Finite Group Databasebeta

← #13332394

Subgroups — #13332394

Order 1792.

1138 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×D4×Q8C211yesmaximal, characteristic
2#11740669C211yesmaximal, characteristic, complemented
3#11739892C211yesmaximal, characteristic, complemented
4#11739741C211yesmaximal, characteristic
5#11729670C214yesmaximal, complemented
6#11729669C214yesmaximal, complemented
7#11729529C211yesmaximal, characteristic, complemented
8#11729376C211yesmaximal, characteristic, complemented
9#11729351C211yesmaximal, characteristic
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
22#90161C2211yescharacteristic
23#90000—21no—
24#89987—21no—
25#89865C2212yescomplemented
26#89863—22no—
27#89850C2212yescomplemented
28#89803C2212yescomplemented
29C4×#1144—22no—
30#89793C2212yescomplemented
31#89781—22no—
32#89780C2212yes—
33#89779C2212yes—
34#89742—21no—
35#89742—22no—
36#89736—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89342C2214yescomplemented
41Q8⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yes—
43#18759—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—
53C7×C4⋊Q8—24no—
54C7×C4⋊Q8—24no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8D412yes—
58C7×C4⋊Q8C2312yes—
59C7×C22⋊Q8—24no—
60C7×C22⋊Q8—28no—
61C7×C22⋊Q8—24no—
62C7×C22⋊Q8—28no—
63C4×C7×Q8—21no—
64C4×C7×Q8—21no—
65C4×C7×Q8D412yes—
66C4×C7×Q8C2312yes—
67C4×C7×D4—24no—
68C4×C7×D4—24no—
69C4×C7×D4—21no—
70C4×C7×D4—21no—
71C4×C7×D4D412yes—
72C4×C7×D4C2312yes—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4D412yes—
79C2×C7×C22⋊C4—22no—
80C2×C7×C22⋊C4—21no—
81C2×C42×C7—21no—
82C2×C42×C7C2311yescharacteristic
83C2×#1144—22no—
84#6489—21no—
85#6487—21no—
86#6487—21no—
87C2×#1136—21no—
88#6433—21no—
89#6432C2311yescharacteristic
90#6432C2311yescharacteristic
91C4×Dic14—21no—
92C4×Dic14—21no—
93#6409—22no—
94#6406—22no—
95#6406—22no—
96C22⋊(C7⋊C8)—21no—
97C22⋊(C7⋊C8)—21no—
98#6384—24no—
99#6384—22no—
100#6382—22no—
101#6381—24no—
102#6381—22no—
103#6381—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×D4×Q8D711yescharacteristic
111#3230—71no—
112#3104—71no—
113#3077—71no—
114#1649—74no—
115#1637—74no—
116#1549—71no—
117#1487—71no—
118#1478—71no—
119C24×C7—21no—
120C2×C7×Q8—24no—
121C2×C7×Q8—28no—
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—48no—
126C2×C7×Q8—44no—
127C2×C7×Q8—416no—
128C2×C7×Q8—48no—
129C2×C7×Q8—48no—
130C2×C7×Q8C2×D412yes—
131C2×C7×D4—21no—
132C2×C7×D4—22no—
133C2×C7×D4Q1614yescomplemented
134C2×C7×D4C2×D412yes—
135C22×C4×C7—21no—
136C22×C4×C7—28no—
137C22×C4×C7—21no—
138C22×C4×C7—22no—
139C22×C4×C7—21no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—48no—
143C22×C4×C7—44no—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C2×Dic14—21no—
148C2×Dic14—21no—
149C7×C4⋊C4—28no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4C2×D412yes—
158C7×C4⋊C4C2×D412yes—
159C7×C4⋊C4C2×D414yes—
160C7×C22⋊C4—28no—
161C7×C22⋊C4—22no—
162C7×C22⋊C4—22no—
163C42×C7—22no—
164C42×C7—22no—
165C42×C7C2×D411yescharacteristic
166C42×C7C2×D411yescharacteristic
167C42×C7C2×D411yescharacteristic
168C42×C7C2411yescharacteristic
169C22⋊Dic7—41no—
170#1144—44no—
171C4⋊Dic7—21no—
172C4⋊Dic7—21no—
173C4⋊Dic7—21no—
174C4⋊Dic7—21no—
175Dic7⋊C4—41no—
176Dic7⋊C4—41no—
177C4×Dic7—41no—
178#1136—42no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—22no—
182C23×Q8—21no—
183D4×Q8—22no—
184D4×Q8—22no—
185D4×Q8—22no—
186D4×Q8C2×D714yes—
187#533—71no—
188C2×C4⋊Q8—21no—
189C2×C4⋊Q8C2×D711yescharacteristic
190C2×C22⋊Q8—21no—
191C2×C22⋊Q8—22no—
192C2×C4×Q8C2×D711yescharacteristic
193C2×C4×D4—21no—
194C2×C4×D4C2×D711yescharacteristic
195#500—72no—
196#487—72no—
197#484—141no—
198#484—142no—
199#478—142no—
200#477—72no—
201#477—142no—
202#476—72no—
203#474—72no—
204#462—72no—
205#451—141no—
206C4×Q16—142no—
207#431—71no—
208#431—71no—
209#428—141no—
210#423—71no—
211#418—141no—
212#332—74no—
213Q8⋊C8—74no—
214D4⋊C8—74no—
215C23×C7—22no—
216C23×C7—24no—
217C23×C7—42no—
218C23×C7—42no—
219C23×C7#12011yescharacteristic
220C7×Q8—24no—
221C7×Q8—22no—
222C7×Q8—416no—
223C7×Q8—44no—
224C7×Q8—44no—
225C7×Q8AGL(1,ℤ/8)14yes—
226C7×D4—22no—
227C7×D4—24no—
228C7×D4C2×Q1614yes—
229C2×C4×C7—24no—
230C2×C4×C7—22no—
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—24no—
234C2×C4×C7—22no—
235C2×C4×C7—41no—
236C2×C4×C7—416no—
237C2×C4×C7—44no—
238C2×C4×C7—42no—
239C2×C4×C7—44no—
240C2×C4×C7—42no—
241C2×C4×C7—44no—
242C2×C4×C7#12011yescharacteristic
243C2×C4×C7#12011yescharacteristic
244C2×C4×C7#12011yescharacteristic
245C2×C4×C7C2×Q1612yes—
246C2×C4×C7#13712yes—
247C2×C4×C7C22×D411yescharacteristic
248C2×C4×C7C22×D411yescharacteristic
249C2×C4×C7C22×D411yescharacteristic
250C2×Dic7—41no—
251C2×Dic7—41no—
252Dic14—42no—
253Dic14—42no—
254C7⋊C8—42no—
255C7⋊C8—42no—
256C7⋊C8—42no—
257C22×Q8—22no—
258C22×Q8—24no—
259C22×Q8—24no—
260C22×Q8—21no—
261C22×Q8—48no—
262C22×Q8C22⋊D711yescharacteristic
263C22×D4C22⋊D711yescharacteristic
264C23×C4—22no—
265C23×C4—21no—
266C2×Q16—142no—
267C2×M16—141no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—21no—
271C4⋊Q8—21no—
272C4⋊Q8—71no—
273C4⋊Q8—71no—
274C4⋊Q8C22⋊D712yes—
275C4⋊Q8C22×D712yes—
276#125—141no—
277C22⋊Q8—24no—
278C22⋊Q8—28no—
279C22⋊Q8—24no—
280C22⋊Q8—28no—
281C22⋊Q8—141no—
282C22⋊Q8—141no—
283C4×Q8—21no—
284C4×Q8—21no—
285C4×Q8—141no—
286C4×Q8—141no—
287C4×Q8C22⋊D712yes—
288C4×Q8C22×D712yes—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4C22⋊D712yes—
294C4×D4C22×D712yes—
295#117—141no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4C22⋊D711yescharacteristic
300C2×C4⋊C4C22⋊D711yescharacteristic
301C2×C4⋊C4C22⋊D712yes—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C42—21no—
305C2×C42C22×D711yescharacteristic
306#107—142no—
307#107—142no—
308#107—144no—
309#106—142no—
310C4⋊C8—72no—
311C4⋊C8—72no—
312C4⋊C8—141no—
313C4⋊C8—141no—
314#103—142no—
315#103—142no—
316#103—144no—
317#102—142no—
318#102—142no—
319C22⋊C8—141no—
320C22⋊C8—141no—
321C4×C8—72no—
322C4×C8—141no—
323C22×C7—21no—
324C22×C7—21no—
325C22×C7—24no—
326C22×C7—48no—
327C22×C7—42no—
328C22×C7—42no—
329C22×C7#45112yes—
330C22×C7C2×#12011yescharacteristic
331C4×C7—24no—
332C4×C7—21no—
333C4×C7—21no—
334C4×C7—21no—
335C4×C7—44no—
336C4×C7—42no—
337C4×C7—42no—
338C4×C7#45112yes—
339C4×C7C22×Q1611yescharacteristic
340C4×C7C2×AGL(1,ℤ/8)12yes—
341C4×C7C2×#13711yescharacteristic
342Dic7—81no—
343Dic7—81no—
344C24—21no—
345C2×Q8—24no—
346C2×Q8—28no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—48no—
351C2×Q8—44no—
352C2×Q8—416no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—141no—
356C2×Q8—141no—
357C2×Q8C2×C22⋊D712yes—
358C2×D4—22no—
359C2×D4—21no—
360C2×D4#114414yescomplemented
361C2×D4C2×C22⋊D712yes—
362C22×C4—21no—
363C22×C4—28no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—48no—
370C22×C4—44no—
371C22×C4C2×C22⋊D711yescharacteristic
372C22×C4C2×C22⋊D711yescharacteristic
373C22×C4C2×C22⋊D711yescharacteristic
374Q16—284no—
375M16—282no—
376C2×C8—142no—
377C2×C8—142no—
378C2×C8—142no—
379C4⋊C4—28no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—141no—
388C4⋊C4—141no—
389C4⋊C4—141no—
390C4⋊C4—141no—
391C4⋊C4—281no—
392C4⋊C4—281no—
393C4⋊C4C2×C22⋊D712yes—
394C4⋊C4C2×C22⋊D714yes—
395C4⋊C4C2×C22⋊D712yes—
396C22⋊C4—28no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—281no—
400C42—22no—
401C42—22no—
402C42—281no—
403C42C2×C22⋊D711yescharacteristic
404C42C2×C22⋊D711yescharacteristic
405C42C2×C22⋊D711yescharacteristic
406C42C23×D711yescharacteristic
407C2×C7—22no—
408C2×C7—44no—
409C2×C7C2×#45111yescharacteristic
410C2×C7#300311yescharacteristic
411C2×C7#302011yescharacteristic
412C23—24no—
413C23—22no—
414C23—42no—
415C23—42no—
416C23#651611yescharacteristic
417Q8—24no—
418Q8—22no—
419Q8—416no—
420Q8—44no—
421Q8—44no—
422Q8—282no—
423Q8—282no—
424Q8#649514yes—
425D4—22no—
426D4—24no—
427D4C2×#114414yes—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—41no—
435C2×C4—416no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—281no—
442C2×C4—281no—
443C2×C4#650512yes—
444C2×C4C2×#114412yes—
445C2×C4#651611yescharacteristic
446C2×C4#651611yescharacteristic
447C2×C4#651611yescharacteristic
448C2×C4C22×C22⋊D711yescharacteristic
449C2×C4C22×C22⋊D711yescharacteristic
450C2×C4C22×C22⋊D711yescharacteristic
451C8—282no—
452C8—282no—
453C8—282no—
454C7#1875911yescharacteristic, complemented, Sylow
455C22—21no—
456C22—21no—
457C22—24no—
458C22—48no—
459C22—42no—
460C22—42no—
461C22#9000012yes—
462C22C2×#651611yescharacteristic
463C4—24no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—44no—
468C4—42no—
469C4—42no—
470C4—561no—
471C4—561no—
472C4#9000012yes—
473C4C2×#649512yes—
474C4C2×#650511yescharacteristic
475C4C22×#114411yescharacteristic
476C2—22no—
477C2—44no—
478C2#1174148311yescharacteristic
479C2C2×#9000011yescharacteristic
480C2#1174156111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×D4×Q8C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenterC2×D4×Q8D7
hypocenterC7#18759
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
2. #11740669C2semidirect
3. #11739892C2semidirect
5. #11729670C2semidirect
6. #11729669C2semidirect
7. #11729529C2semidirect
8. #11729376C2semidirect
25. #89865C22semidirect
27. #89850C22semidirect
28. #89803C22semidirect
30. #89793C22semidirect
40. #89342C22semidirect
41. Q8⋊(C7⋊C8)C22semidirect
133. C2×C7×D4Q16semidirect
360. C2×D4#1144semidirect
454. C7#18759semidirect

Maximal subgroups

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

subgroupindex
C2×C7×D4×Q82
#117406692
#117398922
#117397412
#117296702
#117296692
#117295292
#117293762
#117293512
#187597