Finite Group Databasebeta

← #13321510

Subgroups — #13321510

Order 1792.

1070 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11743348C211yesmaximal, characteristic
2#11736731C211yesmaximal, characteristic
3#11731673C211yesmaximal, characteristic
4C7×#1518C211yesmaximal, characteristic, complemented
5#11729589C211yesmaximal, characteristic, complemented
6#11728546C211yesmaximal, characteristic, complemented
7#11727300C211yesmaximal, characteristic, complemented
8#11726093C214yesmaximal, complemented
9D7×#328C214yesmaximal, complemented
10C7×#533C2211yescharacteristic
11#90421C412yescomplemented
12#90418C412yescomplemented
13#90381—22no—
14#90325—22no—
15D7×C22⋊Q8C2214yes—
16#90270—22no—
17#90270C412yescomplemented
18Q8×D28C412yescomplemented
19#90212—21no—
20#90210—21no—
21C2×#6456—21no—
22C2×C4×D28C2211yescharacteristic
23#90161C2211yescharacteristic
24C7×#431C2211yescharacteristic
25C7×#382C2211yescharacteristic
26#89745—21no—
27C4×C4⋊Dic7C2211yescharacteristic
28#89697C2211yescharacteristic
29#89606—22no—
30D7×C22⋊C8C2214yes—
31C8×D28—22no—
32#89440—22no—
33#89439C2212yes—
34D7×#95C2212yes—
35C7×#328C2214yes—
36#89333C2212yes—
37#89333C2212yes—
38#89262C2212yes—
39#89262C2212yes—
40#89247C2214yes—
41#17614—71nomaximal, Sylow
42C2×Q8×D7C2×C414yescomplemented
43C22×D28D411yescharacteristic
44C22×C4×D7—22no—
45C22×C4×D7C2312yes—
46C7×C4⋊Q8C2×C411yescharacteristic, complemented
47C7×C4⋊Q8C2×C411yescharacteristic, complemented
48C7×#125—21no—
49C7×C22⋊Q8—21no—
50C7×C22⋊Q8C2312yes—
51C4×C7×Q8—21no—
52C4×C7×Q8C2×C412yescomplemented
53C7×#117—21no—
54C2×C7×C4⋊C4—21no—
55C2×C42×C7C2311yescharacteristic
56#6509—24no—
57#6509C2×C414yescomplemented
58C2×(C2×D7)⋊C4—22no—
59C2×(C2×D7)⋊C4D412yes—
60#6489—21no—
61C2×C4⋊Dic7—21no—
62C2×C4⋊Dic7D411yescharacteristic
63#6487—21no—
64#6487C2312yes—
65C2×Dic7⋊C4—22no—
66C2×C4×Dic7—21no—
67C2×C8×D7—24no—
68#6460—24no—
69#6460C2×C414yescomplemented
70#6459—24no—
71#6459—24no—
72#6459—24no—
73#6456—22no—
74#6456—22no—
75#6456—44no—
76D7×C4⋊C4—24no—
77D7×C4⋊C4C2×C414yescomplemented
78#6453—22no—
79#6451—22no—
80D7×C22⋊C4—24no—
81#6441—24no—
82C4×D28—22no—
83C4×D28—22no—
84C4×D28C2×C412yes—
85C4×D28C2×C412yes—
86C4×D28D412yes—
87C4×D28D412yes—
88#6433—21no—
89#6433—22no—
90#6432—22no—
91#6432C2×C411yescharacteristic, complemented
92#6432C2×C411yescharacteristic, complemented
93C4×Dic14—21no—
94C4×Dic14C2×C412yescomplemented
95C7×C4⋊C8—21no—
96C7×C22⋊C8C2312yes—
97C4×C7×C8—21no—
98C7×#95—21no—
99C7×#95C2312yes—
100C7⋊#95—22no—
101C7⋊#95C2311yescharacteristic
102C7⋊#95C2311yescharacteristic
103C22⋊(C7⋊C8)C2312yes—
104(C2×D7)⋊C8—24no—
105C4⋊(C7⋊C8)—21no—
106C4×C7⋊C8—21no—
107#3501—71no—
108#2972—71no—
109#2349—71no—
110#1520—74no—
111#1518—71no—
112#1518D711yescharacteristic, complemented
113#1503—71no—
114#1503—71no—
115C2×#328—74no—
116C23×D7C2×D412yes—
117C2×C7×Q8—21no—
118C2×C7×Q8C22×C412yes—
119C22×C4×C7—22no—
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2411yescharacteristic
123C22×Dic7—21no—
124C22×Dic7—21no—
125C22×Dic7C2×D412yes—
126Q8×D7—84no—
127C2×D28C22⋊C412yes—
128C2×D28C22⋊C412yes—
129C2×D28SD1614yes—
130C2×D28Q1614yes—
131C2×C4×D7—24no—
132C2×C4×D7—24no—
133C2×C4×D7—44no—
134C2×C4×D7—44no—
135C2×C4×D7—82no—
136C2×C4×D7—82no—
137C2×C4×D7C22×C414yes—
138C2×C4×D7C2×D414yes—
139C2×Dic14—21no—
140C2×Dic14—24no—
141C2×Dic14C22×C412yes—
142C2×C7×C8—22no—
143C7×C4⋊C4—21no—
144C7×C4⋊C4—21no—
145C7×C4⋊C4—21no—
146C7×C4⋊C4—22no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—42no—
149C7×C4⋊C4C22×C412yes—
150C7×C22⋊C4—22no—
151C42×C7—22no—
152C42×C7C22×C411yescharacteristic
153C42×C7C22×C411yescharacteristic
154C42×C7C2×D411yescharacteristic
155C42×C7C2×D411yescharacteristic
156C22⋊Dic7—22no—
157(C2×D7)⋊C4—24no—
158(C2×D7)⋊C4—24no—
159(C2×D7)⋊C4—44no—
160(C2×D7)⋊C4C22⋊C414yes—
161(C2×D7)⋊C4C22⋊C414yes—
162C4⋊Dic7—21no—
163C4⋊Dic7—21no—
164C4⋊Dic7—21no—
165C4⋊Dic7—24no—
166C4⋊Dic7—42no—
167C4⋊Dic7C22⋊C412yes—
168C4⋊Dic7C22⋊C412yes—
169C4⋊Dic7C22×C412yes—
170Dic7⋊C4—22no—
171Dic7⋊C4—22no—
172Dic7⋊C4—24no—
173Dic7⋊C4—24no—
174Dic7⋊C4—82no—
175C4×Dic7—22no—
176C4×Dic7—42no—
177C2×C7⋊C8—22no—
178C8×D7—84no—
179#554—72no—
180#554—72no—
181#554—142no—
182#554—142no—
183D4×Q8—72no—
184D4×Q8—72no—
185#544—141no—
186#543—142no—
187#533—71no—
188#533C2×D711yescharacteristic
189C2×C22⋊Q8—74no—
190#518—141no—
191C2×C4×D4—71no—
192C2×C4×D4—141no—
193C8×D4—142no—
194C8×D4—142no—
195#431—71no—
196#431C2×D711yescharacteristic
197C2×C22⋊C8—74no—
198#389—141no—
199#385—142no—
200#382C2×D711yescharacteristic
201#380—72no—
202C4×C4⋊C4—71no—
203C2×#95—72no—
204#328—72no—
205#328—72no—
206#328—72no—
207#328—72no—
208#328—74no—
209#328C2×D714yes—
210C23×C7C22×D411yescharacteristic
211C22×D7—42no—
212C22×D7—42no—
213C22×D7C2×C22⋊C414yes—
214C22×D7C2×SD1614yes—
215C22×D7C2×Q1614yes—
216C7×Q8—42no—
217C7×Q8—81no—
218C2×C4×C7—22no—
219C2×C4×C7—22no—
220C2×C4×C7—41no—
221C2×C4×C7—41no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7C2×C22⋊C411yescharacteristic
226C2×C4×C7C2×C22⋊C411yescharacteristic
227C2×C4×C7C2×C22⋊C411yescharacteristic
228C2×C4×C7C2×C22⋊C411yescharacteristic
229C2×C4×C7C2×SD1612yes—
230C2×C4×C7C2×Q1612yes—
231C2×C4×C7C23×C411yescharacteristic
232C2×C4×C7C22×D411yescharacteristic
233C2×Dic7—22no—
234C2×Dic7—22no—
235C2×Dic7—22no—
236C2×Dic7—42no—
237C2×Dic7—41no—
238C2×Dic7—41no—
239C2×Dic7—41no—
240C2×Dic7—41no—
241C2×Dic7—41no—
242C2×Dic7C2×C22⋊C414yes—
243D28#10318yes—
244D28#10318yes—
245C4×D7—44no—
246C4×D7—44no—
247C4×D7—84no—
248C4×D7—84no—
249C4×D7—88no—
250Dic14—42no—
251Dic14—84no—
252Dic14—81no—
253C7×C8—42no—
254C7⋊C8—42no—
255C22×Q8—74no—
256C22×D4—71no—
257C23×C4—72no—
258C23×C4—142no—
259C22×C8—144no—
260C4⋊Q8—71no—
261C4⋊Q8—71no—
262C4⋊Q8—142no—
263C4⋊Q8—142no—
264C4⋊Q8C4×D711yescharacteristic, complemented
265C4⋊Q8C4×D711yescharacteristic, complemented
266#125—21no—
267#125—142no—
268#125—142no—
269#125—141no—
270C22⋊Q8—21no—
271C22⋊Q8—74no—
272C22⋊Q8—72no—
273C22⋊Q8—74no—
274C22⋊Q8—144no—
275C22⋊Q8—144no—
276C22⋊Q8—144no—
277C22⋊Q8—144no—
278C22⋊Q8—144no—
279C22⋊Q8—144no—
280C22⋊Q8—141no—
281C22⋊Q8C22×D712yes—
282C4×Q8—21no—
283C4×Q8—72no—
284C4×Q8—141no—
285C4×Q8C4×D712yescomplemented
286C4×D4—72no—
287C4×D4—72no—
288C4×D4—72no—
289C4×D4—72no—
290C4×D4—142no—
291C4×D4—142no—
292C4×D4—142no—
293C4×D4—142no—
294C4×D4—284no—
295#117—21no—
296#117—141no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—71no—
299C2×C4⋊C4—74no—
300C2×C4⋊C4—144no—
301C2×C4⋊C4—141no—
302C2×C4⋊C4—142no—
303C2×C22⋊C4—72no—
304C2×C22⋊C4—144no—
305C2×C22⋊C4—142no—
306C2×C42—141no—
307C2×C42C22×D711yescharacteristic
308C4⋊C8—21no—
309C4⋊C8—141no—
310C22⋊C8—72no—
311C22⋊C8—144no—
312C22⋊C8C22×D712yes—
313C4×C8—21no—
314C4×C8—141no—
315#95—21no—
316#95—71no—
317#95—71no—
318#95—142no—
319#95C22×D712yes—
320C22×C7—21no—
321C22×C7—21no—
322C22×C7C22×C22⋊C411yescharacteristic
323C22×C7C22×SD1611yescharacteristic
324C22×C7C22×Q1611yescharacteristic
325C2×D7—24no—
326C2×D7—24no—
327C2×D7—44no—
328C2×D7—44no—
329C2×D7C2×#10318yes—
330C4×C7—21no—
331C4×C7—21no—
332C4×C7—21no—
333C4×C7—21no—
334C4×C7—42no—
335C4×C7—42no—
336C4×C7—82no—
337C4×C7C2×#10312yes—
338C4×C7C2×#10312yes—
339Dic7—24no—
340Dic7—42no—
341Dic7—42no—
342Dic7—42no—
343Dic7—81no—
344C24—72no—
345C2×Q8—21no—
346C2×Q8—72no—
347C2×Q8—144no—
348C2×Q8—141no—
349C2×Q8—564no—
350C2×Q8C2×C4×D712yes—
351C2×D4—72no—
352C2×D4—72no—
353C2×D4—74no—
354C2×D4—74no—
355C22×C4—22no—
356C22×C4—72no—
357C22×C4—74no—
358C22×C4—74no—
359C22×C4—144no—
360C22×C4—144no—
361C22×C4—141no—
362C22×C4—141no—
363C22×C4—284no—
364C22×C4—284no—
365C22×C4—562no—
366C22×C4—562no—
367C22×C4D4×D711yescharacteristic
368C22×C4D4×D711yescharacteristic
369C22×C4C23×D711yescharacteristic
370C2×C8—22no—
371C2×C8—142no—
372C2×C8—564no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—42no—
379C4⋊C4—72no—
380C4⋊C4—72no—
381C4⋊C4—72no—
382C4⋊C4—144no—
383C4⋊C4—144no—
384C4⋊C4—144no—
385C4⋊C4—142no—
386C4⋊C4—141no—
387C4⋊C4—141no—
388C4⋊C4—141no—
389C4⋊C4—142no—
390C4⋊C4—282no—
391C4⋊C4—562no—
392C4⋊C4C2×C4×D712yes—
393C22⋊C4—22no—
394C22⋊C4—74no—
395C22⋊C4—74no—
396C22⋊C4—144no—
397C22⋊C4—144no—
398C22⋊C4—142no—
399C22⋊C4—284no—
400C42—22no—
401C42—142no—
402C42—282no—
403C42C2×C4×D711yescharacteristic
404C42C2×C4×D711yescharacteristic
405C42D4×D711yescharacteristic
406C42D4×D711yescharacteristic
407C2×C7—21no—
408C2×C7—21no—
409C2×C7#288311yescharacteristic
410C2×C7C22×#10311yescharacteristic
411C2×C7#290111yescharacteristic
412D7—28no—
413C23—74no—
414C23—74no—
415C23—74no—
416C23—282no—
417C23—282no—
418C23C2×D4×D711yescharacteristic
419Q8—42no—
420Q8—81no—
421Q8—282no—
422Q8—561no—
423Q8—564no—
424D4—78no—
425D4—78no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—74no—
434C2×C4—142no—
435C2×C4—142no—
436C2×C4—142no—
437C2×C4—284no—
438C2×C4—284no—
439C2×C4—282no—
440C2×C4—281no—
441C2×C4—281no—
442C2×C4—281no—
443C2×C4—281no—
444C2×C4—281no—
445C2×C4—564no—
446C2×C4—564no—
447C2×C4—568no—
448C2×C4D7×C22⋊C411yescharacteristic
449C2×C4D7×C22⋊C411yescharacteristic
450C2×C4D7×C22⋊C411yescharacteristic
451C2×C4D7×C22⋊C411yescharacteristic
452C2×C4D7×SD1612yes—
453C2×C4D7×Q1612yes—
454C2×C4C22×C4×D711yescharacteristic
455C2×C4C2×D4×D711yescharacteristic
456C8—42no—
457C8—282no—
458C7#1761411yescharacteristic, complemented, Sylow
459C22—21no—
460C22—21no—
461C22—78no—
462C22—144no—
463C22—144no—
464C22—284no—
465C22—284no—
466C22C2×D7×C22⋊C411yescharacteristic
467C22C2×D7×SD1611yescharacteristic
468C22C2×D7×Q1611yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4—144no—
477C4—282no—
478C4—282no—
479C4—282no—
480C4—561no—
481C4D7×#10312yes—
482C4D7×#10312yes—
483C2—21no—
484C2—21no—
485C2—148no—
486C2#1173613911yescharacteristic
487C2C2×D7×#10311yescharacteristic
488C2#1173643711yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1518C2
socleC22×C7C22×C22⋊C4
prime radicalC22×D28D4
cosocleC22×C4×C7C24
hypercenter#1518D7
hypocenterC7#17614
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
4. C7×#1518C2semidirect
5. #11729589C2semidirect
6. #11728546C2semidirect
7. #11727300C2semidirect
8. #11726093C2semidirect
9. D7×#328C2semidirect
11. #90421C4semidirect
12. #90418C4semidirect
17. #90270C4semidirect
18. Q8×D28C4semidirect
42. C2×Q8×D7C2×C4semidirect
46. C7×C4⋊Q8C2×C4semidirect
47. C7×C4⋊Q8C2×C4semidirect
52. C4×C7×Q8C2×C4semidirect
57. #6509C2×C4semidirect
69. #6460C2×C4semidirect
77. D7×C4⋊C4C2×C4semidirect
91. #6432C2×C4semidirect
92. #6432C2×C4semidirect
94. C4×Dic14C2×C4semidirect
112. #1518D7semidirect
264. C4⋊Q8C4×D7semidirect
265. C4⋊Q8C4×D7semidirect
285. C4×Q8C4×D7semidirect
458. C7#17614semidirect

Maximal subgroups

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

subgroupindex
#117433482
#117367312
#117316732
C7×#15182
#117295892
#117285462
#117273002
#117260932
D7×#3282
#176147