Finite Group Databasebeta

← #14219640

Subgroups — #14219640

Order 1792.

1892 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11743307C211yesmaximal, characteristic
2C7×#2931C211yesmaximal, characteristic, complemented
3#11741240C214yesmaximal, complemented
4#11740723C211yesmaximal, characteristic, complemented
5#11736988C214yesmaximal, complemented
6#11736971C214yesmaximal, complemented
7D4×#1131C214yesmaximal, complemented
8#11736848C214yesmaximal, complemented
9#11736497C212yesmaximal, complemented
10#11736111C214yesmaximal, complemented
11#11735892C212yesmaximal, complemented
12C2×C7×#131C2212yescomplemented
13C7×#532C2211yescharacteristic
14#90519C2214yes—
15C22×#1131C2214yescomplemented
16#90421C412yes—
17#90408C411yescharacteristic
18#90404C411yescharacteristic
19#90370C414yes—
20#90305C414yes—
21#90303C414yes—
22#90248C2214yes—
23C4×D4×D7C2214yes—
24C4×#1157C2212yes—
25C7×#436C2214yescomplemented
26C7×C8×D4C2214yescomplemented
27C7×#423C2211yescharacteristic, complemented
28C7×#408C2212yescomplemented
29C2×C4×C7×C8C2211yescharacteristic, complemented
30#90006C2214yescomplemented
31#89900C2218yescomplemented
32#89900C2218yescomplemented
33#89898C2214yescomplemented
34#89891C2214yescomplemented
35#89883C2218yescomplemented
36#89883C2214yescomplemented
37#89883C2214yescomplemented
38#89873C2214yescomplemented
39#89784C2214yescomplemented
40D4×C7⋊C8C2214yescomplemented
41#89697C2212yescomplemented
42#89610C2214yes—
43#89609C2214yescomplemented
44#89605C2214yescomplemented
45#89602C2214yescomplemented
46#89502C2218yescomplemented
47#89501C2218yescomplemented
48#89498C2218yescomplemented
49#89492C2218yescomplemented
50#89460C2212yes—
51#89460C2212yes—
52C4×#1131C2214yescomplemented
53#89457C2214yescomplemented
54#89451C2212yescomplemented
55#89451C2212yescomplemented
56#60317—71nomaximal, Sylow
57C2×C7×#41C2312yes—
58C2×D4×D7C2×C414yes—
59C2×#1157C2×C412yes—
60C22×C4×D7C2314yes—
61C7×#131—44no—
62C2×C7×M16C2314yes—
63C2×C7×M16C2312yes—
64C22×C7×C8C2314yes—
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C7×C4⋊Q8C2×C411yescharacteristic
68C7×#127C2×C411yescharacteristic
69C7×#124C2×C412yes—
70C7×#121C2×C414yes—
71C4×C7×D4C2314yes—
72C2×C42×C7C2311yescharacteristic
73#6509C2×C418yes—
74#6501C2×C414yes—
75#6501C2×C418yes—
76#6500C2×C414yes—
77#6500C2×C414yes—
78#6499C2×C414yes—
79#6498C2×C414yes—
80#6498C2×C414yes—
81D4×Dic7C2×C414yes—
82C4×C22⋊D7C2×C414yes—
83C4×C22⋊D7C2314yes—
84C4×C22⋊D7C2318yes—
85#6489C2×C412yes—
86#6489C2314yes—
87C2×#1136C2314yes—
88C22×C7⋊C8C2314yes—
89C2×#1131—48no—
90C2×#1131C2314yes—
91C2×#1131C2318yes—
92C2×#1131C2314yes—
93#6460C2×C414yes—
94#6456C2×C414yes—
95D7×C4⋊C4C2×C414yes—
96#6453C2×C414yes—
97#6446C2×C418yes—
98(C22⋊D7)⋊C4C2×C418yes—
99D7×C22⋊C4C2×C418yes—
100#6442C2×C418yes—
101C4×D28C2×C412yes—
102C4×D28C2312yes—
103#6435C2×C414yes—
104C42×D7C2314yes—
105C4×Dic14C2×C412yes—
106C4×Dic14C2312yes—
107C7×C4⋊C8C2314yes—
108C7×C22⋊C8C2318yes—
109C4×C7×C8C2312yes—
110C4×C7×C8C2312yes—
111C22⋊(C7⋊C8)C2314yes—
112C22⋊(C7⋊C8)C2318yes—
113(C2×D7)⋊C8C2318yes—
114(C2×D7)⋊C8C2314yes—
115(C2×D7)⋊C8C2314yes—
116#6389C2314yes—
117#6389C2314yes—
118Dic7⋊C8C2318yes—
119Dic7⋊C8C2314yes—
120Dic7⋊C8C2314yes—
121C8×Dic7C2318yes—
122C4⋊(C7⋊C8)C2312yes—
123C4⋊(C7⋊C8)C2312yes—
124C4×C7⋊C8C2314yes—
125#3481—71no—
126#2993—74no—
127#2991—72no—
128#2979—71no—
129#2978—74no—
130#2975—74no—
131#2969—74no—
132#2965—72no—
133D4×M16—74no—
134#2931D711yescharacteristic, complemented
135#2930—74no—
136C23×D7C22×C414yes—
137C7×#41—42no—
138C7×#41—42no—
139C2×C7×Q8C22×C412yes—
140C2×C7×D4C22×C414yes—
141C2×C7×D4C22×C412yes—
142C2×C7×D4C22×C414yes—
143C22×C4×C7C22×C412yes—
144C22×C4×C7C2411yescharacteristic
145C22×C4×C7C2414yes—
146C2×C22⋊D7C22×C418yes—
147C2×C22⋊D7C22×C414yes—
148C22×Dic7C22×C414yes—
149D4×D7—48no—
150#1157—44no—
151C2×D28C22×C412yes—
152C2×C4×D7—44no—
153C2×C4×D7—44no—
154C2×C4×D7C22×C414yes—
155C2×C4×D7C22×C414yes—
156C2×C4×D7C22×C418yes—
157C2×C4×D7C2414yes—
158C2×Dic14C22×C412yes—
159C7×M16—42no—
160C7×M16—44no—
161C2×C7×C8—28no—
162C2×C7×C8—22no—
163C2×C7×C8—41no—
164C2×C7×C8C2412yes—
165C2×C7×C8C2411yescharacteristic
166C2×C7×C8C2414yes—
167C2×C7×C8C2411yescharacteristic
168C7×C4⋊C4C22×C414yes—
169C7×C22⋊C4C22×C418yes—
170C42×C7C22×C411yescharacteristic
171C42×C7C22×C411yescharacteristic
172C42×C7C2412yes—
173C22⋊Dic7C22×C414yes—
174C22⋊Dic7C22×C418yes—
175C22⋊Dic7C22×C414yes—
176C22⋊Dic7C22×C414yes—
177(C2×D7)⋊C4C22×C414yes—
178(C2×D7)⋊C4C22×C414yes—
179(C2×D7)⋊C4C22×C418yes—
180C4⋊Dic7C22×C412yes—
181C4⋊Dic7C22×C412yes—
182C4⋊Dic7C22×C418yes—
183Dic7⋊C4C22×C414yes—
184Dic7⋊C4C22×C414yes—
185Dic7⋊C4C22×C418yes—
186C4×Dic7C22×C414yes—
187C4×Dic7C22×C414yes—
188C4×Dic7C2414yes—
189#1136—44no—
190C2×C7⋊C8—44no—
191C2×C7⋊C8C2414yes—
192C2×C7⋊C8C2414yes—
193#1131—28no—
194#1131—48no—
195#1131—44no—
196C2×#131C2×D712yescomplemented
197C22×M16—74no—
198#559—71no—
199#554—72no—
200#553—74no—
201#551—74no—
202#546—74no—
203D42—71no—
204#532C2×D711yescharacteristic
205#529—74no—
206#525—74no—
207C4×#41—72no—
208C2×C4×D4—74no—
209#446—72no—
210#446—72no—
211C8×Q8—74no—
212#436—78no—
213#436—78no—
214#436—74no—
215#436—74no—
216#436—78no—
217#436—72no—
218#436—72no—
219#436C2×D714yescomplemented
220#435—74no—
221#435—78no—
222#435—74no—
223#435—78no—
224C8×D4—74no—
225C8×D4C2×D714yescomplemented
226#433—78no—
227#431—72no—
228#431—74no—
229#431—74no—
230#424—74no—
231#423—74no—
232#423C2×D711yescharacteristic, complemented
233#408—74no—
234#408C2×D712yescomplemented
235#407—74no—
236#407—78no—
237#405—74no—
238C4×M16—74no—
239C2×C4×C8C2×D711yescharacteristic, complemented
240C23×C7C23×C414yes—
241C23×C7C23×C411yescharacteristic
242C22×D7—44no—
243C22×D7—48no—
244C22×D7C23×C414yes—
245C7×Q8—42no—
246C7×D4—28no—
247C7×D4—44no—
248C7×D4—42no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—21no—
252C2×C4×C7—24no—
253C2×C4×C7—21no—
254C2×C4×C7C23×C412yes—
255C2×C4×C7C23×C411yescharacteristic
256C2×C4×C7C23×C414yes—
257C2×C4×C7C23×C412yes—
258C2×C4×C7C2511yescharacteristic
259C22⋊D7—44no—
260C22⋊D7—48no—
261C2×Dic7—44no—
262C2×Dic7C23×C414yes—
263C2×Dic7C23×C414yes—
264C2×Dic7C23×C414yes—
265D28—24no—
266C4×D7—28no—
267C4×D7—24no—
268C4×D7—24no—
269C4×D7—48no—
270C4×D7—44no—
271Dic14—24no—
272C7×C8—24no—
273C7×C8—21no—
274C7×C8—22no—
275C7×C8H(5,𝔽2)12yes—
276C7⋊C8—24no—
277C7⋊C8—24no—
278C2×#41—72no—
279C2×#41C22×D712yes—
280C22×D4—74no—
281C23×C4—74no—
282#131—44no—
283C2×M16—74no—
284C2×M16—74no—
285C2×M16—78no—
286C2×M16—74no—
287C2×M16—288no—
288C2×M16C22×D714yes—
289C2×M16C22×D712yes—
290C22×C8—74no—
291C22×C8C22×D714yes—
292C22×C8C22×D711yescharacteristic
293C22×C8C22×D711yescharacteristic
294C4⋊Q8C4×D711yescharacteristic
295#127C4×D711yescharacteristic
296#126—78no—
297#125—74no—
298#124—74no—
299#124C4×D712yes—
300#123—74no—
301#123—74no—
302#123—78no—
303C22⋊Q8—78no—
304C22⋊Q8—74no—
305#121—78no—
306#121—74no—
307#121C4×D714yes—
308#120—74no—
309#120—74no—
310C4×Q8—72no—
311C4×Q8—72no—
312C4×D4—74no—
313C4×D4—74no—
314C4×D4—78no—
315C4×D4—74no—
316C4×D4—74no—
317C4×D4—78no—
318C4×D4—72no—
319C4×D4—72no—
320C4×D4C22×D714yes—
321#117—72no—
322#117—74no—
323#117—74no—
324C2×C4⋊C4—74no—
325C2×C22⋊C4—78no—
326C2×C42—74no—
327C2×C42C22×D711yescharacteristic
328C4⋊C8—78no—
329C4⋊C8—74no—
330C4⋊C8—74no—
331C4⋊C8—72no—
332C4⋊C8—72no—
333C4⋊C8C22×D714yes—
334C22⋊C8—78no—
335C22⋊C8—74no—
336C22⋊C8—74no—
337C22⋊C8—78no—
338C22⋊C8—74no—
339C22⋊C8C22×D718yes—
340#97—74no—
341#97—74no—
342C4×C8—78no—
343C4×C8—74no—
344C4×C8C22×D712yes—
345C4×C8C22×D712yes—
346C22×C7—24no—
347C22×C7—21no—
348C22×C7—22no—
349C22×C7—28no—
350C22×C7C24×C411yescharacteristic
351C2×D7—24no—
352C2×D7—24no—
353C2×D7—24no—
354C2×D7—88no—
355C4×C7—24no—
356C4×C7—21no—
357C4×C7—22no—
358C4×C7#56814yes—
359C4×C7C2×H(5,𝔽2)11yescharacteristic
360C4×C7#58511yescharacteristic
361Dic7—24no—
362Dic7—24no—
363Dic7—24no—
364C24—74no—
365#41—42no—
366#41—42no—
367#41—284no—
368C2×Q8—72no—
369C2×Q8C2×C4×D712yes—
370C2×D4—74no—
371C2×D4—78no—
372C2×D4—72no—
373C2×D4—288no—
374C2×D4C2×C4×D714yes—
375C2×D4C2×C4×D712yes—
376C2×D4C2×C4×D714yes—
377C22×C4—74no—
378C22×C4—74no—
379C22×C4—74no—
380C22×C4—74no—
381C22×C4—78no—
382C22×C4—284no—
383C22×C4—284no—
384C22×C4C2×C4×D712yes—
385C22×C4C23×D711yescharacteristic
386C22×C4C23×D714yes—
387M16—44no—
388M16—42no—
389M16—148no—
390M16—284no—
391M16—288no—
392M16—284no—
393C2×C8—28no—
394C2×C8—22no—
395C2×C8—41no—
396C2×C8—74no—
397C2×C8—74no—
398C2×C8—284no—
399C2×C8C23×D714yes—
400C2×C8C23×D712yes—
401C2×C8C23×D711yescharacteristic
402C2×C8C23×D711yescharacteristic
403C4⋊C4—78no—
404C4⋊C4—74no—
405C4⋊C4—74no—
406C4⋊C4—72no—
407C4⋊C4—72no—
408C4⋊C4—78no—
409C4⋊C4C2×C4×D714yes—
410C22⋊C4—78no—
411C22⋊C4—74no—
412C22⋊C4—74no—
413C22⋊C4—74no—
414C22⋊C4—74no—
415C22⋊C4—78no—
416C22⋊C4—74no—
417C22⋊C4C2×C4×D718yes—
418C42—74no—
419C42—74no—
420C42—74no—
421C42C2×C4×D711yescharacteristic
422C42C2×C4×D711yescharacteristic
423C42C23×D712yes—
424C2×C7—44no—
425C2×C7—41no—
426C2×C7C4×H(5,𝔽2)11yescharacteristic
427C2×C7C2×#56812yes—
428D7—44no—
429C23—74no—
430C23—284no—
431C23—288no—
432C23C22×C4×D714yes—
433C23C22×C4×D711yescharacteristic
434Q8—42no—
435Q8—144no—
436D4—28no—
437D4—44no—
438D4—42no—
439D4—144no—
440D4—288no—
441D4—284no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—74no—
448C2×C4—74no—
449C2×C4—74no—
450C2×C4—148no—
451C2×C4—144no—
452C2×C4—144no—
453C2×C4—288no—
454C2×C4—284no—
455C2×C4—284no—
456C2×C4C22×C4×D712yes—
457C2×C4C22×C4×D711yescharacteristic
458C2×C4C22×C4×D714yes—
459C2×C4C22×C4×D712yes—
460C2×C4C24×D711yescharacteristic
461C8—21no—
462C8—24no—
463C8—22no—
464C8—144no—
465C8—144no—
466C8#654812yes—
467C7#6031711yescharacteristic, complemented, Sylow
468C22—24no—
469C22—21no—
470C22—22no—
471C22—28no—
472C22—144no—
473C22—144no—
474C22—144no—
475C22—568no—
476C22C23×C4×D711yescharacteristic
477C4—24no—
478C4—21no—
479C4—22no—
480C4—144no—
481C4—144no—
482C4—144no—
483C4#9044114yes—
484C4C2×#654811yescharacteristic
485C4#9061611yescharacteristic
486C2—44no—
487C2—41no—
488C2—284no—
489C2C4×#654811yescharacteristic
490C2C2×#9044112yes—
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
FittingC7×#2931C2
prime radical#11743307C2
cosocleC2×C4×C7C25
hypercenter#2931D7
hypocenterC7#60317
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #2931

Complements

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

NH ≅ G/Ntype
2. C7×#2931C2semidirect
3. #11741240C2semidirect
4. #11740723C2semidirect
5. #11736988C2semidirect
6. #11736971C2semidirect
7. D4×#1131C2semidirect
8. #11736848C2semidirect
9. #11736497C2semidirect
10. #11736111C2semidirect
11. #11735892C2semidirect
12. C2×C7×#131C22semidirect
15. C22×#1131C22semidirect
25. C7×#436C22semidirect
26. C7×C8×D4C22semidirect
27. C7×#423C22semidirect
28. C7×#408C22semidirect
29. C2×C4×C7×C8C22semidirect
30. #90006C22semidirect
31. #89900C22semidirect
32. #89900C22semidirect
33. #89898C22semidirect
34. #89891C22semidirect
35. #89883C22semidirect
36. #89883C22semidirect
37. #89883C22semidirect
38. #89873C22semidirect
39. #89784C22semidirect
40. D4×C7⋊C8C22semidirect
41. #89697C22semidirect
43. #89609C22semidirect
44. #89605C22semidirect
45. #89602C22semidirect
46. #89502C22semidirect
47. #89501C22semidirect
48. #89498C22semidirect
49. #89492C22semidirect
52. C4×#1131C22semidirect
53. #89457C22semidirect
54. #89451C22semidirect
55. #89451C22semidirect
134. #2931D7semidirect
196. C2×#131C2×D7semidirect
219. #436C2×D7semidirect
225. C8×D4C2×D7semidirect
232. #423C2×D7semidirect
234. #408C2×D7semidirect
239. C2×C4×C8C2×D7semidirect
467. C7#60317semidirect

Maximal subgroups

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

subgroupindex
#117433072
C7×#29312
#117412402
#117407232
#117369882
#117369712
D4×#11312
#117368482
#117364972
#117361112
#117358922
#603177