Finite Group Databasebeta

← #13332385

Subgroups — #13332385

Order 1792.

1192 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×D4×Q8C211yesmaximal, characteristic, complemented
2#11740661C211yesmaximal, characteristic, complemented
3#11739887C211yesmaximal, characteristic, complemented
4#11739752C211yesmaximal, characteristic
5#11729671C214yesmaximal, complemented
6#11729668C214yesmaximal, complemented
7#11729529C211yesmaximal, characteristic, complemented
8#11729376C211yesmaximal, characteristic, complemented
9#11729351C211yesmaximal, characteristic, complemented
10C23×C7×Q8—21no—
11C7×D4×Q8—22no—
12C7×D4×Q8—22no—
13C7×D4×Q8—22no—
14C7×D4×Q8C2214yes—
15C2×C7×C4⋊Q8—21no—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8—22no—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2211yescharacteristic
22C2×#6495—21no—
23#90168C2211yescharacteristic
24#89999—21no—
25#89860C2212yescomplemented
26#89856—22no—
27#89851C2212yescomplemented
28#89799C2212yescomplemented
29C4×#1143—22no—
30#89792C2212yescomplemented
31#89791C2212yes—
32#89788—22no—
33#89787C2212yes—
34#89740—21no—
35#89740—22no—
36#89738—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89342C2214yescomplemented
41Q8⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yes—
43#18758—71nomaximal, Sylow
44C22×C7×Q8—22no—
45C22×C7×Q8—24no—
46C22×C7×Q8—24no—
47C22×C7×Q8—21no—
48C22×C7×Q8—48no—
49C22×C7×Q8D411yescharacteristic
50C22×C7×D4D411yescharacteristic
51C23×C4×C7—22no—
52C23×C4×C7—21no—
53C2×#1157—21no—
54C7×C4⋊Q8—24no—
55C7×C4⋊Q8—24no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8D412yes—
59C7×C4⋊Q8C2312yes—
60C7×C22⋊Q8—24no—
61C7×C22⋊Q8—28no—
62C7×C22⋊Q8—24no—
63C7×C22⋊Q8—28no—
64C4×C7×Q8—21no—
65C4×C7×Q8—21no—
66C4×C7×Q8D412yes—
67C4×C7×Q8C2312yes—
68C4×C7×D4—24no—
69C4×C7×D4—24no—
70C4×C7×D4—21no—
71C4×C7×D4—21no—
72C4×C7×D4D412yes—
73C4×C7×D4C2312yes—
74C2×C7×C4⋊C4—22no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4D412yes—
80C2×C7×C22⋊C4—22no—
81C2×C7×C22⋊C4—21no—
82C2×C42×C7—21no—
83C2×C42×C7C2311yescharacteristic
84C2×#1143—22no—
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—
97C22⋊(C7⋊C8)—21no—
98C22⋊(C7⋊C8)—21no—
99#6384—22no—
100#6383—24no—
101#6383—22no—
102#6383—22no—
103#6382—24no—
104#6382—22no—
105C4⋊(C7⋊C8)—21no—
106C4⋊(C7⋊C8)—21no—
107C4⋊(C7⋊C8)C2312yes—
108C4⋊(C7⋊C8)C2312yes—
109C4×C7⋊C8—21no—
110C4×C7⋊C8C2312yes—
111C2×D4×Q8D711yescharacteristic, complemented
112#3227—71no—
113#3103—71no—
114#3036—71no—
115#1651—74no—
116#1627—74no—
117#1549—71no—
118#1487—71no—
119#1478—71no—
120C24×C7—21no—
121C2×C7×Q8—24no—
122C2×C7×Q8—28no—
123C2×C7×Q8—21no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—48no—
127C2×C7×Q8—44no—
128C2×C7×Q8—416no—
129C2×C7×Q8—48no—
130C2×C7×Q8—48no—
131C2×C7×Q8C2×D412yes—
132C2×C7×D4—21no—
133C2×C7×D4—22no—
134C2×C7×D4SD1614yescomplemented
135C2×C7×D4C2×D412yes—
136C22×C4×C7—21no—
137C22×C4×C7—28no—
138C22×C4×C7—21no—
139C22×C4×C7—22no—
140C22×C4×C7—21no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—48no—
144C22×C4×C7—44no—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C2×C22⋊D7—41no—
149#1157—42no—
150#1157—42no—
151C2×D28—21no—
152C2×D28—21no—
153C2×D28—21no—
154C2×C4×D7—41no—
155C2×Dic14—21no—
156C7×C4⋊C4—28no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4C2×D412yes—
165C7×C4⋊C4C2×D412yes—
166C7×C4⋊C4C2×D414yes—
167C7×C22⋊C4—28no—
168C7×C22⋊C4—22no—
169C7×C22⋊C4—22no—
170C42×C7—22no—
171C42×C7—22no—
172C42×C7C2×D411yescharacteristic
173C42×C7C2×D411yescharacteristic
174C42×C7C2×D411yescharacteristic
175C42×C7C2411yescharacteristic
176#1143—44no—
177#1142—44no—
178#1141—44no—
179(C2×D7)⋊C4—41no—
180(C2×D7)⋊C4—41no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183#1136—42no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C23×Q8—21no—
188C2×AGL(1,ℤ/8)—141no—
189D4×Q8—22no—
190D4×Q8—22no—
191D4×Q8—22no—
192D4×Q8C2×D714yes—
193#532—71no—
194C2×C4⋊Q8—21no—
195C2×C4⋊Q8C2×D711yescharacteristic
196C2×C22⋊Q8—21no—
197C2×C22⋊Q8—22no—
198C2×C4×Q8C2×D711yescharacteristic
199C2×C4×D4—21no—
200C2×C4×D4C2×D711yescharacteristic
201#498—72no—
202#486—72no—
203#481—141no—
204#481—142no—
205#476—142no—
206#475—72no—
207#463—142no—
208#461—72no—
209#460—72no—
210#459—72no—
211#448—141no—
212C4×SD16—142no—
213#431—71no—
214#431—71no—
215#423—71no—
216#417—141no—
217#332—74no—
218Q8⋊C8—74no—
219D4⋊C8—74no—
220C23×C7—22no—
221C23×C7—24no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7#12011yescharacteristic
225C22×D7—41no—
226C7×Q8—24no—
227C7×Q8—22no—
228C7×Q8—416no—
229C7×Q8—44no—
230C7×Q8—44no—
231C7×Q8AGL(1,ℤ/8)14yescomplemented
232C7×D4—22no—
233C7×D4—24no—
234C7×D4C2×SD1614yes—
235C2×C4×C7—24no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—24no—
240C2×C4×C7—22no—
241C2×C4×C7—41no—
242C2×C4×C7—416no—
243C2×C4×C7—44no—
244C2×C4×C7—42no—
245C2×C4×C7—44no—
246C2×C4×C7—42no—
247C2×C4×C7—44no—
248C2×C4×C7#12011yescharacteristic
249C2×C4×C7#12011yescharacteristic
250C2×C4×C7#12011yescharacteristic
251C2×C4×C7C2×SD1612yes—
252C2×C4×C7#13712yes—
253C2×C4×C7C22×D411yescharacteristic
254C2×C4×C7C22×D411yescharacteristic
255C2×C4×C7C22×D411yescharacteristic
256C22⋊D7—82no—
257C2×Dic7—41no—
258D28—42no—
259D28—41no—
260D28—42no—
261D28—41no—
262C4×D7—82no—
263Dic14—41no—
264Dic14—41no—
265C7⋊C8—42no—
266C7⋊C8—42no—
267C7⋊C8—42no—
268C2×#41—141no—
269C22×Q8—22no—
270C22×Q8—24no—
271C22×Q8—24no—
272C22×Q8—21no—
273C22×Q8—48no—
274C22×Q8C22⋊D711yescharacteristic
275C22×D4C22⋊D711yescharacteristic
276C23×C4—22no—
277C23×C4—21no—
278AGL(1,ℤ/8)—284no—
279C2×SD16—142no—
280C2×SD16—142no—
281C2×D8—142no—
282C2×M16—141no—
283C4⋊Q8—24no—
284C4⋊Q8—24no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—71no—
288C4⋊Q8C22⋊D712yes—
289C4⋊Q8C22×D712yes—
290#127—71no—
291#124—141no—
292C22⋊Q8—24no—
293C22⋊Q8—28no—
294C22⋊Q8—24no—
295C22⋊Q8—28no—
296#121—141no—
297#121—141no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8C22⋊D712yes—
301C4×Q8C22×D712yes—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4—141no—
307C4×D4—141no—
308C4×D4C22⋊D712yes—
309C4×D4C22×D712yes—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4C22⋊D711yescharacteristic
314C2×C4⋊C4C22⋊D711yescharacteristic
315C2×C4⋊C4C22⋊D712yes—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C42—21no—
319C2×C42C22×D711yescharacteristic
320#106—142no—
321#106—144no—
322C4⋊C8—72no—
323C4⋊C8—72no—
324C4⋊C8—141no—
325C4⋊C8—141no—
326#103—142no—
327#103—142no—
328#102—142no—
329#102—142no—
330#102—144no—
331C22⋊C8—141no—
332C22⋊C8—141no—
333C4×C8—72no—
334C4×C8—141no—
335C22×C7—21no—
336C22×C7—21no—
337C22×C7—24no—
338C22×C7—48no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7#44812yes—
342C22×C7C2×#12011yescharacteristic
343C2×D7—81no—
344C2×D7—81no—
345C2×D7—81no—
346C4×C7—24no—
347C4×C7—21no—
348C4×C7—21no—
349C4×C7—21no—
350C4×C7—44no—
351C4×C7—42no—
352C4×C7—42no—
353C4×C7#44812yes—
354C4×C7C22×SD1611yescharacteristic
355C4×C7C2×AGL(1,ℤ/8)12yes—
356C4×C7C2×#13711yescharacteristic
357Dic7—81no—
358C24—21no—
359#41—282no—
360#41—282no—
361C2×Q8—24no—
362C2×Q8—28no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—48no—
367C2×Q8—44no—
368C2×Q8—416no—
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—141no—
372C2×Q8C2×C22⋊D712yes—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—141no—
376C2×D4—141no—
377C2×D4—141no—
378C2×D4—281no—
379C2×D4#114314yescomplemented
380C2×D4C2×C22⋊D712yes—
381C22×C4—21no—
382C22×C4—28no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—48no—
389C22×C4—44no—
390C22×C4—281no—
391C22×C4C2×C22⋊D711yescharacteristic
392C22×C4C2×C22⋊D711yescharacteristic
393C22×C4C2×C22⋊D711yescharacteristic
394SD16—284no—
395SD16—284no—
396D8—284no—
397M16—282no—
398C2×C8—142no—
399C2×C8—142no—
400C2×C8—142no—
401C4⋊C4—28no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—141no—
410C4⋊C4—141no—
411C4⋊C4C2×C22⋊D712yes—
412C4⋊C4C2×C22⋊D712yes—
413C4⋊C4C2×C22⋊D714yes—
414C22⋊C4—28no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—281no—
418C22⋊C4—281no—
419C42—22no—
420C42—22no—
421C42C2×C22⋊D711yescharacteristic
422C42C2×C22⋊D711yescharacteristic
423C42C2×C22⋊D711yescharacteristic
424C42C23×D711yescharacteristic
425C2×C7—22no—
426C2×C7—44no—
427C2×C7C2×#44811yescharacteristic
428C2×C7#300311yescharacteristic
429C2×C7#302011yescharacteristic
430D7—161no—
431C23—24no—
432C23—22no—
433C23—42no—
434C23—42no—
435C23—281no—
436C23#651611yescharacteristic
437Q8—24no—
438Q8—22no—
439Q8—416no—
440Q8—44no—
441Q8—44no—
442Q8—281no—
443Q8—281no—
444Q8#649514yescomplemented
445D4—22no—
446D4—24no—
447D4—282no—
448D4—282no—
449D4—281no—
450D4—281no—
451D4—562no—
452D4C2×#114314yes—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—416no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—281no—
467C2×C4—562no—
468C2×C4C2×#114312yes—
469C2×C4#650512yes—
470C2×C4#651611yescharacteristic
471C2×C4#651611yescharacteristic
472C2×C4#651611yescharacteristic
473C2×C4C22×C22⋊D711yescharacteristic
474C2×C4C22×C22⋊D711yescharacteristic
475C2×C4C22×C22⋊D711yescharacteristic
476C8—282no—
477C8—282no—
478C8—282no—
479C7#1875811yescharacteristic, complemented, Sylow
480C22—21no—
481C22—21no—
482C22—24no—
483C22—48no—
484C22—42no—
485C22—42no—
486C22—561no—
487C22—561no—
488C22—561no—
489C22#8999912yes—
490C22C2×#651611yescharacteristic
491C4—24no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—44no—
496C4—42no—
497C4—42no—
498C4—561no—
499C4#8999912yes—
500C4C2×#649512yes—
501C4C22×#114311yescharacteristic
502C4C2×#650511yescharacteristic
503C2—22no—
504C2—44no—
505C2—1121no—
506C2#1174148311yescharacteristic
507C2C2×#8999911yescharacteristic
508C2#1174156111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×D4×Q8C2
socleC22×C7C2×#120
prime radical#11739752C2
hypercenterC2×D4×Q8D7
hypocenterC7#18758
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C7×D4×Q8C2semidirect
2. #11740661C2semidirect
3. #11739887C2semidirect
5. #11729671C2semidirect
6. #11729668C2semidirect
7. #11729529C2semidirect
8. #11729376C2semidirect
9. #11729351C2semidirect
25. #89860C22semidirect
27. #89851C22semidirect
28. #89799C22semidirect
30. #89792C22semidirect
40. #89342C22semidirect
41. Q8⋊(C7⋊C8)C22semidirect
111. C2×D4×Q8D7semidirect
134. C2×C7×D4SD16semidirect
231. C7×Q8AGL(1,ℤ/8)semidirect
379. C2×D4#1143semidirect
444. Q8#6495semidirect
479. C7#18758semidirect

Maximal subgroups

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

subgroupindex
C2×C7×D4×Q82
#117406612
#117398872
#117397522
#117296712
#117296682
#117295292
#117293762
#117293512
#187587