Finite Group Databasebeta

← #13332283

Subgroups — #13332283

Order 1792.

1020 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic, complemented
2#11740658C211yesmaximal, characteristic, complemented
3#11739887C211yesmaximal, characteristic, complemented
4#11739752C211yesmaximal, characteristic
5#11729679C214yesmaximal, complemented
6#11729677C214yesmaximal, complemented
7#11729528C211yesmaximal, characteristic, complemented
8#11729376C211yesmaximal, characteristic, complemented
9#11729351C211yesmaximal, characteristic, complemented
10C7×#554—22no—
11C7×#554—22no—
12C7×#554—22no—
13C7×#554C2214yes—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#125—21no—
16C2×C7×C22⋊Q8—21no—
17C2×C7×C22⋊Q8—22no—
18C2×C4×C7×Q8C2211yescharacteristic
19C2×C4×C7×D4—21no—
20C2×C4×C7×D4C2211yescharacteristic
21C22×C7×C4⋊C4—21no—
22C2×#6495—21no—
23#90168C2211yescharacteristic
24#89999—21no—
25#89999—22no—
26#89857C2212yescomplemented
27#89855—22no—
28#89853C2212yescomplemented
29#89799C2212yescomplemented
30C4×#1143—22no—
31#89792C2212yescomplemented
32#89791C2212yes—
33#89788—22no—
34#89787C2212yes—
35#89740—21no—
36#89738—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89341C2214yescomplemented
41Q8⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yes—
43#18748—71nomaximal, Sylow
44C22×C7×Q8D411yescharacteristic
45C22×C7×Q8D412yes—
46C22×C7×D4D411yescharacteristic
47C23×C4×C7—22no—
48C23×C4×C7—21no—
49C2×#1157—21no—
50C7×C4⋊Q8—21no—
51C7×C4⋊Q8—21no—
52C7×C4⋊Q8D412yes—
53C7×C4⋊Q8C2312yes—
54C7×#125—24no—
55C7×#125—24no—
56C7×C22⋊Q8—24no—
57C7×C22⋊Q8—28no—
58C7×C22⋊Q8—24no—
59C7×C22⋊Q8—28no—
60C4×C7×Q8—21no—
61C4×C7×Q8—21no—
62C4×C7×Q8D412yes—
63C4×C7×Q8C2312yes—
64C4×C7×D4—24no—
65C4×C7×D4—24no—
66C4×C7×D4—21no—
67C4×C7×D4—21no—
68C4×C7×D4D412yes—
69C4×C7×D4C2312yes—
70C2×C7×C4⋊C4—22no—
71C2×C7×C4⋊C4—24no—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—24no—
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C22⋊C4—22no—
80C2×C7×C22⋊C4—21no—
81C2×C42×C7—21no—
82C2×C42×C7C2311yescharacteristic
83C2×#1143—22no—
84C2×#1143—24no—
85C2×#1142—22no—
86#6495—44no—
87C2×#1141—22no—
88#6493—21no—
89#6493—21no—
90C2×#1136—21no—
91#6438—21no—
92#6437C2311yescharacteristic
93C4×D28—21no—
94C4×D28—21no—
95#6432C2311yescharacteristic
96#6409—22no—
97#6409—24no—
98C22⋊(C7⋊C8)—21no—
99C22⋊(C7⋊C8)—21no—
100#6384—22no—
101#6383—22no—
102#6383—22no—
103#6382—22no—
104C4⋊(C7⋊C8)—21no—
105C4⋊(C7⋊C8)—21no—
106C4⋊(C7⋊C8)C2312yes—
107C4⋊(C7⋊C8)C2312yes—
108C4×C7⋊C8—21no—
109C4×C7⋊C8C2312yes—
110C2×#554D711yescharacteristic, complemented
111#3206—71no—
112#3103—71no—
113#3036—71no—
114#1643—74no—
115#1622—74no—
116#1548—71no—
117#1487—71no—
118#1478—71no—
119C24×C7—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—22no—
122C2×C7×Q8—42no—
123C2×C7×Q8—82no—
124C2×C7×Q8C2×D412yes—
125C2×C7×Q8C2×D414yes—
126C2×C7×D4—21no—
127C2×C7×D4—22no—
128C2×C7×D4SD1614yescomplemented
129C2×C7×D4C2×D412yes—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7—28no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—44no—
138C22×C4×C7—84no—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C2×C22⋊D7—41no—
143#1157—42no—
144#1157—42no—
145C2×D28—21no—
146C2×D28—21no—
147C2×D28—21no—
148C2×C4×D7—41no—
149C2×Dic14—21no—
150C7×C4⋊C4—28no—
151C7×C4⋊C4—28no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4C2×D412yes—
161C7×C4⋊C4C2×D412yes—
162C7×C22⋊C4—28no—
163C7×C22⋊C4—22no—
164C7×C22⋊C4—22no—
165C42×C7—22no—
166C42×C7—22no—
167C42×C7C2×D411yescharacteristic
168C42×C7C2×D411yescharacteristic
169C42×C7C2×D411yescharacteristic
170C42×C7C2411yescharacteristic
171#1143—44no—
172#1143—84no—
173#1142—44no—
174#1141—44no—
175(C2×D7)⋊C4—41no—
176(C2×D7)⋊C4—41no—
177C4⋊Dic7—21no—
178C4⋊Dic7—21no—
179#1136—42no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×AGL(1,ℤ/8)—141no—
184#554—22no—
185#554—22no—
186#554—22no—
187#554C2×D714yes—
188#532—71no—
189C2×C4⋊Q8C2×D711yescharacteristic
190C2×#125—21no—
191C2×C22⋊Q8—21no—
192C2×C22⋊Q8—22no—
193C2×C4×Q8C2×D711yescharacteristic
194C2×C4×D4—21no—
195C2×C4×D4C2×D711yescharacteristic
196C22×C4⋊C4—21no—
197#492—72no—
198#487—72no—
199#481—141no—
200#475—72no—
201#463—142no—
202#461—72no—
203#461—142no—
204#460—72no—
205#459—72no—
206#448—141no—
207#448—142no—
208C4×SD16—142no—
209#431—71no—
210#431—71no—
211#423—71no—
212#417—141no—
213#333—74no—
214Q8⋊C8—74no—
215D4⋊C8—74no—
216C23×C7—22no—
217C23×C7—24no—
218C23×C7—42no—
219C23×C7—42no—
220C23×C7#12011yescharacteristic
221C22×D7—41no—
222C7×Q8—22no—
223C7×Q8—22no—
224C7×Q8—44no—
225C7×Q8—42no—
226C7×Q8—82no—
227C7×D4—22no—
228C7×D4—24no—
229C7×D4C2×SD1614yes—
230C2×C4×C7—22no—
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—22no—
234C2×C4×C7—24no—
235C2×C4×C7—41no—
236C2×C4×C7—44no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—44no—
240C2×C4×C7—42no—
241C2×C4×C7—44no—
242C2×C4×C7—88no—
243C2×C4×C7#12011yescharacteristic
244C2×C4×C7#12011yescharacteristic
245C2×C4×C7#12011yescharacteristic
246C2×C4×C7C2×SD1612yes—
247C2×C4×C7#13712yes—
248C2×C4×C7C22×D411yescharacteristic
249C2×C4×C7C22×D411yescharacteristic
250C2×C4×C7C22×D411yescharacteristic
251C22⋊D7—82no—
252C2×Dic7—41no—
253D28—42no—
254D28—41no—
255D28—41no—
256D28—42no—
257C4×D7—82no—
258Dic14—41no—
259Dic14—41no—
260C7⋊C8—42no—
261C7⋊C8—42no—
262C7⋊C8—42no—
263C2×#41—141no—
264C22×Q8C22⋊D711yescharacteristic
265C22×Q8C22⋊D712yes—
266C22×D4C22⋊D711yescharacteristic
267C23×C4—22no—
268C23×C4—21no—
269AGL(1,ℤ/8)—284no—
270C2×SD16—142no—
271C2×SD16—142no—
272C2×SD16—144no—
273C2×D8—142no—
274C2×M16—141no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8—71no—
278C4⋊Q8C22⋊D712yes—
279C4⋊Q8C22×D712yes—
280#127—71no—
281#125—24no—
282#125—24no—
283#124—141no—
284C22⋊Q8—24no—
285C22⋊Q8—28no—
286C22⋊Q8—24no—
287C22⋊Q8—28no—
288#121—141no—
289#121—141no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×Q8C22⋊D712yes—
293C4×Q8C22×D712yes—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—141no—
299C4×D4—141no—
300C4×D4C22⋊D712yes—
301C4×D4C22×D712yes—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4C22⋊D711yescharacteristic
310C2×C4⋊C4C22⋊D711yescharacteristic
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C42—21no—
314C2×C42C22×D711yescharacteristic
315#106—142no—
316C4⋊C8—72no—
317C4⋊C8—72no—
318C4⋊C8—141no—
319C4⋊C8—141no—
320#103—142no—
321#103—144no—
322#103—142no—
323#102—142no—
324#102—142no—
325C22⋊C8—141no—
326C22⋊C8—141no—
327C4×C8—72no—
328C4×C8—141no—
329C22×C7—21no—
330C22×C7—21no—
331C22×C7—24no—
332C22×C7—48no—
333C22×C7—42no—
334C22×C7—42no—
335C22×C7#44812yes—
336C22×C7C2×#12011yescharacteristic
337C2×D7—81no—
338C2×D7—81no—
339C2×D7—81no—
340C4×C7—21no—
341C4×C7—21no—
342C4×C7—21no—
343C4×C7—42no—
344C4×C7—42no—
345C4×C7—42no—
346C4×C7—44no—
347C4×C7#44812yes—
348C4×C7C22×SD1611yescharacteristic
349C4×C7C2×#13711yescharacteristic
350C4×C7#57512yes—
351Dic7—81no—
352C24—21no—
353#41—282no—
354#41—282no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—42no—
358C2×Q8—82no—
359C2×Q8—141no—
360C2×Q8C2×C22⋊D712yes—
361C2×Q8C2×C22⋊D714yes—
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—141no—
365C2×D4—141no—
366C2×D4—141no—
367C2×D4—281no—
368C2×D4#114314yescomplemented
369C2×D4C2×C22⋊D712yes—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—44no—
378C22×C4—84no—
379C22×C4—281no—
380C22×C4C2×C22⋊D711yescharacteristic
381C22×C4C2×C22⋊D711yescharacteristic
382C22×C4C2×C22⋊D711yescharacteristic
383SD16—284no—
384SD16—284no—
385SD16—564no—
386D8—284no—
387M16—282no—
388C2×C8—142no—
389C2×C8—142no—
390C2×C8—142no—
391C4⋊C4—28no—
392C4⋊C4—28no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—141no—
402C4⋊C4—141no—
403C4⋊C4C2×C22⋊D712yes—
404C4⋊C4C2×C22⋊D712yes—
405C22⋊C4—28no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—281no—
409C22⋊C4—281no—
410C42—22no—
411C42—22no—
412C42C2×C22⋊D711yescharacteristic
413C42C2×C22⋊D711yescharacteristic
414C42C2×C22⋊D711yescharacteristic
415C42C23×D711yescharacteristic
416C2×C7—22no—
417C2×C7—44no—
418C2×C7C2×#44811yescharacteristic
419C2×C7#300511yescharacteristic
420C2×C7#301911yescharacteristic
421D7—161no—
422C23—24no—
423C23—22no—
424C23—42no—
425C23—42no—
426C23—281no—
427C23#651611yescharacteristic
428Q8—22no—
429Q8—22no—
430Q8—44no—
431Q8—42no—
432Q8—82no—
433Q8—281no—
434Q8—281no—
435D4—22no—
436D4—24no—
437D4—282no—
438D4—281no—
439D4—281no—
440D4—282no—
441D4—562no—
442D4C2×#114314yes—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—41no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—88no—
456C2×C4—281no—
457C2×C4—562no—
458C2×C4C2×#114312yes—
459C2×C4#650512yes—
460C2×C4#651611yescharacteristic
461C2×C4#651611yescharacteristic
462C2×C4#651611yescharacteristic
463C2×C4C22×C22⋊D711yescharacteristic
464C2×C4C22×C22⋊D711yescharacteristic
465C2×C4C22×C22⋊D711yescharacteristic
466C8—282no—
467C8—282no—
468C8—282no—
469C7#1874811yescharacteristic, complemented, Sylow
470C22—21no—
471C22—21no—
472C22—24no—
473C22—48no—
474C22—42no—
475C22—42no—
476C22—561no—
477C22—561no—
478C22—561no—
479C22#8999912yes—
480C22C2×#651611yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—44no—
488C4—561no—
489C4#8999912yes—
490C4C22×#114311yescharacteristic
491C4C2×#650511yescharacteristic
492C4#9051312yes—
493C2—22no—
494C2—44no—
495C2—1121no—
496C2C2×#8999911yescharacteristic
497C2#1174152611yescharacteristic
498C2#1174154911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#554C2
socleC22×C7C2×#120
prime radical#11739752C2
hypercenterC2×#554D7
hypocenterC7#18748
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×C7×#554C2semidirect
2. #11740658C2semidirect
3. #11739887C2semidirect
5. #11729679C2semidirect
6. #11729677C2semidirect
7. #11729528C2semidirect
8. #11729376C2semidirect
9. #11729351C2semidirect
26. #89857C22semidirect
28. #89853C22semidirect
29. #89799C22semidirect
31. #89792C22semidirect
40. #89341C22semidirect
41. Q8⋊(C7⋊C8)C22semidirect
110. C2×#554D7semidirect
128. C2×C7×D4SD16semidirect
368. C2×D4#1143semidirect
469. C7#18748semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5542
#117406582
#117398872
#117397522
#117296792
#117296772
#117295282
#117293762
#117293512
#187487