Finite Group Databasebeta

← #13282986

Subgroups — #13282986

Order 1792.

1406 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89715C211yesmaximal, characteristic
2C7×#1437C211yesmaximal, characteristic
3#11729301C212yesmaximal
4#11729299C211yesmaximal, characteristic
5#11729298C212yesmaximal
6#11729288C212yesmaximal, complemented
7#11727171C212yesmaximal, complemented
8#11727167C212yesmaximal, complemented
9#11727160C212yesmaximal, complemented
10C24×Dic7C2211yescharacteristic
11C22×C7×C22⋊C4C2212yes—
12C22×C7×C22⋊C4C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×C22⋊Dic7C2212yes—
15C22×C22⋊Dic7C2211yescharacteristic
16C22×Dic7⋊C4C2212yes—
17C22×C4×Dic7C2211yescharacteristic
18C2×C7×#95C2212yes—
19C2×C7×#95C2211yescharacteristic
20C2×C7⋊#95C2212yes—
21C2×C7⋊#95C2212yes—
22C2×C7⋊#95C2212yes—
23C2×C7⋊#95C2212yes—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2211yescharacteristic
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31#89715C418yescomplemented
32#14507—71nomaximal, Sylow
33C25×C7C2311yescharacteristic
34C23×C4×C7C2312yes—
35C23×C4×C7C2311yescharacteristic
36C23×C4×C7C2311yescharacteristic
37C23×C4×C7C2312yes—
38C23×Dic7—24no—
39C23×Dic7—42no—
40C23×Dic7—41no—
41C23×Dic7—41no—
42C23×Dic7C2×C414yes—
43C23×Dic7C2312yes—
44C23×Dic7C2312yes—
45C23×Dic7C2312yes—
46C23×Dic7C2312yes—
47C2×C7×C22⋊C4—24no—
48C2×C7×C22⋊C4—24no—
49C2×C7×C22⋊C4—41no—
50C2×C7×C22⋊C4—41no—
51C2×C7×C22⋊C4—42no—
52C2×C7×C22⋊C4C2×C414yes—
53C2×C42×C7—24no—
54C2×C42×C7—41no—
55C2×C22⋊Dic7—24no—
56C2×C22⋊Dic7—24no—
57C2×C22⋊Dic7—42no—
58C2×C22⋊Dic7—42no—
59C2×C22⋊Dic7C2×C414yes—
60C2×Dic7⋊C4—24no—
61C2×Dic7⋊C4—42no—
62C2×Dic7⋊C4—42no—
63C2×C4×Dic7—41no—
64C2×C4×Dic7—42no—
65C7×#95—42no—
66C7×#95—41no—
67C7⋊#95—24no—
68C7⋊#95—24no—
69C7⋊#95—42no—
70C7⋊#95—42no—
71C7⋊#95—42no—
72C7⋊#95—42no—
73C7⋊#95—42no—
74C7⋊#95—42no—
75C7⋊#95—41no—
76C7⋊#95C2×C418yes—
77C7⋊#95C2×C418yes—
78C2×#381—71no—
79#1443—72no—
80#1442—72no—
81#1439—72no—
82#1437—71no—
83#1437D711yescharacteristic
84#1436—72no—
85#1434—72no—
86C22⋊#95—72no—
87C24×C7—22no—
88C24×C7—22no—
89C24×C7—22no—
90C24×C7—22no—
91C24×C7—21no—
92C24×C7—21no—
93C24×C7—22no—
94C24×C7C22×C412yes—
95C24×C7#4114yes—
96C24×C7C2411yescharacteristic
97C22×C4×C7—22no—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—22no—
103C22×C4×C7—21no—
104C22×C4×C7—22no—
105C22×C4×C7—21no—
106C22×C4×C7—22no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
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×C7C22×C412yes—
119C22×C4×C7C22×C412yes—
120C22×C4×C7#4114yes—
121C22×Dic7—28no—
122C22×Dic7—22no—
123C22×Dic7—22no—
124C22×Dic7—22no—
125C22×Dic7—22no—
126C22×Dic7—22no—
127C22×Dic7—22no—
128C22×Dic7—22no—
129C22×Dic7—22no—
130C22×Dic7—22no—
131C22×Dic7—22no—
132C22×Dic7—22no—
133C22×Dic7—22no—
134C22×Dic7—22no—
135C22×Dic7—22no—
136C22×Dic7—22no—
137C22×Dic7—22no—
138C22×Dic7—22no—
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—44no—
148C22×Dic7—44no—
149C22×Dic7—48no—
150C22×Dic7—48no—
151C22×Dic7C22×C414yes—
152C22×Dic7C22×C414yes—
153C22×Dic7#4114yes—
154C7×C22⋊C4—44no—
155C7×C22⋊C4—48no—
156C42×C7—44no—
157C22⋊Dic7—44no—
158C22⋊Dic7—48no—
159Dic7⋊C4—48no—
160C4×Dic7—44no—
161C24×C4—71no—
162C22×C4⋊C4—72no—
163C22×C22⋊C4—71no—
164C22×C22⋊C4—72no—
165C22×C22⋊C4C2×D712yes—
166C22×C22⋊C4C2×D711yescharacteristic
167C22×C42—71no—
168C22×C42C2×D711yescharacteristic
169#381—78no—
170C2×#95—72no—
171C2×#95—72no—
172C2×#95—72no—
173C2×#95—72no—
174C2×#95—72no—
175C2×#95—72no—
176C2×#95—72no—
177C2×#95—72no—
178C2×#95—72no—
179C2×#95—71no—
180C2×#95—72no—
181C2×#95C2×D712yes—
182C2×#95C2×D711yescharacteristic
183C23×C7—24no—
184C23×C7—22no—
185C23×C7—22no—
186C23×C7—24no—
187C23×C7—24no—
188C23×C7—24no—
189C23×C7—24no—
190C23×C7—24no—
191C23×C7—24no—
192C23×C7—24no—
193C23×C7—42no—
194C23×C7—42no—
195C23×C7—42no—
196C23×C7—42no—
197C23×C7—41no—
198C23×C7—42no—
199C23×C7—41no—
200C23×C7—41no—
201C23×C7—42no—
202C23×C7—41no—
203C23×C7#11714yes—
204C23×C7C23×C411yescharacteristic
205C23×C7C2×#4112yes—
206C23×C7C2×#4112yes—
207C23×C7C2×#4112yes—
208C23×C7H(5,𝔽2)11yescharacteristic
209C23×C7H(5,𝔽2)11yescharacteristic
210C23×C7H(5,𝔽2)12yes—
211C23×C7H(5,𝔽2)12yes—
212C23×C72−1+412yes—
213C2×C4×C7—24no—
214C2×C4×C7—22no—
215C2×C4×C7—22no—
216C2×C4×C7—24no—
217C2×C4×C7—24no—
218C2×C4×C7—24no—
219C2×C4×C7—24no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—41no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7—41no—
226C2×C4×C7—41no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7—41no—
230C2×C4×C7—42no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×Dic7—24no—
238C2×Dic7—24no—
239C2×Dic7—24no—
240C2×Dic7—24no—
241C2×Dic7—24no—
242C2×Dic7—24no—
243C2×Dic7—24no—
244C2×Dic7—24no—
245C2×Dic7—24no—
246C2×Dic7—416no—
247C2×Dic7—42no—
248C2×Dic7—42no—
249C2×Dic7—42no—
250C2×Dic7—42no—
251C2×Dic7—42no—
252C2×Dic7—42no—
253C2×Dic7—42no—
254C2×Dic7—42no—
255C2×Dic7—42no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7—42no—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—42no—
265C2×Dic7—42no—
266C25C22×D711yescharacteristic
267C23×C4—72no—
268C23×C4—72no—
269C23×C4—72no—
270C23×C4—74no—
271C23×C4—72no—
272C23×C4—144no—
273C23×C4—281no—
274C23×C4—281no—
275C23×C4—282no—
276C23×C4C22×D712yes—
277C23×C4C22×D711yescharacteristic
278C23×C4C22×D711yescharacteristic
279C23×C4C22×D712yes—
280C2×C4⋊C4—144no—
281C2×C4⋊C4—282no—
282C2×C4⋊C4—282no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—74no—
289C2×C22⋊C4—144no—
290C2×C22⋊C4—144no—
291C2×C22⋊C4—282no—
292C2×C22⋊C4—282no—
293C2×C22⋊C4C4×D714yes—
294C2×C42—24no—
295C2×C42—41no—
296C2×C42—282no—
297C2×C42—281no—
298#95—42no—
299#95—41no—
300#95—78no—
301#95—78no—
302#95—144no—
303#95—144no—
304#95—282no—
305#95—281no—
306#95—282no—
307#95—282no—
308#95—282no—
309#95—282no—
310#95—282no—
311C22×C7—24no—
312C22×C7—28no—
313C22×C7—44no—
314C22×C7—42no—
315C22×C7—42no—
316C22×C7—44no—
317C22×C7—44no—
318C22×C7—44no—
319C22×C7—44no—
320C22×C7C2×#11711yescharacteristic
321C22×C7C4×#4112yes—
322C22×C7#51811yescharacteristic
323C22×C7#51811yescharacteristic
324C22×C7#52012yes—
325C22×C7#53812yes—
326C22×C7#53812yes—
327C22×C7#53812yes—
328C22×C7#53912yes—
329C22×C7#54012yes—
330C22×C7#54212yes—
331C22×C7#55112yes—
332C22×C7#55112yes—
333C22×C7#55111yescharacteristic
334C22×C7#55212yes—
335C22×C7#55212yes—
336C22×C7#55312yes—
337C22×C7#55911yescharacteristic
338C22×C7#56011yescharacteristic
339C22×C7#56212yes—
340C22×C7#56311yescharacteristic
341C4×C7—42no—
342C4×C7—42no—
343C4×C7—44no—
344C4×C7—82no—
345Dic7—44no—
346Dic7—44no—
347Dic7—44no—
348Dic7—82no—
349C24—22no—
350C24—22no—
351C24—21no—
352C24—21no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24C2×C4×D712yes—
357C24#115714yes—
358C24C23×D711yescharacteristic
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—74no—
381C22×C4—74no—
382C22×C4—74no—
383C22×C4—148no—
384C22×C4—142no—
385C22×C4—142no—
386C22×C4—142no—
387C22×C4—142no—
388C22×C4—142no—
389C22×C4—142no—
390C22×C4—142no—
391C22×C4—142no—
392C22×C4—142no—
393C22×C4—142no—
394C22×C4—142no—
395C22×C4—142no—
396C22×C4—142no—
397C22×C4—142no—
398C22×C4—142no—
399C22×C4—142no—
400C22×C4—142no—
401C22×C4—142no—
402C22×C4—142no—
403C22×C4—142no—
404C22×C4—142no—
405C22×C4—142no—
406C22×C4—142no—
407C22×C4—142no—
408C22×C4—142no—
409C22×C4—284no—
410C22×C4—284no—
411C22×C4—288no—
412C22×C4—288no—
413C22×C4C2×C4×D712yes—
414C22×C4C2×C4×D712yes—
415C22×C4#115914yes—
416C4⋊C4—288no—
417C22⋊C4—44no—
418C22⋊C4—48no—
419C22⋊C4—284no—
420C22⋊C4—288no—
421C42—44no—
422C42—284no—
423C2×C7—44no—
424C2×C7#231211yescharacteristic
425C2×C7#233512yes—
426C2×C7#234312yes—
427C2×C7#237411yescharacteristic
428C2×C7#237611yescharacteristic
429C2×C7#274312yes—
430C2×C7#274912yes—
431C2×C7#275112yes—
432C2×C7#276512yes—
433C23—22no—
434C23—24no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—42no—
444C23—42no—
445C23—41no—
446C23—41no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—41no—
453C23#643514yes—
454C23C22×C4×D711yescharacteristic
455C23C2×#115712yes—
456C23C2×#115912yes—
457C23#654812yes—
458C23#654811yescharacteristic
459C23#654811yescharacteristic
460C23D7×#4112yes—
461C23#655312yes—
462C23#655412yes—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—144no—
488C2×C4—144no—
489C2×C4—144no—
490C2×C4—144no—
491C2×C4—144no—
492C2×C4—144no—
493C2×C4—144no—
494C2×C4—144no—
495C2×C4—144no—
496C2×C4—2816no—
497C2×C4—282no—
498C2×C4—282no—
499C2×C4—282no—
500C2×C4—282no—
501C2×C4—282no—
502C2×C4—282no—
503C2×C4—282no—
504C2×C4—282no—
505C2×C4—282no—
506C2×C4—282no—
507C2×C4—282no—
508C2×C4—282no—
509C2×C4—282no—
510C2×C4—282no—
511C2×C4—282no—
512C2×C4—282no—
513C2×C4—282no—
514C2×C4—282no—
515C2×C4—282no—
516C7#1450711yescharacteristic, complemented, Sylow
517C22—24no—
518C22—28no—
519C22—44no—
520C22—44no—
521C22—42no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—42no—
526C22C2×#643511yescharacteristic
527C22#9017711yescharacteristic
528C22#9018612yes—
529C22#9021211yescharacteristic
530C22#9022212yes—
531C22C4×#115912yes—
532C22#9023212yes—
533C22#9023712yes—
534C22#9024712yes—
535C22#9025112yes—
536C22#9025112yes—
537C22#9027811yescharacteristic
538C22#9030512yes—
539C22#9031012yes—
540C22#9031212yes—
541C22#9035511yescharacteristic
542C22#9035612yes—
543C22#9036311yescharacteristic
544C22#9036912yes—
545C22#9040112yes—
546C22#9041111yescharacteristic
547C4—44no—
548C4—42no—
549C4—42no—
550C4—82no—
551C4—284no—
552C4—284no—
553C4—284no—
554C4—562no—
555C2—44no—
556C2#1173090012yes—
557C2#1173101211yescharacteristic
558C2#1173212611yescharacteristic
559C2#1173227912yes—
560C2#1173291412yes—
561C2#1173317712yes—
562C2#1173319612yes—
563C2#1173375411yescharacteristic
564C2#1173385112yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1437C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenter#1437D7
hypocenterC7#14507
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
6. #11729288C2semidirect
7. #11727171C2semidirect
8. #11727167C2semidirect
9. #11727160C2semidirect
31. #89715C4semidirect
516. C7#14507semidirect

Maximal subgroups

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

subgroupindex
C2×#897152
C7×#14372
#117293012
#117292992
#117292982
#117292882
#117271712
#117271672
#117271602
#145077