Finite Group Databasebeta

← #13282298

Subgroups — #13282298

Order 1792.

1810 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6487C211yesmaximal, characteristic
2C2×#89418C212yesmaximal, complemented
3C7×C22⋊#95C211yesmaximal, characteristic
4#11729881C211yesmaximal, characteristic
5#11729301C212yesmaximal
6#11729299C212yesmaximal
7#11727177C212yesmaximal, complemented
8#11727163C212yesmaximal, complemented
9#11727160C212yesmaximal, complemented
10C24×C4×C7C2211yescharacteristic
11C23×Dic14C2211yescharacteristic
12C22×C7×C22⋊C4C2211yescharacteristic
13C22×C7×C22⋊C4C2211yescharacteristic
14C22×C22⋊Dic7C2212yes—
15C22×C22⋊Dic7C2212yes—
16C22×C4⋊Dic7C2211yescharacteristic
17C2×#6487—28no—
18C2×#6487C418yescomplemented
19C22×Dic7⋊C4C2212yes—
20C22×Dic7⋊C4C2212yes—
21C22×C4×Dic7C2212yes—
22C2×C7×#95C2212yes—
23C2×C7×#95C2212yes—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2211yescharacteristic
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33#89418—28no—
34#14449—71nomaximal, Sylow
35C25×C7C2311yescharacteristic
36C23×C4×C7—24no—
37C23×C4×C7—41no—
38C23×C4×C7—41no—
39C23×C4×C7—42no—
40C23×C4×C7C2×C414yes—
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×Dic7C2312yes—
46C23×Dic7C2312yes—
47C23×Dic7C2312yes—
48C23×Dic7C2312yes—
49C22×Dic14—41no—
50C22×Dic14—42no—
51C22×Dic14—41no—
52C22×Dic14C2×C414yes—
53C22×Dic14D418yes—
54C2×C7×C22⋊C4—22no—
55C2×C7×C22⋊C4—41no—
56C2×C7×C22⋊C4—41no—
57C2×C7×C22⋊C4—41no—
58C2×C7×C22⋊C4—41no—
59C2×C7×C22⋊C4Q814yescomplemented
60C2×C22⋊Dic7—24no—
61C2×C22⋊Dic7—24no—
62C2×C22⋊Dic7—24no—
63C2×C22⋊Dic7—24no—
64C2×C22⋊Dic7—42no—
65C2×C22⋊Dic7C2×C418yes—
66C2×C4⋊Dic7—24no—
67C2×C4⋊Dic7C2×C414yes—
68#6487—232no—
69C2×Dic7⋊C4—24no—
70C2×Dic7⋊C4—24no—
71C2×Dic7⋊C4—24no—
72C2×Dic7⋊C4—42no—
73C2×Dic7⋊C4—42no—
74C2×Dic7⋊C4C2×C418yes—
75C2×C4×Dic7—24no—
76C2×C4×Dic7—42no—
77C2×C4×Dic7—42no—
78C7×#95—24no—
79C7×#95—42no—
80C7⋊#95—24no—
81C7⋊#95—24no—
82C7⋊#95—24no—
83C7⋊#95—24no—
84C7⋊#95—24no—
85C7⋊#95—42no—
86C7⋊#95—42no—
87C7⋊#95—41no—
88C7⋊#95—41no—
89C22×C22⋊Q8—71no—
90C2×#391—72no—
91#1445—72no—
92#1441—72no—
93#1439—72no—
94#1437—72no—
95#1436—72no—
96C22⋊#95—71no—
97C22⋊#95D711yescharacteristic
98C24×C7—22no—
99C24×C7—22no—
100C24×C7—22no—
101C24×C7—21no—
102C24×C7—21no—
103C24×C7—21no—
104C24×C7—22no—
105C24×C7—21no—
106C24×C7C22×C412yes—
107C24×C7C2×Q812yes—
108C24×C7#4112yes—
109C24×C7C2411yescharacteristic
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—28no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—44no—
132C22×C4×C7—44no—
133C22×C4×C7—48no—
134C22×C4×C7—48no—
135C22×C4×C7C22×C412yes—
136C22×C4×C7C22×C412yes—
137C22×C4×C7C2×D414yes—
138C22×C4×C7C2×Q814yes—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7C22×C414yes—
163C22×Dic7C22×C414yes—
164C22×Dic7C2×D414yes—
165C22×Dic7C2×D414yes—
166C22×Dic7#4114yes—
167C2×Dic14—44no—
168C2×Dic14—44no—
169C2×Dic14—48no—
170C2×Dic14—48no—
171C2×Dic14C22⋊C4116yescomplemented
172C7×C22⋊C4—44no—
173C7×C22⋊C4—44no—
174C22⋊Dic7—216no—
175C22⋊Dic7—48no—
176C4⋊Dic7—28no—
177Dic7⋊C4—216no—
178Dic7⋊C4—48no—
179C4×Dic7—48no—
180C23×Q8—71no—
181C24×C4C2×D711yescharacteristic
182C2×C22⋊Q8—78no—
183C2×C22⋊Q8—148no—
184C22×C4⋊C4—72no—
185C22×C4⋊C4—71no—
186C22×C4⋊C4—72no—
187C22×C22⋊C4—72no—
188C22×C22⋊C4—72no—
189C22×C22⋊C4C2×D711yescharacteristic
190C22×C22⋊C4C2×D711yescharacteristic
191C22×C42—72no—
192#391—148no—
193C2×#95—72no—
194C2×#95—72no—
195C2×#95—72no—
196C2×#95—72no—
197C2×#95—72no—
198C2×#95—71no—
199C2×#95—71no—
200C2×#95—72no—
201C2×#95—72no—
202C2×#95C2×D712yes—
203C2×#95C2×D712yes—
204C23×C7—22no—
205C23×C7—22no—
206C23×C7—22no—
207C23×C7—22no—
208C23×C7—22no—
209C23×C7—22no—
210C23×C7—22no—
211C23×C7—22no—
212C23×C7—22no—
213C23×C7—22no—
214C23×C7—24no—
215C23×C7—24no—
216C23×C7—24no—
217C23×C7—24no—
218C23×C7—42no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7—42no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7C4×Q814yes—
227C23×C7C23×C411yescharacteristic
228C23×C7C22×D412yes—
229C23×C7C22×Q811yescharacteristic
230C23×C7C2×#4111yescharacteristic
231C23×C7C2×#4112yes—
232C23×C7H(5,𝔽2)11yescharacteristic
233C23×C7H(5,𝔽2)11yescharacteristic
234C23×C7H(5,𝔽2)12yes—
235C23×C7H(5,𝔽2)11yescharacteristic
236C23×C72−1+411yescharacteristic
237C23×C72−1+412yes—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—24no—
241C2×C4×C7—24no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—24no—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—416no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7C2×C22⋊C414yes—
263C2×Dic7—24no—
264C2×Dic7—24no—
265C2×Dic7—24no—
266C2×Dic7—24no—
267C2×Dic7—24no—
268C2×Dic7—24no—
269C2×Dic7—24no—
270C2×Dic7—24no—
271C2×Dic7—24no—
272C2×Dic7—24no—
273C2×Dic7—24no—
274C2×Dic7—24no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7C2×C22⋊C418yes—
291Dic14—416no—
292C25C22×D711yescharacteristic
293C22×Q8—74no—
294C22×Q8—78no—
295C22×Q8—281no—
296C22×Q8—282no—
297C22×Q8—281no—
298C23×C4—24no—
299C23×C4—42no—
300C23×C4—41no—
301C23×C4—41no—
302C23×C4—72no—
303C23×C4—72no—
304C23×C4—72no—
305C23×C4—72no—
306C23×C4C4×D714yes—
307C23×C4C22×D712yes—
308C23×C4C22×D712yes—
309C23×C4C22×D711yescharacteristic
310C23×C4C22×D711yescharacteristic
311C22⋊Q8—1432no—
312C2×C4⋊C4—78no—
313C2×C4⋊C4—74no—
314C2×C4⋊C4—144no—
315C2×C4⋊C4—144no—
316C2×C4⋊C4—144no—
317C2×C4⋊C4—144no—
318C2×C4⋊C4—282no—
319C2×C4⋊C4—282no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—41no—
322C2×C22⋊C4—41no—
323C2×C22⋊C4—41no—
324C2×C22⋊C4—41no—
325C2×C22⋊C4—78no—
326C2×C22⋊C4—144no—
327C2×C22⋊C4—144no—
328C2×C22⋊C4—144no—
329C2×C22⋊C4—144no—
330C2×C22⋊C4—282no—
331C2×C22⋊C4Dic1414yescomplemented
332C2×C42—144no—
333C2×C42—282no—
334C2×C42—282no—
335#95—24no—
336#95—42no—
337#95—144no—
338#95—144no—
339#95—144no—
340#95—144no—
341#95—144no—
342#95—282no—
343#95—281no—
344#95—281no—
345#95—282no—
346C22×C7—24no—
347C22×C7—24no—
348C22×C7—24no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—44no—
354C22×C7—44no—
355C22×C7—44no—
356C22×C7—44no—
357C22×C7C22×C22⋊C411yescharacteristic
358C22×C7C2×C4×Q811yescharacteristic
359C22×C7C4×#4111yescharacteristic
360C22×C7#51811yescharacteristic
361C22×C7#51811yescharacteristic
362C22×C7#51911yescharacteristic
363C22×C7#52011yescharacteristic
364C22×C7#53412yes—
365C22×C7#53412yes—
366C22×C7#53811yescharacteristic
367C22×C7#53911yescharacteristic
368C22×C7#53912yes—
369C22×C7#54212yes—
370C22×C7#54311yescharacteristic
371C22×C7#54411yescharacteristic
372C22×C7#54812yes—
373C22×C7#54812yes—
374C22×C7D4×Q812yes—
375C22×C7#55012yes—
376C22×C7#55112yes—
377C22×C7#55412yes—
378C22×C7#56212yes—
379C22×C7#56312yes—
380C4×C7—24no—
381C4×C7—42no—
382C4×C7—44no—
383C4×C7—82no—
384Dic7—28no—
385Dic7—44no—
386Dic7—44no—
387Dic7—82no—
388C24—22no—
389C24—21no—
390C24—21no—
391C24—21no—
392C24—22no—
393C24—21no—
394C24—22no—
395C24—22no—
396C24C2×Dic1412yes—
397C24C2×C4×D712yes—
398C24#115712yes—
399C24C23×D711yescharacteristic
400C2×Q8—716no—
401C2×Q8—284no—
402C2×Q8—284no—
403C2×Q8—288no—
404C2×Q8—288no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—74no—
431C22×C4—74no—
432C22×C4—74no—
433C22×C4—74no—
434C22×C4—74no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—142no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—142no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4C2×Dic1414yes—
459C22×C4C2×C4×D712yes—
460C22×C4C2×C4×D712yes—
461C22×C4D4×D714yes—
462C4⋊C4—1416no—
463C4⋊C4—148no—
464C4⋊C4—288no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—1416no—
468C22⋊C4—288no—
469C42—288no—
470C2×C7—44no—
471C2×C7#230811yescharacteristic
472C2×C7#232911yescharacteristic
473C2×C7#233211yescharacteristic
474C2×C7Q8×C22⋊C411yescharacteristic
475C2×C7#234111yescharacteristic
476C2×C7#236811yescharacteristic
477C2×C7#237811yescharacteristic
478C2×C7#272812yes—
479C2×C7#272912yes—
480C2×C7#273012yes—
481C2×C7#273212yes—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23C4×Dic1414yes—
505C23C22×Dic1411yescharacteristic
506C23C22×C4×D711yescharacteristic
507C23C2×#115711yescharacteristic
508C23C2×D4×D712yes—
509C23#654811yescharacteristic
510C23#654811yescharacteristic
511C23#654811yescharacteristic
512C23#654812yes—
513C23#655111yescharacteristic
514C23D7×#4112yes—
515C23#655412yes—
516Q8—2816no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—416no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
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—78no—
542C2×C4—144no—
543C2×C4—144no—
544C2×C4—144no—
545C2×C4—144no—
546C2×C4—144no—
547C2×C4—144no—
548C2×C4—144no—
549C2×C4—144no—
550C2×C4—144no—
551C2×C4—144no—
552C2×C4—144no—
553C2×C4—144no—
554C2×C4—282no—
555C2×C4—282no—
556C2×C4—282no—
557C2×C4—282no—
558C2×C4—282no—
559C2×C4—282no—
560C2×C4—282no—
561C2×C4—282no—
562C2×C4—282no—
563C2×C4—282no—
564C2×C4—282no—
565C2×C4—282no—
566C2×C4—282no—
567C2×C4—282no—
568C2×C4—282no—
569C2×C4D7×C22⋊C414yes—
570C7#1444911yescharacteristic, complemented, Sylow
571C22—24no—
572C22—24no—
573C22—24no—
574C22—44no—
575C22—42no—
576C22—44no—
577C22—44no—
578C22—42no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22C2×C4×Dic1411yescharacteristic
583C22#9017411yescharacteristic
584C22C2×D7×C22⋊C411yescharacteristic
585C22#9017711yescharacteristic
586C22#9017711yescharacteristic
587C22#9018511yescharacteristic
588C22#9019111yescharacteristic
589C22#9019511yescharacteristic
590C22#9020611yescharacteristic
591C22#9020711yescharacteristic
592C22#9021311yescharacteristic
593C22D4×Dic1412yes—
594C22#9023012yes—
595C22#9024312yes—
596C22#9024712yes—
597C22#9028012yes—
598C22#9028012yes—
599C22#9029012yes—
600C22#9031212yes—
601C22#9032412yes—
602C22#9033712yes—
603C22#9035112yes—
604C22#9035312yes—
605C4—24no—
606C4—44no—
607C4—42no—
608C4—82no—
609C4—148no—
610C4—284no—
611C4—284no—
612C4—562no—
613C2—44no—
614C2#1173077911yescharacteristic
615C2C22⋊C4×Dic1411yescharacteristic
616C2#1173091811yescharacteristic
617C2#1173129411yescharacteristic
618C2#1173132211yescharacteristic
619C2#1173133111yescharacteristic
620C2#1173150211yescharacteristic
621C2#1173198512yes—
622C2#1173208612yes—
623C2#1173227412yes—
624C2#1173227512yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×C22⋊#95C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC22⋊#95D7
hypocenterC7#14449
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#89418C2semidirect
7. #11727177C2semidirect
8. #11727163C2semidirect
9. #11727160C2semidirect
18. C2×#6487C4semidirect
59. C2×C7×C22⋊C4Q8semidirect
171. C2×Dic14C22⋊C4semidirect
331. C2×C22⋊C4Dic14semidirect
570. C7#14449semidirect

Maximal subgroups

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

subgroupindex
C22×#64872
C2×#894182
C7×C22⋊#952
#117298812
#117293012
#117292992
#117271772
#117271632
#117271602
#144497