Finite Group Databasebeta

← #13367616

Subgroups — #13367616

Order 1792.

906 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#497C211yesmaximal, characteristic
2Dic7×C4⋊Q8C211yesmaximal, characteristic, complemented
3Dic7×#124C211yesmaximal, characteristic
4#11729819C212yesmaximal, complemented
5#11729819C212yesmaximal, complemented
6#11729802C212yesmaximal
7#11729801C212yesmaximal
8#11729540C211yesmaximal, characteristic, complemented
9#11729480C211yesmaximal, characteristic
10#11727569C211yesmaximal, characteristic, complemented
11#11727559C211yesmaximal, characteristic, complemented
12C22×C7×Q16C2212yes—
13C22×C7×SD16C2212yes—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#124C2211yescharacteristic
16C2×Q8×Dic7C2212yes—
17C2×Q8×Dic7C2211yescharacteristic
18C2×D4×Dic7C2211yescharacteristic
19C7×#497C418yescomplemented
20C2×C7×#97C2211yescharacteristic
21#90002C2212yes—
22#90002C2211yescharacteristic
23C2×#6409C2212yes—
24C2×#6409C2212yes—
25C2×#6409C2212yes—
26#89993C2211yescharacteristic
27C2×#6406C2212yes—
28#89757—21no—
29#89757—21no—
30C4⋊C4×Dic7—21no—
31C4⋊C4×Dic7—21no—
32#89723—21no—
33#89723—21no—
34C22⋊C4×Dic7—21no—
35C22⋊C4×Dic7—21no—
36#89708C2211yescharacteristic
37#89707C2211yescharacteristic
38C42×Dic7C2211yescharacteristic
39C2×#6377C2211yescharacteristic
40#89346C2212yes—
41#89346C2212yes—
42#89345C2212yes—
43#89345C2212yes—
44#22273—71nomaximal, Sylow
45C22×C7×Q8C2311yescharacteristic
46C22×C7×Q8C2312yes—
47C22×C7×D4C2311yescharacteristic
48C23×Dic7—21no—
49C2×C7×Q16—42no—
50C2×C7×Q16—42no—
51C2×C7×Q16C2×C418yescomplemented
52C2×C7×SD16—42no—
53C2×C7×SD16—42no—
54C2×C7×SD16C2×C418yes—
55C22×C7×C8C2312yes—
56C7×C4⋊Q8—22no—
57C7×C4⋊Q8C2×C414yescomplemented
58C7×#124—22no—
59C7×#124C2×C414yes—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4—21no—
64C2×C42×C7C2311yescharacteristic
65C2×C22⋊Dic7—21no—
66C2×C22⋊Dic7—21no—
67C2×C22⋊Dic7—21no—
68Q8×Dic7—42no—
69Q8×Dic7—42no—
70Q8×Dic7—42no—
71D4×Dic7—42no—
72C2×C4⋊Dic7—21no—
73C2×C4⋊Dic7—21no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7—22no—
76C2×C4⋊Dic7C2312yes—
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4×Dic7—21no—
80C2×C4×Dic7—22no—
81C2×C4×Dic7D411yescharacteristic
82C2×C4×Dic7D411yescharacteristic
83C2×C4×Dic7D411yescharacteristic
84C2×C4×Dic7D411yescharacteristic
85C2×C4×Dic7C2311yescharacteristic
86C2×C4×Dic7C2311yescharacteristic
87C22×C7⋊C8C2312yes—
88C7×#97C2×C414yescomplemented
89#6409—42no—
90#6409—42no—
91#6409—42no—
92#6406—42no—
93C7⋊#95—21no—
94C7⋊#95—21no—
95C7⋊#95—21no—
96C7⋊#95—21no—
97C7⋊#95—22no—
98C7⋊#95—22no—
99#6377—22no—
100C2×#497D711yescharacteristic
101C4×C4⋊Q8—71no—
102C4×#124—71no—
103#1960—71no—
104#1959—71no—
105#1880—72no—
106#1878—72no—
107#1878—72no—
108#1877—72no—
109#1836—71no—
110#1745—71no—
111C24×C7—21no—
112C2×C7×Q8—21no—
113C2×C7×Q8—22no—
114C2×C7×Q8—22no—
115C2×C7×Q8—22no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8C22×C412yes—
121C2×C7×Q8C22×C414yes—
122C2×C7×D4—21no—
123C2×C7×D4—22no—
124C2×C7×D4—22no—
125C2×C7×D4C22×C412yes—
126C22×C4×C7—21no—
127C22×C4×C7—22no—
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2×D411yescharacteristic
130C22×C4×C7C2411yescharacteristic
131C22×Dic7—21no—
132C22×Dic7—22no—
133C22×Dic7—21no—
134C22×Dic7—42no—
135C22×Dic7—42no—
136C22×Dic7—42no—
137C22×Dic7C2×D411yescharacteristic
138C22×Dic7C2×D411yescharacteristic
139C22×Dic7C2×D411yescharacteristic
140C22×Dic7C2×D411yescharacteristic
141C7×Q16—48no—
142C7×SD16—48no—
143C2×C7×C8—22no—
144C2×C7×C8—22no—
145C2×C7×C8C22×C414yes—
146C7×C4⋊C4—24no—
147C7×C4⋊C4—24no—
148C7×C22⋊C4—24no—
149C7×C22⋊C4—24no—
150C42×C7C22×C412yes—
151C42×C7#4112yes—
152C22⋊Dic7—42no—
153C22⋊Dic7—42no—
154C22⋊Dic7—42no—
155C4⋊Dic7—22no—
156C4⋊Dic7—22no—
157C4⋊Dic7—22no—
158C4⋊Dic7—22no—
159C4⋊Dic7—44no—
160C4⋊Dic7—42no—
161C4⋊Dic7—42no—
162C4⋊Dic7—42no—
163C4×Dic7—22no—
164C4×Dic7—22no—
165C4×Dic7—22no—
166C4×Dic7—22no—
167C4×Dic7—41no—
168C4×Dic7—41no—
169C4×Dic7—44no—
170C4×Dic7—42no—
171C2×C7⋊C8—22no—
172C2×C7⋊C8—22no—
173C2×C7⋊C8—22no—
174C22×Q16C2×D712yes—
175C22×SD16C2×D712yes—
176C2×C4⋊Q8C2×D711yescharacteristic
177C2×#124C2×D711yescharacteristic
178C2×C4×Q8—72no—
179C2×C4×Q8—71no—
180C2×C4×D4—71no—
181#497Dic718yescomplemented
182C2×#103—72no—
183C2×#103—72no—
184C2×#103—72no—
185C2×#102—72no—
186C2×#97—71no—
187C2×#97C2×D711yescharacteristic
188#391—72no—
189#391—71no—
190#390—71no—
191#389—141no—
192#389—141no—
193#388—141no—
194#388—141no—
195#384—71no—
196#382—71no—
197C4×C4⋊C4—141no—
198C4×C4⋊C4—141no—
199C4×C22⋊C4—141no—
200C4×C22⋊C4—141no—
201C43—71no—
202#340—72no—
203#340—72no—
204#336—72no—
205#336—72no—
206C23×C7—22no—
207C23×C7—41no—
208C23×C7—41no—
209C23×C7—41no—
210C23×C7—41no—
211C23×C7—41no—
212C23×C7—41no—
213C23×C7C22×D411yescharacteristic
214C7×Q8—24no—
215C7×Q8—24no—
216C7×Q8—24no—
217C7×Q8—42no—
218C7×Q8—42no—
219C7×Q8—42no—
220C7×D4—24no—
221C7×D4—42no—
222C2×C4×C7—21no—
223C2×C4×C7—21no—
224C2×C4×C7—21no—
225C2×C4×C7—21no—
226C2×C4×C7—22no—
227C2×C4×C7—24no—
228C2×C4×C7—41no—
229C2×C4×C7—42no—
230C2×C4×C7—42no—
231C2×C4×C7—41no—
232C2×C4×C7C4×D412yes—
233C2×C4×C7C4×D412yes—
234C2×C4×C7C23×C411yescharacteristic
235C2×C4×C7C2×#4111yescharacteristic
236C2×C4×C7H(5,𝔽2)12yes—
237C2×C4×C7H(5,𝔽2)12yes—
238C2×Dic7—21no—
239C2×Dic7—21no—
240C2×Dic7—21no—
241C2×Dic7—21no—
242C2×Dic7—21no—
243C2×Dic7—21no—
244C2×Dic7—21no—
245C2×Dic7—21no—
246C2×Dic7—21no—
247C2×Dic7—21no—
248C2×Dic7—41no—
249C2×Dic7—42no—
250C2×Dic7—41no—
251C2×Dic7—41no—
252C2×Dic7—41no—
253C2×Dic7—42no—
254C2×Dic7—42no—
255C2×Dic7—41no—
256C2×Dic7—41no—
257C2×Dic7—84no—
258C2×Dic7#13712yes—
259C2×Dic7#13712yes—
260C7×C8—24no—
261C7⋊C8—42no—
262C22×Q8C22×D711yescharacteristic
263C22×Q8C22×D712yes—
264C22×D4C22×D711yescharacteristic
265C23×C4—141no—
266C2×Q16—42no—
267C2×Q16—42no—
268C2×Q16C2×Dic718yescomplemented
269C2×SD16—42no—
270C2×SD16—42no—
271C2×SD16C2×Dic718yes—
272C22×C8—72no—
273C22×C8C22×D712yes—
274C4⋊Q8—22no—
275C4⋊Q8C2×Dic714yescomplemented
276#124—22no—
277#124C2×Dic714yes—
278C4×Q8—282no—
279C4×Q8—282no—
280C4×Q8—282no—
281C4×D4—282no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—71no—
285C2×C4⋊C4—71no—
286C2×C4⋊C4—72no—
287C2×C4⋊C4—141no—
288C2×C4⋊C4—142no—
289C2×C4⋊C4—141no—
290C2×C4⋊C4—141no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—141no—
294C2×C22⋊C4—141no—
295C2×C22⋊C4—141no—
296C2×C42—71no—
297C2×C42—71no—
298C2×C42—71no—
299C2×C42—71no—
300C2×C42—71no—
301C2×C42—71no—
302C2×C42—141no—
303C2×C42—142no—
304C2×C42C22×D711yescharacteristic
305#103—282no—
306#103—282no—
307#103—282no—
308#102—282no—
309#97—142no—
310#97C2×Dic714yescomplemented
311#95—141no—
312#95—141no—
313#95—142no—
314#95—142no—
315#95—141no—
316#95—141no—
317C22×C7—42no—
318C22×C7—42no—
319C22×C7—42no—
320C22×C7—81no—
321C22×C7—81no—
322C22×C7—81no—
323C22×C7—81no—
324C22×C7C2×C4×D411yescharacteristic
325C22×C7#53511yescharacteristic
326C22×C7#53511yescharacteristic
327C22×C7C2×#13711yescharacteristic
328C22×C7C2×#13711yescharacteristic
329C22×C7#57511yescharacteristic
330C22×C7#57511yescharacteristic
331C4×C7—22no—
332C4×C7—22no—
333C4×C7—42no—
334C4×C7—44no—
335C4×C7#51812yes—
336C4×C7#54012yes—
337Dic7—22no—
338Dic7—22no—
339Dic7—41no—
340Dic7—41no—
341Dic7—81no—
342Dic7—81no—
343Dic7—82no—
344C24—21no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8C22×Dic712yes—
354C2×Q8C22×Dic714yes—
355C2×D4—21no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4C22×Dic712yes—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—71no—
362C22×C4—71no—
363C22×C4—71no—
364C22×C4—71no—
365C22×C4—141no—
366C22×C4—141no—
367C22×C4—142no—
368C22×C4—282no—
369C22×C4—282no—
370C22×C4—282no—
371C22×C4D4×D711yescharacteristic
372C22×C4D4×D711yescharacteristic
373C22×C4C23×D711yescharacteristic
374Q16—48no—
375SD16—48no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—142no—
379C2×C8—142no—
380C2×C8—142no—
381C2×C8C22×Dic714yes—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—142no—
385C4⋊C4—142no—
386C4⋊C4—142no—
387C4⋊C4—142no—
388C4⋊C4—282no—
389C4⋊C4—282no—
390C4⋊C4—282no—
391C4⋊C4—284no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—282no—
395C22⋊C4—282no—
396C22⋊C4—282no—
397C42—142no—
398C42—142no—
399C42—142no—
400C42—142no—
401C42—281no—
402C42—281no—
403C42—284no—
404C42—282no—
405C42#115912yes—
406C42C22×Dic712yes—
407C2×C7—82no—
408C2×C7#232811yescharacteristic
409C2×C7C4×#13711yescharacteristic
410C2×C7C4×#13711yescharacteristic
411C2×C7#321511yescharacteristic
412C2×C7#321611yescharacteristic
413C2×C7#321811yescharacteristic
414C2×C7#321911yescharacteristic
415C23—22no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23C2×D4×D711yescharacteristic
423Q8—24no—
424Q8—24no—
425Q8—24no—
426Q8—42no—
427Q8—42no—
428Q8—42no—
429D4—24no—
430D4—42no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—72no—
442C2×C4—72no—
443C2×C4—141no—
444C2×C4—141no—
445C2×C4—141no—
446C2×C4—141no—
447C2×C4—141no—
448C2×C4—141no—
449C2×C4—141no—
450C2×C4—141no—
451C2×C4—141no—
452C2×C4—141no—
453C2×C4—281no—
454C2×C4—282no—
455C2×C4—281no—
456C2×C4—281no—
457C2×C4—281no—
458C2×C4—281no—
459C2×C4—282no—
460C2×C4—282no—
461C2×C4—281no—
462C2×C4—564no—
463C2×C4D4×Dic712yes—
464C2×C4D4×Dic712yes—
465C2×C4C2×#115911yescharacteristic
466C2×C4#654812yes—
467C2×C4#654812yes—
468C2×C4C23×Dic711yescharacteristic
469C8—24no—
470C8—282no—
471C7#2227311yescharacteristic, complemented, Sylow
472C22—42no—
473C22—42no—
474C22—42no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22#9040611yescharacteristic
480C22#9040611yescharacteristic
481C22D7×#13711yescharacteristic
482C22D7×#13711yescharacteristic
483C22#9046811yescharacteristic
484C22#9046811yescharacteristic
485C22C2×D4×Dic711yescharacteristic
486C4—22no—
487C4—22no—
488C4—42no—
489C4—44no—
490C4—142no—
491C4—142no—
492C4—281no—
493C4—281no—
494C4—561no—
495C4—561no—
496C4—562no—
497C4#9041012yes—
498C4#9048912yes—
499C2—82no—
500C2#1173931911yescharacteristic
501C2#1173932011yescharacteristic
502C2#1173932111yescharacteristic
503C2#1173932211yescharacteristic
504C2#1174061711yescharacteristic
505C2Dic7×#13711yescharacteristic
506C2Dic7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#497C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC2×#497D7
hypocenterC7#22273
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
2. Dic7×C4⋊Q8C2semidirect
4. #11729819C2semidirect
5. #11729819C2semidirect
8. #11729540C2semidirect
10. #11727569C2semidirect
11. #11727559C2semidirect
19. C7×#497C4semidirect
51. C2×C7×Q16C2×C4semidirect
57. C7×C4⋊Q8C2×C4semidirect
88. C7×#97C2×C4semidirect
181. #497Dic7semidirect
268. C2×Q16C2×Dic7semidirect
275. C4⋊Q8C2×Dic7semidirect
310. #97C2×Dic7semidirect
471. C7#22273semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4972
Dic7×C4⋊Q82
Dic7×#1242
#117298192
#117298022
#117298012
#117295402
#117294802
#117275692
#117275592
#222737