Finite Group Databasebeta

← #13319834

Subgroups — #13319834

Order 1792.

976 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90225C211yesmaximal, characteristic
2#11736733C211yesmaximal, characteristic, complemented
3#11736731C211yesmaximal, characteristic, complemented
4C7×#1458C211yesmaximal, characteristic, complemented
5#11729325C211yesmaximal, characteristic, complemented
6#11728540C211yesmaximal, characteristic, complemented
7#11728540C211yesmaximal, characteristic, complemented
8#11725901C214yesmaximal, complemented
9#11725876C214yesmaximal, complemented
10C2×C7×#117C2211yescharacteristic
11C2×#6493C411yescharacteristic
12C2×#6487C411yescharacteristic
13#90225—22no—
14#90225—22no—
15#90225—22no—
16#90225—22no—
17C2×#6461C412yes—
18C2×#6459C2212yes—
19C2×#6448C412yes—
20C2×#6446C2212yes—
21C2×#6438C411yescharacteristic
22C2×C4×D28C2211yescharacteristic
23C2×#6432C411yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25C7×#431C2211yescharacteristic
26C7×#431C2211yescharacteristic
27#89697C2211yescharacteristic
28#89697C2211yescharacteristic
29#89610—22no—
30#89606—22no—
31#89498C2214yes—
32D7×C22⋊C8C2214yes—
33#89460—22no—
34C8×D28—22no—
35C7×#324C2212yes—
36C7×#323C2212yes—
37#89323C2212yes—
38#89320C2212yes—
39#89264C2214yes—
40#89264C2214yes—
41#17388—71nomaximal, Sylow
42C23×C4×C7C2×C411yescharacteristic
43C22×C22⋊D7C2×C412yes—
44C22×D28C2×C411yescharacteristic
45C22×C4×D7C2312yes—
46C22×Dic14C2×C411yescharacteristic
47C22×Dic14C2×C412yes—
48C7×#117—21no—
49C7×#117—21no—
50C7×#117—21no—
51C7×#117—21no—
52C2×C7×C4⋊C4C2×C411yescharacteristic
53C2×C7×C4⋊C4C2311yescharacteristic
54C2×C7×C22⋊C4C2×C411yescharacteristic
55C2×C7×C22⋊C4C2311yescharacteristic
56C2×C42×C7C2×C411yescharacteristic
57C2×C42×C7C2311yescharacteristic
58C2×C22⋊Dic7C2×C412yes—
59#6493—22no—
60#6493—22no—
61C2×(C2×D7)⋊C4C2×C412yes—
62C2×(C2×D7)⋊C4C2×C412yes—
63C2×(C2×D7)⋊C4C2×C412yes—
64C2×(C2×D7)⋊C4C2×C412yes—
65C2×C4⋊Dic7C2×C411yescharacteristic
66C2×C4⋊Dic7C2×C412yes—
67C2×C4⋊Dic7C2×C411yescharacteristic
68#6487—22no—
69#6487—22no—
70C2×Dic7⋊C4C2×C412yes—
71C2×Dic7⋊C4C2312yes—
72C2×C4×Dic7C2×C412yes—
73C2×#1131—24no—
74C2×C8×D7—24no—
75#6461—24no—
76#6461—24no—
77#6459—24no—
78#6459—24no—
79#6448—24no—
80#6448—24no—
81#6446—24no—
82#6446—24no—
83#6438—22no—
84#6438—22no—
85C4×D28D412yes—
86C4×D28D412yes—
87C4×D28D412yes—
88C4×D28D412yes—
89#6432—22no—
90#6432—22no—
91C4×Dic14—21no—
92C4×Dic14—21no—
93C4×Dic14—21no—
94C4×Dic14—21no—
95C7×C4⋊C8—21no—
96C7×C4⋊C8—21no—
97C7×C22⋊C8C2312yes—
98C7×C22⋊C8C2312yes—
99C4×C7×C8—21no—
100C4×C7×C8—21no—
101C22⋊(C7⋊C8)C2312yes—
102C22⋊(C7⋊C8)C2312yes—
103(C2×D7)⋊C8—24no—
104(C2×D7)⋊C8—24no—
105C4⋊(C7⋊C8)—21no—
106C4⋊(C7⋊C8)—21no—
107C4×C7⋊C8—21no—
108C4×C7⋊C8—21no—
109C2×#542—71no—
110#2975—71no—
111#2972—71no—
112#1472—74no—
113#1468—74no—
114#1461—71no—
115#1461—71no—
116#1458—71no—
117#1458D711yescharacteristic, complemented
118C24×C7C22×C411yescharacteristic
119C23×D7C22×C412yes—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7C22×C411yescharacteristic
125C22×C4×C7C22×C411yescharacteristic
126C22×C4×C7C22×C411yescharacteristic
127C22×C4×C7C22×C411yescharacteristic
128C22×C4×C7C22×C411yescharacteristic
129C22×C4×C7C2411yescharacteristic
130C2×C22⋊D7—24no—
131C2×C22⋊D7—82no—
132C22×Dic7C22×C412yes—
133C22×Dic7C22×C412yes—
134C22×Dic7C22×C412yes—
135C2×D28—22no—
136C2×D28—22no—
137C2×D28C22⋊C412yes—
138C2×D28C22⋊C412yes—
139C2×C4×D7—82no—
140C2×C4×D7C2×D414yes—
141C2×C4×D7C2×D414yes—
142C2×Dic14—21no—
143C2×Dic14—21no—
144C2×Dic14—22no—
145C2×Dic14—22no—
146C2×Dic14—24no—
147C2×Dic14—82no—
148C2×C7×C8—22no—
149C2×C7×C8—22no—
150C7×C4⋊C4—21no—
151C7×C4⋊C4—21no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—21no—
154C7×C22⋊C4—21no—
155C7×C22⋊C4—21no—
156C7×C22⋊C4—21no—
157C7×C22⋊C4—21no—
158C42×C7—21no—
159C42×C7—21no—
160C42×C7C2×D411yescharacteristic
161C42×C7C2×D411yescharacteristic
162C42×C7C2×D411yescharacteristic
163C42×C7C2×D411yescharacteristic
164C22⋊Dic7—24no—
165(C2×D7)⋊C4—24no—
166(C2×D7)⋊C4—24no—
167(C2×D7)⋊C4—24no—
168(C2×D7)⋊C4C22⋊C414yes—
169(C2×D7)⋊C4C22⋊C414yes—
170C4⋊Dic7—21no—
171C4⋊Dic7—21no—
172C4⋊Dic7—24no—
173C4⋊Dic7C22⋊C412yes—
174C4⋊Dic7C22⋊C412yes—
175Dic7⋊C4—22no—
176Dic7⋊C4—22no—
177Dic7⋊C4—24no—
178C4×Dic7—22no—
179C4×Dic7—22no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—22no—
182#1131—84no—
183C8×D7—84no—
184#542—142no—
185#542—142no—
186#542—142no—
187#542—142no—
188C2×C4⋊Q8—71no—
189C2×#126—72no—
190C2×#124—72no—
191C2×#124—71no—
192C2×#123—72no—
193C2×C22⋊Q8—71no—
194C2×C22⋊Q8—72no—
195C2×#121—71no—
196C2×C4×Q8—71no—
197C2×C4×D4—71no—
198C2×#117C2×D711yescharacteristic
199#436—142no—
200#436—142no—
201C8×D4—142no—
202C8×D4—142no—
203#431—71no—
204#431—71no—
205#431C2×D711yescharacteristic
206#431C2×D711yescharacteristic
207#407—74no—
208C2×C22⋊C8—74no—
209#324—72no—
210#324—74no—
211#324—74no—
212#324C2×D712yes—
213#323—72no—
214#323C2×D712yes—
215C23×C7—21no—
216C23×C7—21no—
217C23×C7—21no—
218C23×C7—41no—
219C23×C7—41no—
220C23×C7C23×C411yescharacteristic
221C22×D7—22no—
222C22×D7—22no—
223C22×D7—42no—
224C22×D7—42no—
225C22×D7C2×C22⋊C414yes—
226C2×C4×C7—21no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—21no—
230C2×C4×C7—21no—
231C2×C4×C7—41no—
232C2×C4×C7—41no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—41no—
236C2×C4×C7—41no—
237C2×C4×C7C2×C22⋊C411yescharacteristic
238C2×C4×C7C2×C22⋊C411yescharacteristic
239C2×C4×C7C2×C22⋊C411yescharacteristic
240C2×C4×C7C2×C22⋊C411yescharacteristic
241C2×C4×C7#13112yes—
242C2×C4×C7#13112yes—
243C2×C4×C7C22×D411yescharacteristic
244C2×C4×C7C22×D411yescharacteristic
245C22⋊D7—84no—
246C2×Dic7—22no—
247C2×Dic7—22no—
248C2×Dic7—22no—
249C2×Dic7—22no—
250C2×Dic7—42no—
251C2×Dic7—42no—
252C2×Dic7C2×C22⋊C414yes—
253D28—24no—
254D28—24no—
255C4×D7—44no—
256C4×D7—44no—
257Dic14—42no—
258Dic14—42no—
259Dic14—84no—
260C7×C8—42no—
261C7×C8—42no—
262C7⋊C8—42no—
263C7⋊C8—42no—
264C22×Q8—71no—
265C22×Q8—72no—
266C22×D4—72no—
267C22×D4—71no—
268C23×C4—72no—
269C23×C4C4×D711yescharacteristic
270C2×M16—144no—
271C22×C8—144no—
272C4⋊Q8—142no—
273C4⋊Q8—142no—
274#126—144no—
275#126—144no—
276#124—142no—
277#124—142no—
278#124—144no—
279#124—144no—
280#123—144no—
281#123—144no—
282C22⋊Q8—144no—
283C22⋊Q8—144no—
284C22⋊Q8—142no—
285C22⋊Q8—142no—
286#121—142no—
287#121—142no—
288C4×Q8—141no—
289C4×Q8—141no—
290C4×Q8—141no—
291C4×Q8—141no—
292C4×D4—72no—
293C4×D4—72no—
294C4×D4—72no—
295C4×D4—72no—
296#117—21no—
297#117—21no—
298#117—21no—
299#117—21no—
300C2×C4⋊C4—71no—
301C2×C4⋊C4—71no—
302C2×C4⋊C4—72no—
303C2×C4⋊C4—72no—
304C2×C4⋊C4—72no—
305C2×C4⋊C4C4×D711yescharacteristic
306C2×C4⋊C4C22×D711yescharacteristic
307C2×C22⋊C4—72no—
308C2×C22⋊C4—72no—
309C2×C22⋊C4—72no—
310C2×C22⋊C4—72no—
311C2×C22⋊C4—72no—
312C2×C22⋊C4C4×D711yescharacteristic
313C2×C22⋊C4C22×D711yescharacteristic
314C2×C42—72no—
315C2×C42C4×D711yescharacteristic
316C2×C42C22×D711yescharacteristic
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—141no—
320C4⋊C8—141no—
321C22⋊C8—72no—
322C22⋊C8—72no—
323C22⋊C8—144no—
324C22⋊C8—144no—
325C22⋊C8C22×D712yes—
326C22⋊C8C22×D712yes—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—141no—
330C4×C8—141no—
331C22×C7—21no—
332C22×C7—21no—
333C22×C7—41no—
334C22×C7—41no—
335C22×C7—41no—
336C22×C7—81no—
337C22×C7—81no—
338C22×C7C22×C22⋊C411yescharacteristic
339C22×C7C2×#13111yescharacteristic
340C22×C7C2×#13111yescharacteristic
341C2×D7—24no—
342C2×D7—24no—
343C2×D7—24no—
344C2×D7—82no—
345C2×D7—82no—
346C4×C7—21no—
347C4×C7—21no—
348C4×C7—21no—
349C4×C7—21no—
350C4×C7—41no—
351C4×C7—41no—
352C4×C7—41no—
353C4×C7#40812yes—
354C4×C7#40812yes—
355Dic7—24no—
356Dic7—42no—
357Dic7—42no—
358C24—72no—
359C24C2×C4×D711yescharacteristic
360C2×Q8—144no—
361C2×Q8—141no—
362C2×Q8—141no—
363C2×Q8—142no—
364C2×Q8—142no—
365C2×Q8—562no—
366C2×D4—72no—
367C2×D4—72no—
368C2×D4—144no—
369C2×D4—142no—
370C2×D4—142no—
371C2×D4—562no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—72no—
377C22×C4—74no—
378C22×C4—74no—
379C22×C4—72no—
380C22×C4—72no—
381C22×C4—562no—
382C22×C4C2×C4×D711yescharacteristic
383C22×C4C2×C4×D711yescharacteristic
384C22×C4C2×C4×D711yescharacteristic
385C22×C4C2×C4×D711yescharacteristic
386C22×C4C2×C4×D711yescharacteristic
387C22×C4C23×D711yescharacteristic
388M16—564no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—142no—
392C2×C8—142no—
393C2×C8—564no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—72no—
399C4⋊C4—72no—
400C4⋊C4—144no—
401C4⋊C4—144no—
402C4⋊C4—141no—
403C4⋊C4—141no—
404C4⋊C4—142no—
405C4⋊C4—142no—
406C22⋊C4—21no—
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—74no—
411C22⋊C4—74no—
412C22⋊C4—144no—
413C22⋊C4—144no—
414C22⋊C4—144no—
415C22⋊C4—144no—
416C42—21no—
417C42—21no—
418C42—142no—
419C42—142no—
420C42D4×D711yescharacteristic
421C42D4×D711yescharacteristic
422C42D4×D711yescharacteristic
423C42D4×D711yescharacteristic
424C2×C7—21no—
425C2×C7—21no—
426C2×C7—81no—
427C2×C7C2×#40811yescharacteristic
428C2×C7#288311yescharacteristic
429C2×C7#288911yescharacteristic
430D7—44no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—41no—
435C23—41no—
436C23—74no—
437C23—142no—
438C23—142no—
439C23—282no—
440C23—282no—
441C23C22×C4×D711yescharacteristic
442Q8—282no—
443Q8—282no—
444Q8—564no—
445D4—144no—
446D4—144no—
447D4—564no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—74no—
460C2×C4—142no—
461C2×C4—142no—
462C2×C4—142no—
463C2×C4—142no—
464C2×C4—284no—
465C2×C4—284no—
466C2×C4—282no—
467C2×C4—282no—
468C2×C4D7×C22⋊C411yescharacteristic
469C2×C4D7×C22⋊C411yescharacteristic
470C2×C4D7×C22⋊C411yescharacteristic
471C2×C4D7×C22⋊C411yescharacteristic
472C2×C4#646412yes—
473C2×C4#647012yes—
474C2×C4C2×D4×D711yescharacteristic
475C2×C4C2×D4×D711yescharacteristic
476C8—42no—
477C8—42no—
478C8—282no—
479C8—282no—
480C7#1738811yescharacteristic, complemented, Sylow
481C22—21no—
482C22—21no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—81no—
487C22—81no—
488C22—144no—
489C22—144no—
490C22—144no—
491C22—562no—
492C22—562no—
493C22C2×D7×C22⋊C411yescharacteristic
494C22C2×#646411yescharacteristic
495C22C2×#647011yescharacteristic
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—144no—
504C4—282no—
505C4—282no—
506C4#8949912yes—
507C4#8949912yes—
508C2—21no—
509C2—21no—
510C2—81no—
511C2—284no—
512C2C2×#8949911yescharacteristic
513C2#1173613911yescharacteristic
514C2#1173616011yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1458C2
socleC22×C7C22×C22⋊C4
prime radicalC2×#6493C4
cosocleC22×C4×C7C24
hypercenter#1458D7
hypocenterC7#17388
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #11736733C2semidirect
3. #11736731C2semidirect
4. C7×#1458C2semidirect
5. #11729325C2semidirect
6. #11728540C2semidirect
7. #11728540C2semidirect
8. #11725901C2semidirect
9. #11725876C2semidirect
117. #1458D7semidirect
480. C7#17388semidirect

Maximal subgroups

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

subgroupindex
C2×#902252
#117367332
#117367312
C7×#14582
#117293252
#117285402
#117259012
#117258762
#173887