Finite Group Databasebeta

← #13276297

Subgroups — #13276297

Order 1792.

1794 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic14C211yesmaximal, characteristic
2C2×#89418C212yesmaximal
3C2×#89419C212yesmaximal
4C7×#1434C211yesmaximal, characteristic
5#11729289C211yesmaximal, characteristic
6#11727165C212yesmaximal
7#11727164C212yesmaximal
8Dic7⋊#95C211yesmaximal, characteristic
9Dic7⋊#95C211yesmaximal, characteristic
10#11727155C211yesmaximal, characteristic
11Dic7×#95C211yesmaximal, characteristic
12C23×Dic14C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×C4⋊Dic7C2212yes—
15C22×C4⋊Dic7C2211yescharacteristic
16C22×Dic7⋊C4C2212yes—
17C22×Dic7⋊C4C2212yes—
18C22×Dic7⋊C4C2211yescharacteristic
19C22×Dic7⋊C4C2211yescharacteristic
20C22×C4×Dic7C2212yes—
21C22×C4×Dic7C2211yescharacteristic
22C22×C4×Dic7C2211yescharacteristic
23C2×C4×Dic14—28no—
24C2×C4×Dic14C418yescomplemented
25C2×C7×#95C2211yescharacteristic
26C2×C7×#95C2212yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2211yescharacteristic
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2211yescharacteristic
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37#89419—28no—
38#89418—28no—
39#13935—71nomaximal, Sylow
40C23×C4×C7C2312yes—
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×Dic7C2312yes—
46C23×Dic7C2311yescharacteristic
47C23×Dic7C2312yes—
48C23×Dic7C2311yescharacteristic
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2311yescharacteristic
51C22×Dic14—24no—
52C22×Dic14—24no—
53C22×Dic14C2×C414yes—
54C22×Dic14D414yes—
55C22×Dic14Q814yes—
56C2×C42×C7—24no—
57C2×C42×C7C2×C414yes—
58C2×C4⋊Dic7—24no—
59C2×C4⋊Dic7—24no—
60C2×C4⋊Dic7—24no—
61C2×C4⋊Dic7—42no—
62C2×C4⋊Dic7C2×C414yes—
63C2×Dic7⋊C4—24no—
64C2×Dic7⋊C4—24no—
65C2×Dic7⋊C4—24no—
66C2×Dic7⋊C4—24no—
67C2×Dic7⋊C4—24no—
68C2×Dic7⋊C4—24no—
69C2×Dic7⋊C4—42no—
70C2×Dic7⋊C4C2×C414yes—
71C2×Dic7⋊C4C2×C414yes—
72C2×Dic7⋊C4D418yes—
73C2×C4×Dic7—24no—
74C2×C4×Dic7—24no—
75C2×C4×Dic7—24no—
76C2×C4×Dic7—24no—
77C2×C4×Dic7—24no—
78C2×C4×Dic7C2×C414yes—
79C2×C4×Dic7C2×C414yes—
80C4×Dic14—232no—
81C7×#95—24no—
82C7×#95—24no—
83C7×#95—22no—
84C7×#95Q814yescomplemented
85C7⋊#95—24no—
86C7⋊#95—22no—
87C7⋊#95—22no—
88C7⋊#95—24no—
89C7⋊#95—24no—
90C7⋊#95—24no—
91C7⋊#95—24no—
92C7⋊#95—42no—
93C22×C4×Q8—71no—
94C2×#391—72no—
95C2×#389—72no—
96#1441—72no—
97#1441—72no—
98C4⋊#95—71no—
99C4⋊#95—71no—
100C4⋊#95—71no—
101#1434—71no—
102#1434D711yescharacteristic
103C4×#95—71no—
104C24×C7C2411yescharacteristic
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—21no—
110C22×C4×C7—22no—
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—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×C7C22×C412yes—
126C22×C4×C7C22×C412yes—
127C22×C4×C7C22×C412yes—
128C22×C4×C7C2×D414yes—
129C22×C4×C7C2×D412yes—
130C22×C4×C7C2×Q814yes—
131C22×C4×C7C2×Q812yes—
132C22×C4×C7C2×Q812yes—
133C22×C4×C7#4112yes—
134C22×Dic7—22no—
135C22×Dic7—22no—
136C22×Dic7—22no—
137C22×Dic7—21no—
138C22×Dic7—21no—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—21no—
146C22×Dic7—22no—
147C22×Dic7—21no—
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×Dic7C22×C412yes—
157C22×Dic7C22×C412yes—
158C22×Dic7C22×C412yes—
159C22×Dic7C22×C412yes—
160C22×Dic7C2×D414yes—
161C22×Dic7C2×D412yes—
162C22×Dic7C2×D414yes—
163C22×Dic7C2×D412yes—
164C22×Dic7C2×Q812yes—
165C22×Dic7C2×Q812yes—
166C22×Dic7#4114yes—
167C22×Dic7#4114yes—
168C2×Dic14—28no—
169C2×Dic14—28no—
170C2×Dic14—216no—
171C2×Dic14—216no—
172C2×Dic14C4⋊C4116yescomplemented
173C42×C7—28no—
174C4⋊Dic7—28no—
175C4⋊Dic7—48no—
176Dic7⋊C4—28no—
177Dic7⋊C4—48no—
178Dic7⋊C4—48no—
179Dic7⋊C4C22⋊C4116yes—
180C4×Dic7—28no—
181C4×Dic7—28no—
182C4×Dic7—48no—
183C23×Q8—71no—
184C2×C4×Q8—78no—
185C2×C4×Q8—148no—
186C22×C4⋊C4—71no—
187C22×C4⋊C4—71no—
188C22×C4⋊C4—71no—
189C22×C4⋊C4—72no—
190C22×C4⋊C4—72no—
191C22×C4⋊C4—72no—
192C22×C42—71no—
193C22×C42—71no—
194C22×C42—72no—
195C22×C42C2×D711yescharacteristic
196#391—148no—
197#389—148no—
198C2×#95—71no—
199C2×#95—72no—
200C2×#95—71no—
201C2×#95—72no—
202C2×#95—72no—
203C2×#95—72no—
204C2×#95—72no—
205C2×#95—72no—
206C2×#95C2×D711yescharacteristic
207C2×#95C2×D712yes—
208C2×#95C2×D712yes—
209C2×#95C2×D711yescharacteristic
210C23×C7C23×C411yescharacteristic
211C23×C7C22×D412yes—
212C23×C7C22×D411yescharacteristic
213C23×C7C22×Q811yescharacteristic
214C23×C7C22×Q811yescharacteristic
215C23×C7C2×#4111yescharacteristic
216C23×C7C2×#4112yes—
217C23×C7C2×#4112yes—
218C23×C7H(5,𝔽2)11yescharacteristic
219C23×C72−1+412yes—
220C23×C72−1+411yescharacteristic
221C2×C4×C7—22no—
222C2×C4×C7—22no—
223C2×C4×C7—22no—
224C2×C4×C7—22no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—24no—
228C2×C4×C7—24no—
229C2×C4×C7—24no—
230C2×C4×C7—24no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—24no—
235C2×C4×C7—24no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
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×C7C2×C22⋊C414yes—
248C2×C4×C7C2×C4⋊C414yes—
249C2×C4×C7C4×Q814yes—
250C2×Dic7—22no—
251C2×Dic7—22no—
252C2×Dic7—22no—
253C2×Dic7—22no—
254C2×Dic7—24no—
255C2×Dic7—24no—
256C2×Dic7—24no—
257C2×Dic7—24no—
258C2×Dic7—22no—
259C2×Dic7—22no—
260C2×Dic7—22no—
261C2×Dic7—24no—
262C2×Dic7—24no—
263C2×Dic7—24no—
264C2×Dic7—42no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7C2×C22⋊C414yes—
278C2×Dic7C2×C22⋊C414yes—
279C2×Dic7C2×C4⋊C414yes—
280C2×Dic7C2×C4⋊C414yes—
281C2×Dic7#11714yes—
282C2×Dic7#12118yes—
283C2×Dic7C22⋊Q818yes—
284Dic14—232no—
285C22×Q8—74no—
286C22×Q8—74no—
287C22×Q8—74no—
288C22×Q8—144no—
289C22×Q8—144no—
290C23×C4—71no—
291C23×C4—71no—
292C23×C4—71no—
293C23×C4—71no—
294C23×C4—72no—
295C23×C4—72no—
296C23×C4C22×D712yes—
297C23×C4C22×D712yes—
298C23×C4C22×D711yescharacteristic
299C23×C4C22×D711yescharacteristic
300C23×C4C22×D711yescharacteristic
301C4×Q8—1432no—
302C2×C4⋊C4—74no—
303C2×C4⋊C4—78no—
304C2×C4⋊C4—74no—
305C2×C4⋊C4—74no—
306C2×C4⋊C4—144no—
307C2×C4⋊C4—144no—
308C2×C4⋊C4—144no—
309C2×C4⋊C4—144no—
310C2×C4⋊C4—144no—
311C2×C4⋊C4—144no—
312C2×C4⋊C4—144no—
313C2×C4⋊C4—144no—
314C2×C4⋊C4—144no—
315C2×C4⋊C4—282no—
316C2×C4⋊C4—282no—
317C2×C42—24no—
318C2×C42—74no—
319C2×C42—74no—
320C2×C42—144no—
321C2×C42—144no—
322C2×C42—144no—
323C2×C42—144no—
324C2×C42—144no—
325C2×C42C4×D714yes—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—144no—
330#95—144no—
331#95—144no—
332#95—144no—
333#95—142no—
334#95—142no—
335#95—144no—
336#95—282no—
337#95Dic1414yescomplemented
338C22×C7C22×C22⋊C411yescharacteristic
339C22×C7C22×C4⋊C411yescharacteristic
340C22×C7C2×#11711yescharacteristic
341C22×C7C2×C4×Q811yescharacteristic
342C22×C7C4×#4111yescharacteristic
343C22×C7#51911yescharacteristic
344C22×C7#52011yescharacteristic
345C22×C7C2×#12112yes—
346C22×C7C2×C22⋊Q812yes—
347C22×C7#52912yes—
348C22×C7#53312yes—
349C22×C7#54112yes—
350C22×C7#54212yes—
351C22×C7#54612yes—
352C22×C7#54712yes—
353C22×C7#54812yes—
354C22×C7#54812yes—
355C22×C7D4×Q812yes—
356C22×C7D4×Q811yescharacteristic
357C22×C7#55011yescharacteristic
358C22×C7#55712yes—
359C22×C7#55711yescharacteristic
360C22×C7Q8211yescharacteristic
361C4×C7—24no—
362C4×C7—24no—
363C4×C7—24no—
364C4×C7—44no—
365C4×C7—82no—
366Dic7—24no—
367Dic7—24no—
368Dic7—42no—
369Dic7—44no—
370Dic7—82no—
371Dic7#38018yes—
372C24C23×D711yescharacteristic
373C2×Q8—716no—
374C2×Q8—148no—
375C2×Q8—148no—
376C2×Q8—1416no—
377C2×Q8—1416no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—72no—
399C22×C4—74no—
400C22×C4—72no—
401C22×C4—72no—
402C22×C4—74no—
403C22×C4—72no—
404C22×C4—72no—
405C22×C4—72no—
406C22×C4—74no—
407C22×C4—72no—
408C22×C4—72no—
409C22×C4—74no—
410C22×C4—142no—
411C22×C4—142no—
412C22×C4—142no—
413C22×C4—141no—
414C22×C4—141no—
415C22×C4—142no—
416C22×C4—142no—
417C22×C4—142no—
418C22×C4—142no—
419C22×C4—142no—
420C22×C4—142no—
421C22×C4—141no—
422C22×C4—142no—
423C22×C4—142no—
424C22×C4—141no—
425C22×C4—142no—
426C22×C4—142no—
427C22×C4—142no—
428C22×C4—142no—
429C22×C4—142no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4C2×Dic1414yes—
433C22×C4C2×Dic1412yes—
434C22×C4C2×C4×D712yes—
435C22×C4C2×C4×D712yes—
436C22×C4C2×C4×D712yes—
437C22×C4#115712yes—
438C22×C4D4×D714yes—
439C22×C4D4×D712yes—
440C22×C4Q8×D712yes—
441C4⋊C4—716no—
442C4⋊C4—148no—
443C4⋊C4—148no—
444C4⋊C4—288no—
445C4⋊C4—288no—
446C4⋊C4—288no—
447C42—28no—
448C42—148no—
449C42—148no—
450C42—288no—
451C2×C7C2×#38011yescharacteristic
452C2×C7#229511yescharacteristic
453C2×C7#232011yescharacteristic
454C2×C7#233911yescharacteristic
455C2×C7Q8×C22⋊C411yescharacteristic
456C2×C7Q8×C4⋊C411yescharacteristic
457C2×C7#235111yescharacteristic
458C2×C7#242612yes—
459C2×C7#242912yes—
460C2×C7#245112yes—
461C2×C7#245312yes—
462C23C22×Dic1411yescharacteristic
463C23C22×C4×D711yescharacteristic
464C23C2×#115711yescharacteristic
465C23C2×D4×D711yescharacteristic
466C23C2×D4×D712yes—
467C23C2×Q8×D711yescharacteristic
468C23D7×#4112yes—
469C23D7×#4112yes—
470C23#655311yescharacteristic
471C23#655412yes—
472C23#655411yescharacteristic
473Q8—1432no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—74no—
501C2×C4—74no—
502C2×C4—74no—
503C2×C4—78no—
504C2×C4—78no—
505C2×C4—74no—
506C2×C4—74no—
507C2×C4—142no—
508C2×C4—142no—
509C2×C4—142no—
510C2×C4—142no—
511C2×C4—144no—
512C2×C4—144no—
513C2×C4—144no—
514C2×C4—144no—
515C2×C4—144no—
516C2×C4—142no—
517C2×C4—142no—
518C2×C4—142no—
519C2×C4—144no—
520C2×C4—144no—
521C2×C4—282no—
522C2×C4—282no—
523C2×C4—282no—
524C2×C4—282no—
525C2×C4—282no—
526C2×C4—282no—
527C2×C4—282no—
528C2×C4—282no—
529C2×C4—282no—
530C2×C4—282no—
531C2×C4—282no—
532C2×C4—282no—
533C2×C4—282no—
534C2×C4C4×Dic1414yes—
535C2×C4D7×C22⋊C414yes—
536C2×C4D7×C4⋊C414yes—
537C7#1393511yescharacteristic, complemented, Sylow
538C22C2×C4×Dic1411yescharacteristic
539C22C2×D7×C22⋊C411yescharacteristic
540C22C2×D7×C4⋊C411yescharacteristic
541C22#9020711yescharacteristic
542C22#9020812yes—
543C22D7×#11711yescharacteristic
544C22#9021311yescharacteristic
545C22#9021411yescharacteristic
546C22#9021912yes—
547C22D4×Dic1411yescharacteristic
548C22D4×Dic1412yes—
549C22#9024312yes—
550C22#9024411yescharacteristic
551C22Q8×Dic1411yescharacteristic
552C22#9025811yescharacteristic
553C22D7×#12112yes—
554C22D7×C22⋊Q812yes—
555C22#9034512yes—
556C22#9034612yes—
557C22#9036812yes—
558C22#9037612yes—
559C22#9039212yes—
560C22#9040112yes—
561C4—24no—
562C4—24no—
563C4—24no—
564C4—44no—
565C4—78no—
566C4—82no—
567C4—144no—
568C4—144no—
569C4—282no—
570C4—284no—
571C4—562no—
572C2#1173082911yescharacteristic
573C2C22⋊C4×Dic1411yescharacteristic
574C2C4⋊C4×Dic1411yescharacteristic
575C2D7×#38011yescharacteristic
576C2#1173156611yescharacteristic
577C2#1173160211yescharacteristic
578C2#1173170511yescharacteristic
579C2#1173243212yes—
580C2#1173243312yes—
581C2#1173294412yes—
582C2#1173294512yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1434C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1434D7
hypocenterC7#13935
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
24. C2×C4×Dic14C4semidirect
84. C7×#95Q8semidirect
172. C2×Dic14C4⋊C4semidirect
337. #95Dic14semidirect
537. C7#13935semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic142
C2×#894182
C2×#894192
C7×#14342
#117292892
#117271652
#117271642
Dic7⋊#952
#117271552
Dic7×#952
#139357