Finite Group Databasebeta

← #13280520

Subgroups — #13280520

Order 1792.

1262 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×Dic7C211yesmaximal, characteristic
2C2×#89423C211yesmaximal, characteristic
3C2×#89422C211yesmaximal, characteristic
4C7×#1444C211yesmaximal, characteristic
5#11729321C211yesmaximal, characteristic
6#11729320C212yesmaximal
7#11729318C212yesmaximal
8#11727174C211yesmaximal, characteristic
9#11727166C211yesmaximal, characteristic
10#11727163C212yesmaximal
11#11727160C212yesmaximal
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C4⋊C4C2212yes—
14C22×C4⋊Dic7C2211yescharacteristic
15C22×Dic7⋊C4C2212yes—
16C22×Dic7⋊C4C2212yes—
17C22×C4×Dic7C2212yes—
18C22×C4×Dic7C2211yescharacteristic
19C2×C7×#95C2211yescharacteristic
20C2×C7×#95C2212yes—
21C2×C7×#95C2211yescharacteristic
22C2×C7⋊#95C2211yescharacteristic
23C2×C7⋊#95C2212yes—
24C2×C7⋊#95C2211yescharacteristic
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2211yescharacteristic
28C2×C7⋊#95C2211yescharacteristic
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2211yescharacteristic
35C4⋊C4×Dic7—24no—
36#89423—24no—
37#89422C418yescomplemented
38#14296—71nomaximal, Sylow
39C23×C4×C7C2311yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×C4×C7C2312yes—
42C23×C4×C7C2312yes—
43C23×C4×C7C2311yescharacteristic
44C23×Dic7C2311yescharacteristic
45C23×Dic7C2311yescharacteristic
46C23×Dic7C2312yes—
47C23×Dic7C2312yes—
48C23×Dic7C2311yescharacteristic
49C23×Dic7C2311yescharacteristic
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4—24no—
52C2×C7×C4⋊C4—24no—
53C2×C7×C4⋊C4—42no—
54C2×C7×C4⋊C4—42no—
55C2×C4⋊Dic7—22no—
56C2×C4⋊Dic7—24no—
57C2×Dic7⋊C4—24no—
58C2×Dic7⋊C4—42no—
59C2×Dic7⋊C4—42no—
60C2×Dic7⋊C4—42no—
61C2×Dic7⋊C4—42no—
62C2×Dic7⋊C4C2×C418yes—
63C2×C4×Dic7—24no—
64C2×C4×Dic7—42no—
65C2×C4×Dic7—42no—
66C2×C4×Dic7C2×C414yes—
67C2×C4×Dic7D414yes—
68C2×C4×Dic7Q814yes—
69C7×#95—22no—
70C7×#95—42no—
71C7×#95C2×C414yes—
72C7⋊#95—22no—
73C7⋊#95—24no—
74C7⋊#95—22no—
75C7⋊#95—22no—
76C7⋊#95—22no—
77C7⋊#95—22no—
78C7⋊#95—24no—
79C7⋊#95—24no—
80C7⋊#95—42no—
81C7⋊#95—42no—
82C7⋊#95—42no—
83C7⋊#95C2×C414yes—
84C7⋊#95C2×C418yes—
85C2×#383—71no—
86C2×#382—71no—
87C2×C4×C4⋊C4—71no—
88#1444—71no—
89#1444D711yescharacteristic
90#1442—71no—
91#1442—71no—
92#1441—72no—
93#1441—72no—
94#1436—72no—
95#1436—72no—
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—22no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
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—21no—
120C22×C4×C7—21no—
121C22×C4×C7C22×C414yes—
122C22×C4×C7C22×C412yes—
123C22×C4×C7C2×D412yes—
124C22×C4×C7C2×Q812yes—
125C22×C4×C7#4114yes—
126C22×Dic7—22no—
127C22×Dic7—21no—
128C22×Dic7—21no—
129C22×Dic7—21no—
130C22×Dic7—21no—
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—21no—
145C22×Dic7—21no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7C22×C414yes—
155C22×Dic7C22×C412yes—
156C22×Dic7C22×C412yes—
157C22×Dic7C2×D412yes—
158C22×Dic7C2×D412yes—
159C22×Dic7C2×Q812yes—
160C22×Dic7C2×Q812yes—
161C22×Dic7#4114yes—
162C7×C4⋊C4—28no—
163C7×C4⋊C4—48no—
164C4⋊Dic7—28no—
165Dic7⋊C4—48no—
166Dic7⋊C4—48no—
167C4×Dic7—48no—
168C4×Dic7C4⋊C4116yescomplemented
169C22×C4⋊C4—72no—
170C22×C4⋊C4—71no—
171C22×C4⋊C4—72no—
172C22×C4⋊C4C2×D711yescharacteristic
173C22×C4⋊C4C2×D712yes—
174C22×C42—71no—
175C22×C42—72no—
176#383—144no—
177#382—78no—
178C4×C4⋊C4—144no—
179C2×#95—72no—
180C2×#95—71no—
181C2×#95—71no—
182C2×#95—71no—
183C2×#95—72no—
184C2×#95—72no—
185C2×#95—72no—
186C2×#95—71no—
187C2×#95—71no—
188C2×#95—71no—
189C2×#95—72no—
190C2×#95—72no—
191C2×#95—72no—
192C2×#95C2×D711yescharacteristic
193C2×#95C2×D712yes—
194C2×#95C2×D711yescharacteristic
195C23×C7C23×C411yescharacteristic
196C23×C7C22×D411yescharacteristic
197C23×C7C22×Q811yescharacteristic
198C23×C7C2×#4112yes—
199C23×C7C2×#4112yes—
200C23×C7H(5,𝔽2)11yescharacteristic
201C23×C7H(5,𝔽2)12yes—
202C23×C7H(5,𝔽2)11yescharacteristic
203C23×C72−1+411yescharacteristic
204C23×C72−1+412yes—
205C23×C72−1+411yescharacteristic
206C2×C4×C7—24no—
207C2×C4×C7—22no—
208C2×C4×C7—22no—
209C2×C4×C7—24no—
210C2×C4×C7—24no—
211C2×C4×C7—24no—
212C2×C4×C7—24no—
213C2×C4×C7—24no—
214C2×C4×C7—24no—
215C2×C4×C7—24no—
216C2×C4×C7—42no—
217C2×C4×C7—42no—
218C2×C4×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7—41no—
225C2×C4×C7—41no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7—41no—
229C2×C4×C7—41no—
230C2×C4×C7—42no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×C4×C7C2×C4⋊C414yes—
235C2×Dic7—22no—
236C2×Dic7—22no—
237C2×Dic7—24no—
238C2×Dic7—22no—
239C2×Dic7—22no—
240C2×Dic7—22no—
241C2×Dic7—22no—
242C2×Dic7—22no—
243C2×Dic7—22no—
244C2×Dic7—24no—
245C2×Dic7—24no—
246C2×Dic7—24no—
247C2×Dic7—24no—
248C2×Dic7—24no—
249C2×Dic7—24no—
250C2×Dic7—24no—
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—41no—
260C2×Dic7—41no—
261C2×Dic7—41no—
262C2×Dic7—41no—
263C2×Dic7—41no—
264C2×Dic7—41no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7C2×C4⋊C414yes—
270C2×Dic7C2×C4⋊C414yes—
271C23×C4—71no—
272C23×C4—71no—
273C23×C4—72no—
274C23×C4—72no—
275C23×C4—71no—
276C23×C4—71no—
277C23×C4C22×D711yescharacteristic
278C23×C4C22×D711yescharacteristic
279C23×C4C22×D712yes—
280C23×C4C22×D712yes—
281C23×C4C22×D711yescharacteristic
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—78no—
288C2×C4⋊C4—144no—
289C2×C4⋊C4—142no—
290C2×C4⋊C4—144no—
291C2×C4⋊C4—282no—
292C2×C4⋊C4—282no—
293C2×C4⋊C4—282no—
294C2×C4⋊C4—282no—
295C2×C42—74no—
296C2×C42—74no—
297C2×C42—74no—
298C2×C42—144no—
299C2×C42—282no—
300C2×C42—282no—
301#95—22no—
302#95—42no—
303#95—78no—
304#95—74no—
305#95—144no—
306#95—144no—
307#95—142no—
308#95—142no—
309#95—142no—
310#95—142no—
311#95—142no—
312#95—144no—
313#95—282no—
314#95—282no—
315#95—282no—
316#95C4×D714yes—
317C22×C7C22×C4⋊C411yescharacteristic
318C22×C7C4×#4112yes—
319C22×C7#51811yescharacteristic
320C22×C7#51911yescharacteristic
321C22×C7#52012yes—
322C22×C7#52511yescharacteristic
323C22×C7#52511yescharacteristic
324C22×C7#53312yes—
325C22×C7#53511yescharacteristic
326C22×C7#53611yescharacteristic
327C22×C7#53711yescharacteristic
328C22×C7#53711yescharacteristic
329C22×C7#53812yes—
330C22×C7#53812yes—
331C22×C7#53912yes—
332C22×C7#53912yes—
333C22×C7#54112yes—
334C22×C7#54112yes—
335C22×C7#54212yes—
336C22×C7#54212yes—
337C22×C7#54411yescharacteristic
338C22×C7#54411yescharacteristic
339C22×C7#54412yes—
340C4×C7—24no—
341C4×C7—44no—
342C4×C7—44no—
343C4×C7—81no—
344C4×C7—81no—
345Dic7—24no—
346Dic7—24no—
347Dic7—44no—
348Dic7—44no—
349Dic7—81no—
350Dic7—81no—
351C24C23×D711yescharacteristic
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—72no—
377C22×C4—72no—
378C22×C4—72no—
379C22×C4—72no—
380C22×C4—72no—
381C22×C4—72no—
382C22×C4—74no—
383C22×C4—74no—
384C22×C4—142no—
385C22×C4—141no—
386C22×C4—142no—
387C22×C4—141no—
388C22×C4—141no—
389C22×C4—141no—
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—141no—
403C22×C4—141no—
404C22×C4—142no—
405C22×C4—141no—
406C22×C4—142no—
407C22×C4—142no—
408C22×C4—141no—
409C22×C4—142no—
410C22×C4—141no—
411C22×C4—141no—
412C22×C4C2×C4×D714yes—
413C22×C4C2×C4×D712yes—
414C22×C4D4×D712yes—
415C22×C4#115914yes—
416C22×C4Q8×D712yes—
417C4⋊C4—28no—
418C4⋊C4—48no—
419C4⋊C4—148no—
420C4⋊C4—288no—
421C4⋊C4—288no—
422C42—716no—
423C42—288no—
424C2×C7#229611yescharacteristic
425C2×C7#231611yescharacteristic
426C2×C7#231711yescharacteristic
427C2×C7#233612yes—
428C2×C7#233712yes—
429C2×C7#263611yescharacteristic
430C2×C7#263611yescharacteristic
431C2×C7#263711yescharacteristic
432C2×C7#263711yescharacteristic
433C2×C7#266612yes—
434C2×C7#266612yes—
435C23C22×C4×D711yescharacteristic
436C23C2×D4×D711yescharacteristic
437C23C2×#115912yes—
438C23#654812yes—
439C23#654811yescharacteristic
440C23C2×Q8×D711yescharacteristic
441C23#655112yes—
442C23#655111yescharacteristic
443C23D7×#4112yes—
444C23#655311yescharacteristic
445C23#655411yescharacteristic
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—74no—
475C2×C4—74no—
476C2×C4—142no—
477C2×C4—142no—
478C2×C4—142no—
479C2×C4—144no—
480C2×C4—142no—
481C2×C4—144no—
482C2×C4—142no—
483C2×C4—142no—
484C2×C4—142no—
485C2×C4—142no—
486C2×C4—144no—
487C2×C4—144no—
488C2×C4—144no—
489C2×C4—144no—
490C2×C4—144no—
491C2×C4—144no—
492C2×C4—282no—
493C2×C4—282no—
494C2×C4—282no—
495C2×C4—282no—
496C2×C4—282no—
497C2×C4—282no—
498C2×C4—282no—
499C2×C4—282no—
500C2×C4—282no—
501C2×C4—281no—
502C2×C4—281no—
503C2×C4—282no—
504C2×C4—281no—
505C2×C4—281no—
506C2×C4—281no—
507C2×C4—282no—
508C2×C4—282no—
509C2×C4—281no—
510C2×C4D7×C4⋊C414yes—
511C7#1429611yescharacteristic, complemented, Sylow
512C22C2×D7×C4⋊C411yescharacteristic
513C22#9019512yes—
514C22C4×#115912yes—
515C22#9023611yescharacteristic
516C22#9026311yescharacteristic
517C22#9027711yescharacteristic
518C22#9028412yes—
519C22#9029312yes—
520C22#9029911yescharacteristic
521C22#9031312yes—
522C22#9032011yescharacteristic
523C22#9032912yes—
524C22#9033112yes—
525C22#9033412yes—
526C22#9034711yescharacteristic
527C22#9034812yes—
528C22#9038011yescharacteristic
529C22#9038712yes—
530C22#9040611yescharacteristic
531C22#9041511yescharacteristic
532C22#9042012yes—
533C22#9042011yescharacteristic
534C22#9042212yes—
535C4—24no—
536C4—44no—
537C4—44no—
538C4—81no—
539C4—81no—
540C4—144no—
541C4—144no—
542C4—284no—
543C4—284no—
544C4—561no—
545C4—561no—
546C2#1173180411yescharacteristic
547C2#1173182311yescharacteristic
548C2#1173183612yes—
549C2#1173342312yes—
550C2#1173364011yescharacteristic
551C2#1173414911yescharacteristic
552C2#1173465011yescharacteristic
553C2#1173465712yes—
554C2#1173527011yescharacteristic
555C2#1173528912yes—
556C2#1173549811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1444C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1444D7
hypocenterC7#14296
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
37. #89422C4semidirect
168. C4×Dic7C4⋊C4semidirect
511. C7#14296semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×Dic72
C2×#894232
C2×#894222
C7×#14442
#117293212
#117293202
#117293182
#117271742
#117271662
#117271632
#117271602
#142967