612yes—
515C23(C22⋊D71)⋊C414yes—
516C23(C22⋊D71)⋊C414yes—
517C23#1455609014yes—
518C23#1455609014yes—
519C23C22×C4×D7111yescharacteristic
520C23C22×C4×D7112yes—
521C23C2×D4×D7112yes—
522C23C2×#1176586411yescharacteristic
523C23C2×#1176586611yescharacteristic
524D4—14216no—
525D4—1428no—
526D4—1428no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—416no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—2848no—
540C2×C4—2848no—
541C2×C4—2844no—
542C2×C4—2844no—
543C2×C4—28416no—
544C2×C4—2844no—
545C2×C4—2844no—
546C2×C4—2842no—
547C2×C4—2842no—
548C2×C4—712no—
549C2×C4—714no—
550C2×C4—714no—
551C2×C4—712no—
552C2×C4—714no—
553C2×C4—1428no—
554C2×C4—1428no—
555C2×C4—1424no—
556C2×C4—1424no—
557C2×C4—1428no—
558C2×C4—1424no—
559C2×C4—1424no—
560C2×C4—1422no—
561C2×C4—1422no—
562C2×C4—1422no—
563C2×C4—1424no—
564C22—22no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—22no—
569C22—21no—
570C22—21no—
571C22—22no—
572C22—28no—
573C22—28no—
574C22—24no—
575C22—24no—
576C22—42no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—44no—
582C22—44no—
583C22—2844no—
584C22—2848no—
585C22—2844no—
586C22—2842no—
587C22—2842no—
588C22—2848no—
589C22—1424no—
590C22#1798584612yes—
591C22#1798584614yes—
592C22#1798584612yes—
593C22#99×D7112yes—
594C22#1798597112yes—
595C22C2×C42×D7111yescharacteristic
596C22C2×C22⋊C4×D7111yescharacteristic
597C22C2×#1455608611yescharacteristic
598C22C2×(C22⋊D71)⋊C412yes—
599C22C2×#1455609012yes—
600C4—48no—
601C4—42no—
602C4—42no—
603C4—2848no—
604C4—714no—
605C4—1422no—
606C4—1424no—
607C4—1424no—
608C2—22no—
609C2—22no—
610C2—22no—
611C2—21no—
612C2—21no—
613C2—24no—
614C2—2844no—
615C2#2258812714yesdirect product factor
616C2C2×#1798584611yescharacteristic
617C2C2×#99×D7111yescharacteristic
618C2C2×#1798597111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D71
commutatorC22×C71C22×C42
FrattiniC24C23×D71
FittingC2×#342×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC23×C22⋊D71C22
cosocleC24×C71C24
hypercenterC2×#342D71
hypocenterC71C2×C4×#99
LCSG ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C25 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
3. C2×C22⋊C4×Dic71C2semidirect
4. #22588127C2direct (N × H)
5. C2×#17985766C2semidirect
6. C2×#342×C71C2semidirect
9. C23×C4×D71C4semidirect
13. C2×(C22⋊D71)⋊C4C4semidirect
15. C22×Dic71⋊C4C4semidirect
17. C22×(C2×D71)⋊C4C4semidirect
19. C22×C4×Dic71C4semidirect
20. C2×#14556042C4semidirect
23. C22⋊C4×Dic71C22semidirect
25. #17985766C22semidirect
28. #342×C71C22semidirect
51. C22×C4×D71C2×C4semidirect
58. C2×Dic71⋊C4C2×C4semidirect
61. C2×C22⋊Dic71C2×C4semidirect
65. C2×(C2×D71)⋊C4C2×C4semidirect
71. C2×C4×Dic71C2×C4semidirect
72. #14556042C2×C4semidirect
83. C2×C22⋊C4×C71C2×C4semidirect
91. C23×D71C42semidirect
102. C2×C22⋊D71C42semidirect
118. C22×Dic71C42semidirect
140. C22⋊Dic71C22×C4semidirect
171. C22⋊C4×C71C22×C4semidirect
213. C22×D71C2×C42semidirect
247. C2×Dic71#99semidirect
248. C2×Dic71C2×C42semidirect
297. Dic71C2×#99semidirect
305. C2×C71C4×#99semidirect
315. C2×#342D71semidirect
316. C71C2×C4×#99semidirect
338. #342C2×D71semidirect
385. C2×C22⋊C4C4×D71semidirect
478. C22⋊C4C2×C4×D71semidirect
615. C2#22588127direct (N × H)

Maximal subgroups

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

subgroupindex
C22×(C22⋊D71)⋊C42
C22×#145560422
C2×C22⋊C4×Dic712
#225881272
C2×#179857662
C2×#342×C712
C2×C4×#9971