Finite Group Databasebeta

← #13360990

Subgroups — #13360990

Order 1792.

964 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6487C211yesmaximal, characteristic
2C2×C4×C4⋊Dic7C211yesmaximal, characteristic
3C2×#89493C211yesmaximal, characteristic
4C7×#1749C211yesmaximal, characteristic
5#11727684C211yesmaximal, characteristic
6#11727681C211yesmaximal, characteristic
7#11727678C211yesmaximal, characteristic
8#11727568C212yesmaximal, complemented
9#11727567C212yesmaximal, complemented
10#11726382C212yesmaximal, complemented
11#11726376C212yesmaximal, complemented
12C22×C42×C7C2211yescharacteristic
13C22×C4⋊Dic7C2211yescharacteristic
14C22×C4⋊Dic7C2211yescharacteristic
15C2×#6487C2211yescharacteristic
16C22×C4×Dic7—21no—
17C2×C4×Dic14C2211yescharacteristic
18C2×C7×C22⋊C8C2211yescharacteristic
19C2×C7×C22⋊C8C2211yescharacteristic
20C2×C7×#97C2212yes—
21#89984C2211yescharacteristic
22#89982—21no—
23C4×C22⋊Dic7—21no—
24C2×C7⋊#95—21no—
25C2×#6392C2212yes—
26C2×#6391C2212yes—
27C2×#6390C2212yes—
28C2×#6390C2212yes—
29C4×C4⋊Dic7—21no—
30C4×C4⋊Dic7—21no—
31C4×C4⋊Dic7C2212yes—
32C4×C4⋊Dic7C2212yes—
33C4×C4⋊Dic7C2211yescharacteristic
34#89705—21no—
35C4×Dic7⋊C4—21no—
36#89701C2211yescharacteristic
37#89700C2211yescharacteristic
38#89493C418yescomplemented
39C7×#336C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#89346C2212yes—
43#89346C2212yes—
44#21579—71nomaximal, Sylow
45C23×C4×C7D411yescharacteristic
46C23×C4×C7D411yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C23×Dic7—21no—
49C23×Dic7—21no—
50C22×Dic14C2311yescharacteristic
51C22×C7×C8C2312yes—
52C22×C7×C8C2312yes—
53C2×C42×C7—24no—
54C2×C42×C7D411yescharacteristic
55C2×C42×C7D411yescharacteristic
56C2×C42×C7C2311yescharacteristic
57C2×C42×C7C2311yescharacteristic
58C2×C22⋊Dic7—21no—
59C2×C22⋊Dic7—21no—
60C2×C4⋊Dic7—21no—
61C2×C4⋊Dic7—21no—
62C2×C4⋊Dic7—22no—
63C2×C4⋊Dic7—24no—
64C2×C4⋊Dic7C2×C414yes—
65C2×C4⋊Dic7C2311yescharacteristic
66C2×C4⋊Dic7C2311yescharacteristic
67C2×C4⋊Dic7C2312yes—
68C2×C4⋊Dic7C2311yescharacteristic
69C2×C4⋊Dic7C2312yes—
70#6487—22no—
71#6487C2×C414yes—
72C2×Dic7⋊C4—21no—
73C2×Dic7⋊C4—21no—
74C2×Dic7⋊C4—21no—
75C2×C4×Dic7—21no—
76C2×C4×Dic7—21no—
77C2×C4×Dic7—22no—
78C2×C4×Dic7—22no—
79C2×C4×Dic7—44no—
80C4×Dic14—24no—
81C7×C22⋊C8—22no—
82C7×C22⋊C8C2×C414yes—
83C7×#97—24no—
84C7⋊#95—21no—
85C7⋊#95—21no—
86C7⋊#95—21no—
87C7⋊#95—21no—
88C7⋊#95—21no—
89C7⋊#95—22no—
90C7⋊#95—22no—
91#6392—24no—
92#6391C2×C418yescomplemented
93#6390—24no—
94#6390C2×C418yescomplemented
95C2×#483—71no—
96C4×C22⋊Q8—71no—
97C2×C4×C4⋊C4—71no—
98#1928—72no—
99#1919—72no—
100#1872—71no—
101#1852—71no—
102#1823—71no—
103#1776—72no—
104#1763—72no—
105#1749D711yescharacteristic
106C24×C7C2×D411yescharacteristic
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—41no—
115C22×C4×C7C22×C412yes—
116C22×C4×C7C2×D411yescharacteristic
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7#4112yes—
122C22×C4×C7C2411yescharacteristic
123C22×Dic7—24no—
124C22×Dic7—24no—
125C22×Dic7—22no—
126C22×Dic7—22no—
127C22×Dic7—21no—
128C22×Dic7—21no—
129C22×Dic7—21no—
130C22×Dic7—21no—
131C22×Dic7—42no—
132C22×Dic7—42no—
133C22×Dic7—42no—
134C22×Dic7—42no—
135C2×Dic14—22no—
136C2×Dic14—22no—
137C2×Dic14—21no—
138C2×Dic14C22×C412yes—
139C2×C7×C8—22no—
140C2×C7×C8—22no—
141C2×C7×C8—22no—
142C2×C7×C8—22no—
143C2×C7×C8—22no—
144C2×C7×C8C22×C414yes—
145C42×C7—22no—
146C42×C7—22no—
147C42×C7—22no—
148C42×C7#4114yes—
149C22⋊Dic7—24no—
150C22⋊Dic7—42no—
151C4⋊Dic7—24no—
152C4⋊Dic7—24no—
153C4⋊Dic7—21no—
154C4⋊Dic7—22no—
155C4⋊Dic7—22no—
156C4⋊Dic7—22no—
157C4⋊Dic7—22no—
158C4⋊Dic7C22×C412yes—
159C4⋊Dic7C22×C414yes—
160C4⋊Dic7#4114yes—
161Dic7⋊C4—24no—
162Dic7⋊C4—42no—
163Dic7⋊C4—42no—
164C4×Dic7—42no—
165C4×Dic7—44no—
166C4×Dic7—44no—
167C4×Dic7—42no—
168C2×C22⋊Q8—71no—
169C2×C4×Q8—71no—
170C22×C4⋊C4—71no—
171C22×C4⋊C4—71no—
172C22×C42—141no—
173C22×C42C2×D711yescharacteristic
174#483—78no—
175C2×#107—72no—
176C2×#106—72no—
177C2×#103—72no—
178C2×#103—72no—
179C2×C22⋊C8C2×D711yescharacteristic
180C2×C22⋊C8C2×D711yescharacteristic
181C2×#97C2×D712yes—
182#391—71no—
183#389—71no—
184#387—141no—
185#385—141no—
186#380—71no—
187C4×C4⋊C4—72no—
188C4×C4⋊C4—72no—
189C4×C4⋊C4—71no—
190C4×C4⋊C4—141no—
191C4×C4⋊C4—141no—
192C4×C4⋊C4—141no—
193C4×C22⋊C4—141no—
194C2×#95—141no—
195#340—72no—
196#340—72no—
197#340—72no—
198#340—72no—
199#336C2×D712yes—
200C23×C7—21no—
201C23×C7—21no—
202C23×C7—21no—
203C23×C7—21no—
204C23×C7C4×D412yes—
205C23×C7AGL(1,ℤ/8)12yes—
206C23×C7#13712yes—
207C23×C7C22×D411yescharacteristic
208C2×C4×C7—28no—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—21no—
212C2×C4×C7—21no—
213C2×C4×C7—21no—
214C2×C4×C7—21no—
215C2×C4×C7—21no—
216C2×C4×C7—21no—
217C2×C4×C7—24no—
218C2×C4×C7—24no—
219C2×C4×C7—21no—
220C2×C4×C7—21no—
221C2×C4×C7—21no—
222C2×C4×C7—44no—
223C2×C4×C7C4×D412yes—
224C2×C4×C7#12312yes—
225C2×C4×C7#12312yes—
226C2×C4×C7#13712yes—
227C2×C4×C7C23×C411yescharacteristic
228C2×C4×C7C2×#4111yescharacteristic
229C2×C4×C7C2×#4112yes—
230C2×C4×C7C2×#4112yes—
231C2×Dic7—22no—
232C2×Dic7—22no—
233C2×Dic7—24no—
234C2×Dic7—24no—
235C2×Dic7—48no—
236C2×Dic7—41no—
237C2×Dic7—41no—
238C2×Dic7—42no—
239C2×Dic7—42no—
240C2×Dic7—41no—
241C2×Dic7—42no—
242C2×Dic7—41no—
243C2×Dic7—42no—
244C2×Dic7—48no—
245C2×Dic7—41no—
246C2×Dic7—41no—
247C2×Dic7—41no—
248C2×Dic7—41no—
249C2×Dic7—41no—
250C2×Dic7—41no—
251Dic14—24no—
252Dic14—42no—
253C7×C8—24no—
254C7×C8—42no—
255C22×Q8—71no—
256C23×C4—141no—
257C23×C4—141no—
258C23×C4D2811yescharacteristic
259C23×C4D2811yescharacteristic
260C23×C4C22×D711yescharacteristic
261C22×C8C22×D712yes—
262C22×C8C22×D712yes—
263C22⋊Q8—74no—
264C22⋊Q8—142no—
265C4×Q8—144no—
266C2×C4⋊C4—74no—
267C2×C4⋊C4—72no—
268C2×C4⋊C4—72no—
269C2×C4⋊C4—71no—
270C2×C4⋊C4—71no—
271C2×C4⋊C4—71no—
272C2×C4⋊C4—144no—
273C2×C4⋊C4—141no—
274C2×C4⋊C4—141no—
275C2×C4⋊C4—141no—
276C2×C4⋊C4—142no—
277C2×C4⋊C4—141no—
278C2×C4⋊C4—141no—
279C2×C22⋊C4—141no—
280C2×C22⋊C4—141no—
281C2×C42—24no—
282C2×C42—141no—
283C2×C42—141no—
284C2×C42—142no—
285C2×C42—142no—
286C2×C42—284no—
287C2×C42D2811yescharacteristic
288C2×C42D2811yescharacteristic
289C2×C42C22×D711yescharacteristic
290C2×C42C22×D711yescharacteristic
291#107—144no—
292#106—78no—
293#103—78no—
294#103—144no—
295C22⋊C8—22no—
296C22⋊C8C4×D714yes—
297#97—24no—
298#95—142no—
299#95—142no—
300#95—141no—
301#95—141no—
302#95—141no—
303#95—141no—
304#95—141no—
305C22×C7—22no—
306C22×C7—22no—
307C22×C7—22no—
308C22×C7—22no—
309C22×C7—22no—
310C22×C7—22no—
311C22×C7#44014yes—
312C22×C7C2×C4×D411yescharacteristic
313C22×C7C2×#12311yescharacteristic
314C22×C7#52511yescharacteristic
315C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
316C22×C7C2×#13711yescharacteristic
317C22×C7C2×#13711yescharacteristic
318C22×C7#57511yescharacteristic
319C4×C7—22no—
320C4×C7—22no—
321C4×C7—22no—
322C4×C7—22no—
323C4×C7—41no—
324C4×C7C4×#4112yes—
325C4×C7#52912yes—
326Dic7—44no—
327Dic7—44no—
328Dic7—42no—
329Dic7—42no—
330Dic7—81no—
331Dic7—81no—
332C24C2×D2811yescharacteristic
333C2×Q8—72no—
334C2×Q8—141no—
335C2×Q8—142no—
336C2×Q8—142no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—41no—
345C22×C4—144no—
346C22×C4—144no—
347C22×C4—142no—
348C22×C4—142no—
349C22×C4—141no—
350C22×C4—141no—
351C22×C4—141no—
352C22×C4—141no—
353C22×C4—282no—
354C22×C4—282no—
355C22×C4—282no—
356C22×C4—282no—
357C22×C4C2×C4×D712yes—
358C22×C4C2×D2811yescharacteristic
359C22×C4C2×D2811yescharacteristic
360C22×C4C2×D2811yescharacteristic
361C22×C4C2×D2811yescharacteristic
362C22×C4C2×D2811yescharacteristic
363C22×C4#115712yes—
364C22×C4C23×D711yescharacteristic
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8C2×C4×D714yes—
371C4⋊C4—74no—
372C4⋊C4—72no—
373C4⋊C4—74no—
374C4⋊C4—144no—
375C4⋊C4—144no—
376C4⋊C4—144no—
377C4⋊C4—142no—
378C4⋊C4—141no—
379C4⋊C4—142no—
380C4⋊C4—142no—
381C4⋊C4—142no—
382C4⋊C4—282no—
383C4⋊C4—282no—
384C22⋊C4—144no—
385C22⋊C4—282no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—282no—
390C42—284no—
391C42—284no—
392C42—282no—
393C42#115914yes—
394C2×C7—24no—
395C2×C7C4×#12311yescharacteristic
396C2×C7C2×#44011yescharacteristic
397C2×C7C4×#13711yescharacteristic
398C2×C7#309211yescharacteristic
399C2×C7#309511yescharacteristic
400C2×C7#311311yescharacteristic
401C2×C7#311411yescharacteristic
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23C4×D2812yes—
407C23#647112yes—
408C23#647212yes—
409C23C22×D2811yescharacteristic
410Q8—144no—
411Q8—282no—
412C2×C4—28no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—44no—
427C2×C4—142no—
428C2×C4—142no—
429C2×C4—144no—
430C2×C4—144no—
431C2×C4—288no—
432C2×C4—282no—
433C2×C4—282no—
434C2×C4—281no—
435C2×C4—282no—
436C2×C4—281no—
437C2×C4—281no—
438C2×C4—282no—
439C2×C4—281no—
440C2×C4—288no—
441C2×C4—281no—
442C2×C4—281no—
443C2×C4—281no—
444C2×C4—281no—
445C2×C4—281no—
446C2×C4—281no—
447C2×C4C4×D2812yes—
448C2×C4#644912yes—
449C2×C4#644912yes—
450C2×C4#647212yes—
451C2×C4C22×C4×D711yescharacteristic
452C2×C4C2×#115711yescharacteristic
453C2×C4C2×#115912yes—
454C2×C4D7×#4112yes—
455C8—24no—
456C8—42no—
457C7#2157911yescharacteristic, complemented, Sylow
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22#8948214yes—
465C22C2×C4×D2811yescharacteristic
466C22C2×#644911yescharacteristic
467C22#9021511yescharacteristic
468C22C2×#647111yescharacteristic
469C22C2×#647211yescharacteristic
470C22C2×#647211yescharacteristic
471C22#9043911yescharacteristic
472C4—22no—
473C4—22no—
474C4—22no—
475C4—22no—
476C4—41no—
477C4—284no—
478C4—282no—
479C4—282no—
480C4—284no—
481C4—561no—
482C4—561no—
483C4C4×#115912yes—
484C4#9022912yes—
485C2—24no—
486C2C4×#644911yescharacteristic
487C2C2×#8948211yescharacteristic
488C2C4×#647211yescharacteristic
489C2#1173606711yescharacteristic
490C2#1173607011yescharacteristic
491C2#1173679111yescharacteristic
492C2#1173680111yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1749C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1749D7
hypocenterC7#21579
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
8. #11727568C2semidirect
9. #11727567C2semidirect
10. #11726382C2semidirect
11. #11726376C2semidirect
38. #89493C4semidirect
92. #6391C2×C4semidirect
94. #6390C2×C4semidirect
457. C7#21579semidirect

Maximal subgroups

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

subgroupindex
C4×#64872
C2×C4×C4⋊Dic72
C2×#894932
C7×#17492
#117276842
#117276812
#117276782
#117275682
#117275672
#117263822
#117263762
#215797