Finite Group Databasebeta

← #14218137

Subgroups — #14218137

Order 1792.

2604 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11744365C211yesmaximal, characteristic
2#11744364C211yesmaximal, characteristic, complemented
3#11744234C211yesmaximal, characteristic, complemented
4C7×#2901C211yesmaximal, characteristic, complemented
5#11741255C211yesmaximal, characteristic, complemented
6#11739981C211yesmaximal, characteristic, complemented
7#11739956C211yesmaximal, characteristic, complemented
8#11736435C216yesmaximal, complemented
9#11736434C216yesmaximal, complemented
10#11736425C216yesmaximal, complemented
11C2×#89595C216yesmaximal, complemented
12C7×#585C2211yescharacteristic
13#90616C2211yescharacteristic
14C7×#568C2211yescharacteristic
15C4×C7×#41C2211yescharacteristic
16#90529C414yescomplemented
17#90528C414yescomplemented
18#90527C414yescomplemented
19#90526C414yescomplemented
20#90513C2216yes—
21C2×#6513C2216yes—
22#90510—24no—
23#90510C2211yescharacteristic
24#90510C2216yescomplemented
25C2×#6511—24no—
26C2×#6511C2211yescharacteristic
27C2×#6511C2211yescharacteristic
28C2×#6511C2216yescomplemented
29#90441—24no—
30D7×#131—24no—
31#90436C2216yescomplemented
32C2×#6470C2216yescomplemented
33C7×#421C2216yescomplemented
34C2×C7×#104C2216yescomplemented
35#89896C2216yescomplemented
36C2×#6396C2216yescomplemented
37#89777C2216yescomplemented
38#89769C2213yescomplemented
39#89769C2213yescomplemented
40C2×#6379C2216yescomplemented
41C4×#1136C2213yescomplemented
42C2×#6377C2213yescomplemented
43#89596C22124yescomplemented
44#89595C2214yescomplemented
45#89595C22112yescomplemented
46#89595C22112yescomplemented
47#89595C22112yescomplemented
48#60184—71nomaximal, Sylow
49C7×2−1+4—21no—
50C7×2−1+4C2×C414yes—
51C7×H(5,𝔽2)—23no—
52C7×H(5,𝔽2)C2×C414yes—
53C2×C7×#41—24no—
54C2×C7×#41C2316yes—
55C2×C7×#41C2311yescharacteristic
56#6554C2×C414yes—
57#6553C2×C414yes—
58D7×#41—24no—
59#6551—21no—
60#6548—23no—
61C2×#1157C2316yes—
62C7×#131—24no—
63C2×C7×M16C2316yes—
64C4×C7×Q8—21no—
65C4×C7×D4—23no—
66C7×#117C2313yes—
67C2×C42×C7C2313yes—
68#6514C2×C4112yescomplemented
69#6513C2×C4112yescomplemented
70#6513C2×C4112yescomplemented
71#6513C2314yes—
72#6513C23112yes—
73#6512C2×C4112yescomplemented
74#6511—24no—
75#6511—212no—
76#6511—26no—
77#6511—26no—
78#6511C2×C414yes—
79#6511C2×C414yes—
80#6511D414yes—
81#6511D414yes—
82#6511C2314yes—
83#6511C23112yes—
84C2×#1144C2×C416yescomplemented
85#6505C2×C4112yescomplemented
86C2×#1143C2×C416yescomplemented
87C2×#1142C2×C416yescomplemented
88#6495C2×C4112yescomplemented
89C2×#1141C2×C416yescomplemented
90C2×#1136—212no—
91C2×#1136—26no—
92C2×#1136C2316yes—
93C2×#1136C2313yes—
94C2×#1136C2313yes—
95C22×C7⋊C8—26no—
96C22×C7⋊C8C2313yes—
97C22×C7⋊C8C2313yes—
98#6470—26no—
99#6470—26no—
100#6470C2314yes—
101#6470C23112yes—
102D7×M16—212no—
103#6464—212no—
104C2×#1131—26no—
105C2×C8×D7—26no—
106C7×#104C2314yes—
107C7×#104C23112yes—
108#6396C2314yes—
109#6396C23112yes—
110#6379C2314yes—
111#6379C23112yes—
112#6377C2311yescharacteristic
113#6377C2313yes—
114#6377C2313yes—
115#6377C2313yes—
116C4×C7⋊C8C2316yes—
117#3584—71no—
118#3573—71no—
119#3573—71no—
120#2957—76no—
121#2956—76no—
122#2955—76no—
123C2×#444—76no—
124#2916—71no—
125#2901—71no—
126#2901D711yescharacteristic, complemented
127#2875—71no—
128C7×#41—24no—
129C7×#41—212no—
130C7×#41—26no—
131C7×#41—26no—
132C7×#41C22×C4112yes—
133C7×#41C22×C414yes—
134C7×#41C2×D414yes—
135C7×#41C2414yes—
136C2×C7×Q8—24no—
137C2×C7×Q8C22×C416yes—
138C2×C7×Q8#4111yescharacteristic
139C2×C7×D4—26no—
140C2×C7×D4—212no—
141C2×C7×D4C22×C416yes—
142C2×C7×D4#4113yes—
143C22×C4×C7—26no—
144C22×C4×C7—23no—
145C22×C4×C7C2413yes—
146C2×C22⋊D7—26no—
147#1161—24no—
148Q8×D7—24no—
149#1159—212no—
150D4×D7—212no—
151#1157—26no—
152#1157—26no—
153#1157C22×C4112yes—
154#1157C2414yes—
155C2×D28C22×C416yes—
156C2×C4×D7—26no—
157C2×Dic14C22×C416yes—
158C7×M16—26no—
159C7×M16C2414yes—
160C2×C7×C8—26no—
161C7×C4⋊C4—23no—
162C7×C22⋊C4—23no—
163C42×C7—23no—
164C42×C7C2411yescharacteristic
165C42×C7C2413yes—
166#1144C22×C4112yes—
167#1144C22×C414yes—
168#1143C22×C414yes—
169#1143C22×C4112yes—
170#1142C22×C4112yes—
171#1142C22×C414yes—
172#1141C22×C414yes—
173#1141C22×C4112yes—
174#1136—26no—
175#1136—24no—
176#1136—212no—
177#1136C22×C4112yes—
178#1136C2×D4112yes—
179#1136C2414yes—
180C2×C7⋊C8—26no—
181C2×C7⋊C8—24no—
182C2×C7⋊C8—212no—
183C2×C7⋊C8C22×C416yes—
184C2×C7⋊C8C22×C416yes—
185C2×C7⋊C8C2×D416yes—
186C2×C7⋊C8C2×D416yes—
187C2×C7⋊C8C2411yescharacteristic
188C2×C7⋊C8C2413yes—
189C2×C7⋊C8C2411yescharacteristic
190C2×C7⋊C8C2413yes—
191#1131—24no—
192#1131—212no—
193C8×D7—24no—
194C8×D7—212no—
195#585—71no—
196#585C2×D711yescharacteristic
197#578—74no—
198#577—74no—
199#577—74no—
200#576—74no—
201#575—76no—
202C2×#135—76no—
203#568—71no—
204#568—76no—
205#568—76no—
206#568—144no—
207#568—144no—
208#568C2×D711yescharacteristic
209C2×#131—71no—
210C2×#131—71no—
211C2×#131—76no—
212C2×#131—76no—
213C2×#131—144no—
214C2×#131—144no—
215C4×#41C2×D711yescharacteristic
216#444—74no—
217#444—712no—
218#444—712no—
219#444—712no—
220#443—724no—
221#430—76no—
222C2×#108—76no—
223#421—76no—
224#421C2×D716yescomplemented
225C2×#104—76no—
226C2×#104C2×D716yescomplemented
227#405—73no—
228#405—73no—
229C4×M16—73no—
230C2×#97—73no—
231C23×C7—26no—
232C23×C7C2×#4113yes—
233C22×D7—26no—
234C7×Q8—26no—
235C7×Q8C4×D414yes—
236C7×Q8C23×C414yes—
237C7×D4—212no—
238C7×D4—24no—
239C7×D4—26no—
240C7×D4C4×D4112yes—
241C7×D4C23×C414yes—
242C2×C4×C7—26no—
243C2×C4×C7—212no—
244C2×C4×C7—24no—
245C2×C4×C7—21no—
246C2×C4×C7—23no—
247C2×C4×C7—83no—
248C2×C4×C7C23×C416yes—
249C2×C4×C7C22×D416yes—
250C2×C4×C7C2×#4113yes—
251C2×C4×C7C2511yescharacteristic
252C22⋊D7—212no—
253C22⋊D7—24no—
254C2×Dic7—26no—
255D28—26no—
256D28C23×C414yes—
257C4×D7—212no—
258C4×D7—24no—
259Dic14—26no—
260Dic14C23×C414yes—
261C7×C8—24no—
262C7⋊C8—24no—
263C7⋊C8C23×C411yescharacteristic
264C7⋊C8C23×C411yescharacteristic
265C7⋊C8C23×C416yes—
266C7⋊C8C22×D411yescharacteristic
267C7⋊C8C22×D416yes—
268C7⋊C8C22×D411yescharacteristic
2692−1+4—21no—
2702−1+4—74no—
2712−1+4—141no—
2722−1+4C4×D714yes—
273H(5,𝔽2)—23no—
274H(5,𝔽2)—74no—
275H(5,𝔽2)—143no—
276H(5,𝔽2)C4×D714yes—
277C2×#41—24no—
278C2×#41—76no—
279C2×#41—144no—
280C2×#41C22×D716yes—
281C2×#41C22×D711yescharacteristic
282#137—712no—
283#137—712no—
284AGL(1,ℤ/8)—712no—
285AGL(1,ℤ/8)—712no—
286#135—712no—
287#135—712no—
288#135—74no—
289#135—712no—
290C2×Q16—76no—
291C2×SD16—76no—
292C2×SD16—76no—
293C2×D8—76no—
294#131—24no—
295#131—74no—
296#131—74no—
297#131—74no—
298#131—74no—
299#131—74no—
300#131—712no—
301#131—74no—
302#131—712no—
303#131—1412no—
304#131—146no—
305#131—146no—
306#131—146no—
307#131—146no—
308#131—1412no—
309#131—144no—
310C2×M16—73no—
311C2×M16—73no—
312C2×M16—76no—
313C2×M16—146no—
314C2×M16—146no—
315C2×M16—1412no—
316C2×M16—1412no—
317C2×M16C22×D716yes—
318C22×C8—73no—
319C22×C8—73no—
320C22×C8—146no—
321C22×C8—146no—
322C4×Q8—21no—
323C4×D4—23no—
324#117C22×D713yes—
325C2×C42C22×D713yes—
326#108—74no—
327#108—712no—
328#104—74no—
329#104—712no—
330#104C22×D714yes—
331#104C22×D7112yes—
332#97—71no—
333#97—73no—
334#97—73no—
335#97—73no—
336C4×C8—76no—
337C22×C7—24no—
338C22×C7—43no—
339C22×C7—86no—
340C22×C7C2×C4×D416yes—
341C22×C7C22×#4111yescharacteristic
342C2×D7—24no—
343C2×D7—86no—
344C4×C7—24no—
345C4×C7—43no—
346C4×C7—41no—
347C4×C7C2×C4×D416yes—
348C4×C7C24×C411yescharacteristic
349C4×C7C23×D411yescharacteristic
350Dic7—24no—
351#41—24no—
352#41—212no—
353#41—26no—
354#41—26no—
355#41—712no—
356#41—74no—
357#41—144no—
358#41—1412no—
359#41—146no—
360#41—146no—
361#41C2×C4×D7112yes—
362#41C2×C4×D714yes—
363#41D4×D714yes—
364#41C23×D714yes—
365C2×Q8—24no—
366C2×Q8—76no—
367C2×Q8—144no—
368C2×Q8C2×C4×D716yes—
369C2×Q8#115911yescharacteristic
370C2×D4—26no—
371C2×D4—212no—
372C2×D4—76no—
373C2×D4—146no—
374C2×D4—1412no—
375C2×D4C2×C4×D716yes—
376C2×D4#115913yes—
377C22×C4—26no—
378C22×C4—23no—
379C22×C4—146no—
380C22×C4C23×D713yes—
381Q16—74no—
382Q16—712no—
383SD16—74no—
384SD16—712no—
385SD16—74no—
386SD16—712no—
387D8—74no—
388D8—712no—
389M16—26no—
390M16—712no—
391M16—712no—
392M16—74no—
393M16—144no—
394M16—1412no—
395M16—144no—
396M16—1412no—
397M16—146no—
398M16C23×D714yes—
399C2×C8—26no—
400C2×C8—76no—
401C2×C8—76no—
402C2×C8—76no—
403C2×C8—76no—
404C2×C8—73no—
405C2×C8—73no—
406C2×C8—71no—
407C2×C8—71no—
408C2×C8—144no—
409C2×C8—1412no—
410C2×C8—144no—
411C2×C8—1412no—
412C2×C8—146no—
413C4⋊C4—23no—
414C22⋊C4—23no—
415C42—23no—
416C42C23×D713yes—
417C42C23×D711yescharacteristic
418C2×C7—21no—
419C2×C7—26no—
420C2×C7—44no—
421C2×C7C22×C4×D411yescharacteristic
422D7—44no—
423C23—26no—
424C23—146no—
425C23C2×#115913yes—
426Q8—26no—
427Q8—74no—
428Q8—146no—
429Q8(C22⋊D7)⋊C414yes—
430Q8C22×C4×D714yes—
431D4—212no—
432D4—24no—
433D4—26no—
434D4—74no—
435D4—1412no—
436D4—144no—
437D4—146no—
438D4(C22⋊D7)⋊C4112yes—
439D4C22×C4×D714yes—
440C2×C4—26no—
441C2×C4—212no—
442C2×C4—24no—
443C2×C4—21no—
444C2×C4—23no—
445C2×C4—83no—
446C2×C4—146no—
447C2×C4—1412no—
448C2×C4—144no—
449C2×C4C22×C4×D716yes—
450C2×C4C2×D4×D716yes—
451C2×C4C2×#115913yes—
452C2×C4C24×D711yescharacteristic
453C8—24no—
454C8—71no—
455C8—71no—
456C8—76no—
457C8—71no—
458C8—71no—
459C8—76no—
460C8—144no—
461C7#6018411yescharacteristic, complemented, Sylow
462C22—24no—
463C22—43no—
464C22—86no—
465C22—144no—
466C22—566no—
467C22C2×(C22⋊D7)⋊C416yes—
468C22C22×#115911yescharacteristic
469C4—24no—
470C4—41no—
471C4—43no—
472C4—144no—
473C4C2×(C22⋊D7)⋊C416yes—
474C4C23×C4×D711yescharacteristic
475C4C22×D4×D711yescharacteristic
476C2—21no—
477C2—26no—
478C2—44no—
479C2—284no—
480C2C22×(C22⋊D7)⋊C411yescharacteristic
subgroupG/H
centerC4C22×D4×D7
commutatorC4×C7C24×C4
FrattiniC2×C4C24×D7
FittingC7×#2901C2
socleC2×C7C22×C4×D4
prime radical#11744365C2
cosocleC2×C4×C7C25
hypercenter#2901D7
hypocenterC7#60184
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×#41 ◁ #2901

Complements

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

NH ≅ G/Ntype
2. #11744364C2semidirect
3. #11744234C2semidirect
4. C7×#2901C2semidirect
5. #11741255C2semidirect
6. #11739981C2semidirect
7. #11739956C2semidirect
8. #11736435C2semidirect
9. #11736434C2semidirect
10. #11736425C2semidirect
11. C2×#89595C2semidirect
16. #90529C4semidirect
17. #90528C4semidirect
18. #90527C4semidirect
19. #90526C4semidirect
24. #90510C22semidirect
28. C2×#6511C22semidirect
31. #90436C22semidirect
32. C2×#6470C22semidirect
33. C7×#421C22semidirect
34. C2×C7×#104C22semidirect
35. #89896C22semidirect
36. C2×#6396C22semidirect
37. #89777C22semidirect
38. #89769C22semidirect
39. #89769C22semidirect
40. C2×#6379C22semidirect
41. C4×#1136C22semidirect
42. C2×#6377C22semidirect
43. #89596C22semidirect
44. #89595C22semidirect
45. #89595C22semidirect
46. #89595C22semidirect
47. #89595C22semidirect
68. #6514C2×C4semidirect
69. #6513C2×C4semidirect
70. #6513C2×C4semidirect
73. #6512C2×C4semidirect
84. C2×#1144C2×C4semidirect
85. #6505C2×C4semidirect
86. C2×#1143C2×C4semidirect
87. C2×#1142C2×C4semidirect
88. #6495C2×C4semidirect
89. C2×#1141C2×C4semidirect
126. #2901D7semidirect
224. #421C2×D7semidirect
226. C2×#104C2×D7semidirect
461. C7#60184semidirect

Maximal subgroups

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

subgroupindex
#117443652
#117443642
#117442342
C7×#29012
#117412552
#117399812
#117399562
#117364352
#117364342
#117364252
C2×#895952
#601847