Finite Group Databasebeta

← #13249372

Subgroups — #13249372

Order 1792.

952 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2848C211yesmaximal, characteristic
2C2×#89979C211yesmaximal, characteristic
3#11729881C211yesmaximal, characteristic
4C2×#89333C212yesmaximal, complemented
5#11725430C212yesmaximal, complemented
6C24×C4×C7C2211yescharacteristic
7C2×C7×C22⋊Q8C411yescharacteristic, complemented
8C2×C7×C22⋊Q8C411yescharacteristic, complemented
9C2×C7×C22⋊Q8C2211yescharacteristic
10C22×C22⋊Dic7—21no—
11C22×#1136—21no—
12C7×#385—21no—
13C7×#385—21no—
14C7×#385—21no—
15C7×#380C411yescharacteristic, complemented
16C7×#380C411yescharacteristic, complemented
17C7×#380C2211yescharacteristic
18C7×#379—21no—
19C2×C7⋊#95—21no—
20C2×C7⋊#95C2212yes—
21#89979—24no—
22C2×C22⋊(C7⋊C8)—21no—
23C2×C22⋊(C7⋊C8)C2212yes—
24#89333—24no—
25#9115—71nomaximal, Sylow
26C25×C7D411yescharacteristic
27C22×C7×Q8C2×C411yescharacteristic
28C23×C4×C7—22no—
29C23×C4×C7—22no—
30C23×C4×C7—42no—
31C23×C4×C7—42no—
32C23×C4×C7C2×C411yescharacteristic
33C23×C4×C7C2×C411yescharacteristic
34C23×C4×C7D411yescharacteristic
35C23×C4×C7D411yescharacteristic
36C23×C4×C7D411yescharacteristic
37C23×C4×C7C2311yescharacteristic
38C23×Dic7—22no—
39C7×C22⋊Q8—22no—
40C7×C22⋊Q8—22no—
41C7×C22⋊Q8—42no—
42C7×C22⋊Q8—42no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4C2×C411yescharacteristic
47C2×C7×C4⋊C4C2×C411yescharacteristic
48C2×C7×C4⋊C4C2×C411yescharacteristic
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C7×C22⋊C4—21no—
53C2×C7×C22⋊C4—21no—
54C2×C7×C22⋊C4—21no—
55C2×C22⋊Dic7—24no—
56C2×C22⋊Dic7—42no—
57C2×C22⋊Dic7—42no—
58C2×#1136—24no—
59C2×#1136—42no—
60C2×#1136—42no—
61C22×C7⋊C8—22no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95—21no—
65C7⋊#95—22no—
66C7⋊#95—22no—
67C7⋊#95—42no—
68C22⋊(C7⋊C8)—24no—
69C22⋊(C7⋊C8)—22no—
70C22⋊(C7⋊C8)D414yes—
71C2×#407—71no—
72#2848D711yescharacteristic
73C2×#328—72no—
74C22⋊#95—71no—
75#1294—72no—
76C24×C7—22no—
77C24×C7—22no—
78C24×C7—21no—
79C24×C7—21no—
80C24×C7—22no—
81C24×C7—22no—
82C24×C7—41no—
83C24×C7—41no—
84C24×C7C22⋊C411yescharacteristic
85C24×C7C22⋊C411yescharacteristic
86C24×C7C2×D411yescharacteristic
87C2×C7×Q8—22no—
88C2×C7×Q8—41no—
89C2×C7×Q8—41no—
90C22×C4×C7—22no—
91C22×C4×C7—22no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—22no—
95C22×C4×C7—22no—
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—21no—
101C22×C4×C7—21no—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—41no—
107C22×C4×C7—41no—
108C22×C4×C7—41no—
109C22×C4×C7—41no—
110C22×C4×C7—41no—
111C22×C4×C7—41no—
112C22×C4×C7—41no—
113C22×C4×C7—48no—
114C22×C4×C7—48no—
115C22×C4×C7C22⋊C411yescharacteristic
116C22×C4×C7C22⋊C411yescharacteristic
117C22×C4×C7C22×C411yescharacteristic
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2×D412yes—
120C22×C4×C7#4112yes—
121C22×Dic7—24no—
122C22×Dic7—42no—
123C22×Dic7—42no—
124C22×Dic7—42no—
125C22×Dic7—42no—
126C22×Dic7—42no—
127C22×Dic7—42no—
128C7×C4⋊C4—22no—
129C7×C4⋊C4—41no—
130C7×C4⋊C4—41no—
131C7×C4⋊C4—42no—
132C7×C4⋊C4—42no—
133C7×C4⋊C4—41no—
134C7×C4⋊C4—41no—
135C7×C22⋊C4—42no—
136C7×C22⋊C4—42no—
137C7×C22⋊C4—42no—
138C7×C22⋊C4—42no—
139C7×C22⋊C4—42no—
140C7×C22⋊C4—41no—
141C7×C22⋊C4—41no—
142C22⋊Dic7—48no—
143#1136—48no—
144C2×C7⋊C8—24no—
145C2×C7⋊C8—42no—
146C2×C7⋊C8—42no—
147C24×C4C2×D711yescharacteristic
148C22×M16—141no—
149C2×C22⋊Q8Dic711yescharacteristic, complemented
150C2×C22⋊Q8Dic711yescharacteristic, complemented
151C2×C22⋊Q8C2×D711yescharacteristic
152C22×C22⋊C4—141no—
153#407—144no—
154C2×C22⋊C8—72no—
155C2×C22⋊C8—141no—
156#385—21no—
157#385—21no—
158#385—21no—
159#380Dic711yescharacteristic, complemented
160#380Dic711yescharacteristic, complemented
161#380C2×D711yescharacteristic
162#379—21no—
163C2×#95—72no—
164C2×#95—141no—
165#328—144no—
166C23×C7—21no—
167C23×C7—21no—
168C23×C7—21no—
169C23×C7—21no—
170C23×C7—24no—
171C23×C7—24no—
172C23×C7—21no—
173C23×C7—21no—
174C23×C7—21no—
175C23×C7—21no—
176C23×C7—21no—
177C23×C7—21no—
178C23×C7—21no—
179C23×C7—21no—
180C23×C7—21no—
181C23×C7—21no—
182C23×C7—21no—
183C23×C7—21no—
184C23×C7—44no—
185C23×C7—44no—
186C23×C7—44no—
187C23×C7—44no—
188C23×C7—42no—
189C23×C7—42no—
190C23×C7—42no—
191C23×C7—42no—
192C23×C7#9912yes—
193C23×C7#10412yes—
194C23×C7C2×C22⋊C411yescharacteristic
195C23×C7#12112yes—
196C23×C7AGL(1,ℤ/8)11yescharacteristic
197C23×C7#13712yes—
198C23×C7#13711yescharacteristic
199C7×Q8—82no—
200C2×C4×C7—24no—
201C2×C4×C7—24no—
202C2×C4×C7—21no—
203C2×C4×C7—21no—
204C2×C4×C7—21no—
205C2×C4×C7—21no—
206C2×C4×C7—21no—
207C2×C4×C7—21no—
208C2×C4×C7—21no—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—42no—
212C2×C4×C7—42no—
213C2×C4×C7—44no—
214C2×C4×C7—44no—
215C2×C4×C7—44no—
216C2×C4×C7—42no—
217C2×C4×C7—42no—
218C2×C4×C7—41no—
219C2×C4×C7—41no—
220C2×C4×C7—41no—
221C2×C4×C7—41no—
222C2×C4×C7—41no—
223C2×C4×C7—41no—
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#9912yes—
231C2×C4×C7C4×D412yes—
232C2×C4×C7#12011yescharacteristic
233C2×C4×C7#12311yescharacteristic
234C2×Dic7—44no—
235C2×Dic7—44no—
236C2×Dic7—44no—
237C2×Dic7—82no—
238C2×Dic7—82no—
239C2×Dic7—82no—
240C2×Dic7—82no—
241C7⋊C8—44no—
242C25C22⋊D711yescharacteristic
243C22×Q8C2×Dic711yescharacteristic
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—42no—
247C23×C4—42no—
248C23×C4—142no—
249C23×C4C2×Dic711yescharacteristic
250C23×C4C2×Dic711yescharacteristic
251C23×C4C22⋊D711yescharacteristic
252C23×C4C22⋊D711yescharacteristic
253C23×C4C22⋊D711yescharacteristic
254C23×C4C22×D711yescharacteristic
255C2×M16—144no—
256C2×M16—282no—
257C2×M16—282no—
258C22×C8—142no—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261C22⋊Q8—42no—
262C22⋊Q8—42no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C2×Dic711yescharacteristic
267C2×C4⋊C4C2×Dic711yescharacteristic
268C2×C4⋊C4C2×Dic711yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—144no—
276C2×C22⋊C4—282no—
277C2×C22⋊C4—282no—
278C22⋊C8—74no—
279C22⋊C8—144no—
280C22⋊C8—142no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—142no—
285#95—142no—
286#95—282no—
287C22×C7—22no—
288C22×C7—22no—
289C22×C7—21no—
290C22×C7—21no—
291C22×C7—22no—
292C22×C7—22no—
293C22×C7—21no—
294C22×C7—21no—
295C22×C7—22no—
296C22×C7—22no—
297C22×C7—22no—
298C22×C7—22no—
299C22×C7—22no—
300C22×C7—22no—
301C22×C7—48no—
302C22×C7—48no—
303C22×C7—41no—
304C22×C7—41no—
305C22×C7—41no—
306C22×C7—41no—
307C22×C7—41no—
308C22×C7—41no—
309C22×C7—41no—
310C22×C7—41no—
311C22×C7#38611yescharacteristic
312C22×C7C2×#9911yescharacteristic
313C22×C7C2×#9911yescharacteristic
314C22×C7#41711yescharacteristic
315C22×C7#41911yescharacteristic
316C22×C7C2×#10411yescharacteristic
317C22×C7#42111yescharacteristic
318C22×C7#46812yes—
319C22×C7#47012yes—
320C4×C7—21no—
321C4×C7—22no—
322C4×C7—21no—
323C4×C7—22no—
324C4×C7—41no—
325C4×C7—81no—
326C4×C7—81no—
327C4×C7—81no—
328C4×C7—81no—
329Dic7—84no—
330C24—22no—
331C24—22no—
332C24—21no—
333C24—21no—
334C24—22no—
335C24—22no—
336C24—41no—
337C24—41no—
338C24C22⋊Dic711yescharacteristic
339C24C22⋊Dic711yescharacteristic
340C24C2×C22⋊D711yescharacteristic
341C2×Q8—22no—
342C2×Q8—41no—
343C2×Q8—41no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—144no—
370C22×C4—282no—
371C22×C4—282no—
372C22×C4—282no—
373C22×C4—282no—
374C22×C4—282no—
375C22×C4—282no—
376C22×C4C22⋊Dic711yescharacteristic
377C22×C4C22⋊Dic711yescharacteristic
378C22×C4D4×D712yes—
379C22×C4#115912yes—
380C22×C4C22×Dic711yescharacteristic
381C22×C4C2×C22⋊D711yescharacteristic
382M16—288no—
383C2×C8—144no—
384C2×C8—282no—
385C2×C8—282no—
386C4⋊C4—22no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—288no—
401C2×C7—22no—
402C2×C7—22no—
403C2×C7—42no—
404C2×C7—42no—
405C2×C7#150411yescharacteristic
406C2×C7#150811yescharacteristic
407C2×C7#151111yescharacteristic
408C2×C7#151311yescharacteristic
409C2×C7#187111yescharacteristic
410C2×C7#189411yescharacteristic
411C2×C7#189811yescharacteristic
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—24no—
417C23—24no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23#640812yes—
439C23#641112yes—
440C23#650212yes—
441C23#650512yes—
442C23#651211yescharacteristic
443C23#651411yescharacteristic
444C23C2×C22⋊Dic711yescharacteristic
445Q8—82no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—284no—
477C2×C4—284no—
478C2×C4—284no—
479C2×C4—562no—
480C2×C4—562no—
481C2×C4—562no—
482C2×C4—562no—
483C2×C4#640812yes—
484C2×C4D4×Dic712yes—
485C2×C4#649811yescharacteristic
486C2×C4#650011yescharacteristic
487C8—284no—
488C7#911511yescharacteristic, complemented, Sylow
489C22—22no—
490C22—22no—
491C22—22no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—48no—
504C22—48no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22#8982112yes—
514C22#8982412yes—
515C22C2×#640811yescharacteristic
516C22C2×#640811yescharacteristic
517C22#8999211yescharacteristic
518C22#8999711yescharacteristic
519C22#9000411yescharacteristic
520C22C2×#641111yescharacteristic
521C22#9000811yescharacteristic
522C4—21no—
523C4—22no—
524C4—21no—
525C4—22no—
526C4—41no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4—564no—
532C2—22no—
533C2—22no—
534C2—42no—
535C2—42no—
536C2#1172942411yescharacteristic
537C2#1172944211yescharacteristic
538C2#1172944311yescharacteristic
539C2#1172945111yescharacteristic
540C2#1172946211yescharacteristic
541C2#1172946311yescharacteristic
542C2#1172946711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2848C2
socleC23×C7C2×C22⋊C4
prime radicalC25×C7D4
cosocleC23×C4×C7C23
hypercenter#2848D7
hypocenterC7#9115
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
4. C2×#89333C2semidirect
5. #11725430C2semidirect
7. C2×C7×C22⋊Q8C4semidirect
8. C2×C7×C22⋊Q8C4semidirect
15. C7×#380C4semidirect
16. C7×#380C4semidirect
149. C2×C22⋊Q8Dic7semidirect
150. C2×C22⋊Q8Dic7semidirect
159. #380Dic7semidirect
160. #380Dic7semidirect
488. C7#9115semidirect

Maximal subgroups

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

subgroupindex
C7×#28482
C2×#899792
#117298812
C2×#893332
#117254302
#91157