Finite Group Databasebeta

← #31920060

Subgroups — #31920060

Order 18176.

1486 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Dic71C211yesmaximal, characteristic
2#22586600C212yesmaximal, complemented
3#22586587C212yesmaximal
4#22586577C212yesmaximal
5#22586566C211yesmaximal, characteristic, complemented
6#22586566C211yesmaximal, characteristic, complemented
7#22586566C212yesmaximal, complemented
8#22586565C211yesmaximal, characteristic, complemented
9#22586563C211yesmaximal, characteristic
10#95×Dic71C211yesmaximal, characteristic, complemented
11C71×C22⋊#95C211yesmaximal, characteristic
12C22×C22⋊Dic71C2212yes—
13C22×C22⋊Dic71C2211yescharacteristic
14C22×C22⋊Dic71C2211yescharacteristic
15C22×C4×Dic71C2212yes—
16C22×C4×Dic71C2211yescharacteristic
17C22×C4×Dic71C2211yescharacteristic
18C4×C22⋊Dic71C418yescomplemented
19C2×C71⋊#95C2211yescharacteristic
20C2×C71⋊#95C2212yes—
21C2×C71⋊#95C2211yescharacteristic
22C2×C71⋊#95C2212yes—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2212yes—
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2211yescharacteristic
30C2×C71⋊#95C2211yescharacteristic
31C24×C4×C71C2211yescharacteristic
32C22×C22⋊C4×C71C2212yes—
33C2×#95×C71C2211yescharacteristic
34C2×#95×C71C2211yescharacteristic
35C2×#95×C71C2212yes—
36C23×Dic71C2312yes—
37C23×Dic71C2311yescharacteristic
38C23×Dic71C2311yescharacteristic
39C23×Dic71C2311yescharacteristic
40C23×Dic71C2312yes—
41C23×Dic71C2311yescharacteristic
42C2×C22⋊Dic71—22no—
43C2×C22⋊Dic71—22no—
44C2×C22⋊Dic71—24no—
45C2×C22⋊Dic71—22no—
46C2×C22⋊Dic71—22no—
47C2×C22⋊Dic71—24no—
48C2×C22⋊Dic71—42no—
49C2×C22⋊Dic71C2×C414yes—
50C2×C22⋊Dic71C2×C414yes—
51C2×C4×Dic71—24no—
52C2×C4×Dic71—24no—
53C2×C4×Dic71—42no—
54C2×C4×Dic71—42no—
55C2×C4×Dic71—42no—
56C2×C4×Dic71C2×C414yes—
57C2×C4×Dic71C2×C414yes—
58C71⋊#95—24no—
59C71⋊#95—24no—
60C71⋊#95—22no—
61C71⋊#95—22no—
62C71⋊#95—24no—
63C71⋊#95—24no—
64C71⋊#95—42no—
65C71⋊#95—42no—
66C71⋊#95—42no—
67C71⋊#95—42no—
68C71⋊#95C2×C414yes—
69C71⋊#95C2×C414yes—
70C25×C71C2311yescharacteristic
71C23×C4×C71—22no—
72C23×C4×C71—22no—
73C23×C4×C71—42no—
74C23×C4×C71—42no—
75C23×C4×C71C2×C414yes—
76C23×C4×C71C2312yes—
77C23×C4×C71C2311yescharacteristic
78C23×C4×C71C2312yes—
79C23×C4×C71C2311yescharacteristic
80C2×C22⋊C4×C71—24no—
81C2×C22⋊C4×C71—24no—
82C2×C22⋊C4×C71—42no—
83#95×C71—22no—
84#95×C71—22no—
85#95×C71—42no—
86C22×Dic71—22no—
87C22×Dic71—22no—
88C22×Dic71—22no—
89C22×Dic71—22no—
90C22×Dic71—22no—
91C22×Dic71—22no—
92C22×Dic71—22no—
93C22×Dic71—22no—
94C22×Dic71—21no—
95C22×Dic71—21no—
96C22×Dic71—21no—
97C22×Dic71—21no—
98C22×Dic71—21no—
99C22×Dic71—21no—
100C22×Dic71—22no—
101C22×Dic71—22no—
102C22×Dic71—22no—
103C22×Dic71—22no—
104C22×Dic71—22no—
105C22×Dic71—22no—
106C22×Dic71—22no—
107C22×Dic71—22no—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—22no—
111C22×Dic71—22no—
112C22×Dic71—22no—
113C22×Dic71C22×C412yes—
114C22×Dic71C22×C412yes—
115C22×Dic71C22×C412yes—
116C22×Dic71C22×C412yes—
117C22×Dic71#4114yes—
118C22×Dic71#4112yes—
119C22×Dic71#4112yes—
120C22⋊Dic71—28no—
121C22⋊Dic71—28no—
122C22⋊Dic71—48no—
123C4×Dic71—44no—
124C4×Dic71—44no—
125C4×Dic71—48no—
126C24×C71—22no—
127C24×C71—21no—
128C24×C71—21no—
129C24×C71—21no—
130C24×C71—21no—
131C24×C71—22no—
132C24×C71—22no—
133C24×C71—22no—
134C24×C71C22×C412yes—
135C24×C71#4112yes—
136C24×C71#4112yes—
137C24×C71C2411yescharacteristic
138C22×C4×C71—22no—
139C22×C4×C71—22no—
140C22×C4×C71—22no—
141C22×C4×C71—22no—
142C22×C4×C71—21no—
143C22×C4×C71—21no—
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—22no—
147C22×C4×C71—22no—
148C22×C4×C71—22no—
149C22×C4×C71—22no—
150C22×C4×C71—28no—
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—22no—
157C22×C4×C71—22no—
158C22×C4×C71—44no—
159C22×C4×C71—44no—
160C22×C4×C71—48no—
161C22×C4×C71—48no—
162C22×C4×C71C22×C412yes—
163C22×C4×C71C22×C412yes—
164C22×C4×C71#4112yes—
165C22×C4×C71#4112yes—
166C22×C4×C71#4114yes—
167C22⋊C4×C71—48no—
168C23×C71—22no—
169C23×C71—22no—
170C23×C71—22no—
171C23×C71—22no—
172C23×C71—22no—
173C23×C71—22no—
174C23×C71—22no—
175C23×C71—22no—
176C23×C71—22no—
177C23×C71—22no—
178C23×C71—24no—
179C23×C71—24no—
180C23×C71—24no—
181C23×C71—24no—
182C23×C71—42no—
183C23×C71—42no—
184C23×C71—42no—
185C23×C71—42no—
186C23×C71—42no—
187C23×C71—42no—
188C23×C71—42no—
189C23×C71—42no—
190C23×C71#11714yes—
191C23×C71C23×C411yescharacteristic
192C23×C71C2×#4111yescharacteristic
193C23×C71C2×#4111yescharacteristic
194C23×C71C2×#4112yes—
195C23×C71C2×#4111yescharacteristic
196C23×C71C2×#4111yescharacteristic
197C23×C71C2×#4111yescharacteristic
198C23×C71C2×#4112yes—
199C23×C71C2×#4111yescharacteristic
200C23×C71H(5,𝔽2)12yes—
201C23×C71H(5,𝔽2)12yes—
202C2×C4×C71—22no—
203C2×C4×C71—22no—
204C2×C4×C71—24no—
205C2×C4×C71—24no—
206C2×C4×C71—22no—
207C2×C4×C71—22no—
208C2×C4×C71—22no—
209C2×C4×C71—24no—
210C2×C4×C71—24no—
211C2×C4×C71—24no—
212C2×C4×C71—416no—
213C2×C4×C71—42no—
214C2×C4×C71—42no—
215C2×C4×C71—42no—
216C2×C4×C71—42no—
217C2×C4×C71—42no—
218C2×C4×C71—42no—
219C2×C4×C71—42no—
220C2×C4×C71—42no—
221C2×C4×C71—42no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71—42no—
226C2×C4×C71#11714yes—
227C2×Dic71—22no—
228C2×Dic71—22no—
229C2×Dic71—24no—
230C2×Dic71—24no—
231C2×Dic71—22no—
232C2×Dic71—22no—
233C2×Dic71—22no—
234C2×Dic71—22no—
235C2×Dic71—22no—
236C2×Dic71—22no—
237C2×Dic71—22no—
238C2×Dic71—22no—
239C2×Dic71—22no—
240C2×Dic71—24no—
241C2×Dic71—24no—
242C2×Dic71—24no—
243C2×Dic71—42no—
244C2×Dic71—42no—
245C2×Dic71—42no—
246C2×Dic71—42no—
247C2×Dic71—42no—
248C2×Dic71—42no—
249C2×Dic71—42no—
250C2×Dic71—42no—
251C2×Dic71—42no—
252C2×Dic71—42no—
253C2×Dic71—42no—
254C2×Dic71—42no—
255C2×Dic71—42no—
256C2×Dic71—42no—
257C2×Dic71—42no—
258C2×Dic71—42no—
259C2×Dic71—42no—
260C2×Dic71#11714yes—
261C22×C71—24no—
262C22×C71—24no—
263C22×C71—24no—
264C22×C71—42no—
265C22×C71—44no—
266C22×C71—44no—
267C22×C71—44no—
268C22×C71—44no—
269C22×C71—42no—
270C22×C71—42no—
271C22×C71—42no—
272C22×C71C2×#11711yescharacteristic
273C22×C71C2×#11711yescharacteristic
274C22×C71C2×#11711yescharacteristic
275C22×C71C4×#4111yescharacteristic
276C22×C71C4×#4111yescharacteristic
277C22×C71#51811yescharacteristic
278C22×C71#51811yescharacteristic
279C22×C71#52912yes—
280C22×C71#52911yescharacteristic
281C22×C71#52911yescharacteristic
282C22×C71#52912yes—
283C22×C71#52911yescharacteristic
284C22×C71#52911yescharacteristic
285C22×C71#53812yes—
286C22×C71#53812yes—
287C22×C71#55112yes—
288C22×C71#55112yes—
289C22×C71#55112yes—
290C22×C71#55312yes—
291C22×C71#55312yes—
292C22×C71#55312yes—
293C22×C71#55312yes—
294C22×C71#55912yes—
295C4×C71—24no—
296C4×C71—42no—
297C4×C71—44no—
298C4×C71—82no—
299Dic71—24no—
300Dic71—44no—
301Dic71—42no—
302Dic71—42no—
303Dic71—42no—
304Dic71—82no—
305#13947—711nomaximal, Sylow
306C2×C71—44no—
307C2×C71#228311yescharacteristic
308C2×C71#229811yescharacteristic
309C2×C71#231211yescharacteristic
310C2×C71#235311yescharacteristic
311C2×C71#235311yescharacteristic
312C2×C71#235311yescharacteristic
313C2×C71#235711yescharacteristic
314C2×C71#257612yes—
315C2×C71#257812yes—
316C2×C71#259512yes—
317C2×C71#259712yes—
318C2×C4×C22⋊C4—711no—
319#1443—712no—
320#1440—712no—
321#1437—712no—
322#1434—711no—
323#1434—712no—
324#1434—711no—
325#1434—711no—
326C22⋊#95—711no—
327C22⋊#95D7111yescharacteristic
328C4×#95—711no—
329C71#1394711yescharacteristic, complemented, Sylow
330C24×C4C2×D7111yescharacteristic
331C22×C22⋊C4—712no—
332C22×C22⋊C4—711no—
333C22×C22⋊C4—711no—
334C22×C22⋊C4C2×D7112yes—
335C22×C42—711no—
336C22×C42—711no—
337C22×C42—712no—
338C4×C22⋊C4—718no—
339C2×#95—712no—
340C2×#95—712no—
341C2×#95—712no—
342C2×#95—712no—
343C2×#95—711no—
344C2×#95—711no—
345C2×#95—711no—
346C2×#95—712no—
347C2×#95—711no—
348C2×#95—712no—
349C2×#95—712no—
350C2×#95—712no—
351C2×#95C2×D7111yescharacteristic
352C2×#95C2×D7111yescharacteristic
353C2×#95C2×D7112yes—
354C25C22×D7111yescharacteristic
355C23×C4—22no—
356C23×C4—22no—
357C23×C4—42no—
358C23×C4—42no—
359C23×C4—711no—
360C23×C4—711no—
361C23×C4—711no—
362C23×C4—711no—
363C23×C4—712no—
364C23×C4—712no—
365C23×C4C4×D7114yes—
366C23×C4C22×D7112yes—
367C23×C4C22×D7111yescharacteristic
368C23×C4C22×D7112yes—
369C23×C4C22×D7111yescharacteristic
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—42no—
373C2×C22⋊C4—2842no—
374C2×C22⋊C4—714no—
375C2×C22⋊C4—714no—
376C2×C22⋊C4—1424no—
377C2×C22⋊C4—1424no—
378C2×C22⋊C4—1422no—
379C2×C22⋊C4—1422no—
380C2×C22⋊C4—1422no—
381C2×C22⋊C4—1422no—
382C2×C42—2842no—
383C2×C42—2842no—
384C2×C42—2842no—
385C2×C42—714no—
386C2×C42—714no—
387C2×C42—1424no—
388C2×C42—1424no—
389#95—22no—
390#95—22no—
391#95—42no—
392#95—2842no—
393#95—2842no—
394#95—2842no—
395#95—2842no—
396#95—714no—
397#95—714no—
398#95—1424no—
399#95—1424no—
400#95—1422no—
401#95—1422no—
402#95—1424no—
403#95—1424no—
404C24—22no—
405C24—21no—
406C24—21no—
407C24—21no—
408C24—21no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24C2×C4×D7112yes—
413C24#1176586412yes—
414C24#1176586612yes—
415C24C23×D7111yescharacteristic
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—28no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—712no—
441C22×C4—712no—
442C22×C4—712no—
443C22×C4—712no—
444C22×C4—712no—
445C22×C4—712no—
446C22×C4—714no—
447C22×C4—1422no—
448C22×C4—1422no—
449C22×C4—1422no—
450C22×C4—1422no—
451C22×C4—1422no—
452C22×C4—1422no—
453C22×C4—1422no—
454C22×C4—1422no—
455C22×C4—1422no—
456C22×C4—1421no—
457C22×C4—1421no—
458C22×C4—1422no—
459C22×C4—1421no—
460C22×C4—1421no—
461C22×C4—1422no—
462C22×C4—1422no—
463C22×C4—1422no—
464C22×C4—1421no—
465C22×C4—1421no—
466C22×C4—1422no—
467C22×C4—1422no—
468C22×C4—1422no—
469C22×C4—1422no—
470C22×C4—1422no—
471C22×C4—1422no—
472C22×C4—1422no—
473C22×C4—1422no—
474C22×C4C2×C4×D7112yes—
475C22×C4C2×C4×D7112yes—
476C22×C4#1176586412yes—
477C22×C4#1176586412yes—
478C22×C4#1176586514yes—
479C22⋊C4—48no—
480C22⋊C4—2848no—
481C22⋊C4—1428no—
482C22⋊C4—1428no—
483C42—2844no—
484C42—2844no—
485C42—2848no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23#1455608614yes—
509C23C22×C4×D7111yescharacteristic
510C23#41×D7112yes—
511C23#41×D7111yescharacteristic
512C23#41×D7111yescharacteristic
513C23C2×#1176586411yescharacteristic
514C23C2×#1176586411yescharacteristic
515C23C2×#1176586411yescharacteristic
516C23C2×#1176586512yes—
517C23C2×#1176586611yescharacteristic
518C23#1455617312yes—
519C23#1455617312yes—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—416no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—2842no—
545C2×C4—2842no—
546C2×C4—2842no—
547C2×C4—2842no—
548C2×C4—2842no—
549C2×C4—2842no—
550C2×C4—2842no—
551C2×C4—2842no—
552C2×C4—2842no—
553C2×C4—2842no—
554C2×C4—2842no—
555C2×C4—2842no—
556C2×C4—2842no—
557C2×C4—2842no—
558C2×C4—2842no—
559C2×C4—2842no—
560C2×C4—2842no—
561C2×C4—714no—
562C2×C4—1424no—
563C2×C4—1424no—
564C2×C4—1422no—
565C2×C4—1422no—
566C2×C4—1422no—
567C2×C4—1422no—
568C2×C4—1422no—
569C2×C4—1422no—
570C2×C4—1422no—
571C2×C4—1422no—
572C2×C4—1422no—
573C2×C4—1424no—
574C2×C4—1424no—
575C2×C4—1424no—
576C2×C4—1422no—
577C2×C4—1422no—
578C2×C4#1455608414yes—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—42no—
583C22—44no—
584C22—42no—
585C22—44no—
586C22—44no—
587C22—42no—
588C22—44no—
589C22—42no—
590C22#117×D7111yescharacteristic
591C22C2×#1455608411yescharacteristic
592C22C2×#1455608611yescharacteristic
593C22#1798646711yescharacteristic
594C22C4×#1176586511yescharacteristic
595C22#1798647211yescharacteristic
596C22#1798647211yescharacteristic
597C22#1798652912yes—
598C22#1798653011yescharacteristic
599C22#1798653011yescharacteristic
600C22#1798653112yes—
601C22#1798653511yescharacteristic
602C22#1798653511yescharacteristic
603C22#1798658412yes—
604C22#1798658412yes—
605C22#1798667512yes—
606C22#1798667612yes—
607C22#1798667712yes—
608C22#1798669212yes—
609C22#1798669412yes—
610C22#1798669712yes—
611C22#1798669812yes—
612C22#1798673212yes—
613C4—24no—
614C4—44no—
615C4—42no—
616C4—82no—
617C4—2842no—
618C4—2842no—
619C4—2842no—
620C4—2844no—
621C4—5682no—
622C4—1424no—
623C2—44no—
624C2#2259073511yescharacteristic
625C2#2259083611yescharacteristic
626C2#2259093611yescharacteristic
627C2#2259127611yescharacteristic
628C2#2259127711yescharacteristic
629C2#2259128211yescharacteristic
630C2#2259131411yescharacteristic
631C2#2259364312yes—
632C2#2259367112yes—
633C2#2259386012yes—
634C2#2259388812yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC71×C22⋊#95C2
socle, cosocleC24×C71C24
prime radicalC25×C71C23
hypercenterC22⋊#95D71
hypocenterC71#13947
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. #22586600C2semidirect
5. #22586566C2semidirect
6. #22586566C2semidirect
7. #22586566C2semidirect
8. #22586565C2semidirect
10. #95×Dic71C2semidirect
18. C4×C22⋊Dic71C4semidirect
329. C71#13947semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Dic712
#225866002
#225865872
#225865772
#225865662
#225865652
#225865632
#95×Dic712
C71×C22⋊#952
#1394771