Finite Group Databasebeta

← #14218331

Subgroups — #14218331

Order 1792.

1878 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11743341C211yesmaximal, characteristic
2C7×#2960C211yesmaximal, characteristic, complemented
3#11740774C211yesmaximal, characteristic, complemented
4#11737385C218yesmaximal, complemented
5#11737348C218yesmaximal, complemented
6#11736519C212yesmaximal, complemented
7#11736516C212yesmaximal, complemented
8#11736487C211yesmaximal, characteristic, complemented
9#11736484C211yesmaximal, characteristic, complemented
10#11735998C212yesmaximal, complemented
11#11735994C212yesmaximal, complemented
12#11735903C211yesmaximal, characteristic, complemented
13C2×#89461C211yesmaximal, characteristic, complemented
14C7×#533C2211yescharacteristic
15C2×#6470C2212yes—
16C2×#6464C2212yes—
17C22×#1131C2211yescharacteristic
18C22×C8×D7C2211yescharacteristic
19#90424C414yes—
20#90389C412yes—
21#90388C412yes—
22#90336C414yes—
23#90335C414yes—
24#90271C2218yes—
25#90215C2212yes—
26#90213C2212yes—
27#90168C2211yescharacteristic
28C2×#6435C2211yescharacteristic
29C7×#446C2214yescomplemented
30C7×C8×Q8C2214yescomplemented
31C7×#424C2212yes—
32C7×#423C2211yescharacteristic
33C7×#405C2212yes—
34C2×C4×C7×C8C2211yescharacteristic
35#89901C2212yes—
36#89898C2212yes—
37M16×Dic7C2212yes—
38#89882C2212yes—
39C2×#6389C2211yescharacteristic
40C2×C8×Dic7C2211yescharacteristic
41#89796C2218yescomplemented
42#89771C2212yes—
43#89769C2212yes—
44#89696C2211yescharacteristic
45C2×#6377C2211yescharacteristic
46#89612C2218yes—
47#89612C2218yescomplemented
48#89610C2218yes—
49#89608C2218yes—
50#89608C2218yescomplemented
51#89607C2218yescomplemented
52#89480C2214yescomplemented
53#89479C2214yescomplemented
54#89478C2218yes—
55#89461C2214yes—
56#89461C2214yes—
57#89461C2214yescomplemented
58#89460C2214yescomplemented
59C8×D28C2218yes—
60C8×D28C2214yescomplemented
61C4×C8×D7C2214yescomplemented
62#60206—71nomaximal, Sylow
63C2×#1157C2312yes—
64C22×C4×D7C2311yescharacteristic
65C2×C7×M16C2312yes—
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C7×C4⋊Q8C2×C412yes—
69C7×#125C2×C412yes—
70C7×C22⋊Q8C2×C414yes—
71C4×C7×Q8C2314yes—
72C7×#117C2312yes—
73C2×C42×C7C2311yescharacteristic
74#6510C2×C418yes—
75#6509C2×C418yes—
76#6493C2×C414yes—
77C2×(C2×D7)⋊C4C2×C412yes—
78C2×Dic7⋊C4C2×C412yes—
79C2×C4×Dic7C2311yescharacteristic
80C2×#1136C2312yes—
81C22×C7⋊C8C2311yescharacteristic
82C22×C7⋊C8C2311yescharacteristic
83#6470—28no—
84#6464—28no—
85C2×#1131—24no—
86C2×#1131C2314yes—
87C2×#1131C2314yes—
88C2×#1131C2314yes—
89C2×C8×D7—24no—
90C2×C8×D7C2314yes—
91C2×C8×D7C2312yes—
92C2×C8×D7C2312yes—
93C2×C8×D7C2314yes—
94#6461C2×C418yes—
95#6461C2×C418yes—
96#6460C2×C414yes—
97#6458C2×C414yes—
98#6458C2×C418yes—
99#6457C2×C418yes—
100#6457C2×C418yes—
101#6456C2×C414yes—
102#6453C2×C418yes—
103#6450C2×C414yes—
104#6449C2×C414yes—
105#6445C2×C414yes—
106(C22⋊D7)⋊C4C2×C418yes—
107#6438C2×C412yes—
108#6437C2×C411yescharacteristic
109C4×D28C2318yes—
110C4×D28C2314yes—
111#6435C2×C412yes—
112#6435C2×C412yes—
113#6435C2314yes—
114#6435C2314yes—
115C42×D7C2314yes—
116#6432C2×C411yescharacteristic
117C7×C4⋊C8C2318yes—
118C7×C4⋊C8C2314yes—
119C7×#97C2314yes—
120C4×C7×C8C2314yes—
121C4×C7×C8C2312yes—
122C4×C7×C8C2312yes—
123(C2×D7)⋊C8C2318yes—
124(C2×D7)⋊C8C2318yes—
125(C2×D7)⋊C8C2318yes—
126#6389C2314yes—
127#6389C2314yes—
128C8×Dic7C2314yes—
129C8×Dic7C2312yes—
130C8×Dic7C2312yes—
131C4⋊(C7⋊C8)C2318yes—
132C4⋊(C7⋊C8)C2314yes—
133#6377C2314yes—
134#6377C2314yes—
135C4×C7⋊C8C2314yes—
136#3455—71no—
137#2967—78no—
138#2966—78no—
139#2961—71no—
140#2960D711yescharacteristic, complemented
141#2932—71no—
142#2931—71no—
143#2930—72no—
144#2929—72no—
145#2923—72no—
146#2921—71no—
147C4×#131—72no—
148C2×#405—71no—
149C23×D7C22×C411yescharacteristic
150C2×C7×Q8C22×C414yes—
151C22×C4×C7C22×C412yes—
152C22×C4×C7C2411yescharacteristic
153C2×C22⋊D7C22×C414yes—
154C22×Dic7C22×C411yescharacteristic
155C22×Dic7C22×C412yes—
156#1157—24no—
157#1157—24no—
158C2×D28C22×C412yes—
159C2×D28C22×C414yes—
160C2×D28C22×C418yes—
161C2×C4×D7—22no—
162C2×C4×D7—22no—
163C2×C4×D7C22×C412yes—
164C2×C4×D7C22×C418yes—
165C2×C4×D7C2414yes—
166C2×C4×D7C2412yes—
167C2×Dic14C22×C412yes—
168C7×M16—24no—
169C2×C7×C8—22no—
170C2×C7×C8#4114yes—
171C2×C7×C8C2412yes—
172C2×C7×C8C2414yes—
173C2×C7×C8C2411yescharacteristic
174C2×C7×C8C2411yescharacteristic
175C7×C4⋊C4C22×C414yes—
176C7×C4⋊C4C22×C418yes—
177C7×C4⋊C4C22×C414yes—
178C7×C22⋊C4C22×C414yes—
179C42×C7C22×C411yescharacteristic
180C42×C7C22×C411yescharacteristic
181C42×C7C2412yes—
182C42×C7C2414yes—
183(C2×D7)⋊C4C22×C414yes—
184(C2×D7)⋊C4C22×C414yes—
185(C2×D7)⋊C4C22×C418yes—
186(C2×D7)⋊C4C22×C418yes—
187(C2×D7)⋊C4C22×C418yes—
188(C2×D7)⋊C4C22×C418yes—
189C4⋊Dic7C22×C414yes—
190C4⋊Dic7C22×C418yes—
191Dic7⋊C4C22×C414yes—
192Dic7⋊C4C22×C414yes—
193Dic7⋊C4C22×C418yes—
194C4×Dic7C22×C418yes—
195C4×Dic7C22×C412yes—
196C4×Dic7C2412yes—
197#1136—24no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8#4114yes—
200C2×C7⋊C8C2412yes—
201C2×C7⋊C8C2414yes—
202C2×C7⋊C8C2412yes—
203#1131—28no—
204#1131—28no—
205#1131—28no—
206C8×D7—24no—
207C8×D7—24no—
208C8×D7—28no—
209C8×D7—28no—
210C2×#131—72no—
211C2×#131—72no—
212C22×M16—71no—
213C23×C8—71no—
214#542—74no—
215#541—72no—
216#540—72no—
217#539—74no—
218#538—74no—
219#533C2×D711yescharacteristic
220#532—71no—
221#529—78no—
222#525—72no—
223C4×#41—72no—
224C2×#117—71no—
225#446—78no—
226#446C2×D714yescomplemented
227C8×Q8C2×D714yescomplemented
228#436—78no—
229#436—74no—
230#435—78no—
231#435—78no—
232#435—78no—
233C8×D4—78no—
234C8×D4—74no—
235#433—78no—
236#433—78no—
237#432—78no—
238#424—72no—
239#424C2×D712yes—
240#423—71no—
241#423C2×D711yescharacteristic
242#408—72no—
243#408—72no—
244#407—72no—
245#405—72no—
246#405—74no—
247#405—74no—
248#405—74no—
249#405—74no—
250#405C2×D712yes—
251C4×M16—72no—
252C4×M16—74no—
253C2×#97—71no—
254C2×#97—71no—
255C2×C4×C8—71no—
256C2×C4×C8—74no—
257C2×C4×C8C2×D711yescharacteristic
258C23×C7C23×C411yescharacteristic
259C22×D7—22no—
260C22×D7—42no—
261C22×D7C23×C412yes—
262C22×D7C23×C414yes—
263C7×Q8—44no—
264C2×C4×C7—42no—
265C2×C4×C7C23×C418yes—
266C2×C4×C7C23×C412yes—
267C2×C4×C7C23×C412yes—
268C2×C4×C7C23×C411yescharacteristic
269C2×C4×C7C2×#4112yes—
270C2×C4×C7C2×#4112yes—
271C2×C4×C7C2511yescharacteristic
272C22⋊D7—28no—
273C2×Dic7—22no—
274C2×Dic7—42no—
275C2×Dic7C23×C414yes—
276C2×Dic7C23×C414yes—
277C2×Dic7C23×C412yes—
278D28—24no—
279D28—48no—
280D28—44no—
281C4×D7—24no—
282C4×D7—22no—
283C4×D7—24no—
284C4×D7—24no—
285C4×D7—22no—
286C4×D7—48no—
287Dic14—24no—
288C7×C8—24no—
289C7×C8—22no—
290C7×C8C2×#4112yes—
291C7×C8C2×#4112yes—
292C7⋊C8—22no—
293C7⋊C8—24no—
294C7⋊C8C2×#4114yes—
295C2×#41—72no—
296C23×C4—71no—
297#131—148no—
298#131—148no—
299C2×M16—72no—
300C2×M16—74no—
301C2×M16—74no—
302C2×M16—74no—
303C2×M16—144no—
304C2×M16C22×D712yes—
305C22×C8—71no—
306C22×C8—71no—
307C22×C8—72no—
308C22×C8—74no—
309C22×C8—72no—
310C22×C8—74no—
311C22×C8—144no—
312C22×C8C22×D711yescharacteristic
313C22×C8C22×D711yescharacteristic
314C4⋊Q8—71no—
315C4⋊Q8C4×D712yes—
316#127—71no—
317#126—78no—
318#126—78no—
319#125—78no—
320#125C4×D712yes—
321#124—78no—
322#124—72no—
323#123—78no—
324#123—78no—
325#123—74no—
326C22⋊Q8—78no—
327C22⋊Q8—74no—
328C22⋊Q8C4×D714yes—
329#121—74no—
330#121—74no—
331#121—78no—
332#120—74no—
333C4×Q8—74no—
334C4×Q8C22×D714yes—
335C4×D4—74no—
336C4×D4—78no—
337C4×D4—78no—
338C4×D4—74no—
339#117—72no—
340#117—72no—
341#117—74no—
342#117—74no—
343#117C22×D712yes—
344C2×C4⋊C4—72no—
345C2×C22⋊C4—72no—
346C2×C42—71no—
347C2×C42—74no—
348C2×C42C22×D711yescharacteristic
349C4⋊C8—74no—
350C4⋊C8—78no—
351C4⋊C8C22×D714yes—
352C4⋊C8C22×D718yes—
353C22⋊C8—78no—
354C22⋊C8—78no—
355C22⋊C8—78no—
356#97—74no—
357#97—74no—
358#97—74no—
359#97—74no—
360#97C22×D714yes—
361C4×C8—72no—
362C4×C8—74no—
363C4×C8—72no—
364C4×C8—74no—
365C4×C8C22×D712yes—
366C4×C8C22×D714yes—
367C4×C8C22×D712yes—
368C22×C7—22no—
369C22×C7C4×#4112yes—
370C22×C7C24×C411yescharacteristic
371C2×D7—22no—
372C2×D7—24no—
373C2×D7—28no—
374C2×D7—24no—
375C2×D7—44no—
376C4×C7—28no—
377C4×C7—22no—
378C4×C7—22no—
379C4×C7C4×#4112yes—
380C4×C7C22×#4111yescharacteristic
381C4×C7C22×#4111yescharacteristic
382Dic7—24no—
383Dic7—22no—
384Dic7—24no—
385C24—71no—
386#41—144no—
387#41—144no—
388C2×Q8—72no—
389C2×Q8C2×C4×D714yes—
390C2×D4—74no—
391C2×D4—72no—
392C2×D4—78no—
393C2×D4—74no—
394C22×C4—71no—
395C22×C4—72no—
396C22×C4—72no—
397C22×C4—74no—
398C22×C4—72no—
399C22×C4—78no—
400C22×C4—142no—
401C22×C4—142no—
402C22×C4C2×C4×D712yes—
403C22×C4C23×D711yescharacteristic
404M16—24no—
405M16—148no—
406M16—148no—
407M16—144no—
408M16—148no—
409C2×C8—22no—
410C2×C8—74no—
411C2×C8—72no—
412C2×C8—74no—
413C2×C8—72no—
414C2×C8—148no—
415C2×C8—142no—
416C2×C8—144no—
417C2×C8—144no—
418C2×C8—148no—
419C2×C8#116114yes—
420C2×C8C23×D714yes—
421C2×C8C23×D712yes—
422C2×C8C23×D711yescharacteristic
423C2×C8C23×D711yescharacteristic
424C4⋊C4—74no—
425C4⋊C4—74no—
426C4⋊C4—78no—
427C4⋊C4—74no—
428C4⋊C4—78no—
429C4⋊C4C2×C4×D714yes—
430C4⋊C4C2×C4×D718yes—
431C4⋊C4C2×C4×D714yes—
432C22⋊C4—74no—
433C22⋊C4—74no—
434C22⋊C4—78no—
435C22⋊C4—78no—
436C22⋊C4—78no—
437C22⋊C4—78no—
438C22⋊C4C2×C4×D714yes—
439C42—72no—
440C42—78no—
441C42—72no—
442C42C2×C4×D711yescharacteristic
443C42C2×C4×D711yescharacteristic
444C42C23×D714yes—
445C42C23×D712yes—
446C2×C7—22no—
447C2×C7C2×C4×#4111yescharacteristic
448C2×C7#357312yes—
449D7—44no—
450D7—42no—
451C23—72no—
452C23—74no—
453C23—142no—
454C23—282no—
455C23C22×C4×D711yescharacteristic
456Q8—44no—
457Q8—144no—
458D4—148no—
459D4—144no—
460D4—284no—
461D4—288no—
462C2×C4—42no—
463C2×C4—74no—
464C2×C4—74no—
465C2×C4—72no—
466C2×C4—142no—
467C2×C4—144no—
468C2×C4—142no—
469C2×C4—144no—
470C2×C4—144no—
471C2×C4—142no—
472C2×C4—288no—
473C2×C4—282no—
474C2×C4C22×C4×D718yes—
475C2×C4C22×C4×D712yes—
476C2×C4C22×C4×D712yes—
477C2×C4C22×C4×D711yescharacteristic
478C2×C4C2×#116112yes—
479C2×C4D7×#4112yes—
480C2×C4C24×D711yescharacteristic
481C8—22no—
482C8—24no—
483C8—74no—
484C8—142no—
485C8—144no—
486C8C2×#116112yes—
487C8C2×#116112yes—
488C7#6020611yescharacteristic, complemented, Sylow
489C22—22no—
490C22—142no—
491C22—144no—
492C22—148no—
493C22—144no—
494C22—284no—
495C22C4×#116112yes—
496C22C23×C4×D711yescharacteristic
497C4—22no—
498C4—22no—
499C4—28no—
500C4—142no—
501C4—144no—
502C4—144no—
503C4C4×#116112yes—
504C4C22×#116111yescharacteristic
505C4C2×D7×#4111yescharacteristic
506C2—22no—
507C2—282no—
508C2—284no—
509C2C2×C4×#116111yescharacteristic
510C2#1174422812yes—
subgroupG/H
centerC2×C8C23×D7
commutator, socleC22×C7C24×C4
FrattiniC2×C4C24×D7
FittingC7×#2960C2
prime radical#11743341C2
cosocleC2×C4×C7C25
hypercenter#2960D7
hypocenterC7#60206
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C8 ◁ #2960

Complements

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

NH ≅ G/Ntype
2. C7×#2960C2semidirect
3. #11740774C2semidirect
4. #11737385C2semidirect
5. #11737348C2semidirect
6. #11736519C2semidirect
7. #11736516C2semidirect
8. #11736487C2semidirect
9. #11736484C2semidirect
10. #11735998C2semidirect
11. #11735994C2semidirect
12. #11735903C2semidirect
13. C2×#89461C2semidirect
29. C7×#446C22semidirect
30. C7×C8×Q8C22semidirect
41. #89796C22semidirect
47. #89612C22semidirect
50. #89608C22semidirect
51. #89607C22semidirect
52. #89480C22semidirect
53. #89479C22semidirect
57. #89461C22semidirect
58. #89460C22semidirect
60. C8×D28C22semidirect
61. C4×C8×D7C22semidirect
140. #2960D7semidirect
226. #446C2×D7semidirect
227. C8×Q8C2×D7semidirect
488. C7#60206semidirect

Maximal subgroups

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

subgroupindex
#117433412
C7×#29602
#117407742
#117373852
#117373482
#117365192
#117365162
#117364872
#117364842
#117359982
#117359942
#117359032
C2×#894612
#602067