Finite Group Databasebeta

← #13403934

Subgroups — #13403934

Order 1792.

894 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
1C2×C7×C8×D4C211yesmaximal, characteristic
2#11739827C211yesmaximal, characteristic
3#11739821C211yesmaximal, characteristic
4C2×#89623C211yesmaximal, characteristic, complemented
5#11729726C212yesmaximal, complemented
6#11729724C212yesmaximal, complemented
7#11728208C211yesmaximal, characteristic, complemented
8#11727694C212yesmaximal, complemented
9#11727690C212yesmaximal, complemented
10#11727531C211yesmaximal, characteristic, complemented
11C4×#6390C211yesmaximal, characteristic, complemented
12C23×C7×C8C2212yes—
13C2×C4×C7×D4C2211yescharacteristic
14C2×#6499—21no—
15C2×#6487C2212yes—
16C2×#6432C2211yescharacteristic
17C2×C4×Dic14C2211yescharacteristic
18C7×C8×D4—24no—
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×C22⋊C8C2212yes—
21C2×C4×C7×C8C2211yescharacteristic
22#89993—21no—
23#89984C2212yes—
24#89984C2212yes—
25#89984C2212yes—
26C2×#6391C2212yes—
27C2×#6391C2212yes—
28C2×#6390C2211yescharacteristic
29C2×#6390C2212yes—
30C2×#6390C2211yescharacteristic
31#89746—21no—
32#89745—21no—
33#89725—22no—
34#89720—22no—
35#89707C2211yescharacteristic
36C4×C4⋊Dic7C2211yescharacteristic
37#89623—24no—
38#89346C2211yescharacteristic
39#89346C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#89346C2211yescharacteristic
43#25341—71nomaximal, Sylow
44C22×C7×D4D411yescharacteristic
45C23×C4×C7C2312yes—
46C22×Dic14—21no—
47C22×Dic14C2311yescharacteristic
48C22×C7×C8—24no—
49C22×C7×C8—24no—
50C22×C7×C8—42no—
51C22×C7×C8C2311yescharacteristic
52C22×C7×C8C2312yes—
53C22×C7×C8C2311yescharacteristic
54C4×C7×D4—22no—
55C4×C7×D4—22no—
56C2×C7×C4⋊C4D411yescharacteristic
57C2×C7×C22⋊C4D412yes—
58C2×C42×C7C2311yescharacteristic
59C2×C22⋊Dic7—22no—
60C2×C22⋊Dic7—22no—
61C2×C22⋊Dic7—22no—
62C2×C22⋊Dic7—22no—
63#6499—44no—
64C2×C4⋊Dic7—21no—
65C2×C4⋊Dic7—21no—
66C2×C4⋊Dic7—21no—
67C2×C4⋊Dic7C2311yescharacteristic
68C2×C4⋊Dic7C2312yes—
69C2×C4⋊Dic7C2312yes—
70C2×C4⋊Dic7C2311yescharacteristic
71C2×C4⋊Dic7C2311yescharacteristic
72#6487—24no—
73#6487—42no—
74C2×Dic7⋊C4—21no—
75C2×Dic7⋊C4—22no—
76C2×C4×Dic7—21no—
77C2×C4×Dic7—21no—
78#6432—22no—
79#6432—22no—
80C4×Dic14—22no—
81C4×Dic14—22no—
82C7×C4⋊C8D414yes—
83C7×C22⋊C8—24no—
84C4×C7×C8—22no—
85C7⋊#95—21no—
86C7⋊#95—22no—
87C7⋊#95—22no—
88C7⋊#95—22no—
89#6391—24no—
90#6391—42no—
91#6390—22no—
92#6390—24no—
93#6390—22no—
94C2×#475—71no—
95C2×C8×D4D711yescharacteristic
96#2547—71no—
97#2538—71no—
98#2077—72no—
99#2035—72no—
100#1990—71no—
101#1948—71no—
102#1892—72no—
103#1807—72no—
104C4×#103—71no—
105C24×C7C2×D412yes—
106C2×C7×D4—22no—
107C2×C7×D4—22no—
108C2×C7×D4SD1614yes—
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×C7C2×D412yes—
116C22×C4×C7C2×D411yescharacteristic
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7C2411yescharacteristic
119C22×Dic7—21no—
120C22×Dic7—21no—
121C22×Dic7—21no—
122C22×Dic7—21no—
123C22×Dic7—22no—
124C22×Dic7—22no—
125C2×Dic14—21no—
126C2×Dic14—21no—
127C2×Dic14—21no—
128C2×Dic14—21no—
129C2×Dic14—21no—
130C2×Dic14—42no—
131C2×Dic14—42no—
132C2×Dic14—42no—
133C2×Dic14#4112yes—
134C2×C7×C8—21no—
135C2×C7×C8—22no—
136C2×C7×C8—21no—
137C2×C7×C8—22no—
138C2×C7×C8—21no—
139C2×C7×C8—21no—
140C2×C7×C8—21no—
141C2×C7×C8—48no—
142C2×C7×C8C2×D414yes—
143C2×C7×C8#4112yes—
144C7×C4⋊C4—22no—
145C7×C22⋊C4—24no—
146C42×C7C2×D412yes—
147C42×C7#4112yes—
148C22⋊Dic7—44no—
149C22⋊Dic7—44no—
150C22⋊Dic7—44no—
151C22⋊Dic7—82no—
152C4⋊Dic7—21no—
153C4⋊Dic7—22no—
154C4⋊Dic7—21no—
155C4⋊Dic7—22no—
156C4⋊Dic7—22no—
157C4⋊Dic7—21no—
158C4⋊Dic7—21no—
159C4⋊Dic7—22no—
160C4⋊Dic7—21no—
161C4⋊Dic7—21no—
162C4⋊Dic7—42no—
163C4⋊Dic7—42no—
164C4⋊Dic7—81no—
165Dic7⋊C4—44no—
166Dic7⋊C4—42no—
167C4×Dic7—42no—
168C4×Dic7—42no—
169C23×C8C2×D712yes—
170C2×C4⋊Q8—71no—
171C2×#124—141no—
172C2×C22⋊Q8—72no—
173C2×C4×Q8—71no—
174C2×C4×D4C2×D711yescharacteristic
175#475—144no—
176C8×D4—24no—
177C2×#106—72no—
178C2×#106—72no—
179C2×C4⋊C8C2×D711yescharacteristic
180C2×#103—72no—
181C2×#103—71no—
182C2×#103—71no—
183C2×C22⋊C8C2×D712yes—
184C2×C4×C8C2×D711yescharacteristic
185#397—142no—
186#396—142no—
187#391—141no—
188#390—141no—
189#389—141no—
190#384—71no—
191#380—72no—
192#380—72no—
193#380—72no—
194C4×C4⋊C4—71no—
195#340—71no—
196#340—72no—
197#340—71no—
198#340—72no—
199#340—72no—
200C23×C7—22no—
201C23×C7—22no—
202C23×C7—22no—
203C23×C7—22no—
204C23×C7—22no—
205C23×C7—22no—
206C23×C7C2×SD1614yes—
207C23×C7C22×D411yescharacteristic
208C7×D4—28no—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—22no—
212C2×C4×C7—22no—
213C2×C4×C7—22no—
214C2×C4×C7—42no—
215C2×C4×C7—42no—
216C2×C4×C7—42no—
217C2×C4×C7—42no—
218C2×C4×C7#12112yes—
219C2×C4×C7#12112yes—
220C2×C4×C7C2×SD1612yes—
221C2×C4×C7#13512yes—
222C2×C4×C7C22×D411yescharacteristic
223C2×C4×C7C2×#4111yescharacteristic
224C2×C4×C7C2×#4111yescharacteristic
225C2×C4×C7C2×#4111yescharacteristic
226C2×C4×C7H(5,𝔽2)11yescharacteristic
227C2×C4×C72−1+411yescharacteristic
228C2×Dic7—22no—
229C2×Dic7—41no—
230C2×Dic7—41no—
231C2×Dic7—41no—
232C2×Dic7—42no—
233C2×Dic7—42no—
234C2×Dic7—41no—
235C2×Dic7—41no—
236C2×Dic7—41no—
237C2×Dic7—41no—
238C2×Dic7—42no—
239C2×Dic7—42no—
240C2×Dic7—42no—
241C2×Dic7—42no—
242C2×Dic7—41no—
243C2×Dic7—41no—
244C2×Dic7—41no—
245C2×Dic7—41no—
246Dic14—22no—
247Dic14—22no—
248Dic14—41no—
249Dic14—41no—
250Dic14—84no—
251C7×C8—22no—
252C7×C8—24no—
253C7×C8—41no—
254C22×Q8—71no—
255C22×Q8—141no—
256C22×D4D2811yescharacteristic
257C23×C4C22×D712yes—
258C22×C8—24no—
259C22×C8—24no—
260C22×C8—42no—
261C22×C8C22×D711yescharacteristic
262C22×C8C22×D711yescharacteristic
263C22×C8C22×D712yes—
264C4⋊Q8—142no—
265C4⋊Q8—142no—
266#124—284no—
267C22⋊Q8—144no—
268C22⋊Q8—282no—
269C4×Q8—142no—
270C4×Q8—142no—
271C4×D4—22no—
272C4×D4—22no—
273C2×C4⋊C4—71no—
274C2×C4⋊C4—72no—
275C2×C4⋊C4—72no—
276C2×C4⋊C4—71no—
277C2×C4⋊C4—71no—
278C2×C4⋊C4—141no—
279C2×C4⋊C4—142no—
280C2×C4⋊C4—141no—
281C2×C4⋊C4—141no—
282C2×C4⋊C4—141no—
283C2×C4⋊C4D2811yescharacteristic
284C2×C22⋊C4—142no—
285C2×C22⋊C4—142no—
286C2×C22⋊C4—142no—
287C2×C22⋊C4—142no—
288C2×C22⋊C4D2812yes—
289C2×C42—141no—
290C2×C42—141no—
291C2×C42C22×D711yescharacteristic
292#106—144no—
293#106—282no—
294C4⋊C8C22⋊D714yes—
295#103—142no—
296#103—142no—
297#103—144no—
298C22⋊C8—24no—
299C4×C8—22no—
300#95—141no—
301#95—142no—
302#95—142no—
303#95—142no—
304C22×C7—24no—
305C22×C7—24no—
306C22×C7—24no—
307C22×C7—42no—
308C22×C7—42no—
309C22×C7—42no—
310C22×C7—42no—
311C22×C7C2×#12111yescharacteristic
312C22×C7#54611yescharacteristic
313C22×C7#54711yescharacteristic
314C22×C7C22×SD1611yescharacteristic
315C22×C7C2×#13511yescharacteristic
316C22×C7#57711yescharacteristic
317C22×C7#57811yescharacteristic
318C4×C7—22no—
319C4×C7—42no—
320C4×C7#46514yes—
321C4×C7#52911yescharacteristic
322C4×C7#54211yescharacteristic
323C4×C7#54611yescharacteristic
324C4×C7#54811yescharacteristic
325Dic7—42no—
326Dic7—81no—
327Dic7—81no—
328Dic7—82no—
329Dic7—82no—
330Dic7—81no—
331C24C2×D2812yes—
332C2×Q8—72no—
333C2×Q8—141no—
334C2×Q8—141no—
335C2×Q8—141no—
336C2×Q8—141no—
337C2×Q8—141no—
338C2×Q8—282no—
339C2×Q8—282no—
340C2×Q8—282no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4#113214yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—141no—
351C22×C4—141no—
352C22×C4—141no—
353C22×C4—141no—
354C22×C4—142no—
355C22×C4—142no—
356C22×C4C2×D2812yes—
357C22×C4C2×D2811yescharacteristic
358C22×C4C2×D2811yescharacteristic
359C22×C4C23×D711yescharacteristic
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—48no—
368C2×C8#115912yes—
369C2×C8C2×C22⋊D714yes—
370C4⋊C4—22no—
371C4⋊C4—141no—
372C4⋊C4—142no—
373C4⋊C4—141no—
374C4⋊C4—141no—
375C4⋊C4—142no—
376C4⋊C4—141no—
377C4⋊C4—142no—
378C4⋊C4—141no—
379C4⋊C4—142no—
380C4⋊C4—141no—
381C4⋊C4—284no—
382C4⋊C4—282no—
383C4⋊C4—282no—
384C4⋊C4—282no—
385C4⋊C4—561no—
386C22⋊C4—24no—
387C22⋊C4—284no—
388C22⋊C4—284no—
389C22⋊C4—284no—
390C22⋊C4—562no—
391C42—282no—
392C42—282no—
393C42#115712yes—
394C42C2×C22⋊D712yes—
395C2×C7—44no—
396C2×C7#242711yescharacteristic
397C2×C7C2×#46511yescharacteristic
398C2×C7#306311yescharacteristic
399C2×C7#329811yescharacteristic
400C2×C7#330411yescharacteristic
401C2×C7#333511yescharacteristic
402C2×C7#334211yescharacteristic
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23C2×#113214yes—
410C23C22×D2811yescharacteristic
411Q8—142no—
412Q8—142no—
413Q8—281no—
414Q8—281no—
415Q8—564no—
416D4—28no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—142no—
427C2×C4—281no—
428C2×C4—281no—
429C2×C4—281no—
430C2×C4—281no—
431C2×C4—282no—
432C2×C4—282no—
433C2×C4—282no—
434C2×C4—281no—
435C2×C4—282no—
436C2×C4—281no—
437C2×C4—281no—
438C2×C4—281no—
439C2×C4—281no—
440C2×C4—282no—
441C2×C4—282no—
442C2×C4—281no—
443C2×C4—281no—
444C2×C4C2×#113212yes—
445C2×C4#646712yes—
446C2×C4#649312yes—
447C2×C4#649312yes—
448C2×C4C2×#115711yescharacteristic
449C2×C4C2×#115911yescharacteristic
450C2×C4#654811yescharacteristic
451C2×C4D7×#4111yescharacteristic
452C2×C4#655411yescharacteristic
453C2×C4C22×C22⋊D711yescharacteristic
454C8—22no—
455C8—24no—
456C8—41no—
457C7#2534111yescharacteristic, complemented, Sylow
458C22—24no—
459C22—24no—
460C22—24no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22#9024511yescharacteristic
466C22#9024611yescharacteristic
467C22C22×#113211yescharacteristic
468C22C2×#646711yescharacteristic
469C22#9044211yescharacteristic
470C22#9044411yescharacteristic
471C22C2×#649311yescharacteristic
472C4—22no—
473C4—42no—
474C4—282no—
475C4—562no—
476C4—561no—
477C4—561no—
478C4—562no—
479C4—561no—
480C4#8988614yes—
481C4#9024811yescharacteristic
482C4#9025211yescharacteristic
483C4#9049711yescharacteristic
484C4#9052211yescharacteristic
485C2—44no—
486C2#1173685711yescharacteristic
487C2#1173686411yescharacteristic
488C2#1173689211yescharacteristic
489C2#1173689611yescharacteristic
490C2#1173985311yescharacteristic
491C2C2×#8988611yescharacteristic
492C2#1174125311yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×C8×D4C2
socleC23×C7C22×D4
prime radicalC22×C7×D4D4
hypercenterC2×C8×D4D7
hypocenterC7#25341
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
4. C2×#89623C2semidirect
5. #11729726C2semidirect
6. #11729724C2semidirect
7. #11728208C2semidirect
8. #11727694C2semidirect
9. #11727690C2semidirect
10. #11727531C2semidirect
11. C4×#6390C2semidirect
457. C7#25341semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C8×D42
#117398272
#117398212
C2×#896232
#117297262
#117297242
#117282082
#117276942
#117276902
#117275312
C4×#63902
#253417