Finite Group Databasebeta

← #31921167

Subgroups — #31921167

Order 18176.

2006 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985872C211yesmaximal, characteristic
2C2×#17985872C211yesmaximal, characteristic
3C2×#17985872C211yesmaximal, characteristic
4C2×C4×C22⋊Dic71C211yesmaximal, characteristic
5#22586595C212yesmaximal, complemented
6#22586591C212yesmaximal, complemented
7#22586575C212yesmaximal, complemented
8Dic71⋊#95C212yesmaximal, complemented
9#22586563C211yesmaximal, characteristic
10#22586563C211yesmaximal, characteristic
11C22×#95×C71C211yesmaximal, characteristic
12C22×Dic71⋊C4C2211yescharacteristic
13C22×Dic71⋊C4C2212yes—
14C22×Dic71⋊C4C2211yescharacteristic
15C22×Dic71⋊C4C2212yes—
16C22×C22⋊Dic71C2211yescharacteristic
17C22×C22⋊Dic71C2211yescharacteristic
18C22×C22⋊Dic71C2211yescharacteristic
19C22×C22⋊Dic71C2211yescharacteristic
20C22×C4×Dic71C2212yes—
21#17985872—24no—
22#17985872—24no—
23#17985872C418yescomplemented
24C4×C22⋊Dic71—24no—
25C2×C71⋊#95C2211yescharacteristic
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2211yescharacteristic
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2212yes—
31C2×C71⋊#95C2212yes—
32C2×C71⋊#95C2212yes—
33C2×C71⋊#95C2212yes—
34C24×C4×C71C2211yescharacteristic
35C24×C4×C71C2211yescharacteristic
36C24×C4×C71C2211yescharacteristic
37C2×#95×C71—24no—
38C2×#95×C71C2212yes—
39C2×#95×C71C2212yes—
40C23×Dic71C2311yescharacteristic
41C23×Dic71C2311yescharacteristic
42C23×Dic71C2311yescharacteristic
43C23×Dic71C2311yescharacteristic
44C23×Dic71C2312yes—
45C23×Dic71C2312yes—
46C2×Dic71⋊C4—24no—
47C2×Dic71⋊C4—22no—
48C2×Dic71⋊C4—22no—
49C2×Dic71⋊C4—24no—
50C2×Dic71⋊C4—24no—
51C2×Dic71⋊C4—42no—
52C2×Dic71⋊C4—42no—
53C2×Dic71⋊C4—42no—
54C2×Dic71⋊C4—42no—
55C2×Dic71⋊C4C2×C418yescomplemented
56C2×C22⋊Dic71—22no—
57C2×C22⋊Dic71—22no—
58C2×C22⋊Dic71—22no—
59C2×C22⋊Dic71—22no—
60C2×C22⋊Dic71—22no—
61C2×C22⋊Dic71—22no—
62C2×C22⋊Dic71—22no—
63C2×C22⋊Dic71C2×C414yes—
64C2×C22⋊Dic71C2×C414yes—
65C2×C22⋊Dic71D414yes—
66C2×C22⋊Dic71Q814yes—
67C2×C22⋊Dic71Q814yes—
68C2×C4×Dic71—24no—
69C2×C4×Dic71—24no—
70C2×C4×Dic71—42no—
71C71⋊#95—24no—
72C71⋊#95—24no—
73C71⋊#95—22no—
74C71⋊#95—22no—
75C71⋊#95—24no—
76C71⋊#95—42no—
77C71⋊#95—42no—
78C71⋊#95—42no—
79C71⋊#95C2×C418yescomplemented
80C25×C71C2311yescharacteristic
81C23×C4×C71—22no—
82C23×C4×C71—22no—
83C23×C4×C71—22no—
84C23×C4×C71—22no—
85C23×C4×C71—22no—
86C23×C4×C71—24no—
87C23×C4×C71—24no—
88C23×C4×C71—22no—
89C23×C4×C71—22no—
90C23×C4×C71—42no—
91C23×C4×C71—42no—
92C23×C4×C71—42no—
93C23×C4×C71—42no—
94C23×C4×C71C2×C414yes—
95C23×C4×C71D414yes—
96C23×C4×C71C2312yes—
97C23×C4×C71C2311yescharacteristic
98C23×C4×C71C2312yes—
99C23×C4×C71C2311yescharacteristic
100#95×C71—24no—
101#95×C71—24no—
102C22×Dic71—22no—
103C22×Dic71—22no—
104C22×Dic71—22no—
105C22×Dic71—22no—
106C22×Dic71—22no—
107C22×Dic71—21no—
108C22×Dic71—21no—
109C22×Dic71—22no—
110C22×Dic71—22no—
111C22×Dic71—22no—
112C22×Dic71—21no—
113C22×Dic71—21no—
114C22×Dic71—22no—
115C22×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—21no—
121C22×Dic71—21no—
122C22×Dic71—21no—
123C22×Dic71—21no—
124C22×Dic71—21no—
125C22×Dic71—21no—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—22no—
129C22×Dic71—22no—
130C22×Dic71C22×C412yes—
131C22×Dic71C22×C412yes—
132C22×Dic71C22×C414yes—
133C22×Dic71C2×D414yes—
134C22×Dic71C2×Q814yes—
135C22×Dic71C2×Q814yes—
136Dic71⋊C4—48no—
137Dic71⋊C4—48no—
138Dic71⋊C4—44no—
139Dic71⋊C4—44no—
140C22⋊Dic71—28no—
141C22⋊Dic71—28no—
142C22⋊Dic71—44no—
143C22⋊Dic71C4⋊C4116yescomplemented
144C4×Dic71—48no—
145C24×C71—22no—
146C24×C71—22no—
147C24×C71—22no—
148C24×C71—22no—
149C24×C71C22×C412yes—
150C24×C71C2×D412yes—
151C24×C71C2×D412yes—
152C24×C71C2×Q812yes—
153C24×C71C2×Q812yes—
154C24×C71#4112yes—
155C24×C71#4112yes—
156C24×C71C2411yescharacteristic
157C22×C4×C71—21no—
158C22×C4×C71—21no—
159C22×C4×C71—22no—
160C22×C4×C71—28no—
161C22×C4×C71—28no—
162C22×C4×C71—28no—
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—21no—
166C22×C4×C71—22no—
167C22×C4×C71—21no—
168C22×C4×C71—22no—
169C22×C4×C71—22no—
170C22×C4×C71—22no—
171C22×C4×C71—22no—
172C22×C4×C71—22no—
173C22×C4×C71—22no—
174C22×C4×C71—22no—
175C22×C4×C71—22no—
176C22×C4×C71—22no—
177C22×C4×C71—22no—
178C22×C4×C71—22no—
179C22×C4×C71—28no—
180C22×C4×C71—48no—
181C22×C4×C71—44no—
182C22×C4×C71—48no—
183C22×C4×C71—48no—
184C22×C4×C71—48no—
185C22×C4×C71—48no—
186C22×C4×C71—44no—
187C22×C4×C71—44no—
188C22×C4×C71—44no—
189C22×C4×C71—48no—
190C22×C4×C71—44no—
191C22×C4×C71C22×C414yes—
192C22×C4×C71C2×D412yes—
193C22×C4×C71C2×D412yes—
194C22×C4×C71#4114yes—
195C22×C4×C71#4114yes—
196C23×C71—24no—
197C23×C71—24no—
198C23×C71—22no—
199C23×C71—22no—
200C23×C71—24no—
201C23×C71—22no—
202C23×C71—22no—
203C23×C71—24no—
204C23×C71—24no—
205C23×C71—24no—
206C23×C71—24no—
207C23×C71—24no—
208C23×C71—24no—
209C23×C71—24no—
210C23×C71—24no—
211C23×C71—24no—
212C23×C71C2×C4⋊C414yes—
213C23×C71C4×D414yes—
214C23×C71C4×Q814yes—
215C23×C71#12114yes—
216C23×C71C22⋊Q814yes—
217C23×C71#12514yes—
218C23×C71C4⋊Q814yes—
219C23×C71C23×C411yescharacteristic
220C23×C71C22×D411yescharacteristic
221C23×C71C22×D411yescharacteristic
222C23×C71C22×Q811yescharacteristic
223C23×C71C22×Q811yescharacteristic
224C23×C71C2×#4111yescharacteristic
225C23×C71C2×#4111yescharacteristic
226C23×C71C2×#4112yes—
227C23×C71C2×#4112yes—
228C23×C71H(5,𝔽2)12yes—
229C23×C71H(5,𝔽2)12yes—
230C2×C4×C71—22no—
231C2×C4×C71—22no—
232C2×C4×C71—24no—
233C2×C4×C71—24no—
234C2×C4×C71—24no—
235C2×C4×C71—24no—
236C2×C4×C71—24no—
237C2×C4×C71—22no—
238C2×C4×C71—22no—
239C2×C4×C71—24no—
240C2×C4×C71—22no—
241C2×C4×C71—24no—
242C2×C4×C71—24no—
243C2×C4×C71—416no—
244C2×C4×C71—416no—
245C2×C4×C71—416no—
246C2×C4×C71—42no—
247C2×C4×C71—42no—
248C2×C4×C71—42no—
249C2×C4×C71—42no—
250C2×C4×C71—42no—
251C2×C4×C71—42no—
252C2×C4×C71—42no—
253C2×C4×C71—42no—
254C2×C4×C71—42no—
255C2×C4×C71—42no—
256C2×C4×C71#12314yes—
257C2×Dic71—24no—
258C2×Dic71—24no—
259C2×Dic71—24no—
260C2×Dic71—24no—
261C2×Dic71—24no—
262C2×Dic71—24no—
263C2×Dic71—22no—
264C2×Dic71—22no—
265C2×Dic71—24no—
266C2×Dic71—24no—
267C2×Dic71—24no—
268C2×Dic71—24no—
269C2×Dic71—24no—
270C2×Dic71—41no—
271C2×Dic71—41no—
272C2×Dic71—42no—
273C2×Dic71—42no—
274C2×Dic71—41no—
275C2×Dic71—41no—
276C2×Dic71—42no—
277C2×Dic71—42no—
278C2×Dic71—42no—
279C2×Dic71—42no—
280C2×Dic71—41no—
281C2×Dic71—41no—
282C2×Dic71—41no—
283C2×Dic71—41no—
284C2×Dic71—41no—
285C2×Dic71—41no—
286C2×Dic71—42no—
287C2×Dic71—42no—
288C2×Dic71—42no—
289C2×Dic71—42no—
290C2×Dic71C2×C4⋊C418yes—
291C22×C71—24no—
292C22×C71—24no—
293C22×C71—28no—
294C22×C71—28no—
295C22×C71—24no—
296C22×C71—24no—
297C22×C71—28no—
298C22×C71—24no—
299C22×C71—28no—
300C22×C71—24no—
301C22×C71#38918yes—
302C22×C71C22×C4⋊C411yescharacteristic
303C22×C71C2×C4×D411yescharacteristic
304C22×C71C2×C4×Q811yescharacteristic
305C22×C71C4×#4112yes—
306C22×C71#51812yes—
307C22×C71C2×#12111yescharacteristic
308C22×C71C2×C22⋊Q811yescharacteristic
309C22×C71C2×#12311yescharacteristic
310C22×C71#52511yescharacteristic
311C22×C71C2×#12511yescharacteristic
312C22×C71#52912yes—
313C22×C71C2×C4⋊Q811yescharacteristic
314C22×C71#53411yescharacteristic
315C22×C71#53511yescharacteristic
316C22×C71#53812yes—
317C22×C71#54612yes—
318C22×C71#54612yes—
319C22×C71#55112yes—
320C22×C71#55112yes—
321C22×C71#55412yes—
322C22×C71#55412yes—
323C22×C71#55412yes—
324C22×C71#55412yes—
325C4×C71—24no—
326C4×C71—44no—
327C4×C71—44no—
328C4×C71—42no—
329Dic71—28no—
330Dic71—42no—
331Dic71—44no—
332Dic71—42no—
333Dic71—81no—
334Dic71—81no—
335#14044—711nomaximal, Sylow
336C2×C71—28no—
337C2×C71C2×#38911yescharacteristic
338C2×C71C4×#12311yescharacteristic
339C2×C71#232111yescharacteristic
340C2×C71#234912yes—
341C2×C71#236812yes—
342C2×C71#241111yescharacteristic
343C2×C71#241211yescharacteristic
344C2×C71#245012yes—
345C2×C71#250211yescharacteristic
346C2×C71#250512yes—
347C2×C71#258311yescharacteristic
348C2×#385—711no—
349C2×#385—711no—
350C2×#385—711no—
351C2×C4×C22⋊C4—711no—
352C22×#95D7111yescharacteristic
353#1442—712no—
354#1441—712no—
355#1436—712no—
356C4⋊#95—712no—
357C22⋊#95—711no—
358C22⋊#95—711no—
359C71#1404411yescharacteristic, complemented, Sylow
360C24×C4C2×D7111yescharacteristic
361C24×C4C2×D7111yescharacteristic
362C24×C4C2×D7111yescharacteristic
363C22×C4⋊C4—712no—
364C22×C4⋊C4—711no—
365C22×C4⋊C4—712no—
366C22×C4⋊C4—711no—
367C22×C22⋊C4—711no—
368C22×C22⋊C4—711no—
369C22×C22⋊C4—711no—
370C22×C22⋊C4—711no—
371C22×C42—712no—
372#385—718no—
373#385—1424no—
374#385—1424no—
375C4×C22⋊C4—1424no—
376C2×#95—24no—
377C2×#95—712no—
378C2×#95—712no—
379C2×#95—712no—
380C2×#95—712no—
381C2×#95—712no—
382C2×#95—711no—
383C2×#95—712no—
384C2×#95—712no—
385C2×#95—711no—
386C2×#95C2×D7112yes—
387C2×#95C2×D7112yes—
388C25C22×D7111yescharacteristic
389C23×C4—22no—
390C23×C4—22no—
391C23×C4—22no—
392C23×C4—22no—
393C23×C4—22no—
394C23×C4—24no—
395C23×C4—24no—
396C23×C4—22no—
397C23×C4—22no—
398C23×C4—42no—
399C23×C4—42no—
400C23×C4—42no—
401C23×C4—42no—
402C23×C4—712no—
403C23×C4—711no—
404C23×C4—711no—
405C23×C4—712no—
406C23×C4—711no—
407C23×C4—711no—
408C23×C4C4×D7114yes—
409C23×C4C22⋊D7114yes—
410C23×C4C22×D7112yes—
411C23×C4C22×D7111yescharacteristic
412C23×C4C22×D7112yes—
413C23×C4C22×D7111yescharacteristic
414C2×C4⋊C4—2842no—
415C2×C4⋊C4—2842no—
416C2×C4⋊C4—2842no—
417C2×C4⋊C4—2842no—
418C2×C4⋊C4—718no—
419C2×C4⋊C4—1424no—
420C2×C4⋊C4—1422no—
421C2×C4⋊C4—1422no—
422C2×C4⋊C4—1424no—
423C2×C4⋊C4—1424no—
424C2×C22⋊C4—714no—
425C2×C22⋊C4—714no—
426C2×C22⋊C4—714no—
427C2×C22⋊C4—714no—
428C2×C22⋊C4—714no—
429C2×C22⋊C4—1422no—
430C2×C22⋊C4—1422no—
431C2×C22⋊C4—1422no—
432C2×C22⋊C4—1422no—
433C2×C22⋊C4—1422no—
434C2×C22⋊C4—1422no—
435C2×C22⋊C4—1422no—
436C2×C42—2842no—
437C2×C42—1424no—
438C2×C42—1424no—
439#95—24no—
440#95—24no—
441#95—2842no—
442#95—2842no—
443#95—2842no—
444#95—718no—
445#95—1424no—
446#95—1424no—
447#95—1422no—
448#95—1422no—
449#95—1424no—
450C24—22no—
451C24—22no—
452C24—22no—
453C24—22no—
454C24C2×C4×D7112yes—
455C24D4×D7112yes—
456C24C2×C22⋊D7112yes—
457C24Q8×D7112yes—
458C24Q8×D7112yes—
459C24#1176586412yes—
460C24#1176586512yes—
461C24C23×D7111yescharacteristic
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—28no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—28no—
483C22×C4—28no—
484C22×C4—28no—
485C22×C4—48no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—48no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—714no—
497C22×C4—714no—
498C22×C4—714no—
499C22×C4—712no—
500C22×C4—712no—
501C22×C4—714no—
502C22×C4—1421no—
503C22×C4—1421no—
504C22×C4—1421no—
505C22×C4—1421no—
506C22×C4—1421no—
507C22×C4—1421no—
508C22×C4—1422no—
509C22×C4—1422no—
510C22×C4—1422no—
511C22×C4—1422no—
512C22×C4—1422no—
513C22×C4—1421no—
514C22×C4—1421no—
515C22×C4—1422no—
516C22×C4—1422no—
517C22×C4—1422no—
518C22×C4—1421no—
519C22×C4—1422no—
520C22×C4—1421no—
521C22×C4—1422no—
522C22×C4—1422no—
523C22×C4—1422no—
524C22×C4—1422no—
525C22×C4—1422no—
526C22×C4—1422no—
527C22×C4—1422no—
528C22×C4—1422no—
529C22×C4—1422no—
530C22×C4C2×C4×D7114yes—
531C22×C4C2×C22⋊D7112yes—
532C22×C4C2×C22⋊D7112yes—
533C22×C4#1176586514yes—
534C22×C4#1176586514yes—
535C4⋊C4—2844no—
536C4⋊C4—2848no—
537C4⋊C4—2848no—
538C4⋊C4—2844no—
539C22⋊C4—2844no—
540C22⋊C4—7116no—
541C22⋊C4—1428no—
542C22⋊C4—1428no—
543C42—2848no—
544C23—22no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—24no—
559C23—22no—
560C23C4⋊C4×D7114yes—
561C23C4×C22⋊D7114yes—
562C23#1455609314yes—
563C23#1455610014yes—
564C23#1455610614yes—
565C23#1455611814yes—
566C23#1455612614yes—
567C23C22×C4×D7111yescharacteristic
568C23C2×D4×D7111yescharacteristic
569C23C22×C22⋊D7111yescharacteristic
570C23C2×Q8×D7111yescharacteristic
571C23C2×Q8×D7111yescharacteristic
572C23C2×#1176586411yescharacteristic
573C23C2×#1176586511yescharacteristic
574C23C2×#1176586512yes—
575C23C2×#1176586512yes—
576C23#1455617312yes—
577C23#1455617312yes—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—416no—
592C2×C4—416no—
593C2×C4—416no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—2841no—
605C2×C4—2842no—
606C2×C4—2842no—
607C2×C4—2842no—
608C2×C4—2842no—
609C2×C4—2841no—
610C2×C4—2841no—
611C2×C4—2842no—
612C2×C4—2842no—
613C2×C4—2841no—
614C2×C4—2841no—
615C2×C4—2842no—
616C2×C4—2842no—
617C2×C4—2842no—
618C2×C4—2842no—
619C2×C4—2841no—
620C2×C4—2841no—
621C2×C4—2841no—
622C2×C4—2841no—
623C2×C4—2841no—
624C2×C4—718no—
625C2×C4—1424no—
626C2×C4—1424no—
627C2×C4—1422no—
628C2×C4—1422no—
629C2×C4—1424no—
630C2×C4—1424no—
631C2×C4—1424no—
632C2×C4—1424no—
633C2×C4—1424no—
634C2×C4—1424no—
635C2×C4—1424no—
636C2×C4—1424no—
637C2×C4—1424no—
638C2×C4#1455611114yes—
639C22—28no—
640C22—24no—
641C22—24no—
642C22—28no—
643C22—24no—
644C22—28no—
645C22—24no—
646C22—28no—
647C22—24no—
648C22—24no—
649C22#1798589418yes—
650C22C2×C4⋊C4×D7111yescharacteristic
651C22C2×C4×C22⋊D7111yescharacteristic
652C22C2×#1455609311yescharacteristic
653C22C4×#1176586512yes—
654C22#1798647212yes—
655C22C2×#1455610011yescharacteristic
656C22C2×#1455610611yescharacteristic
657C22C2×#1455611111yescharacteristic
658C22#1798651011yescharacteristic
659C22C2×#1455611811yescharacteristic
660C22#1798653312yes—
661C22C2×#1455612611yescharacteristic
662C22#1798655811yescharacteristic
663C22#1798656611yescharacteristic
664C22#1798658712yes—
665C22#1798663912yes—
666C22#1798663912yes—
667C22#1798667512yes—
668C22#1798667512yes—
669C22#1798670412yes—
670C22#1798670412yes—
671C22#1798670412yes—
672C22#1798670412yes—
673C4—24no—
674C4—44no—
675C4—44no—
676C4—42no—
677C4—2842no—
678C4—2842no—
679C4—2844no—
680C4—5681no—
681C4—5681no—
682C4—1428no—
683C2—28no—
684C2C2×#1798589411yescharacteristic
685C2C4×#1455611111yescharacteristic
686C2#2259099811yescharacteristic
687C2#2259124712yes—
688C2#2259142012yes—
689C2#2259172511yescharacteristic
690C2#2259173611yescharacteristic
691C2#2259216412yes—
692C2#2259281411yescharacteristic
693C2#2259284112yes—
694C2#2259372011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC22×#95×C71C2
socle, cosocleC24×C71C24
prime radicalC25×C71C23
hypercenterC22×#95D71
hypocenterC71#14044
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
5. #22586595C2semidirect
6. #22586591C2semidirect
7. #22586575C2semidirect
8. Dic71⋊#95C2semidirect
23. #17985872C4semidirect
55. C2×Dic71⋊C4C2×C4semidirect
79. C71⋊#95C2×C4semidirect
143. C22⋊Dic71C4⋊C4semidirect
359. C71#14044semidirect

Maximal subgroups

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

subgroupindex
C2×#179858722
C2×C4×C22⋊Dic712
#225865952
#225865912
#225865752
Dic71⋊#952
#225865632
C22×#95×C712
#1404471