Finite Group Databasebeta

← #13360985

Subgroups — #13360985

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
1C2×C4⋊C4×Dic7C211yesmaximal, characteristic
2C2×#89493C211yesmaximal, characteristic
3#11731311C211yesmaximal, characteristic
4C7×#1852C211yesmaximal, characteristic
5#11729313C211yesmaximal, characteristic
6#11728378C212yesmaximal, complemented
7#11728359C212yesmaximal, complemented
8#11727660C211yesmaximal, characteristic
9#11727255C211yesmaximal, characteristic
10#11726570C212yesmaximal, complemented
11#11726537C212yesmaximal, complemented
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic7C2211yescharacteristic
14C2×#6487C2211yescharacteristic
15C22×C4×Dic7—21no—
16C22×C4×Dic7C2211yescharacteristic
17C2×#6450C2211yescharacteristic
18C2×C7×#107C2212yes—
19C2×C7×C22⋊C8C2211yescharacteristic
20C7×#380C2211yescharacteristic
21C2×C7⋊#95—21no—
22C2×C22⋊(C7⋊C8)C2211yescharacteristic
23C2×#6391C2212yes—
24C2×#6390C2212yes—
25C2×#6389C2212yes—
26C4⋊C4×Dic7—21no—
27C4⋊C4×Dic7—21no—
28C4⋊C4×Dic7C2211yescharacteristic
29C4⋊C4×Dic7C2212yes—
30C4⋊C4×Dic7C2212yes—
31#89746C2211yescharacteristic
32#89745C2211yescharacteristic
33C2×#6384C2212yes—
34#89718—21no—
35C22⋊C4×Dic7—21no—
36#89493C418yescomplemented
37#89424—21no—
38#89421—21no—
39C7×#340C2212yes—
40#89345C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#21579—71nomaximal, Sylow
45C23×C4×C7—21no—
46C23×C4×C7C2311yescharacteristic
47C23×Dic7—21no—
48C23×Dic7D411yescharacteristic
49C23×Dic7D411yescharacteristic
50C22×Dic14C2311yescharacteristic
51C22×C7×C8C2312yes—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—24no—
54C2×C7×C4⋊C4C2311yescharacteristic
55C2×C7×C4⋊C4C2311yescharacteristic
56C2×C7×C4⋊C4C2312yes—
57C2×C7×C22⋊C4—21no—
58C2×C22⋊Dic7—21no—
59C2×C4⋊Dic7—21no—
60C2×C4⋊Dic7—22no—
61C2×C4⋊Dic7C2×C414yes—
62C2×C4⋊Dic7C2312yes—
63C2×C4⋊Dic7C2311yescharacteristic
64#6487—22no—
65#6487C2×C414yes—
66C2×Dic7⋊C4—21no—
67C2×Dic7⋊C4—21no—
68C2×Dic7⋊C4—21no—
69C2×C4×Dic7—24no—
70C2×C4×Dic7—21no—
71C2×C4×Dic7—21no—
72C2×C4×Dic7—22no—
73C2×C4×Dic7—22no—
74C2×C4×Dic7—44no—
75C2×C4×Dic7D411yescharacteristic
76C2×C4×Dic7D411yescharacteristic
77C2×C4×Dic7C2311yescharacteristic
78C2×C4×Dic7C2311yescharacteristic
79C22×C7⋊C8C2312yes—
80#6450—24no—
81C7×#107—24no—
82C7×C22⋊C8C2×C414yes—
83C7×#95—21no—
84C7⋊#95—21no—
85C7⋊#95—21no—
86C7⋊#95—21no—
87C7⋊#95—21no—
88C7⋊#95—22no—
89C7⋊#95—22no—
90C22⋊(C7⋊C8)—22no—
91#6391C2×C418yescomplemented
92#6390C2×C418yescomplemented
93#6389—24no—
94#6384—24no—
95C2×#483—71no—
96C4×C22⋊Q8—71no—
97C2×C4×C4⋊C4—71no—
98#1928—72no—
99#1919—72no—
100#1872—71no—
101#1852D711yescharacteristic
102#1823—71no—
103#1776—72no—
104#1763—72no—
105#1749—71no—
106C24×C7C2×D411yescharacteristic
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—24no—
110C22×C4×C7—22no—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—42no—
114C22×C4×C7—42no—
115C22×C4×C7C22×C412yes—
116C22×C4×C7C2×D411yescharacteristic
117C22×C4×C7#4112yes—
118C22×C4×C7C2411yescharacteristic
119C22×Dic7—22no—
120C22×Dic7—22no—
121C22×Dic7—22no—
122C22×Dic7—24no—
123C22×Dic7—22no—
124C22×Dic7—22no—
125C22×Dic7—21no—
126C22×Dic7—21no—
127C22×Dic7—22no—
128C22×Dic7—41no—
129C22×Dic7—42no—
130C22×Dic7—42no—
131C22×Dic7C2×D411yescharacteristic
132C22×Dic7C2×D411yescharacteristic
133C22×Dic7C2×D411yescharacteristic
134C22×Dic7C2×D411yescharacteristic
135C2×Dic14—22no—
136C2×Dic14—22no—
137C2×Dic14—21no—
138C2×Dic14C22×C412yes—
139C2×C7×C8—22no—
140C2×C7×C8—22no—
141C2×C7×C8C22×C414yes—
142C7×C4⋊C4—24no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4—22no—
146C7×C4⋊C4#4114yes—
147C7×C22⋊C4—42no—
148C22⋊Dic7—24no—
149C4⋊Dic7—24no—
150C4⋊Dic7—21no—
151C4⋊Dic7—22no—
152C4⋊Dic7C22×C412yes—
153C4⋊Dic7C22×C414yes—
154Dic7⋊C4—24no—
155Dic7⋊C4—42no—
156Dic7⋊C4—42no—
157C4×Dic7—22no—
158C4×Dic7—22no—
159C4×Dic7—22no—
160C4×Dic7—42no—
161C4×Dic7—44no—
162C4×Dic7—44no—
163C4×Dic7—42no—
164C4×Dic7#4114yes—
165C2×C7⋊C8—22no—
166C2×C7⋊C8—22no—
167C2×C7⋊C8—22no—
168C2×C22⋊Q8—71no—
169C2×C4×Q8—71no—
170C22×C4⋊C4—71no—
171C22×C4⋊C4C2×D711yescharacteristic
172C22×C42—71no—
173C22×C42—141no—
174#483—78no—
175C2×#107C2×D712yes—
176C2×#106—72no—
177C2×#103—72no—
178C2×#103—72no—
179C2×C22⋊C8—71no—
180C2×C22⋊C8C2×D711yescharacteristic
181C2×#97—72no—
182#391—71no—
183#389—71no—
184#387—141no—
185#385—141no—
186#380C2×D711yescharacteristic
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#340C2×D712yes—
199#336—72no—
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—21no—
209C2×C4×C7—21no—
210C2×C4×C7—24no—
211C2×C4×C7—24no—
212C2×C4×C7—24no—
213C2×C4×C7—48no—
214C2×C4×C7—42no—
215C2×C4×C7—42no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—41no—
219C2×C4×C7—41no—
220C2×C4×C7—41no—
221C2×C4×C7—41no—
222C2×C4×C7C4×D412yes—
223C2×C4×C7C23×C411yescharacteristic
224C2×C4×C7C2×#4112yes—
225C2×C4×C7C2×#4111yescharacteristic
226C2×C4×C7C2×#4112yes—
227C2×Dic7—28no—
228C2×Dic7—21no—
229C2×Dic7—21no—
230C2×Dic7—21no—
231C2×Dic7—21no—
232C2×Dic7—21no—
233C2×Dic7—21no—
234C2×Dic7—22no—
235C2×Dic7—22no—
236C2×Dic7—24no—
237C2×Dic7—21no—
238C2×Dic7—21no—
239C2×Dic7—21no—
240C2×Dic7—44no—
241C2×Dic7—41no—
242C2×Dic7—41no—
243C2×Dic7—42no—
244C2×Dic7—41no—
245C2×Dic7—42no—
246C2×Dic7—41no—
247C2×Dic7—48no—
248C2×Dic7#12312yes—
249C2×Dic7#12312yes—
250C2×Dic7#13712yes—
251Dic14—24no—
252Dic14—42no—
253C7×C8—24no—
254C7⋊C8—42no—
255C22×Q8—71no—
256C23×C4—21no—
257C23×C4—71no—
258C23×C4—71no—
259C23×C4—141no—
260C23×C4C22×D711yescharacteristic
261C22×C8—72no—
262C22×C8C22×D712yes—
263C22⋊Q8—74no—
264C22⋊Q8—142no—
265C4×Q8—144no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—74no—
269C2×C4⋊C4—72no—
270C2×C4⋊C4—71no—
271C2×C4⋊C4—141no—
272C2×C4⋊C4—141no—
273C2×C4⋊C4—142no—
274C2×C4⋊C4—141no—
275C2×C4⋊C4—141no—
276C2×C4⋊C4C22×D711yescharacteristic
277C2×C4⋊C4C22×D711yescharacteristic
278C2×C4⋊C4C22×D712yes—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—141no—
281C2×C42—71no—
282C2×C42—71no—
283C2×C42—71no—
284C2×C42—71no—
285C2×C42—144no—
286C2×C42—141no—
287C2×C42—141no—
288C2×C42—142no—
289C2×C42—142no—
290C2×C42—284no—
291#107—24no—
292#106—78no—
293#103—78no—
294#103—144no—
295C22⋊C8—142no—
296C22⋊C8C4×D714yes—
297#97—144no—
298#95—21no—
299#95—142no—
300#95—142no—
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—44no—
321C4×C7—81no—
322C4×C7—81no—
323C4×C7C4×#4112yes—
324C4×C7#52912yes—
325Dic7—22no—
326Dic7—22no—
327Dic7—22no—
328Dic7—41no—
329Dic7—44no—
330Dic7—42no—
331Dic7—42no—
332C24D4×D711yescharacteristic
333C2×Q8—72no—
334C2×Q8—141no—
335C2×Q8—142no—
336C2×Q8—142no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—71no—
346C22×C4—71no—
347C22×C4—71no—
348C22×C4—71no—
349C22×C4—142no—
350C22×C4—142no—
351C22×C4—142no—
352C22×C4—142no—
353C22×C4—144no—
354C22×C4—142no—
355C22×C4—142no—
356C22×C4—141no—
357C22×C4—141no—
358C22×C4—281no—
359C22×C4—282no—
360C22×C4—282no—
361C22×C4C2×C4×D712yes—
362C22×C4D4×D711yescharacteristic
363C22×C4#116112yes—
364C22×C4C23×D711yescharacteristic
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—142no—
368C2×C8—142no—
369C2×C8—142no—
370C2×C8C2×C4×D714yes—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—74no—
376C4⋊C4—72no—
377C4⋊C4—144no—
378C4⋊C4—144no—
379C4⋊C4—142no—
380C4⋊C4—141no—
381C4⋊C4—282no—
382C4⋊C4—282no—
383C4⋊C4#115914yes—
384C22⋊C4—42no—
385C22⋊C4—144no—
386C42—74no—
387C42—142no—
388C42—142no—
389C42—142no—
390C42—282no—
391C42—282no—
392C42—284no—
393C42—284no—
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—
406C23#645612yes—
407C23#647412yes—
408C23#648112yes—
409C23C2×D4×D711yescharacteristic
410Q8—144no—
411Q8—282no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—48no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—72no—
427C2×C4—72no—
428C2×C4—72no—
429C2×C4—148no—
430C2×C4—141no—
431C2×C4—141no—
432C2×C4—141no—
433C2×C4—141no—
434C2×C4—141no—
435C2×C4—141no—
436C2×C4—142no—
437C2×C4—142no—
438C2×C4—144no—
439C2×C4—141no—
440C2×C4—141no—
441C2×C4—141no—
442C2×C4—284no—
443C2×C4—288no—
444C2×C4—282no—
445C2×C4—282no—
446C2×C4—281no—
447C2×C4—281no—
448C2×C4—281no—
449C2×C4—281no—
450C2×C4#645612yes—
451C2×C4C22×C4×D711yescharacteristic
452C2×C4C2×#115912yes—
453C2×C4C2×#116111yescharacteristic
454C2×C4D7×#4112yes—
455C8—24no—
456C8—282no—
457C7#2157911yescharacteristic, complemented, Sylow
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22#8966114yes—
465C22C2×#645611yescharacteristic
466C22#9027711yescharacteristic
467C22D7×#12311yescharacteristic
468C22C2×#647411yescharacteristic
469C22C2×#648111yescharacteristic
470C22#9046411yescharacteristic
471C22D7×#13711yescharacteristic
472C4—22no—
473C4—44no—
474C4—81no—
475C4—81no—
476C4—142no—
477C4—142no—
478C4—142no—
479C4—284no—
480C4—281no—
481C4—282no—
482C4—282no—
483C4C4×#115912yes—
484C4#9031512yes—
485C2—24no—
486C2#1173182511yescharacteristic
487C2C2×#8966111yescharacteristic
488C2#1173664511yescharacteristic
489C2#1173743611yescharacteristic
490C2#1173761911yescharacteristic
491C2#1173872711yescharacteristic
492C2#1173891411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1852C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1852D7
hypocenterC7#21579
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
6. #11728378C2semidirect
7. #11728359C2semidirect
10. #11726570C2semidirect
11. #11726537C2semidirect
36. #89493C4semidirect
91. #6391C2×C4semidirect
92. #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
C2×C4⋊C4×Dic72
C2×#894932
#117313112
C7×#18522
#117293132
#117283782
#117283592
#117276602
#117272552
#117265702
#117265372
#215797