Finite Group Databasebeta

← #14218315

Subgroups — #14218315

Order 1792.

1950 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11743287C211yesmaximal, characteristic
2C7×#2932C211yesmaximal, characteristic, complemented
3#11741240C212yesmaximal, complemented
4Dic7×#131C212yesmaximal, complemented
5#11740721C211yesmaximal, characteristic, complemented
6#11736901C218yesmaximal, complemented
7#11736898C218yesmaximal, complemented
8#11736490C211yesmaximal, characteristic, complemented
9#11736483C211yesmaximal, characteristic, complemented
10#11736101C212yesmaximal, complemented
11#11736093C212yesmaximal, complemented
12C2×#89479C211yesmaximal, characteristic, complemented
13#11735925C211yesmaximal, characteristic, complemented
14C2×C7×#131C2212yes—
15C7×#532C2211yescharacteristic
16#90519C2212yes—
17#41×Dic7C2212yes—
18C2×#6511C2212yes—
19C22×#1131C2211yescharacteristic
20C22×#1131C2211yescharacteristic
21#90419C412yes—
22#90407C412yes—
23#90371C414yes—
24#90297C414yes—
25#90291C414yes—
26#90248C2214yes—
27C4×#1159C2214yes—
28C2×C42×D7C2211yescharacteristic
29#90161C2211yescharacteristic
30C7×#435C2218yescomplemented
31C7×#423C2211yescharacteristic
32C7×#408C2212yes—
33C2×C7×#97C2211yescharacteristic
34#90006C2212yes—
35#89900C2218yes—
36#89898C2212yes—
37#89894C2212yes—
38#89894C2214yescomplemented
39#89891C2212yes—
40M16×Dic7C2212yes—
41M16×Dic7C2214yescomplemented
42#89883C2218yes—
43#89874C2212yes—
44#89873C2212yes—
45#89873C2214yescomplemented
46C2×C8×Dic7C2211yescharacteristic
47C2×C8×Dic7C2211yescharacteristic
48C2×C8×Dic7C2214yescomplemented
49#89783C2218yescomplemented
50#89696C2211yescharacteristic
51C2×#6377C2211yescharacteristic
52#89607C2214yes—
53#89605C2214yes—
54#89600C2218yescomplemented
55#89502C2218yes—
56#89501C2218yes—
57#89492C2218yescomplemented
58#89490C2218yescomplemented
59#89479C2214yes—
60#89479C2214yes—
61#89473C2218yescomplemented
62#60205—71nomaximal, Sylow
63C2×C7×#41C2312yes—
64C2×#1159C2×C414yes—
65C22×C4×D7C2×C412yes—
66C22×C4×D7C2311yescharacteristic
67C7×#131—28no—
68C2×C7×M16C2312yes—
69C2×C7×M16C2314yes—
70C22×C7×C8C2311yescharacteristic
71C22×C7×C8C2314yes—
72C22×C7×C8C2311yescharacteristic
73C7×C4⋊Q8C2×C411yescharacteristic
74C7×#127C2×C411yescharacteristic
75C7×#124C2×C412yes—
76C7×#121C2×C414yes—
77C4×C7×D4C2314yes—
78C2×C42×C7C2311yescharacteristic
79#6511—28no—
80#6509C2×C414yes—
81Q8×Dic7C2×C414yes—
82#6501C2×C414yes—
83#6501C2×C414yes—
84#6500C2×C414yes—
85#6499C2×C414yes—
86#6498C2×C414yes—
87D4×Dic7C2×C414yes—
88D4×Dic7C2×C414yes—
89D4×Dic7C2×C418yes—
90C4×C22⋊D7C2318yes—
91#6489C2312yes—
92#6489C2314yes—
93#6487C2×C414yes—
94C2×C4×Dic7C2×C412yes—
95C2×C4×Dic7C2311yescharacteristic
96C2×C4×Dic7C2314yes—
97C2×#1136C2312yes—
98C2×#1136C2314yes—
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2314yes—
101C22×C7⋊C8C2311yescharacteristic
102C2×#1131—24no—
103C2×#1131—24no—
104C2×#1131C2314yes—
105C2×#1131C2314yes—
106#6460C2×C414yes—
107#6455C2×C414yes—
108#6453C2×C414yes—
109#6450C2×C414yes—
110#6446C2×C418yes—
111(C22⋊D7)⋊C4C2×C418yes—
112#6442C2×C418yes—
113#6440C2×C418yes—
114C42×D7C2×C412yes—
115C42×D7C2×C412yes—
116C42×D7C2312yes—
117C42×D7C2312yes—
118#6433C2×C412yes—
119#6432C2×C412yes—
120C4×Dic14C2314yes—
121C7×C4⋊C8C2314yes—
122C7×C22⋊C8C2318yes—
123C7×#97C2314yes—
124C22⋊(C7⋊C8)C2318yes—
125(C2×D7)⋊C8C2318yes—
126#6389C2314yes—
127#6389C2314yes—
128#6389C2314yes—
129Dic7⋊C8C2318yes—
130Dic7⋊C8C2318yes—
131C8×Dic7C2314yes—
132C8×Dic7C2314yes—
133C8×Dic7C2312yes—
134C8×Dic7C2312yes—
135C8×Dic7C2312yes—
136C8×Dic7C2312yes—
137C8×Dic7C2314yes—
138C4⋊(C7⋊C8)C2314yes—
139#6377C2314yes—
140#3444—71no—
141#2967—78no—
142#2965—78no—
143#2961—71no—
144#2961—71no—
145#2932—71no—
146#2932D711yescharacteristic, complemented
147#2930—72no—
148#2930—72no—
149#2922—72no—
150#2920—71no—
151C4×#131—72no—
152C2×C4×M16—71no—
153C23×D7C22×C411yescharacteristic
154C7×#41—24no—
155C7×#41—24no—
156C2×C7×Q8C22×C412yes—
157C2×C7×D4C22×C414yes—
158C2×C7×D4C22×C412yes—
159C2×C7×D4C22×C414yes—
160C22×C4×C7C22×C412yes—
161C22×C4×C7C2414yes—
162C22×C4×C7C2411yescharacteristic
163C2×C22⋊D7C22×C418yes—
164C22×Dic7C22×C418yes—
165C22×Dic7C22×C411yescharacteristic
166C22×Dic7C22×C412yes—
167#1159—48no—
168C2×C4×D7—22no—
169C2×C4×D7—22no—
170C2×C4×D7—44no—
171C2×C4×D7C22×C412yes—
172C2×C4×D7C22×C412yes—
173C2×C4×D7C22×C412yes—
174C2×C4×D7C22×C412yes—
175C2×C4×D7C22×C412yes—
176C2×C4×D7C2412yes—
177C2×Dic14C22×C414yes—
178C7×M16—24no—
179C7×M16—28no—
180C2×C7×C8—22no—
181C2×C7×C8—28no—
182C2×C7×C8#4114yescomplemented
183C2×C7×C8C2412yes—
184C2×C7×C8C2414yes—
185C2×C7×C8C2412yes—
186C7×C4⋊C4C22×C414yes—
187C7×C22⋊C4C22×C418yes—
188C42×C7C22×C411yescharacteristic
189C42×C7C22×C411yescharacteristic
190C42×C7C2412yes—
191C22⋊Dic7C22×C418yes—
192C22⋊Dic7C22×C418yes—
193C22⋊Dic7C22×C414yes—
194(C2×D7)⋊C4C22×C418yes—
195C4⋊Dic7C22×C414yes—
196C4⋊Dic7C22×C414yes—
197C4⋊Dic7C22×C418yes—
198Dic7⋊C4C22×C418yes—
199Dic7⋊C4C22×C418yes—
200C4×Dic7C22×C412yes—
201C4×Dic7C22×C412yes—
202C4×Dic7C22×C412yes—
203C4×Dic7C22×C412yes—
204C4×Dic7C22×C412yes—
205C4×Dic7C22×C418yes—
206C4×Dic7C2414yes—
207C4×Dic7C2411yescharacteristic
208C4×Dic7C2411yescharacteristic
209#1136—24no—
210#1136—28no—
211C2×C7⋊C8—22no—
212C2×C7⋊C8—28no—
213C2×C7⋊C8#4114yescomplemented
214C2×C7⋊C8C2412yes—
215C2×C7⋊C8C2414yes—
216C2×C7⋊C8C2412yes—
217#1131—28no—
218#1131—28no—
219C2×#131—72no—
220C2×#131C2×D712yes—
221C22×M16—71no—
222C22×M16—71no—
223#533—72no—
224#533—71no—
225#532—72no—
226#532C2×D711yescharacteristic
227#529—74no—
228#529—74no—
229#529—74no—
230#525—72no—
231#525—74no—
232C4×#41—72no—
233C4×#41—74no—
234C22×C42—71no—
235#446—78no—
236#446—78no—
237#436—78no—
238#436—78no—
239#435—78no—
240#435—78no—
241#435—78no—
242#435C2×D718yescomplemented
243#433—78no—
244#432—74no—
245#431—74no—
246#431—78no—
247#424—72no—
248#424—72no—
249#423—71no—
250#423C2×D711yescharacteristic
251#408—72no—
252#408C2×D712yes—
253#407—72no—
254#405—72no—
255#405—74no—
256#405—72no—
257#405—74no—
258C4×M16—72no—
259C4×M16—74no—
260C4×M16—74no—
261C4×M16—74no—
262C2×#97—71no—
263C2×#97C2×D711yescharacteristic
264C2×C4×C8—71no—
265C2×C4×C8—71no—
266C2×C4×C8—74no—
267C23×C7C23×C414yes—
268C23×C7C23×C411yescharacteristic
269C22×D7—22no—
270C22×D7—42no—
271C22×D7C23×C412yes—
272C7×Q8—24no—
273C7×D4—28no—
274C7×D4—24no—
275C7×D4—44no—
276C2×C4×C7—24no—
277C2×C4×C7—24no—
278C2×C4×C7—42no—
279C2×C4×C7C23×C412yes—
280C2×C4×C7C23×C414yes—
281C2×C4×C7C23×C412yes—
282C2×C4×C7C23×C411yescharacteristic
283C2×C4×C7C2×#4112yes—
284C2×C4×C7C2×#4112yes—
285C2×C4×C7C2511yescharacteristic
286C22⋊D7—48no—
287C2×Dic7—22no—
288C2×Dic7—24no—
289C2×Dic7—216no—
290C2×Dic7C23×C412yes—
291C2×Dic7C23×C412yes—
292C2×Dic7C23×C418yes—
293C2×Dic7C23×C412yes—
294C4×D7—24no—
295C4×D7—22no—
296C4×D7—22no—
297C4×D7—24no—
298C4×D7—24no—
299C4×D7—42no—
300C4×D7—42no—
301Dic14—44no—
302C7×C8—22no—
303C7×C8—24no—
304C7×C8C2×#4114yes—
305C7⋊C8—22no—
306C7⋊C8—24no—
307C7⋊C8C2×#4114yes—
308C2×#41—74no—
309C2×#41C22×D712yes—
310C23×C4—71no—
311C23×C4—72no—
312#131—28no—
313#131—148no—
314C2×M16—72no—
315C2×M16—74no—
316C2×M16—74no—
317C2×M16—74no—
318C2×M16—144no—
319C2×M16—144no—
320C2×M16C22×D712yes—
321C2×M16C22×D714yes—
322C22×C8—71no—
323C22×C8—71no—
324C22×C8—74no—
325C22×C8C22×D711yescharacteristic
326C22×C8C22×D714yes—
327C22×C8C22×D711yescharacteristic
328C4⋊Q8—72no—
329C4⋊Q8C4×D711yescharacteristic
330#127C4×D711yescharacteristic
331#126—78no—
332#125—74no—
333#125—72no—
334#124—74no—
335#124C4×D712yes—
336#123—74no—
337#123—78no—
338C22⋊Q8—74no—
339C22⋊Q8—74no—
340C22⋊Q8—74no—
341#121—74no—
342#121—74no—
343#121C4×D714yes—
344#120—74no—
345C4×Q8—74no—
346C4×Q8—74no—
347C4×Q8—74no—
348C4×D4—78no—
349C4×D4—74no—
350C4×D4—74no—
351C4×D4—78no—
352C4×D4—78no—
353C4×D4C22×D714yes—
354#117—72no—
355#117—74no—
356#117—74no—
357#117—78no—
358C2×C42—72no—
359C2×C42—74no—
360C2×C42—71no—
361C2×C42—72no—
362C2×C42—72no—
363C2×C42—72no—
364C2×C42—72no—
365C2×C42C22×D711yescharacteristic
366C4⋊C8—78no—
367C4⋊C8—78no—
368C4⋊C8—74no—
369C4⋊C8C22×D714yes—
370C22⋊C8—78no—
371C22⋊C8—78no—
372C22⋊C8C22×D718yes—
373#97—74no—
374#97—74no—
375#97—74no—
376#97—74no—
377#97C22×D714yes—
378C4×C8—72no—
379C4×C8—72no—
380C4×C8—74no—
381C4×C8—72no—
382C4×C8—72no—
383C4×C8—74no—
384C4×C8—74no—
385C22×C7—22no—
386C22×C7—24no—
387C22×C7—28no—
388C22×C7C4×#4112yes—
389C22×C7C24×C411yescharacteristic
390C2×D7—22no—
391C2×D7—22no—
392C2×D7—22no—
393C2×D7—44no—
394C4×C7—24no—
395C4×C7—22no—
396C4×C7—22no—
397C4×C7C4×#4112yes—
398C4×C7C22×#4111yescharacteristic
399C4×C7C22×#4111yescharacteristic
400Dic7—22no—
401Dic7—28no—
402Dic7—22no—
403Dic7#56814yes—
404C24—71no—
405#41—24no—
406#41—24no—
407#41—288no—
408C2×Q8—74no—
409C2×Q8C2×C4×D712yes—
410C2×D4—78no—
411C2×D4C2×C4×D714yes—
412C2×D4C2×C4×D712yes—
413C2×D4C2×C4×D714yes—
414C22×C4—72no—
415C22×C4—71no—
416C22×C4—78no—
417C22×C4—72no—
418C22×C4—72no—
419C22×C4—72no—
420C22×C4—72no—
421C22×C4—72no—
422C22×C4—72no—
423C22×C4—142no—
424C22×C4—142no—
425C22×C4—284no—
426C22×C4C2×C4×D712yes—
427C22×C4C23×D714yes—
428C22×C4C23×D711yescharacteristic
429M16—24no—
430M16—28no—
431M16—148no—
432M16—148no—
433M16—144no—
434M16—148no—
435C2×C8—22no—
436C2×C8—28no—
437C2×C8—74no—
438C2×C8—72no—
439C2×C8—72no—
440C2×C8—74no—
441C2×C8—148no—
442C2×C8—142no—
443C2×C8#115914yescomplemented
444C2×C8C23×D714yes—
445C2×C8C23×D712yes—
446C2×C8C23×D712yes—
447C4⋊C4—78no—
448C4⋊C4—78no—
449C4⋊C4—74no—
450C4⋊C4—74no—
451C4⋊C4—78no—
452C4⋊C4C2×C4×D714yes—
453C22⋊C4—74no—
454C22⋊C4—78no—
455C22⋊C4—78no—
456C22⋊C4—78no—
457C22⋊C4C2×C4×D718yes—
458C42—72no—
459C42—72no—
460C42—72no—
461C42—72no—
462C42—72no—
463C42—78no—
464C42—71no—
465C42—71no—
466C42—74no—
467C42C2×C4×D711yescharacteristic
468C42C2×C4×D711yescharacteristic
469C42C23×D712yes—
470C2×C7—22no—
471C2×C7—44no—
472C2×C7C2×C4×#4111yescharacteristic
473C2×C7C2×#56812yes—
474D7—42no—
475C23—72no—
476C23—142no—
477C23—282no—
478C23C22×C4×D714yes—
479C23C22×C4×D711yescharacteristic
480Q8—24no—
481Q8—284no—
482D4—28no—
483D4—24no—
484D4—44no—
485D4—288no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—72no—
490C2×C4—72no—
491C2×C4—72no—
492C2×C4—78no—
493C2×C4—144no—
494C2×C4—144no—
495C2×C4—144no—
496C2×C4—1416no—
497C2×C4—142no—
498C2×C4—144no—
499C2×C4—142no—
500C2×C4—142no—
501C2×C4—282no—
502C2×C4—282no—
503C2×C4C22×C4×D712yes—
504C2×C4C22×C4×D712yes—
505C2×C4C22×C4×D714yes—
506C2×C4C22×C4×D711yescharacteristic
507C2×C4C2×#115912yes—
508C2×C4D7×#4112yes—
509C2×C4C24×D711yescharacteristic
510C8—24no—
511C8—22no—
512C8—74no—
513C8—142no—
514C8—144no—
515C8C2×#115914yes—
516C7#6020511yescharacteristic, complemented, Sylow
517C22—22no—
518C22—24no—
519C22—28no—
520C22—142no—
521C22—142no—
522C22—142no—
523C22—284no—
524C22C4×#115912yes—
525C22C23×C4×D711yescharacteristic
526C4—24no—
527C4—22no—
528C4—22no—
529C4—74no—
530C4—142no—
531C4—142no—
532C4—148no—
533C4C4×#115912yes—
534C4C22×#115911yescharacteristic
535C4C2×D7×#4111yescharacteristic
536C2—22no—
537C2—44no—
538C2—282no—
539C2C2×C4×#115911yescharacteristic
540C2D7×#56812yes—
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
FittingC7×#2932C2
prime radical#11743287C2
cosocleC2×C4×C7C25
hypercenter#2932D7
hypocenterC7#60205
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #2932

Complements

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

NH ≅ G/Ntype
2. C7×#2932C2semidirect
3. #11741240C2semidirect
4. Dic7×#131C2semidirect
5. #11740721C2semidirect
6. #11736901C2semidirect
7. #11736898C2semidirect
8. #11736490C2semidirect
9. #11736483C2semidirect
10. #11736101C2semidirect
11. #11736093C2semidirect
12. C2×#89479C2semidirect
13. #11735925C2semidirect
30. C7×#435C22semidirect
38. #89894C22semidirect
41. M16×Dic7C22semidirect
45. #89873C22semidirect
48. C2×C8×Dic7C22semidirect
49. #89783C22semidirect
54. #89600C22semidirect
57. #89492C22semidirect
58. #89490C22semidirect
61. #89473C22semidirect
146. #2932D7semidirect
182. C2×C7×C8#41semidirect
213. C2×C7⋊C8#41semidirect
242. #435C2×D7semidirect
443. C2×C8#1159semidirect
516. C7#60205semidirect

Maximal subgroups

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

subgroupindex
#117432872
C7×#29322
#117412402
Dic7×#1312
#117407212
#117369012
#117368982
#117364902
#117364832
#117361012
#117360932
C2×#894792
#117359252
#602057