Finite Group Databasebeta

← #13281016

Subgroups — #13281016

Order 1792.

1542 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6450C211yesmaximal, characteristic
2C2×#89749C211yesmaximal, characteristic
3C2×#89423C211yesmaximal, characteristic
4C7×#1441C211yesmaximal, characteristic
5#11729320C211yesmaximal, characteristic
6#11729319C211yesmaximal, characteristic
7#11727178C212yesmaximal
8#11727175C212yesmaximal
9#11727164C211yesmaximal, characteristic
10#11727157C212yesmaximal
11#11727155C212yesmaximal
12C23×Dic14C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C4⋊Dic7C2211yescharacteristic
16C22×Dic7⋊C4C2211yescharacteristic
17C22×Dic7⋊C4C2211yescharacteristic
18C22×C4×Dic7C2211yescharacteristic
19C22×C4×Dic7C2211yescharacteristic
20C22×C4×Dic7C2211yescharacteristic
21C2×#6450—24no—
22C2×#6450—24no—
23C2×#6450C418yescomplemented
24C2×C7×#95C2212yes—
25C2×C7×#95C2211yescharacteristic
26C2×C7×#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2211yescharacteristic
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38#89749—24no—
39#89423—24no—
40#14346—71nomaximal, Sylow
41C23×C4×C7C2312yes—
42C23×C4×C7C2312yes—
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C23×Dic7C2312yes—
47C23×Dic7C2312yes—
48C23×Dic7C2311yescharacteristic
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2311yescharacteristic
51C23×Dic7C2311yescharacteristic
52C22×Dic14—22no—
53C22×Dic14—22no—
54C22×Dic14—22no—
55C22×Dic14—22no—
56C22×Dic14—22no—
57C22×Dic14—22no—
58C22×Dic14C2×C414yes—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—42no—
63C2×C7×C4⋊C4C2×C414yes—
64C2×C4⋊Dic7—22no—
65C2×C4⋊Dic7—42no—
66C2×Dic7⋊C4—22no—
67C2×Dic7⋊C4—22no—
68C2×Dic7⋊C4—22no—
69C2×Dic7⋊C4—22no—
70C2×Dic7⋊C4C2×C414yes—
71C2×Dic7⋊C4C2×C414yes—
72C2×C4×Dic7—22no—
73C2×C4×Dic7—22no—
74C2×C4×Dic7—22no—
75C2×C4×Dic7—22no—
76C2×C4×Dic7C2×C414yes—
77C2×C4×Dic7C2×C414yes—
78C2×C4×Dic7C2×C414yes—
79C2×C4×Dic7D414yes—
80C2×C4×Dic7D414yes—
81#6450—232no—
82C7×#95—22no—
83C7×#95—24no—
84C7×#95—42no—
85C7⋊#95—22no—
86C7⋊#95—24no—
87C7⋊#95—24no—
88C7⋊#95—24no—
89C7⋊#95—24no—
90C7⋊#95—24no—
91C7⋊#95—24no—
92C7⋊#95—24no—
93C7⋊#95—42no—
94C7⋊#95—42no—
95C7⋊#95—42no—
96C22×C4×Q8—71no—
97C2×#383—71no—
98C2×#382—71no—
99#1446—72no—
100#1445—72no—
101#1441—71no—
102#1441—71no—
103#1441—71no—
104#1441D711yescharacteristic
105#1434—72no—
106#1434—72no—
107C24×C7C2411yescharacteristic
108C22×C4×C7—21no—
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—21no—
115C22×C4×C7—21no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
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—22no—
132C22×C4×C7C22×C412yes—
133C22×C4×C7C22×C412yes—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C2×D412yes—
136C22×C4×C7C2×D412yes—
137C22×C4×C7#4112yes—
138C22×C4×C7#4112yes—
139C22×Dic7—21no—
140C22×Dic7—21no—
141C22×Dic7—21no—
142C22×Dic7—21no—
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—21no—
150C22×Dic7—21no—
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×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7C22×C412yes—
168C22×Dic7C22×C412yes—
169C22×Dic7C22×C412yes—
170C22×Dic7C22×C412yes—
171C22×Dic7C2×D412yes—
172C22×Dic7C2×D412yes—
173C22×Dic7C2×D412yes—
174C22×Dic7C2×D412yes—
175C22×Dic7#4112yes—
176C22×Dic7#4112yes—
177C2×Dic14—28no—
178C2×Dic14—28no—
179C2×Dic14—28no—
180C2×Dic14—28no—
181C2×Dic14—28no—
182C2×Dic14—28no—
183C2×Dic14—28no—
184C7×C4⋊C4—28no—
185C7×C4⋊C4—44no—
186C4⋊Dic7—44no—
187Dic7⋊C4—28no—
188Dic7⋊C4—28no—
189C4×Dic7—28no—
190C4×Dic7—28no—
191C4×Dic7C22⋊C4116yes—
192C23×Q8—71no—
193C2×C4×Q8—78no—
194C2×C4×Q8—144no—
195C2×C4×Q8—144no—
196C22×C4⋊C4—71no—
197C22×C4⋊C4—71no—
198C22×C4⋊C4—71no—
199C22×C4⋊C4C2×D711yescharacteristic
200C22×C4⋊C4C2×D711yescharacteristic
201C22×C42—71no—
202C22×C42—71no—
203C22×C42—71no—
204#383—144no—
205#382—144no—
206C2×#95—72no—
207C2×#95—72no—
208C2×#95—72no—
209C2×#95—72no—
210C2×#95—72no—
211C2×#95—71no—
212C2×#95—72no—
213C2×#95—72no—
214C2×#95—72no—
215C2×#95—72no—
216C2×#95—72no—
217C2×#95C2×D712yes—
218C2×#95C2×D711yescharacteristic
219C2×#95C2×D712yes—
220C23×C7C23×C411yescharacteristic
221C23×C7C22×D411yescharacteristic
222C23×C7C22×D411yescharacteristic
223C23×C7C2×#4111yescharacteristic
224C23×C7C2×#4111yescharacteristic
225C23×C7C2×#4111yescharacteristic
226C23×C7C2×#4111yescharacteristic
227C23×C7H(5,𝔽2)12yes—
228C23×C72−1+412yes—
229C23×C72−1+412yes—
230C23×C72−1+412yes—
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—24no—
234C2×C4×C7—22no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—24no—
241C2×C4×C7—22no—
242C2×C4×C7—24no—
243C2×C4×C7—41no—
244C2×C4×C7—41no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
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×C7C2×C22⋊C414yes—
261C2×C4×C7#11714yes—
262C2×Dic7—22no—
263C2×Dic7—22no—
264C2×Dic7—22no—
265C2×Dic7—22no—
266C2×Dic7—24no—
267C2×Dic7—22no—
268C2×Dic7—22no—
269C2×Dic7—22no—
270C2×Dic7—22no—
271C2×Dic7—22no—
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—22no—
275C2×Dic7—22no—
276C2×Dic7—24no—
277C2×Dic7—22no—
278C2×Dic7—22no—
279C2×Dic7—24no—
280C2×Dic7—41no—
281C2×Dic7—41no—
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×Dic7—42no—
291C2×Dic7—42no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7C2×C22⋊C414yes—
298C2×Dic7C2×C22⋊C414yes—
299C2×Dic7#11714yes—
300Dic14—232no—
301C22×Q8—74no—
302C22×Q8—142no—
303C22×Q8—142no—
304C22×Q8—142no—
305C22×Q8—142no—
306C22×Q8—142no—
307C22×Q8—142no—
308C23×C4—71no—
309C23×C4—71no—
310C23×C4—71no—
311C23×C4—71no—
312C23×C4—72no—
313C23×C4—72no—
314C23×C4C22×D712yes—
315C23×C4C22×D712yes—
316C23×C4C22×D711yescharacteristic
317C23×C4C22×D711yescharacteristic
318C23×C4C22×D711yescharacteristic
319C4×Q8—1432no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—74no—
325C2×C4⋊C4—74no—
326C2×C4⋊C4—142no—
327C2×C4⋊C4—142no—
328C2×C4⋊C4—142no—
329C2×C4⋊C4—142no—
330C2×C4⋊C4—142no—
331C2×C4⋊C4—282no—
332C2×C4⋊C4C4×D714yes—
333C2×C42—74no—
334C2×C42—74no—
335C2×C42—74no—
336C2×C42—74no—
337C2×C42—74no—
338C2×C42—142no—
339C2×C42—142no—
340C2×C42—142no—
341C2×C42—142no—
342#95—22no—
343#95—24no—
344#95—42no—
345#95—144no—
346#95—144no—
347#95—144no—
348#95—144no—
349#95—144no—
350#95—144no—
351#95—142no—
352#95—144no—
353#95—282no—
354#95—282no—
355#95—282no—
356C22×C7C22×C22⋊C411yescharacteristic
357C22×C7C2×#11711yescharacteristic
358C22×C7C2×#11711yescharacteristic
359C22×C7#51912yes—
360C22×C7#52012yes—
361C22×C7#52511yescharacteristic
362C22×C7#52511yescharacteristic
363C22×C7#52511yescharacteristic
364C22×C7#52511yescharacteristic
365C22×C7#53511yescharacteristic
366C22×C7#53611yescharacteristic
367C22×C7#53612yes—
368C22×C7#53611yescharacteristic
369C22×C7#53611yescharacteristic
370C22×C7#53712yes—
371C22×C7#54112yes—
372C22×C7#54112yes—
373C22×C7#54112yes—
374C22×C7#54112yes—
375C22×C7#54212yes—
376C22×C7#54212yes—
377C22×C7#54212yes—
378C22×C7#54212yes—
379C4×C7—24no—
380C4×C7—24no—
381C4×C7—42no—
382C4×C7—82no—
383C4×C7—82no—
384Dic7—24no—
385Dic7—24no—
386Dic7—24no—
387Dic7—42no—
388Dic7—82no—
389Dic7—82no—
390C24C23×D711yescharacteristic
391C2×Q8—148no—
392C2×Q8—148no—
393C2×Q8—148no—
394C2×Q8—148no—
395C2×Q8—148no—
396C2×Q8—148no—
397C2×Q8—148no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—72no—
423C22×C4—72no—
424C22×C4—72no—
425C22×C4—72no—
426C22×C4—72no—
427C22×C4—72no—
428C22×C4—72no—
429C22×C4—72no—
430C22×C4—72no—
431C22×C4—72no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4—141no—
436C22×C4—141no—
437C22×C4—141no—
438C22×C4—142no—
439C22×C4—141no—
440C22×C4—141no—
441C22×C4—142no—
442C22×C4—141no—
443C22×C4—141no—
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×C4—142no—
459C22×C4—142no—
460C22×C4C2×C4×D712yes—
461C22×C4C2×C4×D712yes—
462C22×C4C2×C4×D712yes—
463C22×C4D4×D712yes—
464C22×C4D4×D712yes—
465C22×C4#115912yes—
466C22×C4#116112yes—
467C4⋊C4—28no—
468C4⋊C4—44no—
469C4⋊C4—148no—
470C4⋊C4—148no—
471C4⋊C4—284no—
472C42—716no—
473C42—148no—
474C42—148no—
475C2×C7#228011yescharacteristic
476C2×C7#231011yescharacteristic
477C2×C7#231111yescharacteristic
478C2×C7#232012yes—
479C2×C7#232012yes—
480C2×C7#263211yescharacteristic
481C2×C7#263211yescharacteristic
482C2×C7#263411yescharacteristic
483C2×C7#263411yescharacteristic
484C2×C7#263712yes—
485C2×C7#263712yes—
486C23C22×C4×D711yescharacteristic
487C23C2×D4×D711yescharacteristic
488C23C2×D4×D711yescharacteristic
489C23C2×#115911yescharacteristic
490C23#654812yes—
491C23C2×#116111yescharacteristic
492C23#655112yes—
493C23D7×#4111yescharacteristic
494C23D7×#4111yescharacteristic
495C23#655412yes—
496C23#655412yes—
497Q8—1432no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—74no—
528C2×C4—74no—
529C2×C4—74no—
530C2×C4—142no—
531C2×C4—142no—
532C2×C4—142no—
533C2×C4—142no—
534C2×C4—142no—
535C2×C4—144no—
536C2×C4—142no—
537C2×C4—142no—
538C2×C4—142no—
539C2×C4—142no—
540C2×C4—142no—
541C2×C4—142no—
542C2×C4—142no—
543C2×C4—142no—
544C2×C4—144no—
545C2×C4—142no—
546C2×C4—142no—
547C2×C4—144no—
548C2×C4—281no—
549C2×C4—282no—
550C2×C4—281no—
551C2×C4—282no—
552C2×C4—282no—
553C2×C4—282no—
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×C4D7×C22⋊C414yes—
566C2×C4#645514yes—
567C7#1434611yescharacteristic, complemented, Sylow
568C22C2×D7×C22⋊C411yescharacteristic
569C22C2×#645511yescharacteristic
570C22D7×#11711yescharacteristic
571C22#9023712yes—
572C22#9026312yes—
573C22#9028311yescharacteristic
574C22#9028812yes—
575C22#9029412yes—
576C22#9029711yescharacteristic
577C22#9029711yescharacteristic
578C22#9031612yes—
579C22#9031811yescharacteristic
580C22#9031811yescharacteristic
581C22#9031912yes—
582C22#9033112yes—
583C22#9033911yescharacteristic
584C22#9035312yes—
585C22#9036611yescharacteristic
586C22#9038712yes—
587C22#9038812yes—
588C22#9039212yes—
589C22#9041511yescharacteristic
590C22#9042412yes—
591C4—24no—
592C4—24no—
593C4—42no—
594C4—82no—
595C4—82no—
596C4—144no—
597C4—144no—
598C4—144no—
599C4—282no—
600C4—562no—
601C4—562no—
602C2#1173130711yescharacteristic
603C2#1173210711yescharacteristic
604C2#1173239112yes—
605C2#1173341512yes—
606C2#1173405511yescharacteristic
607C2#1173442211yescharacteristic
608C2#1173464811yescharacteristic
609C2#1173478112yes—
610C2#1173526811yescharacteristic
611C2#1173526912yes—
612C2#1173565611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1441C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1441D7
hypocenterC7#14346
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
23. C2×#6450C4semidirect
567. C7#14346semidirect

Maximal subgroups

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

subgroupindex
C22×#64502
C2×#897492
C2×#894232
C7×#14412
#117293202
#117293192
#117271782
#117271752
#117271642
#117271572
#117271552
#143467