Finite Group Databasebeta

← #13332430

Subgroups — #13332430

Order 1792.

910 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#556C211yesmaximal, characteristic
2#11740669C211yesmaximal, characteristic, complemented
3#11739876C211yesmaximal, characteristic, complemented
4#11739759C211yesmaximal, characteristic
5#11729687C214yesmaximal, complemented
6#11729686C214yesmaximal, complemented
7#11729529C211yesmaximal, characteristic, complemented
8#11729375C211yesmaximal, characteristic, complemented
9#11729352C211yesmaximal, characteristic
10C7×#556—22no—
11C7×#556—22no—
12C7×#556—22no—
13C7×#556C2214yes—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#126—22no—
16C2×C7×#124—21no—
17C2×C7×C22⋊Q8—21no—
18C2×C4×C7×Q8—21no—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4C2211yescharacteristic
21C2×C7×#117—21no—
22#90161C2211yescharacteristic
23#89997—21no—
24#89990—21no—
25#89865C2212yescomplemented
26#89863—22no—
27#89850C2212yescomplemented
28#89794—22no—
29#89793C2212yescomplemented
30#89792C2212yescomplemented
31#89791C2212yes—
32C4×#1142—22no—
33#89780C2212yes—
34#89743—21no—
35#89742—22no—
36#89736—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89342C2214yescomplemented
41Q8⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yes—
43#18763—71nomaximal, Sylow
44C22×C7×Q8—21no—
45C22×C7×Q8D411yescharacteristic
46C22×C7×D4D411yescharacteristic
47C23×C4×C7—21no—
48C7×C4⋊Q8—21no—
49C7×C4⋊Q8—21no—
50C7×C4⋊Q8D412yes—
51C7×C4⋊Q8C2312yes—
52C7×#126—28no—
53C7×#126—28no—
54C7×#124—24no—
55C7×#124—24no—
56C7×C22⋊Q8—24no—
57C7×C22⋊Q8—24no—
58C4×C7×Q8—24no—
59C4×C7×Q8—24no—
60C4×C7×Q8—21no—
61C4×C7×Q8—21no—
62C4×C7×Q8D412yes—
63C4×C7×Q8C2312yes—
64C4×C7×D4—21no—
65C4×C7×D4—21no—
66C4×C7×D4D412yes—
67C4×C7×D4C2312yes—
68C7×#117—24no—
69C7×#117—24no—
70C2×C7×C4⋊C4—22no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C7×C4⋊C4D412yes—
75C2×C7×C4⋊C4D411yescharacteristic
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4—22no—
78C2×C7×C22⋊C4—21no—
79C2×C42×C7—22no—
80C2×C42×C7—21no—
81C2×C42×C7C2311yescharacteristic
82C2×#1142—22no—
83#6489—21no—
84#6487—21no—
85#6487—21no—
86C2×#1136—21no—
87#6433—21no—
88#6432C2311yescharacteristic
89#6432C2311yescharacteristic
90C4×Dic14—21no—
91C4×Dic14—21no—
92#6409—22no—
93#6409—22no—
94#6406—22no—
95C22⋊(C7⋊C8)—21no—
96C22⋊(C7⋊C8)—21no—
97#6384—24no—
98#6384—22no—
99#6382—22no—
100#6382—22no—
101#6381—24no—
102#6381—22no—
103C4⋊(C7⋊C8)—21no—
104C4⋊(C7⋊C8)—21no—
105C4⋊(C7⋊C8)C2312yes—
106C4⋊(C7⋊C8)C2312yes—
107C4×C7⋊C8—21no—
108C4×C7⋊C8C2312yes—
109C2×#556D711yescharacteristic
110#3230—71no—
111#3101—71no—
112#3078—71no—
113#1650—74no—
114#1638—74no—
115#1549—71no—
116#1486—71no—
117#1479—71no—
118C24×C7—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8—44no—
125C2×C7×Q8C2×D412yes—
126C2×C7×D4—21no—
127C2×C7×D4—42no—
128C2×C7×D4C2×D412yes—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—42no—
136C22×C4×C7—42no—
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C2×Dic14—21no—
141C2×Dic14—21no—
142C7×C4⋊C4—28no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4—21no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4C2×D412yes—
151C7×C4⋊C4C2×D412yes—
152C7×C4⋊C4C2×D414yes—
153C7×C22⋊C4—28no—
154C7×C22⋊C4—28no—
155C7×C22⋊C4—22no—
156C7×C22⋊C4—22no—
157C42×C7—28no—
158C42×C7—22no—
159C42×C7—22no—
160C42×C7C2×D411yescharacteristic
161C42×C7C2×D411yescharacteristic
162C42×C7C2×D411yescharacteristic
163C42×C7C2411yescharacteristic
164C22⋊Dic7—41no—
165#1142—44no—
166C4⋊Dic7—21no—
167C4⋊Dic7—21no—
168C4⋊Dic7—21no—
169C4⋊Dic7—21no—
170Dic7⋊C4—41no—
171Dic7⋊C4—41no—
172C4×Dic7—41no—
173#1136—42no—
174C2×C7⋊C8—22no—
175C2×C7⋊C8—22no—
176C2×C7⋊C8—22no—
177#556—22no—
178#556—22no—
179#556—22no—
180#556C2×D714yes—
181#533—71no—
182C2×C4⋊Q8C2×D711yescharacteristic
183C2×#126—22no—
184C2×#124—21no—
185C2×C22⋊Q8—21no—
186C2×C4×Q8—21no—
187C2×C4×Q8C2×D711yescharacteristic
188C2×C4×D4C2×D711yescharacteristic
189C2×#117—21no—
190#500—72no—
191#487—72no—
192#484—142no—
193#483—141no—
194#479—142no—
195#477—72no—
196#477—142no—
197#476—72no—
198#475—72no—
199#461—72no—
200#450—141no—
201C4×SD16—142no—
202#431—71no—
203#431—71no—
204#428—141no—
205#423—71no—
206#419—141no—
207#332—74no—
208Q8⋊C8—74no—
209D4⋊C8—74no—
210C23×C7—22no—
211C23×C7—42no—
212C23×C7—42no—
213C23×C7—42no—
214C23×C7#12011yescharacteristic
215C7×Q8—22no—
216C7×Q8—22no—
217C7×Q8—44no—
218C7×Q8—44no—
219C7×Q8—42no—
220C7×D4—22no—
221C7×D4—22no—
222C7×D4—42no—
223C2×C4×C7—22no—
224C2×C4×C7—22no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—24no—
228C2×C4×C7—41no—
229C2×C4×C7—42no—
230C2×C4×C7—44no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—44no—
236C2×C4×C7#12011yescharacteristic
237C2×C4×C7#12011yescharacteristic
238C2×C4×C7#12011yescharacteristic
239C2×C4×C7#13512yes—
240C2×C4×C7#13712yes—
241C2×C4×C7C22×D411yescharacteristic
242C2×C4×C7C22×D411yescharacteristic
243C2×C4×C7C22×D411yescharacteristic
244C2×Dic7—41no—
245C2×Dic7—41no—
246Dic14—42no—
247Dic14—42no—
248C7⋊C8—42no—
249C7⋊C8—42no—
250C7⋊C8—42no—
251C22×Q8—21no—
252C22×Q8C22⋊D711yescharacteristic
253C22×D4C22⋊D711yescharacteristic
254C23×C4—21no—
255C2×SD16—142no—
256C2×M16—141no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8—71no—
260C4⋊Q8—71no—
261C4⋊Q8C22⋊D712yes—
262C4⋊Q8C22×D712yes—
263#126—28no—
264#126—28no—
265#125—141no—
266#124—24no—
267#124—24no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—141no—
271C22⋊Q8—141no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—141no—
277C4×Q8—141no—
278C4×Q8C22⋊D712yes—
279C4×Q8C22×D712yes—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4C22⋊D712yes—
283C4×D4C22×D712yes—
284#117—24no—
285#117—24no—
286#117—141no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4C22⋊D711yescharacteristic
291C2×C4⋊C4C22⋊D711yescharacteristic
292C2×C4⋊C4C22⋊D712yes—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—21no—
296C2×C42—22no—
297C2×C42—21no—
298C2×C42C22×D711yescharacteristic
299#107—142no—
300#107—144no—
301#106—142no—
302#106—142no—
303C4⋊C8—72no—
304C4⋊C8—72no—
305C4⋊C8—141no—
306C4⋊C8—141no—
307#103—142no—
308#103—142no—
309#103—142no—
310#103—144no—
311#102—142no—
312C22⋊C8—141no—
313C22⋊C8—141no—
314C4×C8—72no—
315C4×C8—141no—
316C22×C7—21no—
317C22×C7—21no—
318C22×C7—42no—
319C22×C7—42no—
320C22×C7—42no—
321C22×C7—84no—
322C22×C7#45212yes—
323C22×C7C2×#12011yescharacteristic
324C4×C7—21no—
325C4×C7—21no—
326C4×C7—21no—
327C4×C7—42no—
328C4×C7—42no—
329C4×C7—44no—
330C4×C7—42no—
331C4×C7#45212yes—
332C4×C7C2×#13511yescharacteristic
333C4×C7C2×#13711yescharacteristic
334C4×C7#57512yes—
335Dic7—81no—
336Dic7—81no—
337C24—21no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—44no—
344C2×Q8—141no—
345C2×Q8—141no—
346C2×Q8C2×C22⋊D712yes—
347C2×D4—21no—
348C2×D4—42no—
349C2×D4C2×C22⋊D712yes—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4C2×C22⋊D711yescharacteristic
359C22×C4C2×C22⋊D711yescharacteristic
360C22×C4C2×C22⋊D711yescharacteristic
361SD16—284no—
362M16—282no—
363C2×C8—142no—
364C2×C8—142no—
365C2×C8—142no—
366C4⋊C4—28no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—141no—
375C4⋊C4—141no—
376C4⋊C4—141no—
377C4⋊C4—141no—
378C4⋊C4—281no—
379C4⋊C4—281no—
380C4⋊C4C2×C22⋊D712yes—
381C4⋊C4C2×C22⋊D714yes—
382C4⋊C4C2×C22⋊D712yes—
383C22⋊C4—28no—
384C22⋊C4—28no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—281no—
388C42—28no—
389C42—22no—
390C42—22no—
391C42—281no—
392C42C2×C22⋊D711yescharacteristic
393C42C2×C22⋊D711yescharacteristic
394C42C2×C22⋊D711yescharacteristic
395C42C23×D711yescharacteristic
396C2×C7—22no—
397C2×C7—82no—
398C2×C7C2×#45211yescharacteristic
399C2×C7#300511yescharacteristic
400C2×C7#302011yescharacteristic
401C23—22no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23#651611yescharacteristic
406Q8—22no—
407Q8—22no—
408Q8—44no—
409Q8—44no—
410Q8—42no—
411Q8—282no—
412Q8—282no—
413D4—22no—
414D4—22no—
415D4—42no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—281no—
430C2×C4—281no—
431C2×C4#651312yes—
432C2×C4#651412yes—
433C2×C4#651611yescharacteristic
434C2×C4#651611yescharacteristic
435C2×C4#651611yescharacteristic
436C2×C4C22×C22⋊D711yescharacteristic
437C2×C4C22×C22⋊D711yescharacteristic
438C2×C4C22×C22⋊D711yescharacteristic
439C8—282no—
440C8—282no—
441C8—282no—
442C7#1876311yescharacteristic, complemented, Sylow
443C22—21no—
444C22—21no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—84no—
449C22#9001112yes—
450C22C2×#651611yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—44no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—561no—
459C4—561no—
460C4#9001112yes—
461C4C2×#651311yescharacteristic
462C4#9051312yes—
463C4C2×#651411yescharacteristic
464C2—22no—
465C2—82no—
466C2#1174152611yescharacteristic
467C2C2×#9001111yescharacteristic
468C2#1174156111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#556C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenterC2×#556D7
hypocenterC7#18763
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. #11740669C2semidirect
3. #11739876C2semidirect
5. #11729687C2semidirect
6. #11729686C2semidirect
7. #11729529C2semidirect
8. #11729375C2semidirect
25. #89865C22semidirect
27. #89850C22semidirect
29. #89793C22semidirect
30. #89792C22semidirect
40. #89342C22semidirect
41. Q8⋊(C7⋊C8)C22semidirect
442. C7#18763semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5562
#117406692
#117398762
#117397592
#117296872
#117296862
#117295292
#117293752
#117293522
#187637