Finite Group Databasebeta

← #13410769

Subgroups — #13410769

Order 1792.

1054 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×Q82C211yesmaximal, characteristic, complemented
2C2×#89861C211yesmaximal, characteristic
3#11739929C211yesmaximal, characteristic
4C2×#89800C211yesmaximal, characteristic
5#11729548C212yesmaximal, complemented
6#11729546C212yesmaximal
7#11729534C211yesmaximal, characteristic, complemented
8#11729389C211yesmaximal, characteristic, complemented
9#11729387C212yesmaximal
10#11729386C212yesmaximal, complemented
11#11729385C211yesmaximal, characteristic, complemented
12C7×Q82—24no—
13C7×Q82—24no—
14C2×C7×C4⋊Q8—22no—
15C2×C7×C4⋊Q8—21no—
16C2×C7×C4⋊Q8C2212yes—
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C4×C7×Q8—21no—
19C2×C4×C7×Q8C2212yes—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×Q8C2211yescharacteristic
22C22×#1144C2211yescharacteristic
23C22×#1143C2211yescharacteristic
24C2×#6438C2211yescharacteristic
25#90003—21no—
26C2×#6409C2212yes—
27#89861—24no—
28#89800—24no—
29#89765—21no—
30#89762—22no—
31C2×#6384C2211yescharacteristic
32C2×#6384C2212yes—
33C2×#6383C2211yescharacteristic
34C2×#6383C2212yes—
35#89711C2212yes—
36C4×C4⋊Dic7C2211yescharacteristic
37#89701C2212yes—
38C2×C4⋊(C7⋊C8)C2211yescharacteristic
39C2×#6377C2211yescharacteristic
40#89324C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89319C2212yes—
44#89319C2212yes—
45#25942—71nomaximal, Sylow
46C22×C7×Q8—22no—
47C22×C7×Q8—21no—
48C22×C7×Q8D411yescharacteristic
49C22×C7×Q8C2311yescharacteristic
50C22×D28C2311yescharacteristic
51C22×Dic14C2311yescharacteristic
52C7×C4⋊Q8—216no—
53C7×C4⋊Q8—28no—
54C7×C4⋊Q8—24no—
55C7×C4⋊Q8—24no—
56C7×C4⋊Q8—22no—
57C7×C4⋊Q8D414yes—
58C4×C7×Q8—28no—
59C4×C7×Q8—24no—
60C4×C7×Q8—24no—
61C4×C7×Q8—22no—
62C4×C7×Q8—22no—
63C4×C7×Q8—22no—
64C4×C7×Q8D414yes—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4D412yes—
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4C2312yes—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C42×C7—22no—
74C2×C42×C7—21no—
75C2×C42×C7C2312yes—
76C2×C42×C7C2311yescharacteristic
77C2×#1144—22no—
78C2×#1144—24no—
79C2×#1143—22no—
80C2×#1143—24no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—22no—
84C2×(C2×D7)⋊C4—22no—
85C2×C4⋊Dic7C2312yes—
86C2×C4×Dic7—21no—
87C22×C7⋊C8C2312yes—
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2311yescharacteristic
90#6438—22no—
91#6438—22no—
92#6409—24no—
93C7⋊#95—22no—
94C7⋊#95—22no—
95C7⋊#95—21no—
96#6384—24no—
97#6384—22no—
98#6383—24no—
99#6383—22no—
100C4⋊(C7⋊C8)—22no—
101#6377—22no—
102C2×Q82D711yescharacteristic, complemented
103C2×#489—71no—
104C2×#464—71no—
105#2592—71no—
106#2024—72no—
107#2023—72no—
108#1988—71no—
109#1985—71no—
110#1880—72no—
111#1878—72no—
112#1763—71no—
113C23×D7—21no—
114C2×C7×Q8—28no—
115C2×C7×Q8—22no—
116C2×C7×Q8—24no—
117C2×C7×Q8—24no—
118C2×C7×Q8—21no—
119C2×C7×Q8—22no—
120C2×C7×Q8—81no—
121C2×C7×Q8—81no—
122C2×C7×Q8—82no—
123C2×C7×Q8—82no—
124C2×C7×Q8C2×D412yes—
125C2×C7×Q8#4114yes—
126C22×C4×C7—21no—
127C22×C4×C7—22no—
128C22×C4×C7—21no—
129C22×C4×C7C2×D412yes—
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D412yes—
133C22×C4×C7C2411yescharacteristic
134C22×Dic7—21no—
135C22×Dic7—22no—
136C2×D28—21no—
137C2×D28—21no—
138C2×D28—22no—
139C2×D28—22no—
140C2×Dic14—21no—
141C2×Dic14—21no—
142C2×Dic14—22no—
143C2×Dic14—22no—
144C7×C4⋊C4—28no—
145C7×C4⋊C4—28no—
146C7×C4⋊C4—24no—
147C7×C4⋊C4—24no—
148C7×C4⋊C4—24no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—21no—
152C7×C4⋊C4C2×D414yes—
153C7×C4⋊C4C2×D412yes—
154C42×C7—28no—
155C42×C7—24no—
156C42×C7—22no—
157C42×C7—22no—
158C42×C7C2×D412yes—
159C42×C7C2×D412yes—
160#1144—44no—
161#1143—44no—
162(C2×D7)⋊C4—42no—
163(C2×D7)⋊C4—42no—
164(C2×D7)⋊C4—44no—
165(C2×D7)⋊C4—82no—
166C4⋊Dic7—22no—
167C4⋊Dic7—22no—
168C4×Dic7—81no—
169C2×C7⋊C8—21no—
170C2×C7⋊C8—21no—
171C2×C7⋊C8—22no—
172C2×C7⋊C8—22no—
173C2×C7⋊C8—22no—
174C2×C7⋊C8—22no—
175C2×C7⋊C8—22no—
176C22×Q16—71no—
177C22×SD16—71no—
178Q82—24no—
179Q82—24no—
180C2×C4⋊Q8—22no—
181C2×C4⋊Q8—21no—
182C2×C4⋊Q8C2×D712yes—
183C2×C4⋊Q8C2×D711yescharacteristic
184C2×#124—71no—
185C2×C4×Q8—21no—
186C2×C4×Q8C2×D712yes—
187C2×C4×Q8C2×D711yescharacteristic
188C2×C4×Q8C2×D711yescharacteristic
189#489—144no—
190#464—144no—
191C2×C4⋊C8—71no—
192C2×#103—72no—
193C2×#103—72no—
194C2×#103—71no—
195C2×#102—72no—
196C2×#102—71no—
197C2×#97—71no—
198#397—141no—
199#393—141no—
200#391—72no—
201#390—72no—
202#388—142no—
203C4×C4⋊C4—71no—
204#340—72no—
205#340—72no—
206#340—72no—
207#336—72no—
208#336—72no—
209C23×C7C22×D411yescharacteristic
210C22×D7—41no—
211C22×D7—42no—
212C22×D7—42no—
213C22×D7—41no—
214C22×D7—41no—
215C7×Q8—28no—
216C7×Q8—22no—
217C7×Q8—24no—
218C7×Q8—84no—
219C7×Q8—88no—
220C7×Q8#12414yes—
221C2×C4×C7—22no—
222C2×C4×C7—22no—
223C2×C4×C7—24no—
224C2×C4×C7—22no—
225C2×C4×C7—22no—
226C2×C4×C7—21no—
227C2×C4×C7—21no—
228C2×C4×C7—22no—
229C2×C4×C7—24no—
230C2×C4×C7—22no—
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—42no—
234C2×C4×C7—41no—
235C2×C4×C7—42no—
236C2×C4×C7—41no—
237C2×C4×C7#12012yes—
238C2×C4×C7#12012yes—
239C2×C4×C7#13714yes—
240C2×C4×C7C22×D411yescharacteristic
241C2×C4×C7C22×D411yescharacteristic
242C2×C4×C7C2×#4112yes—
243C2×C4×C72−1+412yes—
244C2×Dic7—41no—
245C2×Dic7—42no—
246C2×Dic7—42no—
247C2×Dic7—42no—
248C2×Dic7—42no—
249D28—41no—
250D28—41no—
251D28—42no—
252Dic14—41no—
253Dic14—41no—
254Dic14—42no—
255C7⋊C8—41no—
256C7⋊C8—41no—
257C7⋊C8—42no—
258C22×Q8—22no—
259C22×Q8—21no—
260C22×Q8—71no—
261C22×Q8C22⋊D711yescharacteristic
262C22×Q8C22×D711yescharacteristic
263C22×D4—71no—
264C2×Q16—144no—
265C2×Q16—142no—
266C2×SD16—144no—
267C2×SD16—142no—
268C22×C8—71no—
269C22×C8—71no—
270C22×C8—72no—
271C4⋊Q8—216no—
272C4⋊Q8—28no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—22no—
276C4⋊Q8C22⋊D714yes—
277#124—142no—
278#124—142no—
279C4×Q8—28no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8C22⋊D714yes—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—72no—
291C2×C4⋊C4C22⋊D712yes—
292C2×C4⋊C4C22⋊D711yescharacteristic
293C2×C4⋊C4C22×D712yes—
294C2×C4⋊C4C22×D711yescharacteristic
295C2×C22⋊C4—141no—
296C2×C22⋊C4—141no—
297C2×C22⋊C4—142no—
298C2×C22⋊C4—142no—
299C2×C42—22no—
300C2×C42—21no—
301C2×C42—141no—
302C2×C42C22×D712yes—
303C2×C42C22×D711yescharacteristic
304C4⋊C8—142no—
305#103—144no—
306#103—142no—
307#103—144no—
308#102—142no—
309#102—144no—
310#97—142no—
311#95—141no—
312#95—142no—
313#95—142no—
314C22×C7C2×#12011yescharacteristic
315C22×C7#54812yes—
316C22×C7C2×#13712yes—
317C22×C7#57611yescharacteristic
318C22×C7#57711yescharacteristic
319C2×D7—82no—
320C2×D7—82no—
321C2×D7—81no—
322C2×D7—81no—
323C2×D7—81no—
324C4×C7—24no—
325C4×C7—22no—
326C4×C7—24no—
327C4×C7—22no—
328C4×C7—42no—
329C4×C7—44no—
330C4×C7—42no—
331C4×C7C2×#12411yescharacteristic
332C4×C7#53611yescharacteristic
333C4×C7#54812yes—
334Dic7—82no—
335Dic7—81no—
336C24—141no—
337C2×Q8—28no—
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—24no—
341C2×Q8—22no—
342C2×Q8—21no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×Q8—82no—
346C2×Q8—82no—
347C2×Q8—142no—
348C2×Q8—142no—
349C2×Q8—141no—
350C2×Q8—141no—
351C2×Q8#116114yes—
352C2×Q8C2×C22⋊D712yes—
353C2×D4—142no—
354C2×D4—142no—
355C2×D4—141no—
356C2×D4—141no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—141no—
361C22×C4—142no—
362C22×C4C2×C22⋊D712yes—
363C22×C4C2×C22⋊D712yes—
364C22×C4C2×C22⋊D711yescharacteristic
365C22×C4C2×C22⋊D711yescharacteristic
366C22×C4C23×D711yescharacteristic
367Q16—284no—
368SD16—284no—
369C2×C8—142no—
370C2×C8—142no—
371C2×C8—142no—
372C2×C8—142no—
373C2×C8—142no—
374C2×C8—141no—
375C2×C8—141no—
376C4⋊C4—28no—
377C4⋊C4—28no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—142no—
385C4⋊C4—142no—
386C4⋊C4C2×C22⋊D714yes—
387C4⋊C4C2×C22⋊D712yes—
388C22⋊C4—284no—
389C22⋊C4—282no—
390C22⋊C4—282no—
391C22⋊C4—562no—
392C42—28no—
393C42—24no—
394C42—22no—
395C42—22no—
396C42—561no—
397C42C2×C22⋊D712yes—
398C42C2×C22⋊D712yes—
399C2×C7#243611yescharacteristic
400C2×C7#300411yescharacteristic
401C2×C7#301211yescharacteristic
402C2×C7#336912yes—
403C2×C7#337512yes—
404D7—161no—
405C23—282no—
406C23—282no—
407C23—281no—
408C23—281no—
409C23—281no—
410C23C22×C22⋊D711yescharacteristic
411Q8—28no—
412Q8—22no—
413Q8—24no—
414Q8—84no—
415Q8—88no—
416Q8—281no—
417Q8—281no—
418Q8—282no—
419Q8#651014yes—
420D4—281no—
421D4—281no—
422D4—282no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—282no—
440C2×C4—282no—
441C2×C4—282no—
442C2×C4—282no—
443C2×C4—281no—
444C2×C4#650514yes—
445C2×C4#651612yes—
446C2×C4#651612yes—
447C2×C4C2×#116112yes—
448C2×C4#655112yes—
449C2×C4C22×C22⋊D711yescharacteristic
450C2×C4C22×C22⋊D711yescharacteristic
451C8—281no—
452C8—281no—
453C8—282no—
454C7#2594211yescharacteristic, complemented, Sylow
455C22—561no—
456C22—561no—
457C22—561no—
458C22—562no—
459C22—562no—
460C22C2×#650512yes—
461C22#9050812yes—
462C22#9052611yescharacteristic
463C22#9052711yescharacteristic
464C22C2×#651611yescharacteristic
465C4—24no—
466C4—22no—
467C4—24no—
468C4—22no—
469C4—42no—
470C4—44no—
471C4—42no—
472C4—561no—
473C4—562no—
474C4C2×#651011yescharacteristic
475C4#9050711yescharacteristic
476C4#9050812yes—
477C2—1121no—
478C2#1174114912yes—
479C2#1174115312yes—
480C2#1174150411yescharacteristic
481C2#1174151111yescharacteristic
482C2#1174154711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×Q82C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenterC2×Q82D7
hypocenterC7#25942
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
1. C2×C7×Q82C2semidirect
5. #11729548C2semidirect
7. #11729534C2semidirect
8. #11729389C2semidirect
10. #11729386C2semidirect
11. #11729385C2semidirect
102. C2×Q82D7semidirect
454. C7#25942semidirect

Maximal subgroups

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

subgroupindex
C2×C7×Q822
C2×#898612
#117399292
C2×#898002
#117295482
#117295462
#117295342
#117293892
#117293872
#117293862
#117293852
#259427