Finite Group Databasebeta

← #13281015

Subgroups — #13281015

Order 1792.

1542 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic14C211yesmaximal, characteristic
2C2×#89423C211yesmaximal, characteristic
3C2×#89422C211yesmaximal, characteristic
4C7×#1434C211yesmaximal, characteristic
5#11729288C211yesmaximal, characteristic
6#11727178C212yesmaximal
7#11727175C212yesmaximal
8#11727165C212yesmaximal
9#11727163C212yesmaximal
10#11727156C211yesmaximal, characteristic
11#11727156C211yesmaximal, characteristic
12C23×Dic14C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×C4⋊Dic7C2211yescharacteristic
15C22×Dic7⋊C4C2211yescharacteristic
16C22×Dic7⋊C4C2211yescharacteristic
17C22×Dic7⋊C4C2212yes—
18C22×C4×Dic7C2211yescharacteristic
19C22×C4×Dic7C2211yescharacteristic
20C2×C4×Dic14—24no—
21C2×C4×Dic14—24no—
22C2×C4×Dic14C418yescomplemented
23C2×C7×#95C2211yescharacteristic
24C2×C7×#95C2211yescharacteristic
25C2×C7×#95C2212yes—
26C2×C7×#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2211yescharacteristic
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38#89423—24no—
39#89422—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—24no—
53C22×Dic14—22no—
54C22×Dic14—22no—
55C22×Dic14—24no—
56C22×Dic14C2×C414yes—
57C2×C42×C7—22no—
58C2×C42×C7—22no—
59C2×C42×C7C2×C414yes—
60C2×C4⋊Dic7—22no—
61C2×C4⋊Dic7—22no—
62C2×C4⋊Dic7C2×C414yes—
63C2×Dic7⋊C4—22no—
64C2×Dic7⋊C4—22no—
65C2×Dic7⋊C4—22no—
66C2×Dic7⋊C4—22no—
67C2×Dic7⋊C4—24no—
68C2×Dic7⋊C4—42no—
69C2×Dic7⋊C4—42no—
70C2×Dic7⋊C4C2×C414yes—
71C2×Dic7⋊C4C2×C414yes—
72C2×C4×Dic7—22no—
73C2×C4×Dic7—22no—
74C2×C4×Dic7C2×C414yes—
75C2×C4×Dic7C2×C414yes—
76C2×C4×Dic7D414yes—
77C2×C4×Dic7D414yes—
78C4×Dic14—232no—
79C7×#95—22no—
80C7×#95—22no—
81C7×#95—24no—
82C7×#95—42no—
83C7⋊#95—22no—
84C7⋊#95—22no—
85C7⋊#95—24no—
86C7⋊#95—24no—
87C7⋊#95—24no—
88C7⋊#95—24no—
89C7⋊#95—24no—
90C7⋊#95—24no—
91C7⋊#95—42no—
92C7⋊#95—42no—
93C7⋊#95—42no—
94C22×C4×Q8—71no—
95C2×#383—71no—
96C2×#382—71no—
97#1446—72no—
98#1445—72no—
99#1441—72no—
100#1441—72no—
101#1434—71no—
102#1434—71no—
103#1434—71no—
104#1434D711yescharacteristic
105C24×C7C2411yescharacteristic
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—21no—
116C22×C4×C7—21no—
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×C7C22×C412yes—
132C22×C4×C7C22×C412yes—
133C22×C4×C7C22×C412yes—
134C22×C4×C7C2×D412yes—
135C22×C4×C7C2×D412yes—
136C22×C4×C7#4114yes—
137C22×Dic7—21no—
138C22×Dic7—21no—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—21no—
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
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×Dic7C22×C412yes—
167C22×Dic7C22×C412yes—
168C22×Dic7C22×C412yes—
169C22×Dic7C22×C412yes—
170C22×Dic7C2×D412yes—
171C22×Dic7C2×D412yes—
172C22×Dic7C2×D412yes—
173C22×Dic7C2×D412yes—
174C22×Dic7#4114yes—
175C2×Dic14—216no—
176C2×Dic14—216no—
177C2×Dic14—28no—
178C2×Dic14—28no—
179C2×Dic14—28no—
180C42×C7—28no—
181C4⋊Dic7—28no—
182Dic7⋊C4—28no—
183Dic7⋊C4—28no—
184Dic7⋊C4—48no—
185C4×Dic7—28no—
186C4×Dic7C22⋊C4116yes—
187C23×Q8—71no—
188C2×C4×Q8—78no—
189C2×C4×Q8—144no—
190C2×C4×Q8—144no—
191C22×C4⋊C4—71no—
192C22×C4⋊C4—71no—
193C22×C4⋊C4—71no—
194C22×C4⋊C4—72no—
195C22×C42—71no—
196C22×C42—71no—
197C22×C42C2×D711yescharacteristic
198#383—144no—
199#382—144no—
200C2×#95—71no—
201C2×#95—71no—
202C2×#95—72no—
203C2×#95—72no—
204C2×#95—72no—
205C2×#95—72no—
206C2×#95—72no—
207C2×#95—72no—
208C2×#95—72no—
209C2×#95—72no—
210C2×#95—72no—
211C2×#95C2×D711yescharacteristic
212C2×#95C2×D711yescharacteristic
213C2×#95C2×D712yes—
214C2×#95C2×D712yes—
215C23×C7C23×C411yescharacteristic
216C23×C7C22×D411yescharacteristic
217C23×C7C22×D411yescharacteristic
218C23×C7C2×#4112yes—
219C23×C7C2×#4112yes—
220C23×C7H(5,𝔽2)11yescharacteristic
221C23×C7H(5,𝔽2)11yescharacteristic
222C23×C72−1+411yescharacteristic
223C23×C72−1+412yes—
224C23×C72−1+412yes—
225C23×C72−1+411yescharacteristic
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—22no—
229C2×C4×C7—22no—
230C2×C4×C7—24no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—22no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—41no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—41no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
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×C7C2×C22⋊C414yes—
256C2×C4×C7#11714yes—
257C2×Dic7—22no—
258C2×Dic7—22no—
259C2×Dic7—22no—
260C2×Dic7—22no—
261C2×Dic7—22no—
262C2×Dic7—22no—
263C2×Dic7—24no—
264C2×Dic7—24no—
265C2×Dic7—24no—
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—41no—
275C2×Dic7—41no—
276C2×Dic7—41no—
277C2×Dic7—41no—
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×Dic7—42no—
291C2×Dic7—42no—
292C2×Dic7C2×C22⋊C414yes—
293C2×Dic7C2×C22⋊C414yes—
294C2×Dic7#11714yes—
295Dic14—232no—
296C22×Q8—74no—
297C22×Q8—142no—
298C22×Q8—142no—
299C22×Q8—144no—
300C22×Q8—144no—
301C23×C4—71no—
302C23×C4—71no—
303C23×C4—71no—
304C23×C4—71no—
305C23×C4—72no—
306C23×C4—72no—
307C23×C4C22×D712yes—
308C23×C4C22×D712yes—
309C23×C4C22×D711yescharacteristic
310C23×C4C22×D711yescharacteristic
311C23×C4C22×D711yescharacteristic
312C4×Q8—1432no—
313C2×C4⋊C4—74no—
314C2×C4⋊C4—74no—
315C2×C4⋊C4—74no—
316C2×C4⋊C4—142no—
317C2×C4⋊C4—142no—
318C2×C4⋊C4—142no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4—142no—
322C2×C4⋊C4—144no—
323C2×C4⋊C4—282no—
324C2×C4⋊C4—282no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—74no—
328C2×C42—74no—
329C2×C42—74no—
330C2×C42—74no—
331C2×C42—142no—
332C2×C42—142no—
333C2×C42C4×D714yes—
334#95—24no—
335#95—22no—
336#95—22no—
337#95—42no—
338#95—144no—
339#95—144no—
340#95—144no—
341#95—144no—
342#95—144no—
343#95—144no—
344#95—142no—
345#95—142no—
346#95—282no—
347#95—282no—
348#95—282no—
349C22×C7C22×C22⋊C411yescharacteristic
350C22×C7C2×#11711yescharacteristic
351C22×C7C2×#11711yescharacteristic
352C22×C7#51911yescharacteristic
353C22×C7#51911yescharacteristic
354C22×C7#52011yescharacteristic
355C22×C7#52011yescharacteristic
356C22×C7#52512yes—
357C22×C7#52512yes—
358C22×C7#53511yescharacteristic
359C22×C7#53611yescharacteristic
360C22×C7#53612yes—
361C22×C7#53612yes—
362C22×C7#53711yescharacteristic
363C22×C7#53711yescharacteristic
364C22×C7#54112yes—
365C22×C7#54112yes—
366C22×C7#54112yes—
367C22×C7#54112yes—
368C22×C7#54212yes—
369C22×C7#54212yes—
370C22×C7#54212yes—
371C22×C7#54212yes—
372C4×C7—24no—
373C4×C7—24no—
374C4×C7—42no—
375C4×C7—82no—
376C4×C7—82no—
377Dic7—24no—
378Dic7—24no—
379Dic7—24no—
380Dic7—42no—
381Dic7—82no—
382Dic7—82no—
383C24C23×D711yescharacteristic
384C2×Q8—148no—
385C2×Q8—148no—
386C2×Q8—1416no—
387C2×Q8—1416no—
388C2×Q8—148no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
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—72no—
415C22×C4—72no—
416C22×C4—72no—
417C22×C4—72no—
418C22×C4—72no—
419C22×C4—72no—
420C22×C4—72no—
421C22×C4—74no—
422C22×C4—72no—
423C22×C4—142no—
424C22×C4—142no—
425C22×C4—141no—
426C22×C4—141no—
427C22×C4—141no—
428C22×C4—141no—
429C22×C4—142no—
430C22×C4—141no—
431C22×C4—141no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4—141no—
436C22×C4—141no—
437C22×C4—141no—
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×C4C2×C4×D712yes—
453C22×C4C2×C4×D712yes—
454C22×C4C2×C4×D712yes—
455C22×C4#115714yes—
456C22×C4D4×D712yes—
457C22×C4D4×D712yes—
458C4⋊C4—148no—
459C4⋊C4—148no—
460C4⋊C4—148no—
461C4⋊C4—288no—
462C42—28no—
463C42—716no—
464C42—148no—
465C2×C7#228011yescharacteristic
466C2×C7#231011yescharacteristic
467C2×C7#231111yescharacteristic
468C2×C7#232011yescharacteristic
469C2×C7#232011yescharacteristic
470C2×C7#232011yescharacteristic
471C2×C7#232011yescharacteristic
472C2×C7#263212yes—
473C2×C7#263412yes—
474C2×C7#263712yes—
475C2×C7#263712yes—
476C23C22×C4×D711yescharacteristic
477C23C2×#115712yes—
478C23C2×D4×D711yescharacteristic
479C23C2×D4×D711yescharacteristic
480C23#654811yescharacteristic
481C23#655111yescharacteristic
482C23D7×#4112yes—
483C23#655311yescharacteristic
484C23#655412yes—
485C23#655411yescharacteristic
486C23#655412yes—
487Q8—1432no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—74no—
518C2×C4—74no—
519C2×C4—74no—
520C2×C4—142no—
521C2×C4—142no—
522C2×C4—142no—
523C2×C4—142no—
524C2×C4—142no—
525C2×C4—142no—
526C2×C4—144no—
527C2×C4—144no—
528C2×C4—144no—
529C2×C4—144no—
530C2×C4—142no—
531C2×C4—142no—
532C2×C4—142no—
533C2×C4—142no—
534C2×C4—142no—
535C2×C4—142no—
536C2×C4—142no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—281no—
540C2×C4—281no—
541C2×C4—282no—
542C2×C4—282no—
543C2×C4—282no—
544C2×C4—282no—
545C2×C4—282no—
546C2×C4—282no—
547C2×C4—282no—
548C2×C4—282no—
549C2×C4—282no—
550C2×C4—282no—
551C2×C4—282no—
552C2×C4—282no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4#643514yes—
556C2×C4D7×C22⋊C414yes—
557C7#1434611yescharacteristic, complemented, Sylow
558C22C2×#643511yescharacteristic
559C22C2×D7×C22⋊C411yescharacteristic
560C22#9019511yescharacteristic
561C22#9020711yescharacteristic
562C22#9020711yescharacteristic
563C22#9020912yes—
564C22D7×#11711yescharacteristic
565C22#9021411yescharacteristic
566C22#9022512yes—
567C22#9023812yes—
568C22#9023812yes—
569C22#9025212yes—
570C22#9027412yes—
571C22#9029811yescharacteristic
572C22#9029911yescharacteristic
573C22#9031911yescharacteristic
574C22#9032011yescharacteristic
575C22#9033912yes—
576C22#9033912yes—
577C22#9036012yes—
578C22#9038712yes—
579C22#9039212yes—
580C22#9040112yes—
581C4—24no—
582C4—24no—
583C4—42no—
584C4—82no—
585C4—82no—
586C4—144no—
587C4—144no—
588C4—144no—
589C4—282no—
590C4—562no—
591C4—562no—
592C2#1173083111yescharacteristic
593C2#1173100611yescharacteristic
594C2#1173107011yescharacteristic
595C2#1173144811yescharacteristic
596C2#1173149811yescharacteristic
597C2#1173159611yescharacteristic
598C2#1173170511yescharacteristic
599C2#1173258012yes—
600C2#1173282312yes—
601C2#1173309712yes—
602C2#1173309812yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1434C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1434D7
hypocenterC7#14346
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
22. C2×C4×Dic14C4semidirect
557. C7#14346semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic142
C2×#894232
C2×#894222
C7×#14342
#117292882
#117271782
#117271752
#117271652
#117271632
#117271562
#143467