Finite Group Databasebeta

← #13363696

Subgroups — #13363696

Order 1792.

1194 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90315C211yesmaximal, characteristic
2C22×Dic7⋊C8C211yesmaximal, characteristic, complemented
3#11731309C211yesmaximal, characteristic, complemented
4C7×#1892C211yesmaximal, characteristic, complemented
5#11729727C211yesmaximal, characteristic
6#11729450C211yesmaximal, characteristic, complemented
7#11727984C212yesmaximal, complemented
8#11727975C212yesmaximal, complemented
9#11727269C211yesmaximal, characteristic
10#11726646C212yesmaximal, complemented
11#11726556C212yesmaximal, complemented
12C23×C7×C8C2211yescharacteristic
13C2×C7×C22⋊Q8C2211yescharacteristic
14C2×#6510C2211yescharacteristic
15C2×Q8×Dic7C2211yescharacteristic
16C2×#6493C2211yescharacteristic
17C22×C4×Dic7C2211yescharacteristic
18C23×C7⋊C8C2211yescharacteristic
19#90315—24no—
20#90315C418yescomplemented
21C2×#6461—21no—
22C2×#6456C2211yescharacteristic
23C2×#6455—21no—
24C2×#6453C2211yescharacteristic
25C2×#6449—21no—
26C2×(C22⋊D7)⋊C4—21no—
27C2×#6440—21no—
28C2×C7×#103C2212yes—
29C7×#380C2211yescharacteristic
30C2×#6409C2212yes—
31#89984C2211yescharacteristic
32C2×#6395C2212yes—
33C2×Dic7⋊C8—24no—
34C2×Dic7⋊C8C2212yes—
35C2×Dic7⋊C8C2212yes—
36#89748C2212yes—
37C4⋊C4×Dic7C2212yes—
38C2×#6383C2212yes—
39#89714—21no—
40#89422—21no—
41#89421—21no—
42C7×#340C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21887—71nomaximal, Sylow
47C22×C7×Q8C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C22×C22⋊D7—21no—
50C23×Dic7D411yescharacteristic
51C23×Dic7D411yescharacteristic
52C22×D28C2311yescharacteristic
53C22×C4×D7—21no—
54C22×C7×C8—22no—
55C22×C7×C8—42no—
56C22×C7×C8C2312yes—
57C7×C22⋊Q8—22no—
58C7×C22⋊Q8C2×C414yes—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4C2312yes—
61C2×C7×C4⋊C4C2311yescharacteristic
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4—21no—
64C2×C22⋊Dic7—21no—
65#6510—22no—
66#6510C2×C414yes—
67Q8×Dic7—22no—
68Q8×Dic7C2×C414yes—
69#6493—22no—
70#6493C2×C414yes—
71C2×(C2×D7)⋊C4—21no—
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7C2312yes—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—22no—
81C2×Dic7⋊C4—22no—
82C2×C4×Dic7—21no—
83C2×C4×Dic7—22no—
84C2×C4×Dic7C2×C414yes—
85C2×C4×Dic7D411yescharacteristic
86C2×C4×Dic7D411yescharacteristic
87C2×C4×Dic7D414yes—
88C2×C4×Dic7C2311yescharacteristic
89C2×C4×Dic7C2311yescharacteristic
90C22×C7⋊C8—22no—
91C22×C7⋊C8—42no—
92C22×C7⋊C8C2312yes—
93#6461—28no—
94#6456—22no—
95#6456C2×C414yes—
96#6455—28no—
97#6453—22no—
98#6453C2×C414yes—
99#6449—28no—
100(C22⋊D7)⋊C4—28no—
101#6440—44no—
102C7×#103—42no—
103C7×#95—21no—
104#6409—42no—
105C7⋊#95—21no—
106C7⋊#95—22no—
107#6395—42no—
108Dic7⋊C8—24no—
109Dic7⋊C8—24no—
110#6383—42no—
111C2×#529—71no—
112C22×C4⋊C8—71no—
113#2285—71no—
114#1989—72no—
115#1983—72no—
116#1893—71no—
117#1893—71no—
118#1892—71no—
119#1892D711yescharacteristic, complemented
120#1803—72no—
121#1802—72no—
122C24×C7C2×D411yescharacteristic
123C23×D7—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8C22×C412yes—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7C22×C412yes—
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D412yes—
136C22×C4×C7C2411yescharacteristic
137C2×C22⋊D7—24no—
138C2×C22⋊D7—42no—
139C2×C22⋊D7—42no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—24no—
143C22×Dic7—21no—
144C22×Dic7—22no—
145C22×Dic7C22⋊C414yes—
146C22×Dic7C22⋊C414yes—
147C22×Dic7C2×D411yescharacteristic
148C22×Dic7C2×D411yescharacteristic
149C22×Dic7C2×D411yescharacteristic
150C22×Dic7C2×D411yescharacteristic
151C2×D28—22no—
152C2×D28—22no—
153C2×D28—21no—
154C2×D28C22×C412yes—
155C2×C4×D7—24no—
156C2×C4×D7—44no—
157C2×C7×C8—22no—
158C2×C7×C8—22no—
159C2×C7×C8—22no—
160C2×C7×C8—44no—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4C22×C412yes—
166C7×C22⋊C4—24no—
167C7×C22⋊C4—42no—
168C22⋊Dic7—42no—
169(C2×D7)⋊C4—24no—
170(C2×D7)⋊C4—24no—
171(C2×D7)⋊C4—24no—
172(C2×D7)⋊C4—24no—
173C4⋊Dic7—21no—
174C4⋊Dic7—24no—
175C4⋊Dic7—24no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178C4⋊Dic7C22×C412yes—
179Dic7⋊C4—24no—
180Dic7⋊C4—44no—
181Dic7⋊C4—44no—
182C4×Dic7—24no—
183C4×Dic7—44no—
184C4×Dic7C22⋊C414yes—
185C4×Dic7C22⋊C414yes—
186C4×Dic7C22×C412yes—
187C4×Dic7C22×C412yes—
188C4×Dic7C2×D412yes—
189C4×Dic7C2×D412yes—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—44no—
194C23×C8—71no—
195C23×C8C2×D711yescharacteristic
196#529—78no—
197#529—144no—
198C2×#126—141no—
199C2×#125—71no—
200C2×#124—71no—
201C2×#123—141no—
202C2×C22⋊Q8C2×D711yescharacteristic
203C2×#121—71no—
204C2×C4×Q8—71no—
205C2×C4×D4—71no—
206C2×C4×D4—141no—
207C2×#117—141no—
208C2×#117—141no—
209C22×C42—71no—
210C2×C4⋊C8—72no—
211C2×C4⋊C8—72no—
212C2×C4⋊C8—144no—
213C2×#103—72no—
214C2×#103C2×D712yes—
215C2×#102—72no—
216C2×#102—72no—
217#384—72no—
218#382—141no—
219#380—71no—
220#380—141no—
221#380C2×D711yescharacteristic
222C4×C4⋊C4—72no—
223C4×C4⋊C4—141no—
224#340—72no—
225#340—72no—
226#340—72no—
227#340C2×D712yes—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7C2×C22⋊C412yes—
231C23×C7AGL(1,ℤ/8)12yes—
232C23×C7#13712yes—
233C23×C7C22×D411yescharacteristic
234C22×D7—22no—
235C22×D7—42no—
236C22×D7—42no—
237C22×D7—41no—
238C22×D7—41no—
239C7×Q8—24no—
240C7×Q8—41no—
241C7×Q8—41no—
242C2×C4×C7—22no—
243C2×C4×C7—24no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7C2×C22⋊C412yes—
254C2×C4×C7C23×C411yescharacteristic
255C2×C4×C7C22×D411yescharacteristic
256C2×C4×C7H(5,𝔽2)12yes—
257C2×C4×C72−1+412yes—
258C22⋊D7—48no—
259C2×Dic7—22no—
260C2×Dic7—21no—
261C2×Dic7—21no—
262C2×Dic7—21no—
263C2×Dic7—21no—
264C2×Dic7—28no—
265C2×Dic7—28no—
266C2×Dic7—22no—
267C2×Dic7—41no—
268C2×Dic7—41no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7C2×C22⋊C412yes—
273C2×Dic7C2×C22⋊C412yes—
274C2×Dic7C2×C22⋊C412yes—
275C2×Dic7C2×C22⋊C412yes—
276C2×Dic7#13514yes—
277D28—24no—
278D28—41no—
279D28—41no—
280C4×D7—48no—
281C7×C8—42no—
282C7⋊C8—42no—
283C22×Q8C22×D711yescharacteristic
284C22×D4—71no—
285C22×D4—141no—
286C23×C4—71no—
287C23×C4—71no—
288C23×C4—141no—
289C23×C4C22×D711yescharacteristic
290C22×C8—22no—
291C22×C8—42no—
292C22×C8—72no—
293C22×C8—142no—
294C22×C8—282no—
295C22×C8C22×D712yes—
296#126—148no—
297#125—74no—
298#125—142no—
299#124—74no—
300#124—142no—
301#123—148no—
302C22⋊Q8—22no—
303C22⋊Q8C4×D714yes—
304#121—74no—
305#121—142no—
306C4×Q8—74no—
307C4×Q8—142no—
308C4×D4—74no—
309C4×D4—142no—
310C4×D4—148no—
311#117—148no—
312#117—284no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—71no—
315C2×C4⋊C4—72no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4—141no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4C22×D712yes—
322C2×C4⋊C4C22×D711yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—141no—
326C2×C22⋊C4—141no—
327C2×C22⋊C4—141no—
328C2×C22⋊C4—141no—
329C2×C22⋊C4—141no—
330C2×C42—74no—
331C2×C42—71no—
332C2×C42—71no—
333C2×C42—74no—
334C2×C42—71no—
335C2×C42—71no—
336C2×C42—141no—
337C2×C42—142no—
338C4⋊C8—144no—
339C4⋊C8—144no—
340#103—42no—
341#103—282no—
342#102—282no—
343#102—282no—
344#95—21no—
345#95—142no—
346#95—141no—
347C22×C7—22no—
348C22×C7—22no—
349C22×C7—24no—
350C22×C7—24no—
351C22×C7#41714yes—
352C22×C7C22×C22⋊C411yescharacteristic
353C22×C7#53712yes—
354C22×C7C2×#13512yes—
355C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
356C22×C7C2×#13711yescharacteristic
357C2×D7—42no—
358C2×D7—42no—
359C2×D7—42no—
360C2×D7—82no—
361C2×D7—82no—
362C4×C7—22no—
363C4×C7—42no—
364C4×C7—42no—
365C4×C7—82no—
366C4×C7#52012yes—
367C4×C7#53511yescharacteristic
368C4×C7#53611yescharacteristic
369Dic7—22no—
370Dic7—22no—
371Dic7—22no—
372Dic7—42no—
373Dic7—82no—
374Dic7#41614yes—
375C24—141no—
376C24D4×D711yescharacteristic
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8C2×C4×D712yes—
381C2×D4—72no—
382C2×D4—144no—
383C2×D4—141no—
384C2×D4—142no—
385C2×D4—142no—
386C2×D4—282no—
387C2×D4—282no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—71no—
394C22×C4—74no—
395C22×C4—74no—
396C22×C4—71no—
397C22×C4—71no—
398C22×C4—71no—
399C22×C4—144no—
400C22×C4—144no—
401C22×C4—142no—
402C22×C4—142no—
403C22×C4—141no—
404C22×C4—142no—
405C22×C4—284no—
406C22×C4C2×C4×D712yes—
407C22×C4D4×D711yescharacteristic
408C22×C4D4×D712yes—
409C22×C4C23×D711yescharacteristic
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—44no—
414C2×C8—142no—
415C2×C8—142no—
416C2×C8—142no—
417C2×C8—284no—
418C4⋊C4—24no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—72no—
423C4⋊C4—144no—
424C4⋊C4—144no—
425C4⋊C4—144no—
426C4⋊C4—141no—
427C4⋊C4—142no—
428C4⋊C4—142no—
429C4⋊C4—284no—
430C4⋊C4—284no—
431C4⋊C4C2×C4×D712yes—
432C22⋊C4—24no—
433C22⋊C4—42no—
434C22⋊C4—144no—
435C22⋊C4—144no—
436C22⋊C4—144no—
437C22⋊C4—144no—
438C22⋊C4—282no—
439C42—74no—
440C42—74no—
441C42—72no—
442C42—72no—
443C42—72no—
444C42—72no—
445C42—144no—
446C42—284no—
447C2×C7—24no—
448C2×C7#231111yescharacteristic
449C2×C7C2×#41611yescharacteristic
450C2×C7C2×#41711yescharacteristic
451C2×C7#324712yes—
452C2×C7#324812yes—
453D7—82no—
454C23—22no—
455C23—22no—
456C23—142no—
457C23—281no—
458C23—281no—
459C23—282no—
460C23—282no—
461C23D7×C22⋊C412yes—
462C23#647712yes—
463C23#648112yes—
464C23C2×D4×D711yescharacteristic
465Q8—24no—
466Q8—41no—
467Q8—41no—
468D4—144no—
469D4—288no—
470D4—281no—
471D4—281no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—72no—
484C2×C4—74no—
485C2×C4—72no—
486C2×C4—72no—
487C2×C4—72no—
488C2×C4—148no—
489C2×C4—141no—
490C2×C4—141no—
491C2×C4—141no—
492C2×C4—141no—
493C2×C4—142no—
494C2×C4—148no—
495C2×C4—142no—
496C2×C4—288no—
497C2×C4—281no—
498C2×C4—281no—
499C2×C4—282no—
500C2×C4—282no—
501C2×C4—282no—
502C2×C4D7×C22⋊C412yes—
503C2×C4C22×C4×D711yescharacteristic
504C2×C4C2×D4×D711yescharacteristic
505C2×C4#654812yes—
506C2×C4#655412yes—
507C8—42no—
508C8—282no—
509C7#2188711yescharacteristic, complemented, Sylow
510C22—24no—
511C22—22no—
512C22—22no—
513C22—24no—
514C22—282no—
515C22—282no—
516C22—282no—
517C22—562no—
518C22—562no—
519C22#8957514yes—
520C22C2×D7×C22⋊C411yescharacteristic
521C22#9028812yes—
522C22C2×#647711yescharacteristic
523C22C2×#648111yescharacteristic
524C22D7×#13512yes—
525C4—22no—
526C4—42no—
527C4—42no—
528C4—74no—
529C4—82no—
530C4—142no—
531C4—142no—
532C4—142no—
533C4—282no—
534C4—562no—
535C4#9023712yes—
536C4#9028311yescharacteristic
537C4#9033911yescharacteristic
538C2—24no—
539C2—562no—
540C2#1173210711yescharacteristic
541C2C2×#8957511yescharacteristic
542C2D7×#41611yescharacteristic
543C2#1173808612yes—
544C2#1173819112yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1892C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
cosocleC22×C4×C7C24
hypercenter#1892D7
hypocenterC7#21887
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
2. C22×Dic7⋊C8C2semidirect
3. #11731309C2semidirect
4. C7×#1892C2semidirect
6. #11729450C2semidirect
7. #11727984C2semidirect
8. #11727975C2semidirect
10. #11726646C2semidirect
11. #11726556C2semidirect
20. #90315C4semidirect
119. #1892D7semidirect
509. C7#21887semidirect

Maximal subgroups

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

subgroupindex
C2×#903152
C22×Dic7⋊C82
#117313092
C7×#18922
#117297272
#117294502
#117279842
#117279752
#117272692
#117266462
#117265562
#218877