Finite Group Databasebeta

← #13337824

Subgroups — #13337824

Order 1792.

684 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3167C211yesmaximal, characteristic, complemented
2#11740798C211yesmaximal, characteristic, complemented
3#11740745C211yesmaximal, characteristic
4C2×#89460C211yesmaximal, characteristic, complemented
5#11729585C211yesmaximal, characteristic, complemented
6#11729561C211yesmaximal, characteristic, complemented
7#11729099C212yesmaximal, complemented
8#11729022C212yesmaximal, complemented
9#11728989C212yesmaximal, complemented
10#11728957C212yesmaximal, complemented
11#11727486C211yesmaximal, characteristic, complemented
12C7×#533C2211yescharacteristic
13C7×#532C2211yescharacteristic
14C22×#1131—21no—
15C2×C4×D28C2211yescharacteristic
16C7×#498C2211yescharacteristic, complemented
17C7×#492C2211yescharacteristic, complemented
18C7×#487C2211yescharacteristic, complemented
19C7×#486C2211yescharacteristic, complemented
20C7×#465—21no—
21C7×#465—21no—
22C4×C7×SD16—21no—
23C4×C7×SD16—21no—
24C2×C4×C7×C8C2211yescharacteristic
25C2×(C2×D7)⋊C8—21no—
26#89859C2212yes—
27#89856C2212yes—
28#89855C2212yes—
29#89844C2212yes—
30#89818—21no—
31#89816—21no—
32#89808—21no—
33#89806—21no—
34#89798—22no—
35#89786—22no—
36#89698C2211yescharacteristic
37#89698C2211yescharacteristic
38C2×C4⋊(C7⋊C8)C2211yescharacteristic
39#89460—21no—
40#89460—21no—
41#89460C2212yes—
42#89460C2212yes—
43#89342C2212yescomplemented
44#89342C2212yescomplemented
45#89341C2212yescomplemented
46#89339C2212yescomplemented
47#89253C2212yescomplemented
48#89252C2212yescomplemented
49#19282—71nomaximal, Sylow
50C2×C7×#41—21no—
51C22×D28D411yescharacteristic
52C22×C4×D7—21no—
53C2×C7×SD16—21no—
54C2×C7×SD16—21no—
55C22×C7×C8D411yescharacteristic
56C22×C7×C8D411yescharacteristic
57C7×C4⋊Q8C2311yescharacteristic
58C7×C4⋊Q8C2311yescharacteristic
59C7×C4⋊Q8C2311yescharacteristic
60C7×#127C2311yescharacteristic
61C7×#125—21no—
62C7×#124—21no—
63C7×C22⋊Q8—21no—
64C7×C22⋊Q8—21no—
65C7×#121—21no—
66C7×#121—21no—
67C4×C7×Q8—21no—
68C4×C7×Q8—21no—
69C4×C7×D4—21no—
70C4×C7×D4—21no—
71C7×#117—21no—
72C2×C42×C7C2311yescharacteristic
73C2×#1143—22no—
74C2×#1141—22no—
75C2×(C2×D7)⋊C4—21no—
76C2×C4⋊Dic7D411yescharacteristic
77C22×C7⋊C8—21no—
78C2×#1131—22no—
79C2×#1131—22no—
80C2×#1131—44no—
81C4×D28—21no—
82C4×D28—21no—
83C4×D28D412yes—
84C4×D28C2312yes—
85C7×#106—21no—
86C7×#106—21no—
87C7×#103—21no—
88C7×#103—21no—
89C7×#102—21no—
90C7×#102—21no—
91C4×C7×C8D411yescharacteristic
92C4×C7×C8D411yescharacteristic
93C4×C7×C8C2311yescharacteristic
94C4×C7×C8C2311yescharacteristic
95#6409—22no—
96#6406—22no—
97C22⋊(C7⋊C8)—21no—
98C22⋊(C7⋊C8)—21no—
99(C2×D7)⋊C8—22no—
100(C2×D7)⋊C8—22no—
101#6383—22no—
102#6383—22no—
103#6382—22no—
104#6381—22no—
105C4⋊(C7⋊C8)—21no—
106C4⋊(C7⋊C8)C2312yes—
107C4⋊(C7⋊C8)C2312yes—
108C4⋊(C7⋊C8)C2312yes—
109#6377—21no—
110#6377—21no—
111#3167D711yescharacteristic, complemented
112#3110—71no—
113#3109—71no—
114C2×#436—71no—
115#1703—72no—
116#1701—72no—
117#1691—72no—
118#1685—72no—
119#1568—71no—
120#1552—71no—
121#1551—71no—
122C23×D7—21no—
123C7×#41—82no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—41no—
131C22×C4×C7—41no—
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7#4111yescharacteristic
135C22×Dic7—21no—
136C2×D28—21no—
137C2×D28—42no—
138C2×D28C2×D412yes—
139C2×C4×D7—22no—
140C2×C4×D7—22no—
141C2×C4×D7—42no—
142C2×C4×D7—42no—
143C7×SD16—42no—
144C7×SD16—42no—
145C2×C7×C8—22no—
146C2×C7×C8—41no—
147C2×C7×C8C2×D411yescharacteristic
148C2×C7×C8C2×D411yescharacteristic
149C2×C7×C8C2×D411yescharacteristic
150C2×C7×C8C2×D411yescharacteristic
151C7×C4⋊C4—21no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—21no—
155C7×C4⋊C4—21no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—41no—
158C7×C4⋊C4—41no—
159C7×C22⋊C4—41no—
160C7×C22⋊C4—41no—
161C7×C22⋊C4—41no—
162C42×C7—41no—
163C42×C7C2×D411yescharacteristic
164C42×C7C2×D411yescharacteristic
165C42×C7#4111yescharacteristic
166C42×C7C2411yescharacteristic
167#1143—44no—
168#1141—44no—
169(C2×D7)⋊C4—22no—
170(C2×D7)⋊C4—22no—
171C4⋊Dic7—21no—
172C4⋊Dic7C2×D412yes—
173C2×C7⋊C8—22no—
174C2×C7⋊C8—22no—
175C2×C7⋊C8—22no—
176C2×C7⋊C8—42no—
177#1131—44no—
178#1131—44no—
179C22×M16—141no—
180#533C2×D711yescharacteristic
181#532C2×D711yescharacteristic
182C2×C4×D4—71no—
183#498C2×D711yescharacteristic, complemented
184#492C2×D711yescharacteristic, complemented
185#487C2×D711yescharacteristic, complemented
186#486C2×D711yescharacteristic, complemented
187#484—141no—
188#481—141no—
189#476—72no—
190#474—72no—
191#465—21no—
192#465—21no—
193#461—72no—
194#459—72no—
195#450—141no—
196#447—141no—
197#442—142no—
198#440—142no—
199C4×SD16—21no—
200C4×SD16—21no—
201#436—72no—
202#436—72no—
203#436—141no—
204#436—141no—
205#432—71no—
206#432—71no—
207C2×C4⋊C8—71no—
208C2×C22⋊C8—141no—
209C2×C4×C8C2×D711yescharacteristic
210#335—72no—
211#334—72no—
212#333—72no—
213#332—72no—
214#332—72no—
215#331—72no—
216C23×C7—41no—
217C23×C7#12111yescharacteristic
218C22×D7—22no—
219C22×D7—42no—
220C22×D7—42no—
221C22×D7—42no—
222C7×Q8—41no—
223C7×Q8—42no—
224C7×Q8—41no—
225C7×Q8—81no—
226C7×D4—41no—
227C7×D4—42no—
228C7×D4—41no—
229C7×D4—81no—
230C7×D4—82no—
231C2×C4×C7—22no—
232C2×C4×C7—21no—
233C2×C4×C7—21no—
234C2×C4×C7—22no—
235C2×C4×C7—41no—
236C2×C4×C7—41no—
237C2×C4×C7—41no—
238C2×C4×C7—82no—
239C2×C4×C7#12111yescharacteristic
240C2×C4×C7#12111yescharacteristic
241C2×C4×C7#12111yescharacteristic
242C2×C4×C7C22×D411yescharacteristic
243C2×C4×C7C22×D411yescharacteristic
244C2×C4×C7C2×#4111yescharacteristic
245C2×Dic7—22no—
246C2×Dic7—42no—
247D28—22no—
248D28—22no—
249D28—42no—
250C4×D7—44no—
251C4×D7—42no—
252C4×D7—42no—
253C7×C8—21no—
254C7×C8—21no—
255C7×C8—21no—
256C7×C8AGL(1,ℤ/8)12yes—
257C7⋊C8—42no—
258C7⋊C8—42no—
259C7⋊C8—42no—
260C2×#41—21no—
261C22×D4—71no—
262C23×C4—141no—
263C2×SD16—21no—
264C2×SD16—21no—
265C2×SD16—142no—
266C2×D8—142no—
267C2×M16—142no—
268C2×M16—142no—
269C2×M16—284no—
270C22×C8—141no—
271C22×C8C22⋊D711yescharacteristic
272C22×C8C22⋊D711yescharacteristic
273C4⋊Q8C22×D711yescharacteristic
274C4⋊Q8C22×D711yescharacteristic
275C4⋊Q8C22×D711yescharacteristic
276#127C22×D711yescharacteristic
277#125—21no—
278#124—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281#121—21no—
282#121—21no—
283C4×Q8—21no—
284C4×Q8—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—72no—
288C4×D4—72no—
289C4×D4—141no—
290C4×D4—141no—
291#117—21no—
292C2×C4⋊C4—71no—
293C2×C22⋊C4—141no—
294C2×C42C22×D711yescharacteristic
295#107—142no—
296#106—21no—
297#106—21no—
298#106—142no—
299C4⋊C8—72no—
300C4⋊C8—72no—
301C4⋊C8—72no—
302C4⋊C8—141no—
303#103—21no—
304#103—21no—
305#103—142no—
306#102—21no—
307#102—21no—
308#102—142no—
309#102—142no—
310#102—142no—
311C22⋊C8—141no—
312C22⋊C8—141no—
313C22⋊C8—142no—
314C22⋊C8—142no—
315#97—141no—
316#97—141no—
317C4×C8C22⋊D711yescharacteristic
318C4×C8C22⋊D711yescharacteristic
319C4×C8C22×D711yescharacteristic
320C4×C8C22×D711yescharacteristic
321C22×C7—21no—
322C22×C7—21no—
323C22×C7—81no—
324C22×C7—81no—
325C22×C7—81no—
326C22×C7#47112yes—
327C22×C7C2×#12111yescharacteristic
328C2×D7—42no—
329C2×D7—42no—
330C2×D7—42no—
331C2×D7—84no—
332C4×C7—21no—
333C4×C7—21no—
334C4×C7—21no—
335C4×C7—21no—
336C4×C7—81no—
337C4×C7—81no—
338C4×C7—81no—
339C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
340C4×C7#57512yes—
341C4×C7#57511yescharacteristic
342Dic7—42no—
343C24—141no—
344#41—82no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—41no—
352C2×D4—72no—
353C2×D4—141no—
354C2×D4—282no—
355C22×C4—41no—
356C22×C4—141no—
357C22×C4—142no—
358C22×C4—142no—
359C22×C4—282no—
360C22×C4—282no—
361C22×C4D4×D711yescharacteristic
362C22×C4#115911yescharacteristic
363C22×C4C2×C22⋊D711yescharacteristic
364SD16—42no—
365SD16—42no—
366SD16—284no—
367D8—284no—
368M16—284no—
369M16—284no—
370C2×C8—22no—
371C2×C8—41no—
372C2×C8—142no—
373C2×C8—142no—
374C2×C8—142no—
375C2×C8—282no—
376C2×C8C2×C22⋊D711yescharacteristic
377C2×C8C2×C22⋊D711yescharacteristic
378C2×C8C2×C22⋊D711yescharacteristic
379C2×C8C2×C22⋊D711yescharacteristic
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—72no—
389C4⋊C4—141no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—142no—
394C22⋊C4—142no—
395C42—41no—
396C42D4×D711yescharacteristic
397C42#115911yescharacteristic
398C42C2×C22⋊D711yescharacteristic
399C42C23×D711yescharacteristic
400C2×C7—22no—
401C2×C7—161no—
402C2×C7#303611yescharacteristic
403C2×C7#305411yescharacteristic
404C2×C7C2×#47111yescharacteristic
405D7—82no—
406C23—41no—
407C23—142no—
408C23—282no—
409C23—282no—
410C23—282no—
411C23#650111yescharacteristic
412Q8—42no—
413Q8—41no—
414Q8—41no—
415Q8—81no—
416D4—42no—
417D4—41no—
418D4—41no—
419D4—81no—
420D4—82no—
421D4—142no—
422D4—142no—
423D4—282no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—82no—
432C2×C4—142no—
433C2×C4—284no—
434C2×C4—282no—
435C2×C4—282no—
436C2×C4—282no—
437C2×C4#650111yescharacteristic
438C2×C4#650111yescharacteristic
439C2×C4#650111yescharacteristic
440C2×C4C2×D4×D711yescharacteristic
441C2×C4C2×#115911yescharacteristic
442C2×C4C22×C22⋊D711yescharacteristic
443C8—21no—
444C8—21no—
445C8—21no—
446C8—282no—
447C8—282no—
448C8—282no—
449C8#651212yes—
450C7#1928211yescharacteristic, complemented, Sylow
451C22—21no—
452C22—21no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—282no—
457C22—282no—
458C22—282no—
459C22—564no—
460C22#8994912yes—
461C22C2×#650111yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—282no—
470C4#9045912yes—
471C4C2×#651211yescharacteristic
472C4#9051311yescharacteristic
473C2—22no—
474C2—161no—
475C2—562no—
476C2#1174074211yescharacteristic
477C2C2×#8994911yescharacteristic
478C2#1174136011yescharacteristic
subgroupG/H
centerC22C2×#6501
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3167C2
socleC22×C7C2×#121
prime radical#11740745C2
hypercenter#3167D7
hypocenterC7#19282
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. C7×#3167C2semidirect
2. #11740798C2semidirect
4. C2×#89460C2semidirect
5. #11729585C2semidirect
6. #11729561C2semidirect
7. #11729099C2semidirect
8. #11729022C2semidirect
9. #11728989C2semidirect
10. #11728957C2semidirect
11. #11727486C2semidirect
16. C7×#498C22semidirect
17. C7×#492C22semidirect
18. C7×#487C22semidirect
19. C7×#486C22semidirect
43. #89342C22semidirect
44. #89342C22semidirect
45. #89341C22semidirect
46. #89339C22semidirect
47. #89253C22semidirect
48. #89252C22semidirect
111. #3167D7semidirect
183. #498C2×D7semidirect
184. #492C2×D7semidirect
185. #487C2×D7semidirect
186. #486C2×D7semidirect
450. C7#19282semidirect

Maximal subgroups

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

subgroupindex
C7×#31672
#117407982
#117407452
C2×#894602
#117295852
#117295612
#117290992
#117290222
#117289892
#117289572
#117274862
#192827