Finite Group Databasebeta

← #31918822

Subgroups — #31918822

Order 18176.

2194 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985901C211yesmaximal, characteristic
2C2×#17985877C212yesmaximal
3C2×C4×(C2×D71)⋊C4C212yesmaximal, complemented
4#22586578C212yesmaximal, complemented
5#22586576C212yesmaximal, complemented
6#22586574C212yesmaximal, complemented
7#22586567C211yesmaximal, characteristic, complemented
8(C2×D71)⋊#95C212yesmaximal, complemented
9C4×#95×C71C211yesmaximal, characteristic, complemented
10C23×D284C2211yescharacteristic
11C23×C4×D71C2212yes—
12C22×C4⋊Dic71C2211yescharacteristic
13C22×(C2×D71)⋊C4C2212yes—
14C22×(C2×D71)⋊C4C2212yes—
15C22×(C2×D71)⋊C4C2212yes—
16C22×(C2×D71)⋊C4C2212yes—
17C22×(C2×D71)⋊C4C2212yes—
18C22×C4×Dic71C2212yes—
19#17985901C418yescomplemented
20#17985877C4116yescomplemented
21C4×(C2×D71)⋊C4—28no—
22C2×C71⋊#95C2212yes—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2211yescharacteristic
25C2×C71⋊#95C2212yes—
26C2×C71⋊#95C2211yescharacteristic
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2212yes—
29C22×C42×C71C2211yescharacteristic
30C22×C42×C71C2211yescharacteristic
31C22×C42×C71C2211yescharacteristic
32C2×#95×C71C2212yes—
33C2×#95×C71C2212yes—
34C24×D71C2312yes—
35C22×D284—41no—
36C22×D284—41no—
37C22×D284—42no—
38C22×D284C2×C418yes—
39C22×D284C2×C414yes—
40C23×Dic71C2312yes—
41C23×Dic71C2312yes—
42C23×Dic71C2312yes—
43C22×C4×D71—24no—
44C22×C4×D71—24no—
45C22×C4×D71—42no—
46C22×C4×D71—42no—
47C22×C4×D71—42no—
48C22×C4×D71—42no—
49C22×C4×D71C2×C418yescomplemented
50C2×C4⋊Dic71—42no—
51C2×C4⋊Dic71C2×C414yescomplemented
52C2×(C2×D71)⋊C4—24no—
53C2×(C2×D71)⋊C4—24no—
54C2×(C2×D71)⋊C4—24no—
55C2×(C2×D71)⋊C4—24no—
56C2×(C2×D71)⋊C4—42no—
57C2×(C2×D71)⋊C4—42no—
58C2×(C2×D71)⋊C4—42no—
59C2×(C2×D71)⋊C4—42no—
60C2×(C2×D71)⋊C4—42no—
61C2×(C2×D71)⋊C4—42no—
62C2×(C2×D71)⋊C4C2×C418yes—
63C2×(C2×D71)⋊C4C2×C418yes—
64C2×(C2×D71)⋊C4C2×C418yescomplemented
65C2×(C2×D71)⋊C4C2×C418yescomplemented
66C2×(C2×D71)⋊C4C2×C418yescomplemented
67C2×C4×Dic71—24no—
68C2×C4×Dic71—42no—
69C2×C4×Dic71—42no—
70C71⋊#95—24no—
71C71⋊#95—24no—
72C71⋊#95—41no—
73C71⋊#95—41no—
74C71⋊#95—42no—
75C71⋊#95—42no—
76C71⋊#95C2×C418yescomplemented
77C23×C4×C71C2311yescharacteristic
78C23×C4×C71C2312yes—
79C23×C4×C71C2311yescharacteristic
80C23×C4×C71C2312yes—
81C23×C4×C71C2311yescharacteristic
82C2×C42×C71—22no—
83C2×C42×C71—41no—
84C2×C42×C71—41no—
85C2×C42×C71—41no—
86C2×C42×C71—41no—
87C2×C42×C71—41no—
88C2×C42×C71—41no—
89C2×C42×C71C2×C414yescomplemented
90C2×C42×C71D414yescomplemented
91#95×C71—24no—
92#95×C71C2×C418yescomplemented
93C23×D71—22no—
94C23×D71—22no—
95C23×D71—22no—
96C23×D71—22no—
97C23×D71—22no—
98C23×D71—22no—
99C23×D71—22no—
100C23×D71—22no—
101C23×D71—22no—
102C23×D71—22no—
103C23×D71—22no—
104C23×D71—22no—
105C23×D71C22×C414yes—
106C23×D71C22×C414yes—
107C23×D71C22×C414yes—
108C2×D284—44no—
109C2×D284—44no—
110C2×D284—48no—
111C2×D284—48no—
112C2×D284C42116yescomplemented
113C22×Dic71—22no—
114C22×Dic71—22no—
115C22×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—22no—
121C22×Dic71—22no—
122C22×Dic71—22no—
123C22×Dic71—22no—
124C22×Dic71—22no—
125C22×Dic71—22no—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—22no—
129C22×Dic71—22no—
130C22×Dic71—22no—
131C22×Dic71C22×C414yes—
132C22×Dic71C22×C414yes—
133C22×Dic71C22×C414yes—
134C2×C4×D71—216no—
135C2×C4×D71—48no—
136C2×C4×D71—48no—
137C2×C4×D71—48no—
138C2×C4×D71—48no—
139C2×C4×D71—48no—
140C2×C4×D71—48no—
141C4⋊Dic71—44no—
142(C2×D71)⋊C4—216no—
143(C2×D71)⋊C4—216no—
144(C2×D71)⋊C4—48no—
145(C2×D71)⋊C4—48no—
146(C2×D71)⋊C4—48no—
147C4×Dic71—48no—
148C24×C71C2411yescharacteristic
149C22×C4×C71—22no—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—21no—
154C22×C4×C71—21no—
155C22×C4×C71—22no—
156C22×C4×C71—21no—
157C22×C4×C71—21no—
158C22×C4×C71—21no—
159C22×C4×C71—21no—
160C22×C4×C71—22no—
161C22×C4×C71—22no—
162C22×C4×C71—22no—
163C22×C4×C71—21no—
164C22×C4×C71—21no—
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71—22no—
168C22×C4×C71—22no—
169C22×C4×C71—22no—
170C22×C4×C71C22×C414yes—
171C22×C4×C71C22×C412yes—
172C22×C4×C71C22×C414yes—
173C22×C4×C71C22×C412yes—
174C22×C4×C71C22×C414yes—
175C22×C4×C71C22×C412yes—
176C22×C4×C71C2×D414yes—
177C22×C4×C71C2×D412yes—
178C22×C4×C71#4114yes—
179C22×C4×C71#4112yes—
180C42×C71—44no—
181C42×C71—44no—
182C42×C71—44no—
183C23×C71C23×C412yes—
184C23×C71C23×C411yescharacteristic
185C23×C71C22×D411yescharacteristic
186C23×C71C2×#4112yes—
187C23×C71C2×#4111yescharacteristic
188C23×C71H(5,𝔽2)11yescharacteristic
189C23×C71H(5,𝔽2)11yescharacteristic
190C23×C71H(5,𝔽2)11yescharacteristic
191C23×C71H(5,𝔽2)12yes—
192C23×C71H(5,𝔽2)12yes—
193C23×C712−1+411yescharacteristic
194C22×D71—24no—
195C22×D71—24no—
196C22×D71—24no—
197C22×D71—24no—
198C22×D71—24no—
199C22×D71—24no—
200C22×D71—24no—
201C22×D71—24no—
202C22×D71—24no—
203C22×D71—24no—
204C22×D71—24no—
205C22×D71—24no—
206C22×D71—24no—
207C22×D71—24no—
208C22×D71—24no—
209C22×D71—24no—
210C22×D71—24no—
211C22×D71—24no—
212C22×D71—42no—
213C22×D71—42no—
214C22×D71—42no—
215C22×D71—42no—
216C22×D71—42no—
217C22×D71—42no—
218C22×D71—42no—
219C22×D71—42no—
220C22×D71—42no—
221C22×D71—42no—
222C22×D71—42no—
223C22×D71—42no—
224C22×D71—42no—
225C22×D71—42no—
226C22×D71—42no—
227C22×D71—42no—
228C22×D71C2×C4218yes—
229C2×C4×C71—22no—
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—22no—
237C2×C4×C71—22no—
238C2×C4×C71—24no—
239C2×C4×C71—24no—
240C2×C4×C71—22no—
241C2×C4×C71—22no—
242C2×C4×C71—24no—
243C2×C4×C71—24no—
244C2×C4×C71—22no—
245C2×C4×C71—24no—
246C2×C4×C71—22no—
247C2×C4×C71—22no—
248C2×C4×C71—24no—
249C2×C4×C71—24no—
250C2×C4×C71—22no—
251C2×C4×C71—42no—
252C2×C4×C71—42no—
253C2×C4×C71—42no—
254C2×C4×C71—42no—
255C2×C4×C71—42no—
256C2×C4×C71—42no—
257C2×C4×C71C2×C4214yes—
258C2×C4×C71C4×D414yes—
259C2×C4×C71C4×D418yes—
260C2×Dic71—24no—
261C2×Dic71—24no—
262C2×Dic71—24no—
263C2×Dic71—24no—
264C2×Dic71—24no—
265C2×Dic71—24no—
266C2×Dic71—24no—
267C2×Dic71—24no—
268C2×Dic71—24no—
269C2×Dic71—42no—
270C2×Dic71—42no—
271C2×Dic71—42no—
272C2×Dic71—42no—
273C2×Dic71—42no—
274C2×Dic71—42no—
275C2×Dic71—42no—
276C2×Dic71—42no—
277C2×Dic71—42no—
278C2×Dic71—42no—
279C2×Dic71—42no—
280C2×Dic71—42no—
281D284—416no—
282C4×D71—432no—
283C22×C71C22×C4211yescharacteristic
284C22×C71C2×C4×D412yes—
285C22×C71C2×C4×D411yescharacteristic
286C22×C71C4×#4112yes—
287C22×C71C4×#4111yescharacteristic
288C22×C71#51812yes—
289C22×C71#51812yes—
290C22×C71#51812yes—
291C22×C71#51811yescharacteristic
292C22×C71#51811yescharacteristic
293C22×C71#51811yescharacteristic
294C22×C71#52011yescharacteristic
295C22×C71#52012yes—
296C22×C71#53411yescharacteristic
297C22×C71#53512yes—
298C22×C71#53711yescharacteristic
299C22×C71#53812yes—
300C22×C71#53812yes—
301C22×C71#53811yescharacteristic
302C22×C71#53911yescharacteristic
303C22×C71#54012yes—
304C22×C71#54012yes—
305C22×C71#54212yes—
306C2×D71—28no—
307C2×D71—28no—
308C2×D71—28no—
309C2×D71—44no—
310C2×D71—44no—
311C2×D71—44no—
312C2×D71—44no—
313C2×D71—44no—
314C2×D71—44no—
315C2×D71—44no—
316C2×D71—44no—
317C2×D71—44no—
318C2×D71—44no—
319C2×D71—44no—
320C2×D71—44no—
321C4×C71—28no—
322C4×C71—24no—
323C4×C71—24no—
324C4×C71—44no—
325C4×C71—42no—
326Dic71—44no—
327Dic71—44no—
328Dic71—44no—
329#13791—711nomaximal, Sylow
330C2×C71C42×D411yescharacteristic
331C2×C71#227311yescharacteristic
332C2×C71#227511yescharacteristic
333C2×C71#232112yes—
334C2×C71#232412yes—
335C2×C71#232511yescharacteristic
336C2×C71#232811yescharacteristic
337C2×C71#233312yes—
338C2×C71#233311yescharacteristic
339C2×C71#233512yes—
340C2×C71#233711yescharacteristic
341D71—48no—
342C2×#390—711no—
343C2×#386—712no—
344C2×C4×C22⋊C4—712no—
345#1437—712no—
346#1437—712no—
347#1436—712no—
348#1434—711no—
349C22⋊#95—712no—
350C4×#95D7111yescharacteristic, complemented
351C71#1379111yescharacteristic, complemented, Sylow
352C23×D4—711no—
353C24×C4—712no—
354C22×C4⋊C4—711no—
355C22×C22⋊C4—712no—
356C22×C22⋊C4—712no—
357C22×C22⋊C4—712no—
358C22×C22⋊C4—712no—
359C22×C22⋊C4—712no—
360C22×C42—712no—
361C22×C42C2×D7111yescharacteristic
362C22×C42C2×D7111yescharacteristic
363C22×C42C2×D7111yescharacteristic
364#390—718no—
365#386—7116no—
366C4×C22⋊C4—1428no—
367C2×#95—711no—
368C2×#95—712no—
369C2×#95—711no—
370C2×#95—712no—
371C2×#95—712no—
372C2×#95—712no—
373C2×#95—712no—
374C2×#95C2×D7112yes—
375C2×#95C2×D7112yes—
376C25—712no—
377C22×D4—2841no—
378C22×D4—2841no—
379C22×D4—2842no—
380C22×D4—718no—
381C22×D4—714no—
382C23×C4—2842no—
383C23×C4—2842no—
384C23×C4—2842no—
385C23×C4—2842no—
386C23×C4—718no—
387C23×C4—712no—
388C23×C4—712no—
389C23×C4—712no—
390C23×C4—1424no—
391C23×C4—1424no—
392C23×C4C22×D7111yescharacteristic
393C23×C4C22×D7112yes—
394C23×C4C22×D7111yescharacteristic
395C23×C4C22×D7111yescharacteristic
396C23×C4C22×D7112yes—
397C2×C4⋊C4—2842no—
398C2×C4⋊C4—714no—
399C2×C22⋊C4—2842no—
400C2×C22⋊C4—2842no—
401C2×C22⋊C4—2842no—
402C2×C22⋊C4—2842no—
403C2×C22⋊C4—2842no—
404C2×C22⋊C4—2842no—
405C2×C22⋊C4—718no—
406C2×C22⋊C4—718no—
407C2×C22⋊C4—718no—
408C2×C22⋊C4—718no—
409C2×C22⋊C4—718no—
410C2×C22⋊C4—1424no—
411C2×C22⋊C4—1424no—
412C2×C22⋊C4—1424no—
413C2×C22⋊C4—1424no—
414C2×C42—22no—
415C2×C42—41no—
416C2×C42—41no—
417C2×C42—41no—
418C2×C42—41no—
419C2×C42—41no—
420C2×C42—41no—
421C2×C42—2842no—
422C2×C42—2842no—
423C2×C42—1424no—
424C2×C42C4×D7114yescomplemented
425C2×C42D28414yescomplemented
426#95—24no—
427#95—2841no—
428#95—2841no—
429#95—2842no—
430#95—2842no—
431#95—718no—
432#95—1424no—
433#95—1424no—
434#95C4×D7118yescomplemented
435C24—714no—
436C24—714no—
437C24—714no—
438C24—1422no—
439C24—1422no—
440C24—1422no—
441C24—1422no—
442C24—1422no—
443C24—1422no—
444C24—1422no—
445C24—1422no—
446C24—1422no—
447C24—1422no—
448C24—1422no—
449C24—1422no—
450C24C23×D7111yescharacteristic
451C2×D4—2848no—
452C2×D4—2848no—
453C2×D4—2844no—
454C2×D4—2844no—
455C2×D4—7116no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—2848no—
478C22×C4—2848no—
479C22×C4—2848no—
480C22×C4—2848no—
481C22×C4—2848no—
482C22×C4—2848no—
483C22×C4—714no—
484C22×C4—714no—
485C22×C4—714no—
486C22×C4—1422no—
487C22×C4—1422no—
488C22×C4—1422no—
489C22×C4—1422no—
490C22×C4—1422no—
491C22×C4—1422no—
492C22×C4—1422no—
493C22×C4—1422no—
494C22×C4—1422no—
495C22×C4—1422no—
496C22×C4—1422no—
497C22×C4—1422no—
498C22×C4—1422no—
499C22×C4—1422no—
500C22×C4—1422no—
501C22×C4—1422no—
502C22×C4—1422no—
503C22×C4—1422no—
504C22×C4—14216no—
505C22×C4C2×C4×D7114yes—
506C22×C4C2×C4×D7112yes—
507C22×C4C2×C4×D7114yes—
508C22×C4C2×C4×D7112yes—
509C22×C4C2×C4×D7114yes—
510C22×C4C2×C4×D7112yes—
511C22×C4C2×D28412yes—
512C22×C4C2×D28414yes—
513C22×C4#1176586514yes—
514C22×C4#1176586512yes—
515C4⋊C4—2844no—
516C22⋊C4—2848no—
517C22⋊C4—2848no—
518C22⋊C4—2848no—
519C22⋊C4—14216no—
520C22⋊C4—14216no—
521C42—44no—
522C42—44no—
523C42—44no—
524C42—2848no—
525C23—2842no—
526C23—2842no—
527C23—2842no—
528C23—2842no—
529C23—2842no—
530C23—2842no—
531C23—2842no—
532C23—2842no—
533C23—2842no—
534C23—2842no—
535C23—2842no—
536C23—2842no—
537C23—2842no—
538C23—2842no—
539C23—2842no—
540C23—2842no—
541C23—718no—
542C23—1424no—
543C23—1424no—
544C23—1424no—
545C23—1424no—
546C23—1424no—
547C23—1424no—
548C23—1424no—
549C23—1424no—
550C23—1424no—
551C23—1424no—
552C23—1424no—
553C23—1424no—
554C23—1424no—
555C23—1424no—
556C23—1424no—
557C23—1424no—
558C23—1424no—
559C23—1424no—
560C23C22×C4×D7112yes—
561C23C22×C4×D7111yescharacteristic
562C23C22×D28411yescharacteristic
563C23C2×#1176586512yes—
564C23C2×#1176586511yescharacteristic
565C23#1455617311yescharacteristic
566C23#1455617311yescharacteristic
567C23#1455617311yescharacteristic
568C23#1455617412yes—
569C23#1455617412yes—
570C23#1455617611yescharacteristic
571D4—28416no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—28432no—
601C2×C4—2842no—
602C2×C4—2842no—
603C2×C4—2842no—
604C2×C4—2842no—
605C2×C4—2842no—
606C2×C4—2842no—
607C2×C4—2842no—
608C2×C4—2842no—
609C2×C4—2842no—
610C2×C4—2842no—
611C2×C4—2842no—
612C2×C4—2842no—
613C2×C4—1424no—
614C2×C4—1424no—
615C2×C4—1424no—
616C2×C4—1424no—
617C2×C4—1424no—
618C2×C4—1424no—
619C2×C4—1424no—
620C2×C4—1424no—
621C2×C4—1424no—
622C2×C4C42×D7114yes—
623C2×C4C4×D28414yes—
624C2×C4C4×D28418yes—
625C22—2844no—
626C22—2844no—
627C22—2844no—
628C22—2844no—
629C22—2844no—
630C22—2844no—
631C22—2844no—
632C22—2844no—
633C22—2844no—
634C22—2844no—
635C22—2844no—
636C22—2844no—
637C22—1428no—
638C22—1428no—
639C22—1428no—
640C22C2×C42×D7111yescharacteristic
641C22C2×C4×D28412yes—
642C22C2×C4×D28411yescharacteristic
643C22C4×#1176586512yes—
644C22C4×#1176586511yescharacteristic
645C22#1798647112yes—
646C22#1798647112yes—
647C22#1798647112yes—
648C22#1798647211yescharacteristic
649C22#1798647311yescharacteristic
650C22#1798647311yescharacteristic
651C22#1798648011yescharacteristic
652C22#1798648112yes—
653C22#1798656111yescharacteristic
654C22#1798656812yes—
655C22#1798658011yescharacteristic
656C22#1798658212yes—
657C22#1798658212yes—
658C22#1798658711yescharacteristic
659C22#1798659611yescharacteristic
660C22#1798660012yes—
661C22#1798660412yes—
662C22#1798662112yes—
663C4—28no—
664C4—24no—
665C4—24no—
666C4—44no—
667C4—42no—
668C4—2844no—
669C4—2844no—
670C4—2844no—
671C2—2848no—
672C2C42×D28411yescharacteristic
673C2#2259069211yescharacteristic
674C2#2259070011yescharacteristic
675C2#2259100012yes—
676C2#2259103412yes—
677C2#2259104311yescharacteristic
678C2#2259106511yescharacteristic
679C2#2259110012yes—
680C2#2259110511yescharacteristic
681C2#2259112412yes—
682C2#2259115311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC22×C71C22×C42
FittingC4×#95×C71C2
socle, cosocleC24×C71C24
prime radicalC23×D284C22
hypercenterC4×#95D71
hypocenterC71#13791
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
3. C2×C4×(C2×D71)⋊C4C2semidirect
4. #22586578C2semidirect
5. #22586576C2semidirect
6. #22586574C2semidirect
7. #22586567C2semidirect
8. (C2×D71)⋊#95C2semidirect
9. C4×#95×C71C2semidirect
19. #17985901C4semidirect
20. #17985877C4semidirect
49. C22×C4×D71C2×C4semidirect
51. C2×C4⋊Dic71C2×C4semidirect
64. C2×(C2×D71)⋊C4C2×C4semidirect
65. C2×(C2×D71)⋊C4C2×C4semidirect
66. C2×(C2×D71)⋊C4C2×C4semidirect
76. C71⋊#95C2×C4semidirect
89. C2×C42×C71C2×C4semidirect
90. C2×C42×C71D4semidirect
92. #95×C71C2×C4semidirect
112. C2×D284C42semidirect
350. C4×#95D71semidirect
351. C71#13791semidirect
424. C2×C42C4×D71semidirect
425. C2×C42D284semidirect
434. #95C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179859012
C2×#179858772
C2×C4×(C2×D71)⋊C42
#225865782
#225865762
#225865742
#225865672
(C2×D71)⋊#952
C4×#95×C712
#1379171